author | huffman |
Mon, 14 May 2007 17:37:31 +0200 | |
changeset 22966 | 9dc4f5048353 |
parent 21865 | 55cc354fd2d9 |
child 27435 | b3f8e9bdf9a7 |
permissions | -rw-r--r-- |
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(* Title : Star.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
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*) |
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|
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header{*Star-Transforms in Non-Standard Analysis*} |
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theory Star |
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imports NSA |
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begin |
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definition |
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(* internal sets *) |
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starset_n :: "(nat => 'a set) => 'a star set" ("*sn* _" [80] 80) where |
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"*sn* As = Iset (star_n As)" |
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definition |
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InternalSets :: "'a star set set" where |
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"InternalSets = {X. \<exists>As. X = *sn* As}" |
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definition |
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(* nonstandard extension of function *) |
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is_starext :: "['a star => 'a star, 'a => 'a] => bool" where |
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"is_starext F f = (\<forall>x y. \<exists>X \<in> Rep_star(x). \<exists>Y \<in> Rep_star(y). |
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((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))" |
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definition |
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(* internal functions *) |
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starfun_n :: "(nat => ('a => 'b)) => 'a star => 'b star" ("*fn* _" [80] 80) where |
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"*fn* F = Ifun (star_n F)" |
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definition |
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InternalFuns :: "('a star => 'b star) set" where |
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"InternalFuns = {X. \<exists>F. X = *fn* F}" |
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(*-------------------------------------------------------- |
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Preamble - Pulling "EX" over "ALL" |
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---------------------------------------------------------*) |
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(* This proof does not need AC and was suggested by the |
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referee for the JCM Paper: let f(x) be least y such |
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that Q(x,y) |
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*) |
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lemma no_choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>(f :: 'a => nat). \<forall>x. Q x (f x)" |
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apply (rule_tac x = "%x. LEAST y. Q x y" in exI) |
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apply (blast intro: LeastI) |
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done |
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subsection{*Properties of the Star-transform Applied to Sets of Reals*} |
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lemma STAR_star_of_image_subset: "star_of ` A <= *s* A" |
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by auto |
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lemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X" |
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by (auto simp add: SReal_def) |
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lemma STAR_star_of_Int: "*s* X Int Standard = star_of ` X" |
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by (auto simp add: Standard_def) |
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lemma lemma_not_hyprealA: "x \<notin> hypreal_of_real ` A ==> \<forall>y \<in> A. x \<noteq> hypreal_of_real y" |
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by auto |
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lemma lemma_not_starA: "x \<notin> star_of ` A ==> \<forall>y \<in> A. x \<noteq> star_of y" |
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by auto |
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lemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n \<noteq> xa}" |
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by auto |
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lemma STAR_real_seq_to_hypreal: |
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"\<forall>n. (X n) \<notin> M ==> star_n X \<notin> *s* M" |
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apply (unfold starset_def star_of_def) |
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apply (simp add: Iset_star_n) |
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done |
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lemma STAR_singleton: "*s* {x} = {star_of x}" |
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by simp |
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lemma STAR_not_mem: "x \<notin> F ==> star_of x \<notin> *s* F" |
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by transfer |
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lemma STAR_subset_closed: "[| x : *s* A; A <= B |] ==> x : *s* B" |
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by (erule rev_subsetD, simp) |
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text{*Nonstandard extension of a set (defined using a constant |
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sequence) as a special case of an internal set*} |
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lemma starset_n_starset: "\<forall>n. (As n = A) ==> *sn* As = *s* A" |
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apply (drule expand_fun_eq [THEN iffD2]) |
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apply (simp add: starset_n_def starset_def star_of_def) |
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done |
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(*----------------------------------------------------------------*) |
