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(* Title: NSComplex.thy
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Author: Jacques D. Fleuriot
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Copyright: 2001 University of Edinburgh
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Description: Nonstandard Complex numbers
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*)
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theory NSComplex = NSInduct:
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(* Move to one of the hyperreal theories *)
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lemma hypreal_of_nat: "hypreal_of_nat m = Abs_hypreal(hyprel `` {%n. real m})"
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apply (induct_tac "m")
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apply (auto simp add: hypreal_zero_def hypreal_of_nat_Suc hypreal_zero_num hypreal_one_num hypreal_add real_of_nat_Suc)
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done
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constdefs
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hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
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"hcomplexrel == {p. EX 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 ..
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instance hcomplex :: one ..
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instance hcomplex :: plus ..
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instance hcomplex :: times ..
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instance hcomplex :: minus ..
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instance hcomplex :: inverse ..
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instance hcomplex :: 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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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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hcinv :: "hcomplex => hcomplex"
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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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(*--- 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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(* 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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(*----- 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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(*------------ 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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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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primrec
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hcomplexpow_0: "z ^ 0 = 1"
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hcomplexpow_Suc: "z ^ (Suc n) = (z::hcomplex) * (z ^ 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_iff:
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"((X,Y): hcomplexrel) = ({n. X n = Y n}: FreeUltrafilterNat)"
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apply (unfold hcomplexrel_def)
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apply fast
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done
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lemma hcomplexrel_refl: "(x,x): hcomplexrel"
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apply (simp add: hcomplexrel_iff)
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done
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lemma hcomplexrel_sym: "(x,y): hcomplexrel ==> (y,x):hcomplexrel"
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apply (auto simp add: hcomplexrel_iff eq_commute)
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done
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lemma hcomplexrel_trans:
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"[|(x,y): hcomplexrel; (y,z):hcomplexrel|] ==> (x,z):hcomplexrel"
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apply (simp add: hcomplexrel_iff)
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apply ultra
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done
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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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apply (unfold hcomplex_def hcomplexrel_def quotient_def)
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apply blast
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done
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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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declare hcomplexrel_iff [iff]
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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: "x: hcomplexrel `` {x}"
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apply (unfold hcomplexrel_def)
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apply (safe)
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apply auto
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done
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declare lemma_hcomplexrel_refl [simp]
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lemma hcomplex_empty_not_mem: "{} ~: hcomplex"
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apply (unfold hcomplex_def)
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apply (auto elim!: quotientE)
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done
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declare hcomplex_empty_not_mem [simp]
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lemma Rep_hcomplex_nonempty: "Rep_hcomplex x ~= {}"
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apply (cut_tac x = "x" in Rep_hcomplex)
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apply auto
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done
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declare Rep_hcomplex_nonempty [simp]
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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)
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done
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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 (unfold hRe_def)
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apply (rule_tac f = "Abs_hypreal" in arg_cong)
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apply (auto , 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 (unfold hIm_def)
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apply (rule_tac f = "Abs_hypreal" in arg_cong)
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apply (auto , ultra)
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done
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lemma hcomplex_hRe_hIm_cancel_iff: "(w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
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apply (rule_tac z = "z" in eq_Abs_hcomplex)
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apply (rule_tac z = "w" in eq_Abs_hcomplex)
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apply (auto simp add: hRe hIm complex_Re_Im_cancel_iff)
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apply (ultra+)
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done
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lemma hcomplex_hRe_zero: "hRe 0 = 0"
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apply (unfold hcomplex_zero_def)
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apply (simp (no_asm) add: hRe hypreal_zero_num)
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done
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declare hcomplex_hRe_zero [simp]
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lemma hcomplex_hIm_zero: "hIm 0 = 0"
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apply (unfold hcomplex_zero_def)
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apply (simp (no_asm) add: hIm hypreal_zero_num)
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done
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declare hcomplex_hIm_zero [simp]
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lemma hcomplex_hRe_one: "hRe 1 = 1"
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apply (unfold hcomplex_one_def)
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apply (simp (no_asm) add: hRe hypreal_one_num)
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done
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declare hcomplex_hRe_one [simp]
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lemma hcomplex_hIm_one: "hIm 1 = 0"
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apply (unfold hcomplex_one_def)
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apply (simp (no_asm) add: hIm hypreal_one_def hypreal_zero_num)
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done
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declare hcomplex_hIm_one [simp]
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(*-----------------------------------------------------------------------*)
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(* hcomplex_of_complex: the injection from complex to hcomplex *)
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(* ----------------------------------------------------------------------*)
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lemma inj_hcomplex_of_complex: "inj(hcomplex_of_complex)"
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apply (rule inj_onI , rule ccontr)
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apply (auto simp add: hcomplex_of_complex_def)
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done
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lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex ii"
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apply (unfold iii_def hcomplex_of_complex_def)
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apply (simp (no_asm))
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done
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(*-----------------------------------------------------------------------*)
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(* Addition for nonstandard complex numbers: hcomplex_add *)
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(* ----------------------------------------------------------------------*)
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lemma hcomplex_add_congruent2:
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"congruent2 hcomplexrel (%X Y. hcomplexrel `` {%n. X n + Y n})"
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apply (unfold congruent2_def)
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apply safe
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apply (ultra+)
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done
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lemma hcomplex_add:
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"Abs_hcomplex(hcomplexrel``{%n. X n}) + Abs_hcomplex(hcomplexrel``{%n. Y n}) =
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Abs_hcomplex(hcomplexrel``{%n. X n + Y n})"
