(* Title : NSCA.thy Author : Jacques D. Fleuriot Copyright : 2001,2002 University of Edinburgh *) header{*Non-Standard Complex Analysis*} theory NSCA imports NSComplex begin constdefs CInfinitesimal :: "hcomplex set" "CInfinitesimal == {x. ∀r ∈ Reals. 0 < r --> hcmod x < r}" capprox :: "[hcomplex,hcomplex] => bool" (infixl "@c=" 50) --{*the ``infinitely close'' relation*} "x @c= y == (x - y) ∈ CInfinitesimal" (* standard complex numbers reagarded as an embedded subset of NS complex *) SComplex :: "hcomplex set" "SComplex == {x. ∃r. x = hcomplex_of_complex r}" CFinite :: "hcomplex set" "CFinite == {x. ∃r ∈ Reals. hcmod x < r}" CInfinite :: "hcomplex set" "CInfinite == {x. ∀r ∈ Reals. r < hcmod x}" stc :: "hcomplex => hcomplex" --{* standard part map*} "stc x == (@r. x ∈ CFinite & r:SComplex & r @c= x)" cmonad :: "hcomplex => hcomplex set" "cmonad x == {y. x @c= y}" cgalaxy :: "hcomplex => hcomplex set" "cgalaxy x == {y. (x - y) ∈ CFinite}" subsection{*Closure Laws for SComplex, the Standard Complex Numbers*} lemma SComplex_add: "[| x ∈ SComplex; y ∈ SComplex |] ==> x + y ∈ SComplex" apply (simp add: SComplex_def, safe) apply (rule_tac x = "r + ra" in exI, simp) done lemma SComplex_mult: "[| x ∈ SComplex; y ∈ SComplex |] ==> x * y ∈ SComplex" apply (simp add: SComplex_def, safe) apply (rule_tac x = "r * ra" in exI, simp) done lemma SComplex_inverse: "x ∈ SComplex ==> inverse x ∈ SComplex" apply (simp add: SComplex_def) apply (blast intro: star_of_inverse [symmetric]) done lemma SComplex_divide: "[| x ∈ SComplex; y ∈ SComplex |] ==> x/y ∈ SComplex" by (simp add: SComplex_mult SComplex_inverse divide_inverse) lemma SComplex_minus: "x ∈ SComplex ==> -x ∈ SComplex" apply (simp add: SComplex_def) apply (blast intro: star_of_minus [symmetric]) done lemma SComplex_minus_iff [simp]: "(-x ∈ SComplex) = (x ∈ SComplex)" apply auto apply (erule_tac [2] SComplex_minus) apply (drule SComplex_minus, auto) done lemma SComplex_diff: "[| x ∈ SComplex; y ∈ SComplex |] ==> x - y ∈ SComplex" by (simp add: diff_minus SComplex_add) lemma SComplex_add_cancel: "[| x + y ∈ SComplex; y ∈ SComplex |] ==> x ∈ SComplex" by (drule SComplex_diff, assumption, simp) lemma SReal_hcmod_hcomplex_of_complex [simp]: "hcmod (hcomplex_of_complex r) ∈ Reals" by (auto simp add: hcmod SReal_def star_of_def) lemma SReal_hcmod_number_of [simp]: "hcmod (number_of w ::hcomplex) ∈ Reals" apply (subst star_of_number_of [symmetric]) apply (rule SReal_hcmod_hcomplex_of_complex) done lemma SReal_hcmod_SComplex: "x ∈ SComplex ==> hcmod x ∈ Reals" by (auto simp add: SComplex_def) lemma SComplex_hcomplex_of_complex [simp]: "hcomplex_of_complex x ∈ SComplex" by (simp add: SComplex_def) lemma SComplex_number_of [simp]: "(number_of w ::hcomplex) ∈ SComplex" apply (subst star_of_number_of [symmetric]) apply (rule SComplex_hcomplex_of_complex) done lemma SComplex_divide_number_of: "r ∈ SComplex ==> r/(number_of w::hcomplex) ∈ SComplex" apply (simp only: divide_inverse) apply (blast intro!: SComplex_number_of SComplex_mult SComplex_inverse) done lemma SComplex_UNIV_complex: "{x. hcomplex_of_complex x ∈ SComplex} = (UNIV::complex set)" by (simp add: SComplex_def) lemma SComplex_iff: "(x ∈ SComplex) = (∃y. x = hcomplex_of_complex y)" by (simp add: SComplex_def) lemma hcomplex_of_complex_image: "hcomplex_of_complex `(UNIV::complex set) = SComplex" by (auto simp add: SComplex_def) lemma inv_hcomplex_of_complex_image: "inv hcomplex_of_complex `SComplex = UNIV" apply (auto simp add: SComplex_def) apply (rule inj_hcomplex_of_complex [THEN inv_f_f, THEN subst], blast) done lemma SComplex_hcomplex_of_complex_image: "[| ∃x. x: P; P ≤ SComplex |] ==> ∃Q. P = hcomplex_of_complex ` Q" apply (simp add: SComplex_def, blast) done lemma SComplex_SReal_dense: "[| x ∈ SComplex; y ∈ SComplex; hcmod x < hcmod y |] ==> ∃r ∈ Reals. hcmod x< r & r < hcmod y" apply (auto intro: SReal_dense simp add: SReal_hcmod_SComplex) done lemma SComplex_hcmod_SReal: "z ∈ SComplex ==> hcmod z ∈ Reals" by (auto simp add: SComplex_def SReal_def hcmod_def) lemma SComplex_zero [simp]: "0 ∈ SComplex" by (simp add: SComplex_def) lemma SComplex_one [simp]: "1 ∈ SComplex" by (simp add: SComplex_def) (* Goalw [SComplex_def,SReal_def] "hcmod z ∈ Reals ==> z ∈ SComplex" by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1); by (auto_tac (claset(),simpset() addsimps [hcmod,hypreal_of_real_def, hcomplex_of_complex_def,cmod_def])); *) subsection{*The Finite Elements form a Subring*} lemma CFinite_add: "[|x ∈ CFinite; y ∈ CFinite|] ==> (x+y) ∈ CFinite" apply (simp add: CFinite_def) apply (blast intro!: SReal_add hcmod_add_less) done lemma CFinite_mult: "[|x ∈ CFinite; y ∈ CFinite|] ==> x*y ∈ CFinite" apply (simp add: CFinite_def) apply (blast intro!: SReal_mult hcmod_mult_less) done lemma CFinite_minus_iff [simp]: "(-x ∈ CFinite) = (x ∈ CFinite)" by (simp add: CFinite_def) lemma SComplex_subset_CFinite [simp]: "SComplex ≤ CFinite" apply (auto simp add: SComplex_def CFinite_def) apply (rule_tac x = "1 + hcmod (hcomplex_of_complex r) " in bexI) apply (auto intro: SReal_add) done lemma HFinite_hcmod_hcomplex_of_complex [simp]: "hcmod (hcomplex_of_complex r) ∈ HFinite" by (auto intro!: SReal_subset_HFinite [THEN subsetD]) lemma CFinite_hcomplex_of_complex [simp]: "hcomplex_of_complex x ∈ CFinite" by (auto intro!: SComplex_subset_CFinite [THEN subsetD]) lemma CFiniteD: "x ∈ CFinite ==> ∃t ∈ Reals. hcmod x < t" by (simp add: CFinite_def) lemma CFinite_hcmod_iff: "(x ∈ CFinite) = (hcmod x ∈ HFinite)" by (simp add: CFinite_def HFinite_def) lemma CFinite_number_of [simp]: "number_of w ∈ CFinite" by (rule SComplex_number_of [THEN SComplex_subset_CFinite [THEN subsetD]]) lemma CFinite_bounded: "[|x ∈ CFinite; y ≤ hcmod x; 0 ≤ y |] ==> y: HFinite" by (auto intro: HFinite_bounded simp add: CFinite_hcmod_iff) subsection{*The Complex Infinitesimals form a Subring*} lemma CInfinitesimal_zero [iff]: "0 ∈ CInfinitesimal" by (simp add: CInfinitesimal_def) lemma hcomplex_sum_of_halves: "x/(2::hcomplex) + x/(2::hcomplex) = x" by auto lemma CInfinitesimal_hcmod_iff: "(z ∈ CInfinitesimal) = (hcmod z ∈ Infinitesimal)" by (simp add: CInfinitesimal_def Infinitesimal_def) lemma one_not_CInfinitesimal [simp]: "1 ∉ CInfinitesimal" by (simp add: CInfinitesimal_hcmod_iff) lemma CInfinitesimal_add: "[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> (x+y) ∈ CInfinitesimal" apply (auto simp add: CInfinitesimal_hcmod_iff) apply (rule hrabs_le_Infinitesimal) apply (rule_tac y = "hcmod y" in Infinitesimal_add, auto) done lemma CInfinitesimal_minus_iff [simp]: "(-x:CInfinitesimal) = (x:CInfinitesimal)" by (simp add: CInfinitesimal_def) lemma CInfinitesimal_diff: "[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x-y ∈ CInfinitesimal" by (simp add: diff_minus CInfinitesimal_add) lemma CInfinitesimal_mult: "[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x * y ∈ CInfinitesimal" by (auto intro: Infinitesimal_mult simp add: CInfinitesimal_hcmod_iff hcmod_mult) lemma CInfinitesimal_CFinite_mult: "[| x ∈ CInfinitesimal; y ∈ CFinite |] ==> (x * y) ∈ CInfinitesimal" by (auto intro: Infinitesimal_HFinite_mult simp add: CInfinitesimal_hcmod_iff CFinite_hcmod_iff hcmod_mult) lemma CInfinitesimal_CFinite_mult2: "[| x ∈ CInfinitesimal; y ∈ CFinite |] ==> (y * x) ∈ CInfinitesimal" by (auto dest: CInfinitesimal_CFinite_mult simp add: mult_commute) lemma CInfinite_hcmod_iff: "(z ∈ CInfinite) = (hcmod z ∈ HInfinite)" by (simp add: CInfinite_def HInfinite_def) lemma CInfinite_inverse_CInfinitesimal: "x ∈ CInfinite ==> inverse x ∈ CInfinitesimal" by (auto intro: HInfinite_inverse_Infinitesimal simp add: CInfinitesimal_hcmod_iff CInfinite_hcmod_iff hcmod_hcomplex_inverse) lemma CInfinite_mult: "[|x ∈ CInfinite; y ∈ CInfinite|] ==> (x*y): CInfinite" by (auto intro: HInfinite_mult simp add: CInfinite_hcmod_iff hcmod_mult) lemma CInfinite_minus_iff [simp]: "(-x ∈ CInfinite) = (x ∈ CInfinite)" by (simp add: CInfinite_def) lemma CFinite_sum_squares: "[|a ∈ CFinite; b ∈ CFinite; c ∈ CFinite|] ==> a*a + b*b + c*c ∈ CFinite" by (auto intro: CFinite_mult CFinite_add) lemma not_CInfinitesimal_not_zero: "x ∉ CInfinitesimal ==> x ≠ 0" by auto lemma not_CInfinitesimal_not_zero2: "x ∈ CFinite - CInfinitesimal ==> x ≠ 0" by auto lemma CFinite_diff_CInfinitesimal_hcmod: "x ∈ CFinite - CInfinitesimal ==> hcmod x ∈ HFinite - Infinitesimal" by (simp add: CFinite_hcmod_iff CInfinitesimal_hcmod_iff) lemma hcmod_less_CInfinitesimal: "[| e ∈ CInfinitesimal; hcmod x < hcmod e |] ==> x ∈ CInfinitesimal" by (auto intro: hrabs_less_Infinitesimal simp add: CInfinitesimal_hcmod_iff) lemma hcmod_le_CInfinitesimal: "[| e ∈ CInfinitesimal; hcmod x ≤ hcmod e |] ==> x ∈ CInfinitesimal" by (auto intro: hrabs_le_Infinitesimal simp add: CInfinitesimal_hcmod_iff) lemma CInfinitesimal_interval: "[| e ∈ CInfinitesimal; e' ∈ CInfinitesimal; hcmod e' < hcmod x ; hcmod x < hcmod e |] ==> x ∈ CInfinitesimal" by (auto intro: Infinitesimal_interval simp add: CInfinitesimal_hcmod_iff) lemma CInfinitesimal_interval2: "[| e ∈ CInfinitesimal; e' ∈ CInfinitesimal; hcmod e' ≤ hcmod x ; hcmod x ≤ hcmod e |] ==> x ∈ CInfinitesimal" by (auto intro: Infinitesimal_interval2 simp add: CInfinitesimal_hcmod_iff) lemma not_CInfinitesimal_mult: "[| x ∉ CInfinitesimal; y ∉ CInfinitesimal|] ==> (x*y) ∉ CInfinitesimal" apply (auto simp add: CInfinitesimal_hcmod_iff hcmod_mult) apply (drule not_Infinitesimal_mult, auto) done lemma CInfinitesimal_mult_disj: "x*y ∈ CInfinitesimal ==> x ∈ CInfinitesimal | y ∈ CInfinitesimal" by (auto dest: Infinitesimal_mult_disj simp add: CInfinitesimal_hcmod_iff hcmod_mult) lemma CFinite_CInfinitesimal_diff_mult: "[| x ∈ CFinite - CInfinitesimal; y ∈ CFinite - CInfinitesimal |] ==> x*y ∈ CFinite - CInfinitesimal" by (blast dest: CFinite_mult not_CInfinitesimal_mult) lemma CInfinitesimal_subset_CFinite: "CInfinitesimal ≤ CFinite" by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: CInfinitesimal_hcmod_iff CFinite_hcmod_iff) lemma CInfinitesimal_hcomplex_of_complex_mult: "x ∈ CInfinitesimal ==> x * hcomplex_of_complex r ∈ CInfinitesimal" by (auto intro!: Infinitesimal_HFinite_mult simp add: CInfinitesimal_hcmod_iff hcmod_mult) lemma CInfinitesimal_hcomplex_of_complex_mult2: "x ∈ CInfinitesimal ==> hcomplex_of_complex r * x ∈ CInfinitesimal" by (auto intro!: Infinitesimal_HFinite_mult2 simp add: CInfinitesimal_hcmod_iff hcmod_mult) subsection{*The ``Infinitely Close'' Relation*} (* Goalw [capprox_def,approx_def] "(z @c= w) = (hcmod z @= hcmod w)" by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff])); *) lemma mem_cinfmal_iff: "x:CInfinitesimal = (x @c= 0)" by (simp add: CInfinitesimal_hcmod_iff capprox_def) lemma capprox_minus_iff: "(x @c= y) = (x + -y @c= 0)" by (simp add: capprox_def diff_minus) lemma capprox_minus_iff2: "(x @c= y) = (-y + x @c= 0)" by (simp add: capprox_def diff_minus add_commute) lemma capprox_refl [simp]: "x @c= x" by (simp add: capprox_def) lemma capprox_sym: "x @c= y ==> y @c= x" by (simp add: capprox_def CInfinitesimal_def hcmod_diff_commute) lemma capprox_trans: "[| x @c= y; y @c= z |] ==> x @c= z" apply (simp add: capprox_def) apply (drule CInfinitesimal_add, assumption) apply (simp add: diff_minus) done lemma capprox_trans2: "[| r @c= x; s @c= x |] ==> r @c= s" by (blast intro: capprox_sym capprox_trans) lemma capprox_trans3: "[| x @c= r; x @c= s|] ==> r @c= s" by (blast intro: capprox_sym capprox_trans) lemma number_of_capprox_reorient [simp]: "(number_of w @c= x) = (x @c= number_of w)" by (blast intro: capprox_sym) lemma CInfinitesimal_capprox_minus: "(x-y ∈ CInfinitesimal) = (x @c= y)" by (simp add: diff_minus capprox_minus_iff [symmetric] mem_cinfmal_iff) lemma capprox_monad_iff: "(x @c= y) = (cmonad(x)=cmonad(y))" by (auto simp add: cmonad_def dest: capprox_sym elim!: capprox_trans equalityCE) lemma Infinitesimal_capprox: "[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x @c= y" apply (simp add: mem_cinfmal_iff) apply (blast intro: capprox_trans capprox_sym) done lemma capprox_add: "[| a @c= b; c @c= d |] ==> a+c @c= b+d" apply (simp add: capprox_def diff_minus) apply (rule minus_add_distrib [THEN ssubst]) apply (rule add_assoc [THEN ssubst]) apply (rule_tac b1 = c in add_left_commute [THEN subst]) apply (rule add_assoc [THEN subst]) apply (blast intro: CInfinitesimal_add) done lemma capprox_minus: "a @c= b ==> -a @c= -b" apply (rule capprox_minus_iff [THEN iffD2, THEN capprox_sym]) apply (drule capprox_minus_iff [THEN iffD1]) apply (simp add: add_commute) done lemma capprox_minus2: "-a @c= -b ==> a @c= b" by (auto dest: capprox_minus) lemma capprox_minus_cancel [simp]: "(-a @c= -b) = (a @c= b)" by (blast intro: capprox_minus capprox_minus2) lemma capprox_add_minus: "[| a @c= b; c @c= d |] ==> a + -c @c= b + -d" by (blast intro!: capprox_add capprox_minus) lemma capprox_mult1: "[| a @c= b; c ∈ CFinite|] ==> a*c @c= b*c" apply (simp add: capprox_def diff_minus) apply (simp only: CInfinitesimal_CFinite_mult minus_mult_left left_distrib [symmetric]) done lemma capprox_mult2: "[|a @c= b; c ∈ CFinite|] ==> c*a @c= c*b" by (simp add: capprox_mult1 mult_commute) lemma capprox_mult_subst: "[|u @c= v*x; x @c= y; v ∈ CFinite|] ==> u @c= v*y" by (blast intro: capprox_mult2 capprox_trans) lemma capprox_mult_subst2: "[| u @c= x*v; x @c= y; v ∈ CFinite |] ==> u @c= y*v" by (blast intro: capprox_mult1 capprox_trans) lemma capprox_mult_subst_SComplex: "[| u @c= x*hcomplex_of_complex v; x @c= y |] ==> u @c= y*hcomplex_of_complex v" by (auto intro: capprox_mult_subst2) lemma capprox_eq_imp: "a = b ==> a @c= b" by (simp add: capprox_def) lemma CInfinitesimal_minus_capprox: "x ∈ CInfinitesimal ==> -x @c= x" by (blast intro: CInfinitesimal_minus_iff [THEN iffD2] mem_cinfmal_iff [THEN iffD1] capprox_trans2) lemma bex_CInfinitesimal_iff: "(∃y ∈ CInfinitesimal. x - z = y) = (x @c= z)" by (unfold capprox_def, blast) lemma bex_CInfinitesimal_iff2: "(∃y ∈ CInfinitesimal. x = z + y) = (x @c= z)" by (simp add: bex_CInfinitesimal_iff [symmetric], force) lemma CInfinitesimal_add_capprox: "[| y ∈ CInfinitesimal; x + y = z |] ==> x @c= z" apply (rule bex_CInfinitesimal_iff [THEN iffD1]) apply (drule CInfinitesimal_minus_iff [THEN iffD2]) apply (simp add: eq_commute compare_rls) done lemma CInfinitesimal_add_capprox_self: "y ∈ CInfinitesimal ==> x @c= x + y" apply (rule bex_CInfinitesimal_iff [THEN iffD1]) apply (drule CInfinitesimal_minus_iff [THEN iffD2]) apply (simp add: eq_commute compare_rls) done lemma CInfinitesimal_add_capprox_self2: "y ∈ CInfinitesimal ==> x @c= y + x" by (auto dest: CInfinitesimal_add_capprox_self simp add: add_commute) lemma CInfinitesimal_add_minus_capprox_self: "y ∈ CInfinitesimal ==> x @c= x + -y" by (blast intro!: CInfinitesimal_add_capprox_self CInfinitesimal_minus_iff [THEN iffD2]) lemma CInfinitesimal_add_cancel: "[| y ∈ CInfinitesimal; x+y @c= z|] ==> x @c= z" apply (drule_tac x = x in CInfinitesimal_add_capprox_self [THEN capprox_sym]) apply (erule capprox_trans3 [THEN capprox_sym], assumption) done lemma CInfinitesimal_add_right_cancel: "[| y ∈ CInfinitesimal; x @c= z + y|] ==> x @c= z" apply (drule_tac x = z in CInfinitesimal_add_capprox_self2 [THEN capprox_sym]) apply (erule capprox_trans3 [THEN capprox_sym]) apply (simp add: add_commute) apply (erule capprox_sym) done lemma capprox_add_left_cancel: "d + b @c= d + c ==> b @c= c" apply (drule capprox_minus_iff [THEN iffD1]) apply (simp add: minus_add_distrib capprox_minus_iff [symmetric] add_ac) done lemma capprox_add_right_cancel: "b + d @c= c + d ==> b @c= c" apply (rule capprox_add_left_cancel) apply (simp add: add_commute) done lemma capprox_add_mono1: "b @c= c ==> d + b @c= d + c" apply (rule capprox_minus_iff [THEN iffD2]) apply (simp add: capprox_minus_iff [symmetric] add_ac) done lemma capprox_add_mono2: "b @c= c ==> b + a @c= c + a" apply (simp (no_asm_simp) add: add_commute capprox_add_mono1) done lemma capprox_add_left_iff [iff]: "(a + b @c= a + c) = (b @c= c)" by (fast elim: capprox_add_left_cancel capprox_add_mono1) lemma capprox_add_right_iff [iff]: "(b + a @c= c + a) = (b @c= c)" by (simp add: add_commute) lemma capprox_CFinite: "[| x ∈ CFinite; x @c= y |] ==> y ∈ CFinite" apply (drule bex_CInfinitesimal_iff2 [THEN iffD2], safe) apply (drule CInfinitesimal_subset_CFinite [THEN subsetD, THEN CFinite_minus_iff [THEN iffD2]]) apply (drule CFinite_add) apply (assumption, auto) done lemma capprox_hcomplex_of_complex_CFinite: "x @c= hcomplex_of_complex D ==> x ∈ CFinite" by (rule capprox_sym [THEN [2] capprox_CFinite], auto) lemma capprox_mult_CFinite: "[|a @c= b; c @c= d; b ∈ CFinite; d ∈ CFinite|] ==> a*c @c= b*d" apply (rule capprox_trans) apply (rule_tac [2] capprox_mult2) apply (rule capprox_mult1) prefer 2 apply (blast intro: capprox_CFinite capprox_sym, auto) done lemma capprox_mult_hcomplex_of_complex: "[|a @c= hcomplex_of_complex b; c @c= hcomplex_of_complex d |] ==> a*c @c= hcomplex_of_complex b * hcomplex_of_complex d" apply (blast intro!: capprox_mult_CFinite capprox_hcomplex_of_complex_CFinite CFinite_hcomplex_of_complex) done lemma capprox_SComplex_mult_cancel_zero: "[| a ∈ SComplex; a ≠ 0; a*x @c= 0 |] ==> x @c= 0" apply (drule SComplex_inverse [THEN SComplex_subset_CFinite [THEN subsetD]]) apply (auto dest: capprox_mult2 simp add: mult_assoc [symmetric]) done lemma capprox_mult_SComplex1: "[| a ∈ SComplex; x @c= 0 |] ==> x*a @c= 0" by (auto dest: SComplex_subset_CFinite [THEN subsetD] capprox_mult1) lemma capprox_mult_SComplex2: "[| a ∈ SComplex; x @c= 0 |] ==> a*x @c= 0" by (auto dest: SComplex_subset_CFinite [THEN subsetD] capprox_mult2) lemma capprox_mult_SComplex_zero_cancel_iff [simp]: "[|a ∈ SComplex; a ≠ 0 |] ==> (a*x @c= 0) = (x @c= 0)" by (blast intro: capprox_SComplex_mult_cancel_zero capprox_mult_SComplex2) lemma capprox_SComplex_mult_cancel: "[| a ∈ SComplex; a ≠ 0; a* w @c= a*z |] ==> w @c= z" apply (drule SComplex_inverse [THEN SComplex_subset_CFinite [THEN subsetD]]) apply (auto dest: capprox_mult2 simp add: mult_assoc [symmetric]) done lemma capprox_SComplex_mult_cancel_iff1 [simp]: "[| a ∈ SComplex; a ≠ 0|] ==> (a* w @c= a*z) = (w @c= z)" by (auto intro!: capprox_mult2 SComplex_subset_CFinite [THEN subsetD] intro: capprox_SComplex_mult_cancel) lemma capprox_hcmod_approx_zero: "(x @c= y) = (hcmod (y - x) @= 0)" apply (rule capprox_minus_iff [THEN ssubst]) apply (simp add: capprox_def CInfinitesimal_hcmod_iff mem_infmal_iff diff_minus [symmetric] hcmod_diff_commute) done lemma capprox_approx_zero_iff: "(x @c= 0) = (hcmod x @= 0)" by (simp add: capprox_hcmod_approx_zero) lemma capprox_minus_zero_cancel_iff [simp]: "(-x @c= 0) = (x @c= 0)" by (simp add: capprox_hcmod_approx_zero) lemma Infinitesimal_hcmod_add_diff: "u @c= 0 ==> hcmod(x + u) - hcmod x ∈ Infinitesimal" apply (rule_tac e = "hcmod u" and e' = "- hcmod u" in Infinitesimal_interval2) apply (auto dest: capprox_approx_zero_iff [THEN iffD1] simp add: mem_infmal_iff [symmetric] diff_def) apply (rule_tac c1 = "hcmod x" in add_le_cancel_left [THEN iffD1]) apply (auto simp add: diff_minus [symmetric]) done lemma approx_hcmod_add_hcmod: "u @c= 0 ==> hcmod(x + u) @= hcmod x" apply (rule approx_minus_iff [THEN iffD2]) apply (auto intro: Infinitesimal_hcmod_add_diff simp add: mem_infmal_iff [symmetric] diff_minus [symmetric]) done lemma capprox_hcmod_approx: "x @c= y ==> hcmod x @= hcmod y" by (auto intro: approx_hcmod_add_hcmod dest!: bex_CInfinitesimal_iff2 [THEN iffD2] simp add: mem_cinfmal_iff) subsection{*Zero is the Only Infinitesimal Complex Number*} lemma CInfinitesimal_less_SComplex: "[| x ∈ SComplex; y ∈ CInfinitesimal; 0 < hcmod x |] ==> hcmod y < hcmod x" by (auto intro!: Infinitesimal_less_SReal SComplex_hcmod_SReal simp add: CInfinitesimal_hcmod_iff) lemma SComplex_Int_CInfinitesimal_zero: "SComplex Int CInfinitesimal = {0}" apply (auto simp add: SComplex_def CInfinitesimal_hcmod_iff) apply (cut_tac r = r in SReal_hcmod_hcomplex_of_complex) apply (drule_tac A = Reals in IntI, assumption) apply (subgoal_tac "hcmod (hcomplex_of_complex r) = 0") apply simp apply (simp add: SReal_Int_Infinitesimal_zero) done lemma SComplex_CInfinitesimal_zero: "[| x ∈ SComplex; x ∈ CInfinitesimal|] ==> x = 0" by (cut_tac SComplex_Int_CInfinitesimal_zero, blast) lemma SComplex_CFinite_diff_CInfinitesimal: "[| x ∈ SComplex; x ≠ 0 |] ==> x ∈ CFinite - CInfinitesimal" by (auto dest: SComplex_CInfinitesimal_zero SComplex_subset_CFinite [THEN subsetD]) lemma hcomplex_of_complex_CFinite_diff_CInfinitesimal: "hcomplex_of_complex x ≠ 0 ==> hcomplex_of_complex x ∈ CFinite - CInfinitesimal" by (rule SComplex_CFinite_diff_CInfinitesimal, auto) lemma hcomplex_of_complex_CInfinitesimal_iff_0 [iff]: "(hcomplex_of_complex x ∈ CInfinitesimal) = (x=0)" apply (auto) apply (rule ccontr) apply (rule hcomplex_of_complex_CFinite_diff_CInfinitesimal [THEN DiffD2], auto) done lemma number_of_not_CInfinitesimal [simp]: "number_of w ≠ (0::hcomplex) ==> number_of w ∉ CInfinitesimal" by (fast dest: SComplex_number_of [THEN SComplex_CInfinitesimal_zero]) lemma capprox_SComplex_not_zero: "[| y ∈ SComplex; x @c= y; y≠ 0 |] ==> x ≠ 0" by (auto dest: SComplex_CInfinitesimal_zero capprox_sym [THEN mem_cinfmal_iff [THEN iffD2]]) lemma CFinite_diff_CInfinitesimal_capprox: "[| x @c= y; y ∈ CFinite - CInfinitesimal |] ==> x ∈ CFinite - CInfinitesimal" apply (auto intro: capprox_sym [THEN [2] capprox_CFinite] simp add: mem_cinfmal_iff) apply (drule capprox_trans3, assumption) apply (blast dest: capprox_sym) done lemma CInfinitesimal_ratio: "[| y ≠ 0; y ∈ CInfinitesimal; x/y ∈ CFinite |] ==> x ∈ CInfinitesimal" apply (drule CInfinitesimal_CFinite_mult2, assumption) apply (simp add: divide_inverse mult_assoc) done lemma SComplex_capprox_iff: "[|x ∈ SComplex; y ∈ SComplex|] ==> (x @c= y) = (x = y)" apply auto apply (simp add: capprox_def) apply (subgoal_tac "x-y = 0", simp) apply (rule SComplex_CInfinitesimal_zero) apply (simp add: SComplex_diff, assumption) done lemma number_of_capprox_iff [simp]: "(number_of v @c= number_of w) = (number_of v = (number_of w :: hcomplex))" by (rule SComplex_capprox_iff, auto) lemma number_of_CInfinitesimal_iff [simp]: "(number_of w ∈ CInfinitesimal) = (number_of w = (0::hcomplex))" apply (rule iffI) apply (fast dest: SComplex_number_of [THEN SComplex_CInfinitesimal_zero]) apply (simp (no_asm_simp)) done lemma hcomplex_of_complex_approx_iff [simp]: "(hcomplex_of_complex k @c= hcomplex_of_complex m) = (k = m)" apply auto apply (rule inj_hcomplex_of_complex [THEN injD]) apply (rule SComplex_capprox_iff [THEN iffD1], auto) done lemma hcomplex_of_complex_capprox_number_of_iff [simp]: "(hcomplex_of_complex k @c= number_of w) = (k = number_of w)" by (subst hcomplex_of_complex_approx_iff [symmetric], auto) lemma capprox_unique_complex: "[| r ∈ SComplex; s ∈ SComplex; r @c= x; s @c= x|] ==> r = s" by (blast intro: SComplex_capprox_iff [THEN iffD1] capprox_trans2) lemma hcomplex_capproxD1: "star_n X @c= star_n Y ==> star_n (%n. Re(X n)) @= star_n (%n. Re(Y n))" apply (auto simp add: approx_FreeUltrafilterNat_iff) apply (drule capprox_minus_iff [THEN iffD1]) apply (auto simp add: star_n_minus star_n_add mem_cinfmal_iff [symmetric] CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff2) apply (drule_tac x = m in spec, ultra) apply (rename_tac Z x) apply (case_tac "X x") apply (case_tac "Y x") apply (auto simp add: complex_minus complex_add complex_mod simp del: realpow_Suc) apply (rule_tac y="abs(Z x)" in order_le_less_trans) apply (drule_tac t = "Z x" in sym) apply (auto simp del: realpow_Suc) done (* same proof *) lemma hcomplex_capproxD2: "star_n X @c= star_n Y ==> star_n (%n. Im(X n)) @= star_n (%n. Im(Y n))" apply (auto simp add: approx_FreeUltrafilterNat_iff) apply (drule capprox_minus_iff [THEN iffD1]) apply (auto simp add: star_n_minus star_n_add mem_cinfmal_iff [symmetric] CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff2) apply (drule_tac x = m in spec, ultra) apply (rename_tac Z x) apply (case_tac "X x") apply (case_tac "Y x") apply (auto simp add: complex_minus complex_add complex_mod simp del: realpow_Suc) apply (rule_tac y="abs(Z x)" in order_le_less_trans) apply (drule_tac t = "Z x" in sym) apply (auto simp del: realpow_Suc) done lemma hcomplex_capproxI: "[| star_n (%n. Re(X n)) @= star_n (%n. Re(Y n)); star_n (%n. Im(X n)) @= star_n (%n. Im(Y n)) |] ==> star_n X @c= star_n Y" apply (drule approx_minus_iff [THEN iffD1]) apply (drule approx_minus_iff [THEN iffD1]) apply (rule capprox_minus_iff [THEN iffD2]) apply (auto simp add: mem_cinfmal_iff [symmetric] mem_infmal_iff [symmetric] star_n_add star_n_minus CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff) apply (rule bexI [OF _ Rep_star_star_n], auto) apply (drule_tac x = "u/2" in spec) apply (drule_tac x = "u/2" in spec, auto, ultra) apply (drule sym, drule sym) apply (case_tac "X x") apply (case_tac "Y x") apply (auto simp add: complex_minus complex_add complex_mod snd_conv fst_conv numeral_2_eq_2) apply (rename_tac a b c d) apply (subgoal_tac "sqrt (abs (a + - c) ^ 2 + abs (b + - d) ^ 2) < u") apply (rule_tac [2] lemma_sqrt_hcomplex_capprox, auto) apply (simp add: power2_eq_square) done lemma capprox_approx_iff: "(star_n X @c= star_n Y) = (star_n (%n. Re(X n)) @= star_n (%n. Re(Y n)) & star_n (%n. Im(X n)) @= star_n (%n. Im(Y n)))" apply (blast intro: hcomplex_capproxI hcomplex_capproxD1 hcomplex_capproxD2) done lemma hcomplex_of_hypreal_capprox_iff [simp]: "(hcomplex_of_hypreal x @c= hcomplex_of_hypreal z) = (x @= z)" apply (cases x, cases z) apply (simp add: hcomplex_of_hypreal capprox_approx_iff) done lemma CFinite_HFinite_Re: "star_n X ∈ CFinite ==> star_n (%n. Re(X n)) ∈ HFinite" apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff) apply (rule bexI [OF _ Rep_star_star_n]) apply (rule_tac x = u in exI, ultra) apply (drule sym, case_tac "X x") apply (auto simp add: complex_mod numeral_2_eq_2 simp del: realpow_Suc) apply (rule ccontr, drule linorder_not_less [THEN iffD1]) apply (drule order_less_le_trans, assumption) apply (drule real_sqrt_ge_abs1 [THEN [2] order_less_le_trans]) apply (auto simp add: numeral_2_eq_2 [symmetric]) done lemma CFinite_HFinite_Im: "star_n X ∈ CFinite ==> star_n (%n. Im(X n)) ∈ HFinite" apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff) apply (rule bexI [OF _ Rep_star_star_n]) apply (rule_tac x = u in exI, ultra) apply (drule sym, case_tac "X x") apply (auto simp add: complex_mod simp del: realpow_Suc) apply (rule ccontr, drule linorder_not_less [THEN iffD1]) apply (drule order_less_le_trans, assumption) apply (drule real_sqrt_ge_abs2 [THEN [2] order_less_le_trans], auto) done lemma HFinite_Re_Im_CFinite: "[| star_n (%n. Re(X n)) ∈ HFinite; star_n (%n. Im(X n)) ∈ HFinite |] ==> star_n X ∈ CFinite" apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff) apply (rename_tac Y Z u v) apply (rule bexI [OF _ Rep_star_star_n]) apply (rule_tac x = "2* (u + v) " in exI) apply ultra apply (drule sym, case_tac "X x") apply (auto simp add: complex_mod numeral_2_eq_2 simp del: realpow_Suc) apply (subgoal_tac "0 < u") prefer 2 apply arith apply (subgoal_tac "0 < v") prefer 2 apply arith apply (subgoal_tac "sqrt (abs (Y x) ^ 2 + abs (Z x) ^ 2) < 2*u + 2*v") apply (rule_tac [2] lemma_sqrt_hcomplex_capprox, auto) apply (simp add: power2_eq_square) done lemma CFinite_HFinite_iff: "(star_n X ∈ CFinite) = (star_n (%n. Re(X n)) ∈ HFinite & star_n (%n. Im(X n)) ∈ HFinite)" by (blast intro: HFinite_Re_Im_CFinite