(* Title : CLim.thy
Author : Jacques D. Fleuriot
Copyright : 2001 University of Edinburgh
Description : A first theory of limits, continuity and
differentiation for complex functions
*)
theory CLim = CSeries:
(*not in simpset?*)
declare hypreal_epsilon_not_zero [simp]
(*??generalize*)
lemma lemma_complex_mult_inverse_squared [simp]:
"x \<noteq> (0::complex) \<Longrightarrow> (x * inverse(x) ^ 2) = inverse x"
by (auto simp add: numeral_2_eq_2)
text{*Changing the quantified variable. Install earlier?*}
lemma all_shift: "(\<forall>x::'a::ring. P x) = (\<forall>x. P (x-a))";
apply auto
apply (drule_tac x="x+a" in spec)
apply (simp add: diff_minus add_assoc)
done
lemma complex_add_minus_iff [simp]: "(x + - a = (0::complex)) = (x=a)"
by (simp add: diff_eq_eq diff_minus [symmetric])
lemma complex_add_eq_0_iff [iff]: "(x+y = (0::complex)) = (y = -x)"
apply auto
apply (drule sym [THEN diff_eq_eq [THEN iffD2]], auto)
done
constdefs
CLIM :: "[complex=>complex,complex,complex] => bool"
("((_)/ -- (_)/ --C> (_))" [60, 0, 60] 60)
"f -- a --C> L ==
\<forall>r. 0 < r -->
(\<exists>s. 0 < s & (\<forall>x. (x \<noteq> a & (cmod(x - a) < s)
--> cmod(f x - L) < r)))"
NSCLIM :: "[complex=>complex,complex,complex] => bool"
("((_)/ -- (_)/ --NSC> (_))" [60, 0, 60] 60)
"f -- a --NSC> L == (\<forall>x. (x \<noteq> hcomplex_of_complex a &
x @c= hcomplex_of_complex a
--> ( *fc* f) x @c= hcomplex_of_complex L))"
(* f: C --> R *)
CRLIM :: "[complex=>real,complex,real] => bool"
("((_)/ -- (_)/ --CR> (_))" [60, 0, 60] 60)
"f -- a --CR> L ==
\<forall>r. 0 < r -->
(\<exists>s. 0 < s & (\<forall>x. (x \<noteq> a & (cmod(x - a) < s)
--> abs(f x - L) < r)))"
NSCRLIM :: "[complex=>real,complex,real] => bool"
("((_)/ -- (_)/ --NSCR> (_))" [60, 0, 60] 60)
"f -- a --NSCR> L == (\<forall>x. (x \<noteq> hcomplex_of_complex a &
x @c= hcomplex_of_complex a
--> ( *fcR* f) x @= hypreal_of_real L))"
isContc :: "[complex=>complex,complex] => bool"
"isContc f a == (f -- a --C> (f a))"
(* NS definition dispenses with limit notions *)
isNSContc :: "[complex=>complex,complex] => bool"
"isNSContc f a == (\<forall>y. y @c= hcomplex_of_complex a -->
( *fc* f) y @c= hcomplex_of_complex (f a))"
isContCR :: "[complex=>real,complex] => bool"
"isContCR f a == (f -- a --CR> (f a))"
(* NS definition dispenses with limit notions *)
isNSContCR :: "[complex=>real,complex] => bool"
"isNSContCR f a == (\<forall>y. y @c= hcomplex_of_complex a -->
( *fcR* f) y @= hypreal_of_real (f a))"
(* differentiation: D is derivative of function f at x *)
cderiv:: "[complex=>complex,complex,complex] => bool"
("(CDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
"CDERIV f x :> D == ((%h. (f(x + h) - f(x))/h) -- 0 --C> D)"
nscderiv :: "[complex=>complex,complex,complex] => bool"
("(NSCDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
"NSCDERIV f x :> D == (\<forall>h \<in> CInfinitesimal - {0}.
(( *fc* f)(hcomplex_of_complex x + h)
- hcomplex_of_complex (f x))/h @c= hcomplex_of_complex D)"
cdifferentiable :: "[complex=>complex,complex] => bool"
(infixl "cdifferentiable" 60)
"f cdifferentiable x == (\<exists>D. CDERIV f x :> D)"
NSCdifferentiable :: "[complex=>complex,complex] => bool"
(infixl "NSCdifferentiable" 60)
"f NSCdifferentiable x == (\<exists>D. NSCDERIV f x :> D)"
isUContc :: "(complex=>complex) => bool"
"isUContc f == (\<forall>r. 0 < r -->
(\<exists>s. 0 < s & (\<forall>x y. cmod(x - y) < s
--> cmod(f x - f y) < r)))"
isNSUContc :: "(complex=>complex) => bool"
"isNSUContc f == (\<forall>x y. x @c= y --> ( *fc* f) x @c= ( *fc* f) y)"
subsection{*Limit of Complex to Complex Function*}
lemma NSCLIM_NSCRLIM_Re: "f -- a --NSC> L ==> (%x. Re(f x)) -- a --NSCR> Re(L)"
apply (unfold NSCLIM_def NSCRLIM_def)
apply (rule eq_Abs_hcomplex [of x])
apply (auto simp add: starfunC_approx_Re_Im_iff hRe_hcomplex_of_complex)
done
lemma NSCLIM_NSCRLIM_Im:
"f -- a --NSC> L ==> (%x. Im(f x)) -- a --NSCR> Im(L)"
apply (unfold NSCLIM_def NSCRLIM_def)
apply (rule eq_Abs_hcomplex [of x])
apply (auto simp add: starfunC_approx_Re_Im_iff hIm_hcomplex_of_complex)
done
lemma CLIM_NSCLIM:
"f -- x --C> L ==> f -- x --NSC> L"
apply (unfold CLIM_def NSCLIM_def capprox_def, auto)
apply (rule_tac z = xa in eq_Abs_hcomplex)
apply (auto simp add: hcomplex_of_complex_def starfunC hcomplex_diff
CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff)
apply (rule bexI, rule_tac [2] lemma_hyprel_refl, safe)
apply (drule_tac x = u in spec, auto)
apply (drule_tac x = s in spec, auto, ultra)
apply (drule sym, auto)
done
lemma eq_Abs_hcomplex_ALL:
"(\<forall>t. P t) = (\<forall>X. P (Abs_hcomplex(hcomplexrel `` {X})))"
apply auto
apply (rule_tac z = t in eq_Abs_hcomplex, auto)
done
lemma lemma_CLIM:
"\<forall>s. 0 < s --> (\<exists>xa. xa \<noteq> x &
cmod (xa - x) < s & r \<le> cmod (f xa - L))
==> \<forall>(n::nat). \<exists>xa. xa \<noteq> x &
cmod(xa - x) < inverse(real(Suc n)) & r \<le> cmod(f xa - L)"
apply clarify
apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
done
lemma lemma_skolemize_CLIM2:
"\<forall>s. 0 < s --> (\<exists>xa. xa \<noteq> x &
cmod (xa - x) < s & r \<le> cmod (f xa - L))
==> \<exists>X. \<forall>(n::nat). X n \<noteq> x &
cmod(X n - x) < inverse(real(Suc n)) & r \<le> cmod(f (X n) - L)"
apply (drule lemma_CLIM)
apply (drule choice, blast)
done
lemma lemma_csimp:
"\<forall>n. X n \<noteq> x &