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apply (unfold hcomplex_add_def)
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apply (rule_tac f = "Abs_hcomplex" in arg_cong)
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apply auto
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apply (ultra)
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done
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lemma hcomplex_add_commute: "(z::hcomplex) + w = w + z"
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apply (rule_tac z = "z" in eq_Abs_hcomplex)
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apply (rule_tac z = "w" in eq_Abs_hcomplex)
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apply (simp (no_asm_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 (rule_tac z = "z1" in eq_Abs_hcomplex)
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apply (rule_tac z = "z2" in eq_Abs_hcomplex)
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apply (rule_tac z = "z3" in eq_Abs_hcomplex)
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apply (simp (no_asm_simp) add: hcomplex_add complex_add_assoc)
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done
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(*For AC rewriting*)
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lemma hcomplex_add_left_commute: "(x::hcomplex)+(y+z)=y+(x+z)"
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apply (rule hcomplex_add_commute [THEN trans])
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apply (rule hcomplex_add_assoc [THEN trans])
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apply (rule hcomplex_add_commute [THEN arg_cong])
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done
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(* hcomplex addition is an AC operator *)
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lemmas hcomplex_add_ac = hcomplex_add_assoc hcomplex_add_commute
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hcomplex_add_left_commute
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lemma hcomplex_add_zero_left: "(0::hcomplex) + z = z"
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apply (unfold hcomplex_zero_def)
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apply (rule_tac z = "z" in eq_Abs_hcomplex)
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apply (simp add: hcomplex_add)
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done
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lemma hcomplex_add_zero_right: "z + (0::hcomplex) = z"
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apply (simp (no_asm) add: hcomplex_add_zero_left hcomplex_add_commute)
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done
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declare hcomplex_add_zero_left [simp] hcomplex_add_zero_right [simp]
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lemma hRe_add: "hRe(x + y) = hRe(x) + hRe(y)"
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apply (rule_tac z = "x" in eq_Abs_hcomplex)
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apply (rule_tac z = "y" in eq_Abs_hcomplex)
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apply (auto 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 (rule_tac z = "x" in eq_Abs_hcomplex)
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apply (rule_tac z = "y" in eq_Abs_hcomplex)
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apply (auto simp add: hIm hcomplex_add hypreal_add complex_Im_add)
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done
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(*-----------------------------------------------------------------------*)
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(* hypreal_minus: additive inverse on nonstandard complex *)
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(* ----------------------------------------------------------------------*)
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lemma hcomplex_minus_congruent:
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"congruent hcomplexrel (%X. hcomplexrel `` {%n. - (X n)})"
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apply (unfold congruent_def)
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apply safe
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apply (ultra+)
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done
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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 (unfold hcomplex_minus_def)
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apply (rule_tac f = "Abs_hcomplex" in arg_cong)
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apply (auto , ultra)
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done
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lemma hcomplex_add_minus_left: "-z + z = (0::hcomplex)"
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apply (rule_tac z = "z" in eq_Abs_hcomplex)
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369 |
apply (auto simp add: hcomplex_add hcomplex_minus hcomplex_zero_def)
|
|
370 |
done
|
|
371 |
declare hcomplex_add_minus_left [simp]
|
|
372 |
|
|
373 |
subsection{*Multiplication for Nonstandard Complex Numbers*}
|
|
374 |
|
|
375 |
lemma hcomplex_mult:
|
|
376 |
"Abs_hcomplex(hcomplexrel``{%n. X n}) * Abs_hcomplex(hcomplexrel``{%n. Y n}) =
|
|
377 |
Abs_hcomplex(hcomplexrel``{%n. X n * Y n})"
|
|
378 |
|
|
379 |
apply (unfold hcomplex_mult_def)
|
|
380 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
381 |
apply (auto , ultra)
|
|
382 |
done
|
|
383 |
|
|
384 |
lemma hcomplex_mult_commute: "(w::hcomplex) * z = z * w"
|
|
385 |
apply (rule_tac z = "w" in eq_Abs_hcomplex)
|
|
386 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
387 |
apply (auto simp add: hcomplex_mult complex_mult_commute)
|
|
388 |
done
|
|
389 |
|
|
390 |
lemma hcomplex_mult_assoc: "((u::hcomplex) * v) * w = u * (v * w)"
|
|
391 |
apply (rule_tac z = "u" in eq_Abs_hcomplex)
|
|
392 |
apply (rule_tac z = "v" in eq_Abs_hcomplex)
|
|
393 |
apply (rule_tac z = "w" in eq_Abs_hcomplex)
|
|
394 |
apply (auto simp add: hcomplex_mult complex_mult_assoc)
|
|
395 |
done
|
|
396 |
|
|
397 |
lemma hcomplex_mult_one_left: "(1::hcomplex) * z = z"
|
|
398 |
apply (unfold hcomplex_one_def)
|
|
399 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
400 |
apply (auto simp add: hcomplex_mult)
|
|
401 |
done
|
|
402 |
declare hcomplex_mult_one_left [simp]
|
|
403 |
|
|
404 |
lemma hcomplex_mult_zero_left: "(0::hcomplex) * z = 0"
|
|
405 |
apply (unfold hcomplex_zero_def)
|
|
406 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
407 |
apply (auto simp add: hcomplex_mult)
|
|
408 |
done
|
|
409 |
declare hcomplex_mult_zero_left [simp]
|
|
410 |
|
|
411 |
lemma hcomplex_add_mult_distrib: "((z1::hcomplex) + z2) * w = (z1 * w) + (z2 * w)"
|
|
412 |
apply (rule_tac z = "z1" in eq_Abs_hcomplex)
|
|
413 |
apply (rule_tac z = "z2" in eq_Abs_hcomplex)
|
|
414 |
apply (rule_tac z = "w" in eq_Abs_hcomplex)
|
|
415 |
apply (auto simp add: hcomplex_mult hcomplex_add complex_add_mult_distrib)
|
|
416 |
done
|
|
417 |
|
|
418 |
lemma hcomplex_zero_not_eq_one: "(0::hcomplex) ~= (1::hcomplex)"
|
|
419 |
apply (unfold hcomplex_zero_def hcomplex_one_def)
|
|
420 |
apply auto
|
|
421 |
done
|
|
422 |
declare hcomplex_zero_not_eq_one [simp]
|
|
423 |
declare hcomplex_zero_not_eq_one [THEN not_sym, simp]
|
|
424 |
|
|
425 |
|
|
426 |
subsection{*Inverse of Nonstandard Complex Number*}
|
|
427 |
|
|
428 |
lemma hcomplex_inverse:
|
|
429 |
"inverse (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
|
|
430 |
Abs_hcomplex(hcomplexrel `` {%n. inverse (X n)})"
|
|
431 |
apply (unfold hcinv_def)
|
|
432 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
433 |
apply (auto , ultra)
|
|
434 |
done
|
|
435 |
|
|
436 |
lemma hcomplex_mult_inv_left:
|
|
437 |
"z ~= (0::hcomplex) ==> inverse(z) * z = (1::hcomplex)"
|
|
438 |
apply (unfold hcomplex_zero_def hcomplex_one_def)
|
|
439 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
440 |
apply (auto simp add: hcomplex_inverse hcomplex_mult)
|
|
441 |
apply (ultra)
|
|
442 |
apply (rule ccontr)
|
|
443 |
apply (drule complex_mult_inv_left)
|
|
444 |
apply auto
|
|
445 |
done
|
|
446 |
declare hcomplex_mult_inv_left [simp]
|
|
447 |
|
14318
|
448 |
subsection {* The Field of Nonstandard Complex Numbers *}
|
|
449 |
|
|
450 |
instance hcomplex :: field
|
|
451 |
proof
|
|
452 |
fix z u v w :: hcomplex
|
|
453 |
show "(u + v) + w = u + (v + w)"
|
|
454 |
by (simp add: hcomplex_add_assoc)
|
|
455 |
show "z + w = w + z"
|
|
456 |
by (simp add: hcomplex_add_commute)
|
|
457 |
show "0 + z = z"
|
|
458 |
by (simp)
|
|
459 |
show "-z + z = 0"
|
|
460 |
by (simp)
|
|
461 |
show "z - w = z + -w"
|
|
462 |
by (simp add: hcomplex_diff_def)
|
|
463 |
show "(u * v) * w = u * (v * w)"
|
|
464 |
by (simp add: hcomplex_mult_assoc)
|
|
465 |
show "z * w = w * z"
|
|
466 |
by (simp add: hcomplex_mult_commute)
|
|
467 |
show "1 * z = z"
|
|
468 |
by (simp)
|
|
469 |
show "0 \<noteq> (1::hcomplex)"
|
|
470 |
by (rule hcomplex_zero_not_eq_one)
|
|
471 |
show "(u + v) * w = u * w + v * w"
|
|
472 |
by (simp add: hcomplex_add_mult_distrib)
|
|
473 |
assume neq: "w \<noteq> 0"
|
|
474 |
thus "z / w = z * inverse w"
|
|
475 |
by (simp add: hcomplex_divide_def)
|
|
476 |
show "inverse w * w = 1"
|
|
477 |
by (rule hcomplex_mult_inv_left)
|
|
478 |
qed
|
|
479 |
|
|
480 |
lemma HCOMPLEX_INVERSE_ZERO: "inverse (0::hcomplex) = 0"
|
|
481 |
apply (unfold hcomplex_zero_def)
|
|
482 |
apply (auto simp add: hcomplex_inverse)
|
14314
|
483 |
done
|
14318
|
484 |
|
|
485 |
lemma HCOMPLEX_DIVISION_BY_ZERO: "a / (0::hcomplex) = 0"
|
|
486 |
apply (simp (no_asm) add: hcomplex_divide_def HCOMPLEX_INVERSE_ZERO)
|
|
487 |
done
|
|
488 |
|
|
489 |
instance hcomplex :: division_by_zero
|
|
490 |
proof
|
|
491 |
fix x :: hcomplex
|
|
492 |
show "inverse 0 = (0::hcomplex)" by (rule HCOMPLEX_INVERSE_ZERO)
|
|
493 |
show "x/0 = 0" by (rule HCOMPLEX_DIVISION_BY_ZERO)
|
|
494 |
qed
|
14314
|
495 |
|
|
496 |
lemma hcomplex_mult_left_cancel: "(c::hcomplex) ~= (0::hcomplex) ==> (c*a=c*b) = (a=b)"
|
14318
|
497 |
by (simp add: field_mult_cancel_left)
|
|
498 |
|
|
499 |
subsection{*More Minus Laws*}
|
|
500 |
|
|
501 |
lemma inj_hcomplex_minus: "inj(%z::hcomplex. -z)"
|
|
502 |
apply (rule inj_onI)
|
|
503 |
apply (drule_tac f = "uminus" in arg_cong)
|
|
504 |
apply simp
|
|
505 |
done
|
|
506 |
|
|
507 |
lemma hRe_minus: "hRe(-z) = - hRe(z)"
|
|
508 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
509 |
apply (auto simp add: hRe hcomplex_minus hypreal_minus complex_Re_minus)
|
|
510 |
done
|
|
511 |
|
|
512 |
lemma hIm_minus: "hIm(-z) = - hIm(z)"
|
|
513 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
514 |
apply (auto simp add: hIm hcomplex_minus hypreal_minus complex_Im_minus)
|
|
515 |
done
|
|
516 |
|
|
517 |
lemma hcomplex_add_minus_eq_minus:
|
|
518 |
"x + y = (0::hcomplex) ==> x = -y"
|
|
519 |
apply (drule Ring_and_Field.equals_zero_I)
|
|
520 |
apply (simp add: minus_equation_iff [of x y])
|
|
521 |
done
|
|
522 |
|
|
523 |
lemma hcomplex_minus_add_distrib: "-(x + y) = -x + -(y::hcomplex)"
|
|
524 |
apply (rule Ring_and_Field.minus_add_distrib)
|
|
525 |
done
|
|
526 |
|
|
527 |
lemma hcomplex_add_left_cancel: "((x::hcomplex) + y = x + z) = (y = z)"
