(* Title : CLim.thy
Author : Jacques D. Fleuriot
Copyright : 2001 University of Edinburgh
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
header{*Limits, Continuity and Differentiation for Complex Functions*}
theory CLim
imports CSeries
begin
(*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 (simp add: numeral_2_eq_2)
text{*Changing the quantified variable. Install earlier?*}
lemma all_shift: "(\<forall>x::'a::comm_ring_1. 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
abbreviation
CLIM :: "[complex=>complex,complex,complex] => bool"
("((_)/ -- (_)/ --C> (_))" [60, 0, 60] 60)
"CLIM == LIM"
NSCLIM :: "[complex=>complex,complex,complex] => bool"
("((_)/ -- (_)/ --NSC> (_))" [60, 0, 60] 60)
"NSCLIM == NSLIM"
(* f: C --> R *)
CRLIM :: "[complex=>real,complex,real] => bool"
("((_)/ -- (_)/ --CR> (_))" [60, 0, 60] 60)
"CRLIM == LIM"
NSCRLIM :: "[complex=>real,complex,real] => bool"
("((_)/ -- (_)/ --NSCR> (_))" [60, 0, 60] 60)
"NSCRLIM == NSLIM"
isContc :: "[complex=>complex,complex] => bool"
"isContc == isCont"
(* NS definition dispenses with limit notions *)
isNSContc :: "[complex=>complex,complex] => bool"
"isNSContc == isNSCont"
isContCR :: "[complex=>real,complex] => bool"
"isContCR == isCont"
(* NS definition dispenses with limit notions *)
isNSContCR :: "[complex=>real,complex] => bool"
"isNSContCR == isNSCont"
lemma CLIM_def:
"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))))"
by (rule LIM_def)
lemma NSCLIM_def:
"f -- a --NSC> L = (\<forall>x. (x \<noteq> hcomplex_of_complex a &
x @= hcomplex_of_complex a
--> ( *f* f) x @= hcomplex_of_complex L))"
by (rule NSLIM_def)
lemma CRLIM_def:
"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))))"
by (simp add: LIM_def)
lemma NSCRLIM_def:
"f -- a --NSCR> L = (\<forall>x. (x \<noteq> hcomplex_of_complex a &
x @= hcomplex_of_complex a
--> ( *f* f) x @= hypreal_of_real L))"
by (rule NSLIM_def)
lemma isContc_def:
"isContc f a = (f -- a --C> (f a))"
by (rule isCont_def)
lemma isNSContc_def:
"isNSContc f a = (\<forall>y. y @= hcomplex_of_complex a -->
( *f* f) y @= hcomplex_of_complex (f a))"
by (rule isNSCont_def)
lemma isContCR_def:
"isContCR f a = (f -- a --CR> (f a))"
by (rule isCont_def)
lemma isNSContCR_def:
"isNSContCR f a = (\<forall>y. y @= hcomplex_of_complex a -->
( *f* f) y @= hypreal_of_real (f a))"
by (rule isNSCont_def)
definition
(* 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> Infinitesimal - {0}.
(( *f* f)(hcomplex_of_complex x + h)
- hcomplex_of_complex (f x))/h @= 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 @= y --> ( *f* f) x @= ( *f* 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)"
by (simp add: NSLIM_def starfunC_approx_Re_Im_iff
hRe_hcomplex_of_complex)
lemma NSCLIM_NSCRLIM_Im: "f -- a --NSC> L ==> (%x. Im(f x)) -- a --NSCR> Im(L)"
by (simp add: NSLIM_def starfunC_approx_Re_Im_iff
hIm_hcomplex_of_complex)
lemma CLIM_NSCLIM:
"f -- x --C> L ==> f -- x --NSC> L"
by (rule LIM_NSLIM)
lemma eq_Abs_star_ALL: "(\<forall>t. P t) = (\<forall>X. P (star_n X))"
apply auto
apply (rule_tac x = t in star_cases, 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"
by (rule NSLIM_LIM)
text{*First key result*}
theorem CLIM_NSCLIM_iff: "(f -- x --C> L) = (f -- x --NSC> L)"
by (rule LIM_NSLIM_iff)
subsection{*Limit of Complex to Real Function*}
lemma CRLIM_NSCRLIM: "f -- x --CR> L ==> f -- x --NSCR> L"
by (rule LIM_NSLIM)
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"
by (rule NSLIM_LIM)
text{*Second key result*}
theorem CRLIM_NSCRLIM_iff: "(f -- x --CR> L) = (f -- x --NSCR> L)"
by (rule LIM_NSLIM_iff)
(** 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 (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 (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 (rule NSLIM_add)
lemma CLIM_add:
"[| f -- x --C> l; g -- x --C> m |]
==> (%x. f(x) + g(x)) -- x --C> (l + m)"