cmod (X n - x) < inverse (real(Suc n)) &
r \<le> cmod (f (X n) - L) ==>
\<forall>n. cmod (X n - x) < inverse (real(Suc n))"
by auto
lemma NSCLIM_CLIM:
"f -- x --NSC> L ==> f -- x --C> L"
apply (unfold CLIM_def NSCLIM_def)
apply (rule ccontr)
apply (auto simp add: eq_Abs_hcomplex_ALL starfunC CInfinitesimal_capprox_minus [symmetric] hcomplex_diff CInfinitesimal_hcmod_iff hcomplex_of_complex_def Infinitesimal_FreeUltrafilterNat_iff hcmod)
apply (simp add: linorder_not_less)
apply (drule lemma_skolemize_CLIM2, safe)
apply (drule_tac x = X in spec, auto)
apply (drule lemma_csimp [THEN complex_seq_to_hcomplex_CInfinitesimal])
apply (simp add: CInfinitesimal_hcmod_iff hcomplex_of_complex_def Infinitesimal_FreeUltrafilterNat_iff hcomplex_diff hcmod, blast)
apply (drule_tac x = r in spec, clarify)
apply (drule FreeUltrafilterNat_all, ultra, arith)
done
text{*First key result*}
theorem CLIM_NSCLIM_iff: "(f -- x --C> L) = (f -- x --NSC> L)"
by (blast intro: CLIM_NSCLIM NSCLIM_CLIM)
subsection{*Limit of Complex to Real Function*}
lemma CRLIM_NSCRLIM: "f -- x --CR> L ==> f -- x --NSCR> L"
apply (unfold CRLIM_def NSCRLIM_def capprox_def, auto)
apply (rule_tac z = xa in eq_Abs_hcomplex)
apply (auto simp add: hcomplex_of_complex_def starfunCR hcomplex_diff CInfinitesimal_hcmod_iff hcmod hypreal_diff Infinitesimal_FreeUltrafilterNat_iff Infinitesimal_approx_minus [symmetric] hypreal_of_real_def)
apply (rule bexI, rule_tac [2] lemma_hyprel_refl, safe)
apply (drule_tac x = u in spec, auto)
apply (drule_tac x = s in spec, auto, ultra)
apply (drule sym, auto)
done
lemma lemma_CRLIM:
"\<forall>s. 0 < s --> (\<exists>xa. xa \<noteq> x &
cmod (xa - x) < s & r \<le> abs (f xa - L))
==> \<forall>(n::nat). \<exists>xa. xa \<noteq> x &
cmod(xa - x) < inverse(real(Suc n)) & r \<le> abs (f xa - L)"
apply clarify
apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
done
lemma lemma_skolemize_CRLIM2:
"\<forall>s. 0 < s --> (\<exists>xa. xa \<noteq> x &
cmod (xa - x) < s & r \<le> abs (f xa - L))
==> \<exists>X. \<forall>(n::nat). X n \<noteq> x &
cmod(X n - x) < inverse(real(Suc n)) & r \<le> abs (f (X n) - L)"
apply (drule lemma_CRLIM)
apply (drule choice, blast)
done
lemma lemma_crsimp:
"\<forall>n. X n \<noteq> x &
cmod (X n - x) < inverse (real(Suc n)) &
r \<le> abs (f (X n) - L) ==>
\<forall>n. cmod (X n - x) < inverse (real(Suc n))"
by auto
lemma NSCRLIM_CRLIM: "f -- x --NSCR> L ==> f -- x --CR> L"
apply (unfold CRLIM_def NSCRLIM_def capprox_def)
apply (rule ccontr)
apply (auto simp add: eq_Abs_hcomplex_ALL starfunCR hcomplex_diff
hcomplex_of_complex_def hypreal_diff CInfinitesimal_hcmod_iff
hcmod Infinitesimal_approx_minus [symmetric]
Infinitesimal_FreeUltrafilterNat_iff)
apply (simp add: linorder_not_less)
apply (drule lemma_skolemize_CRLIM2, safe)
apply (drule_tac x = X in spec, auto)
apply (drule lemma_crsimp [THEN complex_seq_to_hcomplex_CInfinitesimal])
apply (simp add: CInfinitesimal_hcmod_iff hcomplex_of_complex_def
Infinitesimal_FreeUltrafilterNat_iff hcomplex_diff hcmod)
apply (auto simp add: hypreal_of_real_def hypreal_diff)
apply (drule_tac x = r in spec, clarify)
apply (drule FreeUltrafilterNat_all, ultra)
done
text{*Second key result*}
theorem CRLIM_NSCRLIM_iff: "(f -- x --CR> L) = (f -- x --NSCR> L)"
by (blast intro: CRLIM_NSCRLIM NSCRLIM_CRLIM)
(** get this result easily now **)
lemma CLIM_CRLIM_Re: "f -- a --C> L ==> (%x. Re(f x)) -- a --CR> Re(L)"
by (auto dest: NSCLIM_NSCRLIM_Re simp add: CLIM_NSCLIM_iff CRLIM_NSCRLIM_iff [symmetric])
lemma CLIM_CRLIM_Im: "f -- a --C> L ==> (%x. Im(f x)) -- a --CR> Im(L)"
by (auto dest: NSCLIM_NSCRLIM_Im simp add: CLIM_NSCLIM_iff CRLIM_NSCRLIM_iff [symmetric])
lemma CLIM_cnj: "f -- a --C> L ==> (%x. cnj (f x)) -- a --C> cnj L"
by (auto simp add: CLIM_def complex_cnj_diff [symmetric])
lemma CLIM_cnj_iff: "((%x. cnj (f x)) -- a --C> cnj L) = (f -- a --C> L)"
by (auto simp add: CLIM_def complex_cnj_diff [symmetric])
(*** NSLIM_add hence CLIM_add *)
lemma NSCLIM_add:
"[| f -- x --NSC> l; g -- x --NSC> m |]
==> (%x. f(x) + g(x)) -- x --NSC> (l + m)"
by (auto simp: NSCLIM_def intro!: capprox_add)
lemma CLIM_add:
"[| f -- x --C> l; g -- x --C> m |]
==> (%x. f(x) + g(x)) -- x --C> (l + m)"
by (simp add: CLIM_NSCLIM_iff NSCLIM_add)
(*** NSLIM_mult hence CLIM_mult *)
lemma NSCLIM_mult:
"[| f -- x --NSC> l; g -- x --NSC> m |]
==> (%x. f(x) * g(x)) -- x --NSC> (l * m)"
by (auto simp add: NSCLIM_def intro!: capprox_mult_CFinite)
lemma CLIM_mult:
"[| f -- x --C> l; g -- x --C> m |]
==> (%x. f(x) * g(x)) -- x --C> (l * m)"
by (simp add: CLIM_NSCLIM_iff NSCLIM_mult)
(*** NSCLIM_const and CLIM_const ***)
lemma NSCLIM_const [simp]: "(%x. k) -- x --NSC> k"
by (simp add: NSCLIM_def)
lemma CLIM_const [simp]: "(%x. k) -- x --C> k"
by (simp add: CLIM_def)
(*** NSCLIM_minus and CLIM_minus ***)
lemma NSCLIM_minus: "f -- a --NSC> L ==> (%x. -f(x)) -- a --NSC> -L"
by (simp add: NSCLIM_def)
lemma CLIM_minus: "f -- a --C> L ==> (%x. -f(x)) -- a --C> -L"
by (simp add: CLIM_NSCLIM_iff NSCLIM_minus)
(*** NSCLIM_diff hence CLIM_diff ***)
lemma NSCLIM_diff:
"[| f -- x --NSC> l; g -- x --NSC> m |]
==> (%x. f(x) - g(x)) -- x --NSC> (l - m)"
by (simp add: complex_diff_def NSCLIM_add NSCLIM_minus)
lemma CLIM_diff:
"[| f -- x --C> l; g -- x --C> m |]
==> (%x. f(x) - g(x)) -- x --C> (l - m)"
by (simp add: CLIM_NSCLIM_iff NSCLIM_diff)