|
|
528 |
apply (rule Ring_and_Field.add_left_cancel)
|
|
529 |
done
|
|
530 |
declare hcomplex_add_left_cancel [iff]
|
|
531 |
|
|
532 |
lemma hcomplex_add_right_cancel: "(y + (x::hcomplex)= z + x) = (y = z)"
|
|
533 |
apply (rule Ring_and_Field.add_right_cancel)
|
|
534 |
done
|
|
535 |
declare hcomplex_add_right_cancel [iff]
|
|
536 |
|
|
537 |
subsection{*More Multiplication Laws*}
|
|
538 |
|
|
539 |
lemma hcomplex_mult_left_commute: "(x::hcomplex) * (y * z) = y * (x * z)"
|
|
540 |
apply (rule Ring_and_Field.mult_left_commute)
|
|
541 |
done
|
|
542 |
|
|
543 |
lemmas hcomplex_mult_ac = hcomplex_mult_assoc hcomplex_mult_commute
|
|
544 |
hcomplex_mult_left_commute
|
|
545 |
|
|
546 |
lemma hcomplex_mult_one_right: "z * (1::hcomplex) = z"
|
|
547 |
apply (rule Ring_and_Field.mult_1_right)
|
|
548 |
done
|
|
549 |
|
|
550 |
lemma hcomplex_mult_zero_right: "z * (0::hcomplex) = 0"
|
|
551 |
apply (rule Ring_and_Field.mult_right_zero)
|
|
552 |
done
|
|
553 |
|
|
554 |
lemma hcomplex_minus_mult_eq1: "-(x * y) = -x * (y::hcomplex)"
|
|
555 |
apply (rule Ring_and_Field.minus_mult_left)
|
|
556 |
done
|
|
557 |
|
|
558 |
declare hcomplex_minus_mult_eq1 [symmetric, simp]
|
|
559 |
|
|
560 |
lemma hcomplex_minus_mult_eq2: "-(x * y) = x * -(y::hcomplex)"
|
|
561 |
apply (rule Ring_and_Field.minus_mult_right)
|
|
562 |
done
|
|
563 |
|
|
564 |
declare hcomplex_minus_mult_eq2 [symmetric, simp]
|
|
565 |
|
|
566 |
lemma hcomplex_mult_minus_one: "- 1 * (z::hcomplex) = -z"
|
|
567 |
apply (simp (no_asm))
|
|
568 |
done
|
|
569 |
declare hcomplex_mult_minus_one [simp]
|
|
570 |
|
|
571 |
lemma hcomplex_mult_minus_one_right: "(z::hcomplex) * - 1 = -z"
|
|
572 |
apply (subst hcomplex_mult_commute)
|
|
573 |
apply (simp (no_asm))
|
|
574 |
done
|
|
575 |
declare hcomplex_mult_minus_one_right [simp]
|
|
576 |
|
|
577 |
lemma hcomplex_add_mult_distrib2: "(w::hcomplex) * (z1 + z2) = (w * z1) + (w * z2)"
|
|
578 |
apply (rule Ring_and_Field.right_distrib)
|
14314
|
579 |
done
|
|
580 |
|
|
581 |
lemma hcomplex_mult_right_cancel: "(c::hcomplex) ~= (0::hcomplex) ==> (a*c=b*c) = (a=b)"
|
14318
|
582 |
apply (simp add: Ring_and_Field.field_mult_cancel_right);
|
14314
|
583 |
done
|
|
584 |
|
|
585 |
lemma hcomplex_inverse_not_zero: "z ~= (0::hcomplex) ==> inverse(z) ~= 0"
|
14318
|
586 |
apply (simp add: );
|
14314
|
587 |
done
|
|
588 |
|
|
589 |
lemma hcomplex_mult_not_zero: "[| x ~= (0::hcomplex); y ~= 0 |] ==> x * y ~= 0"
|
14318
|
590 |
apply (simp add: Ring_and_Field.field_mult_eq_0_iff);
|
14314
|
591 |
done
|
|
592 |
|
|
593 |
lemmas hcomplex_mult_not_zeroE = hcomplex_mult_not_zero [THEN notE, standard]
|
|
594 |
|
|
595 |
lemma hcomplex_minus_inverse: "inverse(-x) = -inverse(x::hcomplex)"
|
14318
|
596 |
apply (rule Ring_and_Field.inverse_minus_eq)
|
14314
|
597 |
done
|
|
598 |
|
|
599 |
|
14318
|
600 |
subsection{*Subraction and Division*}
|
14314
|
601 |
|
14318
|
602 |
lemma hcomplex_diff:
|
|
603 |
"Abs_hcomplex(hcomplexrel``{%n. X n}) - Abs_hcomplex(hcomplexrel``{%n. Y n}) =
|
|
604 |
Abs_hcomplex(hcomplexrel``{%n. X n - Y n})"
|
|
605 |
apply (unfold hcomplex_diff_def)
|
|
606 |
apply (auto simp add: hcomplex_minus hcomplex_add complex_diff_def)
|
14314
|
607 |
done
|
|
608 |
|
14318
|
609 |
lemma hcomplex_diff_eq_eq: "((x::hcomplex) - y = z) = (x = z + y)"
|
|
610 |
apply (rule Ring_and_Field.diff_eq_eq)
|
14314
|
611 |
done
|
|
612 |
|
|
613 |
lemma hcomplex_add_divide_distrib: "(x+y)/(z::hcomplex) = x/z + y/z"
|
14318
|
614 |
apply (rule Ring_and_Field.add_divide_distrib)
|
14314
|
615 |
done
|
|
616 |
|
|
617 |
|
|
618 |
subsection{*Embedding Properties for @{term hcomplex_of_hypreal} Map*}
|
|
619 |
|
|
620 |
lemma hcomplex_of_hypreal:
|
|
621 |
"hcomplex_of_hypreal (Abs_hypreal(hyprel `` {%n. X n})) =
|
|
622 |
Abs_hcomplex(hcomplexrel `` {%n. complex_of_real (X n)})"
|
|
623 |
apply (unfold hcomplex_of_hypreal_def)
|
|
624 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
625 |
apply auto
|
|
626 |
apply (ultra)
|
|
627 |
done
|
|
628 |
|
|
629 |
lemma inj_hcomplex_of_hypreal: "inj hcomplex_of_hypreal"
|
|
630 |
apply (rule inj_onI)
|
|
631 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
632 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
633 |
apply (auto simp add: hcomplex_of_hypreal)
|
|
634 |
done
|
|
635 |
|
|
636 |
lemma hcomplex_of_hypreal_cancel_iff: "(hcomplex_of_hypreal x = hcomplex_of_hypreal y) = (x = y)"
|
|
637 |
apply (auto dest: inj_hcomplex_of_hypreal [THEN injD])
|
|
638 |
done
|
|
639 |
declare hcomplex_of_hypreal_cancel_iff [iff]
|
|
640 |
|
|
641 |
lemma hcomplex_of_hypreal_minus: "hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x"
|
|
642 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
643 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_minus hypreal_minus complex_of_real_minus)
|
|
644 |
done
|
|
645 |
|
|
646 |
lemma hcomplex_of_hypreal_inverse: "hcomplex_of_hypreal(inverse x) = inverse(hcomplex_of_hypreal x)"
|
|
647 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
648 |
apply (auto simp add: hcomplex_of_hypreal hypreal_inverse hcomplex_inverse complex_of_real_inverse)
|
|
649 |
done
|
|
650 |
|
|
651 |
lemma hcomplex_of_hypreal_add: "hcomplex_of_hypreal x + hcomplex_of_hypreal y =
|
|
652 |
hcomplex_of_hypreal (x + y)"
|
|
653 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
654 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
655 |
apply (auto simp add: hcomplex_of_hypreal hypreal_add hcomplex_add complex_of_real_add)
|
|
656 |
done
|
|
657 |
|
|
658 |
lemma hcomplex_of_hypreal_diff:
|
|
659 |
"hcomplex_of_hypreal x - hcomplex_of_hypreal y =
|
|
660 |
hcomplex_of_hypreal (x - y)"
|
|
661 |
apply (unfold hcomplex_diff_def)
|
|
662 |
apply (auto simp add: hcomplex_of_hypreal_minus [symmetric] hcomplex_of_hypreal_add hypreal_diff_def)
|
|
663 |
done
|
|
664 |
|
|
665 |
lemma hcomplex_of_hypreal_mult: "hcomplex_of_hypreal x * hcomplex_of_hypreal y =
|
|
666 |
hcomplex_of_hypreal (x * y)"
|
|
667 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
668 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
669 |
apply (auto simp add: hcomplex_of_hypreal hypreal_mult hcomplex_mult complex_of_real_mult)
|
|
670 |
done
|
|
671 |
|
|
672 |
lemma hcomplex_of_hypreal_divide:
|
|
673 |
"hcomplex_of_hypreal x / hcomplex_of_hypreal y = hcomplex_of_hypreal(x/y)"
|
|
674 |
apply (unfold hcomplex_divide_def)
|
|
675 |
apply (case_tac "y=0")
|
|
676 |
apply (simp (no_asm_simp) add: HYPREAL_DIVISION_BY_ZERO HYPREAL_INVERSE_ZERO HCOMPLEX_INVERSE_ZERO)
|
|
677 |
apply (auto simp add: hcomplex_of_hypreal_mult hcomplex_of_hypreal_inverse [symmetric])
|
|
678 |
apply (simp (no_asm) add: hypreal_divide_def)
|
|
679 |
done
|
|
680 |
|
|
681 |
lemma hcomplex_of_hypreal_one [simp]:
|
|
682 |
"hcomplex_of_hypreal 1 = 1"
|
|
683 |
apply (unfold hcomplex_one_def)
|
|
684 |
apply (auto simp add: hcomplex_of_hypreal hypreal_one_num)
|
|
685 |
done
|
|
686 |
|
|
687 |
lemma hcomplex_of_hypreal_zero [simp]:
|
|
688 |
"hcomplex_of_hypreal 0 = 0"
|
|
689 |
apply (unfold hcomplex_zero_def hypreal_zero_def)
|
|
690 |
apply (auto simp add: hcomplex_of_hypreal)
|
|
691 |
done
|
|
692 |
|
|
693 |
lemma hcomplex_of_hypreal_pow:
|
|
694 |
"hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n"
|
|
695 |
apply (induct_tac "n")
|
|
696 |
apply (auto simp add: hcomplex_of_hypreal_mult [symmetric])
|
|
697 |
done
|
|
698 |
|
|
699 |
lemma hRe_hcomplex_of_hypreal: "hRe(hcomplex_of_hypreal z) = z"
|
|
700 |
apply (rule_tac z = "z" in eq_Abs_hypreal)
|
|
701 |
apply (auto simp add: hcomplex_of_hypreal hRe)
|
|
702 |
done
|
|
703 |
declare hRe_hcomplex_of_hypreal [simp]
|
|
704 |
|
|
705 |
lemma hIm_hcomplex_of_hypreal: "hIm(hcomplex_of_hypreal z) = 0"
|
|
706 |
apply (rule_tac z = "z" in eq_Abs_hypreal)
|
|
707 |
apply (auto simp add: hcomplex_of_hypreal hIm hypreal_zero_num)
|
|
708 |
done
|
|
709 |
declare hIm_hcomplex_of_hypreal [simp]
|
|
710 |
|
|
711 |
lemma hcomplex_of_hypreal_epsilon_not_zero: "hcomplex_of_hypreal epsilon ~= 0"
|
|
712 |
apply (auto simp add: hcomplex_of_hypreal epsilon_def hcomplex_zero_def)
|
|
713 |
done
|
|
714 |
declare hcomplex_of_hypreal_epsilon_not_zero [simp]
|
|
715 |
|
14318
|
716 |
|
|
717 |
subsection{*Modulus (Absolute Value) of Nonstandard Complex Number*}
|
14314
|
718 |
|
|
719 |
lemma hcmod:
|
|
720 |
"hcmod (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
|
|
721 |
Abs_hypreal(hyprel `` {%n. cmod (X n)})"
|
|
722 |
|
|
723 |
apply (unfold hcmod_def)
|
|
724 |
apply (rule_tac f = "Abs_hypreal" in arg_cong)
|
|
725 |
apply (auto , ultra)
|
|
726 |
done
|
|
727 |
|
|
728 |
lemma hcmod_zero [simp]:
|
|
729 |
"hcmod(0) = 0"
|
|
730 |
apply (unfold hcomplex_zero_def hypreal_zero_def)
|
|
731 |
apply (auto simp add: hcmod)
|
|
732 |
done
|
|
733 |
|
|
734 |
lemma hcmod_one:
|
|
735 |
"hcmod(1) = 1"
|
|
736 |
apply (unfold hcomplex_one_def)
|
|
737 |
apply (auto simp add: hcmod hypreal_one_num)
|
|
738 |
done
|
|
739 |
declare hcmod_one [simp]
|
|
740 |
|
|
741 |
lemma hcmod_hcomplex_of_hypreal: "hcmod(hcomplex_of_hypreal x) = abs x"
|
|
742 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
743 |
apply (auto simp add: hcmod hcomplex_of_hypreal hypreal_hrabs)
|
|
744 |
done
|
|
745 |
declare hcmod_hcomplex_of_hypreal [simp]
|
|
746 |
|
|
747 |
lemma hcomplex_of_hypreal_abs: "hcomplex_of_hypreal (abs x) =
|
|
748 |
hcomplex_of_hypreal(hcmod(hcomplex_of_hypreal x))"
|
|
749 |
apply (simp (no_asm))
|
|
750 |
done
|
|
751 |
|
|
752 |
|
|
753 |
subsection{*Conjugation*}
|
|
754 |
|
|
755 |
lemma hcnj:
|
|
756 |
"hcnj (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
|
14318
|
757 |
Abs_hcomplex(hcomplexrel `` {%n. cnj(X n)})"
|
14314
|
758 |
apply (unfold hcnj_def)
|
|
759 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
760 |
apply (auto , ultra)
|
|
761 |
done
|
|
762 |
|
|
763 |
lemma inj_hcnj: "inj hcnj"
|
|
764 |
apply (rule inj_onI)
|
|
765 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
766 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
767 |
apply (auto simp add: hcnj)
|
|
768 |
done
|
|
769 |
|
|
770 |
lemma hcomplex_hcnj_cancel_iff: "(hcnj x = hcnj y) = (x = y)"
|
|
771 |
apply (auto dest: inj_hcnj [THEN injD])
|
|
772 |
done
|
|
773 |
declare hcomplex_hcnj_cancel_iff [simp]
|
|
774 |
|
|
775 |
lemma hcomplex_hcnj_hcnj: "hcnj (hcnj z) = z"
|
|
776 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
777 |
apply (auto simp add: hcnj)
|
|
778 |
done
|
|
779 |
declare hcomplex_hcnj_hcnj [simp]
|
|
780 |
|
|
781 |
lemma hcomplex_hcnj_hcomplex_of_hypreal: "hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
|
|
782 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
783 |
apply (auto simp add: hcnj hcomplex_of_hypreal)
|
|
784 |
done
|
|
785 |
declare hcomplex_hcnj_hcomplex_of_hypreal [simp]
|
|
786 |
|
|
787 |
lemma hcomplex_hmod_hcnj: "hcmod (hcnj z) = hcmod z"
|
|
788 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
789 |
apply (auto simp add: hcnj hcmod)
|
|
790 |
done
|
|
791 |
declare hcomplex_hmod_hcnj [simp]
|
|
792 |
|
|
793 |
lemma hcomplex_hcnj_minus: "hcnj (-z) = - hcnj z"
|
|
794 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
795 |
apply (auto simp add: hcnj hcomplex_minus complex_cnj_minus)
|
|
796 |
done
|
|
797 |
|
|
798 |
lemma hcomplex_hcnj_inverse: "hcnj(inverse z) = inverse(hcnj z)"
|
|
799 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