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(* Theorems about nonstandard extensions of functions *) |
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(*----------------------------------------------------------------*) |
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(*----------------------------------------------------------------*) |
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(* Nonstandard extension of a function (defined using a *) |
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(* constant sequence) as a special case of an internal function *) |
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(*----------------------------------------------------------------*) |
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lemma starfun_n_starfun: "\<forall>n. (F n = f) ==> *fn* F = *f* f" |
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apply (drule expand_fun_eq [THEN iffD2]) |
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apply (simp add: starfun_n_def starfun_def star_of_def) |
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done |
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(* |
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Prove that abs for hypreal is a nonstandard extension of abs for real w/o |
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use of congruence property (proved after this for general |
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nonstandard extensions of real valued functions). |
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Proof now Uses the ultrafilter tactic! |
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*) |
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lemma hrabs_is_starext_rabs: "is_starext abs abs" |
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apply (simp add: is_starext_def, safe) |
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apply (rule_tac x=x in star_cases) |
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apply (rule_tac x=y in star_cases) |
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apply (unfold star_n_def, auto) |
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apply (rule bexI, rule_tac [2] lemma_starrel_refl) |
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apply (rule bexI, rule_tac [2] lemma_starrel_refl) |
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apply (fold star_n_def) |
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apply (unfold star_abs_def starfun_def star_of_def) |
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apply (simp add: Ifun_star_n star_n_eq_iff) |
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done |
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text{*Nonstandard extension of functions*} |
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lemma starfun: |
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"( *f* f) (star_n X) = star_n (%n. f (X n))" |
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by (rule starfun_star_n) |
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lemma starfun_if_eq: |
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"!!w. w \<noteq> star_of x |
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==> ( *f* (\<lambda>z. if z = x then a else g z)) w = ( *f* g) w" |
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by (transfer, simp) |
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(*------------------------------------------- |
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multiplication: ( *f) x ( *g) = *(f x g) |
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------------------------------------------*) |
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lemma starfun_mult: "!!x. ( *f* f) x * ( *f* g) x = ( *f* (%x. f x * g x)) x" |
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by (transfer, rule refl) |
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declare starfun_mult [symmetric, simp] |
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(*--------------------------------------- |
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addition: ( *f) + ( *g) = *(f + g) |
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---------------------------------------*) |
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lemma starfun_add: "!!x. ( *f* f) x + ( *f* g) x = ( *f* (%x. f x + g x)) x" |
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by (transfer, rule refl) |
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declare starfun_add [symmetric, simp] |
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(*-------------------------------------------- |
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subtraction: ( *f) + -( *g) = *(f + -g) |
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-------------------------------------------*) |
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lemma starfun_minus: "!!x. - ( *f* f) x = ( *f* (%x. - f x)) x" |
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by (transfer, rule refl) |
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declare starfun_minus [symmetric, simp] |
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(*FIXME: delete*) |
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lemma starfun_add_minus: "!!x. ( *f* f) x + -( *f* g) x = ( *f* (%x. f x + -g x)) x" |
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by (transfer, rule refl) |
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declare starfun_add_minus [symmetric, simp] |
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lemma starfun_diff: "!!x. ( *f* f) x - ( *f* g) x = ( *f* (%x. f x - g x)) x" |
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by (transfer, rule refl) |
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declare starfun_diff [symmetric, simp] |
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(*-------------------------------------- |
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composition: ( *f) o ( *g) = *(f o g) |
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---------------------------------------*) |
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lemma starfun_o2: "(%x. ( *f* f) (( *f* g) x)) = *f* (%x. f (g x))" |
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by (transfer, rule refl) |
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lemma starfun_o: "( *f* f) o ( *f* g) = ( *f* (f o g))" |
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by (transfer o_def, rule refl) |
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text{*NS extension of constant function*} |
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lemma starfun_const_fun [simp]: "!!x. ( *f* (%x. k)) x = star_of k" |
