27468
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(* Title : HSEQ.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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Additional contributions by Jeremy Avigad and Brian Huffman
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*)
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header {* Sequences and Convergence (Nonstandard) *}
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theory HSEQ
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imports SEQ NatStar
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begin
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definition
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NSLIMSEQ :: "[nat => 'a::real_normed_vector, 'a] => bool"
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("((_)/ ----NS> (_))" [60, 60] 60) where
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--{*Nonstandard definition of convergence of sequence*}
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[code func del]: "X ----NS> L = (\<forall>N \<in> HNatInfinite. ( *f* X) N \<approx> star_of L)"
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definition
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nslim :: "(nat => 'a::real_normed_vector) => 'a" where
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--{*Nonstandard definition of limit using choice operator*}
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"nslim X = (THE L. X ----NS> L)"
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definition
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NSconvergent :: "(nat => 'a::real_normed_vector) => bool" where
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--{*Nonstandard definition of convergence*}
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"NSconvergent X = (\<exists>L. X ----NS> L)"
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definition
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NSBseq :: "(nat => 'a::real_normed_vector) => bool" where
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--{*Nonstandard definition for bounded sequence*}
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[code func del]: "NSBseq X = (\<forall>N \<in> HNatInfinite. ( *f* X) N : HFinite)"
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definition
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NSCauchy :: "(nat => 'a::real_normed_vector) => bool" where
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--{*Nonstandard definition*}
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[code func del]: "NSCauchy X = (\<forall>M \<in> HNatInfinite. \<forall>N \<in> HNatInfinite. ( *f* X) M \<approx> ( *f* X) N)"
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subsection {* Limits of Sequences *}
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lemma NSLIMSEQ_iff:
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"(X ----NS> L) = (\<forall>N \<in> HNatInfinite. ( *f* X) N \<approx> star_of L)"
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by (simp add: NSLIMSEQ_def)
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lemma NSLIMSEQ_I:
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"(\<And>N. N \<in> HNatInfinite \<Longrightarrow> starfun X N \<approx> star_of L) \<Longrightarrow> X ----NS> L"
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by (simp add: NSLIMSEQ_def)
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lemma NSLIMSEQ_D:
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"\<lbrakk>X ----NS> L; N \<in> HNatInfinite\<rbrakk> \<Longrightarrow> starfun X N \<approx> star_of L"
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by (simp add: NSLIMSEQ_def)
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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 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 starfun_add [symmetric])
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lemma NSLIMSEQ_add_const: "f ----NS> a ==> (%n.(f n + b)) ----NS> a + b"
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by (simp only: NSLIMSEQ_add NSLIMSEQ_const)
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lemma NSLIMSEQ_mult:
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fixes a b :: "'a::real_normed_algebra"
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shows "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n * Y n) ----NS> a * b"
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by (auto intro!: approx_mult_HFinite simp add: NSLIMSEQ_def)
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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)
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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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(* FIXME: delete *)
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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)
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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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by (simp add: diff_minus NSLIMSEQ_add NSLIMSEQ_minus)
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lemma NSLIMSEQ_diff_const: "f ----NS> a ==> (%n.(f n - b)) ----NS> a - b"
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by (simp add: NSLIMSEQ_diff NSLIMSEQ_const)
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lemma NSLIMSEQ_inverse:
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fixes a :: "'a::real_normed_div_algebra"
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shows "[| X ----NS> a; a ~= 0 |] ==> (%n. inverse(X n)) ----NS> inverse(a)"
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by (simp add: NSLIMSEQ_def star_of_approx_inverse)
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lemma NSLIMSEQ_mult_inverse:
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fixes a b :: "'a::real_normed_field"
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shows
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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 starfun_hnorm: "\<And>x. hnorm (( *f* f) x) = ( *f* (\<lambda>x. norm (f x))) x"
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by transfer simp
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lemma NSLIMSEQ_norm: "X ----NS> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----NS> norm a"
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by (simp add: NSLIMSEQ_def starfun_hnorm [symmetric] approx_hnorm)
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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 NSLIMSEQ_pow [rule_format]:
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fixes a :: "'a::{real_normed_algebra,recpower}"
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shows "(X ----NS> a) --> ((%n. (X n) ^ m) ----NS> a ^ m)"
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apply (induct "m")
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apply (auto simp add: power_Suc intro: NSLIMSEQ_mult NSLIMSEQ_const)
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done
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text{*We can now try and derive a few properties of sequences,
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starting with the limit comparison property for sequences.*}
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lemma NSLIMSEQ_le:
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"[| f ----NS> l; g ----NS> m;
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\<exists>N. \<forall>n \<ge> N. f(n) \<le> g(n)
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|] ==> l \<le> (m::real)"
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apply (simp add: NSLIMSEQ_def, safe)