800 |
apply (auto simp add: hcnj hcomplex_inverse complex_cnj_inverse)
|
|
801 |
done
|
|
802 |
|
|
803 |
lemma hcomplex_hcnj_add: "hcnj(w + z) = hcnj(w) + hcnj(z)"
|
|
804 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
805 |
apply (rule_tac z = "w" in eq_Abs_hcomplex)
|
|
806 |
apply (auto simp add: hcnj hcomplex_add complex_cnj_add)
|
|
807 |
done
|
|
808 |
|
|
809 |
lemma hcomplex_hcnj_diff: "hcnj(w - z) = hcnj(w) - hcnj(z)"
|
|
810 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
811 |
apply (rule_tac z = "w" in eq_Abs_hcomplex)
|
|
812 |
apply (auto simp add: hcnj hcomplex_diff complex_cnj_diff)
|
|
813 |
done
|
|
814 |
|
|
815 |
lemma hcomplex_hcnj_mult: "hcnj(w * z) = hcnj(w) * hcnj(z)"
|
|
816 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
817 |
apply (rule_tac z = "w" in eq_Abs_hcomplex)
|
|
818 |
apply (auto simp add: hcnj hcomplex_mult complex_cnj_mult)
|
|
819 |
done
|
|
820 |
|
|
821 |
lemma hcomplex_hcnj_divide: "hcnj(w / z) = (hcnj w)/(hcnj z)"
|
|
822 |
apply (unfold hcomplex_divide_def)
|
|
823 |
apply (simp (no_asm) add: hcomplex_hcnj_mult hcomplex_hcnj_inverse)
|
|
824 |
done
|
|
825 |
|
|
826 |
lemma hcnj_one: "hcnj 1 = 1"
|
|
827 |
apply (unfold hcomplex_one_def)
|
|
828 |
apply (simp (no_asm) add: hcnj)
|
|
829 |
done
|
|
830 |
declare hcnj_one [simp]
|
|
831 |
|
|
832 |
lemma hcomplex_hcnj_pow: "hcnj(z ^ n) = hcnj(z) ^ n"
|
|
833 |
apply (induct_tac "n")
|
|
834 |
apply (auto simp add: hcomplex_hcnj_mult)
|
|
835 |
done
|
|
836 |
|
|
837 |
(* MOVE to NSComplexBin
|
|
838 |
Goal "z + hcnj z =
|
|
839 |
hcomplex_of_hypreal (2 * hRe(z))"
|
|
840 |
by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
|
|
841 |
by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
|
|
842 |
hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
|
|
843 |
qed "hcomplex_add_hcnj";
|
|
844 |
|
|
845 |
Goal "z - hcnj z = \
|
|
846 |
\ hcomplex_of_hypreal (hypreal_of_real 2 * hIm(z)) * iii";
|
|
847 |
by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
|
|
848 |
by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
|
|
849 |
hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
|
|
850 |
complex_diff_cnj,iii_def,hcomplex_mult]));
|
|
851 |
qed "hcomplex_diff_hcnj";
|
|
852 |
*)
|
|
853 |
|
|
854 |
lemma hcomplex_hcnj_zero:
|
|
855 |
"hcnj 0 = 0"
|
|
856 |
apply (unfold hcomplex_zero_def)
|
|
857 |
apply (auto simp add: hcnj)
|
|
858 |
done
|
|
859 |
declare hcomplex_hcnj_zero [simp]
|
|
860 |
|
|
861 |
lemma hcomplex_hcnj_zero_iff: "(hcnj z = 0) = (z = 0)"
|
|
862 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
863 |
apply (auto simp add: hcomplex_zero_def hcnj)
|
|
864 |
done
|
|
865 |
declare hcomplex_hcnj_zero_iff [iff]
|
|
866 |
|
|
867 |
lemma hcomplex_mult_hcnj: "z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
|
|
868 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
869 |
apply (auto simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add hypreal_mult complex_mult_cnj two_eq_Suc_Suc)
|
|
870 |
done
|
|
871 |
|
|
872 |
|
|
873 |
(*---------------------------------------------------------------------------*)
|
|
874 |
(* some algebra etc. *)
|
|
875 |
(*---------------------------------------------------------------------------*)
|
|
876 |
|
|
877 |
lemma hcomplex_mult_zero_iff: "(x*y = (0::hcomplex)) = (x = 0 | y = 0)"
|
|
878 |
apply auto
|
|
879 |
apply (auto intro: ccontr dest: hcomplex_mult_not_zero)
|
|
880 |
done
|
|
881 |
declare hcomplex_mult_zero_iff [simp]
|
|
882 |
|
|
883 |
lemma hcomplex_add_left_cancel_zero: "(x + y = x) = (y = (0::hcomplex))"
|
|
884 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
885 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
886 |
apply (auto simp add: hcomplex_add hcomplex_zero_def)
|
|
887 |
done
|
|
888 |
declare hcomplex_add_left_cancel_zero [simp]
|
|
889 |
|
|
890 |
lemma hcomplex_diff_mult_distrib:
|
|
891 |
"((z1::hcomplex) - z2) * w = (z1 * w) - (z2 * w)"
|
|
892 |
apply (unfold hcomplex_diff_def)
|
|
893 |
apply (simp (no_asm) add: hcomplex_add_mult_distrib)
|
|
894 |
done
|
|
895 |
|
|
896 |
lemma hcomplex_diff_mult_distrib2:
|
|
897 |
"(w::hcomplex) * (z1 - z2) = (w * z1) - (w * z2)"
|
|
898 |
apply (unfold hcomplex_diff_def)
|
|
899 |
apply (simp (no_asm) add: hcomplex_add_mult_distrib2)
|
|
900 |
done
|
|
901 |
|
|
902 |
(*---------------------------------------------------------------------------*)
|
|
903 |
(* More theorems about hcmod *)
|
|
904 |
(*---------------------------------------------------------------------------*)
|
|
905 |
|
|
906 |
lemma hcomplex_hcmod_eq_zero_cancel: "(hcmod x = 0) = (x = 0)"
|
|
907 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
908 |
apply (auto simp add: hcmod hcomplex_zero_def hypreal_zero_num)
|
|
909 |
done
|
|
910 |
declare hcomplex_hcmod_eq_zero_cancel [simp]
|
|
911 |
|
|
912 |
(* not proved already? strange! *)
|
|
913 |
lemma hypreal_of_nat_le_iff:
|
|
914 |
"(hypreal_of_nat n <= hypreal_of_nat m) = (n <= m)"
|
|
915 |
apply (unfold hypreal_le_def)
|
|
916 |
apply auto
|
|
917 |
done
|
|
918 |
declare hypreal_of_nat_le_iff [simp]
|
|
919 |
|
|
920 |
lemma hypreal_of_nat_ge_zero: "0 <= hypreal_of_nat n"
|
|
921 |
apply (simp (no_asm) add: hypreal_of_nat_zero [symmetric]
|
|
922 |
del: hypreal_of_nat_zero)
|
|
923 |
done
|
|
924 |
declare hypreal_of_nat_ge_zero [simp]
|
|
925 |
|
|
926 |
declare hypreal_of_nat_ge_zero [THEN hrabs_eqI1, simp]
|
|
927 |
|
|
928 |
lemma hypreal_of_hypnat_ge_zero: "0 <= hypreal_of_hypnat n"
|
|
929 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
930 |
apply (simp (no_asm_simp) add: hypreal_of_hypnat hypreal_zero_num hypreal_le)
|
|
931 |
done
|
|
932 |
declare hypreal_of_hypnat_ge_zero [simp]
|
|
933 |
|
|
934 |
declare hypreal_of_hypnat_ge_zero [THEN hrabs_eqI1, simp]
|
|
935 |
|
|
936 |
lemma hcmod_hcomplex_of_hypreal_of_nat: "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n"
|
|
937 |
apply auto
|
|
938 |
done
|
|
939 |
declare hcmod_hcomplex_of_hypreal_of_nat [simp]
|
|
940 |
|
|
941 |
lemma hcmod_hcomplex_of_hypreal_of_hypnat: "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
|
|
942 |
apply auto
|
|
943 |
done
|
|
944 |
declare hcmod_hcomplex_of_hypreal_of_hypnat [simp]
|
|
945 |
|
|
946 |
lemma hcmod_minus: "hcmod (-x) = hcmod(x)"
|
|
947 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
948 |
apply (auto simp add: hcmod hcomplex_minus)
|
|
949 |
done
|
|
950 |
declare hcmod_minus [simp]
|
|
951 |
|
|
952 |
lemma hcmod_mult_hcnj: "hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
|
|
953 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
954 |
apply (auto simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj two_eq_Suc_Suc)
|
|
955 |
done
|
|
956 |
|
|
957 |
lemma hcmod_ge_zero: "(0::hypreal) <= hcmod x"
|
|
958 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
959 |
apply (auto simp add: hcmod hypreal_zero_num hypreal_le)
|
|
960 |
done
|
|
961 |
declare hcmod_ge_zero [simp]
|
|
962 |
|
|
963 |
lemma hrabs_hcmod_cancel: "abs(hcmod x) = hcmod x"
|
|
964 |
apply (auto intro: hrabs_eqI1)
|
|
965 |
done
|
|
966 |
declare hrabs_hcmod_cancel [simp]
|
|
967 |
|
|
968 |
lemma hcmod_mult: "hcmod(x*y) = hcmod(x) * hcmod(y)"
|
|
969 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
970 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
971 |
apply (auto simp add: hcmod hcomplex_mult hypreal_mult complex_mod_mult)
|
|
972 |
done
|
|
973 |
|
|
974 |
lemma hcmod_add_squared_eq:
|
|
975 |
"hcmod(x + y) ^ 2 = hcmod(x) ^ 2 + hcmod(y) ^ 2 + 2 * hRe(x * hcnj y)"
|
|
976 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
977 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
978 |
apply (auto simp add: hcmod hcomplex_add hypreal_mult hRe hcnj hcomplex_mult
|
|
979 |
two_eq_Suc_Suc realpow_two [symmetric]
|
|
980 |
simp del: realpow_Suc)
|
|
981 |
apply (auto simp add: two_eq_Suc_Suc [symmetric] complex_mod_add_squared_eq
|
|
982 |
hypreal_add [symmetric] hypreal_mult [symmetric]
|
|
983 |
hypreal_of_real_def [symmetric])
|
|
984 |
done
|
|
985 |
|
|
986 |
lemma hcomplex_hRe_mult_hcnj_le_hcmod: "hRe(x * hcnj y) <= hcmod(x * hcnj y)"
|
|
987 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
988 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
989 |
apply (auto simp add: hcmod hcnj hcomplex_mult hRe hypreal_le)
|
|
990 |
done
|
|
991 |
declare hcomplex_hRe_mult_hcnj_le_hcmod [simp]
|
|
992 |
|
|
993 |
lemma hcomplex_hRe_mult_hcnj_le_hcmod2: "hRe(x * hcnj y) <= hcmod(x * y)"
|
|
994 |
apply (cut_tac x = "x" and y = "y" in hcomplex_hRe_mult_hcnj_le_hcmod)
|
|
995 |
apply (simp add: hcmod_mult)
|
|
996 |
done
|
|
997 |
declare hcomplex_hRe_mult_hcnj_le_hcmod2 [simp]
|
|
998 |
|
|
999 |
lemma hcmod_triangle_squared: "hcmod (x + y) ^ 2 <= (hcmod(x) + hcmod(y)) ^ 2"
|
|
1000 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1001 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
1002 |
apply (auto simp add: hcmod hcnj hcomplex_add hypreal_mult hypreal_add
|
|
1003 |
hypreal_le realpow_two [symmetric] two_eq_Suc_Suc
|
|
1004 |
simp del: realpow_Suc)
|
|
1005 |
apply (simp (no_asm) add: two_eq_Suc_Suc [symmetric])
|
|
1006 |
done
|
|
1007 |
declare hcmod_triangle_squared [simp]
|
|
1008 |
|
|
1009 |
lemma hcmod_triangle_ineq: "hcmod (x + y) <= hcmod(x) + hcmod(y)"
|
|
1010 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1011 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
1012 |
apply (auto simp add: hcmod hcomplex_add hypreal_add hypreal_le)
|
|
1013 |
done
|
|
1014 |
declare hcmod_triangle_ineq [simp]
|
|
1015 |
|
|
1016 |
lemma hcmod_triangle_ineq2: "hcmod(b + a) - hcmod b <= hcmod a"
|
|
1017 |
apply (cut_tac x1 = "b" and y1 = "a" and c = "-hcmod b" in hcmod_triangle_ineq [THEN add_right_mono])
|
|
1018 |
apply (simp add: hypreal_add_ac)
|
|
1019 |
done
|
|
1020 |
declare hcmod_triangle_ineq2 [simp]
|
|
1021 |
|
|
1022 |
lemma hcmod_diff_commute: "hcmod (x - y) = hcmod (y - x)"
|
|
1023 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1024 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
1025 |
apply (auto simp add: hcmod hcomplex_diff complex_mod_diff_commute)
|
|
1026 |
done
|
|
1027 |
|
|
1028 |
lemma hcmod_add_less: "[| hcmod x < r; hcmod y < s |] ==> hcmod (x + y) < r + s"
|
|
1029 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1030 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
1031 |
apply (rule_tac z = "r" in eq_Abs_hypreal)
|
|
1032 |
apply (rule_tac z = "s" in eq_Abs_hypreal)
|
|
1033 |
apply (auto simp add: hcmod hcomplex_add hypreal_add hypreal_less)
|
|
1034 |
apply ultra
|
|
1035 |
apply (auto intro: complex_mod_add_less)
|
|
1036 |
done
|
|
1037 |
|
|
1038 |
lemma hcmod_mult_less: "[| hcmod x < r; hcmod y < s |] ==> hcmod (x * y) < r * s"
|
|
1039 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1040 |
apply (rule_tac z = "y" in eq_Abs_hcomplex)
|
|
1041 |
apply (rule_tac z = "r" in eq_Abs_hypreal)
|
|
1042 |
apply (rule_tac z = "s" in eq_Abs_hypreal)
|
|
1043 |
apply (auto simp add: hcmod hypreal_mult hypreal_less hcomplex_mult)
|
|
1044 |
apply ultra
|
|
1045 |
apply (auto intro: complex_mod_mult_less)
|
|
1046 |
done
|
|
1047 |
|
|
1048 |
lemma hcmod_diff_ineq: "hcmod(a) - hcmod(b) <= hcmod(a + b)"
|
|
1049 |
apply (rule_tac z = "a" in eq_Abs_hcomplex)
|
|
1050 |
apply (rule_tac z = "b" in eq_Abs_hcomplex)
|
|
1051 |
apply (auto simp add: hcmod hcomplex_add hypreal_diff hypreal_le)