CFinite_HFinite_Im CFinite_HFinite_Re) lemma SComplex_Re_SReal: "star_n X ∈ SComplex ==> star_n (%n. Re(X n)) ∈ Reals" apply (auto simp add: SComplex_def SReal_def star_of_def star_n_eq_iff) apply (rule_tac x = "Re r" in exI, ultra) done lemma SComplex_Im_SReal: "star_n X ∈ SComplex ==> star_n (%n. Im(X n)) ∈ Reals" apply (auto simp add: SComplex_def SReal_def star_of_def star_n_eq_iff) apply (rule_tac x = "Im r" in exI, ultra) done lemma Reals_Re_Im_SComplex: "[| star_n (%n. Re(X n)) ∈ Reals; star_n (%n. Im(X n)) ∈ Reals |] ==> star_n X ∈ SComplex" apply (auto simp add: SComplex_def SReal_def star_of_def star_n_eq_iff) apply (rule_tac x = "Complex r ra" in exI, ultra) done lemma SComplex_SReal_iff: "(star_n X ∈ SComplex) = (star_n (%n. Re(X n)) ∈ Reals & star_n (%n. Im(X n)) ∈ Reals)" by (blast intro: SComplex_Re_SReal SComplex_Im_SReal Reals_Re_Im_SComplex) lemma CInfinitesimal_Infinitesimal_iff: "(star_n X ∈ CInfinitesimal) = (star_n (%n. Re(X n)) ∈ Infinitesimal & star_n (%n. Im(X n)) ∈ Infinitesimal)" by (simp add: mem_cinfmal_iff mem_infmal_iff star_n_zero_num capprox_approx_iff) lemma eq_Abs_star_EX: "(∃t. P t) = (∃X. P (star_n X))" by (rule ex_star_eq) lemma eq_Abs_star_Bex: "(∃t ∈ A. P t) = (∃X. star_n X ∈ A & P (star_n X))" by (simp add: Bex_def ex_star_eq) (* Here we go - easy proof now!! *) lemma stc_part_Ex: "x:CFinite ==> ∃t ∈ SComplex. x @c= t" apply (cases x) apply (auto simp add: CFinite_HFinite_iff eq_Abs_star_Bex SComplex_SReal_iff capprox_approx_iff) apply (drule st_part_Ex, safe)+ apply (rule_tac x = t in star_cases) apply (rule_tac x = ta in star_cases, auto) apply (rule_tac x = "%n. Complex (Xa n) (Xb n) " in exI) apply auto done lemma stc_part_Ex1: "x:CFinite ==> EX! t. t ∈ SComplex & x @c= t" apply (drule stc_part_Ex, safe) apply (drule_tac [2] capprox_sym, drule_tac [2] capprox_sym, drule_tac [2] capprox_sym) apply (auto intro!: capprox_unique_complex) done lemma CFinite_Int_CInfinite_empty: "CFinite Int CInfinite = {}" by (simp add: CFinite_def CInfinite_def, auto) lemma CFinite_not_CInfinite: "x ∈ CFinite ==> x ∉ CInfinite" by (insert CFinite_Int_CInfinite_empty, blast) text{*Not sure this is a good idea!*} declare CFinite_Int_CInfinite_empty [simp] lemma not_CFinite_CInfinite: "x∉ CFinite ==> x ∈ CInfinite" by (auto intro: not_HFinite_HInfinite simp add: CFinite_hcmod_iff CInfinite_hcmod_iff) lemma CInfinite_CFinite_disj: "x ∈ CInfinite | x ∈ CFinite" by (blast intro: not_CFinite_CInfinite) lemma CInfinite_CFinite_iff: "(x ∈ CInfinite) = (x ∉ CFinite)" by (blast dest: CFinite_not_CInfinite not_CFinite_CInfinite) lemma CFinite_CInfinite_iff: "(x ∈ CFinite) = (x ∉ CInfinite)" by (simp add: CInfinite_CFinite_iff) lemma CInfinite_diff_CFinite_CInfinitesimal_disj: "x ∉ CInfinitesimal ==> x ∈ CInfinite | x ∈ CFinite - CInfinitesimal" by (fast intro: not_CFinite_CInfinite) lemma CFinite_inverse: "[| x ∈ CFinite; x ∉ CInfinitesimal |] ==> inverse x ∈ CFinite" apply (cut_tac x = "inverse x" in CInfinite_CFinite_disj) apply (auto dest!: CInfinite_inverse_CInfinitesimal) done lemma CFinite_inverse2: "x ∈ CFinite - CInfinitesimal ==> inverse x ∈ CFinite" by (blast intro: CFinite_inverse) lemma CInfinitesimal_inverse_CFinite: "x ∉ CInfinitesimal ==> inverse(x) ∈ CFinite" apply (drule CInfinite_diff_CFinite_CInfinitesimal_disj) apply (blast intro: CFinite_inverse CInfinite_inverse_CInfinitesimal CInfinitesimal_subset_CFinite [THEN subsetD]) done lemma CFinite_not_CInfinitesimal_inverse: "x ∈ CFinite - CInfinitesimal ==> inverse x ∈ CFinite - CInfinitesimal" apply (auto intro: CInfinitesimal_inverse_CFinite) apply (drule CInfinitesimal_CFinite_mult2, assumption) apply (simp add: not_CInfinitesimal_not_zero) done lemma capprox_inverse: "[| x @c= y; y ∈ CFinite - CInfinitesimal |] ==> inverse x @c= inverse y" apply (frule CFinite_diff_CInfinitesimal_capprox, assumption) apply (frule not_CInfinitesimal_not_zero2) apply (frule_tac x = x in not_CInfinitesimal_not_zero2) apply (drule CFinite_inverse2)+ apply (drule capprox_mult2, assumption, auto) apply (drule_tac c = "inverse x" in capprox_mult1, assumption) apply (auto intro: capprox_sym simp add: mult_assoc) done lemmas hcomplex_of_complex_capprox_inverse = hcomplex_of_complex_CFinite_diff_CInfinitesimal [THEN [2] capprox_inverse] lemma inverse_add_CInfinitesimal_capprox: "[| x ∈ CFinite - CInfinitesimal; h ∈ CInfinitesimal |] ==> inverse(x + h) @c= inverse x" by (auto intro: capprox_inverse capprox_sym CInfinitesimal_add_capprox_self) lemma inverse_add_CInfinitesimal_capprox2: "[| x ∈ CFinite - CInfinitesimal; h ∈ CInfinitesimal |] ==> inverse(h + x) @c= inverse x" apply (rule add_commute [THEN subst]) apply (blast intro: inverse_add_CInfinitesimal_capprox) done lemma inverse_add_CInfinitesimal_approx_CInfinitesimal: "[| x ∈ CFinite - CInfinitesimal; h ∈ CInfinitesimal |] ==> inverse(x + h) - inverse x @c= h" apply (rule capprox_trans2) apply (auto intro: inverse_add_CInfinitesimal_capprox simp add: mem_cinfmal_iff diff_minus capprox_minus_iff [symmetric]) done lemma CInfinitesimal_square_iff [iff]: "(x*x ∈ CInfinitesimal) = (x ∈ CInfinitesimal)" by (simp add: CInfinitesimal_hcmod_iff hcmod_mult) lemma capprox_CFinite_mult_cancel: "[| a ∈ CFinite-CInfinitesimal; a*w @c= a*z |] ==> w @c= z" apply safe apply (frule CFinite_inverse, assumption) apply (drule not_CInfinitesimal_not_zero) apply (auto dest: capprox_mult2 simp add: mult_assoc [symmetric]) done lemma capprox_CFinite_mult_cancel_iff1: "a ∈ CFinite-CInfinitesimal ==> (a * w @c= a * z) = (w @c= z)" by (auto intro: capprox_mult2 capprox_CFinite_mult_cancel) subsection{*Theorems About Monads*} lemma capprox_cmonad_iff: "(x @c= y) = (cmonad(x)=cmonad(y))" apply (simp add: cmonad_def) apply (auto dest: capprox_sym elim!: capprox_trans equalityCE) done lemma CInfinitesimal_cmonad_eq: "e ∈ CInfinitesimal ==> cmonad (x+e) = cmonad x" by (fast intro!: CInfinitesimal_add_capprox_self [THEN capprox_sym] capprox_cmonad_iff [THEN iffD1]) lemma mem_cmonad_iff: "(u ∈ cmonad x) = (-u ∈ cmonad (-x))" by (simp add: cmonad_def) lemma CInfinitesimal_cmonad_zero_iff: "(x:CInfinitesimal) = (x ∈ cmonad 0)" by (auto intro: capprox_sym simp add: mem_cinfmal_iff cmonad_def) lemma cmonad_zero_minus_iff: "(x ∈ cmonad 0) = (-x ∈ cmonad 0)" by (simp add: CInfinitesimal_cmonad_zero_iff [symmetric]) lemma cmonad_zero_hcmod_iff: "(x ∈ cmonad 0) = (hcmod x:monad 0)" by (simp add: CInfinitesimal_cmonad_zero_iff [symmetric] CInfinitesimal_hcmod_iff Infinitesimal_monad_zero_iff [symmetric]) lemma mem_cmonad_self [simp]: "x ∈ cmonad x" by (simp add: cmonad_def) subsection{*Theorems About Standard Part*} lemma stc_capprox_self: "x ∈ CFinite ==> stc x @c= x" apply (simp add: stc_def) apply (frule stc_part_Ex, safe) apply (rule someI2) apply (auto intro: capprox_sym) done lemma stc_SComplex: "x ∈ CFinite ==> stc x ∈ SComplex" apply (simp add: stc_def) apply (frule stc_part_Ex, safe) apply (rule someI2) apply (auto intro: capprox_sym) done lemma stc_CFinite: "x ∈ CFinite ==> stc x ∈ CFinite" by (erule stc_SComplex [THEN SComplex_subset_CFinite [THEN subsetD]]) lemma stc_SComplex_eq [simp]: "x ∈ SComplex ==> stc x = x" apply (simp add: stc_def) apply (rule some_equality) apply (auto intro: SComplex_subset_CFinite [THEN subsetD]) apply (blast dest: SComplex_capprox_iff [THEN iffD1]) done lemma stc_hcomplex_of_complex: "stc (hcomplex_of_complex x) = hcomplex_of_complex x" by auto lemma stc_eq_capprox: "[| x ∈ CFinite; y ∈ CFinite; stc x = stc y |] ==> x @c= y" by (auto dest!: stc_capprox_self elim!: capprox_trans3) lemma capprox_stc_eq: "[| x ∈ CFinite; y ∈ CFinite; x @c= y |] ==> stc x = stc y" by (blast intro: capprox_trans capprox_trans2 SComplex_capprox_iff [THEN iffD1] dest: stc_capprox_self stc_SComplex) lemma stc_eq_capprox_iff: "[| x ∈ CFinite; y ∈ CFinite|] ==> (x @c= y) = (stc x = stc y)" by (blast intro: capprox_stc_eq stc_eq_capprox) lemma stc_CInfinitesimal_add_SComplex: "[| x ∈ SComplex; e ∈ CInfinitesimal |] ==> stc(x + e) = x" apply (frule stc_SComplex_eq [THEN subst]) prefer 2 apply assumption apply (frule SComplex_subset_CFinite [THEN subsetD]) apply (frule CInfinitesimal_subset_CFinite [THEN subsetD]) apply (drule stc_SComplex_eq) apply (rule capprox_stc_eq) apply (auto intro: CFinite_add simp add: CInfinitesimal_add_capprox_self [THEN capprox_sym]) done lemma stc_CInfinitesimal_add_SComplex2: "[| x ∈ SComplex; e ∈ CInfinitesimal |] ==> stc(e + x) = x" apply (rule add_commute [THEN subst]) apply (blast intro!: stc_CInfinitesimal_add_SComplex) done lemma CFinite_stc_CInfinitesimal_add: "x ∈ CFinite ==> ∃e ∈ CInfinitesimal. x = stc(x) + e" by (blast dest!: stc_capprox_self [THEN capprox_sym] bex_CInfinitesimal_iff2 [THEN iffD2]) lemma stc_add: "[| x ∈ CFinite; y ∈ CFinite |] ==> stc (x + y) = stc(x) + stc(y)" apply (frule CFinite_stc_CInfinitesimal_add) apply (frule_tac x = y in CFinite_stc_CInfinitesimal_add, safe) apply (subgoal_tac "stc (x + y) = stc ((stc x + e) + (stc y + ea))") apply (drule_tac [2] sym, drule_tac [2] sym) prefer 2 apply simp apply (simp (no_asm_simp) add: add_ac) apply (drule stc_SComplex)+ apply (drule