src/HOL/Hyperreal/SEQ.thy
author paulson
Tue Oct 19 18:18:45 2004 +0200 (2004-10-19)
changeset 15251 bb6f072c8d10
parent 15241 a3949068537e
child 15312 7d6e12ead964
permissions -rw-r--r--
converted some induct_tac to induct
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(*  Title       : SEQ.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Description : Convergence of sequences and series
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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theory SEQ
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imports NatStar
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begin
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constdefs
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  LIMSEQ :: "[nat=>real,real] => bool"    ("((_)/ ----> (_))" [60, 60] 60)
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    --{*Standard definition of convergence of sequence*}
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  "X ----> L == (\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> \<bar>X n + -L\<bar> < r))"
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  NSLIMSEQ :: "[nat=>real,real] => bool"    ("((_)/ ----NS> (_))" [60, 60] 60)
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    --{*Nonstandard definition of convergence of sequence*}
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  "X ----NS> L == (\<forall>N \<in> HNatInfinite. ( *fNat* X) N \<approx> hypreal_of_real L)"
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  lim :: "(nat => real) => real"
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    --{*Standard definition of limit using choice operator*}
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  "lim X == (@L. (X ----> L))"
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  nslim :: "(nat => real) => real"
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    --{*Nonstandard definition of limit using choice operator*}
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  "nslim X == (@L. (X ----NS> L))"
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  convergent :: "(nat => real) => bool"
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    --{*Standard definition of convergence*}
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  "convergent X == (\<exists>L. (X ----> L))"
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  NSconvergent :: "(nat => real) => bool"
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    --{*Nonstandard definition of convergence*}
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  "NSconvergent X == (\<exists>L. (X ----NS> L))"
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  Bseq :: "(nat => real) => bool"
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    --{*Standard definition for bounded sequence*}
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  "Bseq X == (\<exists>K. (0 < K & (\<forall>n. \<bar>X n\<bar> \<le> K)))"
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  NSBseq :: "(nat=>real) => bool"
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    --{*Nonstandard definition for bounded sequence*}
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  "NSBseq X == (\<forall>N \<in> HNatInfinite. ( *fNat* X) N : HFinite)"
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  monoseq :: "(nat=>real)=>bool"
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    --{*Definition for monotonicity*}
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  "monoseq X == ((\<forall>(m::nat) n. m \<le> n --> X m \<le> X n) |
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                 (\<forall>m n. m \<le> n --> X n \<le> X m))"
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  subseq :: "(nat => nat) => bool"
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    --{*Definition of subsequence*}
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  "subseq f == (\<forall>m n. m < n --> (f m) < (f n))"
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  Cauchy :: "(nat => real) => bool"
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    --{*Standard definition of the Cauchy condition*}
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  "Cauchy X == (\<forall>e. (0 < e -->
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                       (\<exists>M. (\<forall>m n. M \<le> m & M \<le> n
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                             --> abs((X m) + -(X n)) < e))))"
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  NSCauchy :: "(nat => real) => bool"
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    --{*Nonstandard definition*}
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  "NSCauchy X == (\<forall>M \<in> HNatInfinite. \<forall>N \<in> HNatInfinite.
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                      ( *fNat* X) M \<approx> ( *fNat* X) N)"
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text{* Example of an hypersequence (i.e. an extended standard sequence)
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   whose term with an hypernatural suffix is an infinitesimal i.e.
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   the whn'nth term of the hypersequence is a member of Infinitesimal*}
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lemma SEQ_Infinitesimal:
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      "( *fNat* (%n::nat. inverse(real(Suc n)))) whn : Infinitesimal"
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apply (simp add: hypnat_omega_def Infinitesimal_FreeUltrafilterNat_iff starfunNat)
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apply (rule bexI, rule_tac [2] lemma_hyprel_refl)
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apply (simp add: real_of_nat_Suc_gt_zero FreeUltrafilterNat_inverse_real_of_posnat)
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done
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subsection{*LIMSEQ and NSLIMSEQ*}
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lemma LIMSEQ_iff:
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      "(X ----> L) =
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       (\<forall>r. 0 <r --> (\<exists>no. \<forall>n. no \<le> n --> \<bar>X n + -L\<bar> < r))"
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by (simp add: LIMSEQ_def)
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lemma NSLIMSEQ_iff:
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    "(X ----NS> L) = (\<forall>N \<in> HNatInfinite. ( *fNat* X) N \<approx> hypreal_of_real L)"
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by (simp add: NSLIMSEQ_def)
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text{*LIMSEQ ==> NSLIMSEQ*}
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lemma LIMSEQ_NSLIMSEQ:
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      "X ----> L ==> X ----NS> L"
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apply (simp add: LIMSEQ_def NSLIMSEQ_def)
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apply (auto simp add: HNatInfinite_FreeUltrafilterNat_iff)
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apply (rule_tac z = N in eq_Abs_hypnat)
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apply (rule approx_minus_iff [THEN iffD2])
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apply (auto simp add: starfunNat mem_infmal_iff [symmetric] hypreal_of_real_def
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              hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff)
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apply (rule bexI [OF _ lemma_hyprel_refl], safe)
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apply (drule_tac x = u in spec, safe)
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apply (drule_tac x = no in spec, fuf)
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apply (blast dest: less_imp_le)
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done
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text{*NSLIMSEQ ==> LIMSEQ*}
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lemma lemma_NSLIMSEQ1: "!!(f::nat=>nat). \<forall>n. n \<le> f n
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           ==> {n. f n = 0} = {0} | {n. f n = 0} = {}"
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apply auto
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apply (drule_tac x = xa in spec)
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apply (drule_tac [2] x = x in spec, auto)
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done
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lemma lemma_NSLIMSEQ2: "{n. f n \<le> Suc u} = {n. f n \<le> u} Un {n. f n = Suc u}"
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by (auto simp add: le_Suc_eq)
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lemma lemma_NSLIMSEQ3:
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     "!!(f::nat=>nat). \<forall>n. n \<le> f n ==> {n. f n = Suc u} \<le> {n. n \<le> Suc u}"
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apply auto
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apply (drule_tac x = x in spec, auto)
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done
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text{* the following sequence @{term "f(n)"} defines a hypernatural *}
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lemma NSLIMSEQ_finite_set:
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     "!!(f::nat=>nat). \<forall>n. n \<le> f n ==> finite {n. f n \<le> u}"
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apply (induct u)
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apply (auto simp add: lemma_NSLIMSEQ2)
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apply (auto intro: lemma_NSLIMSEQ3 [THEN finite_subset] finite_atMost [unfolded atMost_def])
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apply (drule lemma_NSLIMSEQ1, safe)
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apply (simp_all (no_asm_simp)) 
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done
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lemma Compl_less_set: "- {n. u < (f::nat=>nat) n} = {n. f n \<le> u}"
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by (auto dest: less_le_trans simp add: le_def)
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text{* the index set is in the free ultrafilter *}
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lemma FreeUltrafilterNat_NSLIMSEQ:
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     "!!(f::nat=>nat). \<forall>n. n \<le> f n ==> {n. u < f n} : FreeUltrafilterNat"
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apply (rule FreeUltrafilterNat_Compl_iff2 [THEN iffD2])
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apply (rule FreeUltrafilterNat_finite)
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apply (auto dest: NSLIMSEQ_finite_set simp add: Compl_less_set)
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done
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text{* thus, the sequence defines an infinite hypernatural! *}
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lemma HNatInfinite_NSLIMSEQ: "\<forall>n. n \<le> f n
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          ==> Abs_hypnat (hypnatrel `` {f}) : HNatInfinite"
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apply (auto simp add: HNatInfinite_FreeUltrafilterNat_iff)
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apply (rule bexI [OF _ lemma_hypnatrel_refl], safe)
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apply (erule FreeUltrafilterNat_NSLIMSEQ)
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done
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lemma lemmaLIM:
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     "{n. X (f n) + - L = Y n} Int {n. \<bar>Y n\<bar> < r} \<le>
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      {n. \<bar>X (f n) + - L\<bar> < r}"
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by auto
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lemma lemmaLIM2:
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  "{n. \<bar>X (f n) + - L\<bar> < r} Int {n. r \<le> abs (X (f n) + - (L::real))} = {}"
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by auto
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lemma lemmaLIM3: "[| 0 < r; \<forall>n. r \<le> \<bar>X (f n) + - L\<bar>;
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           ( *fNat* X) (Abs_hypnat (hypnatrel `` {f})) +
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           - hypreal_of_real  L \<approx> 0 |] ==> False"
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apply (auto simp add: starfunNat mem_infmal_iff [symmetric] hypreal_of_real_def hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff)
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apply (rename_tac "Y")
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apply (drule_tac x = r in spec, safe)
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apply (drule FreeUltrafilterNat_Int, assumption)
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apply (drule lemmaLIM [THEN [2] FreeUltrafilterNat_subset])
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apply (drule FreeUltrafilterNat_all)
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apply (erule_tac V = "{n. \<bar>Y n\<bar> < r} : FreeUltrafilterNat" in thin_rl)
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apply (drule FreeUltrafilterNat_Int, assumption)
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apply (simp add: lemmaLIM2 FreeUltrafilterNat_empty)
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done
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lemma NSLIMSEQ_LIMSEQ: "X ----NS> L ==> X ----> L"
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apply (simp add: LIMSEQ_def NSLIMSEQ_def)
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apply (rule ccontr, simp, safe)
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txt{* skolemization step *}
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apply (drule choice, safe)
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apply (drule_tac x = "Abs_hypnat (hypnatrel``{f}) " in bspec)
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apply (drule_tac [2] approx_minus_iff [THEN iffD1])
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apply (simp_all add: linorder_not_less)
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apply (blast intro: HNatInfinite_NSLIMSEQ)
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apply (blast intro: lemmaLIM3)
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done
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text{* Now, the all-important result is trivially proved! *}
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theorem LIMSEQ_NSLIMSEQ_iff: "(f ----> L) = (f ----NS> L)"
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by (blast intro: LIMSEQ_NSLIMSEQ NSLIMSEQ_LIMSEQ)
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subsection{*Theorems About Sequences*}
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lemma NSLIMSEQ_const: "(%n. k) ----NS> k"
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by (simp add: NSLIMSEQ_def)
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lemma LIMSEQ_const: "(%n. k) ----> k"