|
|
1052 |
done
|
|
1053 |
declare hcmod_diff_ineq [simp]
|
|
1054 |
|
|
1055 |
|
|
1056 |
|
|
1057 |
subsection{*A Few Nonlinear Theorems*}
|
|
1058 |
|
|
1059 |
lemma hcpow:
|
|
1060 |
"Abs_hcomplex(hcomplexrel``{%n. X n}) hcpow
|
|
1061 |
Abs_hypnat(hypnatrel``{%n. Y n}) =
|
|
1062 |
Abs_hcomplex(hcomplexrel``{%n. X n ^ Y n})"
|
|
1063 |
apply (unfold hcpow_def)
|
|
1064 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
1065 |
apply (auto , ultra)
|
|
1066 |
done
|
|
1067 |
|
|
1068 |
lemma hcomplex_of_hypreal_hyperpow: "hcomplex_of_hypreal (x pow n) = (hcomplex_of_hypreal x) hcpow n"
|
|
1069 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1070 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1071 |
apply (auto simp add: hcomplex_of_hypreal hyperpow hcpow complex_of_real_pow)
|
|
1072 |
done
|
|
1073 |
|
|
1074 |
lemma hcmod_hcomplexpow: "hcmod(x ^ n) = hcmod(x) ^ n"
|
|
1075 |
apply (induct_tac "n")
|
|
1076 |
apply (auto simp add: hcmod_mult)
|
|
1077 |
done
|
|
1078 |
|
|
1079 |
lemma hcmod_hcpow: "hcmod(x hcpow n) = hcmod(x) pow n"
|
|
1080 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1081 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1082 |
apply (auto simp add: hcpow hyperpow hcmod complex_mod_complexpow)
|
|
1083 |
done
|
|
1084 |
|
|
1085 |
lemma hcomplexpow_minus: "(-x::hcomplex) ^ n = (if even n then (x ^ n) else -(x ^ n))"
|
|
1086 |
apply (induct_tac "n")
|
|
1087 |
apply auto
|
|
1088 |
done
|
|
1089 |
|
|
1090 |
lemma hcpow_minus: "(-x::hcomplex) hcpow n =
|
|
1091 |
(if ( *pNat* even) n then (x hcpow n) else -(x hcpow n))"
|
|
1092 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1093 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1094 |
apply (auto simp add: hcpow hyperpow starPNat hcomplex_minus)
|
|
1095 |
apply ultra
|
|
1096 |
apply (auto simp add: complexpow_minus)
|
|
1097 |
apply ultra
|
|
1098 |
done
|
|
1099 |
|
|
1100 |
lemma hccomplex_inverse_minus: "inverse(-x) = - inverse (x::hcomplex)"
|
|
1101 |
apply (rule_tac z = "x" in eq_Abs_hcomplex)
|
|
1102 |
apply (auto simp add: hcomplex_inverse hcomplex_minus complex_inverse_minus)
|
|
1103 |
done
|
|
1104 |
|
|
1105 |
lemma hcomplex_div_one: "x / (1::hcomplex) = x"
|
|
1106 |
apply (unfold hcomplex_divide_def)
|
|
1107 |
apply (simp (no_asm))
|
|
1108 |
done
|
|
1109 |
declare hcomplex_div_one [simp]
|
|
1110 |
|
|
1111 |
lemma hcmod_hcomplex_inverse: "hcmod(inverse x) = inverse(hcmod x)"
|
|
1112 |
apply (case_tac "x = 0", simp add: HCOMPLEX_INVERSE_ZERO)
|
|
1113 |
apply (rule_tac c1 = "hcmod x" in hypreal_mult_left_cancel [THEN iffD1])
|
|
1114 |
apply (auto simp add: hcmod_mult [symmetric])
|
|
1115 |
done
|
|
1116 |
|
|
1117 |
lemma hcmod_divide:
|
|
1118 |
"hcmod(x/y) = hcmod(x)/(hcmod y)"
|
|
1119 |
apply (unfold hcomplex_divide_def hypreal_divide_def)
|
|
1120 |
apply (auto simp add: hcmod_mult hcmod_hcomplex_inverse)
|
|
1121 |
done
|
|
1122 |
|
|
1123 |
lemma hcomplex_inverse_divide:
|
|
1124 |
"inverse(x/y) = y/(x::hcomplex)"
|
|
1125 |
apply (unfold hcomplex_divide_def)
|
14318
|
1126 |
apply (auto simp add: inverse_mult_distrib hcomplex_mult_commute)
|
14314
|
1127 |
done
|
|
1128 |
declare hcomplex_inverse_divide [simp]
|
|
1129 |
|
|
1130 |
lemma hcomplexpow_mult: "((r::hcomplex) * s) ^ n = (r ^ n) * (s ^ n)"
|
|
1131 |
apply (induct_tac "n")
|
|
1132 |
apply (auto simp add: hcomplex_mult_ac)
|
|
1133 |
done
|
|
1134 |
|
|
1135 |
lemma hcpow_mult: "((r::hcomplex) * s) hcpow n = (r hcpow n) * (s hcpow n)"
|
|
1136 |
apply (rule_tac z = "r" in eq_Abs_hcomplex)
|
|
1137 |
apply (rule_tac z = "s" in eq_Abs_hcomplex)
|
|
1138 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1139 |
apply (auto simp add: hcpow hypreal_mult hcomplex_mult complexpow_mult)
|
|
1140 |
done
|
|
1141 |
|
|
1142 |
lemma hcomplexpow_zero: "(0::hcomplex) ^ (Suc n) = 0"
|
|
1143 |
apply auto
|
|
1144 |
done
|
|
1145 |
declare hcomplexpow_zero [simp]
|
|
1146 |
|
|
1147 |
lemma hcpow_zero:
|
|
1148 |
"0 hcpow (n + 1) = 0"
|
|
1149 |
apply (unfold hcomplex_zero_def hypnat_one_def)
|
|
1150 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1151 |
apply (auto simp add: hcpow hypnat_add)
|
|
1152 |
done
|
|
1153 |
declare hcpow_zero [simp]
|
|
1154 |
|
|
1155 |
lemma hcpow_zero2:
|
|
1156 |
"0 hcpow (hSuc n) = 0"
|
|
1157 |
apply (unfold hSuc_def)
|
|
1158 |
apply (simp (no_asm))
|
|
1159 |
done
|
|
1160 |
declare hcpow_zero2 [simp]
|
|
1161 |
|
|
1162 |
lemma hcomplexpow_not_zero [rule_format (no_asm)]: "r ~= (0::hcomplex) --> r ^ n ~= 0"
|
|
1163 |
apply (induct_tac "n")
|
|
1164 |
apply (auto simp add: hcomplex_mult_not_zero)
|
|
1165 |
done
|
|
1166 |
declare hcomplexpow_not_zero [simp]
|
|
1167 |
declare hcomplexpow_not_zero [intro]
|
|
1168 |
|
|
1169 |
lemma hcpow_not_zero: "r ~= 0 ==> r hcpow n ~= (0::hcomplex)"
|
|
1170 |
apply (rule_tac z = "r" in eq_Abs_hcomplex)
|
|
1171 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1172 |
apply (auto simp add: hcpow hcomplex_zero_def)
|
|
1173 |
apply ultra
|
|
1174 |
apply (auto dest: complexpow_zero_zero)
|
|
1175 |
done
|
|
1176 |
declare hcpow_not_zero [simp]
|
|
1177 |
declare hcpow_not_zero [intro]
|
|
1178 |
|
|
1179 |
lemma hcomplexpow_zero_zero: "r ^ n = (0::hcomplex) ==> r = 0"
|
|
1180 |
apply (blast intro: ccontr dest: hcomplexpow_not_zero)
|
|
1181 |
done
|
|
1182 |
|
|
1183 |
lemma hcpow_zero_zero: "r hcpow n = (0::hcomplex) ==> r = 0"
|
|
1184 |
apply (blast intro: ccontr dest: hcpow_not_zero)
|
|
1185 |
done
|
|
1186 |
|
|
1187 |
lemma hcomplex_i_mult_eq: "iii * iii = - 1"
|
|
1188 |
apply (unfold iii_def)
|
|
1189 |
apply (auto simp add: hcomplex_mult hcomplex_one_def hcomplex_minus)
|
|
1190 |
done
|
|
1191 |
declare hcomplex_i_mult_eq [simp]
|
|
1192 |
|
|
1193 |
lemma hcomplexpow_i_squared: "iii ^ 2 = - 1"
|
|
1194 |
apply (simp (no_asm) add: two_eq_Suc_Suc)
|
|
1195 |
done
|
|
1196 |
declare hcomplexpow_i_squared [simp]
|
|
1197 |
|
|
1198 |
lemma hcomplex_i_not_zero: "iii ~= 0"
|
|
1199 |
apply (unfold iii_def hcomplex_zero_def)
|
|
1200 |
apply auto
|
|
1201 |
done
|
|
1202 |
declare hcomplex_i_not_zero [simp]
|
|
1203 |
|
|
1204 |
lemma hcomplex_mult_eq_zero_cancel1: "x * y ~= (0::hcomplex) ==> x ~= 0"
|
|
1205 |
apply auto
|
|
1206 |
done
|
|
1207 |
|
|
1208 |
lemma hcomplex_mult_eq_zero_cancel2: "x * y ~= (0::hcomplex) ==> y ~= 0"
|
|
1209 |
apply auto
|
|
1210 |
done
|
|
1211 |
|
|
1212 |
lemma hcomplex_mult_not_eq_zero_iff: "(x * y ~= (0::hcomplex)) = (x ~= 0 & y ~= 0)"
|
|
1213 |
apply auto
|
|
1214 |
done
|
|
1215 |
declare hcomplex_mult_not_eq_zero_iff [iff]
|
|
1216 |
|
|
1217 |
lemma hcomplex_divide:
|
|
1218 |
"Abs_hcomplex(hcomplexrel``{%n. X n}) / Abs_hcomplex(hcomplexrel``{%n. Y n}) =
|
|
1219 |
Abs_hcomplex(hcomplexrel``{%n. X n / Y n})"
|
|
1220 |
apply (unfold hcomplex_divide_def complex_divide_def)
|
|
1221 |
apply (auto simp add: hcomplex_inverse hcomplex_mult)
|
|
1222 |
done
|
|
1223 |
|
|
1224 |
|
|
1225 |
subsection{*The Function @{term hsgn}*}
|
|
1226 |
|
|
1227 |
lemma hsgn:
|
|
1228 |
"hsgn (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
|
|
1229 |
Abs_hcomplex(hcomplexrel `` {%n. sgn (X n)})"
|
|
1230 |
apply (unfold hsgn_def)
|
|
1231 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
1232 |
apply (auto , ultra)
|
|
1233 |
done
|
|
1234 |
|
|
1235 |
lemma hsgn_zero: "hsgn 0 = 0"
|
|
1236 |
apply (unfold hcomplex_zero_def)
|
|
1237 |
apply (simp (no_asm) add: hsgn)
|
|
1238 |
done
|
|
1239 |
declare hsgn_zero [simp]
|
|
1240 |
|
|
1241 |
|
|
1242 |
lemma hsgn_one: "hsgn 1 = 1"
|
|
1243 |
apply (unfold hcomplex_one_def)
|
|
1244 |
apply (simp (no_asm) add: hsgn)
|
|
1245 |
done
|
|
1246 |
declare hsgn_one [simp]
|
|
1247 |
|
|
1248 |
lemma hsgn_minus: "hsgn (-z) = - hsgn(z)"
|
|
1249 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
1250 |
apply (auto simp add: hsgn hcomplex_minus sgn_minus)
|
|
1251 |
done
|
|
1252 |
|
|
1253 |
lemma hsgn_eq: "hsgn z = z / hcomplex_of_hypreal (hcmod z)"
|
|
1254 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
1255 |
apply (auto simp add: hsgn hcomplex_divide hcomplex_of_hypreal hcmod sgn_eq)
|
|
1256 |
done
|
|
1257 |
|
|
1258 |
lemma lemma_hypreal_P_EX2: "(EX (x::hypreal) y. P x y) =
|
|
1259 |
(EX f g. P (Abs_hypreal(hyprel `` {f})) (Abs_hypreal(hyprel `` {g})))"
|
|
1260 |
apply auto
|
|
1261 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1262 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1263 |
apply auto
|
|
1264 |
done
|
|
1265 |
|
|
1266 |
lemma complex_split2: "ALL (n::nat). EX x y. (z n) = complex_of_real(x) + ii * complex_of_real(y)"
|
|
1267 |
apply (blast intro: complex_split)
|
|
1268 |
done
|
|
1269 |
|
|
1270 |
(* Interesting proof! *)
|
|
1271 |
lemma hcomplex_split: "EX x y. z = hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)"
|
|
1272 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
1273 |
apply (auto simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def hcomplex_add hcomplex_mult)
|
|
1274 |
apply (cut_tac z = "x" in complex_split2)
|
|
1275 |
apply (drule choice , safe)+
|
|
1276 |
apply (rule_tac x = "f" in exI)
|
|
1277 |
apply (rule_tac x = "fa" in exI)
|
|
1278 |
apply auto
|
|
1279 |
done
|
|
1280 |
|
|
1281 |
lemma hRe_hcomplex_i: "hRe(hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = x"
|
|
1282 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1283 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1284 |
apply (auto simp add: hRe iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
|
|
1285 |
done
|
|
1286 |
declare hRe_hcomplex_i [simp]
|
|
1287 |
|
|
1288 |
lemma hIm_hcomplex_i: "hIm(hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = y"
|
|
1289 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1290 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1291 |
apply (auto simp add: hIm iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
|
|
1292 |
done
|
|
1293 |
declare hIm_hcomplex_i [simp]
|
|
1294 |
|
|
1295 |
lemma hcmod_i: "hcmod (hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) =
|
|
1296 |
( *f* sqrt) (x ^ 2 + y ^ 2)"
|
|
1297 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1298 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1299 |
apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_add hcomplex_mult starfun hypreal_mult hypreal_add hcmod cmod_i two_eq_Suc_Suc)
|
|
1300 |
done
|
|
1301 |
|
|
1302 |
lemma hcomplex_eq_hRe_eq:
|
|
1303 |
"hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
|
|
1304 |
hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb
|
|
1305 |
==> xa = xb"
|
|
1306 |
apply (unfold iii_def)
|
|
1307 |
apply (rule_tac z = "xa" in eq_Abs_hypreal)
|
|
1308 |
apply (rule_tac z = "ya" in eq_Abs_hypreal)
|
|
1309 |
apply (rule_tac z = "xb" in eq_Abs_hypreal)
|
|
1310 |
apply (rule_tac z = "yb" in eq_Abs_hypreal)
|
|
1311 |
apply (auto simp add: hcomplex_mult hcomplex_add hcomplex_of_hypreal)
|
|
1312 |
apply (ultra)
|
|
1313 |
done
|
|
1314 |
|
|
1315 |
lemma hcomplex_eq_hIm_eq:
|
|
1316 |
"hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
|
|
1317 |
hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb
|
|
1318 |
==> ya = yb"
|
|
1319 |
apply (unfold iii_def)
|
|
1320 |
apply (rule_tac z = "xa" in eq_Abs_hypreal)
|
|
1321 |
apply (rule_tac z = "ya" in eq_Abs_hypreal)
|
|
1322 |
apply (rule_tac z = "xb" in eq_Abs_hypreal)
|
|
1323 |
apply (rule_tac z = "yb" in eq_Abs_hypreal)