SComplex_add, assumption) apply (drule CInfinitesimal_add, assumption) apply (rule add_assoc [THEN subst]) apply (blast intro!: stc_CInfinitesimal_add_SComplex2) done lemma stc_number_of [simp]: "stc (number_of w) = number_of w" by (rule SComplex_number_of [THEN stc_SComplex_eq]) lemma stc_zero [simp]: "stc 0 = 0" by simp lemma stc_one [simp]: "stc 1 = 1" by simp lemma stc_minus: "y ∈ CFinite ==> stc(-y) = -stc(y)" apply (frule CFinite_minus_iff [THEN iffD2]) apply (rule hcomplex_add_minus_eq_minus) apply (drule stc_add [symmetric], assumption) apply (simp add: add_commute) done lemma stc_diff: "[| x ∈ CFinite; y ∈ CFinite |] ==> stc (x-y) = stc(x) - stc(y)" apply (simp add: diff_minus) apply (frule_tac y1 = y in stc_minus [symmetric]) apply (drule_tac x1 = y in CFinite_minus_iff [THEN iffD2]) apply (auto intro: stc_add) done lemma lemma_stc_mult: "[| x ∈ CFinite; y ∈ CFinite; e ∈ CInfinitesimal; ea: CInfinitesimal |] ==> e*y + x*ea + e*ea: CInfinitesimal" apply (frule_tac x = e and y = y in CInfinitesimal_CFinite_mult) apply (frule_tac [2] x = ea and y = x in CInfinitesimal_CFinite_mult) apply (drule_tac [3] CInfinitesimal_mult) apply (auto intro: CInfinitesimal_add simp add: add_ac mult_ac) done lemma stc_mult: "[| x ∈ CFinite; y ∈ CFinite |] ==> stc (x * y) = stc(x) * stc(y)" apply (frule CFinite_stc_CInfinitesimal_add) apply (frule_tac x = y in CFinite_stc_CInfinitesimal_add, safe) apply (subgoal_tac "stc (x * y) = stc ((stc x + e) * (stc y + ea))") apply (drule_tac [2] sym, drule_tac [2] sym) prefer 2 apply simp apply (erule_tac V = "x = stc x + e" in thin_rl) apply (erule_tac V = "y = stc y + ea" in thin_rl) apply (simp add: left_distrib right_distrib) apply (drule stc_SComplex)+ apply (simp (no_asm_use) add: add_assoc) apply (rule stc_CInfinitesimal_add_SComplex) apply (blast intro!: SComplex_mult) apply (drule SComplex_subset_CFinite [THEN subsetD])+ apply (rule add_assoc [THEN subst]) apply (blast intro!: lemma_stc_mult) done lemma stc_CInfinitesimal: "x ∈ CInfinitesimal ==> stc x = 0" apply (rule stc_zero [THEN subst]) apply (rule capprox_stc_eq) apply (auto intro: CInfinitesimal_subset_CFinite [THEN subsetD] simp add: mem_cinfmal_iff [symmetric]) done lemma stc_not_CInfinitesimal: "stc(x) ≠ 0 ==> x ∉ CInfinitesimal" by (fast intro: stc_CInfinitesimal) lemma stc_inverse: "[| x ∈ CFinite; stc x ≠ 0 |] ==> stc(inverse x) = inverse (stc x)" apply (rule_tac c1 = "stc x" in hcomplex_mult_left_cancel [THEN iffD1]) apply (auto simp add: stc_mult [symmetric] stc_not_CInfinitesimal CFinite_inverse) apply (subst right_inverse, auto) done lemma stc_divide [simp]: "[| x ∈ CFinite; y ∈ CFinite; stc y ≠ 0 |] ==> stc(x/y) = (stc x) / (stc y)" by (simp add: divide_inverse stc_mult stc_not_CInfinitesimal CFinite_inverse stc_inverse) lemma stc_idempotent [simp]: "x ∈ CFinite ==> stc(stc(x)) = stc(x)" by (blast intro: stc_CFinite stc_capprox_self capprox_stc_eq) lemma CFinite_HFinite_hcomplex_of_hypreal: "z ∈ HFinite ==> hcomplex_of_hypreal z ∈ CFinite" apply (cases z) apply (simp add: hcomplex_of_hypreal CFinite_HFinite_iff star_n_zero_num [symmetric]) done lemma SComplex_SReal_hcomplex_of_hypreal: "x ∈ Reals ==> hcomplex_of_hypreal x ∈ SComplex" by (auto simp add: SReal_def SComplex_def hcomplex_of_hypreal_def) lemma stc_hcomplex_of_hypreal: "z ∈ HFinite ==> stc(hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)" apply (simp add: st_def stc_def) apply (frule st_part_Ex, safe) apply (rule someI2) apply (auto intro: approx_sym) apply (drule CFinite_HFinite_hcomplex_of_hypreal) apply (frule stc_part_Ex, safe) apply (rule someI2) apply (auto intro: capprox_sym intro!: capprox_unique_complex dest: SComplex_SReal_hcomplex_of_hypreal) done (* Goal "x ∈ CFinite ==> hcmod(stc x) = st(hcmod x)" by (dtac stc_capprox_self 1) by (auto_tac (claset(),simpset() addsimps [bex_CInfinitesimal_iff2 RS sym])); approx_hcmod_add_hcmod *) lemma CInfinitesimal_hcnj_iff [simp]: "(hcnj z ∈ CInfinitesimal) = (z ∈ CInfinitesimal)" by (simp add: CInfinitesimal_hcmod_iff) lemma CInfinite_HInfinite_iff: "(star_n X ∈ CInfinite) = (star_n (%n. Re(X n)) ∈ HInfinite | star_n (%n. Im(X n)) ∈ HInfinite)" by (simp add: CInfinite_CFinite_iff HInfinite_HFinite_iff CFinite_HFinite_iff) text{*These theorems should probably be deleted*} lemma hcomplex_split_CInfinitesimal_iff: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ CInfinitesimal) = (x ∈ Infinitesimal & y ∈ Infinitesimal)" apply (cases x, cases y) apply (simp add: iii_def star_of_def star_n_add star_n_mult hcomplex_of_hypreal CInfinitesimal_Infinitesimal_iff) done lemma hcomplex_split_CFinite_iff: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ CFinite) = (x ∈ HFinite & y ∈ HFinite)" apply (cases x, cases y) apply (simp add: iii_def star_of_def star_n_add star_n_mult hcomplex_of_hypreal CFinite_HFinite_iff) done lemma hcomplex_split_SComplex_iff: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ SComplex) = (x ∈ Reals & y ∈ Reals)" apply (cases x, cases y) apply (simp add: iii_def star_of_def star_n_add star_n_mult hcomplex_of_hypreal SComplex_SReal_iff) done lemma hcomplex_split_CInfinite_iff: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ CInfinite) = (x ∈ HInfinite | y ∈ HInfinite)" apply (cases x, cases y) apply (simp add: iii_def star_of_def star_n_add star_n_mult hcomplex_of_hypreal CInfinite_HInfinite_iff) done lemma hcomplex_split_capprox_iff: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y @c= hcomplex_of_hypreal x' + iii * hcomplex_of_hypreal y') = (x @= x' & y @= y')" apply (cases x, cases y, cases x', cases y') apply (simp add: iii_def star_of_def star_n_add star_n_mult hcomplex_of_hypreal capprox_approx_iff) done lemma complex_seq_to_hcomplex_CInfinitesimal: "∀n. cmod (X n - x) < inverse (real (Suc n)) ==> star_n X - hcomplex_of_complex x ∈ CInfinitesimal" apply (simp add: star_n_diff CInfinitesimal_hcmod_iff star_of_def Infinitesimal_FreeUltrafilterNat_iff hcmod) apply (rule bexI, auto) apply (auto dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset) done lemma CInfinitesimal_hcomplex_of_hypreal_epsilon [simp]: "hcomplex_of_hypreal epsilon ∈ CInfinitesimal" by (simp add: CInfinitesimal_hcmod_iff) lemma hcomplex_of_complex_approx_zero_iff [simp]: "(hcomplex_of_complex z @c= 0) = (z = 0)" by (simp add: star_of_zero [symmetric] del: star_of_zero) lemma hcomplex_of_complex_approx_zero_iff2 [simp]: "(0 @c= hcomplex_of_complex z) = (z = 0)" by (simp add: star_of_zero [symmetric] del: star_of_zero) ML {* val SComplex_add = thm "SComplex_add"; val SComplex_mult = thm "SComplex_mult"; val SComplex_inverse = thm "SComplex_inverse"; val SComplex_divide = thm "SComplex_divide"; val SComplex_minus = thm "SComplex_minus"; val SComplex_minus_iff = thm "SComplex_minus_iff"; val SComplex_diff = thm "SComplex_diff"; val SComplex_add_cancel = thm "SComplex_add_cancel"; val SReal_hcmod_hcomplex_of_complex = thm "SReal_hcmod_hcomplex_of_complex"; val SReal_hcmod_number_of = thm "SReal_hcmod_number_of"; val SReal_hcmod_SComplex = thm "SReal_hcmod_SComplex"; val SComplex_hcomplex_of_complex = thm "SComplex_hcomplex_of_complex"; val SComplex_number_of = thm "SComplex_number_of"; val SComplex_divide_number_of = thm "SComplex_divide_number_of"; val SComplex_UNIV_complex = thm "SComplex_UNIV_complex"; val SComplex_iff = thm "SComplex_iff"; val hcomplex_of_complex_image = thm "hcomplex_of_complex_image"; val inv_hcomplex_of_complex_image = thm "inv_hcomplex_of_complex_image"; val SComplex_hcomplex_of_complex_image = thm "SComplex_hcomplex_of_complex_image"; val SComplex_SReal_dense = thm "SComplex_SReal_dense"; val SComplex_hcmod_SReal = thm "SComplex_hcmod_SReal"; val SComplex_zero = thm "SComplex_zero"; val SComplex_one = thm "SComplex_one"; val CFinite_add = thm "CFinite_add"; val CFinite_mult = thm "CFinite_mult"; val CFinite_minus_iff = thm "CFinite_minus_iff"; val SComplex_subset_CFinite = thm "SComplex_subset_CFinite"; val HFinite_hcmod_hcomplex_of_complex = thm "HFinite_hcmod_hcomplex_of_complex"; val CFinite_hcomplex_of_complex = thm "CFinite_hcomplex_of_complex"; val CFiniteD = thm "CFiniteD"; val CFinite_hcmod_iff = thm "CFinite_hcmod_iff"; val CFinite_number_of = thm "CFinite_number_of"; val CFinite_bounded = thm "CFinite_bounded"; val CInfinitesimal_zero = thm "CInfinitesimal_zero"; val hcomplex_sum_of_halves = thm "hcomplex_sum_of_halves"; val CInfinitesimal_hcmod_iff = thm "CInfinitesimal_hcmod_iff"; val one_not_CInfinitesimal = thm "one_not_CInfinitesimal"; val CInfinitesimal_add = thm "CInfinitesimal_add"; val CInfinitesimal_minus_iff = thm "CInfinitesimal_minus_iff"; val CInfinitesimal_diff = thm "CInfinitesimal_diff"; val CInfinitesimal_mult = thm "CInfinitesimal_mult"; val CInfinitesimal_CFinite_mult = thm "CInfinitesimal_CFinite_mult"; val CInfinitesimal_CFinite_mult2 = thm "CInfinitesimal_CFinite_mult2"; val CInfinite_hcmod_iff = thm "CInfinite_hcmod_iff"; val CInfinite_inverse_CInfinitesimal = thm "CInfinite_inverse_CInfinitesimal"; val CInfinite_mult = thm "CInfinite_mult"; val CInfinite_minus_iff = thm "CInfinite_minus_iff"; val CFinite_sum_squares = thm "CFinite_sum_squares"; val