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apply (drule starfun_le_mono)
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apply (drule HNatInfinite_whn [THEN [2] bspec])+
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apply (drule_tac x = whn in spec)
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apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
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apply clarify
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apply (auto intro: hypreal_of_real_le_add_Infininitesimal_cancel2)
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done
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lemma NSLIMSEQ_le_const: "[| X ----NS> (r::real); \<forall>n. a \<le> X n |] ==> a \<le> r"
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by (erule NSLIMSEQ_le [OF NSLIMSEQ_const], auto)
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lemma NSLIMSEQ_le_const2: "[| X ----NS> (r::real); \<forall>n. X n \<le> a |] ==> r \<le> a"
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by (erule NSLIMSEQ_le [OF _ NSLIMSEQ_const], auto)
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text{*Shift a convergent series by 1:
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By the equivalence between Cauchiness and convergence and because
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the successor of an infinite hypernatural is also infinite.*}
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lemma NSLIMSEQ_Suc: "f ----NS> l ==> (%n. f(Suc n)) ----NS> l"
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apply (unfold NSLIMSEQ_def, safe)
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apply (drule_tac x="N + 1" in bspec)
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apply (erule HNatInfinite_add)
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apply (simp add: starfun_shift_one)
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done
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lemma NSLIMSEQ_imp_Suc: "(%n. f(Suc n)) ----NS> l ==> f ----NS> l"
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apply (unfold NSLIMSEQ_def, safe)
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apply (drule_tac x="N - 1" in bspec)
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apply (erule Nats_1 [THEN [2] HNatInfinite_diff])
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apply (simp add: starfun_shift_one one_le_HNatInfinite)
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done
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lemma NSLIMSEQ_Suc_iff: "((%n. f(Suc n)) ----NS> l) = (f ----NS> l)"
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by (blast intro: NSLIMSEQ_imp_Suc NSLIMSEQ_Suc)
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subsubsection {* Equivalence of @{term LIMSEQ} and @{term NSLIMSEQ} *}
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lemma LIMSEQ_NSLIMSEQ:
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assumes X: "X ----> L" shows "X ----NS> L"
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proof (rule NSLIMSEQ_I)
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fix N assume N: "N \<in> HNatInfinite"
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have "starfun X N - star_of L \<in> Infinitesimal"
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proof (rule InfinitesimalI2)
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fix r::real assume r: "0 < r"
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from LIMSEQ_D [OF X r]
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obtain no where "\<forall>n\<ge>no. norm (X n - L) < r" ..
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hence "\<forall>n\<ge>star_of no. hnorm (starfun X n - star_of L) < star_of r"
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by transfer
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thus "hnorm (starfun X N - star_of L) < star_of r"
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using N by (simp add: star_of_le_HNatInfinite)
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qed
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thus "starfun X N \<approx> star_of L"
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by (unfold approx_def)
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qed
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lemma NSLIMSEQ_LIMSEQ:
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assumes X: "X ----NS> L" shows "X ----> L"
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proof (rule LIMSEQ_I)
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fix r::real assume r: "0 < r"
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have "\<exists>no. \<forall>n\<ge>no. hnorm (starfun X n - star_of L) < star_of r"
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proof (intro exI allI impI)
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fix n assume "whn \<le> n"
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with HNatInfinite_whn have "n \<in> HNatInfinite"
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by (rule HNatInfinite_upward_closed)
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with X have "starfun X n \<approx> star_of L"
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by (rule NSLIMSEQ_D)
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hence "starfun X n - star_of L \<in> Infinitesimal"
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by (unfold approx_def)
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thus "hnorm (starfun X n - star_of L) < star_of r"
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using r by (rule InfinitesimalD2)
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qed
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thus "\<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
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by transfer
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qed
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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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subsubsection {* Derived theorems about @{term NSLIMSEQ} *}
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text{*We prove the NS version from the standard one, since the NS proof
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seems more complicated than the standard one above!*}
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lemma NSLIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----NS> 0) = (X ----NS> 0)"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_norm_zero)
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lemma NSLIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----NS> 0) = (f ----NS> (0::real))"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_rabs_zero)
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text{*Generalization to other limits*}
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lemma NSLIMSEQ_imp_rabs: "f ----NS> (l::real) ==> (%n. \<bar>f n\<bar>) ----NS> \<bar>l\<bar>"
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apply (simp add: NSLIMSEQ_def)
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apply (auto intro: approx_hrabs
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simp add: starfun_abs)
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done
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lemma NSLIMSEQ_inverse_zero:
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"\<forall>y::real. \<exists>N. \<forall>n \<ge> N. y < f(n)
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==> (%n. inverse(f n)) ----NS> 0"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_zero)
