src/HOL/Complex/NSComplex.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 15169 2b5da07a0b89
permissions -rw-r--r--
import -> imports
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(*  Title:       NSComplex.thy
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    ID:      $Id$
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    Author:      Jacques D. Fleuriot
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    Copyright:   2001  University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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*)
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header{*Nonstandard Complex Numbers*}
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theory NSComplex
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imports Complex
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begin
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constdefs
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    hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
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    "hcomplexrel == {p. \<exists>X Y. p = ((X::nat=>complex),Y) &
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                        {n::nat. X(n) = Y(n)}: FreeUltrafilterNat}"
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typedef hcomplex = "{x::nat=>complex. True}//hcomplexrel"
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  by (auto simp add: quotient_def)
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instance hcomplex :: "{zero, one, plus, times, minus, inverse, power}" ..
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defs (overloaded)
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  hcomplex_zero_def:
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  "0 == Abs_hcomplex(hcomplexrel `` {%n. (0::complex)})"
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  hcomplex_one_def:
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  "1 == Abs_hcomplex(hcomplexrel `` {%n. (1::complex)})"
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  hcomplex_minus_def:
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  "- z == Abs_hcomplex(UN X: Rep_hcomplex(z).
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                       hcomplexrel `` {%n::nat. - (X n)})"
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  hcomplex_diff_def:
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  "w - z == w + -(z::hcomplex)"
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  hcinv_def:
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  "inverse(P) == Abs_hcomplex(UN X: Rep_hcomplex(P).
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                    hcomplexrel `` {%n. inverse(X n)})"
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constdefs
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  hcomplex_of_complex :: "complex => hcomplex"
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  "hcomplex_of_complex z == Abs_hcomplex(hcomplexrel `` {%n. z})"
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  (*--- real and Imaginary parts ---*)
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  hRe :: "hcomplex => hypreal"
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  "hRe(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Re (X n)})"
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  hIm :: "hcomplex => hypreal"
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  "hIm(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Im (X n)})"
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  (*----------- modulus ------------*)
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  hcmod :: "hcomplex => hypreal"
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  "hcmod z == Abs_hypreal(UN X: Rep_hcomplex(z).
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			  hyprel `` {%n. cmod (X n)})"
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  (*------ imaginary unit ----------*)
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  iii :: hcomplex
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  "iii == Abs_hcomplex(hcomplexrel `` {%n. ii})"
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  (*------- complex conjugate ------*)
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  hcnj :: "hcomplex => hcomplex"
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  "hcnj z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. cnj (X n)})"
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  (*------------ Argand -------------*)
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  hsgn :: "hcomplex => hcomplex"
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  "hsgn z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. sgn(X n)})"
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  harg :: "hcomplex => hypreal"
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  "harg z == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. arg(X n)})"
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  (* abbreviation for (cos a + i sin a) *)
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  hcis :: "hypreal => hcomplex"
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  "hcis a == Abs_hcomplex(UN X:Rep_hypreal(a). hcomplexrel `` {%n. cis (X n)})"
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  (*----- injection from hyperreals -----*)
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  hcomplex_of_hypreal :: "hypreal => hcomplex"
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  "hcomplex_of_hypreal r == Abs_hcomplex(UN X:Rep_hypreal(r).
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			       hcomplexrel `` {%n. complex_of_real (X n)})"
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  (* abbreviation for r*(cos a + i sin a) *)
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  hrcis :: "[hypreal, hypreal] => hcomplex"
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  "hrcis r a == hcomplex_of_hypreal r * hcis a"
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  (*------------ e ^ (x + iy) ------------*)
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  hexpi :: "hcomplex => hcomplex"
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  "hexpi z == hcomplex_of_hypreal(( *f* exp) (hRe z)) * hcis (hIm z)"
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constdefs
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  HComplex :: "[hypreal,hypreal] => hcomplex"
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   "HComplex x y == hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y"
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defs (overloaded)
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  (*----------- division ----------*)
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  hcomplex_divide_def:
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  "w / (z::hcomplex) == w * inverse z"
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  hcomplex_add_def:
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  "w + z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
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		      hcomplexrel `` {%n. X n + Y n})"
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  hcomplex_mult_def:
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  "w * z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
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		      hcomplexrel `` {%n. X n * Y n})"
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consts
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  "hcpow"  :: "[hcomplex,hypnat] => hcomplex"     (infixr 80)
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defs
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  (* hypernatural powers of nonstandard complex numbers *)
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  hcpow_def:
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  "(z::hcomplex) hcpow (n::hypnat)
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      == Abs_hcomplex(UN X:Rep_hcomplex(z). UN Y: Rep_hypnat(n).
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             hcomplexrel `` {%n. (X n) ^ (Y n)})"
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lemma hcomplexrel_refl: "(x,x): hcomplexrel"
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by (simp add: hcomplexrel_def)
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lemma hcomplexrel_sym: "(x,y): hcomplexrel ==> (y,x):hcomplexrel"
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by (auto simp add: hcomplexrel_def eq_commute)
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lemma hcomplexrel_trans:
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      "[|(x,y): hcomplexrel; (y,z):hcomplexrel|] ==> (x,z):hcomplexrel"
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by (simp add: hcomplexrel_def, ultra)
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lemma equiv_hcomplexrel: "equiv UNIV hcomplexrel"
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apply (simp add: equiv_def refl_def sym_def trans_def hcomplexrel_refl)
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apply (blast intro: hcomplexrel_sym hcomplexrel_trans)
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done
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lemmas equiv_hcomplexrel_iff =
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    eq_equiv_class_iff [OF equiv_hcomplexrel UNIV_I UNIV_I, simp]
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lemma hcomplexrel_in_hcomplex [simp]: "hcomplexrel``{x} : hcomplex"
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by (simp add: hcomplex_def hcomplexrel_def quotient_def, blast)
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lemma inj_on_Abs_hcomplex: "inj_on Abs_hcomplex hcomplex"
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apply (rule inj_on_inverseI)
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apply (erule Abs_hcomplex_inverse)
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done
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declare inj_on_Abs_hcomplex [THEN inj_on_iff, simp]
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        Abs_hcomplex_inverse [simp]
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declare equiv_hcomplexrel [THEN eq_equiv_class_iff, simp]
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lemma inj_Rep_hcomplex: "inj(Rep_hcomplex)"
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apply (rule inj_on_inverseI)
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apply (rule Rep_hcomplex_inverse)
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done
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lemma lemma_hcomplexrel_refl [simp]: "x: hcomplexrel `` {x}"
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by (simp add: hcomplexrel_def)
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lemma hcomplex_empty_not_mem [simp]: "{} \<notin> hcomplex"
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apply (simp add: hcomplex_def hcomplexrel_def)
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apply (auto elim!: quotientE)
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done
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lemma Rep_hcomplex_nonempty [simp]: "Rep_hcomplex x \<noteq> {}"
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by (cut_tac x = x in Rep_hcomplex, auto)
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lemma eq_Abs_hcomplex:
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    "(!!x. z = Abs_hcomplex(hcomplexrel `` {x}) ==> P) ==> P"
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apply (rule_tac x1=z in Rep_hcomplex [unfolded hcomplex_def, THEN quotientE])
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apply (drule_tac f = Abs_hcomplex in arg_cong)
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apply (force simp add: Rep_hcomplex_inverse hcomplexrel_def)
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done
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theorem hcomplex_cases [case_names Abs_hcomplex, cases type: hcomplex]:
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    "(!!x. z = Abs_hcomplex(hcomplexrel``{x}) ==> P) ==> P"
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by (rule eq_Abs_hcomplex [of z], blast)
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lemma hcomplexrel_iff [simp]:
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   "((X,Y): hcomplexrel) = ({n. X n = Y n}: FreeUltrafilterNat)"
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by (simp add: hcomplexrel_def)
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subsection{*Properties of Nonstandard Real and Imaginary Parts*}
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lemma hRe:
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     "hRe(Abs_hcomplex (hcomplexrel `` {X})) =
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      Abs_hypreal(hyprel `` {%n. Re(X n)})"
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apply (simp add: hRe_def)