not_CInfinitesimal_not_zero = thm "not_CInfinitesimal_not_zero"; val not_CInfinitesimal_not_zero2 = thm "not_CInfinitesimal_not_zero2"; val CFinite_diff_CInfinitesimal_hcmod = thm "CFinite_diff_CInfinitesimal_hcmod"; val hcmod_less_CInfinitesimal = thm "hcmod_less_CInfinitesimal"; val hcmod_le_CInfinitesimal = thm "hcmod_le_CInfinitesimal"; val CInfinitesimal_interval = thm "CInfinitesimal_interval"; val CInfinitesimal_interval2 = thm "CInfinitesimal_interval2"; val not_CInfinitesimal_mult = thm "not_CInfinitesimal_mult"; val CInfinitesimal_mult_disj = thm "CInfinitesimal_mult_disj"; val CFinite_CInfinitesimal_diff_mult = thm "CFinite_CInfinitesimal_diff_mult"; val CInfinitesimal_subset_CFinite = thm "CInfinitesimal_subset_CFinite"; val CInfinitesimal_hcomplex_of_complex_mult = thm "CInfinitesimal_hcomplex_of_complex_mult"; val CInfinitesimal_hcomplex_of_complex_mult2 = thm "CInfinitesimal_hcomplex_of_complex_mult2"; val mem_cinfmal_iff = thm "mem_cinfmal_iff"; val capprox_minus_iff = thm "capprox_minus_iff"; val capprox_minus_iff2 = thm "capprox_minus_iff2"; val capprox_refl = thm "capprox_refl"; val capprox_sym = thm "capprox_sym"; val capprox_trans = thm "capprox_trans"; val capprox_trans2 = thm "capprox_trans2"; val capprox_trans3 = thm "capprox_trans3"; val number_of_capprox_reorient = thm "number_of_capprox_reorient"; val CInfinitesimal_capprox_minus = thm "CInfinitesimal_capprox_minus"; val capprox_monad_iff = thm "capprox_monad_iff"; val Infinitesimal_capprox = thm "Infinitesimal_capprox"; val capprox_add = thm "capprox_add"; val capprox_minus = thm "capprox_minus"; val capprox_minus2 = thm "capprox_minus2"; val capprox_minus_cancel = thm "capprox_minus_cancel"; val capprox_add_minus = thm "capprox_add_minus"; val capprox_mult1 = thm "capprox_mult1"; val capprox_mult2 = thm "capprox_mult2"; val capprox_mult_subst = thm "capprox_mult_subst"; val capprox_mult_subst2 = thm "capprox_mult_subst2"; val capprox_mult_subst_SComplex = thm "capprox_mult_subst_SComplex"; val capprox_eq_imp = thm "capprox_eq_imp"; val CInfinitesimal_minus_capprox = thm "CInfinitesimal_minus_capprox"; val bex_CInfinitesimal_iff = thm "bex_CInfinitesimal_iff"; val bex_CInfinitesimal_iff2 = thm "bex_CInfinitesimal_iff2"; val CInfinitesimal_add_capprox = thm "CInfinitesimal_add_capprox"; val CInfinitesimal_add_capprox_self = thm "CInfinitesimal_add_capprox_self"; val CInfinitesimal_add_capprox_self2 = thm "CInfinitesimal_add_capprox_self2"; val CInfinitesimal_add_minus_capprox_self = thm "CInfinitesimal_add_minus_capprox_self"; val CInfinitesimal_add_cancel = thm "CInfinitesimal_add_cancel"; val CInfinitesimal_add_right_cancel = thm "CInfinitesimal_add_right_cancel"; val capprox_add_left_cancel = thm "capprox_add_left_cancel"; val capprox_add_right_cancel = thm "capprox_add_right_cancel"; val capprox_add_mono1 = thm "capprox_add_mono1"; val capprox_add_mono2 = thm "capprox_add_mono2"; val capprox_add_left_iff = thm "capprox_add_left_iff"; val capprox_add_right_iff = thm "capprox_add_right_iff"; val capprox_CFinite = thm "capprox_CFinite"; val capprox_hcomplex_of_complex_CFinite = thm "capprox_hcomplex_of_complex_CFinite"; val capprox_mult_CFinite = thm "capprox_mult_CFinite"; val capprox_mult_hcomplex_of_complex = thm "capprox_mult_hcomplex_of_complex"; val capprox_SComplex_mult_cancel_zero = thm "capprox_SComplex_mult_cancel_zero"; val capprox_mult_SComplex1 = thm "capprox_mult_SComplex1"; val capprox_mult_SComplex2 = thm "capprox_mult_SComplex2"; val capprox_mult_SComplex_zero_cancel_iff = thm "capprox_mult_SComplex_zero_cancel_iff"; val capprox_SComplex_mult_cancel = thm "capprox_SComplex_mult_cancel"; val capprox_SComplex_mult_cancel_iff1 = thm "capprox_SComplex_mult_cancel_iff1"; val capprox_hcmod_approx_zero = thm "capprox_hcmod_approx_zero"; val capprox_approx_zero_iff = thm "capprox_approx_zero_iff"; val capprox_minus_zero_cancel_iff = thm "capprox_minus_zero_cancel_iff"; val Infinitesimal_hcmod_add_diff = thm "Infinitesimal_hcmod_add_diff"; val approx_hcmod_add_hcmod = thm "approx_hcmod_add_hcmod"; val capprox_hcmod_approx = thm "capprox_hcmod_approx"; val CInfinitesimal_less_SComplex = thm "CInfinitesimal_less_SComplex"; val SComplex_Int_CInfinitesimal_zero = thm "SComplex_Int_CInfinitesimal_zero"; val SComplex_CInfinitesimal_zero = thm "SComplex_CInfinitesimal_zero"; val SComplex_CFinite_diff_CInfinitesimal = thm "SComplex_CFinite_diff_CInfinitesimal"; val hcomplex_of_complex_CFinite_diff_CInfinitesimal = thm "hcomplex_of_complex_CFinite_diff_CInfinitesimal"; val hcomplex_of_complex_CInfinitesimal_iff_0 = thm "hcomplex_of_complex_CInfinitesimal_iff_0"; val number_of_not_CInfinitesimal = thm "number_of_not_CInfinitesimal"; val capprox_SComplex_not_zero = thm "capprox_SComplex_not_zero"; val CFinite_diff_CInfinitesimal_capprox = thm "CFinite_diff_CInfinitesimal_capprox"; val CInfinitesimal_ratio = thm "CInfinitesimal_ratio"; val SComplex_capprox_iff = thm "SComplex_capprox_iff"; val number_of_capprox_iff = thm "number_of_capprox_iff"; val number_of_CInfinitesimal_iff = thm "number_of_CInfinitesimal_iff"; val hcomplex_of_complex_approx_iff = thm "hcomplex_of_complex_approx_iff"; val hcomplex_of_complex_capprox_number_of_iff = thm "hcomplex_of_complex_capprox_number_of_iff"; val capprox_unique_complex = thm "capprox_unique_complex"; val hcomplex_capproxD1 = thm "hcomplex_capproxD1"; val hcomplex_capproxD2 = thm "hcomplex_capproxD2"; val hcomplex_capproxI = thm "hcomplex_capproxI"; val capprox_approx_iff = thm "capprox_approx_iff"; val hcomplex_of_hypreal_capprox_iff = thm "hcomplex_of_hypreal_capprox_iff"; val CFinite_HFinite_Re = thm "CFinite_HFinite_Re"; val CFinite_HFinite_Im = thm "CFinite_HFinite_Im"; val HFinite_Re_Im_CFinite = thm "HFinite_Re_Im_CFinite"; val CFinite_HFinite_iff = thm "CFinite_HFinite_iff"; val SComplex_Re_SReal = thm "SComplex_Re_SReal"; val SComplex_Im_SReal = thm "SComplex_Im_SReal"; val Reals_Re_Im_SComplex = thm "Reals_Re_Im_SComplex"; val SComplex_SReal_iff = thm "SComplex_SReal_iff"; val CInfinitesimal_Infinitesimal_iff = thm "CInfinitesimal_Infinitesimal_iff"; val eq_Abs_star_Bex = thm "eq_Abs_star_Bex"; val stc_part_Ex = thm "stc_part_Ex"; val stc_part_Ex1 = thm "stc_part_Ex1"; val CFinite_Int_CInfinite_empty = thm "CFinite_Int_CInfinite_empty"; val CFinite_not_CInfinite = thm "CFinite_not_CInfinite"; val not_CFinite_CInfinite = thm "not_CFinite_CInfinite"; val CInfinite_CFinite_disj = thm "CInfinite_CFinite_disj"; val CInfinite_CFinite_iff = thm "CInfinite_CFinite_iff"; val CFinite_CInfinite_iff = thm "CFinite_CInfinite_iff"; val CInfinite_diff_CFinite_CInfinitesimal_disj = thm "CInfinite_diff_CFinite_CInfinitesimal_disj"; val CFinite_inverse = thm "CFinite_inverse"; val CFinite_inverse2 = thm "CFinite_inverse2"; val CInfinitesimal_inverse_CFinite = thm "CInfinitesimal_inverse_CFinite"; val CFinite_not_CInfinitesimal_inverse = thm "CFinite_not_CInfinitesimal_inverse"; val capprox_inverse = thm "capprox_inverse"; val hcomplex_of_complex_capprox_inverse = thms "hcomplex_of_complex_capprox_inverse"; val inverse_add_CInfinitesimal_capprox = thm "inverse_add_CInfinitesimal_capprox"; val inverse_add_CInfinitesimal_capprox2 = thm "inverse_add_CInfinitesimal_capprox2"; val inverse_add_CInfinitesimal_approx_CInfinitesimal = thm "inverse_add_CInfinitesimal_approx_CInfinitesimal"; val CInfinitesimal_square_iff = thm "CInfinitesimal_square_iff"; val capprox_CFinite_mult_cancel = thm "capprox_CFinite_mult_cancel"; val capprox_CFinite_mult_cancel_iff1 = thm "capprox_CFinite_mult_cancel_iff1"; val capprox_cmonad_iff = thm "capprox_cmonad_iff"; val CInfinitesimal_cmonad_eq = thm "CInfinitesimal_cmonad_eq"; val mem_cmonad_iff = thm "mem_cmonad_iff"; val CInfinitesimal_cmonad_zero_iff = thm "CInfinitesimal_cmonad_zero_iff"; val cmonad_zero_minus_iff = thm "cmonad_zero_minus_iff"; val cmonad_zero_hcmod_iff = thm "cmonad_zero_hcmod_iff"; val mem_cmonad_self = thm "mem_cmonad_self"; val stc_capprox_self = thm "stc_capprox_self"; val stc_SComplex = thm "stc_SComplex"; val stc_CFinite = thm "stc_CFinite"; val stc_SComplex_eq = thm "stc_SComplex_eq"; val stc_hcomplex_of_complex = thm "stc_hcomplex_of_complex"; val stc_eq_capprox = thm "stc_eq_capprox"; val capprox_stc_eq = thm "capprox_stc_eq"; val stc_eq_capprox_iff = thm "stc_eq_capprox_iff"; val stc_CInfinitesimal_add_SComplex = thm "stc_CInfinitesimal_add_SComplex"; val stc_CInfinitesimal_add_SComplex2 = thm "stc_CInfinitesimal_add_SComplex2"; val CFinite_stc_CInfinitesimal_add = thm "CFinite_stc_CInfinitesimal_add"; val stc_add = thm "stc_add"; val stc_number_of = thm "stc_number_of"; val stc_zero = thm "stc_zero"; val stc_one = thm "stc_one"; val stc_minus = thm "stc_minus"; val stc_diff = thm "stc_diff"; val lemma_stc_mult = thm "lemma_stc_mult"; val stc_mult = thm "stc_mult"; val stc_CInfinitesimal = thm "stc_CInfinitesimal"; val stc_not_CInfinitesimal = thm "stc_not_CInfinitesimal"; val stc_inverse = thm "stc_inverse"; val stc_divide = thm "stc_divide"; val stc_idempotent = thm "stc_idempotent"; val CFinite_HFinite_hcomplex_of_hypreal = thm "CFinite_HFinite_hcomplex_of_hypreal"; val SComplex_SReal_hcomplex_of_hypreal = thm "SComplex_SReal_hcomplex_of_hypreal"; val stc_hcomplex_of_hypreal = thm "stc_hcomplex_of_hypreal"; val CInfinitesimal_hcnj_iff = thm "CInfinitesimal_hcnj_iff"; val CInfinite_HInfinite_iff = thm "CInfinite_HInfinite_iff"; val hcomplex_split_CInfinitesimal_iff = thm "hcomplex_split_CInfinitesimal_iff"; val hcomplex_split_CFinite_iff = thm "hcomplex_split_CFinite_iff"; val hcomplex_split_SComplex_iff = thm "hcomplex_split_SComplex_iff"; val hcomplex_split_CInfinite_iff = thm "hcomplex_split_CInfinite_iff"; val hcomplex_split_capprox_iff = thm "hcomplex_split_capprox_iff"; val complex_seq_to_hcomplex_CInfinitesimal = thm "complex_seq_to_hcomplex_CInfinitesimal"; val CInfinitesimal_hcomplex_of_hypreal_epsilon = thm "CInfinitesimal_hcomplex_of_hypreal_epsilon"; val hcomplex_of_complex_approx_zero_iff = thm "hcomplex_of_complex_approx_zero_iff"; val hcomplex_of_complex_approx_zero_iff2 = thm "hcomplex_of_complex_approx_zero_iff2"; *} end