by (rule LIM_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 (rule NSLIM_mult)
lemma CLIM_mult:
"[| f -- x --C> l; g -- x --C> m |]
==> (%x. f(x) * g(x)) -- x --C> (l * m)"
by (rule LIM_mult2)
(*** NSCLIM_const and CLIM_const ***)
lemma NSCLIM_const [simp]: "(%x. k) -- x --NSC> k"
by (rule NSLIM_const)
lemma CLIM_const [simp]: "(%x. k) -- x --C> k"
by (rule LIM_const)
(*** NSCLIM_minus and CLIM_minus ***)
lemma NSCLIM_minus: "f -- a --NSC> L ==> (%x. -f(x)) -- a --NSC> -L"
by (rule NSLIM_minus)
lemma CLIM_minus: "f -- a --C> L ==> (%x. -f(x)) -- a --C> -L"
by (rule LIM_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: diff_minus 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 (rule LIM_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)"
by (rule NSLIM_inverse)
lemma CLIM_inverse:
"[| f -- a --C> L; L \<noteq> 0 |]
==> (%x. inverse(f(x))) -- a --C> (inverse L)"
by (rule LIM_inverse)
(*** NSCLIM_zero, CLIM_zero, etc. ***)
lemma NSCLIM_zero: "f -- a --NSC> l ==> (%x. f(x) - l) -- a --NSC> 0"
apply (simp add: diff_minus)
apply (rule_tac a1 = l in right_minus [THEN subst])
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 (rule NSLIM_zero_cancel)
lemma CLIM_zero_cancel: "(%x. f(x) - l) -- x --C> 0 ==> f -- x --C> l"
by (rule LIM_zero_cancel)
(*** NSCLIM_not zero and hence CLIM_not_zero ***)
lemma NSCLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --NSC> 0)"
apply (auto simp del: star_of_zero simp add: NSCLIM_def)
apply (rule_tac x = "hcomplex_of_complex x + hcomplex_of_hypreal epsilon" in exI)
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
simp del: star_of_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 (rule LIM_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 (rule LIM_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 (rule NSLIM_mult_zero)
lemma CLIM_mult_zero:
"[| f -- x --C> 0; g -- x --C> 0 |] ==> (%x. f(x)*g(x)) -- x --C> 0"
by (rule LIM_mult_zero)
(*** NSCLIM_self hence CLIM_self ***)
lemma NSCLIM_self: "(%x. x) -- a --NSC> a"
by (rule NSLIM_self)
lemma CLIM_self: "(%x. x) -- a --C> a"
by (rule LIM_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 Infinitesimal_approx_minus [symmetric] Infinitesimal_hcmod_iff)
apply (auto dest!: spec)
apply (rule_tac [!] x = xa in star_cases)
apply (auto simp add: star_n_diff starfun hcmod mem_infmal_iff star_of_def)
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 (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 x = x in star_cases)
apply (simp add: approx_approx_iff starfun star_of_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 @= hcomplex_of_complex a |]
==> ( *f* f) y @= hcomplex_of_complex (f a)"
by (simp add: isNSContc_def)
lemma isNSContc_NSCLIM: "isNSContc f a ==> f -- a --NSC> (f a) "
by (rule isNSCont_NSLIM)
lemma NSCLIM_isNSContc:
"f -- a --NSC> (f a) ==> isNSContc f a"
by (rule NSLIM_isNSCont)
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 (rule isNSCont_NSLIM_iff)
lemma isNSContc_CLIM_iff: "(isNSContc f a) = (f -- a --C> (f a))"
by (rule isNSCont_LIM_iff)
(*** key result for continuity ***)
lemma isNSContc_isContc_iff: "(isNSContc f a) = (isContc f a)"
by (rule isNSCont_isCont_iff)
lemma isContc_isNSContc: "isContc f a ==> isNSContc f a"
by (rule isCont_isNSCont)
lemma isNSContc_isContc: "isNSContc f a ==> isContc f a"
by (rule isNSCont_isCont)
text{*Alternative definition of continuity*}
lemma NSCLIM_h_iff: "(f -- a --NSC> L) = ((%h. f(a + h)) -- 0 --NSC> L)"
by (rule NSLIM_h_iff)
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 (rule LIM_isCont_iff)
lemma isContc_iff: "(isContc f x) = ((%h. f(x + h)) -- 0 --C> f(x))"
by (rule isCont_iff)
lemma isContc_add:
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) + g(x)) a"
by (rule isCont_add)
lemma isContc_mult:
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) * g(x)) a"