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lemma NSLIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----NS> 0"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat)
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lemma NSLIMSEQ_inverse_real_of_nat_add:
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"(%n. r + inverse(real(Suc n))) ----NS> r"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add)
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lemma NSLIMSEQ_inverse_real_of_nat_add_minus:
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"(%n. r + -inverse(real(Suc n))) ----NS> r"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add_minus)
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lemma NSLIMSEQ_inverse_real_of_nat_add_minus_mult:
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"(%n. r*( 1 + -inverse(real(Suc n)))) ----NS> r"
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by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add_minus_mult)
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subsection {* Convergence *}
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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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lemma NSconvergentD: "NSconvergent X ==> \<exists>L. (X ----NS> L)"
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by (simp add: NSconvergent_def)
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lemma NSconvergentI: "(X ----NS> L) ==> NSconvergent X"
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by (auto simp add: NSconvergent_def)
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lemma convergent_NSconvergent_iff: "convergent X = NSconvergent X"
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by (simp add: convergent_def NSconvergent_def LIMSEQ_NSLIMSEQ_iff)
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lemma NSconvergent_NSLIMSEQ_iff: "NSconvergent X = (X ----NS> nslim X)"
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by (auto intro: theI NSLIMSEQ_unique simp add: NSconvergent_def nslim_def)
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subsection {* Bounded Monotonic Sequences *}
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lemma NSBseqD: "[| NSBseq X; N: HNatInfinite |] ==> ( *f* X) N : HFinite"
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by (simp add: NSBseq_def)
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lemma Standard_subset_HFinite: "Standard \<subseteq> HFinite"
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unfolding Standard_def by auto
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lemma NSBseqD2: "NSBseq X \<Longrightarrow> ( *f* X) N \<in> HFinite"
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apply (cases "N \<in> HNatInfinite")
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apply (erule (1) NSBseqD)
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apply (rule subsetD [OF Standard_subset_HFinite])
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apply (simp add: HNatInfinite_def Nats_eq_Standard)
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done
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lemma NSBseqI: "\<forall>N \<in> HNatInfinite. ( *f* X) N : HFinite ==> NSBseq X"
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by (simp add: NSBseq_def)
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text{*The standard definition implies the nonstandard definition*}
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lemma Bseq_NSBseq: "Bseq X ==> NSBseq X"
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proof (unfold NSBseq_def, safe)
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assume X: "Bseq X"
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fix N assume N: "N \<in> HNatInfinite"
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from BseqD [OF X] obtain K where "\<forall>n. norm (X n) \<le> K" by fast
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hence "\<forall>N. hnorm (starfun X N) \<le> star_of K" by transfer
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hence "hnorm (starfun X N) \<le> star_of K" by simp
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also have "star_of K < star_of (K + 1)" by simp
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finally have "\<exists>x\<in>Reals. hnorm (starfun X N) < x" by (rule bexI, simp)
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thus "starfun X N \<in> HFinite" by (simp add: HFinite_def)
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qed
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text{*The nonstandard definition implies the standard definition*}
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lemma SReal_less_omega: "r \<in> \<real> \<Longrightarrow> r < \<omega>"
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apply (insert HInfinite_omega)
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apply (simp add: HInfinite_def)
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apply (simp add: order_less_imp_le)
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done
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lemma NSBseq_Bseq: "NSBseq X \<Longrightarrow> Bseq X"
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proof (rule ccontr)
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let ?n = "\<lambda>K. LEAST n. K < norm (X n)"
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assume "NSBseq X"
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hence finite: "( *f* X) (( *f* ?n) \<omega>) \<in> HFinite"
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by (rule NSBseqD2)
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assume "\<not> Bseq X"
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hence "\<forall>K>0. \<exists>n. K < norm (X n)"
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by (simp add: Bseq_def linorder_not_le)
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hence "\<forall>K>0. K < norm (X (?n K))"
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by (auto intro: LeastI_ex)
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hence "\<forall>K>0. K < hnorm (( *f* X) (( *f* ?n) K))"
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by transfer
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hence "\<omega> < hnorm (( *f* X) (( *f* ?n) \<omega>))"
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by simp
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hence "\<forall>r\<in>\<real>. r < hnorm (( *f* X) (( *f* ?n) \<omega>))"
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by (simp add: order_less_trans [OF SReal_less_omega])
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hence "( *f* X) (( *f* ?n) \<omega>) \<in> HInfinite"
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by (simp add: HInfinite_def)
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with finite show "False"
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by (simp add: HFinite_HInfinite_iff)
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qed
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text{* Equivalence of nonstandard and standard definitions
|
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330 |
for a bounded sequence*}
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331 |
lemma Bseq_NSBseq_iff: "(Bseq X) = (NSBseq X)"
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by (blast intro!: NSBseq_Bseq Bseq_NSBseq)
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333 |
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text{*A convergent sequence is bounded:
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Boundedness as a necessary condition for convergence.