(*** NSCLIM_inverse and hence CLIM_inverse *)
lemma NSCLIM_inverse:
"[| f -- a --NSC> L; L \<noteq> 0 |]
==> (%x. inverse(f(x))) -- a --NSC> (inverse L)"
apply (unfold NSCLIM_def, clarify)
apply (drule spec)
apply (auto simp add: hcomplex_of_complex_capprox_inverse)
done
lemma CLIM_inverse:
"[| f -- a --C> L; L \<noteq> 0 |]
==> (%x. inverse(f(x))) -- a --C> (inverse L)"
by (simp add: CLIM_NSCLIM_iff NSCLIM_inverse)
(*** NSCLIM_zero, CLIM_zero, etc. ***)
lemma NSCLIM_zero: "f -- a --NSC> l ==> (%x. f(x) - l) -- a --NSC> 0"
apply (rule_tac a1 = l in right_minus [THEN subst])
apply (unfold complex_diff_def)
apply (rule NSCLIM_add, auto)
done
lemma CLIM_zero: "f -- a --C> l ==> (%x. f(x) - l) -- a --C> 0"
by (simp add: CLIM_NSCLIM_iff NSCLIM_zero)
lemma NSCLIM_zero_cancel: "(%x. f(x) - l) -- x --NSC> 0 ==> f -- x --NSC> l"
by (drule_tac g = "%x. l" and m = l in NSCLIM_add, auto)
lemma CLIM_zero_cancel: "(%x. f(x) - l) -- x --C> 0 ==> f -- x --C> l"
by (drule_tac g = "%x. l" and m = l in CLIM_add, auto)
(*** NSCLIM_not zero and hence CLIM_not_zero ***)
lemma NSCLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --NSC> 0)"
apply (auto simp del: hcomplex_of_complex_zero simp add: NSCLIM_def)
apply (rule_tac x = "hcomplex_of_complex x + hcomplex_of_hypreal epsilon" in exI)
apply (auto intro: CInfinitesimal_add_capprox_self [THEN capprox_sym]
simp del: hcomplex_of_complex_zero)
done
(* [| k \<noteq> 0; (%x. k) -- x --NSC> 0 |] ==> R *)
lemmas NSCLIM_not_zeroE = NSCLIM_not_zero [THEN notE, standard]
lemma CLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --C> 0)"
by (simp add: CLIM_NSCLIM_iff NSCLIM_not_zero)
(*** NSCLIM_const hence CLIM_const ***)
lemma NSCLIM_const_eq: "(%x. k) -- x --NSC> L ==> k = L"
apply (rule ccontr)
apply (drule NSCLIM_zero)
apply (rule NSCLIM_not_zeroE [of "k-L"], auto)
done
lemma CLIM_const_eq: "(%x. k) -- x --C> L ==> k = L"
by (simp add: CLIM_NSCLIM_iff NSCLIM_const_eq)
(*** NSCLIM and hence CLIM are unique ***)
lemma NSCLIM_unique: "[| f -- x --NSC> L; f -- x --NSC> M |] ==> L = M"
apply (drule NSCLIM_minus)
apply (drule NSCLIM_add, assumption)
apply (auto dest!: NSCLIM_const_eq [symmetric])
done
lemma CLIM_unique: "[| f -- x --C> L; f -- x --C> M |] ==> L = M"
by (simp add: CLIM_NSCLIM_iff NSCLIM_unique)
(*** NSCLIM_mult_zero and CLIM_mult_zero ***)
lemma NSCLIM_mult_zero:
"[| f -- x --NSC> 0; g -- x --NSC> 0 |] ==> (%x. f(x)*g(x)) -- x --NSC> 0"
by (drule NSCLIM_mult, auto)
lemma CLIM_mult_zero:
"[| f -- x --C> 0; g -- x --C> 0 |] ==> (%x. f(x)*g(x)) -- x --C> 0"
by (drule CLIM_mult, auto)
(*** NSCLIM_self hence CLIM_self ***)
lemma NSCLIM_self: "(%x. x) -- a --NSC> a"
by (auto simp add: NSCLIM_def intro: starfunC_Idfun_capprox)
lemma CLIM_self: "(%x. x) -- a --C> a"
by (simp add: CLIM_NSCLIM_iff NSCLIM_self)
(** another equivalence result **)
lemma NSCLIM_NSCRLIM_iff:
"(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)"
apply (auto simp add: NSCLIM_def NSCRLIM_def CInfinitesimal_capprox_minus [symmetric] CInfinitesimal_hcmod_iff)
apply (auto dest!: spec)
apply (rule_tac [!] z = xa in eq_Abs_hcomplex)
apply (auto simp add: hcomplex_diff starfunC starfunCR hcomplex_of_complex_def hcmod mem_infmal_iff)
done
(** much, much easier standard proof **)
lemma CLIM_CRLIM_iff: "(f -- x --C> L) = ((%y. cmod(f y - L)) -- x --CR> 0)"
by (simp add: CLIM_def CRLIM_def)
(* so this is nicer nonstandard proof *)
lemma NSCLIM_NSCRLIM_iff2:
"(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)"
by (auto simp add: CRLIM_NSCRLIM_iff [symmetric] CLIM_CRLIM_iff CLIM_NSCLIM_iff [symmetric])
lemma NSCLIM_NSCRLIM_Re_Im_iff:
"(f -- a --NSC> L) = ((%x. Re(f x)) -- a --NSCR> Re(L) &
(%x. Im(f x)) -- a --NSCR> Im(L))"
apply (auto intro: NSCLIM_NSCRLIM_Re NSCLIM_NSCRLIM_Im)
apply (auto simp add: NSCLIM_def NSCRLIM_def)
apply (auto dest!: spec)
apply (rule_tac z = x in eq_Abs_hcomplex)
apply (auto simp add: capprox_approx_iff starfunC hcomplex_of_complex_def starfunCR hypreal_of_real_def)
done
lemma CLIM_CRLIM_Re_Im_iff:
"(f -- a --C> L) = ((%x. Re(f x)) -- a --CR> Re(L) &
(%x. Im(f x)) -- a --CR> Im(L))"
by (simp add: CLIM_NSCLIM_iff CRLIM_NSCRLIM_iff NSCLIM_NSCRLIM_Re_Im_iff)
subsection{*Continuity*}
lemma isNSContcD:
"[| isNSContc f a; y @c= hcomplex_of_complex a |]
==> ( *fc* f) y @c= hcomplex_of_complex (f a)"
by (simp add: isNSContc_def)
lemma isNSContc_NSCLIM: "isNSContc f a ==> f -- a --NSC> (f a) "
by (simp add: isNSContc_def NSCLIM_def)
lemma NSCLIM_isNSContc:
"f -- a --NSC> (f a) ==> isNSContc f a"
apply (simp add: isNSContc_def NSCLIM_def, auto)
apply (case_tac "y = hcomplex_of_complex a", auto)
done
text{*Nonstandard continuity can be defined using NS Limit in
similar fashion to standard definition of continuity*}
lemma isNSContc_NSCLIM_iff: "(isNSContc f a) = (f -- a --NSC> (f a))"
by (blast intro: isNSContc_NSCLIM NSCLIM_isNSContc)
lemma isNSContc_CLIM_iff: "(isNSContc f a) = (f -- a --C> (f a))"
by (simp add: CLIM_NSCLIM_iff isNSContc_NSCLIM_iff)
(*** key result for continuity ***)
lemma isNSContc_isContc_iff: "(isNSContc f a) = (isContc f a)"
by (simp add: isContc_def isNSContc_CLIM_iff)
lemma isContc_isNSContc: "isContc f a ==> isNSContc f a"
by (erule isNSContc_isContc_iff [THEN iffD2])
lemma isNSContc_isContc: "isNSContc f a ==> isContc f a"
by (erule isNSContc_isContc_iff [THEN iffD1])