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by (simp add: LIMSEQ_def)
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lemma NSLIMSEQ_add:
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      "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n + Y n) ----NS> a + b"
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by (auto intro: approx_add simp add: NSLIMSEQ_def starfunNat_add [symmetric])
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lemma LIMSEQ_add: "[| X ----> a; Y ----> b |] ==> (%n. X n + Y n) ----> a + b"
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by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_add)
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lemma NSLIMSEQ_mult:
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      "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n * Y n) ----NS> a * b"
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by (auto intro!: approx_mult_HFinite 
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        simp add: NSLIMSEQ_def hypreal_of_real_mult starfunNat_mult [symmetric])
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lemma LIMSEQ_mult: "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
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by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_mult)
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lemma NSLIMSEQ_minus: "X ----NS> a ==> (%n. -(X n)) ----NS> -a"
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by (auto simp add: NSLIMSEQ_def starfunNat_minus [symmetric])
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lemma LIMSEQ_minus: "X ----> a ==> (%n. -(X n)) ----> -a"
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by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_minus)
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lemma LIMSEQ_minus_cancel: "(%n. -(X n)) ----> -a ==> X ----> a"
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by (drule LIMSEQ_minus, simp)
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lemma NSLIMSEQ_minus_cancel: "(%n. -(X n)) ----NS> -a ==> X ----NS> a"
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by (drule NSLIMSEQ_minus, simp)
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lemma NSLIMSEQ_add_minus:
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     "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n + -Y n) ----NS> a + -b"
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by (simp add: NSLIMSEQ_add NSLIMSEQ_minus [of Y])
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lemma LIMSEQ_add_minus:
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     "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
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by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_add_minus)
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lemma LIMSEQ_diff: "[| X ----> a; Y ----> b |] ==> (%n. X n - Y n) ----> a - b"
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apply (simp add: diff_minus)
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apply (blast intro: LIMSEQ_add_minus)
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done
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lemma NSLIMSEQ_diff:
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     "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n - Y n) ----NS> a - b"
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apply (simp add: diff_minus)
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apply (blast intro: NSLIMSEQ_add_minus)
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done
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text{*Proof is like that of @{text NSLIM_inverse}.*}
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lemma NSLIMSEQ_inverse:
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     "[| X ----NS> a;  a ~= 0 |] ==> (%n. inverse(X n)) ----NS> inverse(a)"
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by (simp add: NSLIMSEQ_def starfunNat_inverse [symmetric] 
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              hypreal_of_real_approx_inverse)
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text{*Standard version of theorem*}
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lemma LIMSEQ_inverse:
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     "[| X ----> a; a ~= 0 |] ==> (%n. inverse(X n)) ----> inverse(a)"
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by (simp add: NSLIMSEQ_inverse LIMSEQ_NSLIMSEQ_iff)
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lemma NSLIMSEQ_mult_inverse:
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     "[| X ----NS> a;  Y ----NS> b;  b ~= 0 |] ==> (%n. X n / Y n) ----NS> a/b"
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by (simp add: NSLIMSEQ_mult NSLIMSEQ_inverse divide_inverse)
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lemma LIMSEQ_divide:
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     "[| X ----> a;  Y ----> b;  b ~= 0 |] ==> (%n. X n / Y n) ----> a/b"
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by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
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text{*Uniqueness of limit*}
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lemma NSLIMSEQ_unique: "[| X ----NS> a; X ----NS> b |] ==> a = b"
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apply (simp add: NSLIMSEQ_def)
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apply (drule HNatInfinite_whn [THEN [2] bspec])+
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apply (auto dest: approx_trans3)
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done
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lemma LIMSEQ_unique: "[| X ----> a; X ----> b |] ==> a = b"
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by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_unique)
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subsection{*Nslim and Lim*}
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lemma limI: "X ----> L ==> lim X = L"
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apply (simp add: lim_def)
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apply (blast intro: LIMSEQ_unique)
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done
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lemma nslimI: "X ----NS> L ==> nslim X = L"
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apply (simp add: nslim_def)
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apply (blast intro: NSLIMSEQ_unique)
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done
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lemma lim_nslim_iff: "lim X = nslim X"
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by (simp add: lim_def nslim_def LIMSEQ_NSLIMSEQ_iff)
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subsection{*Convergence*}
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lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
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by (simp add: convergent_def)
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lemma convergentI: "(X ----> L) ==> convergent X"
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by (auto simp add: convergent_def)
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lemma NSconvergentD: "NSconvergent X ==> \<exists>L. (X ----NS> L)"
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by (simp add: NSconvergent_def)
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   307
paulson@15082
   308
lemma NSconvergentI: "(X ----NS> L) ==> NSconvergent X"
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   309
by (auto simp add: NSconvergent_def)
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   310
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   311
lemma convergent_NSconvergent_iff: "convergent X = NSconvergent X"
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   312
by (simp add: convergent_def NSconvergent_def LIMSEQ_NSLIMSEQ_iff)
paulson@15082
   313
paulson@15082
   314
lemma NSconvergent_NSLIMSEQ_iff: "NSconvergent X = (X ----NS> nslim X)"
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   315
by (auto intro: someI simp add: NSconvergent_def nslim_def)
paulson@15082
   316
paulson@15082
   317
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
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   318
by (auto intro: someI simp add: convergent_def lim_def)
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   319
paulson@15082
   320
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
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   321
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   322
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
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   323
apply (simp add: subseq_def)
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   324
apply (auto dest!: less_imp_Suc_add)
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   325
apply (induct_tac k)
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   326
apply (auto intro: less_trans)
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   327
done
paulson@15082
   328
paulson@15082
   329
paulson@15082
   330
subsection{*Monotonicity*}
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   331
paulson@15082
   332
lemma monoseq_Suc:
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   333
   "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
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   334
                 | (\<forall>n. X (Suc n) \<le> X n))"
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   335
apply (simp add: monoseq_def)
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   336
apply (auto dest!: le_imp_less_or_eq)
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   337
apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
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   338
apply (induct_tac "ka")
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   339
apply (auto intro: order_trans)
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   340
apply (erule swap) 
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   341
apply (induct_tac "k")
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   342
apply (auto intro: order_trans)
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   343
done
paulson@15082
   344
paulson@15082
   345
lemma monoI1: "\<forall>m n. m \<le> n --> X m \<le> X n ==> monoseq X"
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   346
by (simp add: monoseq_def)
paulson@15082
   347
paulson@15082
   348
lemma monoI2: "\<forall>m n. m \<le> n --> X n \<le> X m ==> monoseq X"
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   349
by (simp add: monoseq_def)
paulson@15082
   350
paulson@15082
   351
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
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   352
by (simp add: monoseq_Suc)
paulson@15082
   353
paulson@15082
   354
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
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   355
by (simp add: monoseq_Suc)
paulson@15082
   356
paulson@15082
   357
paulson@15082
   358
subsection{*Bounded Sequence*}
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   359
paulson@15082
   360
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. \<bar>X n\<bar> \<le> K)"
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   361
by (simp add: Bseq_def)
paulson@15082
   362
paulson@15082
   363
lemma BseqI: "[| 0 < K; \<forall>n. \<bar>X n\<bar> \<le> K |] ==> Bseq X"
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   364
by (auto simp add: Bseq_def)
paulson@15082
   365
paulson@15082
   366
lemma lemma_NBseq_def:
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   367
     "(\<exists>K. 0 < K & (\<forall>n. \<bar>X n\<bar> \<le> K)) =
paulson@15082
   368
      (\<exists>N. \<forall>n. \<bar>X n\<bar> \<le> real(Suc N))"
paulson@15082
   369
apply auto
paulson@15082
   370
 prefer 2 apply force
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   371
apply (cut_tac x = K in reals_Archimedean2, clarify)
paulson@15082
   372
apply (rule_tac x = n in exI, clarify)
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   373
apply (drule_tac x = na in spec)
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   374
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   375
done
paulson@15082
   376
paulson@15082
   377
text{* alternative definition for Bseq *}
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   378
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. \<bar>X n\<bar> \<le> real(Suc N))"
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   379
apply (simp add: Bseq_def)
paulson@15082
   380
apply (simp (no_asm) add: lemma_NBseq_def)
paulson@15082
   381
done
paulson@15082
   382
paulson@15082
   383
lemma lemma_NBseq_def2:
paulson@15082
   384
     "(\<exists>K. 0 < K & (\<forall>n. \<bar>X n\<bar> \<le> K)) =
paulson@15082
   385
      (\<exists>N. \<forall>n. \<bar>X n\<bar> < real(Suc N))"
paulson@15082
   386
apply (subst lemma_NBseq_def, auto)
paulson@15082
   387
apply (rule_tac x = "Suc N" in exI)
paulson@15082
   388
apply (rule_tac [2] x = N in exI)
paulson@15082
   389
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   390
 prefer 2 apply (blast intro: order_less_imp_le)
paulson@15082
   391
apply (drule_tac x = n in spec, simp)
paulson@15082
   392
done
paulson@15082
   393
paulson@15082
   394
(* yet another definition for Bseq *)
paulson@15082
   395
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. \<bar>X n\<bar> < real(Suc N))"
paulson@15082
   396
by (simp add: Bseq_def lemma_NBseq_def2)
paulson@15082
   397
paulson@15082
   398
lemma NSBseqD: "[| NSBseq X;  N: HNatInfinite |] ==> ( *fNat* X) N : HFinite"
paulson@15082
   399
by (simp add: NSBseq_def)
paulson@15082
   400
paulson@15082
   401
lemma NSBseqI: "\<forall>N \<in> HNatInfinite. ( *fNat* X) N : HFinite ==> NSBseq X"
paulson@15082
   402
by (simp add: NSBseq_def)
paulson@15082
   403
paulson@15082
   404
text{*The standard definition implies the nonstandard definition*}
paulson@15082
   405
paulson@15082
   406
lemma lemma_Bseq: "\<forall>n. \<bar>X n\<bar> \<le> K ==> \<forall>n. abs(X((f::nat=>nat) n)) \<le> K"
paulson@15082
   407
by auto
paulson@15082
   408
paulson@15082
   409
lemma Bseq_NSBseq: "Bseq X ==> NSBseq X"
paulson@15082
   410
apply (simp add: Bseq_def NSBseq_def, safe)
paulson@15082
   411
apply (rule_tac z = N in eq_Abs_hypnat)
paulson@15082
   412
apply (auto simp add: starfunNat HFinite_FreeUltrafilterNat_iff 
paulson@15082
   413
                      HNatInfinite_FreeUltrafilterNat_iff)
paulson@15082
   414
apply (rule bexI [OF _ lemma_hyprel_refl]) 
paulson@15082
   415
apply (drule_tac f = Xa in lemma_Bseq)
paulson@15082
   416
apply (rule_tac x = "K+1" in exI)
paulson@15082
   417
apply (drule_tac P="%n. ?f n \<le> K" in FreeUltrafilterNat_all, ultra)
paulson@15082
   418
done
paulson@15082
   419
paulson@15082
   420
text{*The nonstandard definition implies the standard definition*}
paulson@15082
   421
paulson@15082
   422
(* similar to NSLIM proof in REALTOPOS *)
paulson@15082
   423
paulson@15082
   424
text{* We need to get rid of the real variable and do so by proving the
paulson@15082
   425
   following, which relies on the Archimedean property of the reals.