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apply (rule_tac f = Abs_hypreal in arg_cong)
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apply (auto iff: hcomplexrel_iff, ultra)
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done
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lemma hIm:
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     "hIm(Abs_hcomplex (hcomplexrel `` {X})) =
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      Abs_hypreal(hyprel `` {%n. Im(X n)})"
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apply (simp add: hIm_def)
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apply (rule_tac f = Abs_hypreal in arg_cong)
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apply (auto iff: hcomplexrel_iff, ultra)
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done
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lemma hcomplex_hRe_hIm_cancel_iff:
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     "(w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
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apply (cases z, cases w)
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apply (auto simp add: hRe hIm complex_Re_Im_cancel_iff iff: hcomplexrel_iff)
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apply (ultra+)
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done
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lemma hcomplex_equality [intro?]: "hRe z = hRe w ==> hIm z = hIm w ==> z = w"
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by (simp add: hcomplex_hRe_hIm_cancel_iff) 
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lemma hcomplex_hRe_zero [simp]: "hRe 0 = 0"
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by (simp add: hcomplex_zero_def hRe hypreal_zero_num)
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lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
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by (simp add: hcomplex_zero_def hIm hypreal_zero_num)
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lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
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by (simp add: hcomplex_one_def hRe hypreal_one_num)
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lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
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by (simp add: hcomplex_one_def hIm hypreal_one_def hypreal_zero_num)
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subsection{*Addition for Nonstandard Complex Numbers*}
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lemma hcomplex_add_congruent2:
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    "congruent2 hcomplexrel hcomplexrel (%X Y. hcomplexrel `` {%n. X n + Y n})"
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by (auto simp add: congruent2_def iff: hcomplexrel_iff, ultra) 
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lemma hcomplex_add:
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  "Abs_hcomplex(hcomplexrel``{%n. X n}) + 
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   Abs_hcomplex(hcomplexrel``{%n. Y n}) =
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     Abs_hcomplex(hcomplexrel``{%n. X n + Y n})"
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apply (simp add: hcomplex_add_def)
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apply (rule_tac f = Abs_hcomplex in arg_cong)
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apply (auto simp add: iff: hcomplexrel_iff, ultra) 
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done
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lemma hcomplex_add_commute: "(z::hcomplex) + w = w + z"
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apply (cases z, cases w)
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apply (simp add: complex_add_commute hcomplex_add)
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done
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lemma hcomplex_add_assoc: "((z1::hcomplex) + z2) + z3 = z1 + (z2 + z3)"
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apply (cases z1, cases z2, cases z3)
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apply (simp add: hcomplex_add complex_add_assoc)
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done
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lemma hcomplex_add_zero_left: "(0::hcomplex) + z = z"
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apply (cases z)
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apply (simp add: hcomplex_zero_def hcomplex_add)
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done
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lemma hcomplex_add_zero_right: "z + (0::hcomplex) = z"
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by (simp add: hcomplex_add_zero_left hcomplex_add_commute)
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lemma hRe_add: "hRe(x + y) = hRe(x) + hRe(y)"
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apply (cases x, cases y)
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apply (simp add: hRe hcomplex_add hypreal_add complex_Re_add)
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done
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lemma hIm_add: "hIm(x + y) = hIm(x) + hIm(y)"
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apply (cases x, cases y)
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apply (simp add: hIm hcomplex_add hypreal_add complex_Im_add)
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done
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subsection{*Additive Inverse on Nonstandard Complex Numbers*}
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lemma hcomplex_minus_congruent:
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     "congruent hcomplexrel (%X. hcomplexrel `` {%n. - (X n)})"
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by (simp add: congruent_def)
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lemma hcomplex_minus:
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  "- (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
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      Abs_hcomplex(hcomplexrel `` {%n. -(X n)})"
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apply (simp add: hcomplex_minus_def)
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apply (rule_tac f = Abs_hcomplex in arg_cong)
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apply (auto iff: hcomplexrel_iff, ultra)
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done
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lemma hcomplex_add_minus_left: "-z + z = (0::hcomplex)"
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apply (cases z)
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apply (simp add: hcomplex_add hcomplex_minus hcomplex_zero_def)
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done
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subsection{*Multiplication for Nonstandard Complex Numbers*}
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lemma hcomplex_mult:
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  "Abs_hcomplex(hcomplexrel``{%n. X n}) *
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     Abs_hcomplex(hcomplexrel``{%n. Y n}) =
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     Abs_hcomplex(hcomplexrel``{%n. X n * Y n})"
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apply (simp add: hcomplex_mult_def)
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apply (rule_tac f = Abs_hcomplex in arg_cong)
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apply (auto iff: hcomplexrel_iff, ultra)
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done
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lemma hcomplex_mult_commute: "(w::hcomplex) * z = z * w"
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apply (cases w, cases z)
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apply (simp add: hcomplex_mult complex_mult_commute)
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done
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lemma hcomplex_mult_assoc: "((u::hcomplex) * v) * w = u * (v * w)"
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apply (cases u, cases v, cases w)
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apply (simp add: hcomplex_mult complex_mult_assoc)
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done
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lemma hcomplex_mult_one_left: "(1::hcomplex) * z = z"
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apply (cases z)
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apply (simp add: hcomplex_one_def hcomplex_mult)
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done
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lemma hcomplex_mult_zero_left: "(0::hcomplex) * z = 0"
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apply (cases z)
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apply (simp add: hcomplex_zero_def hcomplex_mult)
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done
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lemma hcomplex_add_mult_distrib:
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     "((z1::hcomplex) + z2) * w = (z1 * w) + (z2 * w)"
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apply (cases z1, cases z2, cases w)
paulson@14374
   337
apply (simp add: hcomplex_mult hcomplex_add left_distrib)
paulson@14314
   338
done
paulson@14314
   339
paulson@14354
   340
lemma hcomplex_zero_not_eq_one: "(0::hcomplex) \<noteq> (1::hcomplex)"
paulson@14374
   341
by (simp add: hcomplex_zero_def hcomplex_one_def)
paulson@14374
   342
paulson@14314
   343
declare hcomplex_zero_not_eq_one [THEN not_sym, simp]
paulson@14314
   344
paulson@14314
   345
paulson@14314
   346
subsection{*Inverse of Nonstandard Complex Number*}
paulson@14314
   347
paulson@14314
   348
lemma hcomplex_inverse:
paulson@14314
   349
  "inverse (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   350
      Abs_hcomplex(hcomplexrel `` {%n. inverse (X n)})"
paulson@14374
   351
apply (simp add: hcinv_def)
paulson@14374
   352
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   353
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   354
done
paulson@14314
   355
paulson@14314
   356
lemma hcomplex_mult_inv_left:
paulson@14354
   357
      "z \<noteq> (0::hcomplex) ==> inverse(z) * z = (1::hcomplex)"
paulson@14469
   358
apply (cases z)
paulson@14374
   359
apply (simp add: hcomplex_zero_def hcomplex_one_def hcomplex_inverse hcomplex_mult, ultra)
paulson@14314
   360
apply (rule ccontr)
paulson@14374
   361
apply (drule left_inverse, auto)
paulson@14314
   362
done
paulson@14314
   363
paulson@14318
   364
subsection {* The Field of Nonstandard Complex Numbers *}
paulson@14318
   365
paulson@14318
   366
instance hcomplex :: field
paulson@14318
   367
proof
paulson@14318
   368
  fix z u v w :: hcomplex
paulson@14318
   369
  show "(u + v) + w = u + (v + w)"
paulson@14318
   370
    by (simp add: hcomplex_add_assoc)
paulson@14318
   371
  show "z + w = w + z"
paulson@14318
   372
    by (simp add: hcomplex_add_commute)
paulson@14318
   373
  show "0 + z = z"
paulson@14335
   374
    by (simp add: hcomplex_add_zero_left)
paulson@14318
   375
  show "-z + z = 0"
paulson@14335
   376
    by (simp add: hcomplex_add_minus_left)
paulson@14318
   377
  show "z - w = z + -w"
paulson@14318
   378
    by (simp add: hcomplex_diff_def)
paulson@14318
   379
  show "(u * v) * w = u * (v * w)"
paulson@14318
   380
    by (simp add: hcomplex_mult_assoc)
paulson@14318
   381
  show "z * w = w * z"
paulson@14318
   382
    by (simp add: hcomplex_mult_commute)
paulson@14318
   383
  show "1 * z = z"
paulson@14335
   384
    by (simp add: hcomplex_mult_one_left)
paulson@14318
   385
  show "0 \<noteq> (1::hcomplex)"
paulson@14318
   386
    by (rule hcomplex_zero_not_eq_one)
paulson@14318
   387
  show "(u + v) * w = u * w + v * w"
paulson@14318
   388
    by (simp add: hcomplex_add_mult_distrib)
paulson@14430
   389
  show "z / w = z * inverse w"
paulson@14318
   390
    by (simp add: hcomplex_divide_def)
paulson@14430
   391
  assume "w \<noteq> 0"
paulson@14430
   392
  thus "inverse w * w = 1"
paulson@14318
   393
    by (rule hcomplex_mult_inv_left)
paulson@14318
   394
qed
paulson@14318
   395
paulson@14318
   396
instance hcomplex :: division_by_zero
paulson@14318
   397
proof
paulson@14430
   398
  show "inverse 0 = (0::hcomplex)"
paulson@14374
   399
    by (simp add: hcomplex_inverse hcomplex_zero_def)
paulson@14318
   400
qed
paulson@14314
   401
paulson@14374
   402
paulson@14318
   403
subsection{*More Minus Laws*}
paulson@14318
   404
paulson@14318
   405
lemma hRe_minus: "hRe(-z) = - hRe(z)"
paulson@14469
   406
apply (cases z)
paulson@14374
   407
apply (simp add: hRe hcomplex_minus hypreal_minus complex_Re_minus)
paulson@14318
   408
done
paulson@14318
   409
paulson@14318
   410
lemma hIm_minus: "hIm(-z) = - hIm(z)"
paulson@14469