text{*Alternative definition of continuity*}
lemma NSCLIM_h_iff: "(f -- a --NSC> L) = ((%h. f(a + h)) -- 0 --NSC> L)"
apply (simp add: NSCLIM_def, auto)
apply (drule_tac x = "hcomplex_of_complex a + x" in spec)
apply (drule_tac [2] x = "- hcomplex_of_complex a + x" in spec, safe, simp)
apply (rule mem_cinfmal_iff [THEN iffD2, THEN CInfinitesimal_add_capprox_self [THEN capprox_sym]])
apply (rule_tac [4] capprox_minus_iff2 [THEN iffD1])
prefer 3 apply (simp add: compare_rls hcomplex_add_commute)
apply (rule_tac [2] z = x in eq_Abs_hcomplex)
apply (rule_tac [4] z = x in eq_Abs_hcomplex)
apply (auto simp add: starfunC hcomplex_of_complex_def hcomplex_minus hcomplex_add)
done
lemma NSCLIM_isContc_iff:
"(f -- a --NSC> f a) = ((%h. f(a + h)) -- 0 --NSC> f a)"
by (rule NSCLIM_h_iff)
lemma CLIM_isContc_iff: "(f -- a --C> f a) = ((%h. f(a + h)) -- 0 --C> f(a))"
by (simp add: CLIM_NSCLIM_iff NSCLIM_isContc_iff)
lemma isContc_iff: "(isContc f x) = ((%h. f(x + h)) -- 0 --C> f(x))"
by (simp add: isContc_def CLIM_isContc_iff)
lemma isContc_add:
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) + g(x)) a"
by (auto intro: capprox_add simp add: isNSContc_isContc_iff [symmetric] isNSContc_def)
lemma isContc_mult:
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) * g(x)) a"
by (auto intro!: starfunC_mult_CFinite_capprox
simp del: starfunC_mult [symmetric]
simp add: isNSContc_isContc_iff [symmetric] isNSContc_def)
lemma isContc_o: "[| isContc f a; isContc g (f a) |] ==> isContc (g o f) a"
by (auto simp add: isNSContc_isContc_iff [symmetric] isNSContc_def starfunC_o [symmetric])
lemma isContc_o2:
"[| isContc f a; isContc g (f a) |] ==> isContc (%x. g (f x)) a"
by (auto dest: isContc_o simp add: o_def)
lemma isNSContc_minus: "isNSContc f a ==> isNSContc (%x. - f x) a"
by (simp add: isNSContc_def)
lemma isContc_minus: "isContc f a ==> isContc (%x. - f x) a"
by (simp add: isNSContc_isContc_iff [symmetric] isNSContc_minus)
lemma isContc_inverse:
"[| isContc f x; f x \<noteq> 0 |] ==> isContc (%x. inverse (f x)) x"
by (simp add: isContc_def CLIM_inverse)
lemma isNSContc_inverse:
"[| isNSContc f x; f x \<noteq> 0 |] ==> isNSContc (%x. inverse (f x)) x"
by (auto intro: isContc_inverse simp add: isNSContc_isContc_iff)
lemma isContc_diff:
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) - g(x)) a"
apply (simp add: complex_diff_def)
apply (auto intro: isContc_add isContc_minus)
done
lemma isContc_const [simp]: "isContc (%x. k) a"
by (simp add: isContc_def)
lemma isNSContc_const [simp]: "isNSContc (%x. k) a"
by (simp add: isNSContc_def)
subsection{*Functions from Complex to Reals*}
lemma isNSContCRD:
"[| isNSContCR f a; y @c= hcomplex_of_complex a |]
==> ( *fcR* f) y @= hypreal_of_real (f a)"
by (simp add: isNSContCR_def)
lemma isNSContCR_NSCRLIM: "isNSContCR f a ==> f -- a --NSCR> (f a) "
by (simp add: isNSContCR_def NSCRLIM_def)
lemma NSCRLIM_isNSContCR: "f -- a --NSCR> (f a) ==> isNSContCR f a"
apply (auto simp add: isNSContCR_def NSCRLIM_def)
apply (case_tac "y = hcomplex_of_complex a", auto)
done
lemma isNSContCR_NSCRLIM_iff: "(isNSContCR f a) = (f -- a --NSCR> (f a))"
by (blast intro: isNSContCR_NSCRLIM NSCRLIM_isNSContCR)
lemma isNSContCR_CRLIM_iff: "(isNSContCR f a) = (f -- a --CR> (f a))"
by (simp add: CRLIM_NSCRLIM_iff isNSContCR_NSCRLIM_iff)
(*** another key result for continuity ***)
lemma isNSContCR_isContCR_iff: "(isNSContCR f a) = (isContCR f a)"
by (simp add: isContCR_def isNSContCR_CRLIM_iff)
lemma isContCR_isNSContCR: "isContCR f a ==> isNSContCR f a"
by (erule isNSContCR_isContCR_iff [THEN iffD2])
lemma isNSContCR_isContCR: "isNSContCR f a ==> isContCR f a"
by (erule isNSContCR_isContCR_iff [THEN iffD1])
lemma isNSContCR_cmod [simp]: "isNSContCR cmod (a)"
by (auto intro: capprox_hcmod_approx
simp add: starfunCR_cmod hcmod_hcomplex_of_complex [symmetric]
isNSContCR_def)
lemma isContCR_cmod [simp]: "isContCR cmod (a)"
by (auto simp add: isNSContCR_isContCR_iff [symmetric])
lemma isContc_isContCR_Re: "isContc f a ==> isContCR (%x. Re (f x)) a"
by (simp add: isContc_def isContCR_def CLIM_CRLIM_Re)
lemma isContc_isContCR_Im: "isContc f a ==> isContCR (%x. Im (f x)) a"
by (simp add: isContc_def isContCR_def CLIM_CRLIM_Im)
subsection{*Derivatives*}
lemma CDERIV_iff: "(CDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --C> D)"
by (simp add: cderiv_def)
lemma CDERIV_NSC_iff:
"(CDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NSC> D)"
by (simp add: cderiv_def CLIM_NSCLIM_iff)
lemma CDERIVD: "CDERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --C> D"
by (simp add: cderiv_def)
lemma NSC_DERIVD: "CDERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NSC> D"
by (simp add: cderiv_def CLIM_NSCLIM_iff)
text{*Uniqueness*}
lemma CDERIV_unique: "[| CDERIV f x :> D; CDERIV f x :> E |] ==> D = E"
by (simp add: cderiv_def CLIM_unique)
(*** uniqueness: a nonstandard proof ***)
lemma NSCDeriv_unique: "[| NSCDERIV f x :> D; NSCDERIV f x :> E |] ==> D = E"
apply (simp add: nscderiv_def)
apply (auto dest!: bspec [where x = "hcomplex_of_hypreal epsilon"]
intro!: inj_hcomplex_of_complex [THEN injD] dest: capprox_trans3)
done
subsection{* Differentiability*}
lemma CDERIV_CLIM_iff:
"((%h. (f(a + h) - f(a))/h) -- 0 --C> D) =
((%x. (f(x) - f(a)) / (x - a)) -- a --C> D)"
apply (simp add: CLIM_def)
apply (rule_tac f=All in arg_cong)
apply (rule ext)