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The nonstandard version has no existential, as usual *}
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337 |
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lemma NSconvergent_NSBseq: "NSconvergent X ==> NSBseq X"
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apply (simp add: NSconvergent_def NSBseq_def NSLIMSEQ_def)
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apply (blast intro: HFinite_star_of approx_sym approx_HFinite)
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341 |
done
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342 |
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text{*Standard Version: easily now proved using equivalence of NS and
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standard definitions *}
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345 |
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lemma convergent_Bseq: "convergent X ==> Bseq X"
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by (simp add: NSconvergent_NSBseq convergent_NSconvergent_iff Bseq_NSBseq_iff)
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348 |
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349 |
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
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lemma NSBseq_isUb: "NSBseq X ==> \<exists>U::real. isUb UNIV {x. \<exists>n. X n = x} U"
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by (simp add: Bseq_NSBseq_iff [symmetric] Bseq_isUb)
|
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353 |
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354 |
lemma NSBseq_isLub: "NSBseq X ==> \<exists>U::real. isLub UNIV {x. \<exists>n. X n = x} U"
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by (simp add: Bseq_NSBseq_iff [symmetric] Bseq_isLub)
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356 |
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357 |
subsubsection{*A Bounded and Monotonic Sequence Converges*}
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358 |
|
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359 |
text{* The best of both worlds: Easier to prove this result as a standard
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360 |
theorem and then use equivalence to "transfer" it into the
|
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361 |
equivalent nonstandard form if needed!*}
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362 |
|
|
363 |
lemma Bmonoseq_NSLIMSEQ: "\<forall>n \<ge> m. X n = X m ==> \<exists>L. (X ----NS> L)"
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364 |
by (auto dest!: Bmonoseq_LIMSEQ simp add: LIMSEQ_NSLIMSEQ_iff)
|
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365 |
|
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366 |
lemma NSBseq_mono_NSconvergent:
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"[| NSBseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> NSconvergent (X::nat=>real)"
|
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368 |
by (auto intro: Bseq_mono_convergent
|
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simp add: convergent_NSconvergent_iff [symmetric]
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370 |
Bseq_NSBseq_iff [symmetric])
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371 |
|
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372 |
|
|
373 |
subsection {* Cauchy Sequences *}
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374 |
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|
375 |
lemma NSCauchyI:
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"(\<And>M N. \<lbrakk>M \<in> HNatInfinite; N \<in> HNatInfinite\<rbrakk> \<Longrightarrow> starfun X M \<approx> starfun X N)
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377 |
\<Longrightarrow> NSCauchy X"
|
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378 |
by (simp add: NSCauchy_def)
|
|
379 |
|
|
380 |
lemma NSCauchyD:
|
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381 |
"\<lbrakk>NSCauchy X; M \<in> HNatInfinite; N \<in> HNatInfinite\<rbrakk>
|
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382 |
\<Longrightarrow> starfun X M \<approx> starfun X N"
|
|
383 |
by (simp add: NSCauchy_def)
|
|
384 |
|
|
385 |
subsubsection{*Equivalence Between NS and Standard*}
|
|
386 |
|
|
387 |
lemma Cauchy_NSCauchy:
|
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388 |
assumes X: "Cauchy X" shows "NSCauchy X"
|
|
389 |
proof (rule NSCauchyI)
|
|
390 |
fix M assume M: "M \<in> HNatInfinite"
|
|
391 |
fix N assume N: "N \<in> HNatInfinite"
|
|
392 |
have "starfun X M - starfun X N \<in> Infinitesimal"
|
|
393 |
proof (rule InfinitesimalI2)
|
|
394 |
fix r :: real assume r: "0 < r"
|
|
395 |
from CauchyD [OF X r]
|
|
396 |
obtain k where "\<forall>m\<ge>k. \<forall>n\<ge>k. norm (X m - X n) < r" ..
