Theory CLim

theory CLim
imports CStar
(*  Title:      HOL/Nonstandard_Analysis/CLim.thy
    Author:     Jacques D. Fleuriot
    Copyright:  2001 University of Edinburgh
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

section ‹Limits, Continuity and Differentiation for Complex Functions›

theory CLim
  imports CStar
begin

(*not in simpset?*)
declare hypreal_epsilon_not_zero [simp]

(*??generalize*)
lemma lemma_complex_mult_inverse_squared [simp]: "x ≠ 0 ⟹ x * (inverse x)2 = inverse x"
  for x :: complex
  by (simp add: numeral_2_eq_2)

text ‹Changing the quantified variable. Install earlier?›
lemma all_shift: "(∀x::'a::comm_ring_1. P x) ⟷ (∀x. P (x - a))"
  apply auto
  apply (drule_tac x = "x + a" in spec)
  apply (simp add: add.assoc)
  done

lemma complex_add_minus_iff [simp]: "x + - a = 0 ⟷ x = a"
  for x a :: complex
  by (simp add: diff_eq_eq)

lemma complex_add_eq_0_iff [iff]: "x + y = 0 ⟷ y = - x"
  for x y :: complex
  apply auto
  apply (drule sym [THEN diff_eq_eq [THEN iffD2]])
  apply auto
  done


subsection ‹Limit of Complex to Complex Function›

lemma NSLIM_Re: "f ─a→NS L ⟹ (λx. Re (f x)) ─a→NS Re L"
  by (simp add: NSLIM_def starfunC_approx_Re_Im_iff hRe_hcomplex_of_complex)

lemma NSLIM_Im: "f ─a→NS L ⟹ (λx. Im (f x)) ─a→NS Im L"
  by (simp add: NSLIM_def starfunC_approx_Re_Im_iff hIm_hcomplex_of_complex)

lemma LIM_Re: "f ─a→ L ⟹ (λx. Re (f x)) ─a→ Re L"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: LIM_NSLIM_iff NSLIM_Re)

lemma LIM_Im: "f ─a→ L ⟹ (λx. Im (f x)) ─a→ Im L"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: LIM_NSLIM_iff NSLIM_Im)

lemma LIM_cnj: "f ─a→ L ⟹ (λx. cnj (f x)) ─a→ cnj L"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: LIM_eq complex_cnj_diff [symmetric] del: complex_cnj_diff)

lemma LIM_cnj_iff: "((λx. cnj (f x)) ─a→ cnj L) ⟷ f ─a→ L"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: LIM_eq complex_cnj_diff [symmetric] del: complex_cnj_diff)

lemma starfun_norm: "( *f* (λx. norm (f x))) = (λx. hnorm (( *f* f) x))"
  by transfer (rule refl)

lemma star_of_Re [simp]: "star_of (Re x) = hRe (star_of x)"
  by transfer (rule refl)

lemma star_of_Im [simp]: "star_of (Im x) = hIm (star_of x)"
  by transfer (rule refl)

text ‹Another equivalence result.›
lemma NSCLIM_NSCRLIM_iff: "f ─x→NS L ⟷ (λy. cmod (f y - L)) ─x→NS 0"
  by (simp add: NSLIM_def starfun_norm
      approx_approx_zero_iff [symmetric] approx_minus_iff [symmetric])

text ‹Much, much easier standard proof.›
lemma CLIM_CRLIM_iff: "f ─x→ L ⟷ (λy. cmod (f y - L)) ─x→ 0"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: LIM_eq)

text ‹So this is nicer nonstandard proof.›
lemma NSCLIM_NSCRLIM_iff2: "f ─x→NS L ⟷ (λy. cmod (f y - L)) ─x→NS 0"
  by (simp add: LIM_NSLIM_iff [symmetric] CLIM_CRLIM_iff)

lemma NSLIM_NSCRLIM_Re_Im_iff:
  "f ─a→NS L ⟷ (λx. Re (f x)) ─a→NS Re L ∧ (λx. Im (f x)) ─a→NS Im L"
  apply (auto intro: NSLIM_Re NSLIM_Im)
  apply (auto simp add: NSLIM_def starfun_Re starfun_Im)
  apply (auto dest!: spec)
  apply (simp add: hcomplex_approx_iff)
  done