|
|
1324 |
apply (auto simp add: hcomplex_mult hcomplex_add hcomplex_of_hypreal)
|
|
1325 |
apply (ultra)
|
|
1326 |
done
|
|
1327 |
|
|
1328 |
lemma hcomplex_eq_cancel_iff: "(hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
|
|
1329 |
hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb) =
|
|
1330 |
((xa = xb) & (ya = yb))"
|
|
1331 |
apply (auto intro: hcomplex_eq_hIm_eq hcomplex_eq_hRe_eq)
|
|
1332 |
done
|
|
1333 |
declare hcomplex_eq_cancel_iff [simp]
|
|
1334 |
|
|
1335 |
lemma hcomplex_eq_cancel_iffA: "(hcomplex_of_hypreal xa + hcomplex_of_hypreal ya * iii =
|
|
1336 |
hcomplex_of_hypreal xb + hcomplex_of_hypreal yb * iii ) = ((xa = xb) & (ya = yb))"
|
|
1337 |
apply (auto simp add: hcomplex_mult_commute)
|
|
1338 |
done
|
|
1339 |
declare hcomplex_eq_cancel_iffA [iff]
|
|
1340 |
|
|
1341 |
lemma hcomplex_eq_cancel_iffB: "(hcomplex_of_hypreal xa + hcomplex_of_hypreal ya * iii =
|
|
1342 |
hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb) = ((xa = xb) & (ya = yb))"
|
|
1343 |
apply (auto simp add: hcomplex_mult_commute)
|
|
1344 |
done
|
|
1345 |
declare hcomplex_eq_cancel_iffB [iff]
|
|
1346 |
|
|
1347 |
lemma hcomplex_eq_cancel_iffC: "(hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
|
|
1348 |
hcomplex_of_hypreal xb + hcomplex_of_hypreal yb * iii) = ((xa = xb) & (ya = yb))"
|
|
1349 |
apply (auto simp add: hcomplex_mult_commute)
|
|
1350 |
done
|
|
1351 |
declare hcomplex_eq_cancel_iffC [iff]
|
|
1352 |
|
|
1353 |
lemma hcomplex_eq_cancel_iff2: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y =
|
|
1354 |
hcomplex_of_hypreal xa) = (x = xa & y = 0)"
|
|
1355 |
apply (cut_tac xa = "x" and ya = "y" and xb = "xa" and yb = "0" in hcomplex_eq_cancel_iff)
|
|
1356 |
apply (simp del: hcomplex_eq_cancel_iff)
|
|
1357 |
done
|
|
1358 |
declare hcomplex_eq_cancel_iff2 [simp]
|
|
1359 |
|
|
1360 |
lemma hcomplex_eq_cancel_iff2a: "(hcomplex_of_hypreal x + hcomplex_of_hypreal y * iii =
|
|
1361 |
hcomplex_of_hypreal xa) = (x = xa & y = 0)"
|
|
1362 |
apply (auto simp add: hcomplex_mult_commute)
|
|
1363 |
done
|
|
1364 |
declare hcomplex_eq_cancel_iff2a [simp]
|
|
1365 |
|
|
1366 |
lemma hcomplex_eq_cancel_iff3: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y =
|
|
1367 |
iii * hcomplex_of_hypreal ya) = (x = 0 & y = ya)"
|
|
1368 |
apply (cut_tac xa = "x" and ya = "y" and xb = "0" and yb = "ya" in hcomplex_eq_cancel_iff)
|
|
1369 |
apply (simp del: hcomplex_eq_cancel_iff)
|
|
1370 |
done
|
|
1371 |
declare hcomplex_eq_cancel_iff3 [simp]
|
|
1372 |
|
|
1373 |
lemma hcomplex_eq_cancel_iff3a: "(hcomplex_of_hypreal x + hcomplex_of_hypreal y * iii =
|
|
1374 |
iii * hcomplex_of_hypreal ya) = (x = 0 & y = ya)"
|
|
1375 |
apply (auto simp add: hcomplex_mult_commute)
|
|
1376 |
done
|
|
1377 |
declare hcomplex_eq_cancel_iff3a [simp]
|
|
1378 |
|
|
1379 |
lemma hcomplex_split_hRe_zero:
|
|
1380 |
"hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = 0
|
|
1381 |
==> x = 0"
|
|
1382 |
apply (unfold iii_def)
|
|
1383 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1384 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1385 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_add hcomplex_mult hcomplex_zero_def hypreal_zero_num)
|
|
1386 |
apply ultra
|
|
1387 |
apply (auto simp add: complex_split_Re_zero)
|
|
1388 |
done
|
|
1389 |
|
|
1390 |
lemma hcomplex_split_hIm_zero:
|
|
1391 |
"hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = 0
|
|
1392 |
==> y = 0"
|
|
1393 |
apply (unfold iii_def)
|
|
1394 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1395 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1396 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_add hcomplex_mult hcomplex_zero_def hypreal_zero_num)
|
|
1397 |
apply ultra
|
|
1398 |
apply (auto simp add: complex_split_Im_zero)
|
|
1399 |
done
|
|
1400 |
|
|
1401 |
lemma hRe_hsgn: "hRe(hsgn z) = hRe(z)/hcmod z"
|
|
1402 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
1403 |
apply (auto simp add: hsgn hcmod hRe hypreal_divide)
|
|
1404 |
done
|
|
1405 |
declare hRe_hsgn [simp]
|
|
1406 |
|
|
1407 |
lemma hIm_hsgn: "hIm(hsgn z) = hIm(z)/hcmod z"
|
|
1408 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
1409 |
apply (auto simp add: hsgn hcmod hIm hypreal_divide)
|
|
1410 |
done
|
|
1411 |
declare hIm_hsgn [simp]
|
|
1412 |
|
|
1413 |
lemma real_two_squares_add_zero_iff: "(x*x + y*y = 0) = ((x::real) = 0 & y = 0)"
|
|
1414 |
apply (auto intro: real_sum_squares_cancel)
|
|
1415 |
done
|
|
1416 |
declare real_two_squares_add_zero_iff [simp]
|
|
1417 |
|
|
1418 |
lemma hcomplex_inverse_complex_split: "inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) =
|
|
1419 |
hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) -
|
|
1420 |
iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
|
|
1421 |
apply (rule_tac z = "x" in eq_Abs_hypreal)
|
|
1422 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1423 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hcomplex_add iii_def starfun hypreal_mult hypreal_add hcomplex_inverse hypreal_divide hcomplex_diff complex_inverse_complex_split two_eq_Suc_Suc)
|
|
1424 |
done
|
|
1425 |
|
|
1426 |
lemma hRe_mult_i_eq:
|
|
1427 |
"hRe (iii * hcomplex_of_hypreal y) = 0"
|
|
1428 |
apply (unfold iii_def)
|
|
1429 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1430 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hRe hypreal_zero_num)
|
|
1431 |
done
|
|
1432 |
declare hRe_mult_i_eq [simp]
|
|
1433 |
|
|
1434 |
lemma hIm_mult_i_eq:
|
|
1435 |
"hIm (iii * hcomplex_of_hypreal y) = y"
|
|
1436 |
apply (unfold iii_def)
|
|
1437 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1438 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hIm hypreal_zero_num)
|
|
1439 |
done
|
|
1440 |
declare hIm_mult_i_eq [simp]
|
|
1441 |
|
|
1442 |
lemma hcmod_mult_i: "hcmod (iii * hcomplex_of_hypreal y) = abs y"
|
|
1443 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1444 |
apply (auto simp add: hcomplex_of_hypreal hcmod hypreal_hrabs iii_def hcomplex_mult)
|
|
1445 |
done
|
|
1446 |
declare hcmod_mult_i [simp]
|
|
1447 |
|
|
1448 |
lemma hcmod_mult_i2: "hcmod (hcomplex_of_hypreal y * iii) = abs y"
|
|
1449 |
apply (auto simp add: hcomplex_mult_commute)
|
|
1450 |
done
|
|
1451 |
declare hcmod_mult_i2 [simp]
|
|
1452 |
|
|
1453 |
(*---------------------------------------------------------------------------*)
|
|
1454 |
(* harg *)
|
|
1455 |
(*---------------------------------------------------------------------------*)
|
|
1456 |
|
|
1457 |
lemma harg:
|
|
1458 |
"harg (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
|
|
1459 |
Abs_hypreal(hyprel `` {%n. arg (X n)})"
|
|
1460 |
|
|
1461 |
apply (unfold harg_def)
|
|
1462 |
apply (rule_tac f = "Abs_hypreal" in arg_cong)
|
|
1463 |
apply (auto , ultra)
|
|
1464 |
done
|
|
1465 |
|
|
1466 |
lemma cos_harg_i_mult_zero: "0 < y ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
|
|
1467 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1468 |
apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_mult hypreal_zero_num hypreal_less starfun harg)
|
|
1469 |
apply (ultra)
|
|
1470 |
done
|
|
1471 |
declare cos_harg_i_mult_zero [simp]
|
|
1472 |
|
|
1473 |
lemma cos_harg_i_mult_zero2: "y < 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
|
|
1474 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1475 |
apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_mult hypreal_zero_num hypreal_less starfun harg)
|
|
1476 |
apply (ultra)
|
|
1477 |
done
|
|
1478 |
declare cos_harg_i_mult_zero2 [simp]
|
|
1479 |
|
|
1480 |
lemma hcomplex_of_hypreal_not_zero_iff: "(hcomplex_of_hypreal y ~= 0) = (y ~= 0)"
|
|
1481 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1482 |
apply (auto simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
|
|
1483 |
done
|
|
1484 |
declare hcomplex_of_hypreal_not_zero_iff [simp]
|
|
1485 |
|
|
1486 |
lemma hcomplex_of_hypreal_zero_iff: "(hcomplex_of_hypreal y = 0) = (y = 0)"
|
|
1487 |
apply (rule_tac z = "y" in eq_Abs_hypreal)
|
|
1488 |
apply (auto simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
|
|
1489 |
done
|
|
1490 |
declare hcomplex_of_hypreal_zero_iff [simp]
|
|
1491 |
|
|
1492 |
lemma cos_harg_i_mult_zero3: "y ~= 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
|
|
1493 |
apply (cut_tac x = "y" and y = "0" in hypreal_linear)
|
|
1494 |
apply auto
|
|
1495 |
done
|
|
1496 |
declare cos_harg_i_mult_zero3 [simp]
|
|
1497 |
|
|
1498 |
(*---------------------------------------------------------------------------*)
|
|
1499 |
(* Polar form for nonstandard complex numbers *)
|
|
1500 |
(*---------------------------------------------------------------------------*)
|
|
1501 |
|
|
1502 |
lemma complex_split_polar2: "ALL n. EX r a. (z n) = complex_of_real r *
|
|
1503 |
(complex_of_real(cos a) + ii * complex_of_real(sin a))"
|
|
1504 |
apply (blast intro: complex_split_polar)
|
|
1505 |
done
|
|
1506 |
|
|
1507 |
lemma hcomplex_split_polar:
|
|
1508 |
"EX r a. z = hcomplex_of_hypreal r *
|
|
1509 |
(hcomplex_of_hypreal(( *f* cos) a) + iii * hcomplex_of_hypreal(( *f* sin) a))"
|
|
1510 |
apply (rule_tac z = "z" in eq_Abs_hcomplex)
|
|
1511 |
apply (auto simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def starfun hcomplex_add hcomplex_mult)
|
|
1512 |
apply (cut_tac z = "x" in complex_split_polar2)
|
|
1513 |
apply (drule choice , safe)+
|
|
1514 |
apply (rule_tac x = "f" in exI)
|
|
1515 |
apply (rule_tac x = "fa" in exI)
|
|
1516 |
apply auto
|
|
1517 |
done
|
|
1518 |
|
|
1519 |
lemma hcis:
|
|
1520 |
"hcis (Abs_hypreal(hyprel `` {%n. X n})) =
|
|
1521 |
Abs_hcomplex(hcomplexrel `` {%n. cis (X n)})"
|
|
1522 |
apply (unfold hcis_def)
|
|
1523 |
apply (rule_tac f = "Abs_hcomplex" in arg_cong)
|
|
1524 |
apply auto
|
|
1525 |
apply (ultra)
|
|
1526 |
done
|
|
1527 |
|
|
1528 |
lemma hcis_eq:
|
|
1529 |
"hcis a =
|
|
1530 |
(hcomplex_of_hypreal(( *f* cos) a) +
|
|
1531 |
iii * hcomplex_of_hypreal(( *f* sin) a))"
|
|
1532 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1533 |
apply (auto simp add: starfun hcis hcomplex_of_hypreal iii_def hcomplex_mult hcomplex_add cis_def)
|
|
1534 |
done
|
|
1535 |
|
|
1536 |
lemma hrcis:
|
|
1537 |
"hrcis (Abs_hypreal(hyprel `` {%n. X n})) (Abs_hypreal(hyprel `` {%n. Y n})) =
|
|
1538 |
Abs_hcomplex(hcomplexrel `` {%n. rcis (X n) (Y n)})"
|
|
1539 |
apply (unfold hrcis_def)
|
|
1540 |
apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hcis rcis_def)
|
|
1541 |
done
|
|
1542 |
|
|
1543 |
lemma hrcis_Ex: "EX r a. z = hrcis r a"
|
|
1544 |
apply (simp (no_asm) add: hrcis_def hcis_eq)
|
|
1545 |
apply (rule hcomplex_split_polar)
|
|
1546 |
done
|
|
1547 |
|
|
1548 |
lemma hRe_hcomplex_polar: "hRe(hcomplex_of_hypreal r *
|
|
1549 |
(hcomplex_of_hypreal(( *f* cos) a) +
|
|
1550 |
iii * hcomplex_of_hypreal(( *f* sin) a))) = r * ( *f* cos) a"
|
|
1551 |
apply (auto simp add: hcomplex_add_mult_distrib2 hcomplex_of_hypreal_mult hcomplex_mult_ac)
|
|
1552 |
done
|
|
1553 |
declare hRe_hcomplex_polar [simp]
|
|
1554 |
|
|
1555 |
lemma hRe_hrcis: "hRe(hrcis r a) = r * ( *f* cos) a"
|
|
1556 |
apply (unfold hrcis_def)
|
|
1557 |
apply (auto simp add: hcis_eq)
|
|
1558 |
done
|
|
1559 |
declare hRe_hrcis [simp]
|
|
1560 |
|
|
1561 |
lemma hIm_hcomplex_polar: "hIm(hcomplex_of_hypreal r *
|
|
1562 |
(hcomplex_of_hypreal(( *f* cos) a) +
|
|
1563 |
iii * hcomplex_of_hypreal(( *f* sin) a))) = r * ( *f* sin) a"