   411
apply (cases z)
paulson@14374
   412
apply (simp add: hIm hcomplex_minus hypreal_minus complex_Im_minus)
paulson@14318
   413
done
paulson@14318
   414
paulson@14318
   415
lemma hcomplex_add_minus_eq_minus:
paulson@14318
   416
      "x + y = (0::hcomplex) ==> x = -y"
obua@14738
   417
apply (drule OrderedGroup.equals_zero_I)
paulson@14374
   418
apply (simp add: minus_equation_iff [of x y])
paulson@14318
   419
done
paulson@14318
   420
paulson@14377
   421
lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
paulson@14377
   422
by (simp add: iii_def hcomplex_mult hcomplex_one_def hcomplex_minus)
paulson@14377
   423
paulson@14377
   424
lemma hcomplex_i_mult_left [simp]: "iii * (iii * z) = -z"
paulson@14377
   425
by (simp add: mult_assoc [symmetric])
paulson@14377
   426
paulson@14377
   427
lemma hcomplex_i_not_zero [simp]: "iii \<noteq> 0"
paulson@14377
   428
by (simp add: iii_def hcomplex_zero_def)
paulson@14377
   429
paulson@14318
   430
paulson@14318
   431
subsection{*More Multiplication Laws*}
paulson@14318
   432
paulson@14318
   433
lemma hcomplex_mult_one_right: "z * (1::hcomplex) = z"
obua@14738
   434
by (rule OrderedGroup.mult_1_right)
paulson@14318
   435
paulson@14374
   436
lemma hcomplex_mult_minus_one [simp]: "- 1 * (z::hcomplex) = -z"
paulson@14374
   437
by simp
paulson@14318
   438
paulson@14374
   439
lemma hcomplex_mult_minus_one_right [simp]: "(z::hcomplex) * - 1 = -z"
paulson@14374
   440
by (subst hcomplex_mult_commute, simp)
paulson@14318
   441
paulson@14335
   442
lemma hcomplex_mult_left_cancel:
paulson@14354
   443
     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (c*a=c*b) = (a=b)"
paulson@14374
   444
by (simp add: field_mult_cancel_left)
paulson@14314
   445
paulson@14335
   446
lemma hcomplex_mult_right_cancel:
paulson@14354
   447
     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (a*c=b*c) = (a=b)"
paulson@14374
   448
by (simp add: Ring_and_Field.field_mult_cancel_right)
paulson@14314
   449
paulson@14314
   450
paulson@14318
   451
subsection{*Subraction and Division*}
paulson@14314
   452
paulson@14318
   453
lemma hcomplex_diff:
paulson@14318
   454
 "Abs_hcomplex(hcomplexrel``{%n. X n}) - Abs_hcomplex(hcomplexrel``{%n. Y n}) =
paulson@14318
   455
  Abs_hcomplex(hcomplexrel``{%n. X n - Y n})"
paulson@14374
   456
by (simp add: hcomplex_diff_def hcomplex_minus hcomplex_add complex_diff_def)
paulson@14314
   457
paulson@14374
   458
lemma hcomplex_diff_eq_eq [simp]: "((x::hcomplex) - y = z) = (x = z + y)"
obua@14738
   459
by (rule OrderedGroup.diff_eq_eq)
paulson@14314
   460
paulson@14314
   461
lemma hcomplex_add_divide_distrib: "(x+y)/(z::hcomplex) = x/z + y/z"
paulson@14374
   462
by (rule Ring_and_Field.add_divide_distrib)
paulson@14314
   463
paulson@14314
   464
paulson@14314
   465
subsection{*Embedding Properties for @{term hcomplex_of_hypreal} Map*}
paulson@14314
   466
paulson@14314
   467
lemma hcomplex_of_hypreal:
paulson@14314
   468
  "hcomplex_of_hypreal (Abs_hypreal(hyprel `` {%n. X n})) =
paulson@14314
   469
      Abs_hcomplex(hcomplexrel `` {%n. complex_of_real (X n)})"
paulson@14374
   470
apply (simp add: hcomplex_of_hypreal_def)
paulson@14377
   471
apply (rule_tac f = Abs_hcomplex in arg_cong, auto iff: hcomplexrel_iff, ultra)
paulson@14314
   472
done
paulson@14314
   473
paulson@14374
   474
lemma hcomplex_of_hypreal_cancel_iff [iff]:
paulson@14374
   475
     "(hcomplex_of_hypreal x = hcomplex_of_hypreal y) = (x = y)"
paulson@14469
   476
apply (cases x, cases y)
paulson@14374
   477
apply (simp add: hcomplex_of_hypreal)
paulson@14314
   478
done
paulson@14314
   479
paulson@14374
   480
lemma hcomplex_of_hypreal_one [simp]: "hcomplex_of_hypreal 1 = 1"
paulson@14374
   481
by (simp add: hcomplex_one_def hcomplex_of_hypreal hypreal_one_num)
paulson@14314
   482
paulson@14374
   483
lemma hcomplex_of_hypreal_zero [simp]: "hcomplex_of_hypreal 0 = 0"
paulson@14374
   484
by (simp add: hcomplex_zero_def hypreal_zero_def hcomplex_of_hypreal)
paulson@14374
   485
paulson@15013
   486
lemma hcomplex_of_hypreal_minus [simp]:
paulson@15013
   487
     "hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x"
paulson@15013
   488
apply (cases x)
paulson@15013
   489
apply (simp add: hcomplex_of_hypreal hcomplex_minus hypreal_minus)
paulson@15013
   490
done
paulson@15013
   491
paulson@15013
   492
lemma hcomplex_of_hypreal_inverse [simp]:
paulson@15013
   493
     "hcomplex_of_hypreal(inverse x) = inverse(hcomplex_of_hypreal x)"
paulson@15013
   494
apply (cases x)
paulson@15013
   495
apply (simp add: hcomplex_of_hypreal hypreal_inverse hcomplex_inverse)
paulson@15013
   496
done
paulson@15013
   497
paulson@15013
   498
lemma hcomplex_of_hypreal_add [simp]:
paulson@15013
   499
  "hcomplex_of_hypreal (x + y) = hcomplex_of_hypreal x + hcomplex_of_hypreal y"
paulson@15013
   500
apply (cases x, cases y)
paulson@15013
   501
apply (simp add: hcomplex_of_hypreal hypreal_add hcomplex_add)
paulson@15013
   502
done
paulson@15013
   503
paulson@15013
   504
lemma hcomplex_of_hypreal_diff [simp]:
paulson@15013
   505
     "hcomplex_of_hypreal (x - y) =
paulson@15013
   506
      hcomplex_of_hypreal x - hcomplex_of_hypreal y "
paulson@15013
   507
by (simp add: hcomplex_diff_def hypreal_diff_def)
paulson@15013
   508
paulson@15013
   509
lemma hcomplex_of_hypreal_mult [simp]:
paulson@15013
   510
  "hcomplex_of_hypreal (x * y) = hcomplex_of_hypreal x * hcomplex_of_hypreal y"
paulson@15013
   511
apply (cases x, cases y)
paulson@15013
   512
apply (simp add: hcomplex_of_hypreal hypreal_mult hcomplex_mult)
paulson@15013
   513
done
paulson@15013
   514
paulson@15013
   515
lemma hcomplex_of_hypreal_divide [simp]:
paulson@15013
   516
  "hcomplex_of_hypreal(x/y) = hcomplex_of_hypreal x / hcomplex_of_hypreal y"
paulson@15013
   517
apply (simp add: hcomplex_divide_def)
paulson@15013
   518
apply (case_tac "y=0", simp)
paulson@15013
   519
apply (simp add: hypreal_divide_def)
paulson@15013
   520
done
paulson@15013
   521
paulson@14374
   522
lemma hRe_hcomplex_of_hypreal [simp]: "hRe(hcomplex_of_hypreal z) = z"
paulson@14469
   523
apply (cases z)
paulson@14314
   524
apply (auto simp add: hcomplex_of_hypreal hRe)
paulson@14314
   525
done
paulson@14314
   526
paulson@14374
   527
lemma hIm_hcomplex_of_hypreal [simp]: "hIm(hcomplex_of_hypreal z) = 0"
paulson@14469
   528
apply (cases z)
paulson@14314
   529
apply (auto simp add: hcomplex_of_hypreal hIm hypreal_zero_num)
paulson@14314
   530
done
paulson@14314
   531
paulson@14374
   532
lemma hcomplex_of_hypreal_epsilon_not_zero [simp]:
paulson@14374
   533
     "hcomplex_of_hypreal epsilon \<noteq> 0"
paulson@14374
   534
by (auto simp add: hcomplex_of_hypreal epsilon_def hcomplex_zero_def)
paulson@14314
   535
paulson@14318
   536
paulson@14377
   537
subsection{*HComplex theorems*}
paulson@14377
   538
paulson@14377
   539
lemma hRe_HComplex [simp]: "hRe (HComplex x y) = x"
paulson@14469
   540
apply (cases x, cases y)
paulson@14377
   541
apply (simp add: HComplex_def hRe iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
paulson@14377
   542
done
paulson@14377
   543
paulson@14377
   544
lemma hIm_HComplex [simp]: "hIm (HComplex x y) = y"
paulson@14469
   545
apply (cases x, cases y)
paulson@14377
   546
apply (simp add: HComplex_def hIm iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
paulson@14377
   547
done
paulson@14377
   548
paulson@14377
   549
text{*Relates the two nonstandard constructions*}
paulson@14377
   550
lemma HComplex_eq_Abs_hcomplex_Complex:
paulson@14377
   551
     "HComplex (Abs_hypreal (hyprel `` {X})) (Abs_hypreal (hyprel `` {Y})) =
paulson@14377
   552
      Abs_hcomplex(hcomplexrel `` {%n::nat. Complex (X n) (Y n)})";
paulson@14377
   553
by (simp add: hcomplex_hRe_hIm_cancel_iff hRe hIm) 
paulson@14377
   554
paulson@14377
   555
lemma hcomplex_surj [simp]: "HComplex (hRe z) (hIm z) = z"
paulson@14377
   556
by (simp add: hcomplex_equality) 
paulson@14377
   557
paulson@14377
   558
lemma hcomplex_induct [case_names rect, induct type: hcomplex]:
paulson@14377
   559
     "(\<And>x y. P (HComplex x y)) ==> P z"
paulson@14377
   560
by (rule hcomplex_surj [THEN subst], blast)
paulson@14377
   561
paulson@14377
   562
paulson@14318
   563
subsection{*Modulus (Absolute Value) of Nonstandard Complex Number*}
paulson@14314
   564
paulson@14314
   565
lemma hcmod:
paulson@14314
   566
  "hcmod (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   567
      Abs_hypreal(hyprel `` {%n. cmod (X n)})"
paulson@14314
   568
paulson@14374
   569
apply (simp add: hcmod_def)
paulson@14374
   570
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14377
   571
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   572
done
paulson@14314
   573
paulson@14374
   574
lemma hcmod_zero [simp]: "hcmod(0) = 0"
paulson@14377
   575
by (simp add: hcomplex_zero_def hypreal_zero_def hcmod)
paulson@14314
   576
paulson@14374
   577
lemma hcmod_one [simp]: "hcmod(1) = 1"
paulson@14374
   578
by (simp add: hcomplex_one_def hcmod hypreal_one_num)
paulson@14314
   579
paulson@14374
   580
lemma hcmod_hcomplex_of_hypreal [simp]: "hcmod(hcomplex_of_hypreal x) = abs x"
paulson@14469
   581
apply (cases x)
paulson@14314
   582
apply (auto simp add: hcmod hcomplex_of_hypreal hypreal_hrabs)
paulson@14314
   583
done
paulson@14314
   584
paulson@14335
   585
lemma hcomplex_of_hypreal_abs:
paulson@14335
   586
     "hcomplex_of_hypreal (abs x) =
paulson@14314
   587
      hcomplex_of_hypreal(hcmod(hcomplex_of_hypreal x))"
paulson@14374
   588
by simp
paulson@14314
   589
paulson@14377
   590
lemma HComplex_inject [simp]: "HComplex x y = HComplex x' y' = (x=x' & y=y')"
paulson@14377
   591
apply (rule iffI) 
paulson@14377
   592
 prefer 2 apply simp 
paulson@14377
   593
apply (simp add: HComplex_def iii_def) 
paulson@14469
   594
apply (cases x, cases y, cases x', cases y')
paulson@14377
   595
apply (auto simp add: iii_def hcomplex_mult hcomplex_add hcomplex_of_hypreal)
paulson@14377
   596
apply (ultra+) 
paulson@14377
   597
done
paulson@14377
   598
paulson@14377
   599
lemma HComplex_add [simp]:
paulson@14377
   600
     "HComplex x1 y1 + HComplex x2 y2 = HComplex (x1+x2) (y1+y2)"
paulson@15013
   601
by (simp add: HComplex_def add_ac right_distrib) 
paulson@14377
   602
paulson@14377
   603
lemma HComplex_minus [simp]: "- HComplex x y = HComplex (-x) (-y)"
paulson@14377
   604
by (simp add: HComplex_def hcomplex_of_hypreal_minus) 
paulson@14377
   605
paulson@14377
   606
lemma HComplex_diff [simp]:
paulson@14377
   607
     "HComplex x1 y1 - HComplex x2 y2 = HComplex (x1-x2) (y1-y2)"
paulson@14377
   608
by (simp add: diff_minus)
paulson@14377
   609
paulson@14377
   610
lemma HComplex_mult [simp]:
paulson@14377
   611
  "HComplex x1 y1 * HComplex x2 y2 = HComplex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
paulson@14377
   612
by (simp add: HComplex_def diff_minus hcomplex_of_hypreal_minus 
paulson@14377
   613
       add_ac mult_ac right_distrib)
paulson@14377
   614
paulson@14377
   615
(*HComplex_inverse is proved below*)
paulson@14377
   616
paulson@14377
   617
lemma hcomplex_of_hypreal_eq: "hcomplex_of_hypreal r = HComplex r 0"
paulson@14377
   618
by (simp add: HComplex_def)
paulson@14377
   619
paulson@14377
   620
lemma HComplex_add_hcomplex_of_hypreal [simp]:
paulson@14377
   621
     "HComplex x y + hcomplex_of_hypreal r = HComplex (x+r) y"
paulson@14377
   622
by (simp add: hcomplex_of_hypreal_eq)
paulson@14377
   623
paulson@14377
   624
lemma hcomplex_of_hypreal_add_HComplex [simp]:
paulson@14377
   625
     "hcomplex_of_hypreal r + HComplex x y = HComplex (r+x) y"
paulson@14377
   626
by (simp add: i_def hcomplex_of_hypreal_eq)
paulson@14377
   627
paulson@14377
   628
lemma HComplex_mult_hcomplex_of_hypreal:
paulson@14377
   629
     "HComplex x y * hcomplex_of_hypreal r = HComplex (x*r) (y*r)"
paulson@14377
   630
by (simp add: hcomplex_of_hypreal_eq)
paulson@14377
   631
paulson@14377
   632
lemma hcomplex_of_hypreal_mult_HComplex:
paulson@14377
   633
     "hcomplex_of_hypreal r * HComplex x y = HComplex (r*x) (r*y)"
paulson@14377
   634
by (simp add: i_def hcomplex_of_hypreal_eq)
paulson@14377
   635
paulson@14377
   636
lemma i_hcomplex_of_hypreal [simp]:
paulson@14377
   637
     "iii * hcomplex_of_hypreal r = HComplex 0 r"
paulson@14377
   638
by (simp add: HComplex_def)
paulson@14377
   639
paulson@14377
   640
lemma hcomplex_of_hypreal_i [simp]:
paulson@14377
   641
     "hcomplex_of_hypreal r * iii = HComplex 0 r"
paulson@14377
   642
by (simp add: mult_commute) 
paulson@14377
   643
paulson@14314
   644
paulson@14314
   645
subsection{*Conjugation*}
paulson@14314
   646
paulson@14314
   647
lemma hcnj:
paulson@14314
   648
  "hcnj (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14318
   649
   Abs_hcomplex(hcomplexrel `` {%n. cnj(X n)})"
paulson@14374
   650
apply (simp add: hcnj_def)
paulson@14374
   651
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   652
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   653
done
paulson@14314
   654
paulson@14374
   655
lemma hcomplex_hcnj_cancel_iff [iff]: "(hcnj x = hcnj y) = (x = y)"