apply (rule imp_cong)
apply (rule refl)
apply (rule_tac f=Ex in arg_cong)
apply (rule ext)
apply (rule conj_cong)
apply (rule refl)
apply (rule trans)
apply (rule all_shift [where a=a], simp)
done
lemma CDERIV_iff2:
"(CDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z - x)) -- x --C> D)"
by (simp add: cderiv_def CDERIV_CLIM_iff)
subsection{* Equivalence of NS and Standard Differentiation*}
(*** first equivalence ***)
lemma NSCDERIV_NSCLIM_iff:
"(NSCDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NSC> D)"
apply (simp add: nscderiv_def NSCLIM_def, auto)
apply (drule_tac x = xa in bspec)
apply (rule_tac [3] ccontr)
apply (drule_tac [3] x = h in spec)
apply (auto simp add: mem_cinfmal_iff starfunC_lambda_cancel)
done
(*** 2nd equivalence ***)
lemma NSCDERIV_NSCLIM_iff2:
"(NSCDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z - x)) -- x --NSC> D)"
by (simp add: NSCDERIV_NSCLIM_iff CDERIV_CLIM_iff CLIM_NSCLIM_iff [symmetric])
lemma NSCDERIV_iff2:
"(NSCDERIV f x :> D) =
(\<forall>xa. xa \<noteq> hcomplex_of_complex x & xa @c= hcomplex_of_complex x -->
( *fc* (%z. (f z - f x) / (z - x))) xa @c= hcomplex_of_complex D)"
by (simp add: NSCDERIV_NSCLIM_iff2 NSCLIM_def)
lemma NSCDERIV_CDERIV_iff: "(NSCDERIV f x :> D) = (CDERIV f x :> D)"
by (simp add: cderiv_def NSCDERIV_NSCLIM_iff CLIM_NSCLIM_iff)
lemma NSCDERIV_isNSContc: "NSCDERIV f x :> D ==> isNSContc f x"
apply (auto simp add: nscderiv_def isNSContc_NSCLIM_iff NSCLIM_def diff_minus)
apply (drule capprox_minus_iff [THEN iffD1])
apply (subgoal_tac "xa + - (hcomplex_of_complex x) \<noteq> 0")
prefer 2 apply (simp add: compare_rls)
apply (drule_tac x = "- hcomplex_of_complex x + xa" in bspec)
prefer 2 apply (simp add: hcomplex_add_assoc [symmetric])
apply (auto simp add: mem_cinfmal_iff [symmetric] hcomplex_add_commute)
apply (drule_tac c = "xa + - hcomplex_of_complex x" in capprox_mult1)
apply (auto intro: CInfinitesimal_subset_CFinite [THEN subsetD]
simp add: mult_assoc)
apply (drule_tac x3 = D in
CFinite_hcomplex_of_complex [THEN [2] CInfinitesimal_CFinite_mult,
THEN mem_cinfmal_iff [THEN iffD1]])
apply (blast intro: capprox_trans mult_commute [THEN subst] capprox_minus_iff [THEN iffD2])
done
lemma CDERIV_isContc: "CDERIV f x :> D ==> isContc f x"
by (simp add: NSCDERIV_CDERIV_iff [symmetric] isNSContc_isContc_iff [symmetric] NSCDERIV_isNSContc)
text{* Differentiation rules for combinations of functions follow by clear,
straightforard algebraic manipulations*}
(* use simple constant nslimit theorem *)
lemma NSCDERIV_const [simp]: "(NSCDERIV (%x. k) x :> 0)"
by (simp add: NSCDERIV_NSCLIM_iff)
lemma CDERIV_const [simp]: "(CDERIV (%x. k) x :> 0)"
by (simp add: NSCDERIV_CDERIV_iff [symmetric])
lemma NSCDERIV_add:
"[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |]
==> NSCDERIV (%x. f x + g x) x :> Da + Db"
apply (simp add: NSCDERIV_NSCLIM_iff NSCLIM_def, clarify)
apply (auto dest!: spec simp add: add_divide_distrib diff_minus)
apply (drule_tac b = "hcomplex_of_complex Da" and d = "hcomplex_of_complex Db" in capprox_add)
apply (auto simp add: add_ac)
done
lemma CDERIV_add:
"[| CDERIV f x :> Da; CDERIV g x :> Db |]
==> CDERIV (%x. f x + g x) x :> Da + Db"
by (simp add: NSCDERIV_add NSCDERIV_CDERIV_iff [symmetric])
subsection{*Lemmas for Multiplication*}
lemma lemma_nscderiv1: "((a::hcomplex)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
by (simp add: right_diff_distrib)
lemma lemma_nscderiv2:
"[| (x + y) / z = hcomplex_of_complex D + yb; z \<noteq> 0;
z : CInfinitesimal; yb : CInfinitesimal |]
==> x + y @c= 0"
apply (frule_tac c1 = z in hcomplex_mult_right_cancel [THEN iffD2], assumption)
apply (erule_tac V = " (x + y) / z = hcomplex_of_complex D + yb" in thin_rl)
apply (auto intro!: CInfinitesimal_CFinite_mult2 CFinite_add
simp add: mem_cinfmal_iff [symmetric])
apply (erule CInfinitesimal_subset_CFinite [THEN subsetD])
done
lemma NSCDERIV_mult:
"[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |]
==> NSCDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
apply (simp add: NSCDERIV_NSCLIM_iff NSCLIM_def, clarify)
apply (auto dest!: spec
simp add: starfunC_lambda_cancel lemma_nscderiv1)
apply (simp (no_asm) add: add_divide_distrib)
apply (drule bex_CInfinitesimal_iff2 [THEN iffD2])+
apply (auto simp del: times_divide_eq_right simp add: times_divide_eq_right [symmetric])
apply (simp add: diff_minus)
apply (drule_tac D = Db in lemma_nscderiv2)
apply (drule_tac [4]
capprox_minus_iff [THEN iffD2, THEN bex_CInfinitesimal_iff2 [THEN iffD2]])
apply (auto intro!: capprox_add_mono1 simp add: left_distrib right_distrib mult_commute add_assoc)
apply (rule_tac b1 = "hcomplex_of_complex Db * hcomplex_of_complex (f x) " in add_commute [THEN subst])
apply (auto intro!: CInfinitesimal_add_capprox_self2 [THEN capprox_sym]
CInfinitesimal_add CInfinitesimal_mult
CInfinitesimal_hcomplex_of_complex_mult
CInfinitesimal_hcomplex_of_complex_mult2
simp add: hcomplex_add_assoc [symmetric])
done
lemma CDERIV_mult:
"[| CDERIV f x :> Da; CDERIV g x :> Db |]
==> CDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
by (simp add: NSCDERIV_mult NSCDERIV_CDERIV_iff [symmetric])
lemma NSCDERIV_cmult: "NSCDERIV f x :> D ==> NSCDERIV (%x. c * f x) x :> c*D"
apply (simp add: times_divide_eq_right [symmetric] NSCDERIV_NSCLIM_iff
minus_mult_right right_distrib [symmetric] complex_diff_def