paulson@15082
   426
   When we skolemize we then get the required function @{term "f::nat=>nat"}.
paulson@15082
   427
   Otherwise, we would be stuck with a skolem function @{term "f::real=>nat"}
paulson@15082
   428
   which woulid be useless.*}
paulson@15082
   429
paulson@15082
   430
lemma lemmaNSBseq:
paulson@15082
   431
     "\<forall>K. 0 < K --> (\<exists>n. K < \<bar>X n\<bar>)
paulson@15082
   432
      ==> \<forall>N. \<exists>n. real(Suc N) < \<bar>X n\<bar>"
paulson@15082
   433
apply safe
paulson@15082
   434
apply (cut_tac n = N in real_of_nat_Suc_gt_zero, blast)
paulson@15082
   435
done
paulson@15082
   436
paulson@15082
   437
lemma lemmaNSBseq2: "\<forall>K. 0 < K --> (\<exists>n. K < \<bar>X n\<bar>)
paulson@15082
   438
                     ==> \<exists>f. \<forall>N. real(Suc N) < \<bar>X (f N)\<bar>"
paulson@15082
   439
apply (drule lemmaNSBseq)
paulson@15082
   440
apply (drule choice, blast)
paulson@15082
   441
done
paulson@15082
   442
paulson@15082
   443
lemma real_seq_to_hypreal_HInfinite:
paulson@15082
   444
     "\<forall>N. real(Suc N) < \<bar>X (f N)\<bar>
paulson@15082
   445
      ==>  Abs_hypreal(hyprel``{X o f}) : HInfinite"
paulson@15082
   446
apply (auto simp add: HInfinite_FreeUltrafilterNat_iff o_def)
paulson@15082
   447
apply (rule bexI [OF _ lemma_hyprel_refl], clarify)  
paulson@15082
   448
apply (cut_tac u = u in FreeUltrafilterNat_nat_gt_real)
paulson@15082
   449
apply (drule FreeUltrafilterNat_all)
paulson@15082
   450
apply (erule FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset])
paulson@15082
   451
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   452
done
paulson@15082
   453
paulson@15082
   454
text{* Now prove that we can get out an infinite hypernatural as well
paulson@15082
   455
     defined using the skolem function  @{term "f::nat=>nat"} above*}
paulson@15082
   456
paulson@15082
   457
lemma lemma_finite_NSBseq:
paulson@15082
   458
     "{n. f n \<le> Suc u & real(Suc n) < \<bar>X (f n)\<bar>} \<le>
paulson@15082
   459
      {n. f n \<le> u & real(Suc n) < \<bar>X (f n)\<bar>} Un
paulson@15082
   460
      {n. real(Suc n) < \<bar>X (Suc u)\<bar>}"
paulson@15082
   461
by (auto dest!: le_imp_less_or_eq)
paulson@15082
   462
paulson@15082
   463
lemma lemma_finite_NSBseq2:
paulson@15082
   464
     "finite {n. f n \<le> (u::nat) &  real(Suc n) < \<bar>X(f n)\<bar>}"
paulson@15251
   465
apply (induct "u")
paulson@15082
   466
apply (rule_tac [2] lemma_finite_NSBseq [THEN finite_subset])
paulson@15082
   467
apply (rule_tac B = "{n. real (Suc n) < \<bar>X 0\<bar> }" in finite_subset)
paulson@15082
   468
apply (auto intro: finite_real_of_nat_less_real 
paulson@15082
   469
            simp add: real_of_nat_Suc less_diff_eq [symmetric])
paulson@15082
   470
done
paulson@15082
   471
paulson@15082
   472
lemma HNatInfinite_skolem_f:
paulson@15082
   473
     "\<forall>N. real(Suc N) < \<bar>X (f N)\<bar>
paulson@15082
   474
      ==> Abs_hypnat(hypnatrel``{f}) : HNatInfinite"
paulson@15082
   475
apply (auto simp add: HNatInfinite_FreeUltrafilterNat_iff)
paulson@15082
   476
apply (rule bexI [OF _ lemma_hypnatrel_refl], safe)
paulson@15082
   477
apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem)
paulson@15082
   478
apply (rule lemma_finite_NSBseq2 [THEN FreeUltrafilterNat_finite, THEN notE]) 
paulson@15082
   479
apply (subgoal_tac "{n. f n \<le> u & real (Suc n) < \<bar>X (f n)\<bar>} =
paulson@15082
   480
                    {n. f n \<le> u} \<inter> {N. real (Suc N) < \<bar>X (f N)\<bar>}")
paulson@15082
   481
apply (erule ssubst) 
paulson@15082
   482
 apply (auto simp add: linorder_not_less Compl_def)
paulson@15082
   483
done
paulson@15082
   484
paulson@15082
   485
lemma NSBseq_Bseq: "NSBseq X ==> Bseq X"
paulson@15082
   486
apply (simp add: Bseq_def NSBseq_def)
paulson@15082
   487
apply (rule ccontr)
paulson@15082
   488
apply (auto simp add: linorder_not_less [symmetric])
paulson@15082
   489
apply (drule lemmaNSBseq2, safe)
paulson@15082
   490
apply (frule_tac X = X and f = f in real_seq_to_hypreal_HInfinite)
paulson@15082
   491
apply (drule HNatInfinite_skolem_f [THEN [2] bspec])
paulson@15082
   492
apply (auto simp add: starfunNat o_def HFinite_HInfinite_iff)
paulson@15082
   493
done
paulson@15082
   494
paulson@15082
   495
text{* Equivalence of nonstandard and standard definitions
paulson@15082
   496
  for a bounded sequence*}
paulson@15082
   497
lemma Bseq_NSBseq_iff: "(Bseq X) = (NSBseq X)"
paulson@15082
   498
by (blast intro!: NSBseq_Bseq Bseq_NSBseq)
paulson@15082
   499
paulson@15082
   500
text{*A convergent sequence is bounded: 
paulson@15082
   501
 Boundedness as a necessary condition for convergence. 
paulson@15082
   502
 The nonstandard version has no existential, as usual *}
paulson@15082
   503
paulson@15082
   504
lemma NSconvergent_NSBseq: "NSconvergent X ==> NSBseq X"
paulson@15082
   505
apply (simp add: NSconvergent_def NSBseq_def NSLIMSEQ_def)
paulson@15082
   506
apply (blast intro: HFinite_hypreal_of_real approx_sym approx_HFinite)
paulson@15082
   507
done
paulson@15082
   508
paulson@15082
   509
text{*Standard Version: easily now proved using equivalence of NS and
paulson@15082
   510
 standard definitions *}
paulson@15082
   511
lemma convergent_Bseq: "convergent X ==> Bseq X"
paulson@15082
   512
by (simp add: NSconvergent_NSBseq convergent_NSconvergent_iff Bseq_NSBseq_iff)
paulson@15082
   513
paulson@15082
   514
paulson@15082
   515
subsection{*Upper Bounds and Lubs of Bounded Sequences*}
paulson@15082
   516
paulson@15082
   517
lemma Bseq_isUb:
paulson@15082
   518
  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
paulson@15082
   519
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_interval_iff)
paulson@15082
   520
paulson@15082
   521
paulson@15082
   522
text{* Use completeness of reals (supremum property)
paulson@15082
   523
   to show that any bounded sequence has a least upper bound*}
paulson@15082
   524
paulson@15082
   525
lemma Bseq_isLub:
paulson@15082
   526
  "!!(X::nat=>real). Bseq X ==>
paulson@15082
   527
   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
paulson@15082
   528
by (blast intro: reals_complete Bseq_isUb)
paulson@15082
   529
paulson@15082
   530
lemma NSBseq_isUb: "NSBseq X ==> \<exists>U. isUb UNIV {x. \<exists>n. X n = x} U"
paulson@15082
   531
by (simp add: Bseq_NSBseq_iff [symmetric] Bseq_isUb)
paulson@15082
   532
paulson@15082
   533
lemma NSBseq_isLub: "NSBseq X ==> \<exists>U. isLub UNIV {x. \<exists>n. X n = x} U"
paulson@15082
   534
by (simp add: Bseq_NSBseq_iff [symmetric] Bseq_isLub)
paulson@15082
   535
paulson@15082
   536
paulson@15082
   537
subsection{*A Bounded and Monotonic Sequence Converges*}
paulson@15082
   538
paulson@15082
   539
lemma lemma_converg1:
paulson@15082
   540
     "!!(X::nat=>real). [| \<forall>m n. m \<le> n -->  X m \<le> X n;
paulson@15082
   541
                  isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
paulson@15082
   542
               |] ==> \<forall>n. ma \<le> n --> X n = X ma"
paulson@15082
   543
apply safe
paulson@15082
   544
apply (drule_tac y = "X n" in isLubD2)
paulson@15082
   545
apply (blast dest: order_antisym)+
paulson@15082
   546
done
paulson@15082
   547
paulson@15082
   548
text{* The best of both worlds: Easier to prove this result as a standard
paulson@15082
   549
   theorem and then use equivalence to "transfer" it into the
paulson@15082
   550
   equivalent nonstandard form if needed!*}
paulson@15082
   551
paulson@15082
   552
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
paulson@15082
   553
apply (simp add: LIMSEQ_def)
paulson@15082
   554
apply (rule_tac x = "X m" in exI, safe)
paulson@15082
   555
apply (rule_tac x = m in exI, safe)
paulson@15082
   556
apply (drule spec, erule impE, auto)
paulson@15082
   557
done
paulson@15082
   558
paulson@15082
   559
text{*Now, the same theorem in terms of NS limit *}
paulson@15082
   560
lemma Bmonoseq_NSLIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----NS> L)"
paulson@15082
   561
by (auto dest!: Bmonoseq_LIMSEQ simp add: LIMSEQ_NSLIMSEQ_iff)
paulson@15082
   562
paulson@15082
   563
lemma lemma_converg2:
paulson@15082
   564
   "!!(X::nat=>real).