paulson@14469
   656
apply (cases x, cases y)
paulson@14374
   657
apply (simp add: hcnj)
paulson@14374
   658
done
paulson@14374
   659
paulson@14374
   660
lemma hcomplex_hcnj_hcnj [simp]: "hcnj (hcnj z) = z"
paulson@14469
   661
apply (cases z)
paulson@14374
   662
apply (simp add: hcnj)
paulson@14314
   663
done
paulson@14314
   664
paulson@14374
   665
lemma hcomplex_hcnj_hcomplex_of_hypreal [simp]:
paulson@14374
   666
     "hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
paulson@14469
   667
apply (cases x)
paulson@14374
   668
apply (simp add: hcnj hcomplex_of_hypreal)
paulson@14314
   669
done
paulson@14314
   670
paulson@14374
   671
lemma hcomplex_hmod_hcnj [simp]: "hcmod (hcnj z) = hcmod z"
paulson@14469
   672
apply (cases z)
paulson@14374
   673
apply (simp add: hcnj hcmod)
paulson@14314
   674
done
paulson@14314
   675
paulson@14314
   676
lemma hcomplex_hcnj_minus: "hcnj (-z) = - hcnj z"
paulson@14469
   677
apply (cases z)
paulson@14374
   678
apply (simp add: hcnj hcomplex_minus complex_cnj_minus)
paulson@14314
   679
done
paulson@14314
   680
paulson@14314
   681
lemma hcomplex_hcnj_inverse: "hcnj(inverse z) = inverse(hcnj z)"
paulson@14469
   682
apply (cases z)
paulson@14374
   683
apply (simp add: hcnj hcomplex_inverse complex_cnj_inverse)
paulson@14314
   684
done
paulson@14314
   685
paulson@14314
   686
lemma hcomplex_hcnj_add: "hcnj(w + z) = hcnj(w) + hcnj(z)"
paulson@14469
   687
apply (cases z, cases w)
paulson@14374
   688
apply (simp add: hcnj hcomplex_add complex_cnj_add)
paulson@14314
   689
done
paulson@14314
   690
paulson@14314
   691
lemma hcomplex_hcnj_diff: "hcnj(w - z) = hcnj(w) - hcnj(z)"
paulson@14469
   692
apply (cases z, cases w)
paulson@14374
   693
apply (simp add: hcnj hcomplex_diff complex_cnj_diff)
paulson@14314
   694
done
paulson@14314
   695
paulson@14314
   696
lemma hcomplex_hcnj_mult: "hcnj(w * z) = hcnj(w) * hcnj(z)"
paulson@14469
   697
apply (cases z, cases w)
paulson@14374
   698
apply (simp add: hcnj hcomplex_mult complex_cnj_mult)
paulson@14314
   699
done
paulson@14314
   700
paulson@14314
   701
lemma hcomplex_hcnj_divide: "hcnj(w / z) = (hcnj w)/(hcnj z)"
paulson@14374
   702
by (simp add: hcomplex_divide_def hcomplex_hcnj_mult hcomplex_hcnj_inverse)
paulson@14314
   703
paulson@14374
   704
lemma hcnj_one [simp]: "hcnj 1 = 1"
paulson@14374
   705
by (simp add: hcomplex_one_def hcnj)
paulson@14314
   706
paulson@14374
   707
lemma hcomplex_hcnj_zero [simp]: "hcnj 0 = 0"
paulson@14374
   708
by (simp add: hcomplex_zero_def hcnj)
paulson@14374
   709
paulson@14374
   710
lemma hcomplex_hcnj_zero_iff [iff]: "(hcnj z = 0) = (z = 0)"
paulson@14469
   711
apply (cases z)
paulson@14374
   712
apply (simp add: hcomplex_zero_def hcnj)
paulson@14314
   713
done
paulson@14314
   714
paulson@14335
   715
lemma hcomplex_mult_hcnj:
paulson@14335
   716
     "z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
paulson@14469
   717
apply (cases z)
paulson@14374
   718
apply (simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add
paulson@14374
   719
                      hypreal_mult complex_mult_cnj numeral_2_eq_2)
paulson@14314
   720
done
paulson@14314
   721
paulson@14314
   722
paulson@14354
   723
subsection{*More Theorems about the Function @{term hcmod}*}
paulson@14314
   724
paulson@14374
   725
lemma hcomplex_hcmod_eq_zero_cancel [simp]: "(hcmod x = 0) = (x = 0)"
paulson@14469
   726
apply (cases x)
paulson@14374
   727
apply (simp add: hcmod hcomplex_zero_def hypreal_zero_num)
paulson@14314
   728
done
paulson@14314
   729
paulson@14374
   730
lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
paulson@14335
   731
     "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n"
paulson@14374
   732
apply (simp add: abs_if linorder_not_less)
paulson@14314
   733
done
paulson@14314
   734
paulson@14374
   735
lemma hcmod_hcomplex_of_hypreal_of_hypnat [simp]:
paulson@14335
   736
     "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
paulson@14374
   737
apply (simp add: abs_if linorder_not_less)
paulson@14314
   738
done
paulson@14314
   739
paulson@14374
   740
lemma hcmod_minus [simp]: "hcmod (-x) = hcmod(x)"
paulson@14469
   741
apply (cases x)
paulson@14374
   742
apply (simp add: hcmod hcomplex_minus)
paulson@14314
   743
done
paulson@14314
   744
paulson@14314
   745
lemma hcmod_mult_hcnj: "hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
paulson@14469
   746
apply (cases z)
paulson@14374
   747
apply (simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj numeral_2_eq_2)
paulson@14314
   748
done
paulson@14314
   749
paulson@14374
   750
lemma hcmod_ge_zero [simp]: "(0::hypreal) \<le> hcmod x"
paulson@14469
   751
apply (cases x)
paulson@14374
   752
apply (simp add: hcmod hypreal_zero_num hypreal_le)
paulson@14314
   753
done
paulson@14314
   754
paulson@14374
   755
lemma hrabs_hcmod_cancel [simp]: "abs(hcmod x) = hcmod x"
paulson@14374
   756
by (simp add: abs_if linorder_not_less)
paulson@14314
   757
paulson@14314
   758
lemma hcmod_mult: "hcmod(x*y) = hcmod(x) * hcmod(y)"
paulson@14469
   759
apply (cases x, cases y)
paulson@14374
   760
apply (simp add: hcmod hcomplex_mult hypreal_mult complex_mod_mult)
paulson@14314
   761
done
paulson@14314
   762
paulson@14314
   763
lemma hcmod_add_squared_eq:
paulson@14314
   764
     "hcmod(x + y) ^ 2 = hcmod(x) ^ 2 + hcmod(y) ^ 2 + 2 * hRe(x * hcnj y)"
paulson@14469
   765
apply (cases x, cases y)
paulson@14374
   766
apply (simp add: hcmod hcomplex_add hypreal_mult hRe hcnj hcomplex_mult
paulson@14374
   767
                      numeral_2_eq_2 realpow_two [symmetric]
paulson@14374
   768
                  del: realpow_Suc)
paulson@14374
   769
apply (simp add: numeral_2_eq_2 [symmetric] complex_mod_add_squared_eq
paulson@14374
   770
                 hypreal_add [symmetric] hypreal_mult [symmetric]
paulson@14314
   771
                 hypreal_of_real_def [symmetric])
paulson@14314
   772
done
paulson@14314
   773
paulson@14374
   774
lemma hcomplex_hRe_mult_hcnj_le_hcmod [simp]: "hRe(x * hcnj y) \<le> hcmod(x * hcnj y)"
paulson@14469
   775
apply (cases x, cases y)
paulson@14374
   776
apply (simp add: hcmod hcnj hcomplex_mult hRe hypreal_le)
paulson@14314
   777
done
paulson@14314
   778
paulson@14374
   779
lemma hcomplex_hRe_mult_hcnj_le_hcmod2 [simp]: "hRe(x * hcnj y) \<le> hcmod(x * y)"
paulson@14374
   780
apply (cut_tac x = x and y = y in hcomplex_hRe_mult_hcnj_le_hcmod)
paulson@14314
   781
apply (simp add: hcmod_mult)
paulson@14314
   782
done
paulson@14314
   783
paulson@14374
   784
lemma hcmod_triangle_squared [simp]: "hcmod (x + y) ^ 2 \<le> (hcmod(x) + hcmod(y)) ^ 2"
paulson@14469
   785
apply (cases x, cases y)
paulson@14374
   786
apply (simp add: hcmod hcnj hcomplex_add hypreal_mult hypreal_add
paulson@14323
   787
                      hypreal_le realpow_two [symmetric] numeral_2_eq_2
paulson@14374
   788
            del: realpow_Suc)
paulson@14374
   789
apply (simp add: numeral_2_eq_2 [symmetric])
paulson@14314
   790
done
paulson@14314
   791
paulson@14374
   792
lemma hcmod_triangle_ineq [simp]: "hcmod (x + y) \<le> hcmod(x) + hcmod(y)"
paulson@14469
   793
apply (cases x, cases y)
paulson@14374
   794
apply (simp add: hcmod hcomplex_add hypreal_add hypreal_le)
paulson@14314
   795
done
paulson@14314
   796
paulson@14374
   797
lemma hcmod_triangle_ineq2 [simp]: "hcmod(b + a) - hcmod b \<le> hcmod a"
paulson@14374
   798
apply (cut_tac x1 = b and y1 = a and c = "-hcmod b" in hcmod_triangle_ineq [THEN add_right_mono])
paulson@14331
   799
apply (simp add: add_ac)
paulson@14314
   800
done
paulson@14314
   801
paulson@14314
   802
lemma hcmod_diff_commute: "hcmod (x - y) = hcmod (y - x)"
paulson@14469
   803
apply (cases x, cases y)
paulson@14374
   804
apply (simp add: hcmod hcomplex_diff complex_mod_diff_commute)
paulson@14314
   805
done
paulson@14314
   806
paulson@14335
   807
lemma hcmod_add_less:
paulson@14335
   808
     "[| hcmod x < r; hcmod y < s |] ==> hcmod (x + y) < r + s"
paulson@14469
   809
apply (cases x, cases y, cases r, cases s)
paulson@14374
   810
apply (simp add: hcmod hcomplex_add hypreal_add hypreal_less, ultra)
paulson@14314
   811
apply (auto intro: complex_mod_add_less)
paulson@14314
   812
done
paulson@14314
   813
paulson@14335
   814
lemma hcmod_mult_less:
paulson@14335
   815
     "[| hcmod x < r; hcmod y < s |] ==> hcmod (x * y) < r * s"
paulson@14469
   816
apply (cases x, cases y, cases r, cases s)
paulson@14374
   817
apply (simp add: hcmod hypreal_mult hypreal_less hcomplex_mult, ultra)
paulson@14314
   818
apply (auto intro: complex_mod_mult_less)
paulson@14314
   819
done
paulson@14314
   820
paulson@14374
   821
lemma hcmod_diff_ineq [simp]: "hcmod(a) - hcmod(b) \<le> hcmod(a + b)"
paulson@14469
   822
apply (cases a, cases b)
paulson@14374
   823
apply (simp add: hcmod hcomplex_add hypreal_diff hypreal_le)
paulson@14314
   824
done
paulson@14314
   825
paulson@14314
   826
paulson@14314
   827
subsection{*A Few Nonlinear Theorems*}
paulson@14314
   828
paulson@14314
   829
lemma hcpow:
paulson@14314
   830
  "Abs_hcomplex(hcomplexrel``{%n. X n}) hcpow
paulson@14314
   831
   Abs_hypnat(hypnatrel``{%n. Y n}) =
paulson@14314
   832
   Abs_hcomplex(hcomplexrel``{%n. X n ^ Y n})"
paulson@14374
   833
apply (simp add: hcpow_def)
paulson@14374
   834
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   835
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   836
done
paulson@14314
   837
paulson@14335
   838
lemma hcomplex_of_hypreal_hyperpow:
paulson@14335
   839
     "hcomplex_of_hypreal (x pow n) = (hcomplex_of_hypreal x) hcpow n"
paulson@14469
   840
apply (cases x, cases n)
paulson@14374
   841
apply (simp add: hcomplex_of_hypreal hyperpow hcpow complex_of_real_pow)
paulson@14314
   842
done
paulson@14314
   843
paulson@14314
   844
lemma hcmod_hcpow: "hcmod(x hcpow n) = hcmod(x) pow n"
paulson@14469
   845
apply (cases x, cases n)
paulson@14374
   846
apply (simp add: hcpow hyperpow hcmod complex_mod_complexpow)
paulson@14314
   847
done
paulson@14314
   848
paulson@14314
   849
lemma hcmod_hcomplex_inverse: "hcmod(inverse x) = inverse(hcmod x)"
paulson@14374
   850
apply (case_tac "x = 0", simp)
paulson@14314
   851
apply (rule_tac c1 = "hcmod x" in hypreal_mult_left_cancel [THEN iffD1])
paulson@14314
   852
apply (auto simp add: hcmod_mult [symmetric])
paulson@14314
   853
done
paulson@14314
   854
paulson@14374
   855
lemma hcmod_divide: "hcmod(x/y) = hcmod(x)/(hcmod y)"
paulson@14374
   856
by (simp add: hcomplex_divide_def hypreal_divide_def hcmod_mult hcmod_hcomplex_inverse)
paulson@14314
   857
paulson@14354
   858
paulson@14354
   859
subsection{*Exponentiation*}
paulson@14354
   860
paulson@14354
   861
primrec
paulson@14354
   862
     hcomplexpow_0:   "z ^ 0       = 1"
paulson@14354
   863
     hcomplexpow_Suc: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
paulson@14354
   864
paulson@15003
   865
instance hcomplex :: recpower
paulson@14354
   866
proof
paulson@14354
   867
  fix z :: hcomplex
paulson@14354
   868
  fix n :: nat
paulson@14354
   869
  show "z^0 = 1" by simp
paulson@14354
   870
  show "z^(Suc n) = z * (z^n)" by simp
paulson@14354
   871
qed
paulson@14354
   872
paulson@14377
   873
lemma hcomplexpow_i_squared [simp]: "iii ^ 2 = - 1"
paulson@14377
   874
by (simp add: power2_eq_square)
paulson@14377
   875
paulson@14354
   876
paulson@14354
   877
lemma hcomplex_of_hypreal_pow:
paulson@14354
   878
     "hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n"
paulson@14354
   879
apply (induct_tac "n")
paulson@14354
   880
apply (auto simp add: hcomplex_of_hypreal_mult [symmetric])
paulson@14354
   881
done
paulson@14354
   882
paulson@14354
   883
lemma hcomplex_hcnj_pow: "hcnj(z ^ n) = hcnj(z) ^ n"
paulson@14314
   884
apply (induct_tac "n")
paulson@14354
   885
apply (auto simp add: hcomplex_hcnj_mult)
paulson@14354
   886
done
paulson@14354
   887
paulson@14354
   888
lemma hcmod_hcomplexpow: "hcmod(x ^ n) = hcmod(x) ^ n"
paulson@14354
   889
apply (induct_tac "n")
paulson@14354
   890
apply (auto simp add: hcmod_mult)
paulson@14354
   891
done
paulson@14354
   892
paulson@14354
   893
lemma hcpow_minus:
paulson@14354
   894
     "(-x::hcomplex) hcpow n =
paulson@14354
   895
      (if ( *pNat* even) n then (x hcpow n) else -(x hcpow n))"
paulson@14469
   896
apply (cases x, cases n)
paulson@14374
   897
apply (auto simp add: hcpow hyperpow starPNat hcomplex_minus, ultra)
paulson@14443
   898
apply (auto simp add: neg_power_if, ultra)
paulson@14314
   899
done
paulson@14314
   900
paulson@14314
   901
lemma hcpow_mult: "((r::hcomplex) * s) hcpow n = (r hcpow n) * (s hcpow n)"
paulson@14469
   902
apply (cases r, cases s, cases n)
paulson@14374
   903
apply (simp add: hcpow hypreal_mult hcomplex_mult power_mult_distrib)
paulson@14314
   904
done
paulson@14314
   905
paulson@14354
   906
lemma hcpow_zero [simp]: "0 hcpow (n + 1) = 0"
paulson@14469
   907
apply (simp add: hcomplex_zero_def hypnat_one_def, cases n)
paulson@14374
   908
apply (simp add: hcpow hypnat_add)
paulson@14314
   909
done
paulson@14314
   910
paulson@14354
   911
lemma hcpow_zero2 [simp]: "0 hcpow (hSuc n) = 0"