lemma SComplex_add:
[| x ∈ SComplex; y ∈ SComplex |] ==> x + y ∈ SComplex
lemma SComplex_mult:
[| x ∈ SComplex; y ∈ SComplex |] ==> x * y ∈ SComplex
lemma SComplex_inverse:
x ∈ SComplex ==> inverse x ∈ SComplex
lemma SComplex_divide:
[| x ∈ SComplex; y ∈ SComplex |] ==> x / y ∈ SComplex
lemma SComplex_minus:
x ∈ SComplex ==> - x ∈ SComplex
lemma SComplex_minus_iff:
(- x ∈ SComplex) = (x ∈ SComplex)
lemma SComplex_diff:
[| x ∈ SComplex; y ∈ SComplex |] ==> x - y ∈ SComplex
lemma SComplex_add_cancel:
[| x + y ∈ SComplex; y ∈ SComplex |] ==> x ∈ SComplex
lemma SReal_hcmod_hcomplex_of_complex:
hcmod (star_of r) ∈ Reals
lemma SReal_hcmod_number_of:
hcmod (number_of w) ∈ Reals
lemma SReal_hcmod_SComplex:
x ∈ SComplex ==> hcmod x ∈ Reals
lemma SComplex_hcomplex_of_complex:
star_of x ∈ SComplex
lemma SComplex_number_of:
number_of w ∈ SComplex
lemma SComplex_divide_number_of:
r ∈ SComplex ==> r / number_of w ∈ SComplex
lemma SComplex_UNIV_complex:
{x. star_of x ∈ SComplex} = UNIV
lemma SComplex_iff:
(x ∈ SComplex) = (∃y. x = star_of y)
lemma hcomplex_of_complex_image:
range star_of = SComplex
lemma inv_hcomplex_of_complex_image:
inv star_of ` SComplex = UNIV
lemma SComplex_hcomplex_of_complex_image:
[| ∃x. x ∈ P; P ⊆ SComplex |] ==> ∃Q. P = star_of ` Q
lemma SComplex_SReal_dense:
[| x ∈ SComplex; y ∈ SComplex; hcmod x < hcmod y |] ==> ∃r∈Reals. hcmod x < r ∧ r < hcmod y
lemma SComplex_hcmod_SReal:
z ∈ SComplex ==> hcmod z ∈ Reals
lemma SComplex_zero:
0 ∈ SComplex
lemma SComplex_one:
1 ∈ SComplex
lemma CFinite_add:
[| x ∈ CFinite; y ∈ CFinite |] ==> x + y ∈ CFinite
lemma CFinite_mult:
[| x ∈ CFinite; y ∈ CFinite |] ==> x * y ∈ CFinite
lemma CFinite_minus_iff:
(- x ∈ CFinite) = (x ∈ CFinite)
lemma SComplex_subset_CFinite:
SComplex ⊆ CFinite
lemma HFinite_hcmod_hcomplex_of_complex:
hcmod (star_of r) ∈ HFinite
lemma CFinite_hcomplex_of_complex:
star_of x ∈ CFinite
lemma CFiniteD:
x ∈ CFinite ==> ∃t∈Reals. hcmod x < t
lemma CFinite_hcmod_iff:
(x ∈ CFinite) = (hcmod x ∈ HFinite)
lemma CFinite_number_of:
number_of w ∈ CFinite
lemma CFinite_bounded:
[| x ∈ CFinite; y ≤ hcmod x; 0 ≤ y |] ==> y ∈ HFinite
lemma CInfinitesimal_zero:
0 ∈ CInfinitesimal
lemma hcomplex_sum_of_halves:
x / 2 + x / 2 = x
lemma CInfinitesimal_hcmod_iff:
(z ∈ CInfinitesimal) = (hcmod z ∈ Infinitesimal)
lemma one_not_CInfinitesimal:
1 ∉ CInfinitesimal
lemma CInfinitesimal_add:
[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x + y ∈ CInfinitesimal
lemma CInfinitesimal_minus_iff:
(- x ∈ CInfinitesimal) = (x ∈ CInfinitesimal)
lemma CInfinitesimal_diff:
[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x - y ∈ CInfinitesimal
lemma CInfinitesimal_mult:
[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x * y ∈ CInfinitesimal
lemma CInfinitesimal_CFinite_mult:
[| x ∈ CInfinitesimal; y ∈ CFinite |] ==> x * y ∈ CInfinitesimal
lemma CInfinitesimal_CFinite_mult2:
[| x ∈ CInfinitesimal; y ∈ CFinite |] ==> y * x ∈ CInfinitesimal
lemma CInfinite_hcmod_iff:
(z ∈ CInfinite) = (hcmod z ∈ HInfinite)
lemma CInfinite_inverse_CInfinitesimal:
x ∈ CInfinite ==> inverse x ∈ CInfinitesimal
lemma CInfinite_mult:
[| x ∈ CInfinite; y ∈ CInfinite |] ==> x * y ∈ CInfinite
lemma CInfinite_minus_iff:
(- x ∈ CInfinite) = (x ∈ CInfinite)
lemma CFinite_sum_squares:
[| a ∈ CFinite; b ∈ CFinite; c ∈ CFinite |] ==> a * a + b * b + c * c ∈ CFinite
lemma not_CInfinitesimal_not_zero:
x ∉ CInfinitesimal ==> x ≠ 0
lemma not_CInfinitesimal_not_zero2:
x ∈ CFinite - CInfinitesimal ==> x ≠ 0
lemma CFinite_diff_CInfinitesimal_hcmod:
x ∈ CFinite - CInfinitesimal ==> hcmod x ∈ HFinite - Infinitesimal
lemma hcmod_less_CInfinitesimal:
[| e ∈ CInfinitesimal; hcmod x < hcmod e |] ==> x ∈ CInfinitesimal
lemma hcmod_le_CInfinitesimal:
[| e ∈ CInfinitesimal; hcmod x ≤ hcmod e |] ==> x ∈ CInfinitesimal
lemma CInfinitesimal_interval:
[| e ∈ CInfinitesimal; e' ∈ CInfinitesimal; hcmod e' < hcmod x; hcmod x < hcmod e |] ==> x ∈ CInfinitesimal
lemma CInfinitesimal_interval2:
[| e ∈ CInfinitesimal; e' ∈ CInfinitesimal; hcmod e' ≤ hcmod x; hcmod x ≤ hcmod e |] ==> x ∈ CInfinitesimal
lemma not_CInfinitesimal_mult:
[| x ∉ CInfinitesimal; y ∉ CInfinitesimal |] ==> x * y ∉ CInfinitesimal
lemma CInfinitesimal_mult_disj:
x * y ∈ CInfinitesimal ==> x ∈ CInfinitesimal ∨ y ∈ CInfinitesimal
lemma CFinite_CInfinitesimal_diff_mult:
[| x ∈ CFinite - CInfinitesimal; y ∈ CFinite - CInfinitesimal |] ==> x * y ∈ CFinite - CInfinitesimal
lemma CInfinitesimal_subset_CFinite:
CInfinitesimal ⊆ CFinite
lemma CInfinitesimal_hcomplex_of_complex_mult:
x ∈ CInfinitesimal ==> x * star_of r ∈ CInfinitesimal
lemma CInfinitesimal_hcomplex_of_complex_mult2:
x ∈ CInfinitesimal ==> star_of r * x ∈ CInfinitesimal
lemma mem_cinfmal_iff:
(x ∈ CInfinitesimal) = (x @c= 0)
lemma capprox_minus_iff:
(x @c= y) = (x + - y @c= 0)
lemma capprox_minus_iff2:
(x @c= y) = (- y + x @c= 0)
lemma capprox_refl:
x @c= x
lemma capprox_sym:
x @c= y ==> y @c= x
lemma capprox_trans:
[| x @c= y; y @c= z |] ==> x @c= z
lemma capprox_trans2:
[| r @c= x; s @c= x |] ==> r @c= s
lemma capprox_trans3:
[| x @c= r; x @c= s |] ==> r @c= s
lemma number_of_capprox_reorient:
(number_of w @c= x) = (x @c= number_of w)
lemma CInfinitesimal_capprox_minus:
(x - y ∈ CInfinitesimal) = (x @c= y)
lemma capprox_monad_iff:
(x @c= y) = (cmonad x = cmonad y)
lemma Infinitesimal_capprox:
[| x ∈ CInfinitesimal; y ∈ CInfinitesimal |] ==> x @c= y
lemma capprox_add:
[| a @c= b; c @c= d |] ==> a + c @c= b + d
lemma capprox_minus:
a @c= b ==> - a @c= - b
lemma capprox_minus2:
- a @c= - b ==> a @c= b
lemma capprox_minus_cancel:
(- a @c= - b) = (a @c= b)
lemma capprox_add_minus:
[| a @c= b; c @c= d |] ==> a + - c @c= b + - d
lemma capprox_mult1:
[| a @c= b; c ∈ CFinite |] ==> a * c @c= b * c
lemma capprox_mult2:
[| a @c= b; c ∈ CFinite |] ==> c * a @c= c * b
lemma capprox_mult_subst:
[| u @c= v * x; x @c= y; v ∈ CFinite |] ==> u @c= v * y
lemma capprox_mult_subst2:
[| u @c= x * v; x @c= y; v ∈ CFinite |] ==> u @c= y * v
lemma capprox_mult_subst_SComplex:
[| u @c= x * star_of v; x @c= y |] ==> u @c= y * star_of v
lemma capprox_eq_imp:
a = b ==> a @c= b
lemma CInfinitesimal_minus_capprox:
x ∈ CInfinitesimal ==> - x @c= x
lemma bex_CInfinitesimal_iff:
(∃y∈CInfinitesimal. x - z = y) = (x @c= z)
lemma bex_CInfinitesimal_iff2:
(∃y∈CInfinitesimal. x = z + y) = (x @c= z)
lemma CInfinitesimal_add_capprox:
[| y ∈ CInfinitesimal; x + y = z |] ==> x @c= z
lemma CInfinitesimal_add_capprox_self:
y ∈ CInfinitesimal ==> x @c= x + y
lemma CInfinitesimal_add_capprox_self2:
y ∈ CInfinitesimal ==> x @c= y + x
lemma CInfinitesimal_add_minus_capprox_self:
y ∈ CInfinitesimal ==> x @c= x + - y
lemma CInfinitesimal_add_cancel:
[| y ∈ CInfinitesimal; x + y @c= z |] ==> x @c= z
lemma CInfinitesimal_add_right_cancel:
[| y ∈ CInfinitesimal; x @c= z + y |] ==> x @c= z
lemma capprox_add_left_cancel:
d + b @c= d + c ==> b @c= c
lemma capprox_add_right_cancel:
b + d @c= c + d ==> b @c= c
lemma capprox_add_mono1:
b @c= c ==> d + b @c= d + c
lemma capprox_add_mono2:
b @c= c ==> b + a @c= c + a
lemma capprox_add_left_iff:
(a + b @c= a + c) = (b @c= c)
lemma capprox_add_right_iff:
(b + a @c= c + a) = (b @c= c)
lemma capprox_CFinite:
[| x ∈ CFinite; x @c= y |] ==> y ∈ CFinite
lemma capprox_hcomplex_of_complex_CFinite:
x @c= star_of D ==> x ∈ CFinite
lemma capprox_mult_CFinite:
[| a @c= b; c @c= d; b ∈ CFinite; d ∈ CFinite |] ==> a * c @c= b * d
lemma capprox_mult_hcomplex_of_complex:
[| a @c= star_of b; c @c= star_of d |] ==> a * c @c= star_of b * star_of d
lemma capprox_SComplex_mult_cancel_zero:
[| a ∈ SComplex; a ≠ 0; a * x @c= 0 |] ==> x @c= 0
lemma capprox_mult_SComplex1:
[| a ∈ SComplex; x @c= 0 |] ==> x * a @c= 0
lemma capprox_mult_SComplex2:
[| a ∈ SComplex; x @c= 0 |] ==> a * x @c= 0
lemma capprox_mult_SComplex_zero_cancel_iff:
[| a ∈ SComplex; a ≠ 0 |] ==> (a * x @c= 0) = (x @c= 0)
lemma capprox_SComplex_mult_cancel:
[| a ∈ SComplex; a ≠ 0; a * w @c= a * z |] ==> w @c= z
lemma capprox_SComplex_mult_cancel_iff1:
[| a ∈ SComplex; a ≠ 0 |] ==> (a * w @c= a * z) = (w @c= z)
lemma capprox_hcmod_approx_zero:
(x @c= y) = (hcmod (y - x) ≈ 0)
lemma capprox_approx_zero_iff:
(x @c= 0) = (hcmod x ≈ 0)