paulson@15082
   565
    [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
paulson@15082
   566
apply safe
paulson@15082
   567
apply (drule_tac y = "X m" in isLubD2)
paulson@15082
   568
apply (auto dest!: order_le_imp_less_or_eq)
paulson@15082
   569
done
paulson@15082
   570
paulson@15082
   571
lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
paulson@15082
   572
by (rule setleI [THEN isUbI], auto)
paulson@15082
   573
paulson@15082
   574
text{* FIXME: @{term "U - T < U"} is redundant *}
paulson@15082
   575
lemma lemma_converg4: "!!(X::nat=> real).
paulson@15082
   576
               [| \<forall>m. X m ~= U;
paulson@15082
   577
                  isLub UNIV {x. \<exists>n. X n = x} U;
paulson@15082
   578
                  0 < T;
paulson@15082
   579
                  U + - T < U
paulson@15082
   580
               |] ==> \<exists>m. U + -T < X m & X m < U"
paulson@15082
   581
apply (drule lemma_converg2, assumption)
paulson@15082
   582
apply (rule ccontr, simp)
paulson@15082
   583
apply (simp add: linorder_not_less)
paulson@15082
   584
apply (drule lemma_converg3)
paulson@15082
   585
apply (drule isLub_le_isUb, assumption)
paulson@15082
   586
apply (auto dest: order_less_le_trans)
paulson@15082
   587
done
paulson@15082
   588
paulson@15082
   589
text{*A standard proof of the theorem for monotone increasing sequence*}
paulson@15082
   590
paulson@15082
   591
lemma Bseq_mono_convergent:
paulson@15082
   592
     "[| Bseq X; \<forall>m n. m \<le> n --> X m \<le> X n |] ==> convergent X"
paulson@15082
   593
apply (simp add: convergent_def)
paulson@15082
   594
apply (frule Bseq_isLub, safe)
paulson@15082
   595
apply (case_tac "\<exists>m. X m = U", auto)
paulson@15082
   596
apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
paulson@15082
   597
(* second case *)
paulson@15082
   598
apply (rule_tac x = U in exI)
paulson@15082
   599
apply (subst LIMSEQ_iff, safe)
paulson@15082
   600
apply (frule lemma_converg2, assumption)
paulson@15082
   601
apply (drule lemma_converg4, auto)
paulson@15082
   602
apply (rule_tac x = m in exI, safe)
paulson@15082
   603
apply (subgoal_tac "X m \<le> X n")
paulson@15082
   604
 prefer 2 apply blast
paulson@15082
   605
apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
paulson@15082
   606
done
paulson@15082
   607
paulson@15082
   608
text{*Nonstandard version of the theorem*}
paulson@15082
   609
paulson@15082
   610
lemma NSBseq_mono_NSconvergent:
paulson@15082
   611
     "[| NSBseq X; \<forall>m n. m \<le> n --> X m \<le> X n |] ==> NSconvergent X"
paulson@15082
   612
by (auto intro: Bseq_mono_convergent 
paulson@15082
   613
         simp add: convergent_NSconvergent_iff [symmetric] 
paulson@15082
   614
                   Bseq_NSBseq_iff [symmetric])
paulson@15082
   615
paulson@15082
   616
lemma convergent_minus_iff: "(convergent X) = (convergent (%n. -(X n)))"
paulson@15082
   617
apply (simp add: convergent_def)
paulson@15082
   618
apply (auto dest: LIMSEQ_minus)
paulson@15082
   619
apply (drule LIMSEQ_minus, auto)
paulson@15082
   620
done
paulson@15082
   621
paulson@15082
   622
lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
paulson@15082
   623
by (simp add: Bseq_def)
paulson@15082
   624
paulson@15082
   625
text{*Main monotonicity theorem*}
paulson@15082
   626
lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
paulson@15082
   627
apply (simp add: monoseq_def, safe)
paulson@15082
   628
apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
paulson@15082
   629
apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
paulson@15082
   630
apply (auto intro!: Bseq_mono_convergent)
paulson@15082
   631
done
paulson@15082
   632
paulson@15082
   633
paulson@15082
   634
subsection{*A Few More Equivalence Theorems for Boundedness*}
paulson@15082
   635
paulson@15082
   636
text{*alternative formulation for boundedness*}
paulson@15082
   637
lemma Bseq_iff2: "Bseq X = (\<exists>k x. 0 < k & (\<forall>n. \<bar>X(n) + -x\<bar> \<le> k))"
paulson@15082
   638
apply (unfold Bseq_def, safe)
paulson@15082
   639
apply (rule_tac [2] x = "k + \<bar>x\<bar> " in exI)
paulson@15082
   640
apply (rule_tac x = K in exI)
paulson@15221
   641
apply (rule exI [where x = 0], auto)
paulson@15221
   642
apply (drule_tac x=n in spec, arith)+
paulson@15082
   643
done
paulson@15082
   644
paulson@15082
   645
text{*alternative formulation for boundedness*}
paulson@15082
   646
lemma Bseq_iff3: "Bseq X = (\<exists>k N. 0 < k & (\<forall>n. abs(X(n) + -X(N)) \<le> k))"
paulson@15082
   647
apply safe
paulson@15082
   648
apply (simp add: Bseq_def, safe)
paulson@15082
   649
apply (rule_tac x = "K + \<bar>X N\<bar> " in exI)
paulson@15082
   650
apply auto
paulson@15082
   651
apply arith
paulson@15082
   652
apply (rule_tac x = N in exI, safe)
paulson@15082
   653
apply (drule_tac x = n in spec, arith)
paulson@15082
   654
apply (auto simp add: Bseq_iff2)
paulson@15082
   655
done
paulson@15082
   656
paulson@15082
   657
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> K) ==> Bseq f"
paulson@15082
   658
apply (simp add: Bseq_def)
paulson@15221
   659
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
paulson@15082
   660
apply (drule_tac [2] x = n in spec, arith+)
paulson@15082
   661
done
paulson@15082
   662
paulson@15082
   663
paulson@15082
   664
subsection{*Equivalence Between NS and Standard Cauchy Sequences*}
paulson@15082
   665
paulson@15082
   666
subsubsection{*Standard Implies Nonstandard*}
paulson@15082
   667
paulson@15082
   668
lemma lemmaCauchy1:
paulson@15082
   669
     "Abs_hypnat (hypnatrel `` {x}) : HNatInfinite
paulson@15082
   670
      ==> {n. M \<le> x n} : FreeUltrafilterNat"
paulson@15082
   671
apply (auto simp add: HNatInfinite_FreeUltrafilterNat_iff)
paulson@15082
   672
apply (drule_tac x = M in spec, ultra)
paulson@15082
   673
done
paulson@15082
   674
paulson@15082
   675
lemma lemmaCauchy2:
paulson@15082
   676
     "{n. \<forall>m n. M \<le> m & M \<le> (n::nat) --> \<bar>X m + - X n\<bar> < u} Int
paulson@15082
   677
      {n. M \<le> xa n} Int {n. M \<le> x n} \<le>
paulson@15082
   678
      {n. abs (X (xa n) + - X (x n)) < u}"
paulson@15082
   679
by blast
paulson@15082
   680
paulson@15082
   681
lemma Cauchy_NSCauchy: "Cauchy X ==> NSCauchy X"
paulson@15082
   682
apply (simp add: Cauchy_def NSCauchy_def, safe)
paulson@15082
   683
apply (rule_tac z = M in eq_Abs_hypnat)
paulson@15082
   684
apply (rule_tac z = N in eq_Abs_hypnat)
paulson@15082
   685
apply (rule approx_minus_iff [THEN iffD2])
paulson@15082
   686
apply (rule mem_infmal_iff [THEN iffD1])
paulson@15082
   687
apply (auto simp add: starfunNat hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff)
paulson@15082
   688
apply (rule bexI, auto)
paulson@15082
   689
apply (drule spec, auto)
paulson@15082
   690
apply (drule_tac M = M in lemmaCauchy1)
paulson@15082
   691
apply (drule_tac M = M in lemmaCauchy1)
paulson@15082
   692
apply (rule_tac x1 = xa in lemmaCauchy2 [THEN [2] FreeUltrafilterNat_subset])
paulson@15082
   693
apply (rule FreeUltrafilterNat_Int)
paulson@15082
   694
apply (auto intro: FreeUltrafilterNat_Int FreeUltrafilterNat_Nat_set)
paulson@15082
   695
done
paulson@15082
   696
paulson@15082
   697
subsubsection{*Nonstandard Implies Standard*}
paulson@15082
   698
paulson@15082
   699
lemma NSCauchy_Cauchy: "NSCauchy X ==> Cauchy X"
paulson@15082
   700
apply (auto simp add: Cauchy_def NSCauchy_def)
paulson@15082
   701
apply (rule ccontr, simp)
paulson@15082
   702
apply (auto dest!: choice HNatInfinite_NSLIMSEQ simp add: all_conj_distrib)  
paulson@15082
   703
apply (drule bspec, assumption)
paulson@15082
   704
apply (drule_tac x = "Abs_hypnat (hypnatrel `` {fa}) " in bspec); 
paulson@15082
   705
apply (auto simp add: starfunNat)
paulson@15082
   706
apply (drule approx_minus_iff [THEN iffD1])
paulson@15082
   707