paulson@14374
   912
by (simp add: hSuc_def)
paulson@14314
   913
paulson@14354
   914
lemma hcpow_not_zero [simp,intro]: "r \<noteq> 0 ==> r hcpow n \<noteq> (0::hcomplex)"
paulson@14469
   915
apply (cases r, cases n)
paulson@14374
   916
apply (auto simp add: hcpow hcomplex_zero_def, ultra)
paulson@14314
   917
done
paulson@14314
   918
paulson@14314
   919
lemma hcpow_zero_zero: "r hcpow n = (0::hcomplex) ==> r = 0"
paulson@14374
   920
by (blast intro: ccontr dest: hcpow_not_zero)
paulson@14314
   921
paulson@14314
   922
lemma hcomplex_divide:
paulson@14314
   923
  "Abs_hcomplex(hcomplexrel``{%n. X n}) / Abs_hcomplex(hcomplexrel``{%n. Y n}) =
paulson@14314
   924
   Abs_hcomplex(hcomplexrel``{%n. X n / Y n})"
paulson@14374
   925
by (simp add: hcomplex_divide_def complex_divide_def hcomplex_inverse hcomplex_mult)
paulson@14374
   926
paulson@14314
   927
paulson@14314
   928
paulson@14377
   929
paulson@14314
   930
subsection{*The Function @{term hsgn}*}
paulson@14314
   931
paulson@14314
   932
lemma hsgn:
paulson@14314
   933
  "hsgn (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   934
      Abs_hcomplex(hcomplexrel `` {%n. sgn (X n)})"
paulson@14374
   935
apply (simp add: hsgn_def)
paulson@14374
   936
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   937
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   938
done
paulson@14314
   939
paulson@14374
   940
lemma hsgn_zero [simp]: "hsgn 0 = 0"
paulson@14374
   941
by (simp add: hcomplex_zero_def hsgn)
paulson@14314
   942
paulson@14374
   943
lemma hsgn_one [simp]: "hsgn 1 = 1"
paulson@14374
   944
by (simp add: hcomplex_one_def hsgn)
paulson@14314
   945
paulson@14314
   946
lemma hsgn_minus: "hsgn (-z) = - hsgn(z)"
paulson@14469
   947
apply (cases z)
paulson@14374
   948
apply (simp add: hsgn hcomplex_minus sgn_minus)
paulson@14314
   949
done
paulson@14314
   950
paulson@14314
   951
lemma hsgn_eq: "hsgn z = z / hcomplex_of_hypreal (hcmod z)"
paulson@14469
   952
apply (cases z)
paulson@14374
   953
apply (simp add: hsgn hcomplex_divide hcomplex_of_hypreal hcmod sgn_eq)
paulson@14314
   954
done
paulson@14314
   955
paulson@14314
   956
paulson@14377
   957
lemma hcmod_i: "hcmod (HComplex x y) = ( *f* sqrt) (x ^ 2 + y ^ 2)"
paulson@14469
   958
apply (cases x, cases y) 
paulson@14377
   959
apply (simp add: HComplex_eq_Abs_hcomplex_Complex starfun 
paulson@14377
   960
                 hypreal_mult hypreal_add hcmod numeral_2_eq_2)
paulson@14314
   961
done
paulson@14314
   962
paulson@14377
   963
lemma hcomplex_eq_cancel_iff1 [simp]:
paulson@14377
   964
     "(hcomplex_of_hypreal xa = HComplex x y) = (xa = x & y = 0)"
paulson@14377
   965
by (simp add: hcomplex_of_hypreal_eq)
paulson@14314
   966
paulson@14374
   967
lemma hcomplex_eq_cancel_iff2 [simp]:
paulson@14377
   968
     "(HComplex x y = hcomplex_of_hypreal xa) = (x = xa & y = 0)"
paulson@14377
   969
by (simp add: hcomplex_of_hypreal_eq)
paulson@14314
   970
paulson@14377
   971
lemma HComplex_eq_0 [simp]: "(HComplex x y = 0) = (x = 0 & y = 0)"
paulson@14377
   972
by (insert hcomplex_eq_cancel_iff2 [of _ _ 0], simp)
paulson@14314
   973
paulson@14377
   974
lemma HComplex_eq_1 [simp]: "(HComplex x y = 1) = (x = 1 & y = 0)"
paulson@14377
   975
by (insert hcomplex_eq_cancel_iff2 [of _ _ 1], simp)
paulson@14314
   976
paulson@14377
   977
lemma i_eq_HComplex_0_1: "iii = HComplex 0 1"
paulson@14377
   978
by (insert hcomplex_of_hypreal_i [of 1], simp)
paulson@14314
   979
paulson@14377
   980
lemma HComplex_eq_i [simp]: "(HComplex x y = iii) = (x = 0 & y = 1)"
paulson@14377
   981
by (simp add: i_eq_HComplex_0_1) 
paulson@14314
   982
paulson@14374
   983
lemma hRe_hsgn [simp]: "hRe(hsgn z) = hRe(z)/hcmod z"
paulson@14469
   984
apply (cases z)
paulson@14374
   985
apply (simp add: hsgn hcmod hRe hypreal_divide)
paulson@14314
   986
done
paulson@14314
   987
paulson@14374
   988
lemma hIm_hsgn [simp]: "hIm(hsgn z) = hIm(z)/hcmod z"
paulson@14469
   989
apply (cases z)
paulson@14374
   990
apply (simp add: hsgn hcmod hIm hypreal_divide)
paulson@14314
   991
done
paulson@14314
   992
paulson@15085
   993
(*????move to RealDef????*)
paulson@14374
   994
lemma real_two_squares_add_zero_iff [simp]: "(x*x + y*y = 0) = ((x::real) = 0 & y = 0)"
paulson@15085
   995
by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)
paulson@14314
   996
paulson@14335
   997
lemma hcomplex_inverse_complex_split:
paulson@14335
   998
     "inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) =
paulson@14314
   999
      hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) -
paulson@14314
  1000
      iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
paulson@14469
  1001
apply (cases x, cases y)
paulson@15013
  1002
apply (simp add: hcomplex_of_hypreal hcomplex_mult hcomplex_add iii_def
paulson@15013
  1003
         starfun hypreal_mult hypreal_add hcomplex_inverse hypreal_divide
paulson@15013
  1004
         hcomplex_diff numeral_2_eq_2 complex_of_real_def i_def)
paulson@14377
  1005
apply (simp add: diff_minus) 
paulson@14374
  1006
done
paulson@14374
  1007
paulson@14377
  1008
lemma HComplex_inverse:
paulson@14377
  1009
     "inverse (HComplex x y) =
paulson@14377
  1010
      HComplex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
paulson@14377
  1011
by (simp only: HComplex_def hcomplex_inverse_complex_split, simp)
paulson@14377
  1012
paulson@14377
  1013
paulson@14377
  1014
paulson@14374
  1015
lemma hRe_mult_i_eq[simp]:
paulson@14374
  1016
    "hRe (iii * hcomplex_of_hypreal y) = 0"
paulson@14469
  1017
apply (simp add: iii_def, cases y)
paulson@14374
  1018
apply (simp add: hcomplex_of_hypreal hcomplex_mult hRe hypreal_zero_num)
paulson@14314
  1019
done
paulson@14314
  1020
paulson@14374
  1021
lemma hIm_mult_i_eq [simp]:
paulson@14314
  1022
    "hIm (iii * hcomplex_of_hypreal y) = y"
paulson@14469
  1023
apply (simp add: iii_def, cases y)
paulson@14374
  1024
apply (simp add: hcomplex_of_hypreal hcomplex_mult hIm hypreal_zero_num)
paulson@14314
  1025
done
paulson@14314
  1026
paulson@14374
  1027
lemma hcmod_mult_i [simp]: "hcmod (iii * hcomplex_of_hypreal y) = abs y"
paulson@14469
  1028
apply (cases y)
paulson@14374
  1029
apply (simp add: hcomplex_of_hypreal hcmod hypreal_hrabs iii_def hcomplex_mult)
paulson@14314
  1030
done
paulson@14314
  1031
paulson@14374
  1032
lemma hcmod_mult_i2 [simp]: "hcmod (hcomplex_of_hypreal y * iii) = abs y"
paulson@14377
  1033
by (simp only: hcmod_mult_i hcomplex_mult_commute)
paulson@14314
  1034
paulson@14314
  1035
(*---------------------------------------------------------------------------*)
paulson@14314
  1036
(*  harg                                                                     *)
paulson@14314
  1037
(*---------------------------------------------------------------------------*)
paulson@14314
  1038
paulson@14314
  1039
lemma harg:
paulson@14314
  1040
  "harg (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
  1041
      Abs_hypreal(hyprel `` {%n. arg (X n)})"
paulson@14374
  1042
apply (simp add: harg_def)
paulson@14374
  1043
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14377
  1044
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
  1045
done
paulson@14314
  1046
paulson@14354
  1047
lemma cos_harg_i_mult_zero_pos:
paulson@14377
  1048
     "0 < y ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
paulson@14469
  1049
apply (cases y)
paulson@14377
  1050
apply (simp add: HComplex_def hcomplex_of_hypreal iii_def hcomplex_mult
paulson@14377
  1051
                hcomplex_add hypreal_zero_num hypreal_less starfun harg, ultra)
paulson@14314
  1052
done
paulson@14314
  1053
paulson@14354
  1054
lemma cos_harg_i_mult_zero_neg:
paulson@14377
  1055
     "y < 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
paulson@14469
  1056
apply (cases y)
paulson@14377
  1057
apply (simp add: HComplex_def hcomplex_of_hypreal iii_def hcomplex_mult
paulson@14377
  1058
                 hcomplex_add hypreal_zero_num hypreal_less starfun harg, ultra)
paulson@14314
  1059
done
paulson@14314
  1060
paulson@14354
  1061
lemma cos_harg_i_mult_zero [simp]:
paulson@14377
  1062
     "y \<noteq> 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
paulson@14377
  1063
by (auto simp add: linorder_neq_iff
paulson@14377
  1064
                   cos_harg_i_mult_zero_pos cos_harg_i_mult_zero_neg)
paulson@14354
  1065
paulson@14354
  1066
lemma hcomplex_of_hypreal_zero_iff [simp]:
paulson@14354
  1067
     "(hcomplex_of_hypreal y = 0) = (y = 0)"
paulson@14469
  1068
apply (cases y)
paulson@14374
  1069
apply (simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
paulson@14314
  1070
done
paulson@14314
  1071
paulson@14314
  1072
paulson@14354
  1073
subsection{*Polar Form for Nonstandard Complex Numbers*}
paulson@14314
  1074
paulson@14335
  1075
lemma complex_split_polar2:
paulson@14377
  1076
     "\<forall>n. \<exists>r a. (z n) =  complex_of_real r * (Complex (cos a) (sin a))"
paulson@14377
  1077
by (blast intro: complex_split_polar)
paulson@14377
  1078
paulson@14377
  1079
lemma lemma_hypreal_P_EX2:
paulson@14377
  1080
     "(\<exists>(x::hypreal) y. P x y) =
paulson@14377
  1081
      (\<exists>f g. P (Abs_hypreal(hyprel `` {f})) (Abs_hypreal(hyprel `` {g})))"
paulson@14377
  1082
apply auto
paulson@14377
  1083
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14377
  1084
apply (rule_tac z = y in eq_Abs_hypreal, auto)
paulson@14314
  1085
done
paulson@14314
  1086
paulson@14314
  1087
lemma hcomplex_split_polar:
paulson@14377
  1088
  "\<exists>r a. z = hcomplex_of_hypreal r * (HComplex(( *f* cos) a)(( *f* sin) a))"
paulson@14469
  1089
apply (cases z)
paulson@14377
  1090
apply (simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def starfun hcomplex_add hcomplex_mult HComplex_def)
paulson@14374
  1091
apply (cut_tac z = x in complex_split_polar2)
paulson@14335
  1092
apply (drule choice, safe)+
paulson@14374
  1093
apply (rule_tac x = f in exI)
paulson@14374
  1094
apply (rule_tac x = fa in exI, auto)
paulson@14314
  1095
done
paulson@14314
  1096
paulson@14314
  1097
lemma hcis:
paulson@14314
  1098
  "hcis (Abs_hypreal(hyprel `` {%n. X n})) =
paulson@14314
  1099
      Abs_hcomplex(hcomplexrel `` {%n. cis (X n)})"
paulson@14374
  1100
apply (simp add: hcis_def)
paulson@14377
  1101
apply (rule_tac f = Abs_hcomplex in arg_cong, auto iff: hcomplexrel_iff, ultra)
paulson@14314
  1102
done
paulson@14314
  1103
paulson@14314
  1104
lemma hcis_eq:
paulson@14314
  1105
   "hcis a =
paulson@14314
  1106
    (hcomplex_of_hypreal(( *f* cos) a) +
paulson@14314
  1107
    iii * hcomplex_of_hypreal(( *f* sin) a))"
paulson@14469
  1108
apply (cases a)
paulson@14374
  1109
apply (simp add: starfun hcis hcomplex_of_hypreal iii_def hcomplex_mult hcomplex_add cis_def)
paulson@14314
  1110
done
paulson@14314
  1111
paulson@14314
  1112
lemma hrcis:
paulson@14314
  1113
  "hrcis (Abs_hypreal(hyprel `` {%n. X n})) (Abs_hypreal(hyprel `` {%n. Y n})) =
paulson@14314
  1114
      Abs_hcomplex(hcomplexrel `` {%n. rcis (X n) (Y n)})"
paulson@14374
  1115
by (simp add: hrcis_def hcomplex_of_hypreal hcomplex_mult hcis rcis_def)
paulson@14314
  1116
paulson@14354
  1117
lemma hrcis_Ex: "\<exists>r a. z = hrcis r a"
paulson@14377
  1118
apply (simp add: hrcis_def hcis_eq hcomplex_of_hypreal_mult_HComplex [symmetric])
paulson@14314
  1119
apply (rule hcomplex_split_polar)
paulson@14314
  1120
done
paulson@14314
  1121
paulson@14374
  1122
lemma hRe_hcomplex_polar [simp]:
paulson@14377
  1123
     "hRe (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = 
paulson@14377
  1124
      r * ( *f* cos) a"
paulson@14377
  1125
by (simp add: hcomplex_of_hypreal_mult_HComplex)
paulson@14314
  1126
paulson@14374
  1127
lemma hRe_hrcis [simp]: "hRe(hrcis r a) = r * ( *f* cos) a"
paulson@14374
  1128
by (simp add: hrcis_def hcis_eq)
paulson@14314
  1129
paulson@14374
  1130
lemma hIm_hcomplex_polar [simp]:
paulson@14377
  1131
     "hIm (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = 
paulson@14377
  1132
      r * ( *f* sin) a"
paulson@14377
  1133
by (simp add: hcomplex_of_hypreal_mult_HComplex)
paulson@14314
  1134
paulson@14374
  1135
lemma hIm_hrcis [simp]: "hIm(hrcis r a) = r * ( *f* sin) a"
paulson@14374
  1136
by (simp add: hrcis_def hcis_eq)
paulson@14314
  1137
paulson@14377
  1138
paulson@14377
  1139
lemma hcmod_unit_one [simp]:
paulson@14377
  1140
     "hcmod (HComplex (( *f* cos) a) (( *f* sin) a)) = 1"
paulson@14469
  1141
apply (cases a) 
paulson@14377
  1142
apply (simp add: HComplex_def iii_def starfun hcomplex_of_hypreal 
paulson@14377
  1143
                 hcomplex_mult hcmod hcomplex_add hypreal_one_def)
paulson@14377
  1144
done
paulson@14377
  1145
paulson@14374
  1146
lemma hcmod_complex_polar [simp]:
paulson@14377
  1147
     "hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) =
paulson@14377
  1148
      abs r"
paulson@14377
  1149
apply (simp only: hcmod_mult hcmod_unit_one, simp)  
paulson@14314
  1150
done
paulson@14314
  1151
paulson@14374
  1152
lemma hcmod_hrcis [simp]: "hcmod(hrcis r a) = abs r"
paulson@14374
  1153
by (simp add: hrcis_def hcis_eq)
paulson@14314
  1154