by (rule isCont_mult)
lemma isContc_o: "[| isContc f a; isContc g (f a) |] ==> isContc (g o f) a"
by (rule isCont_o)
lemma isContc_o2:
"[| isContc f a; isContc g (f a) |] ==> isContc (%x. g (f x)) a"
by (rule isCont_o2)
lemma isNSContc_minus: "isNSContc f a ==> isNSContc (%x. - f x) a"
by (rule isNSCont_minus)
lemma isContc_minus: "isContc f a ==> isContc (%x. - f x) a"
by (rule isCont_minus)
lemma isContc_inverse:
"[| isContc f x; f x \<noteq> 0 |] ==> isContc (%x. inverse (f x)) x"
by (rule isCont_inverse)
lemma isNSContc_inverse:
"[| isNSContc f x; f x \<noteq> 0 |] ==> isNSContc (%x. inverse (f x)) x"
by (rule isNSCont_inverse)
lemma isContc_diff:
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) - g(x)) a"
by (rule isCont_diff)
lemma isContc_const [simp]: "isContc (%x. k) a"
by (rule isCont_const)
lemma isNSContc_const [simp]: "isNSContc (%x. k) a"
by (rule isNSCont_const)
subsection{*Functions from Complex to Reals*}
lemma isNSContCRD:
"[| isNSContCR f a; y @= hcomplex_of_complex a |]
==> ( *f* f) y @= hypreal_of_real (f a)"
by (simp add: isNSContCR_def)
lemma isNSContCR_NSCRLIM: "isNSContCR f a ==> f -- a --NSCR> (f a) "
by (rule isNSCont_NSLIM)
lemma NSCRLIM_isNSContCR: "f -- a --NSCR> (f a) ==> isNSContCR f a"
by (rule NSLIM_isNSCont)
lemma isNSContCR_NSCRLIM_iff: "(isNSContCR f a) = (f -- a --NSCR> (f a))"
by (rule isNSCont_NSLIM_iff)
lemma isNSContCR_CRLIM_iff: "(isNSContCR f a) = (f -- a --CR> (f a))"
by (rule isNSCont_LIM_iff)
(*** another key result for continuity ***)
lemma isNSContCR_isContCR_iff: "(isNSContCR f a) = (isContCR f a)"
by (rule isNSCont_isCont_iff)
lemma isContCR_isNSContCR: "isContCR f a ==> isNSContCR f a"
by (rule isCont_isNSCont)
lemma isNSContCR_isContCR: "isNSContCR f a ==> isContCR f a"
by (rule isNSCont_isCont)
lemma isNSContCR_cmod [simp]: "isNSContCR cmod (a)"
by (auto intro: approx_hcmod_approx
simp add: starfunCR_cmod hcmod_hcomplex_of_complex [symmetric]
isNSContCR_def)
lemma isContCR_cmod [simp]: "isContCR cmod (a)"
by (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: approx_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_infmal_iff starfun_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 @= hcomplex_of_complex x -->
( *f* (%z. (f z - f x) / (z - x))) xa @= 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)
apply (drule approx_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 = "xa - hcomplex_of_complex x" in bspec)
apply (simp add: mem_infmal_iff)
apply (simp add: add_assoc [symmetric])
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
apply (drule_tac c = "xa - hcomplex_of_complex x" in approx_mult1)
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
simp add: mult_assoc)
apply (drule_tac x3 = D in
HFinite_hcomplex_of_complex [THEN [2] Infinitesimal_HFinite_mult,
THEN mem_infmal_iff [THEN iffD1]])
apply (blast intro: approx_trans mult_commute [THEN subst] approx_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 approx_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 : Infinitesimal; yb : Infinitesimal |]
==> x - y @= 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!: Infinitesimal_HFinite_mult2 HFinite_add
simp add: mem_infmal_iff [symmetric])
apply (erule Infinitesimal_subset_HFinite [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 (auto simp add: NSCDERIV_NSCLIM_iff NSCLIM_def)
apply (auto dest!: spec
simp add: starfun_lambda_cancel lemma_nscderiv1)
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
apply (auto simp add: times_divide_eq_right [symmetric]
simp del: times_divide_eq)
apply (drule_tac D = Db in lemma_nscderiv2, assumption+)
apply (drule_tac
approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
apply (auto intro!: approx_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!: Infinitesimal_add_approx_self2 [THEN approx_sym]
Infinitesimal_add Infinitesimal_mult