|
|
397 |
hence "\<forall>m\<ge>star_of k. \<forall>n\<ge>star_of k.
|
|
398 |
hnorm (starfun X m - starfun X n) < star_of r"
|
|
399 |
by transfer
|
|
400 |
thus "hnorm (starfun X M - starfun X N) < star_of r"
|
|
401 |
using M N by (simp add: star_of_le_HNatInfinite)
|
|
402 |
qed
|
|
403 |
thus "starfun X M \<approx> starfun X N"
|
|
404 |
by (unfold approx_def)
|
|
405 |
qed
|
|
406 |
|
|
407 |
lemma NSCauchy_Cauchy:
|
|
408 |
assumes X: "NSCauchy X" shows "Cauchy X"
|
|
409 |
proof (rule CauchyI)
|
|
410 |
fix r::real assume r: "0 < r"
|
|
411 |
have "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. hnorm (starfun X m - starfun X n) < star_of r"
|
|
412 |
proof (intro exI allI impI)
|
|
413 |
fix M assume "whn \<le> M"
|
|
414 |
with HNatInfinite_whn have M: "M \<in> HNatInfinite"
|
|
415 |
by (rule HNatInfinite_upward_closed)
|
|
416 |
fix N assume "whn \<le> N"
|
|
417 |
with HNatInfinite_whn have N: "N \<in> HNatInfinite"
|
|
418 |
by (rule HNatInfinite_upward_closed)
|
|
419 |
from X M N have "starfun X M \<approx> starfun X N"
|
|
420 |
by (rule NSCauchyD)
|
|
421 |
hence "starfun X M - starfun X N \<in> Infinitesimal"
|
|
422 |
by (unfold approx_def)
|
|
423 |
thus "hnorm (starfun X M - starfun X N) < star_of r"
|
|
424 |
using r by (rule InfinitesimalD2)
|
|
425 |
qed
|
|
426 |
thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. norm (X m - X n) < r"
|
|
427 |
by transfer
|
|
428 |
qed
|
|
429 |
|
|
430 |
theorem NSCauchy_Cauchy_iff: "NSCauchy X = Cauchy X"
|
|
431 |
by (blast intro!: NSCauchy_Cauchy Cauchy_NSCauchy)
|
|
432 |
|
|
433 |
subsubsection {* Cauchy Sequences are Bounded *}
|
|
434 |
|
|
435 |
text{*A Cauchy sequence is bounded -- nonstandard version*}
|
|
436 |
|
|
437 |
lemma NSCauchy_NSBseq: "NSCauchy X ==> NSBseq X"
|
|
438 |
by (simp add: Cauchy_Bseq Bseq_NSBseq_iff [symmetric] NSCauchy_Cauchy_iff)
|
|
439 |
|
|
440 |
subsubsection {* Cauchy Sequences are Convergent *}
|
|
441 |
|
|
442 |
text{*Equivalence of Cauchy criterion and convergence:
|
|
443 |
We will prove this using our NS formulation which provides a
|
|
444 |
much easier proof than using the standard definition. We do not
|
|
445 |
need to use properties of subsequences such as boundedness,
|
|
446 |
monotonicity etc... Compare with Harrison's corresponding proof
|
|
447 |
in HOL which is much longer and more complicated. Of course, we do
|
|
448 |
not have problems which he encountered with guessing the right
|
|
449 |
instantiations for his 'espsilon-delta' proof(s) in this case
|
|
450 |
since the NS formulations do not involve existential quantifiers.*}
|
|
451 |
|
|
452 |
lemma NSconvergent_NSCauchy: "NSconvergent X \<Longrightarrow> NSCauchy X"
|
|
453 |
apply (simp add: NSconvergent_def NSLIMSEQ_def NSCauchy_def, safe)
|
|
454 |
apply (auto intro: approx_trans2)
|
|
455 |
done
|
|
456 |
|
|
457 |
lemma real_NSCauchy_NSconvergent:
|
|
458 |
fixes X :: "nat \<Rightarrow> real"
|
|
459 |
shows "NSCauchy X \<Longrightarrow> NSconvergent X"
|
|
460 |
apply (simp add: NSconvergent_def NSLIMSEQ_def)
|
|
461 |
apply (frule NSCauchy_NSBseq)
|
|
462 |
apply (simp add: NSBseq_def NSCauchy_def)