paulson@14314
  1155
(*---------------------------------------------------------------------------*)
paulson@14314
  1156
(*  (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b)                *)
paulson@14314
  1157
(*---------------------------------------------------------------------------*)
paulson@14314
  1158
paulson@14314
  1159
lemma hcis_hrcis_eq: "hcis a = hrcis 1 a"
paulson@14374
  1160
by (simp add: hrcis_def)
paulson@14314
  1161
declare hcis_hrcis_eq [symmetric, simp]
paulson@14314
  1162
paulson@14314
  1163
lemma hrcis_mult:
paulson@14314
  1164
  "hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)"
paulson@14469
  1165
apply (simp add: hrcis_def, cases r1, cases r2, cases a, cases b)
paulson@14374
  1166
apply (simp add: hrcis hcis hypreal_add hypreal_mult hcomplex_of_hypreal
paulson@15013
  1167
                      hcomplex_mult cis_mult [symmetric])
paulson@14314
  1168
done
paulson@14314
  1169
paulson@14314
  1170
lemma hcis_mult: "hcis a * hcis b = hcis (a + b)"
paulson@14469
  1171
apply (cases a, cases b)
paulson@14374
  1172
apply (simp add: hcis hcomplex_mult hypreal_add cis_mult)
paulson@14314
  1173
done
paulson@14314
  1174
paulson@14374
  1175
lemma hcis_zero [simp]: "hcis 0 = 1"
paulson@14374
  1176
by (simp add: hcomplex_one_def hcis hypreal_zero_num)
paulson@14314
  1177
paulson@14374
  1178
lemma hrcis_zero_mod [simp]: "hrcis 0 a = 0"
paulson@14469
  1179
apply (simp add: hcomplex_zero_def, cases a)
paulson@14374
  1180
apply (simp add: hrcis hypreal_zero_num)
paulson@14314
  1181
done
paulson@14314
  1182
paulson@14374
  1183
lemma hrcis_zero_arg [simp]: "hrcis r 0 = hcomplex_of_hypreal r"
paulson@14469
  1184
apply (cases r)
paulson@14374
  1185
apply (simp add: hrcis hypreal_zero_num hcomplex_of_hypreal)
paulson@14314
  1186
done
paulson@14314
  1187
paulson@14374
  1188
lemma hcomplex_i_mult_minus [simp]: "iii * (iii * x) = - x"
paulson@14374
  1189
by (simp add: hcomplex_mult_assoc [symmetric])
paulson@14314
  1190
paulson@14374
  1191
lemma hcomplex_i_mult_minus2 [simp]: "iii * iii * x = - x"
paulson@14374
  1192
by simp
paulson@14314
  1193
paulson@14314
  1194
lemma hcis_hypreal_of_nat_Suc_mult:
paulson@14314
  1195
   "hcis (hypreal_of_nat (Suc n) * a) = hcis a * hcis (hypreal_of_nat n * a)"
paulson@14469
  1196
apply (cases a)
paulson@14374
  1197
apply (simp add: hypreal_of_nat hcis hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
paulson@14314
  1198
done
paulson@14314
  1199
paulson@14314
  1200
lemma NSDeMoivre: "(hcis a) ^ n = hcis (hypreal_of_nat n * a)"
paulson@14314
  1201
apply (induct_tac "n")
paulson@14374
  1202
apply (simp_all add: hcis_hypreal_of_nat_Suc_mult)
paulson@14314
  1203
done
paulson@14314
  1204
paulson@14335
  1205
lemma hcis_hypreal_of_hypnat_Suc_mult:
paulson@14335
  1206
     "hcis (hypreal_of_hypnat (n + 1) * a) =
paulson@14314
  1207
      hcis a * hcis (hypreal_of_hypnat n * a)"
paulson@14469
  1208
apply (cases a, cases n)
paulson@14374
  1209
apply (simp add: hcis hypreal_of_hypnat hypnat_add hypnat_one_def hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
paulson@14314
  1210
done
paulson@14314
  1211
paulson@14314
  1212
lemma NSDeMoivre_ext: "(hcis a) hcpow n = hcis (hypreal_of_hypnat n * a)"
paulson@14469
  1213
apply (cases a, cases n)
paulson@14374
  1214
apply (simp add: hcis hypreal_of_hypnat hypreal_mult hcpow DeMoivre)
paulson@14314
  1215
done
paulson@14314
  1216
paulson@14314
  1217
lemma DeMoivre2:
paulson@14314
  1218
  "(hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
paulson@14374
  1219
apply (simp add: hrcis_def power_mult_distrib NSDeMoivre hcomplex_of_hypreal_pow)
paulson@14314
  1220
done
paulson@14314
  1221
paulson@14314
  1222
lemma DeMoivre2_ext:
paulson@14314
  1223
  "(hrcis r a) hcpow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
paulson@14374
  1224
apply (simp add: hrcis_def hcpow_mult NSDeMoivre_ext hcomplex_of_hypreal_hyperpow)
paulson@14374
  1225
done
paulson@14374
  1226
paulson@14374
  1227
lemma hcis_inverse [simp]: "inverse(hcis a) = hcis (-a)"
paulson@14469
  1228
apply (cases a)
paulson@14374
  1229
apply (simp add: hcomplex_inverse hcis hypreal_minus)
paulson@14314
  1230
done
paulson@14314
  1231
paulson@14374
  1232
lemma hrcis_inverse: "inverse(hrcis r a) = hrcis (inverse r) (-a)"
paulson@14469
  1233
apply (cases a, cases r)
paulson@14374
  1234
apply (simp add: hcomplex_inverse hrcis hypreal_minus hypreal_inverse rcis_inverse, ultra)
paulson@14374
  1235
apply (simp add: real_divide_def)
paulson@14314
  1236
done
paulson@14314
  1237
paulson@14374
  1238
lemma hRe_hcis [simp]: "hRe(hcis a) = ( *f* cos) a"
paulson@14469
  1239
apply (cases a)
paulson@14374
  1240
apply (simp add: hcis starfun hRe)
paulson@14314
  1241
done
paulson@14314
  1242
paulson@14374
  1243
lemma hIm_hcis [simp]: "hIm(hcis a) = ( *f* sin) a"
paulson@14469
  1244
apply (cases a)
paulson@14374
  1245
apply (simp add: hcis starfun hIm)
paulson@14314
  1246
done
paulson@14314
  1247
paulson@14374
  1248
lemma cos_n_hRe_hcis_pow_n: "( *f* cos) (hypreal_of_nat n * a) = hRe(hcis a ^ n)"
paulson@14377
  1249
by (simp add: NSDeMoivre)
paulson@14314
  1250
paulson@14374
  1251
lemma sin_n_hIm_hcis_pow_n: "( *f* sin) (hypreal_of_nat n * a) = hIm(hcis a ^ n)"
paulson@14377
  1252
by (simp add: NSDeMoivre)
paulson@14314
  1253
paulson@14374
  1254
lemma cos_n_hRe_hcis_hcpow_n: "( *f* cos) (hypreal_of_hypnat n * a) = hRe(hcis a hcpow n)"
paulson@14377
  1255
by (simp add: NSDeMoivre_ext)
paulson@14314
  1256
paulson@14374
  1257
lemma sin_n_hIm_hcis_hcpow_n: "( *f* sin) (hypreal_of_hypnat n * a) = hIm(hcis a hcpow n)"
paulson@14377
  1258
by (simp add: NSDeMoivre_ext)
paulson@14314
  1259
paulson@14314
  1260
lemma hexpi_add: "hexpi(a + b) = hexpi(a) * hexpi(b)"
paulson@14469
  1261
apply (simp add: hexpi_def, cases a, cases b)
paulson@14374
  1262
apply (simp add: hcis hRe hIm hcomplex_add hcomplex_mult hypreal_mult starfun hcomplex_of_hypreal cis_mult [symmetric] complex_Im_add complex_Re_add exp_add complex_of_real_mult)
paulson@14314
  1263
done
paulson@14314
  1264
paulson@14314
  1265
paulson@14374
  1266
subsection{*@{term hcomplex_of_complex}: the Injection from
paulson@14354
  1267
  type @{typ complex} to to @{typ hcomplex}*}
paulson@14354
  1268
paulson@14354
  1269
lemma inj_hcomplex_of_complex: "inj(hcomplex_of_complex)"
paulson@14374
  1270
apply (rule inj_onI, rule ccontr)
paulson@14374
  1271
apply (simp add: hcomplex_of_complex_def)
paulson@14354
  1272
done
paulson@14354
  1273
paulson@14354
  1274
lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex ii"
paulson@14374
  1275
by (simp add: iii_def hcomplex_of_complex_def)
paulson@14314
  1276
paulson@14374
  1277
lemma hcomplex_of_complex_add [simp]:
paulson@14314
  1278
     "hcomplex_of_complex (z1 + z2) = hcomplex_of_complex z1 + hcomplex_of_complex z2"
paulson@14374
  1279
by (simp add: hcomplex_of_complex_def hcomplex_add)
paulson@14314
  1280
paulson@14374
  1281
lemma hcomplex_of_complex_mult [simp]:
paulson@14314
  1282
     "hcomplex_of_complex (z1 * z2) = hcomplex_of_complex z1 * hcomplex_of_complex z2"
paulson@14374
  1283
by (simp add: hcomplex_of_complex_def hcomplex_mult)
paulson@14314
  1284
paulson@14374
  1285
lemma hcomplex_of_complex_eq_iff [simp]:
paulson@14374
  1286
     "(hcomplex_of_complex z1 = hcomplex_of_complex z2) = (z1 = z2)"
paulson@14374
  1287
by (simp add: hcomplex_of_complex_def)
paulson@14314
  1288
paulson@14374
  1289
paulson@14374
  1290
lemma hcomplex_of_complex_minus [simp]:
paulson@14335
  1291
     "hcomplex_of_complex (-r) = - hcomplex_of_complex  r"
paulson@14374
  1292
by (simp add: hcomplex_of_complex_def hcomplex_minus)
paulson@14314
  1293
paulson@14374
  1294
lemma hcomplex_of_complex_one [simp]: "hcomplex_of_complex 1 = 1"
paulson@14374
  1295
by (simp add: hcomplex_of_complex_def hcomplex_one_def)
paulson@14314
  1296
paulson@14374
  1297
lemma hcomplex_of_complex_zero [simp]: "hcomplex_of_complex 0 = 0"
paulson@14374
  1298
by (simp add: hcomplex_of_complex_def hcomplex_zero_def)
paulson@14314
  1299
paulson@14387
  1300
lemma hcomplex_of_complex_zero_iff [simp]:
paulson@14387
  1301
     "(hcomplex_of_complex r = 0) = (r = 0)"
paulson@14387
  1302
by (auto intro: FreeUltrafilterNat_P 
paulson@14387
  1303
         simp add: hcomplex_of_complex_def hcomplex_zero_def)
paulson@14314
  1304
paulson@14374
  1305
lemma hcomplex_of_complex_inverse [simp]:
paulson@14335
  1306
     "hcomplex_of_complex (inverse r) = inverse (hcomplex_of_complex r)"
paulson@15013
  1307
proof cases
paulson@15013
  1308
  assume "r=0" thus ?thesis by simp
paulson@15013
  1309
next
paulson@15013
  1310
  assume nz: "r\<noteq>0" 
paulson@15013
  1311
  show ?thesis
paulson@15013
  1312
  proof (rule hcomplex_mult_left_cancel [THEN iffD1]) 
paulson@15013
  1313
    show "hcomplex_of_complex r \<noteq> 0"
paulson@15013
  1314
      by (simp add: nz) 
paulson@15013
  1315
  next
paulson@15013
  1316
    have "hcomplex_of_complex r * hcomplex_of_complex (inverse r) =
paulson@15013
  1317
          hcomplex_of_complex (r * inverse r)"
paulson@15013
  1318
      by simp
paulson@15013
  1319
    also have "... = hcomplex_of_complex r * inverse (hcomplex_of_complex r)" 
paulson@15013
  1320
      by (simp add: nz)
paulson@15013
  1321
    finally show "hcomplex_of_complex r * hcomplex_of_complex (inverse r) =
paulson@15013
  1322
                  hcomplex_of_complex r * inverse (hcomplex_of_complex r)" .
paulson@15013
  1323
  qed
paulson@15013
  1324
qed
paulson@14314
  1325
paulson@14374
  1326
lemma hcomplex_of_complex_divide [simp]:
paulson@15013
  1327
     "hcomplex_of_complex (z1 / z2) = 
paulson@15013
  1328
      hcomplex_of_complex z1 / hcomplex_of_complex z2"
paulson@14374
  1329
by (simp add: hcomplex_divide_def complex_divide_def)
paulson@14314
  1330
paulson@14314
  1331
lemma hRe_hcomplex_of_complex:
paulson@14314
  1332
   "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
paulson@14374
  1333
by (simp add: hcomplex_of_complex_def hypreal_of_real_def hRe)
paulson@14314
  1334
paulson@14314
  1335
lemma hIm_hcomplex_of_complex:
paulson@14314
  1336
   "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
paulson@14374
  1337
by (simp add: hcomplex_of_complex_def hypreal_of_real_def hIm)
paulson@14314
  1338
paulson@14314
  1339
lemma hcmod_hcomplex_of_complex:
paulson@14314
  1340
     "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
paulson@14374
  1341
by (simp add: hypreal_of_real_def hcomplex_of_complex_def hcmod)
paulson@14314
  1342
paulson@14387
  1343
paulson@14387
  1344
subsection{*Numerals and Arithmetic*}
paulson@14387
  1345
paulson@14387
  1346
instance hcomplex :: number ..
paulson@14387
  1347
paulson@15013
  1348
defs (overloaded)
paulson@15013
  1349
  hcomplex_number_of_def: "(number_of w :: hcomplex) == of_int (Rep_Bin w)"
paulson@15013
  1350
    --{*the type constraint is essential!*}
paulson@14387
  1351
paulson@14387
  1352
instance hcomplex :: number_ring
paulson@15013
  1353
by (intro_classes, simp add: hcomplex_number_of_def) 
paulson@15013
  1354
paulson@15013
  1355
paulson@15013
  1356
lemma hcomplex_of_complex_of_nat [simp]:
paulson@15013
  1357
     "hcomplex_of_complex (of_nat n) = of_nat n"
paulson@15013
  1358
by (induct n, simp_all) 
paulson@15013
  1359
paulson@15013
  1360
lemma hcomplex_of_complex_of_int [simp]:
paulson@15013
  1361
     "hcomplex_of_complex (of_int z) = of_int z"
paulson@15013
  1362
proof (cases z)
paulson@15013
  1363
  case (1 n)
paulson@15013
  1364
    thus ?thesis by simp
paulson@15013
  1365
next
paulson@15013
  1366
  case (2 n)
paulson@15013
  1367
    thus ?thesis 
paulson@15013
  1368
      by (simp only: of_int_minus hcomplex_of_complex_minus, simp)
paulson@14387
  1369
qed
paulson@14387
  1370
paulson@14387
  1371
paulson@14387
  1372
text{*Collapse applications of @{term hcomplex_of_complex} to @{term number_of}*}
paulson@14387
  1373
lemma hcomplex_number_of [simp]:
paulson@14387
  1374
     "hcomplex_of_complex (number_of w) = number_of w"
paulson@15013
  1375
by (simp add: hcomplex_number_of_def complex_number_of_def) 
paulson@14387
  1376
paulson@14387
  1377
lemma hcomplex_of_hypreal_eq_hcomplex_of_complex: 
paulson@14387
  1378
     "hcomplex_of_hypreal (hypreal_of_real x) =  
paulson@15013
  1379
      hcomplex_of_complex (complex_of_real x)"
paulson@14387
  1380
by (simp add: hypreal_of_real_def hcomplex_of_hypreal hcomplex_of_complex_def 
paulson@14387
  1381
              complex_of_real_def)
paulson@14387
  1382
paulson@14387
  1383
lemma hcomplex_hypreal_number_of: 
paulson@14387
  1384
  "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)"
paulson@14387
  1385
by (simp only: complex_number_of [symmetric] hypreal_number_of [symmetric] 
paulson@14387
  1386
               hcomplex_of_hypreal_eq_hcomplex_of_complex)
paulson@14387
  1387
paulson@14387
  1388
text{*This theorem is necessary because theorems such as
paulson@14387
  1389
   @{text iszero_number_of_0} only hold for ordered rings. They cannot
paulson@14387
  1390
   be generalized to fields in general because they fail for finite fields.