|
|
1564 |
apply (auto simp add: hcomplex_add_mult_distrib2 hcomplex_of_hypreal_mult hcomplex_mult_ac)
|
|
1565 |
done
|
|
1566 |
declare hIm_hcomplex_polar [simp]
|
|
1567 |
|
|
1568 |
lemma hIm_hrcis: "hIm(hrcis r a) = r * ( *f* sin) a"
|
|
1569 |
apply (unfold hrcis_def)
|
|
1570 |
apply (auto simp add: hcis_eq)
|
|
1571 |
done
|
|
1572 |
declare hIm_hrcis [simp]
|
|
1573 |
|
|
1574 |
lemma hcmod_complex_polar: "hcmod (hcomplex_of_hypreal r *
|
|
1575 |
(hcomplex_of_hypreal(( *f* cos) a) +
|
|
1576 |
iii * hcomplex_of_hypreal(( *f* sin) a))) = abs r"
|
|
1577 |
apply (rule_tac z = "r" in eq_Abs_hypreal)
|
|
1578 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1579 |
apply (auto simp add: iii_def starfun hcomplex_of_hypreal hcomplex_mult hcmod hcomplex_add hypreal_hrabs)
|
|
1580 |
done
|
|
1581 |
declare hcmod_complex_polar [simp]
|
|
1582 |
|
|
1583 |
lemma hcmod_hrcis: "hcmod(hrcis r a) = abs r"
|
|
1584 |
apply (unfold hrcis_def)
|
|
1585 |
apply (auto simp add: hcis_eq)
|
|
1586 |
done
|
|
1587 |
declare hcmod_hrcis [simp]
|
|
1588 |
|
|
1589 |
(*---------------------------------------------------------------------------*)
|
|
1590 |
(* (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b) *)
|
|
1591 |
(*---------------------------------------------------------------------------*)
|
|
1592 |
|
|
1593 |
lemma hcis_hrcis_eq: "hcis a = hrcis 1 a"
|
|
1594 |
apply (unfold hrcis_def)
|
|
1595 |
apply (simp (no_asm))
|
|
1596 |
done
|
|
1597 |
declare hcis_hrcis_eq [symmetric, simp]
|
|
1598 |
|
|
1599 |
lemma hrcis_mult:
|
|
1600 |
"hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)"
|
|
1601 |
apply (unfold hrcis_def)
|
|
1602 |
apply (rule_tac z = "r1" in eq_Abs_hypreal)
|
|
1603 |
apply (rule_tac z = "r2" in eq_Abs_hypreal)
|
|
1604 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1605 |
apply (rule_tac z = "b" in eq_Abs_hypreal)
|
|
1606 |
apply (auto simp add: hrcis hcis hypreal_add hypreal_mult hcomplex_of_hypreal
|
|
1607 |
hcomplex_mult cis_mult [symmetric]
|
|
1608 |
complex_of_real_mult [symmetric])
|
|
1609 |
done
|
|
1610 |
|
|
1611 |
lemma hcis_mult: "hcis a * hcis b = hcis (a + b)"
|
|
1612 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1613 |
apply (rule_tac z = "b" in eq_Abs_hypreal)
|
|
1614 |
apply (auto simp add: hcis hcomplex_mult hypreal_add cis_mult)
|
|
1615 |
done
|
|
1616 |
|
|
1617 |
lemma hcis_zero:
|
|
1618 |
"hcis 0 = 1"
|
|
1619 |
apply (unfold hcomplex_one_def)
|
|
1620 |
apply (auto simp add: hcis hypreal_zero_num)
|
|
1621 |
done
|
|
1622 |
declare hcis_zero [simp]
|
|
1623 |
|
|
1624 |
lemma hrcis_zero_mod:
|
|
1625 |
"hrcis 0 a = 0"
|
|
1626 |
apply (unfold hcomplex_zero_def)
|
|
1627 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1628 |
apply (auto simp add: hrcis hypreal_zero_num)
|
|
1629 |
done
|
|
1630 |
declare hrcis_zero_mod [simp]
|
|
1631 |
|
|
1632 |
lemma hrcis_zero_arg: "hrcis r 0 = hcomplex_of_hypreal r"
|
|
1633 |
apply (rule_tac z = "r" in eq_Abs_hypreal)
|
|
1634 |
apply (auto simp add: hrcis hypreal_zero_num hcomplex_of_hypreal)
|
|
1635 |
done
|
|
1636 |
declare hrcis_zero_arg [simp]
|
|
1637 |
|
|
1638 |
lemma hcomplex_i_mult_minus: "iii * (iii * x) = - x"
|
|
1639 |
apply (simp (no_asm) add: hcomplex_mult_assoc [symmetric])
|
|
1640 |
done
|
|
1641 |
declare hcomplex_i_mult_minus [simp]
|
|
1642 |
|
|
1643 |
lemma hcomplex_i_mult_minus2: "iii * iii * x = - x"
|
|
1644 |
apply (simp (no_asm))
|
|
1645 |
done
|
|
1646 |
declare hcomplex_i_mult_minus2 [simp]
|
|
1647 |
|
|
1648 |
lemma hcis_hypreal_of_nat_Suc_mult:
|
|
1649 |
"hcis (hypreal_of_nat (Suc n) * a) = hcis a * hcis (hypreal_of_nat n * a)"
|
|
1650 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1651 |
apply (auto simp add: hypreal_of_nat hcis hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
|
|
1652 |
done
|
|
1653 |
|
|
1654 |
lemma NSDeMoivre: "(hcis a) ^ n = hcis (hypreal_of_nat n * a)"
|
|
1655 |
apply (induct_tac "n")
|
|
1656 |
apply (auto simp add: hcis_hypreal_of_nat_Suc_mult)
|
|
1657 |
done
|
|
1658 |
|
|
1659 |
lemma hcis_hypreal_of_hypnat_Suc_mult: "hcis (hypreal_of_hypnat (n + 1) * a) =
|
|
1660 |
hcis a * hcis (hypreal_of_hypnat n * a)"
|
|
1661 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1662 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1663 |
apply (auto simp add: hcis hypreal_of_hypnat hypnat_add hypnat_one_def hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
|
|
1664 |
done
|
|
1665 |
|
|
1666 |
lemma NSDeMoivre_ext: "(hcis a) hcpow n = hcis (hypreal_of_hypnat n * a)"
|
|
1667 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1668 |
apply (rule_tac z = "n" in eq_Abs_hypnat)
|
|
1669 |
apply (auto simp add: hcis hypreal_of_hypnat hypreal_mult hcpow DeMoivre)
|
|
1670 |
done
|
|
1671 |
|
|
1672 |
lemma DeMoivre2:
|
|
1673 |
"(hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
|
|
1674 |
apply (unfold hrcis_def)
|
|
1675 |
apply (auto simp add: hcomplexpow_mult NSDeMoivre hcomplex_of_hypreal_pow)
|
|
1676 |
done
|
|
1677 |
|
|
1678 |
lemma DeMoivre2_ext:
|
|
1679 |
"(hrcis r a) hcpow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
|
|
1680 |
apply (unfold hrcis_def)
|
|
1681 |
apply (auto simp add: hcpow_mult NSDeMoivre_ext hcomplex_of_hypreal_hyperpow)
|
|
1682 |
done
|
|
1683 |
|
|
1684 |
lemma hcis_inverse: "inverse(hcis a) = hcis (-a)"
|
|
1685 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1686 |
apply (auto simp add: hcomplex_inverse hcis hypreal_minus)
|
|
1687 |
done
|
|
1688 |
declare hcis_inverse [simp]
|
|
1689 |
|
|
1690 |
lemma hrcis_inverse: "inverse(hrcis r a) = hrcis (inverse r) (-a)"
|
|
1691 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1692 |
apply (rule_tac z = "r" in eq_Abs_hypreal)
|
|
1693 |
apply (auto simp add: hcomplex_inverse hrcis hypreal_minus hypreal_inverse rcis_inverse)
|
|
1694 |
apply (ultra)
|
|
1695 |
apply (unfold real_divide_def)
|
|
1696 |
apply (auto simp add: INVERSE_ZERO)
|
|
1697 |
done
|
|
1698 |
|
|
1699 |
lemma hRe_hcis: "hRe(hcis a) = ( *f* cos) a"
|
|
1700 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1701 |
apply (auto simp add: hcis starfun hRe)
|
|
1702 |
done
|
|
1703 |
declare hRe_hcis [simp]
|
|
1704 |
|
|
1705 |
lemma hIm_hcis: "hIm(hcis a) = ( *f* sin) a"
|
|
1706 |
apply (rule_tac z = "a" in eq_Abs_hypreal)
|
|
1707 |
apply (auto simp add: hcis starfun hIm)
|
|
1708 |
done
|
|
1709 |
declare hIm_hcis [simp]
|
|
1710 |
|
|
1711 |
lemma cos_n_hRe_hcis_pow_n: "( *f* cos) (hypreal_of_nat n * a) = hRe(hcis a ^ n)"
|
|
1712 |
apply (auto simp add: NSDeMoivre)
|
|
1713 |
done
|
|
1714 |
|
|
1715 |
lemma sin_n_hIm_hcis_pow_n: "( *f* sin) (hypreal_of_nat n * a) = hIm(hcis a ^ n)"
|
|
1716 |
apply (auto simp add: NSDeMoivre)
|
|
1717 |
done
|
|
1718 |
|
|
1719 |
lemma cos_n_hRe_hcis_hcpow_n: "( *f* cos) (hypreal_of_hypnat n * a) = hRe(hcis a hcpow n)"
|
|
1720 |
apply (auto simp add: NSDeMoivre_ext)
|
|
1721 |
done
|
|
1722 |
|
|
1723 |
lemma sin_n_hIm_hcis_hcpow_n: "( *f* sin) (hypreal_of_hypnat n * a) = hIm(hcis a hcpow n)"
|
|
1724 |
apply (auto simp add: NSDeMoivre_ext)
|
|
1725 |
done
|
|
1726 |
|
|
1727 |
lemma hexpi_add: "hexpi(a + b) = hexpi(a) * hexpi(b)"
|
|
1728 |
apply (unfold hexpi_def)
|
|
1729 |
apply (rule_tac z = "a" in eq_Abs_hcomplex)
|
|
1730 |
apply (rule_tac z = "b" in eq_Abs_hcomplex)
|
|
1731 |
apply (auto 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)
|
|
1732 |
done
|
|
1733 |
|
|
1734 |
|
|
1735 |
subsection{*@{term hcomplex_of_complex} Preserves Field Properties*}
|
|
1736 |
|
|
1737 |
lemma hcomplex_of_complex_add:
|
|
1738 |
"hcomplex_of_complex (z1 + z2) = hcomplex_of_complex z1 + hcomplex_of_complex z2"
|
|
1739 |
apply (unfold hcomplex_of_complex_def)
|
|
1740 |
apply (simp (no_asm) add: hcomplex_add)
|
|
1741 |
done
|
|
1742 |
declare hcomplex_of_complex_add [simp]
|
|
1743 |
|
|
1744 |
lemma hcomplex_of_complex_mult:
|
|
1745 |
"hcomplex_of_complex (z1 * z2) = hcomplex_of_complex z1 * hcomplex_of_complex z2"
|
|
1746 |
apply (unfold hcomplex_of_complex_def)
|
|
1747 |
apply (simp (no_asm) add: hcomplex_mult)
|
|
1748 |
done
|
|
1749 |
declare hcomplex_of_complex_mult [simp]
|
|
1750 |
|
|
1751 |
lemma hcomplex_of_complex_eq_iff:
|
|
1752 |
"(hcomplex_of_complex z1 = hcomplex_of_complex z2) = (z1 = z2)"
|
|
1753 |
apply (unfold hcomplex_of_complex_def)
|
|
1754 |
apply auto
|
|
1755 |
done
|
|
1756 |
declare hcomplex_of_complex_eq_iff [simp]
|
|
1757 |
|
|
1758 |
lemma hcomplex_of_complex_minus: "hcomplex_of_complex (-r) = - hcomplex_of_complex r"
|
|
1759 |
apply (unfold hcomplex_of_complex_def)
|
|
1760 |
apply (auto simp add: hcomplex_minus)
|
|
1761 |
done
|
|
1762 |
declare hcomplex_of_complex_minus [simp]
|
|
1763 |
|
14320
|
1764 |
lemma hcomplex_of_complex_one [simp]:
|
14314
|
1765 |
"hcomplex_of_complex 1 = 1"
|
|
1766 |
apply (unfold hcomplex_of_complex_def hcomplex_one_def)
|
|
1767 |
apply auto
|
|
1768 |
done
|
|
1769 |
|
14320
|
1770 |
lemma hcomplex_of_complex_zero [simp]:
|
14314
|
1771 |
"hcomplex_of_complex 0 = 0"
|
|
1772 |
apply (unfold hcomplex_of_complex_def hcomplex_zero_def)
|
|
1773 |
apply (simp (no_asm))
|
|
1774 |
done
|
|
1775 |
|
|
1776 |
lemma hcomplex_of_complex_zero_iff: "(hcomplex_of_complex r = 0) = (r = 0)"
|
|
1777 |
apply (auto intro: FreeUltrafilterNat_P simp add: hcomplex_of_complex_def hcomplex_zero_def)
|
|
1778 |
done
|
|
1779 |
|
|
1780 |
lemma hcomplex_of_complex_inverse: "hcomplex_of_complex (inverse r) = inverse (hcomplex_of_complex r)"
|
|
1781 |
apply (case_tac "r=0")
|
|
1782 |
apply (simp (no_asm_simp) add: COMPLEX_INVERSE_ZERO HCOMPLEX_INVERSE_ZERO hcomplex_of_complex_zero COMPLEX_DIVIDE_ZERO)
|
|
1783 |
apply (rule_tac c1 = "hcomplex_of_complex r" in hcomplex_mult_left_cancel [THEN iffD1])
|
|
1784 |
apply (force simp add: hcomplex_of_complex_zero_iff)
|
|
1785 |
apply (subst hcomplex_of_complex_mult [symmetric])
|
|
1786 |
apply (auto simp add: hcomplex_of_complex_one hcomplex_of_complex_zero_iff);
|
|
1787 |
done
|
|
1788 |
declare hcomplex_of_complex_inverse [simp]
|
|
1789 |
|
|
1790 |
lemma hcomplex_of_complex_divide: "hcomplex_of_complex (z1 / z2) = hcomplex_of_complex z1 / hcomplex_of_complex z2"
|
|
1791 |
apply (simp (no_asm) add: hcomplex_divide_def complex_divide_def)
|
|
1792 |
done
|
|
1793 |
declare hcomplex_of_complex_divide [simp]
|
|
1794 |
|
|
1795 |
lemma hRe_hcomplex_of_complex:
|
|
1796 |
"hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
|
|
1797 |
apply (unfold hcomplex_of_complex_def hypreal_of_real_def)
|
|
1798 |
apply (auto simp add: hRe)
|
|
1799 |
done
|
|
1800 |
|
|
1801 |
lemma hIm_hcomplex_of_complex:
|
|
1802 |
"hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
|
|
1803 |
apply (unfold hcomplex_of_complex_def hypreal_of_real_def)
|
|
1804 |
apply (auto simp add: hIm)
|
|
1805 |
done
|
|
1806 |
|
|
1807 |
lemma hcmod_hcomplex_of_complex:
|
|
1808 |
"hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
|
|
1809 |
apply (unfold hypreal_of_real_def hcomplex_of_complex_def)
|
|
1810 |
apply (auto simp add: hcmod)
|
|
1811 |
done
|
|
1812 |
|
|
1813 |
ML
|
|
1814 |
{*
|
|
1815 |
val hcomplex_zero_def = thm"hcomplex_zero_def";
|
|
1816 |
val hcomplex_one_def = thm"hcomplex_one_def";
|
|
1817 |
val hcomplex_minus_def = thm"hcomplex_minus_def";
|
|
1818 |
val hcomplex_diff_def = thm"hcomplex_diff_def";
|
|
1819 |
val hcomplex_divide_def = thm"hcomplex_divide_def";