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by (transfer, rule refl) |
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text{*the NS extension of the identity function*} |
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lemma starfun_Id [simp]: "!!x. ( *f* (%x. x)) x = x" |
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by (transfer, rule refl) |
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(* this is trivial, given starfun_Id *) |
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lemma starfun_Idfun_approx: |
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"x @= star_of a ==> ( *f* (%x. x)) x @= star_of a" |
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by (simp only: starfun_Id) |
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text{*The Star-function is a (nonstandard) extension of the function*} |
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lemma is_starext_starfun: "is_starext ( *f* f) f" |
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apply (simp add: is_starext_def, auto) |
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apply (rule_tac x = x in star_cases) |
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apply (rule_tac x = y in star_cases) |
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apply (auto intro!: bexI [OF _ Rep_star_star_n] |
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simp add: starfun star_n_eq_iff) |
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done |
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text{*Any nonstandard extension is in fact the Star-function*} |
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lemma is_starfun_starext: "is_starext F f ==> F = *f* f" |
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apply (simp add: is_starext_def) |
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apply (rule ext) |
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apply (rule_tac x = x in star_cases) |
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apply (drule_tac x = x in spec) |
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apply (drule_tac x = "( *f* f) x" in spec) |
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apply (auto simp add: starfun_star_n) |
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apply (simp add: star_n_eq_iff [symmetric]) |
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apply (simp add: starfun_star_n [of f, symmetric]) |
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done |
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lemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)" |
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by (blast intro: is_starfun_starext is_starext_starfun) |
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text{*extented function has same solution as its standard |
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version for real arguments. i.e they are the same |
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for all real arguments*} |
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lemma starfun_eq: "( *f* f) (star_of a) = star_of (f a)" |
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by (rule starfun_star_of) |
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lemma starfun_approx: "( *f* f) (star_of a) @= star_of (f a)" |
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by simp |
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|
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(* useful for NS definition of derivatives *) |
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lemma starfun_lambda_cancel: |
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"!!x'. ( *f* (%h. f (x + h))) x' = ( *f* f) (star_of x + x')" |
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by (transfer, rule refl) |
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|
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lemma starfun_lambda_cancel2: |
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"( *f* (%h. f(g(x + h)))) x' = ( *f* (f o g)) (star_of x + x')" |
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by (unfold o_def, rule starfun_lambda_cancel) |
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|
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lemma starfun_mult_HFinite_approx: |
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fixes l m :: "'a::real_normed_algebra star" |
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shows "[| ( *f* f) x @= l; ( *f* g) x @= m; |
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l: HFinite; m: HFinite |
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|] ==> ( *f* (%x. f x * g x)) x @= l * m" |
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apply (drule (3) approx_mult_HFinite) |
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apply (auto intro: approx_HFinite [OF _ approx_sym]) |
246 |
done |
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247 |
||
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lemma starfun_add_approx: "[| ( *f* f) x @= l; ( *f* g) x @= m |
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|] ==> ( *f* (%x. f x + g x)) x @= l + m" |
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by (auto intro: approx_add) |
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|
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text{*Examples: hrabs is nonstandard extension of rabs |
253 |
inverse is nonstandard extension of inverse*} |
|
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|
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(* can be proved easily using theorem "starfun" and *) |
|
256 |
(* properties of ultrafilter as for inverse below we *) |
|
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(* use the theorem we proved above instead *) |
|
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||
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lemma starfun_rabs_hrabs: "*f* abs = abs" |
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by (simp only: star_abs_def) |
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|
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lemma starfun_inverse_inverse [simp]: "( *f* inverse) x = inverse(x)" |
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by (simp only: star_inverse_def) |
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|
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lemma starfun_inverse: "!!x. inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x" |
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by (transfer, rule refl) |