paulson@14387
  1391
   They work for type complex because the reals can be embedded in them.*}
paulson@14387
  1392
lemma iszero_hcomplex_number_of [simp]:
paulson@14387
  1393
     "iszero (number_of w :: hcomplex) = iszero (number_of w :: real)"
paulson@14387
  1394
apply (simp only: iszero_complex_number_of [symmetric])  
paulson@14387
  1395
apply (simp only: hcomplex_of_complex_zero_iff hcomplex_number_of [symmetric] 
paulson@14387
  1396
                  iszero_def)  
paulson@14387
  1397
done
paulson@14387
  1398
paulson@14387
  1399
paulson@14387
  1400
(*
paulson@14387
  1401
Goal "z + hcnj z =  
paulson@14387
  1402
      hcomplex_of_hypreal (2 * hRe(z))"
paulson@14387
  1403
by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
paulson@14387
  1404
by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
paulson@14387
  1405
    hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
paulson@14387
  1406
qed "hcomplex_add_hcnj";
paulson@14387
  1407
paulson@14387
  1408
Goal "z - hcnj z = \
paulson@14387
  1409
\     hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii";
paulson@14387
  1410
by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
paulson@14387
  1411
by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
paulson@14387
  1412
    hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
paulson@14387
  1413
    complex_diff_cnj,iii_def,hcomplex_mult]));
paulson@14387
  1414
qed "hcomplex_diff_hcnj";
paulson@14387
  1415
*)
paulson@14387
  1416
paulson@14387
  1417
paulson@14387
  1418
lemma hcomplex_hcnj_num_zero_iff: "(hcnj z = 0) = (z = 0)"
paulson@14387
  1419
apply (auto simp add: hcomplex_hcnj_zero_iff)
paulson@14387
  1420
done
paulson@14387
  1421
declare hcomplex_hcnj_num_zero_iff [simp]
paulson@14387
  1422
paulson@14387
  1423
lemma hcomplex_zero_num: "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})"
paulson@14387
  1424
apply (simp add: hcomplex_zero_def)
paulson@14387
  1425
done
paulson@14387
  1426
paulson@14387
  1427
lemma hcomplex_one_num: "1 =  Abs_hcomplex (hcomplexrel `` {%n. 1})"
paulson@14387
  1428
apply (simp add: hcomplex_one_def)
paulson@14387
  1429
done
paulson@14387
  1430
paulson@14387
  1431
(*** Real and imaginary stuff ***)
paulson@14387
  1432
paulson@14387
  1433
(*Convert???
paulson@14387
  1434
Goalw [hcomplex_number_of_def] 
paulson@14387
  1435
  "((number_of xa :: hcomplex) + iii * number_of ya =  
paulson@14387
  1436
        number_of xb + iii * number_of yb) =  
paulson@14387
  1437
   (((number_of xa :: hcomplex) = number_of xb) &  
paulson@14387
  1438
    ((number_of ya :: hcomplex) = number_of yb))"
paulson@14387
  1439
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff,
paulson@14387
  1440
     hcomplex_hypreal_number_of]));
paulson@14387
  1441
qed "hcomplex_number_of_eq_cancel_iff";
paulson@14387
  1442
Addsimps [hcomplex_number_of_eq_cancel_iff];
paulson@14387
  1443
paulson@14387
  1444
Goalw [hcomplex_number_of_def] 
paulson@14387
  1445
  "((number_of xa :: hcomplex) + number_of ya * iii = \
paulson@14387
  1446
\       number_of xb + number_of yb * iii) = \
paulson@14387
  1447
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1448
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1449
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA,
paulson@14387
  1450
    hcomplex_hypreal_number_of]));
paulson@14387
  1451
qed "hcomplex_number_of_eq_cancel_iffA";
paulson@14387
  1452
Addsimps [hcomplex_number_of_eq_cancel_iffA];
paulson@14387
  1453
paulson@14387
  1454
Goalw [hcomplex_number_of_def] 
paulson@14387
  1455
  "((number_of xa :: hcomplex) + number_of ya * iii = \
paulson@14387
  1456
\       number_of xb + iii * number_of yb) = \
paulson@14387
  1457
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1458
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1459
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB,
paulson@14387
  1460
    hcomplex_hypreal_number_of]));
paulson@14387
  1461
qed "hcomplex_number_of_eq_cancel_iffB";
paulson@14387
  1462
Addsimps [hcomplex_number_of_eq_cancel_iffB];
paulson@14387
  1463
paulson@14387
  1464
Goalw [hcomplex_number_of_def] 
paulson@14387
  1465
  "((number_of xa :: hcomplex) + iii * number_of ya = \
paulson@14387
  1466
\       number_of xb + number_of yb * iii) = \
paulson@14387
  1467
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1468
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1469
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC,
paulson@14387
  1470
     hcomplex_hypreal_number_of]));
paulson@14387
  1471
qed "hcomplex_number_of_eq_cancel_iffC";
paulson@14387
  1472
Addsimps [hcomplex_number_of_eq_cancel_iffC];
paulson@14387
  1473
paulson@14387
  1474
Goalw [hcomplex_number_of_def] 
paulson@14387
  1475
  "((number_of xa :: hcomplex) + iii * number_of ya = \
paulson@14387
  1476
\       number_of xb) = \
paulson@14387
  1477
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1478
\   ((number_of ya :: hcomplex) = 0))";
paulson@14387
  1479
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2,
paulson@14387
  1480
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1481
qed "hcomplex_number_of_eq_cancel_iff2";
paulson@14387
  1482
Addsimps [hcomplex_number_of_eq_cancel_iff2];
paulson@14387
  1483
paulson@14387
  1484
Goalw [hcomplex_number_of_def] 
paulson@14387
  1485
  "((number_of xa :: hcomplex) + number_of ya * iii = \
paulson@14387
  1486
\       number_of xb) = \
paulson@14387
  1487
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1488
\   ((number_of ya :: hcomplex) = 0))";
paulson@14387
  1489
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a,
paulson@14387
  1490
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1491
qed "hcomplex_number_of_eq_cancel_iff2a";
paulson@14387
  1492
Addsimps [hcomplex_number_of_eq_cancel_iff2a];
paulson@14387
  1493
paulson@14387
  1494
Goalw [hcomplex_number_of_def] 
paulson@14387
  1495
  "((number_of xa :: hcomplex) + iii * number_of ya = \
paulson@14387
  1496
\    iii * number_of yb) = \
paulson@14387
  1497
\  (((number_of xa :: hcomplex) = 0) & \
paulson@14387
  1498
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1499
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3,
paulson@14387
  1500
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1501
qed "hcomplex_number_of_eq_cancel_iff3";
paulson@14387
  1502
Addsimps [hcomplex_number_of_eq_cancel_iff3];
paulson@14387
  1503
paulson@14387
  1504
Goalw [hcomplex_number_of_def] 
paulson@14387
  1505
  "((number_of xa :: hcomplex) + number_of ya * iii= \
paulson@14387
  1506
\    iii * number_of yb) = \
paulson@14387
  1507
\  (((number_of xa :: hcomplex) = 0) & \
paulson@14387
  1508
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1509
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a,
paulson@14387
  1510
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1511
qed "hcomplex_number_of_eq_cancel_iff3a";
paulson@14387
  1512
Addsimps [hcomplex_number_of_eq_cancel_iff3a];
paulson@14387
  1513
*)
paulson@14387
  1514
paulson@14387
  1515
lemma hcomplex_number_of_hcnj [simp]:
paulson@14387
  1516
     "hcnj (number_of v :: hcomplex) = number_of v"
paulson@14387
  1517
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1518
               hcomplex_hcnj_hcomplex_of_hypreal)
paulson@14387
  1519
paulson@14387
  1520
lemma hcomplex_number_of_hcmod [simp]: 
paulson@14387
  1521
      "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)"
paulson@14387
  1522
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1523
               hcmod_hcomplex_of_hypreal)
paulson@14387
  1524
paulson@14387
  1525
lemma hcomplex_number_of_hRe [simp]: 
paulson@14387
  1526
      "hRe(number_of v :: hcomplex) = number_of v"
paulson@14387
  1527
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1528
               hRe_hcomplex_of_hypreal)
paulson@14387
  1529
paulson@14387
  1530
lemma hcomplex_number_of_hIm [simp]: 
paulson@14387
  1531
      "hIm(number_of v :: hcomplex) = 0"
paulson@14387
  1532
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1533
               hIm_hcomplex_of_hypreal)
paulson@14387
  1534
paulson@14387
  1535
paulson@14314
  1536
ML
paulson@14314
  1537
{*
paulson@14314
  1538
val hcomplex_zero_def = thm"hcomplex_zero_def";
paulson@14314
  1539
val hcomplex_one_def = thm"hcomplex_one_def";
paulson@14314
  1540
val hcomplex_minus_def = thm"hcomplex_minus_def";
paulson@14314
  1541
val hcomplex_diff_def = thm"hcomplex_diff_def";
paulson@14314
  1542
val hcomplex_divide_def = thm"hcomplex_divide_def";
paulson@14314
  1543
val hcomplex_mult_def = thm"hcomplex_mult_def";
paulson@14314
  1544
val hcomplex_add_def = thm"hcomplex_add_def";
paulson@14314
  1545
val hcomplex_of_complex_def = thm"hcomplex_of_complex_def";
paulson@14314
  1546
val iii_def = thm"iii_def";
paulson@14314
  1547
paulson@14314
  1548
val hcomplexrel_iff = thm"hcomplexrel_iff";
paulson@14314
  1549
val hcomplexrel_refl = thm"hcomplexrel_refl";
paulson@14314
  1550
val hcomplexrel_sym = thm"hcomplexrel_sym";
paulson@14314
  1551
val hcomplexrel_trans = thm"hcomplexrel_trans";
paulson@14314
  1552
val equiv_hcomplexrel = thm"equiv_hcomplexrel";
paulson@14314
  1553
val equiv_hcomplexrel_iff = thm"equiv_hcomplexrel_iff";
paulson@14314
  1554
val hcomplexrel_in_hcomplex = thm"hcomplexrel_in_hcomplex";
paulson@14314
  1555
val inj_on_Abs_hcomplex = thm"inj_on_Abs_hcomplex";
paulson@14314
  1556
val inj_Rep_hcomplex = thm"inj_Rep_hcomplex";
paulson@14314
  1557
val lemma_hcomplexrel_refl = thm"lemma_hcomplexrel_refl";
paulson@14314
  1558
val hcomplex_empty_not_mem = thm"hcomplex_empty_not_mem";
paulson@14314
  1559
val Rep_hcomplex_nonempty = thm"Rep_hcomplex_nonempty";
paulson@14314
  1560
val eq_Abs_hcomplex = thm"eq_Abs_hcomplex";
paulson@14314
  1561
val hRe = thm"hRe";
paulson@14314
  1562
val hIm = thm"hIm";
paulson@14314
  1563
val hcomplex_hRe_hIm_cancel_iff = thm"hcomplex_hRe_hIm_cancel_iff";
paulson@14314
  1564
val hcomplex_hRe_zero = thm"hcomplex_hRe_zero";
paulson@14314
  1565
val hcomplex_hIm_zero = thm"hcomplex_hIm_zero";
paulson@14314
  1566
val hcomplex_hRe_one = thm"hcomplex_hRe_one";
paulson@14314
  1567
val hcomplex_hIm_one = thm"hcomplex_hIm_one";
paulson@14314
  1568
val inj_hcomplex_of_complex = thm"inj_hcomplex_of_complex";
paulson@14314
  1569
val hcomplex_of_complex_i = thm"hcomplex_of_complex_i";
paulson@14314
  1570
val hcomplex_add = thm"hcomplex_add";
paulson@14314
  1571
val hcomplex_add_commute = thm"hcomplex_add_commute";
paulson@14314
  1572
val hcomplex_add_assoc = thm"hcomplex_add_assoc";
paulson@14314
  1573
val hcomplex_add_zero_left = thm"hcomplex_add_zero_left";
paulson@14314
  1574
val hcomplex_add_zero_right = thm"hcomplex_add_zero_right";
paulson@14314
  1575
val hRe_add = thm"hRe_add";
paulson@14314
  1576
val hIm_add = thm"hIm_add";
paulson@14314
  1577
val hcomplex_minus_congruent = thm"hcomplex_minus_congruent";
paulson@14314
  1578
val hcomplex_minus = thm"hcomplex_minus";
paulson@14314
  1579