lemma LIM_CRLIM_Re_Im_iff: "f ─a→ L ⟷ (λx. Re (f x)) ─a→ Re L ∧ (λx. Im (f x)) ─a→ Im L"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: LIM_NSLIM_iff NSLIM_NSCRLIM_Re_Im_iff)


subsection ‹Continuity›

lemma NSLIM_isContc_iff: "f ─a→NS f a ⟷ (λh. f (a + h)) ─0→NS f a"
  by (rule NSLIM_h_iff)


subsection ‹Functions from Complex to Reals›

lemma isNSContCR_cmod [simp]: "isNSCont cmod a"
  by (auto intro: approx_hnorm
      simp: starfunCR_cmod hcmod_hcomplex_of_complex [symmetric] isNSCont_def)

lemma isContCR_cmod [simp]: "isCont cmod a"
  by (simp add: isNSCont_isCont_iff [symmetric])

lemma isCont_Re: "isCont f a ⟹ isCont (λx. Re (f x)) a"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: isCont_def LIM_Re)

lemma isCont_Im: "isCont f a ⟹ isCont (λx. Im (f x)) a"
  for f :: "'a::real_normed_vector ⇒ complex"
  by (simp add: isCont_def LIM_Im)


subsection ‹Differentiation of Natural Number Powers›

lemma CDERIV_pow [simp]: "DERIV (λx. x ^ n) x :> complex_of_real (real n) * (x ^ (n - Suc 0))"
  apply (induct n)
   apply (drule_tac [2] DERIV_ident [THEN DERIV_mult])
   apply (auto simp add: distrib_right of_nat_Suc)
  apply (case_tac "n")
   apply (auto simp add: ac_simps)
  done

text ‹Nonstandard version.›
lemma NSCDERIV_pow: "NSDERIV (λx. x ^ n) x :> complex_of_real (real n) * (x ^ (n - 1))"
  by (metis CDERIV_pow NSDERIV_DERIV_iff One_nat_def)

text ‹Can't relax the premise @{term "x ≠ 0"}: it isn't continuous at zero.›
lemma NSCDERIV_inverse: "x ≠ 0 ⟹ NSDERIV (λx. inverse x) x :> - (inverse x)2"
  for x :: complex
  unfolding numeral_2_eq_2 by (rule NSDERIV_inverse)

lemma CDERIV_inverse: "x ≠ 0 ⟹ DERIV (λx. inverse x) x :> - (inverse x)2"
  for x :: complex
  unfolding numeral_2_eq_2 by (rule DERIV_inverse)


subsection ‹Derivative of Reciprocals (Function @{term inverse})›

lemma CDERIV_inverse_fun:
  "DERIV f x :> d ⟹ f x ≠ 0 ⟹ DERIV (λx. inverse (f x)) x :> - (d * inverse ((f x)2))"
  for x :: complex
  unfolding numeral_2_eq_2 by (rule DERIV_inverse_fun)

lemma NSCDERIV_inverse_fun:
  "NSDERIV f x :> d ⟹ f x ≠ 0 ⟹ NSDERIV (λx. inverse (f x)) x :> - (d * inverse ((f x)2))"
  for x :: complex
  unfolding numeral_2_eq_2 by (rule NSDERIV_inverse_fun)


subsection ‹Derivative of Quotient›

lemma CDERIV_quotient:
  "DERIV f x :> d ⟹ DERIV g x :> e ⟹ g(x) ≠ 0 ⟹
    DERIV (λy. f y / g y) x :> (d * g x - (e * f x)) / (g x)2"
  for x :: complex
  unfolding numeral_2_eq_2 by (rule DERIV_quotient)

lemma NSCDERIV_quotient:
  "NSDERIV f x :> d ⟹ NSDERIV g x :> e ⟹ g x ≠ (0::complex) ⟹
    NSDERIV (λy. f y / g y) x :> (d * g x - (e * f x)) / (g x)2"
  unfolding numeral_2_eq_2 by (rule NSDERIV_quotient)


subsection ‹Caratheodory Formulation of Derivative at a Point: Standard Proof›

lemma CARAT_CDERIVD:
  "(∀z. f z - f x = g z * (z - x)) ∧ isNSCont g x ∧ g x = l ⟹ NSDERIV f x :> l"
  by clarify (rule CARAT_DERIVD)

end