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declare starfun_inverse [symmetric, simp] |
268 |
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lemma starfun_divide: "!!x. ( *f* f) x / ( *f* g) x = ( *f* (%x. f x / g x)) x" |
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by (transfer, rule refl) |
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declare starfun_divide [symmetric, simp] |
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lemma starfun_inverse2: "!!x. inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x" |
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by (transfer, rule refl) |
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|
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text{*General lemma/theorem needed for proofs in elementary |
277 |
topology of the reals*} |
|
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lemma starfun_mem_starset: |
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"!!x. ( *f* f) x : *s* A ==> x : *s* {x. f x \<in> A}" |
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by (transfer, simp) |
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|
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text{*Alternative definition for hrabs with rabs function |
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applied entrywise to equivalence class representative. |
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This is easily proved using starfun and ns extension thm*} |
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lemma hypreal_hrabs: |
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"abs (star_n X) = star_n (%n. abs (X n))" |
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by (simp only: starfun_rabs_hrabs [symmetric] starfun) |
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|
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text{*nonstandard extension of set through nonstandard extension |
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of rabs function i.e hrabs. A more general result should be |
291 |
where we replace rabs by some arbitrary function f and hrabs |
|
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by its NS extenson. See second NS set extension below.*} |
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lemma STAR_rabs_add_minus: |
294 |
"*s* {x. abs (x + - y) < r} = |
|
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{x. abs(x + -star_of y) < star_of r}" |
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by (transfer, rule refl) |
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|
298 |
lemma STAR_starfun_rabs_add_minus: |
|
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"*s* {x. abs (f x + - y) < r} = |
|
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{x. abs(( *f* f) x + -star_of y) < star_of r}" |
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by (transfer, rule refl) |
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|
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text{*Another characterization of Infinitesimal and one of @= relation. |
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In this theory since @{text hypreal_hrabs} proved here. Maybe |
305 |
move both theorems??*} |
|
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lemma Infinitesimal_FreeUltrafilterNat_iff2: |
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"(star_n X \<in> Infinitesimal) = |
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(\<forall>m. {n. norm(X n) < inverse(real(Suc m))} |
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\<in> FreeUltrafilterNat)" |
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by (simp add: Infinitesimal_hypreal_of_nat_iff star_of_def |
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hnorm_def star_of_nat_def starfun_star_n |
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star_n_inverse star_n_less real_of_nat_def) |
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313 |
|
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lemma HNatInfinite_inverse_Infinitesimal [simp]: |
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"n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal" |
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316 |
apply (cases n) |
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317 |
apply (auto simp add: of_hypnat_def starfun_star_n real_of_nat_def [symmetric] star_n_inverse real_norm_def |
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HNatInfinite_FreeUltrafilterNat_iff |
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319 |
Infinitesimal_FreeUltrafilterNat_iff2) |
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320 |
apply (drule_tac x="Suc m" in spec) |
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321 |
apply (erule ultra, simp) |
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322 |
done |
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|
323 |
|
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lemma approx_FreeUltrafilterNat_iff: "star_n X @= star_n Y = |
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(\<forall>r>0. {n. norm (X n - Y n) < r} : FreeUltrafilterNat)" |
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326 |
apply (subst approx_minus_iff) |
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327 |
apply (rule mem_infmal_iff [THEN subst]) |
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apply (simp add: star_n_diff) |
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329 |
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff) |
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done |
331 |
||
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332 |
lemma approx_FreeUltrafilterNat_iff2: "star_n X @= star_n Y = |
20563 | 333 |
(\<forall>m. {n. norm (X n - Y n) < |
14370 | 334 |
inverse(real(Suc m))} : FreeUltrafilterNat)" |
335 |
apply (subst approx_minus_iff) |
|
336 |
apply (rule mem_infmal_iff [THEN subst]) |
|
20563 | 337 |
apply (simp add: star_n_diff) |
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|
338 |
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff2) |
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done |
340 |
||
341 |
lemma inj_starfun: "inj starfun" |
|
342 |
apply (rule inj_onI) |
|
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343 |
apply (rule ext, rule ccontr) |
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344 |
apply (drule_tac x = "star_n (%n. xa)" in fun_cong) |
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345 |
apply (auto simp add: starfun star_n_eq_iff) |
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done |
347 |
||
348 |
end |