apply (drule mem_infmal_iff [THEN iffD2])
paulson@15082
   708
apply (auto simp add: hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff)
paulson@15082
   709
apply (rename_tac "Y")
paulson@15082
   710
apply (drule_tac x = e in spec, auto)
paulson@15082
   711
apply (drule FreeUltrafilterNat_Int, assumption)
paulson@15082
   712
apply (subgoal_tac "{n. \<bar>X (f n) + - X (fa n)\<bar> < e} \<in> \<U>") 
paulson@15082
   713
 prefer 2 apply (erule FreeUltrafilterNat_subset, force) 
paulson@15082
   714
apply (rule FreeUltrafilterNat_empty [THEN notE]) 
paulson@15082
   715
apply (subgoal_tac
paulson@15082
   716
         "{n. abs (X (f n) + - X (fa n)) < e} Int 
paulson@15082
   717
          {M. ~ abs (X (f M) + - X (fa M)) < e}  =  {}")
paulson@15082
   718
apply auto  
paulson@15082
   719
done
paulson@15082
   720
paulson@15082
   721
paulson@15082
   722
theorem NSCauchy_Cauchy_iff: "NSCauchy X = Cauchy X"
paulson@15082
   723
by (blast intro!: NSCauchy_Cauchy Cauchy_NSCauchy)
paulson@15082
   724
paulson@15082
   725
text{*A Cauchy sequence is bounded -- this is the standard
paulson@15082
   726
  proof mechanization rather than the nonstandard proof*}
paulson@15082
   727
paulson@15082
   728
lemma lemmaCauchy: "\<forall>n. M \<le> n --> \<bar>X M + - X n\<bar> < (1::real)
paulson@15082
   729
          ==>  \<forall>n. M \<le> n --> \<bar>X n\<bar> < 1 + \<bar>X M\<bar>"
paulson@15082
   730
apply safe
paulson@15082
   731
apply (drule spec, auto, arith)
paulson@15082
   732
done
paulson@15082
   733
paulson@15082
   734
lemma less_Suc_cancel_iff: "(n < Suc M) = (n \<le> M)"
paulson@15082
   735
by auto
paulson@15082
   736
paulson@15082
   737
text{* FIXME: Long. Maximal element in subsequence *}
paulson@15082
   738
lemma SUP_rabs_subseq:
paulson@15082
   739
     "\<exists>m. m \<le> M & (\<forall>n. n \<le> M --> \<bar>(X::nat=> real) n\<bar> \<le> \<bar>X m\<bar>)"
paulson@15082
   740
apply (induct M)
paulson@15082
   741
apply (rule_tac x = 0 in exI, simp, safe)
nipkow@15236
   742
apply (cut_tac x = "\<bar>X (Suc M)\<bar>" and y = "\<bar>X m\<bar> " in linorder_less_linear)
paulson@15082
   743
apply safe
paulson@15082
   744
apply (rule_tac x = m in exI)
paulson@15082
   745
apply (rule_tac [2] x = m in exI)
nipkow@15236
   746
apply (rule_tac [3] x = "Suc M" in exI, simp_all, safe)
nipkow@15236
   747
apply (erule_tac [!] m1 = n in le_imp_less_or_eq [THEN disjE]) 
paulson@15082
   748
apply (simp_all add: less_Suc_cancel_iff)
paulson@15082
   749
apply (blast intro: order_le_less_trans [THEN order_less_imp_le])
paulson@15082
   750
done
paulson@15082
   751
paulson@15082
   752
lemma lemma_Nat_covered:
paulson@15082
   753
     "[| \<forall>m::nat. m \<le> M --> P M m;
paulson@15082
   754
         \<forall>m. M \<le> m --> P M m |]
paulson@15082
   755
      ==> \<forall>m. P M m"
paulson@15082
   756
by (auto elim: less_asym simp add: le_def)
paulson@15082
   757
paulson@15082
   758
paulson@15082
   759
lemma lemma_trans1:
paulson@15082
   760
     "[| \<forall>n. n \<le> M --> \<bar>(X::nat=>real) n\<bar> \<le> a;  a < b |]
paulson@15082
   761
      ==> \<forall>n. n \<le> M --> \<bar>X n\<bar> \<le> b"
paulson@15082
   762
by (blast intro: order_le_less_trans [THEN order_less_imp_le])
paulson@15082
   763
paulson@15082
   764
lemma lemma_trans2:
paulson@15082
   765
     "[| \<forall>n. M \<le> n --> \<bar>(X::nat=>real) n\<bar> < a; a < b |]
paulson@15082
   766
      ==> \<forall>n. M \<le> n --> \<bar>X n\<bar>\<le> b"
paulson@15082
   767
by (blast intro: order_less_trans [THEN order_less_imp_le])
paulson@15082
   768
paulson@15082
   769
lemma lemma_trans3:
paulson@15082
   770
     "[| \<forall>n. n \<le> M --> \<bar>X n\<bar> \<le> a; a = b |]
paulson@15082
   771
      ==> \<forall>n. n \<le> M --> \<bar>X n\<bar> \<le> b"
paulson@15082
   772
by auto
paulson@15082
   773
paulson@15082
   774
lemma lemma_trans4: "\<forall>n. M \<le> n --> \<bar>(X::nat=>real) n\<bar> < a
paulson@15082
   775
              ==>  \<forall>n. M \<le> n --> \<bar>X n\<bar> \<le> a"
paulson@15082
   776
by (blast intro: order_less_imp_le)
paulson@15082
   777
paulson@15082
   778
paulson@15082
   779
text{*Proof is more involved than outlines sketched by various authors
paulson@15082
   780
 would suggest*}
paulson@15082
   781
paulson@15082
   782
lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
paulson@15082
   783
apply (simp add: Cauchy_def Bseq_def)
paulson@15082
   784
apply (drule_tac x = 1 in spec)
paulson@15082
   785
apply (erule zero_less_one [THEN [2] impE], safe)
paulson@15082
   786
apply (drule_tac x = M in spec, simp)
paulson@15082
   787
apply (drule lemmaCauchy)
paulson@15082
   788
apply (cut_tac M = M and X = X in SUP_rabs_subseq, safe)
paulson@15082
   789
apply (cut_tac x = "\<bar>X m\<bar> " and y = "1 + \<bar>X M\<bar> " in linorder_less_linear)
paulson@15082
   790
apply safe
paulson@15082
   791
apply (drule lemma_trans1, assumption)
paulson@15082
   792
apply (drule_tac [3] lemma_trans2, erule_tac [3] asm_rl)
paulson@15082
   793
apply (drule_tac [2] lemma_trans3, erule_tac [2] asm_rl)
paulson@15082
   794
apply (drule_tac [3] abs_add_one_gt_zero [THEN order_less_trans])
paulson@15082
   795
apply (drule lemma_trans4)
paulson@15082
   796
apply (drule_tac [2] lemma_trans4)
paulson@15082
   797
apply (rule_tac x = "1 + \<bar>X M\<bar> " in exI)
paulson@15082
   798
apply (rule_tac [2] x = "1 + \<bar>X M\<bar> " in exI)
paulson@15082
   799
apply (rule_tac [3] x = "\<bar>X m\<bar> " in exI)
paulson@15085
   800
apply (auto elim!: lemma_Nat_covered)
paulson@15082
   801
done
paulson@15082
   802
paulson@15082
   803
text{*A Cauchy sequence is bounded -- nonstandard version*}
paulson@15082
   804
paulson@15082
   805
lemma NSCauchy_NSBseq: "NSCauchy X ==> NSBseq X"
paulson@15082
   806
by (simp add: Cauchy_Bseq Bseq_NSBseq_iff [symmetric] NSCauchy_Cauchy_iff)
paulson@15082
   807
paulson@15082
   808
paulson@15082
   809
text{*Equivalence of Cauchy criterion and convergence:
paulson@15082
   810
  We will prove this using our NS formulation which provides a
paulson@15082
   811
  much easier proof than using the standard definition. We do not
paulson@15082
   812
  need to use properties of subsequences such as boundedness,
paulson@15082
   813
  monotonicity etc... Compare with Harrison's corresponding proof
paulson@15082
   814
  in HOL which is much longer and more complicated. Of course, we do
paulson@15082
   815
  not have problems which he encountered with guessing the right
paulson@15082
   816
  instantiations for his 'espsilon-delta' proof(s) in this case
paulson@15082
   817
  since the NS formulations do not involve existential quantifiers.*}
paulson@15082
   818
paulson@15082
   819
lemma NSCauchy_NSconvergent_iff: "NSCauchy X = NSconvergent X"
paulson@15082
   820
apply (simp add: NSconvergent_def NSLIMSEQ_def, safe)
paulson@15082
   821
apply (frule NSCauchy_NSBseq)
paulson@15082
   822
apply (auto intro: approx_trans2 simp add: NSBseq_def NSCauchy_def)
paulson@15082
   823
apply (drule HNatInfinite_whn [THEN [2] bspec])
paulson@15082
   824
apply (drule HNatInfinite_whn [THEN [2] bspec])
paulson@15082
   825
apply (auto dest!: st_part_Ex simp add: SReal_iff)
paulson@15082
   826
apply (blast intro: approx_trans3)
paulson@15082
   827
done
paulson@15082
   828
paulson@15082
   829
text{*Standard proof for free*}
paulson@15082
   830
lemma Cauchy_convergent_iff: "Cauchy X = convergent X"
paulson@15082
   831
by (simp add: NSCauchy_Cauchy_iff [symmetric] convergent_NSconvergent_iff NSCauchy_NSconvergent_iff)
paulson@15082
   832
paulson@15082
   833
paulson@15082
   834
text{*We can now try and derive a few properties of sequences,
paulson@15082
   835
     starting with the limit comparison property for sequences.*}
paulson@15082
   836
paulson@15082
   837
lemma NSLIMSEQ_le:
paulson@15082
   838
       "[| f ----NS> l; g ----NS> m;
paulson@15082
   839
           \<exists>N. \<forall>n. N \<le> n --> f(n) \<le> g(n)
paulson@15082
   840
                |] ==> l \<le> m"
paulson@15082
   841
apply (simp add: NSLIMSEQ_def, safe)
paulson@15082
   842
apply (drule starfun_le_mono)
paulson@15082
   843
apply (drule HNatInfinite_whn [THEN [2] bspec])+
paulson@15082
   844
apply (drule_tac x = whn in spec)
paulson@15082
   845
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
paulson@15082
   846
apply clarify
paulson@15082
   847
apply (auto intro: hypreal_of_real_le_add_Infininitesimal_cancel2)
paulson@15082
   848
done
paulson@15082
   849
paulson@15082
   850
(* standard version *)
paulson@15082
   851
lemma LIMSEQ_le:
paulson@15082
   852
     "[| f ----> l; g ----> m; \<exists>N. \<forall>n. N \<le> n --> f(n) \<le> g(n) |]
paulson@15082
   853
      ==> l \<le> m"
paulson@15082
   854
by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_le)
paulson@15082
   855
paulson@15082
   856
lemma LIMSEQ_le_const: "[| X ----> r; \<forall>n. a \<le> X n |] ==> a \<le> r"
paulson@15082
   857
apply (rule LIMSEQ_le)
paulson@15082
   858
apply (rule LIMSEQ_const, auto)
paulson@15082
   859
done
paulson@15082
   860
paulson@15082
   861
lemma NSLIMSEQ_le_const: "[| X ----NS> r; \<forall>n. a \<le> X n |] ==> a \<le> r"
paulson@15082
   862
by (simp add: LIMSEQ_NSLIMSEQ_iff LIMSEQ_le_const)
paulson@15082
   863
paulson@15082
   864
lemma LIMSEQ_le_const2: "[| X ----> r; \<forall>n. X n \<le> a |] ==> r \<le> a"
paulson@15082
   865
apply (rule LIMSEQ_le)
paulson@15082
   866
apply (rule_tac [2] LIMSEQ_const, auto)
paulson@15082
   867
done
paulson@15082
   868
paulson@15082
   869
lemma NSLIMSEQ_le_const2: "[| X ----NS> r; \<forall>n. X n \<le> a |] ==> r \<le> a"
paulson@15082
   870
by (simp add: LIMSEQ_NSLIMSEQ_iff LIMSEQ_le_const2)
paulson@15082
   871
paulson@15082
   872
text{*Shift a convergent series by 1:
paulson@15082
   873
  By the equivalence between Cauchiness and convergence and because
paulson@15082
   874
  the successor of an infinite hypernatural is also infinite.*}
paulson@15082
   875
paulson@15082
   876
lemma NSLIMSEQ_Suc: "f ----NS> l ==> (%n. f(Suc n)) ----NS> l"
paulson@15082
   877
apply (frule NSconvergentI [THEN NSCauchy_NSconvergent_iff [THEN iffD2]])
paulson@15082
   878
apply (auto simp add: NSCauchy_def NSLIMSEQ_def starfunNat_shift_one)
paulson@15082
   879
apply (drule bspec, assumption)
paulson@15082
   880
apply (drule bspec, assumption)
paulson@15082
   881
apply (drule Nats_1 [THEN [2] HNatInfinite_SHNat_add])
paulson@15082
   882
apply (blast intro: approx_trans3)
paulson@15082
   883
done
paulson@15082
   884
paulson@15082
   885
text{* standard version *}
paulson@15082
   886
lemma LIMSEQ_Suc: "f ----> l ==> (%n. f(Suc n)) ----> l"
paulson@15082
   887
by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_Suc)
paulson@15082
   888
paulson@15082
   889
lemma NSLIMSEQ_imp_Suc: "(%n. f(Suc n)) ----NS> l ==> f ----NS> l"
paulson@15082
   890
apply (frule NSconvergentI [THEN NSCauchy_NSconvergent_iff [THEN iffD2]])
paulson@15082
   891
apply (auto simp add: NSCauchy_def NSLIMSEQ_def starfunNat_shift_one)
paulson@15082
   892
apply (drule bspec, assumption)
paulson@15082
   893
apply (drule bspec, assumption)
paulson@15082
   894
apply (frule Nats_1 [THEN [2] HNatInfinite_SHNat_diff])
paulson@15082
   895
apply (drule_tac x="N - 1" in bspec) 
paulson@15082
   896
apply (auto intro: approx_trans3)
paulson@15082
   897
done
paulson@15082
   898
paulson@15082
   899
lemma LIMSEQ_imp_Suc: "(%n. f(Suc n)) ----> l ==> f ----> l"
paulson@15082
   900
apply (simp add: LIMSEQ_NSLIMSEQ_iff)
paulson@15082
   901
apply (erule NSLIMSEQ_imp_Suc)
paulson@15082
   902
done
paulson@15082
   903
paulson@15082
   904
lemma LIMSEQ_Suc_iff: "((%n. f(Suc n)) ----> l) = (f ----> l)"
paulson@15082
   905
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
paulson@15082
   906
paulson@15082
   907
lemma NSLIMSEQ_Suc_iff: "((%n. f(Suc n)) ----NS> l) = (f ----NS> l)"
paulson@15082
   908
by (blast intro: NSLIMSEQ_imp_Suc NSLIMSEQ_Suc)
paulson@15082
   909
paulson@15082
   910
text{*A sequence tends to zero iff its abs does*}
paulson@15082
   911
lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> 0)"
paulson@15082
   912
by (simp add: LIMSEQ_def)
paulson@15082
   913
paulson@15082
   914
text{*We prove the NS version from the standard one, since the NS proof
paulson@15082
   915
   seems more complicated than the standard one above!*}
paulson@15082
   916
lemma NSLIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----NS> 0) = (f ----NS> 0)"
paulson@15082
   917
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_rabs_zero)
paulson@15082
   918
paulson@15082
   919
text{*Generalization to other limits*}
paulson@15082
   920
lemma NSLIMSEQ_imp_rabs: "f ----NS> l ==> (%n. \<bar>f n\<bar>) ----NS> \<bar>l\<bar>"
paulson@15082
   921
apply (simp add: NSLIMSEQ_def)
paulson@15082
   922
apply (auto intro: approx_hrabs 
paulson@15082
   923
            simp add: starfunNat_rabs hypreal_of_real_hrabs [symmetric])
paulson@15082
   924
done
paulson@15082
   925
paulson@15082
   926
text{* standard version *}
paulson@15082
   927
lemma LIMSEQ_imp_rabs: "f ----> l ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
paulson@15082
   928
by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_imp_rabs)
paulson@15082
   929
paulson@15082
   930
text{*An unbounded sequence's inverse tends to 0*}
paulson@15082
   931
paulson@15082
   932
text{* standard proof seems easier *}
paulson@15082
   933
lemma LIMSEQ_inverse_zero:
paulson@15082
   934
      "\<forall>y. \<exists>N. \<forall>n. N \<le> n --> y < f(n) ==> (%n. inverse(f n)) ----> 0"
paulson@15082
   935
apply (simp add: LIMSEQ_def, safe)
paulson@15082
   936
apply (drule_tac x = "inverse r" in spec, safe)
paulson@15082
   937
apply (rule_tac x = N in exI, safe)
paulson@15082
   938
apply (drule spec, auto)
paulson@15082
   939
apply (frule positive_imp_inverse_positive)
paulson@15082
   940
apply (frule order_less_trans, assumption)
paulson@15082
   941
apply (frule_tac a = "f n" in positive_imp_inverse_positive)
paulson@15082
   942
apply (simp add: abs_if) 
paulson@15082
   943
apply (rule_tac t = r in inverse_inverse_eq [THEN subst])
paulson@15082
   944
apply (auto intro: inverse_less_iff_less [THEN iffD2]
paulson@15082
   945
            simp del: inverse_inverse_eq)
paulson@15082
   946
done
paulson@15082
   947
paulson@15082
   948
lemma NSLIMSEQ_inverse_zero:
paulson@15082
   949
     "\<forall>y. \<exists>N. \<forall>n. N \<le> n --> y < f(n)
paulson@15082
   950
      ==> (%n. inverse(f n)) ----NS> 0"
paulson@15082
   951
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_zero)
paulson@15082
   952
paulson@15082
   953
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
paulson@15082
   954
paulson@15082
   955
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
paulson@15082
   956
apply (rule LIMSEQ_inverse_zero, safe)
paulson@15082
   957
apply (cut_tac x = y in reals_Archimedean2)
paulson@15082
   958
apply (safe, rule_tac x = n in exI)
paulson@15082
   959
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   960
done
paulson@15082
   961
paulson@15082
   962
lemma NSLIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----NS> 0"
paulson@15082
   963
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat)
paulson@15082
   964
paulson@15082
   965
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
paulson@15082
   966
infinity is now easily proved*}
paulson@15082
   967
paulson@15082
   968
lemma LIMSEQ_inverse_real_of_nat_add:
paulson@15082
   969
     "(%n. r + inverse(real(Suc n))) ----> r"
paulson@15082
   970
by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
paulson@15082
   971
paulson@15082
   972