Infinitesimal_hcomplex_of_complex_mult
Infinitesimal_hcomplex_of_complex_mult2
simp add: 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] diff_minus
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 diff_minus)
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: diff_minus 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: diff_minus CDERIV_add_minus)
subsection{*Chain Rule*}
(* lemmas *)
lemma NSCDERIV_zero:
"[| NSCDERIV g x :> D;
( *f* g) (hcomplex_of_complex(x) + xa) = hcomplex_of_complex(g x);
xa : Infinitesimal; xa \<noteq> 0
|] ==> D = 0"
apply (simp add: nscderiv_def)
apply (drule bspec, auto)
done
lemma NSCDERIV_approx:
"[| NSCDERIV f x :> D; h: Infinitesimal; h \<noteq> 0 |]
==> ( *f* f) (hcomplex_of_complex(x) + h) - hcomplex_of_complex(f x) @= 0"
apply (simp add: nscderiv_def mem_infmal_iff [symmetric])
apply (rule Infinitesimal_ratio)
apply (rule_tac [3] approx_hcomplex_of_complex_HFinite, auto)
done
(*--------------------------------------------------*)
(* from one version of differentiability *)
(* *)
(* f(x) - f(a) *)
(* --------------- @= Db *)
(* x - a *)
(* -------------------------------------------------*)
lemma NSCDERIVD1:
"[| NSCDERIV f (g x) :> Da;
( *f* g) (hcomplex_of_complex(x) + xa) \<noteq> hcomplex_of_complex (g x);
( *f* g) (hcomplex_of_complex(x) + xa) @= hcomplex_of_complex (g x)|]
==> (( *f* f) (( *f* g) (hcomplex_of_complex(x) + xa))
- hcomplex_of_complex (f (g x))) /
(( *f* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex (g x))
@= hcomplex_of_complex (Da)"
by (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: Infinitesimal; xa \<noteq> 0 |]
==> (( *f* g) (hcomplex_of_complex x + xa) - hcomplex_of_complex(g x)) / xa
@= hcomplex_of_complex (Db)"
by (simp add: NSCDERIV_NSCLIM_iff NSCLIM_def mem_infmal_iff starfun_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_infmal_iff [symmetric])
apply safe
apply (frule_tac f = g in NSCDERIV_approx)
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
apply (case_tac "( *f* g) (hcomplex_of_complex (x) + xa) = hcomplex_of_complex (g x) ")
apply (drule_tac g = g in NSCDERIV_zero)
apply (auto simp add: divide_inverse)
apply (rule_tac z1 = "( *f* 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 approx_mult_hcomplex_of_complex)
apply (auto intro!: NSCDERIVD1 intro: approx_minus_iff [THEN iffD2]
simp add: diff_minus [symmetric] divide_inverse [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 divide_self del: divide_self_if)
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: 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 Infinitesimal_add_not_zero:
"[| h: Infinitesimal; 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 Infinitesimal_add_not_zero [where x=x])
apply (auto simp add: starfun_inverse_inverse diff_minus
simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])
apply (simp add: 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 approx_trans)
apply (rule inverse_add_Infinitesimal_approx2)
apply (auto dest!: hcomplex_of_complex_HFinite_diff_Infinitesimal
intro: HFinite_mult
simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
apply (rule Infinitesimal_HFinite_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 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: diff_minus)
apply (drule_tac f = g in CDERIV_inverse_fun)
apply (drule_tac [2] CDERIV_mult, assumption+)
apply (simp add: divide_inverse left_distrib power_inverse minus_mult_left
mult_ac
del: minus_mult_right [symmetric] minus_mult_left [symmetric]
complexpow_Suc)
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 (simp add: NSCDERIV_CDERIV_iff isNSContc_isContc_iff CARAT_CDERIV)
lemma CARAT_CDERIVD:
"(\<forall>z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l
==> NSCDERIV f x :> l"
by (auto simp add: NSCDERIV_iff2 isNSContc_def starfun_if_eq);
end