val hcomplex_add_minus_left = thm"hcomplex_add_minus_left";
paulson@14314
  1580
val hRe_minus = thm"hRe_minus";
paulson@14314
  1581
val hIm_minus = thm"hIm_minus";
paulson@14314
  1582
val hcomplex_add_minus_eq_minus = thm"hcomplex_add_minus_eq_minus";
paulson@14314
  1583
val hcomplex_diff = thm"hcomplex_diff";
paulson@14314
  1584
val hcomplex_diff_eq_eq = thm"hcomplex_diff_eq_eq";
paulson@14314
  1585
val hcomplex_mult = thm"hcomplex_mult";
paulson@14314
  1586
val hcomplex_mult_commute = thm"hcomplex_mult_commute";
paulson@14314
  1587
val hcomplex_mult_assoc = thm"hcomplex_mult_assoc";
paulson@14314
  1588
val hcomplex_mult_one_left = thm"hcomplex_mult_one_left";
paulson@14314
  1589
val hcomplex_mult_one_right = thm"hcomplex_mult_one_right";
paulson@14314
  1590
val hcomplex_mult_zero_left = thm"hcomplex_mult_zero_left";
paulson@14314
  1591
val hcomplex_mult_minus_one = thm"hcomplex_mult_minus_one";
paulson@14314
  1592
val hcomplex_mult_minus_one_right = thm"hcomplex_mult_minus_one_right";
paulson@14314
  1593
val hcomplex_add_mult_distrib = thm"hcomplex_add_mult_distrib";
paulson@14314
  1594
val hcomplex_zero_not_eq_one = thm"hcomplex_zero_not_eq_one";
paulson@14314
  1595
val hcomplex_inverse = thm"hcomplex_inverse";
paulson@14314
  1596
val hcomplex_mult_inv_left = thm"hcomplex_mult_inv_left";
paulson@14314
  1597
val hcomplex_mult_left_cancel = thm"hcomplex_mult_left_cancel";
paulson@14314
  1598
val hcomplex_mult_right_cancel = thm"hcomplex_mult_right_cancel";
paulson@14314
  1599
val hcomplex_add_divide_distrib = thm"hcomplex_add_divide_distrib";
paulson@14314
  1600
val hcomplex_of_hypreal = thm"hcomplex_of_hypreal";
paulson@14314
  1601
val hcomplex_of_hypreal_cancel_iff = thm"hcomplex_of_hypreal_cancel_iff";
paulson@14314
  1602
val hcomplex_of_hypreal_minus = thm"hcomplex_of_hypreal_minus";
paulson@14314
  1603
val hcomplex_of_hypreal_inverse = thm"hcomplex_of_hypreal_inverse";
paulson@14314
  1604
val hcomplex_of_hypreal_add = thm"hcomplex_of_hypreal_add";
paulson@14314
  1605
val hcomplex_of_hypreal_diff = thm"hcomplex_of_hypreal_diff";
paulson@14314
  1606
val hcomplex_of_hypreal_mult = thm"hcomplex_of_hypreal_mult";
paulson@14314
  1607
val hcomplex_of_hypreal_divide = thm"hcomplex_of_hypreal_divide";
paulson@14314
  1608
val hcomplex_of_hypreal_one = thm"hcomplex_of_hypreal_one";
paulson@14314
  1609
val hcomplex_of_hypreal_zero = thm"hcomplex_of_hypreal_zero";
paulson@14314
  1610
val hcomplex_of_hypreal_pow = thm"hcomplex_of_hypreal_pow";
paulson@14314
  1611
val hRe_hcomplex_of_hypreal = thm"hRe_hcomplex_of_hypreal";
paulson@14314
  1612
val hIm_hcomplex_of_hypreal = thm"hIm_hcomplex_of_hypreal";
paulson@14314
  1613
val hcomplex_of_hypreal_epsilon_not_zero = thm"hcomplex_of_hypreal_epsilon_not_zero";
paulson@14314
  1614
val hcmod = thm"hcmod";
paulson@14314
  1615
val hcmod_zero = thm"hcmod_zero";
paulson@14314
  1616
val hcmod_one = thm"hcmod_one";
paulson@14314
  1617
val hcmod_hcomplex_of_hypreal = thm"hcmod_hcomplex_of_hypreal";
paulson@14314
  1618
val hcomplex_of_hypreal_abs = thm"hcomplex_of_hypreal_abs";
paulson@14314
  1619
val hcnj = thm"hcnj";
paulson@14314
  1620
val hcomplex_hcnj_cancel_iff = thm"hcomplex_hcnj_cancel_iff";
paulson@14314
  1621
val hcomplex_hcnj_hcnj = thm"hcomplex_hcnj_hcnj";
paulson@14314
  1622
val hcomplex_hcnj_hcomplex_of_hypreal = thm"hcomplex_hcnj_hcomplex_of_hypreal";
paulson@14314
  1623
val hcomplex_hmod_hcnj = thm"hcomplex_hmod_hcnj";
paulson@14314
  1624
val hcomplex_hcnj_minus = thm"hcomplex_hcnj_minus";
paulson@14314
  1625
val hcomplex_hcnj_inverse = thm"hcomplex_hcnj_inverse";
paulson@14314
  1626
val hcomplex_hcnj_add = thm"hcomplex_hcnj_add";
paulson@14314
  1627
val hcomplex_hcnj_diff = thm"hcomplex_hcnj_diff";
paulson@14314
  1628
val hcomplex_hcnj_mult = thm"hcomplex_hcnj_mult";
paulson@14314
  1629
val hcomplex_hcnj_divide = thm"hcomplex_hcnj_divide";
paulson@14314
  1630
val hcnj_one = thm"hcnj_one";
paulson@14314
  1631
val hcomplex_hcnj_pow = thm"hcomplex_hcnj_pow";
paulson@14314
  1632
val hcomplex_hcnj_zero = thm"hcomplex_hcnj_zero";
paulson@14314
  1633
val hcomplex_hcnj_zero_iff = thm"hcomplex_hcnj_zero_iff";
paulson@14314
  1634
val hcomplex_mult_hcnj = thm"hcomplex_mult_hcnj";
paulson@14314
  1635
val hcomplex_hcmod_eq_zero_cancel = thm"hcomplex_hcmod_eq_zero_cancel";
paulson@14371
  1636
paulson@14314
  1637
val hcmod_hcomplex_of_hypreal_of_nat = thm"hcmod_hcomplex_of_hypreal_of_nat";
paulson@14314
  1638
val hcmod_hcomplex_of_hypreal_of_hypnat = thm"hcmod_hcomplex_of_hypreal_of_hypnat";
paulson@14314
  1639
val hcmod_minus = thm"hcmod_minus";
paulson@14314
  1640
val hcmod_mult_hcnj = thm"hcmod_mult_hcnj";
paulson@14314
  1641
val hcmod_ge_zero = thm"hcmod_ge_zero";
paulson@14314
  1642
val hrabs_hcmod_cancel = thm"hrabs_hcmod_cancel";
paulson@14314
  1643
val hcmod_mult = thm"hcmod_mult";
paulson@14314
  1644
val hcmod_add_squared_eq = thm"hcmod_add_squared_eq";
paulson@14314
  1645
val hcomplex_hRe_mult_hcnj_le_hcmod = thm"hcomplex_hRe_mult_hcnj_le_hcmod";
paulson@14314
  1646
val hcomplex_hRe_mult_hcnj_le_hcmod2 = thm"hcomplex_hRe_mult_hcnj_le_hcmod2";
paulson@14314
  1647
val hcmod_triangle_squared = thm"hcmod_triangle_squared";
paulson@14314
  1648
val hcmod_triangle_ineq = thm"hcmod_triangle_ineq";
paulson@14314
  1649
val hcmod_triangle_ineq2 = thm"hcmod_triangle_ineq2";
paulson@14314
  1650
val hcmod_diff_commute = thm"hcmod_diff_commute";
paulson@14314
  1651
val hcmod_add_less = thm"hcmod_add_less";
paulson@14314
  1652
val hcmod_mult_less = thm"hcmod_mult_less";
paulson@14314
  1653
val hcmod_diff_ineq = thm"hcmod_diff_ineq";
paulson@14314
  1654
val hcpow = thm"hcpow";
paulson@14314
  1655
val hcomplex_of_hypreal_hyperpow = thm"hcomplex_of_hypreal_hyperpow";
paulson@14314
  1656
val hcmod_hcomplexpow = thm"hcmod_hcomplexpow";
paulson@14314
  1657
val hcmod_hcpow = thm"hcmod_hcpow";
paulson@14314
  1658
val hcpow_minus = thm"hcpow_minus";
paulson@14314
  1659
val hcmod_hcomplex_inverse = thm"hcmod_hcomplex_inverse";
paulson@14314
  1660
val hcmod_divide = thm"hcmod_divide";
paulson@14314
  1661
val hcpow_mult = thm"hcpow_mult";
paulson@14314
  1662
val hcpow_zero = thm"hcpow_zero";
paulson@14314
  1663
val hcpow_zero2 = thm"hcpow_zero2";
paulson@14314
  1664
val hcpow_not_zero = thm"hcpow_not_zero";
paulson@14314
  1665
val hcpow_zero_zero = thm"hcpow_zero_zero";
paulson@14314
  1666
val hcomplex_i_mult_eq = thm"hcomplex_i_mult_eq";
paulson@14314
  1667
val hcomplexpow_i_squared = thm"hcomplexpow_i_squared";
paulson@14314
  1668
val hcomplex_i_not_zero = thm"hcomplex_i_not_zero";
paulson@14314
  1669
val hcomplex_divide = thm"hcomplex_divide";
paulson@14314
  1670
val hsgn = thm"hsgn";
paulson@14314
  1671
val hsgn_zero = thm"hsgn_zero";
paulson@14314
  1672
val hsgn_one = thm"hsgn_one";
paulson@14314
  1673
val hsgn_minus = thm"hsgn_minus";
paulson@14314
  1674
val hsgn_eq = thm"hsgn_eq";
paulson@14314
  1675
val lemma_hypreal_P_EX2 = thm"lemma_hypreal_P_EX2";
paulson@14314
  1676
val hcmod_i = thm"hcmod_i";
paulson@14314
  1677
val hcomplex_eq_cancel_iff2 = thm"hcomplex_eq_cancel_iff2";
paulson@14314
  1678
val hRe_hsgn = thm"hRe_hsgn";
paulson@14314
  1679
val hIm_hsgn = thm"hIm_hsgn";
paulson@14314
  1680
val real_two_squares_add_zero_iff = thm"real_two_squares_add_zero_iff";
paulson@14314
  1681
val hRe_mult_i_eq = thm"hRe_mult_i_eq";
paulson@14314
  1682
val hIm_mult_i_eq = thm"hIm_mult_i_eq";
paulson@14314
  1683
val hcmod_mult_i = thm"hcmod_mult_i";
paulson@14314
  1684
val hcmod_mult_i2 = thm"hcmod_mult_i2";
paulson@14314
  1685
val harg = thm"harg";
paulson@14314
  1686
val cos_harg_i_mult_zero = thm"cos_harg_i_mult_zero";
paulson@14314
  1687
val hcomplex_of_hypreal_zero_iff = thm"hcomplex_of_hypreal_zero_iff";
paulson@14314
  1688
val complex_split_polar2 = thm"complex_split_polar2";
paulson@14314
  1689
val hcomplex_split_polar = thm"hcomplex_split_polar";
paulson@14314
  1690
val hcis = thm"hcis";
paulson@14314
  1691
val hcis_eq = thm"hcis_eq";
paulson@14314
  1692
val hrcis = thm"hrcis";
paulson@14314
  1693
val hrcis_Ex = thm"hrcis_Ex";
paulson@14314
  1694
val hRe_hcomplex_polar = thm"hRe_hcomplex_polar";
paulson@14314
  1695
val hRe_hrcis = thm"hRe_hrcis";
paulson@14314
  1696
val hIm_hcomplex_polar = thm"hIm_hcomplex_polar";
paulson@14314
  1697
val hIm_hrcis = thm"hIm_hrcis";
paulson@14314
  1698
val hcmod_complex_polar = thm"hcmod_complex_polar";
paulson@14314
  1699
val hcmod_hrcis = thm"hcmod_hrcis";
paulson@14314
  1700
val hcis_hrcis_eq = thm"hcis_hrcis_eq";
paulson@14314
  1701
val hrcis_mult = thm"hrcis_mult";
paulson@14314
  1702
val hcis_mult = thm"hcis_mult";
paulson@14314
  1703
val hcis_zero = thm"hcis_zero";
paulson@14314
  1704
val hrcis_zero_mod = thm"hrcis_zero_mod";
paulson@14314
  1705
val hrcis_zero_arg = thm"hrcis_zero_arg";
paulson@14314
  1706
val hcomplex_i_mult_minus = thm"hcomplex_i_mult_minus";
paulson@14314
  1707
val hcomplex_i_mult_minus2 = thm"hcomplex_i_mult_minus2";
paulson@14314
  1708
val hcis_hypreal_of_nat_Suc_mult = thm"hcis_hypreal_of_nat_Suc_mult";
paulson@14314
  1709
val NSDeMoivre = thm"NSDeMoivre";
paulson@14314
  1710
val hcis_hypreal_of_hypnat_Suc_mult = thm"hcis_hypreal_of_hypnat_Suc_mult";
paulson@14314
  1711
val NSDeMoivre_ext = thm"NSDeMoivre_ext";
paulson@14314
  1712
val DeMoivre2 = thm"DeMoivre2";
paulson@14314
  1713
val DeMoivre2_ext = thm"DeMoivre2_ext";
paulson@14314
  1714
val hcis_inverse = thm"hcis_inverse";
paulson@14314
  1715
val hrcis_inverse = thm"hrcis_inverse";
paulson@14314
  1716
val hRe_hcis = thm"hRe_hcis";
paulson@14314
  1717
val hIm_hcis = thm"hIm_hcis";
paulson@14314
  1718
val cos_n_hRe_hcis_pow_n = thm"cos_n_hRe_hcis_pow_n";
paulson@14314
  1719
val sin_n_hIm_hcis_pow_n = thm"sin_n_hIm_hcis_pow_n";
paulson@14314
  1720
val cos_n_hRe_hcis_hcpow_n = thm"cos_n_hRe_hcis_hcpow_n";
paulson@14314
  1721
val sin_n_hIm_hcis_hcpow_n = thm"sin_n_hIm_hcis_hcpow_n";
paulson@14314
  1722
val hexpi_add = thm"hexpi_add";
paulson@14314
  1723
val hcomplex_of_complex_add = thm"hcomplex_of_complex_add";
paulson@14314
  1724
val hcomplex_of_complex_mult = thm"hcomplex_of_complex_mult";
paulson@14314
  1725
val hcomplex_of_complex_eq_iff = thm"hcomplex_of_complex_eq_iff";
paulson@14314
  1726
val hcomplex_of_complex_minus = thm"hcomplex_of_complex_minus";
paulson@14314
  1727
val hcomplex_of_complex_one = thm"hcomplex_of_complex_one";
paulson@14314
  1728
val hcomplex_of_complex_zero = thm"hcomplex_of_complex_zero";
paulson@14314
  1729
val hcomplex_of_complex_zero_iff = thm"hcomplex_of_complex_zero_iff";
paulson@14314
  1730
val hcomplex_of_complex_inverse = thm"hcomplex_of_complex_inverse";
paulson@14314
  1731
val hcomplex_of_complex_divide = thm"hcomplex_of_complex_divide";
paulson@14314
  1732
val hRe_hcomplex_of_complex = thm"hRe_hcomplex_of_complex";
paulson@14314
  1733
val hIm_hcomplex_of_complex = thm"hIm_hcomplex_of_complex";
paulson@14314
  1734
val hcmod_hcomplex_of_complex = thm"hcmod_hcomplex_of_complex";
paulson@14314
  1735
*}
paulson@14314
  1736
paulson@13957
  1737
end