del: times_divide_eq_right minus_mult_right [symmetric])
apply (erule NSCLIM_const [THEN NSCLIM_mult])
done
lemma CDERIV_cmult: "CDERIV f x :> D ==> CDERIV (%x. c * f x) x :> c*D"
by (simp add: NSCDERIV_cmult NSCDERIV_CDERIV_iff [symmetric])
lemma NSCDERIV_minus: "NSCDERIV f x :> D ==> NSCDERIV (%x. -(f x)) x :> -D"
apply (simp add: NSCDERIV_NSCLIM_iff complex_diff_def)
apply (rule_tac t = "f x" in minus_minus [THEN subst])
apply (simp (no_asm_simp) add: minus_add_distrib [symmetric]
del: minus_add_distrib minus_minus)
apply (erule NSCLIM_minus)
done
lemma CDERIV_minus: "CDERIV f x :> D ==> CDERIV (%x. -(f x)) x :> -D"
by (simp add: NSCDERIV_minus NSCDERIV_CDERIV_iff [symmetric])
lemma NSCDERIV_add_minus:
"[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |]
==> NSCDERIV (%x. f x + -g x) x :> Da + -Db"
by (blast dest: NSCDERIV_add NSCDERIV_minus)
lemma CDERIV_add_minus:
"[| CDERIV f x :> Da; CDERIV g x :> Db |]
==> CDERIV (%x. f x + -g x) x :> Da + -Db"
by (blast dest: CDERIV_add CDERIV_minus)
lemma NSCDERIV_diff:
"[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |]
==> NSCDERIV (%x. f x - g x) x :> Da - Db"
by (simp add: complex_diff_def NSCDERIV_add_minus)
lemma CDERIV_diff:
"[| CDERIV f x :> Da; CDERIV g x :> Db |]
==> CDERIV (%x. f x - g x) x :> Da - Db"
by (simp add: complex_diff_def CDERIV_add_minus)
subsection{*Chain Rule*}
(* lemmas *)
lemma NSCDERIV_zero:
"[| NSCDERIV g x :> D;
( *fc* g) (hcomplex_of_complex(x) + xa) = hcomplex_of_complex(g x);
xa : CInfinitesimal; xa \<noteq> 0
|] ==> D = 0"
apply (simp add: nscderiv_def)
apply (drule bspec, auto)
done
lemma NSCDERIV_capprox:
"[| NSCDERIV f x :> D; h: CInfinitesimal; h \<noteq> 0 |]
==> ( *fc* f) (hcomplex_of_complex(x) + h) - hcomplex_of_complex(f x) @c= 0"
apply (simp add: nscderiv_def mem_cinfmal_iff [symmetric])
apply (rule CInfinitesimal_ratio)
apply (rule_tac [3] capprox_hcomplex_of_complex_CFinite, auto)
done
(*--------------------------------------------------*)
(* from one version of differentiability *)
(* *)
(* f(x) - f(a) *)
(* --------------- @= Db *)
(* x - a *)
(* -------------------------------------------------*)
lemma NSCDERIVD1:
"[| NSCDERIV f (g x) :> Da;
( *fc* g) (hcomplex_of_complex(x) + xa) \<noteq> hcomplex_of_complex (g x);
( *fc* g) (hcomplex_of_complex(x) + xa) @c= hcomplex_of_complex (g x)|]
==> (( *fc* f) (( *fc* g) (hcomplex_of_complex(x) + xa))
- hcomplex_of_complex (f (g x))) /
(( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex (g x))
@c= hcomplex_of_complex (Da)"
by (auto simp add: NSCDERIV_NSCLIM_iff2 NSCLIM_def)
(*--------------------------------------------------*)
(* from other version of differentiability *)
(* *)
(* f(x + h) - f(x) *)
(* ----------------- @= Db *)
(* h *)
(*--------------------------------------------------*)
lemma NSCDERIVD2:
"[| NSCDERIV g x :> Db; xa: CInfinitesimal; xa \<noteq> 0 |]
==> (( *fc* g) (hcomplex_of_complex x + xa) - hcomplex_of_complex(g x)) / xa
@c= hcomplex_of_complex (Db)"
by (auto simp add: NSCDERIV_NSCLIM_iff NSCLIM_def mem_cinfmal_iff starfunC_lambda_cancel)
lemma lemma_complex_chain: "(z::hcomplex) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
by auto
text{*Chain rule*}
theorem NSCDERIV_chain:
"[| NSCDERIV f (g x) :> Da; NSCDERIV g x :> Db |]
==> NSCDERIV (f o g) x :> Da * Db"
apply (simp (no_asm_simp) add: NSCDERIV_NSCLIM_iff NSCLIM_def mem_cinfmal_iff [symmetric])
apply safe
apply (frule_tac f = g in NSCDERIV_capprox)
apply (auto simp add: starfunC_lambda_cancel2 starfunC_o [symmetric])
apply (case_tac "( *fc* g) (hcomplex_of_complex (x) + xa) = hcomplex_of_complex (g x) ")
apply (drule_tac g = g in NSCDERIV_zero)
apply (auto simp add: hcomplex_divide_def)
apply (rule_tac z1 = "( *fc* g) (hcomplex_of_complex (x) + xa) - hcomplex_of_complex (g x) " and y1 = "inverse xa" in lemma_complex_chain [THEN ssubst])
apply (simp (no_asm_simp))
apply (rule capprox_mult_hcomplex_of_complex)
apply (auto intro!: NSCDERIVD1 intro: capprox_minus_iff [THEN iffD2]
simp add: diff_minus [symmetric]
divide_inverse_zero [symmetric])
apply (blast intro: NSCDERIVD2)
done
text{*standard version*}
lemma CDERIV_chain:
"[| CDERIV f (g x) :> Da; CDERIV g x :> Db |]
==> CDERIV (f o g) x :> Da * Db"
by (simp add: NSCDERIV_CDERIV_iff [symmetric] NSCDERIV_chain)
lemma CDERIV_chain2:
"[| CDERIV f (g x) :> Da; CDERIV g x :> Db |]
==> CDERIV (%x. f (g x)) x :> Da * Db"
by (auto dest: CDERIV_chain simp add: o_def)
subsection{* Differentiation of Natural Number Powers*}
lemma NSCDERIV_Id [simp]: "NSCDERIV (%x. x) x :> 1"
by (simp add: NSCDERIV_NSCLIM_iff NSCLIM_def)
lemma CDERIV_Id [simp]: "CDERIV (%x. x) x :> 1"
by (simp add: NSCDERIV_CDERIV_iff [symmetric])
lemmas isContc_Id = CDERIV_Id [THEN CDERIV_isContc, standard]
text{*derivative of linear multiplication*}
lemma CDERIV_cmult_Id [simp]: "CDERIV (op * c) x :> c"
by (cut_tac c = c and x = x in CDERIV_Id [THEN CDERIV_cmult], simp)
lemma NSCDERIV_cmult_Id [simp]: "NSCDERIV (op * c) x :> c"
by (simp add: NSCDERIV_CDERIV_iff)
lemma CDERIV_pow [simp]:
"CDERIV (%x. x ^ n) x :> (complex_of_real (real n)) * (x ^ (n - Suc 0))"
apply (induct_tac "n")
apply (drule_tac [2] CDERIV_Id [THEN CDERIV_mult])
apply (auto simp add: complex_of_real_add [symmetric] left_distrib real_of_nat_Suc)
apply (case_tac "n")
apply (auto simp add: mult_ac add_commute)
done
text{*Nonstandard version*}