lemma capprox_minus_zero_cancel_iff:
(- x @c= 0) = (x @c= 0)
lemma Infinitesimal_hcmod_add_diff:
u @c= 0 ==> hcmod (x + u) - hcmod x ∈ Infinitesimal
lemma approx_hcmod_add_hcmod:
u @c= 0 ==> hcmod (x + u) ≈ hcmod x
lemma capprox_hcmod_approx:
x @c= y ==> hcmod x ≈ hcmod y
lemma CInfinitesimal_less_SComplex:
[| x ∈ SComplex; y ∈ CInfinitesimal; 0 < hcmod x |] ==> hcmod y < hcmod x
lemma SComplex_Int_CInfinitesimal_zero:
SComplex ∩ CInfinitesimal = {0}
lemma SComplex_CInfinitesimal_zero:
[| x ∈ SComplex; x ∈ CInfinitesimal |] ==> x = 0
lemma SComplex_CFinite_diff_CInfinitesimal:
[| x ∈ SComplex; x ≠ 0 |] ==> x ∈ CFinite - CInfinitesimal
lemma hcomplex_of_complex_CFinite_diff_CInfinitesimal:
star_of x ≠ 0 ==> star_of x ∈ CFinite - CInfinitesimal
lemma hcomplex_of_complex_CInfinitesimal_iff_0:
(star_of x ∈ CInfinitesimal) = (x = 0)
lemma number_of_not_CInfinitesimal:
number_of w ≠ 0 ==> number_of w ∉ CInfinitesimal
lemma capprox_SComplex_not_zero:
[| y ∈ SComplex; x @c= y; y ≠ 0 |] ==> x ≠ 0
lemma CFinite_diff_CInfinitesimal_capprox:
[| x @c= y; y ∈ CFinite - CInfinitesimal |] ==> x ∈ CFinite - CInfinitesimal
lemma CInfinitesimal_ratio:
[| y ≠ 0; y ∈ CInfinitesimal; x / y ∈ CFinite |] ==> x ∈ CInfinitesimal
lemma SComplex_capprox_iff:
[| x ∈ SComplex; y ∈ SComplex |] ==> (x @c= y) = (x = y)
lemma number_of_capprox_iff:
(number_of v @c= number_of w) = (number_of v = number_of w)
lemma number_of_CInfinitesimal_iff:
(number_of w ∈ CInfinitesimal) = (number_of w = 0)
lemma hcomplex_of_complex_approx_iff:
(star_of k @c= star_of m) = (k = m)
lemma hcomplex_of_complex_capprox_number_of_iff:
(star_of k @c= number_of w) = (k = number_of w)
lemma capprox_unique_complex:
[| r ∈ SComplex; s ∈ SComplex; r @c= x; s @c= x |] ==> r = s
lemma hcomplex_capproxD1:
star_n X @c= star_n Y ==> star_n (%n. Re (X n)) ≈ star_n (%n. Re (Y n))
lemma hcomplex_capproxD2:
star_n X @c= star_n Y ==> star_n (%n. Im (X n)) ≈ star_n (%n. Im (Y n))
lemma hcomplex_capproxI:
[| star_n (%n. Re (X n)) ≈ star_n (%n. Re (Y n)); star_n (%n. Im (X n)) ≈ star_n (%n. Im (Y n)) |] ==> star_n X @c= star_n Y
lemma capprox_approx_iff:
(star_n X @c= star_n Y) = (star_n (%n. Re (X n)) ≈ star_n (%n. Re (Y n)) ∧ star_n (%n. Im (X n)) ≈ star_n (%n. Im (Y n)))
lemma hcomplex_of_hypreal_capprox_iff:
(hcomplex_of_hypreal x @c= hcomplex_of_hypreal z) = (x ≈ z)
lemma CFinite_HFinite_Re:
star_n X ∈ CFinite ==> star_n (%n. Re (X n)) ∈ HFinite
lemma CFinite_HFinite_Im:
star_n X ∈ CFinite ==> star_n (%n. Im (X n)) ∈ HFinite
lemma HFinite_Re_Im_CFinite:
[| star_n (%n. Re (X n)) ∈ HFinite; star_n (%n. Im (X n)) ∈ HFinite |] ==> star_n X ∈ CFinite
lemma CFinite_HFinite_iff:
(star_n X ∈ CFinite) = (star_n (%n. Re (X n)) ∈ HFinite ∧ star_n (%n. Im (X n)) ∈ HFinite)
lemma SComplex_Re_SReal:
star_n X ∈ SComplex ==> star_n (%n. Re (X n)) ∈ Reals
lemma SComplex_Im_SReal:
star_n X ∈ SComplex ==> star_n (%n. Im (X n)) ∈ Reals
lemma Reals_Re_Im_SComplex:
[| star_n (%n. Re (X n)) ∈ Reals; star_n (%n. Im (X n)) ∈ Reals |] ==> star_n X ∈ SComplex
lemma SComplex_SReal_iff:
(star_n X ∈ SComplex) = (star_n (%n. Re (X n)) ∈ Reals ∧ star_n (%n. Im (X n)) ∈ Reals)
lemma CInfinitesimal_Infinitesimal_iff:
(star_n X ∈ CInfinitesimal) = (star_n (%n. Re (X n)) ∈ Infinitesimal ∧ star_n (%n. Im (X n)) ∈ Infinitesimal)
lemma eq_Abs_star_EX:
(∃t. P t) = (∃X. P (star_n X))
lemma eq_Abs_star_Bex:
(∃t∈A. P t) = (∃X. star_n X ∈ A ∧ P (star_n X))
lemma stc_part_Ex:
x ∈ CFinite ==> ∃t∈SComplex. x @c= t
lemma stc_part_Ex1:
x ∈ CFinite ==> ∃!t. t ∈ SComplex ∧ x @c= t
lemma CFinite_Int_CInfinite_empty:
CFinite ∩ CInfinite = {}
lemma CFinite_not_CInfinite:
x ∈ CFinite ==> x ∉ CInfinite
lemma not_CFinite_CInfinite:
x ∉ CFinite ==> x ∈ CInfinite
lemma CInfinite_CFinite_disj:
x ∈ CInfinite ∨ x ∈ CFinite
lemma CInfinite_CFinite_iff:
(x ∈ CInfinite) = (x ∉ CFinite)
lemma CFinite_CInfinite_iff:
(x ∈ CFinite) = (x ∉ CInfinite)
lemma CInfinite_diff_CFinite_CInfinitesimal_disj:
x ∉ CInfinitesimal ==> x ∈ CInfinite ∨ x ∈ CFinite - CInfinitesimal
lemma CFinite_inverse:
[| x ∈ CFinite; x ∉ CInfinitesimal |] ==> inverse x ∈ CFinite
lemma CFinite_inverse2:
x ∈ CFinite - CInfinitesimal ==> inverse x ∈ CFinite
lemma CInfinitesimal_inverse_CFinite:
x ∉ CInfinitesimal ==> inverse x ∈ CFinite
lemma CFinite_not_CInfinitesimal_inverse:
x ∈ CFinite - CInfinitesimal ==> inverse x ∈ CFinite - CInfinitesimal
lemma capprox_inverse:
[| x @c= y; y ∈ CFinite - CInfinitesimal |] ==> inverse x @c= inverse y
lemmas hcomplex_of_complex_capprox_inverse:
[| x @c= star_of x1; star_of x1 ≠ 0 |] ==> inverse x @c= inverse (star_of x1)
lemmas hcomplex_of_complex_capprox_inverse:
[| x @c= star_of x1; star_of x1 ≠ 0 |] ==> inverse x @c= inverse (star_of x1)
lemma inverse_add_CInfinitesimal_capprox:
[| x ∈ CFinite - CInfinitesimal; h ∈ CInfinitesimal |] ==> inverse (x + h) @c= inverse x
lemma inverse_add_CInfinitesimal_capprox2:
[| x ∈ CFinite - CInfinitesimal; h ∈ CInfinitesimal |] ==> inverse (h + x) @c= inverse x
lemma inverse_add_CInfinitesimal_approx_CInfinitesimal:
[| x ∈ CFinite - CInfinitesimal; h ∈ CInfinitesimal |] ==> inverse (x + h) - inverse x @c= h
lemma CInfinitesimal_square_iff:
(x * x ∈ CInfinitesimal) = (x ∈ CInfinitesimal)
lemma capprox_CFinite_mult_cancel:
[| a ∈ CFinite - CInfinitesimal; a * w @c= a * z |] ==> w @c= z
lemma capprox_CFinite_mult_cancel_iff1:
a ∈ CFinite - CInfinitesimal ==> (a * w @c= a * z) = (w @c= z)
lemma capprox_cmonad_iff:
(x @c= y) = (cmonad x = cmonad y)
lemma CInfinitesimal_cmonad_eq:
e ∈ CInfinitesimal ==> cmonad (x + e) = cmonad x
lemma mem_cmonad_iff:
(u ∈ cmonad x) = (- u ∈ cmonad (- x))
lemma CInfinitesimal_cmonad_zero_iff:
(x ∈ CInfinitesimal) = (x ∈ cmonad 0)
lemma cmonad_zero_minus_iff:
(x ∈ cmonad 0) = (- x ∈ cmonad 0)
lemma cmonad_zero_hcmod_iff:
(x ∈ cmonad 0) = (hcmod x ∈ monad 0)
lemma mem_cmonad_self:
x ∈ cmonad x
lemma stc_capprox_self:
x ∈ CFinite ==> stc x @c= x
lemma stc_SComplex:
x ∈ CFinite ==> stc x ∈ SComplex
lemma stc_CFinite:
x ∈ CFinite ==> stc x ∈ CFinite
lemma stc_SComplex_eq:
x ∈ SComplex ==> stc x = x
lemma stc_hcomplex_of_complex:
stc (star_of x) = star_of x
lemma stc_eq_capprox:
[| x ∈ CFinite; y ∈ CFinite; stc x = stc y |] ==> x @c= y
lemma capprox_stc_eq:
[| x ∈ CFinite; y ∈ CFinite; x @c= y |] ==> stc x = stc y
lemma stc_eq_capprox_iff:
[| x ∈ CFinite; y ∈ CFinite |] ==> (x @c= y) = (stc x = stc y)
lemma stc_CInfinitesimal_add_SComplex:
[| x ∈ SComplex; e ∈ CInfinitesimal |] ==> stc (x + e) = x
lemma stc_CInfinitesimal_add_SComplex2:
[| x ∈ SComplex; e ∈ CInfinitesimal |] ==> stc (e + x) = x
lemma CFinite_stc_CInfinitesimal_add:
x ∈ CFinite ==> ∃e∈CInfinitesimal. x = stc x + e
lemma stc_add:
[| x ∈ CFinite; y ∈ CFinite |] ==> stc (x + y) = stc x + stc y
lemma stc_number_of:
stc (number_of w) = number_of w
lemma stc_zero:
stc 0 = 0
lemma stc_one:
stc 1 = 1
lemma stc_minus:
y ∈ CFinite ==> stc (- y) = - stc y
lemma stc_diff:
[| x ∈ CFinite; y ∈ CFinite |] ==> stc (x - y) = stc x - stc y
lemma lemma_stc_mult:
[| x ∈ CFinite; y ∈ CFinite; e ∈ CInfinitesimal; ea ∈ CInfinitesimal |] ==> e * y + x * ea + e * ea ∈ CInfinitesimal
lemma stc_mult:
[| x ∈ CFinite; y ∈ CFinite |] ==> stc (x * y) = stc x * stc y
lemma stc_CInfinitesimal:
x ∈ CInfinitesimal ==> stc x = 0
lemma stc_not_CInfinitesimal:
stc x ≠ 0 ==> x ∉ CInfinitesimal
lemma stc_inverse:
[| x ∈ CFinite; stc x ≠ 0 |] ==> stc (inverse x) = inverse (stc x)
lemma stc_divide:
[| x ∈ CFinite; y ∈ CFinite; stc y ≠ 0 |] ==> stc (x / y) = stc x / stc y
lemma stc_idempotent:
x ∈ CFinite ==> stc (stc x) = stc x
lemma CFinite_HFinite_hcomplex_of_hypreal:
z ∈ HFinite ==> hcomplex_of_hypreal z ∈ CFinite
lemma SComplex_SReal_hcomplex_of_hypreal:
x ∈ Reals ==> hcomplex_of_hypreal x ∈ SComplex
lemma stc_hcomplex_of_hypreal:
z ∈ HFinite ==> stc (hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)
lemma CInfinitesimal_hcnj_iff:
(hcnj z ∈ CInfinitesimal) = (z ∈ CInfinitesimal)
lemma CInfinite_HInfinite_iff:
(star_n X ∈ CInfinite) = (star_n (%n. Re (X n)) ∈ HInfinite ∨ star_n (%n. Im (X n)) ∈ HInfinite)
lemma hcomplex_split_CInfinitesimal_iff:
(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ CInfinitesimal) = (x ∈ Infinitesimal ∧ y ∈ Infinitesimal)
lemma hcomplex_split_CFinite_iff:
(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ CFinite) = (x ∈ HFinite ∧ y ∈ HFinite)
lemma hcomplex_split_SComplex_iff:
(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ SComplex) = (x ∈ Reals ∧ y ∈ Reals)
lemma hcomplex_split_CInfinite_iff:
(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y ∈ CInfinite) = (x ∈ HInfinite ∨ y ∈ HInfinite)
lemma hcomplex_split_capprox_iff:
(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y @c= hcomplex_of_hypreal x' + iii * hcomplex_of_hypreal y') = (x ≈ x' ∧ y ≈ y')
lemma complex_seq_to_hcomplex_CInfinitesimal:
∀n. cmod (X n - x) < inverse (real (Suc n)) ==> star_n X - star_of x ∈ CInfinitesimal
lemma CInfinitesimal_hcomplex_of_hypreal_epsilon:
hcomplex_of_hypreal ε ∈ CInfinitesimal
lemma hcomplex_of_complex_approx_zero_iff:
(star_of z @c= 0) = (z = 0)
lemma hcomplex_of_complex_approx_zero_iff2:
(0 @c= star_of z) = (z = 0)