lemma NSLIMSEQ_inverse_real_of_nat_add:
paulson@15082
   973
     "(%n. r + inverse(real(Suc n))) ----NS> r"
paulson@15082
   974
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add)
paulson@15082
   975
paulson@15082
   976
lemma LIMSEQ_inverse_real_of_nat_add_minus:
paulson@15082
   977
     "(%n. r + -inverse(real(Suc n))) ----> r"
paulson@15082
   978
by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
paulson@15082
   979
paulson@15082
   980
lemma NSLIMSEQ_inverse_real_of_nat_add_minus:
paulson@15082
   981
     "(%n. r + -inverse(real(Suc n))) ----NS> r"
paulson@15082
   982
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add_minus)
paulson@15082
   983
paulson@15082
   984
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
paulson@15082
   985
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
paulson@15082
   986
by (cut_tac b=1 in
paulson@15082
   987
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
paulson@15082
   988
paulson@15082
   989
lemma NSLIMSEQ_inverse_real_of_nat_add_minus_mult:
paulson@15082
   990
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----NS> r"
paulson@15082
   991
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add_minus_mult)
paulson@15082
   992
paulson@15082
   993
paulson@15082
   994
text{* Real Powers*}
paulson@15082
   995
paulson@15082
   996
lemma NSLIMSEQ_pow [rule_format]:
paulson@15082
   997
     "(X ----NS> a) --> ((%n. (X n) ^ m) ----NS> a ^ m)"
paulson@15251
   998
apply (induct "m")
paulson@15082
   999
apply (auto intro: NSLIMSEQ_mult NSLIMSEQ_const)
paulson@15082
  1000
done
paulson@15082
  1001
paulson@15082
  1002
lemma LIMSEQ_pow: "X ----> a ==> (%n. (X n) ^ m) ----> a ^ m"
paulson@15082
  1003
by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_pow)
paulson@15082
  1004
paulson@15082
  1005
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
paulson@15082
  1006
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
paulson@15082
  1007
  also fact that bounded and monotonic sequence converges.*}
paulson@15082
  1008
paulson@15082
  1009
lemma Bseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> Bseq (%n. x ^ n)"
paulson@15082
  1010
apply (simp add: Bseq_def)
paulson@15082
  1011
apply (rule_tac x = 1 in exI)
paulson@15082
  1012
apply (simp add: power_abs)
paulson@15082
  1013
apply (auto dest: power_mono intro: order_less_imp_le simp add: abs_if)
paulson@15082
  1014
done
paulson@15082
  1015
paulson@15082
  1016
lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
paulson@15082
  1017
apply (clarify intro!: mono_SucI2)
paulson@15082
  1018
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
paulson@15082
  1019
done
paulson@15082
  1020
paulson@15082
  1021
lemma convergent_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> convergent (%n. x ^ n)"
paulson@15082
  1022
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
paulson@15082
  1023
paulson@15082
  1024
text{* We now use NS criterion to bring proof of theorem through *}
paulson@15082
  1025
paulson@15082
  1026
lemma NSLIMSEQ_realpow_zero: "[| 0 \<le> x; x < 1 |] ==> (%n. x ^ n) ----NS> 0"
paulson@15082
  1027
apply (simp add: NSLIMSEQ_def)
paulson@15082
  1028
apply (auto dest!: convergent_realpow simp add: convergent_NSconvergent_iff)
paulson@15082
  1029
apply (frule NSconvergentD)
paulson@15082
  1030
apply (auto simp add: NSLIMSEQ_def NSCauchy_NSconvergent_iff [symmetric] NSCauchy_def starfunNat_pow)
paulson@15082
  1031
apply (frule HNatInfinite_add_one)
paulson@15082
  1032
apply (drule bspec, assumption)
paulson@15082
  1033
apply (drule bspec, assumption)
paulson@15082
  1034
apply (drule_tac x = "N + (1::hypnat) " in bspec, assumption)
paulson@15082
  1035
apply (simp add: hyperpow_add)
paulson@15082
  1036
apply (drule approx_mult_subst_SReal, assumption)
paulson@15082
  1037
apply (drule approx_trans3, assumption)
paulson@15082
  1038
apply (auto simp del: hypreal_of_real_mult simp add: hypreal_of_real_mult [symmetric])
paulson@15082
  1039
done
paulson@15082
  1040
paulson@15082
  1041
text{* standard version *}
paulson@15082
  1042
lemma LIMSEQ_realpow_zero: "[| 0 \<le> x; x < 1 |] ==> (%n. x ^ n) ----> 0"
paulson@15082
  1043
by (simp add: NSLIMSEQ_realpow_zero LIMSEQ_NSLIMSEQ_iff)
paulson@15082
  1044
paulson@15082
  1045
lemma LIMSEQ_divide_realpow_zero: "1 < x ==> (%n. a / (x ^ n)) ----> 0"
paulson@15082
  1046
apply (cut_tac a = a and x1 = "inverse x" in
paulson@15082
  1047
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
paulson@15082
  1048
apply (auto simp add: divide_inverse power_inverse)
paulson@15082
  1049
apply (simp add: inverse_eq_divide pos_divide_less_eq)
paulson@15082
  1050
done
paulson@15082
  1051
paulson@15102
  1052
text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
paulson@15082
  1053
paulson@15082
  1054
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
paulson@15082
  1055
by (blast intro!: LIMSEQ_realpow_zero abs_ge_zero)
paulson@15082
  1056
paulson@15082
  1057
lemma NSLIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 ==> (%n. \<bar>c\<bar> ^ n) ----NS> 0"
paulson@15082
  1058
by (simp add: LIMSEQ_rabs_realpow_zero LIMSEQ_NSLIMSEQ_iff [symmetric])
paulson@15082
  1059
paulson@15082
  1060
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 ==> (%n. c ^ n) ----> 0"
paulson@15082
  1061
apply (rule LIMSEQ_rabs_zero [THEN iffD1])
paulson@15082
  1062
apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
paulson@15082
  1063
done
paulson@15082
  1064
paulson@15082
  1065
lemma NSLIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 ==> (%n. c ^ n) ----NS> 0"
paulson@15082
  1066
by (simp add: LIMSEQ_rabs_realpow_zero2 LIMSEQ_NSLIMSEQ_iff [symmetric])
paulson@15082
  1067
paulson@15082
  1068
subsection{*Hyperreals and Sequences*}
paulson@15082
  1069
paulson@15082
  1070
text{*A bounded sequence is a finite hyperreal*}
paulson@15082
  1071
lemma NSBseq_HFinite_hypreal: "NSBseq X ==> Abs_hypreal(hyprel``{X}) : HFinite"
paulson@15082
  1072
by (auto intro!: bexI lemma_hyprel_refl 
paulson@15082
  1073
            intro: FreeUltrafilterNat_all [THEN FreeUltrafilterNat_subset]
paulson@15082
  1074
            simp add: HFinite_FreeUltrafilterNat_iff Bseq_NSBseq_iff [symmetric]
paulson@15082
  1075
                      Bseq_iff1a)
paulson@15082
  1076
paulson@15082
  1077
text{*A sequence converging to zero defines an infinitesimal*}
paulson@15082
  1078
lemma NSLIMSEQ_zero_Infinitesimal_hypreal:
paulson@15082
  1079
      "X ----NS> 0 ==> Abs_hypreal(hyprel``{X}) : Infinitesimal"
paulson@15082
  1080
apply (simp add: NSLIMSEQ_def)
paulson@15082
  1081
apply (drule_tac x = whn in bspec)
paulson@15082
  1082
apply (simp add: HNatInfinite_whn)
paulson@15082
  1083
apply (auto simp add: hypnat_omega_def mem_infmal_iff [symmetric] starfunNat)
paulson@15082
  1084
done
paulson@15082
  1085
paulson@15082
  1086
(***---------------------------------------------------------------
paulson@15082
  1087
    Theorems proved by Harrison in HOL that we do not need
paulson@15082
  1088
    in order to prove equivalence between Cauchy criterion
paulson@15082
  1089
    and convergence:
paulson@15082
  1090
 -- Show that every sequence contains a monotonic subsequence
paulson@15082
  1091
Goal "\<exists>f. subseq f & monoseq (%n. s (f n))"
paulson@15082
  1092
 -- Show that a subsequence of a bounded sequence is bounded
paulson@15082
  1093
Goal "Bseq X ==> Bseq (%n. X (f n))";
paulson@15082
  1094
 -- Show we can take subsequential terms arbitrarily far
paulson@15082
  1095
    up a sequence
paulson@15082
  1096
Goal "subseq f ==> n \<le> f(n)";
paulson@15082
  1097
Goal "subseq f ==> \<exists>n. N1 \<le> n & N2 \<le> f(n)";
paulson@15082
  1098
 ---------------------------------------------------------------***)
paulson@15082
  1099
paulson@15241
  1100
paulson@15082
  1101
ML
paulson@15082
  1102
{*
paulson@15082