lemma NSCDERIV_pow:
"NSCDERIV (%x. x ^ n) x :> complex_of_real (real n) * (x ^ (n - 1))"
by (simp add: NSCDERIV_CDERIV_iff)
lemma lemma_CDERIV_subst:
"[|CDERIV f x :> D; D = E|] ==> CDERIV f x :> E"
by auto
(*used once, in NSCDERIV_inverse*)
lemma CInfinitesimal_add_not_zero:
"[| h: CInfinitesimal; x \<noteq> 0 |] ==> hcomplex_of_complex x + h \<noteq> 0"
apply clarify
apply (drule equals_zero_I, auto)
done
text{*Can't relax the premise @{term "x \<noteq> 0"}: it isn't continuous at zero*}
lemma NSCDERIV_inverse:
"x \<noteq> 0 ==> NSCDERIV (%x. inverse(x)) x :> (- (inverse x ^ 2))"
apply (simp add: nscderiv_def Ball_def, clarify)
apply (frule CInfinitesimal_add_not_zero [where x=x])
apply (auto simp add: starfunC_inverse_inverse hcomplex_diff_def
simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])
apply (simp add: hcomplex_add_commute numeral_2_eq_2 inverse_add
inverse_mult_distrib [symmetric] inverse_minus_eq [symmetric]
add_ac mult_ac
del: inverse_minus_eq inverse_mult_distrib minus_mult_right [symmetric] minus_mult_left [symmetric])
apply (simp only: mult_assoc [symmetric] right_distrib)
apply (rule_tac y = " inverse (- hcomplex_of_complex x * hcomplex_of_complex x) " in capprox_trans)
apply (rule inverse_add_CInfinitesimal_capprox2)
apply (auto dest!: hcomplex_of_complex_CFinite_diff_CInfinitesimal
intro: CFinite_mult
simp add: inverse_minus_eq [symmetric])
apply (rule CInfinitesimal_CFinite_mult2, auto)
done
lemma CDERIV_inverse:
"x \<noteq> 0 ==> CDERIV (%x. inverse(x)) x :> (-(inverse x ^ 2))"
by (simp add: NSCDERIV_inverse NSCDERIV_CDERIV_iff [symmetric]
del: complexpow_Suc)
subsection{*Derivative of Reciprocals (Function @{term inverse})*}
lemma CDERIV_inverse_fun:
"[| CDERIV f x :> d; f(x) \<noteq> 0 |]
==> CDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"
apply (rule mult_commute [THEN subst])
apply (simp (no_asm_simp) add: minus_mult_left power_inverse del: complexpow_Suc minus_mult_left [symmetric])
apply (fold o_def)
apply (blast intro!: CDERIV_chain CDERIV_inverse)
done
lemma NSCDERIV_inverse_fun:
"[| NSCDERIV f x :> d; f(x) \<noteq> 0 |]
==> NSCDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"
by (simp add: NSCDERIV_CDERIV_iff CDERIV_inverse_fun del: complexpow_Suc)
subsection{* Derivative of Quotient*}
lemma CDERIV_quotient:
"[| CDERIV f x :> d; CDERIV g x :> e; g(x) \<noteq> 0 |]
==> CDERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ 2)"
apply (simp add: complex_diff_def)
apply (drule_tac f = g in CDERIV_inverse_fun)
apply (drule_tac [2] CDERIV_mult, assumption+)
apply (simp add: divide_inverse_zero right_distrib power_inverse minus_mult_left mult_ac
del: complexpow_Suc minus_mult_right [symmetric] minus_mult_left [symmetric])
done
lemma NSCDERIV_quotient:
"[| NSCDERIV f x :> d; NSCDERIV g x :> e; g(x) \<noteq> 0 |]
==> NSCDERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ 2)"
by (simp add: NSCDERIV_CDERIV_iff CDERIV_quotient del: complexpow_Suc)
subsection{*Caratheodory Formulation of Derivative at a Point: Standard Proof*}
lemma CLIM_equal:
"[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --C> l) = (g -- a --C> l)"
by (simp add: CLIM_def complex_add_minus_iff)
lemma CLIM_trans:
"[| (%x. f(x) + -g(x)) -- a --C> 0; g -- a --C> l |] ==> f -- a --C> l"
apply (drule CLIM_add, assumption)
apply (simp add: complex_add_assoc)
done
lemma CARAT_CDERIV:
"(CDERIV f x :> l) =
(\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) & isContc g x & g x = l)"
apply safe
apply (rule_tac x = "%z. if z=x then l else (f (z) - f (x)) / (z-x)" in exI)
apply (auto simp add: mult_assoc isContc_iff CDERIV_iff)
apply (rule_tac [!] CLIM_equal [THEN iffD1], auto)
done
lemma CARAT_NSCDERIV:
"NSCDERIV f x :> l ==>
\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l"
by (auto simp add: NSCDERIV_CDERIV_iff isNSContc_isContc_iff CARAT_CDERIV)
(* FIXME tidy! How about a NS proof? *)
lemma CARAT_CDERIVD:
"(\<forall>z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l
==> NSCDERIV f x :> l"
apply (simp only: NSCDERIV_iff2)
apply (tactic {*(auto_tac (claset(),
simpset() delsimprocs field_cancel_factor
addsimps [thm"NSCDERIV_iff2"])) *})
apply (simp add: isNSContc_def)
done
ML
{*
val complex_add_minus_iff = thm "complex_add_minus_iff";
val complex_add_eq_0_iff = thm "complex_add_eq_0_iff";
val NSCLIM_NSCRLIM_Re = thm "NSCLIM_NSCRLIM_Re";
val NSCLIM_NSCRLIM_Im = thm "NSCLIM_NSCRLIM_Im";
val CLIM_NSCLIM = thm "CLIM_NSCLIM";
val eq_Abs_hcomplex_ALL = thm "eq_Abs_hcomplex_ALL";
val lemma_CLIM = thm "lemma_CLIM";
val lemma_skolemize_CLIM2 = thm "lemma_skolemize_CLIM2";
val lemma_csimp = thm "lemma_csimp";
val NSCLIM_CLIM = thm "NSCLIM_CLIM";
val CLIM_NSCLIM_iff = thm "CLIM_NSCLIM_iff";
val CRLIM_NSCRLIM = thm "CRLIM_NSCRLIM";
val lemma_CRLIM = thm "lemma_CRLIM";
val lemma_skolemize_CRLIM2 = thm "lemma_skolemize_CRLIM2";
val lemma_crsimp = thm "lemma_crsimp";
val NSCRLIM_CRLIM = thm "NSCRLIM_CRLIM";
val CRLIM_NSCRLIM_iff = thm "CRLIM_NSCRLIM_iff";
val CLIM_CRLIM_Re = thm "CLIM_CRLIM_Re";
val CLIM_CRLIM_Im = thm "CLIM_CRLIM_Im";
val CLIM_cnj = thm "CLIM_cnj";
val CLIM_cnj_iff = thm "CLIM_cnj_iff";
val NSCLIM_add = thm "NSCLIM_add";
val CLIM_add = thm "CLIM_add";
val NSCLIM_mult = thm "NSCLIM_mult";
val CLIM_mult = thm "CLIM_mult";
val NSCLIM_const = thm "NSCLIM_const";
val CLIM_const = thm "CLIM_const";