|
|
463 |
apply (drule HNatInfinite_whn [THEN [2] bspec])
|
|
464 |
apply (drule HNatInfinite_whn [THEN [2] bspec])
|
|
465 |
apply (auto dest!: st_part_Ex simp add: SReal_iff)
|
|
466 |
apply (blast intro: approx_trans3)
|
|
467 |
done
|
|
468 |
|
|
469 |
lemma NSCauchy_NSconvergent:
|
|
470 |
fixes X :: "nat \<Rightarrow> 'a::banach"
|
|
471 |
shows "NSCauchy X \<Longrightarrow> NSconvergent X"
|
|
472 |
apply (drule NSCauchy_Cauchy [THEN Cauchy_convergent])
|
|
473 |
apply (erule convergent_NSconvergent_iff [THEN iffD1])
|
|
474 |
done
|
|
475 |
|
|
476 |
lemma NSCauchy_NSconvergent_iff:
|
|
477 |
fixes X :: "nat \<Rightarrow> 'a::banach"
|
|
478 |
shows "NSCauchy X = NSconvergent X"
|
|
479 |
by (fast intro: NSCauchy_NSconvergent NSconvergent_NSCauchy)
|
|
480 |
|
|
481 |
|
|
482 |
subsection {* Power Sequences *}
|
|
483 |
|
|
484 |
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
|
|
485 |
"x<1"}. Proof will use (NS) Cauchy equivalence for convergence and
|
|
486 |
also fact that bounded and monotonic sequence converges.*}
|
|
487 |
|
|
488 |
text{* We now use NS criterion to bring proof of theorem through *}
|
|
489 |
|
|
490 |
lemma NSLIMSEQ_realpow_zero:
|
|
491 |
"[| 0 \<le> (x::real); x < 1 |] ==> (%n. x ^ n) ----NS> 0"
|
|
492 |
apply (simp add: NSLIMSEQ_def)
|
|
493 |
apply (auto dest!: convergent_realpow simp add: convergent_NSconvergent_iff)
|
|
494 |
apply (frule NSconvergentD)
|
|
495 |
apply (auto simp add: NSLIMSEQ_def NSCauchy_NSconvergent_iff [symmetric] NSCauchy_def starfun_pow)
|
|
496 |
apply (frule HNatInfinite_add_one)
|
|
497 |
apply (drule bspec, assumption)
|
|
498 |
apply (drule bspec, assumption)
|
|
499 |
apply (drule_tac x = "N + (1::hypnat) " in bspec, assumption)
|
|
500 |
apply (simp add: hyperpow_add)
|
|
501 |
apply (drule approx_mult_subst_star_of, assumption)
|
|
502 |
apply (drule approx_trans3, assumption)
|
|
503 |
apply (auto simp del: star_of_mult simp add: star_of_mult [symmetric])
|
|
504 |
done
|
|
505 |
|
|
506 |
lemma NSLIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----NS> 0"
|
|
507 |
by (simp add: LIMSEQ_rabs_realpow_zero LIMSEQ_NSLIMSEQ_iff [symmetric])
|
|
508 |
|
|
509 |
lemma NSLIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----NS> 0"
|
|
510 |
by (simp add: LIMSEQ_rabs_realpow_zero2 LIMSEQ_NSLIMSEQ_iff [symmetric])
|
|
511 |
|
|
512 |
(***---------------------------------------------------------------
|
|
513 |
Theorems proved by Harrison in HOL that we do not need
|
|
514 |
in order to prove equivalence between Cauchy criterion
|
|
515 |
and convergence:
|
|
516 |
-- Show that every sequence contains a monotonic subsequence
|
|
517 |
Goal "\<exists>f. subseq f & monoseq (%n. s (f n))"
|
|
518 |
-- Show that a subsequence of a bounded sequence is bounded
|
|
519 |
Goal "Bseq X ==> Bseq (%n. X (f n))";
|
|
520 |
-- Show we can take subsequential terms arbitrarily far
|
|
521 |
up a sequence
|
|
522 |
Goal "subseq f ==> n \<le> f(n)";
|
|
523 |
Goal "subseq f ==> \<exists>n. N1 \<le> n & N2 \<le> f(n)";
|
|
524 |
---------------------------------------------------------------***)
|
|
525 |
|
|
526 |
end
|