  1103
val Cauchy_def = thm"Cauchy_def";
paulson@15082
  1104
val SEQ_Infinitesimal = thm "SEQ_Infinitesimal";
paulson@15082
  1105
val LIMSEQ_iff = thm "LIMSEQ_iff";
paulson@15082
  1106
val NSLIMSEQ_iff = thm "NSLIMSEQ_iff";
paulson@15082
  1107
val LIMSEQ_NSLIMSEQ = thm "LIMSEQ_NSLIMSEQ";
paulson@15082
  1108
val NSLIMSEQ_finite_set = thm "NSLIMSEQ_finite_set";
paulson@15082
  1109
val Compl_less_set = thm "Compl_less_set";
paulson@15082
  1110
val FreeUltrafilterNat_NSLIMSEQ = thm "FreeUltrafilterNat_NSLIMSEQ";
paulson@15082
  1111
val HNatInfinite_NSLIMSEQ = thm "HNatInfinite_NSLIMSEQ";
paulson@15082
  1112
val NSLIMSEQ_LIMSEQ = thm "NSLIMSEQ_LIMSEQ";
paulson@15082
  1113
val LIMSEQ_NSLIMSEQ_iff = thm "LIMSEQ_NSLIMSEQ_iff";
paulson@15082
  1114
val NSLIMSEQ_const = thm "NSLIMSEQ_const";
paulson@15082
  1115
val LIMSEQ_const = thm "LIMSEQ_const";
paulson@15082
  1116
val NSLIMSEQ_add = thm "NSLIMSEQ_add";
paulson@15082
  1117
val LIMSEQ_add = thm "LIMSEQ_add";
paulson@15082
  1118
val NSLIMSEQ_mult = thm "NSLIMSEQ_mult";
paulson@15082
  1119
val LIMSEQ_mult = thm "LIMSEQ_mult";
paulson@15082
  1120
val NSLIMSEQ_minus = thm "NSLIMSEQ_minus";
paulson@15082
  1121
val LIMSEQ_minus = thm "LIMSEQ_minus";
paulson@15082
  1122
val LIMSEQ_minus_cancel = thm "LIMSEQ_minus_cancel";
paulson@15082
  1123
val NSLIMSEQ_minus_cancel = thm "NSLIMSEQ_minus_cancel";
paulson@15082
  1124
val NSLIMSEQ_add_minus = thm "NSLIMSEQ_add_minus";
paulson@15082
  1125
val LIMSEQ_add_minus = thm "LIMSEQ_add_minus";
paulson@15082
  1126
val LIMSEQ_diff = thm "LIMSEQ_diff";
paulson@15082
  1127
val NSLIMSEQ_diff = thm "NSLIMSEQ_diff";
paulson@15082
  1128
val NSLIMSEQ_inverse = thm "NSLIMSEQ_inverse";
paulson@15082
  1129
val LIMSEQ_inverse = thm "LIMSEQ_inverse";
paulson@15082
  1130
val NSLIMSEQ_mult_inverse = thm "NSLIMSEQ_mult_inverse";
paulson@15082
  1131
val LIMSEQ_divide = thm "LIMSEQ_divide";
paulson@15082
  1132
val NSLIMSEQ_unique = thm "NSLIMSEQ_unique";
paulson@15082
  1133
val LIMSEQ_unique = thm "LIMSEQ_unique";
paulson@15082
  1134
val limI = thm "limI";
paulson@15082
  1135
val nslimI = thm "nslimI";
paulson@15082
  1136
val lim_nslim_iff = thm "lim_nslim_iff";
paulson@15082
  1137
val convergentD = thm "convergentD";
paulson@15082
  1138
val convergentI = thm "convergentI";
paulson@15082
  1139
val NSconvergentD = thm "NSconvergentD";
paulson@15082
  1140
val NSconvergentI = thm "NSconvergentI";
paulson@15082
  1141
val convergent_NSconvergent_iff = thm "convergent_NSconvergent_iff";
paulson@15082
  1142
val NSconvergent_NSLIMSEQ_iff = thm "NSconvergent_NSLIMSEQ_iff";
paulson@15082
  1143
val convergent_LIMSEQ_iff = thm "convergent_LIMSEQ_iff";
paulson@15082
  1144
val subseq_Suc_iff = thm "subseq_Suc_iff";
paulson@15082
  1145
val monoseq_Suc = thm "monoseq_Suc";
paulson@15082
  1146
val monoI1 = thm "monoI1";
paulson@15082
  1147
val monoI2 = thm "monoI2";
paulson@15082
  1148
val mono_SucI1 = thm "mono_SucI1";
paulson@15082
  1149
val mono_SucI2 = thm "mono_SucI2";
paulson@15082
  1150
val BseqD = thm "BseqD";
paulson@15082
  1151
val BseqI = thm "BseqI";
paulson@15082
  1152
val Bseq_iff = thm "Bseq_iff";
paulson@15082
  1153
val Bseq_iff1a = thm "Bseq_iff1a";
paulson@15082
  1154
val NSBseqD = thm "NSBseqD";
paulson@15082
  1155
val NSBseqI = thm "NSBseqI";
paulson@15082
  1156
val Bseq_NSBseq = thm "Bseq_NSBseq";
paulson@15082
  1157
val real_seq_to_hypreal_HInfinite = thm "real_seq_to_hypreal_HInfinite";
paulson@15082
  1158
val HNatInfinite_skolem_f = thm "HNatInfinite_skolem_f";
paulson@15082
  1159
val NSBseq_Bseq = thm "NSBseq_Bseq";
paulson@15082
  1160
val Bseq_NSBseq_iff = thm "Bseq_NSBseq_iff";
paulson@15082
  1161
val NSconvergent_NSBseq = thm "NSconvergent_NSBseq";
paulson@15082
  1162
val convergent_Bseq = thm "convergent_Bseq";
paulson@15082
  1163
val Bseq_isUb = thm "Bseq_isUb";
paulson@15082
  1164
val Bseq_isLub = thm "Bseq_isLub";
paulson@15082
  1165
val NSBseq_isUb = thm "NSBseq_isUb";
paulson@15082
  1166
val NSBseq_isLub = thm "NSBseq_isLub";
paulson@15082
  1167
val Bmonoseq_LIMSEQ = thm "Bmonoseq_LIMSEQ";
paulson@15082
  1168
val Bmonoseq_NSLIMSEQ = thm "Bmonoseq_NSLIMSEQ";
paulson@15082
  1169
val Bseq_mono_convergent = thm "Bseq_mono_convergent";
paulson@15082
  1170
val NSBseq_mono_NSconvergent = thm "NSBseq_mono_NSconvergent";
paulson@15082
  1171
val convergent_minus_iff = thm "convergent_minus_iff";
paulson@15082
  1172
val Bseq_minus_iff = thm "Bseq_minus_iff";
paulson@15082
  1173
val Bseq_monoseq_convergent = thm "Bseq_monoseq_convergent";
paulson@15082
  1174
val Bseq_iff2 = thm "Bseq_iff2";
paulson@15082
  1175
val Bseq_iff3 = thm "Bseq_iff3";
paulson@15082
  1176
val BseqI2 = thm "BseqI2";
paulson@15082
  1177
val Cauchy_NSCauchy = thm "Cauchy_NSCauchy";
paulson@15082
  1178
val NSCauchy_Cauchy = thm "NSCauchy_Cauchy";
paulson@15082
  1179
val NSCauchy_Cauchy_iff = thm "NSCauchy_Cauchy_iff";
paulson@15082
  1180
val less_Suc_cancel_iff = thm "less_Suc_cancel_iff";
paulson@15082
  1181
val SUP_rabs_subseq = thm "SUP_rabs_subseq";
paulson@15082
  1182
val Cauchy_Bseq = thm "Cauchy_Bseq";
paulson@15082
  1183
val NSCauchy_NSBseq = thm "NSCauchy_NSBseq";
paulson@15082
  1184
val NSCauchy_NSconvergent_iff = thm "NSCauchy_NSconvergent_iff";
paulson@15082
  1185
val Cauchy_convergent_iff = thm "Cauchy_convergent_iff";
paulson@15082
  1186
val NSLIMSEQ_le = thm "NSLIMSEQ_le";
paulson@15082
  1187
val LIMSEQ_le = thm "LIMSEQ_le";
paulson@15082
  1188
val LIMSEQ_le_const = thm "LIMSEQ_le_const";
paulson@15082
  1189
val NSLIMSEQ_le_const = thm "NSLIMSEQ_le_const";
paulson@15082
  1190
val LIMSEQ_le_const2 = thm "LIMSEQ_le_const2";
paulson@15082
  1191
val NSLIMSEQ_le_const2 = thm "NSLIMSEQ_le_const2";
paulson@15082
  1192
val NSLIMSEQ_Suc = thm "NSLIMSEQ_Suc";
paulson@15082
  1193
val LIMSEQ_Suc = thm "LIMSEQ_Suc";
paulson@15082
  1194
val NSLIMSEQ_imp_Suc = thm "NSLIMSEQ_imp_Suc";
paulson@15082
  1195
val LIMSEQ_imp_Suc = thm "LIMSEQ_imp_Suc";
paulson@15082
  1196
val LIMSEQ_Suc_iff = thm "LIMSEQ_Suc_iff";
paulson@15082
  1197
val NSLIMSEQ_Suc_iff = thm "NSLIMSEQ_Suc_iff";
paulson@15082
  1198
val LIMSEQ_rabs_zero = thm "LIMSEQ_rabs_zero";
paulson@15082
  1199
val NSLIMSEQ_rabs_zero = thm "NSLIMSEQ_rabs_zero";
paulson@15082
  1200
val NSLIMSEQ_imp_rabs = thm "NSLIMSEQ_imp_rabs";
paulson@15082
  1201
val LIMSEQ_imp_rabs = thm "LIMSEQ_imp_rabs";
paulson@15082
  1202
val LIMSEQ_inverse_zero = thm "LIMSEQ_inverse_zero";
paulson@15082
  1203
val NSLIMSEQ_inverse_zero = thm "NSLIMSEQ_inverse_zero";
paulson@15082
  1204
val LIMSEQ_inverse_real_of_nat = thm "LIMSEQ_inverse_real_of_nat";
paulson@15082
  1205
val NSLIMSEQ_inverse_real_of_nat = thm "NSLIMSEQ_inverse_real_of_nat";
paulson@15082
  1206
val LIMSEQ_inverse_real_of_nat_add = thm "LIMSEQ_inverse_real_of_nat_add";
paulson@15082
  1207
val NSLIMSEQ_inverse_real_of_nat_add = thm "NSLIMSEQ_inverse_real_of_nat_add";
paulson@15082
  1208
val LIMSEQ_inverse_real_of_nat_add_minus = thm "LIMSEQ_inverse_real_of_nat_add_minus";
paulson@15082
  1209
val NSLIMSEQ_inverse_real_of_nat_add_minus = thm "NSLIMSEQ_inverse_real_of_nat_add_minus";
paulson@15082
  1210
val LIMSEQ_inverse_real_of_nat_add_minus_mult = thm "LIMSEQ_inverse_real_of_nat_add_minus_mult";
paulson@15082
  1211
val NSLIMSEQ_inverse_real_of_nat_add_minus_mult = thm "NSLIMSEQ_inverse_real_of_nat_add_minus_mult";
paulson@15082
  1212
val NSLIMSEQ_pow = thm "NSLIMSEQ_pow";
paulson@15082
  1213
val LIMSEQ_pow = thm "LIMSEQ_pow";
paulson@15082
  1214
val Bseq_realpow = thm "Bseq_realpow";
paulson@15082
  1215
val monoseq_realpow = thm "monoseq_realpow";
paulson@15082
  1216
val convergent_realpow = thm "convergent_realpow";
paulson@15082
  1217
val NSLIMSEQ_realpow_zero = thm "NSLIMSEQ_realpow_zero";
paulson@15082
  1218
*}
paulson@15082
  1219
paulson@15241
  1220
paulson@10751
  1221
end
paulson@10751
  1222