val NSCLIM_minus = thm "NSCLIM_minus";
val CLIM_minus = thm "CLIM_minus";
val NSCLIM_diff = thm "NSCLIM_diff";
val CLIM_diff = thm "CLIM_diff";
val NSCLIM_inverse = thm "NSCLIM_inverse";
val CLIM_inverse = thm "CLIM_inverse";
val NSCLIM_zero = thm "NSCLIM_zero";
val CLIM_zero = thm "CLIM_zero";
val NSCLIM_zero_cancel = thm "NSCLIM_zero_cancel";
val CLIM_zero_cancel = thm "CLIM_zero_cancel";
val NSCLIM_not_zero = thm "NSCLIM_not_zero";
val NSCLIM_not_zeroE = thms "NSCLIM_not_zeroE";
val CLIM_not_zero = thm "CLIM_not_zero";
val NSCLIM_const_eq = thm "NSCLIM_const_eq";
val CLIM_const_eq = thm "CLIM_const_eq";
val NSCLIM_unique = thm "NSCLIM_unique";
val CLIM_unique = thm "CLIM_unique";
val NSCLIM_mult_zero = thm "NSCLIM_mult_zero";
val CLIM_mult_zero = thm "CLIM_mult_zero";
val NSCLIM_self = thm "NSCLIM_self";
val CLIM_self = thm "CLIM_self";
val NSCLIM_NSCRLIM_iff = thm "NSCLIM_NSCRLIM_iff";
val CLIM_CRLIM_iff = thm "CLIM_CRLIM_iff";
val NSCLIM_NSCRLIM_iff2 = thm "NSCLIM_NSCRLIM_iff2";
val NSCLIM_NSCRLIM_Re_Im_iff = thm "NSCLIM_NSCRLIM_Re_Im_iff";
val CLIM_CRLIM_Re_Im_iff = thm "CLIM_CRLIM_Re_Im_iff";
val isNSContcD = thm "isNSContcD";
val isNSContc_NSCLIM = thm "isNSContc_NSCLIM";
val NSCLIM_isNSContc = thm "NSCLIM_isNSContc";
val isNSContc_NSCLIM_iff = thm "isNSContc_NSCLIM_iff";
val isNSContc_CLIM_iff = thm "isNSContc_CLIM_iff";
val isNSContc_isContc_iff = thm "isNSContc_isContc_iff";
val isContc_isNSContc = thm "isContc_isNSContc";
val isNSContc_isContc = thm "isNSContc_isContc";
val NSCLIM_h_iff = thm "NSCLIM_h_iff";
val NSCLIM_isContc_iff = thm "NSCLIM_isContc_iff";
val CLIM_isContc_iff = thm "CLIM_isContc_iff";
val isContc_iff = thm "isContc_iff";
val isContc_add = thm "isContc_add";
val isContc_mult = thm "isContc_mult";
val isContc_o = thm "isContc_o";
val isContc_o2 = thm "isContc_o2";
val isNSContc_minus = thm "isNSContc_minus";
val isContc_minus = thm "isContc_minus";
val isContc_inverse = thm "isContc_inverse";
val isNSContc_inverse = thm "isNSContc_inverse";
val isContc_diff = thm "isContc_diff";
val isContc_const = thm "isContc_const";
val isNSContc_const = thm "isNSContc_const";
val isNSContCRD = thm "isNSContCRD";
val isNSContCR_NSCRLIM = thm "isNSContCR_NSCRLIM";
val NSCRLIM_isNSContCR = thm "NSCRLIM_isNSContCR";
val isNSContCR_NSCRLIM_iff = thm "isNSContCR_NSCRLIM_iff";
val isNSContCR_CRLIM_iff = thm "isNSContCR_CRLIM_iff";
val isNSContCR_isContCR_iff = thm "isNSContCR_isContCR_iff";
val isContCR_isNSContCR = thm "isContCR_isNSContCR";
val isNSContCR_isContCR = thm "isNSContCR_isContCR";
val isNSContCR_cmod = thm "isNSContCR_cmod";
val isContCR_cmod = thm "isContCR_cmod";
val isContc_isContCR_Re = thm "isContc_isContCR_Re";
val isContc_isContCR_Im = thm "isContc_isContCR_Im";
val CDERIV_iff = thm "CDERIV_iff";
val CDERIV_NSC_iff = thm "CDERIV_NSC_iff";
val CDERIVD = thm "CDERIVD";
val NSC_DERIVD = thm "NSC_DERIVD";
val CDERIV_unique = thm "CDERIV_unique";
val NSCDeriv_unique = thm "NSCDeriv_unique";
val CDERIV_CLIM_iff = thm "CDERIV_CLIM_iff";
val CDERIV_iff2 = thm "CDERIV_iff2";
val NSCDERIV_NSCLIM_iff = thm "NSCDERIV_NSCLIM_iff";
val NSCDERIV_NSCLIM_iff2 = thm "NSCDERIV_NSCLIM_iff2";
val NSCDERIV_iff2 = thm "NSCDERIV_iff2";
val NSCDERIV_CDERIV_iff = thm "NSCDERIV_CDERIV_iff";
val NSCDERIV_isNSContc = thm "NSCDERIV_isNSContc";
val CDERIV_isContc = thm "CDERIV_isContc";
val NSCDERIV_const = thm "NSCDERIV_const";
val CDERIV_const = thm "CDERIV_const";
val NSCDERIV_add = thm "NSCDERIV_add";
val CDERIV_add = thm "CDERIV_add";
val lemma_nscderiv1 = thm "lemma_nscderiv1";
val lemma_nscderiv2 = thm "lemma_nscderiv2";
val NSCDERIV_mult = thm "NSCDERIV_mult";
val CDERIV_mult = thm "CDERIV_mult";
val NSCDERIV_cmult = thm "NSCDERIV_cmult";
val CDERIV_cmult = thm "CDERIV_cmult";
val NSCDERIV_minus = thm "NSCDERIV_minus";
val CDERIV_minus = thm "CDERIV_minus";
val NSCDERIV_add_minus = thm "NSCDERIV_add_minus";
val CDERIV_add_minus = thm "CDERIV_add_minus";
val NSCDERIV_diff = thm "NSCDERIV_diff";
val CDERIV_diff = thm "CDERIV_diff";
val NSCDERIV_zero = thm "NSCDERIV_zero";
val NSCDERIV_capprox = thm "NSCDERIV_capprox";
val NSCDERIVD1 = thm "NSCDERIVD1";
val NSCDERIVD2 = thm "NSCDERIVD2";
val lemma_complex_chain = thm "lemma_complex_chain";
val NSCDERIV_chain = thm "NSCDERIV_chain";
val CDERIV_chain = thm "CDERIV_chain";
val CDERIV_chain2 = thm "CDERIV_chain2";
val NSCDERIV_Id = thm "NSCDERIV_Id";
val CDERIV_Id = thm "CDERIV_Id";
val isContc_Id = thms "isContc_Id";
val CDERIV_cmult_Id = thm "CDERIV_cmult_Id";
val NSCDERIV_cmult_Id = thm "NSCDERIV_cmult_Id";
val CDERIV_pow = thm "CDERIV_pow";
val NSCDERIV_pow = thm "NSCDERIV_pow";
val lemma_CDERIV_subst = thm "lemma_CDERIV_subst";
val CInfinitesimal_add_not_zero = thm "CInfinitesimal_add_not_zero";
val NSCDERIV_inverse = thm "NSCDERIV_inverse";
val CDERIV_inverse = thm "CDERIV_inverse";
val CDERIV_inverse_fun = thm "CDERIV_inverse_fun";
val NSCDERIV_inverse_fun = thm "NSCDERIV_inverse_fun";
val lemma_complex_mult_inverse_squared = thm "lemma_complex_mult_inverse_squared";
val CDERIV_quotient = thm "CDERIV_quotient";
val NSCDERIV_quotient = thm "NSCDERIV_quotient";
val CLIM_equal = thm "CLIM_equal";
val CLIM_trans = thm "CLIM_trans";
val CARAT_CDERIV = thm "CARAT_CDERIV";
val CARAT_NSCDERIV = thm "CARAT_NSCDERIV";
val CARAT_CDERIVD = thm "CARAT_CDERIVD";
*}
end