|
|
1820 |
val hcomplex_mult_def = thm"hcomplex_mult_def";
|
|
1821 |
val hcomplex_add_def = thm"hcomplex_add_def";
|
|
1822 |
val hcomplex_of_complex_def = thm"hcomplex_of_complex_def";
|
|
1823 |
val iii_def = thm"iii_def";
|
|
1824 |
|
|
1825 |
val hcomplexrel_iff = thm"hcomplexrel_iff";
|
|
1826 |
val hcomplexrel_refl = thm"hcomplexrel_refl";
|
|
1827 |
val hcomplexrel_sym = thm"hcomplexrel_sym";
|
|
1828 |
val hcomplexrel_trans = thm"hcomplexrel_trans";
|
|
1829 |
val equiv_hcomplexrel = thm"equiv_hcomplexrel";
|
|
1830 |
val equiv_hcomplexrel_iff = thm"equiv_hcomplexrel_iff";
|
|
1831 |
val hcomplexrel_in_hcomplex = thm"hcomplexrel_in_hcomplex";
|
|
1832 |
val inj_on_Abs_hcomplex = thm"inj_on_Abs_hcomplex";
|
|
1833 |
val inj_Rep_hcomplex = thm"inj_Rep_hcomplex";
|
|
1834 |
val lemma_hcomplexrel_refl = thm"lemma_hcomplexrel_refl";
|
|
1835 |
val hcomplex_empty_not_mem = thm"hcomplex_empty_not_mem";
|
|
1836 |
val Rep_hcomplex_nonempty = thm"Rep_hcomplex_nonempty";
|
|
1837 |
val eq_Abs_hcomplex = thm"eq_Abs_hcomplex";
|
|
1838 |
val hRe = thm"hRe";
|
|
1839 |
val hIm = thm"hIm";
|
|
1840 |
val hcomplex_hRe_hIm_cancel_iff = thm"hcomplex_hRe_hIm_cancel_iff";
|
|
1841 |
val hcomplex_hRe_zero = thm"hcomplex_hRe_zero";
|
|
1842 |
val hcomplex_hIm_zero = thm"hcomplex_hIm_zero";
|
|
1843 |
val hcomplex_hRe_one = thm"hcomplex_hRe_one";
|
|
1844 |
val hcomplex_hIm_one = thm"hcomplex_hIm_one";
|
|
1845 |
val inj_hcomplex_of_complex = thm"inj_hcomplex_of_complex";
|
|
1846 |
val hcomplex_of_complex_i = thm"hcomplex_of_complex_i";
|
|
1847 |
val hcomplex_add_congruent2 = thm"hcomplex_add_congruent2";
|
|
1848 |
val hcomplex_add = thm"hcomplex_add";
|
|
1849 |
val hcomplex_add_commute = thm"hcomplex_add_commute";
|
|
1850 |
val hcomplex_add_assoc = thm"hcomplex_add_assoc";
|
|
1851 |
val hcomplex_add_left_commute = thm"hcomplex_add_left_commute";
|
|
1852 |
val hcomplex_add_zero_left = thm"hcomplex_add_zero_left";
|
|
1853 |
val hcomplex_add_zero_right = thm"hcomplex_add_zero_right";
|
|
1854 |
val hRe_add = thm"hRe_add";
|
|
1855 |
val hIm_add = thm"hIm_add";
|
|
1856 |
val hcomplex_minus_congruent = thm"hcomplex_minus_congruent";
|
|
1857 |
val hcomplex_minus = thm"hcomplex_minus";
|
|
1858 |
val inj_hcomplex_minus = thm"inj_hcomplex_minus";
|
|
1859 |
val hcomplex_add_minus_left = thm"hcomplex_add_minus_left";
|
|
1860 |
val hRe_minus = thm"hRe_minus";
|
|
1861 |
val hIm_minus = thm"hIm_minus";
|
|
1862 |
val hcomplex_add_minus_eq_minus = thm"hcomplex_add_minus_eq_minus";
|
|
1863 |
val hcomplex_minus_add_distrib = thm"hcomplex_minus_add_distrib";
|
|
1864 |
val hcomplex_add_left_cancel = thm"hcomplex_add_left_cancel";
|
|
1865 |
val hcomplex_add_right_cancel = thm"hcomplex_add_right_cancel";
|
|
1866 |
val hcomplex_diff = thm"hcomplex_diff";
|
|
1867 |
val hcomplex_diff_eq_eq = thm"hcomplex_diff_eq_eq";
|
|
1868 |
val hcomplex_mult = thm"hcomplex_mult";
|
|
1869 |
val hcomplex_mult_commute = thm"hcomplex_mult_commute";
|
|
1870 |
val hcomplex_mult_assoc = thm"hcomplex_mult_assoc";
|
|
1871 |
val hcomplex_mult_left_commute = thm"hcomplex_mult_left_commute";
|
|
1872 |
val hcomplex_mult_one_left = thm"hcomplex_mult_one_left";
|
|
1873 |
val hcomplex_mult_one_right = thm"hcomplex_mult_one_right";
|
|
1874 |
val hcomplex_mult_zero_left = thm"hcomplex_mult_zero_left";
|
|
1875 |
val hcomplex_mult_zero_right = thm"hcomplex_mult_zero_right";
|
|
1876 |
val hcomplex_minus_mult_eq1 = thm"hcomplex_minus_mult_eq1";
|
|
1877 |
val hcomplex_minus_mult_eq2 = thm"hcomplex_minus_mult_eq2";
|
|
1878 |
val hcomplex_mult_minus_one = thm"hcomplex_mult_minus_one";
|
|
1879 |
val hcomplex_mult_minus_one_right = thm"hcomplex_mult_minus_one_right";
|
|
1880 |
val hcomplex_add_mult_distrib = thm"hcomplex_add_mult_distrib";
|
|
1881 |
val hcomplex_add_mult_distrib2 = thm"hcomplex_add_mult_distrib2";
|
|
1882 |
val hcomplex_zero_not_eq_one = thm"hcomplex_zero_not_eq_one";
|
|
1883 |
val hcomplex_inverse = thm"hcomplex_inverse";
|
|
1884 |
val HCOMPLEX_INVERSE_ZERO = thm"HCOMPLEX_INVERSE_ZERO";
|
|
1885 |
val HCOMPLEX_DIVISION_BY_ZERO = thm"HCOMPLEX_DIVISION_BY_ZERO";
|
|
1886 |
val hcomplex_mult_inv_left = thm"hcomplex_mult_inv_left";
|
|
1887 |
val hcomplex_mult_left_cancel = thm"hcomplex_mult_left_cancel";
|
|
1888 |
val hcomplex_mult_right_cancel = thm"hcomplex_mult_right_cancel";
|
|
1889 |
val hcomplex_inverse_not_zero = thm"hcomplex_inverse_not_zero";
|
|
1890 |
val hcomplex_mult_not_zero = thm"hcomplex_mult_not_zero";
|
|
1891 |
val hcomplex_mult_not_zeroE = thm"hcomplex_mult_not_zeroE";
|
|
1892 |
val hcomplex_minus_inverse = thm"hcomplex_minus_inverse";
|
|
1893 |
val hcomplex_add_divide_distrib = thm"hcomplex_add_divide_distrib";
|
|
1894 |
val hcomplex_of_hypreal = thm"hcomplex_of_hypreal";
|
|
1895 |
val inj_hcomplex_of_hypreal = thm"inj_hcomplex_of_hypreal";
|
|
1896 |
val hcomplex_of_hypreal_cancel_iff = thm"hcomplex_of_hypreal_cancel_iff";
|
|
1897 |
val hcomplex_of_hypreal_minus = thm"hcomplex_of_hypreal_minus";
|
|
1898 |
val hcomplex_of_hypreal_inverse = thm"hcomplex_of_hypreal_inverse";
|
|
1899 |
val hcomplex_of_hypreal_add = thm"hcomplex_of_hypreal_add";
|
|
1900 |
val hcomplex_of_hypreal_diff = thm"hcomplex_of_hypreal_diff";
|
|
1901 |
val hcomplex_of_hypreal_mult = thm"hcomplex_of_hypreal_mult";
|
|
1902 |
val hcomplex_of_hypreal_divide = thm"hcomplex_of_hypreal_divide";
|
|
1903 |
val hcomplex_of_hypreal_one = thm"hcomplex_of_hypreal_one";
|
|
1904 |
val hcomplex_of_hypreal_zero = thm"hcomplex_of_hypreal_zero";
|
|
1905 |
val hcomplex_of_hypreal_pow = thm"hcomplex_of_hypreal_pow";
|
|
1906 |
val hRe_hcomplex_of_hypreal = thm"hRe_hcomplex_of_hypreal";
|
|
1907 |
val hIm_hcomplex_of_hypreal = thm"hIm_hcomplex_of_hypreal";
|
|
1908 |
val hcomplex_of_hypreal_epsilon_not_zero = thm"hcomplex_of_hypreal_epsilon_not_zero";
|
|
1909 |
val hcmod = thm"hcmod";
|
|
1910 |
val hcmod_zero = thm"hcmod_zero";
|
|
1911 |
val hcmod_one = thm"hcmod_one";
|
|
1912 |
val hcmod_hcomplex_of_hypreal = thm"hcmod_hcomplex_of_hypreal";
|
|
1913 |
val hcomplex_of_hypreal_abs = thm"hcomplex_of_hypreal_abs";
|
|
1914 |
val hcnj = thm"hcnj";
|
|
1915 |
val inj_hcnj = thm"inj_hcnj";
|
|
1916 |
val hcomplex_hcnj_cancel_iff = thm"hcomplex_hcnj_cancel_iff";
|
|
1917 |
val hcomplex_hcnj_hcnj = thm"hcomplex_hcnj_hcnj";
|
|
1918 |
val hcomplex_hcnj_hcomplex_of_hypreal = thm"hcomplex_hcnj_hcomplex_of_hypreal";
|
|
1919 |
val hcomplex_hmod_hcnj = thm"hcomplex_hmod_hcnj";
|
|
1920 |
val hcomplex_hcnj_minus = thm"hcomplex_hcnj_minus";
|
|
1921 |
val hcomplex_hcnj_inverse = thm"hcomplex_hcnj_inverse";
|
|
1922 |
val hcomplex_hcnj_add = thm"hcomplex_hcnj_add";
|
|
1923 |
val hcomplex_hcnj_diff = thm"hcomplex_hcnj_diff";
|
|
1924 |
val hcomplex_hcnj_mult = thm"hcomplex_hcnj_mult";
|
|
1925 |
val hcomplex_hcnj_divide = thm"hcomplex_hcnj_divide";
|
|
1926 |
val hcnj_one = thm"hcnj_one";
|
|
1927 |
val hcomplex_hcnj_pow = thm"hcomplex_hcnj_pow";
|
|
1928 |
val hcomplex_hcnj_zero = thm"hcomplex_hcnj_zero";
|
|
1929 |
val hcomplex_hcnj_zero_iff = thm"hcomplex_hcnj_zero_iff";
|
|
1930 |
val hcomplex_mult_hcnj = thm"hcomplex_mult_hcnj";
|
|
1931 |
val hcomplex_mult_zero_iff = thm"hcomplex_mult_zero_iff";
|
|
1932 |
val hcomplex_add_left_cancel_zero = thm"hcomplex_add_left_cancel_zero";
|
|
1933 |
val hcomplex_diff_mult_distrib = thm"hcomplex_diff_mult_distrib";
|
|
1934 |
val hcomplex_diff_mult_distrib2 = thm"hcomplex_diff_mult_distrib2";
|
|
1935 |
val hcomplex_hcmod_eq_zero_cancel = thm"hcomplex_hcmod_eq_zero_cancel";
|
|
1936 |
val hypreal_of_nat_le_iff = thm"hypreal_of_nat_le_iff";
|
|
1937 |
val hypreal_of_nat_ge_zero = thm"hypreal_of_nat_ge_zero";
|
|
1938 |
val hypreal_of_hypnat_ge_zero = thm"hypreal_of_hypnat_ge_zero";
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|
1939 |
val hcmod_hcomplex_of_hypreal_of_nat = thm"hcmod_hcomplex_of_hypreal_of_nat";
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|
1940 |
val hcmod_hcomplex_of_hypreal_of_hypnat = thm"hcmod_hcomplex_of_hypreal_of_hypnat";
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|
1941 |
val hcmod_minus = thm"hcmod_minus";
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|
1942 |
val hcmod_mult_hcnj = thm"hcmod_mult_hcnj";
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|
1943 |
val hcmod_ge_zero = thm"hcmod_ge_zero";
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|
1944 |
val hrabs_hcmod_cancel = thm"hrabs_hcmod_cancel";
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|
1945 |
val hcmod_mult = thm"hcmod_mult";
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|
1946 |
val hcmod_add_squared_eq = thm"hcmod_add_squared_eq";
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|
1947 |
val hcomplex_hRe_mult_hcnj_le_hcmod = thm"hcomplex_hRe_mult_hcnj_le_hcmod";
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|
1948 |
val hcomplex_hRe_mult_hcnj_le_hcmod2 = thm"hcomplex_hRe_mult_hcnj_le_hcmod2";
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|
1949 |
val hcmod_triangle_squared = thm"hcmod_triangle_squared";
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|
1950 |
val hcmod_triangle_ineq = thm"hcmod_triangle_ineq";
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|
1951 |
val hcmod_triangle_ineq2 = thm"hcmod_triangle_ineq2";
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|
1952 |
val hcmod_diff_commute = thm"hcmod_diff_commute";
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|
1953 |
val hcmod_add_less = thm"hcmod_add_less";
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|
1954 |
val hcmod_mult_less = thm"hcmod_mult_less";
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|
1955 |
val hcmod_diff_ineq = thm"hcmod_diff_ineq";
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|
1956 |
val hcpow = thm"hcpow";
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|
1957 |
val hcomplex_of_hypreal_hyperpow = thm"hcomplex_of_hypreal_hyperpow";
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|
1958 |
val hcmod_hcomplexpow = thm"hcmod_hcomplexpow";
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|
1959 |
val hcmod_hcpow = thm"hcmod_hcpow";
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|
1960 |
val hcomplexpow_minus = thm"hcomplexpow_minus";
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|
1961 |
val hcpow_minus = thm"hcpow_minus";
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|
1962 |
val hccomplex_inverse_minus = thm"hccomplex_inverse_minus";
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|
1963 |
val hcomplex_div_one = thm"hcomplex_div_one";
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|
1964 |
val hcmod_hcomplex_inverse = thm"hcmod_hcomplex_inverse";
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|
1965 |
val hcmod_divide = thm"hcmod_divide";
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|
1966 |
val hcomplex_inverse_divide = thm"hcomplex_inverse_divide";
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|
1967 |
val hcomplexpow_mult = thm"hcomplexpow_mult";
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|
1968 |
val hcpow_mult = thm"hcpow_mult";
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|
1969 |
val hcomplexpow_zero = thm"hcomplexpow_zero";
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|
1970 |
val hcpow_zero = thm"hcpow_zero";
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|
1971 |
val hcpow_zero2 = thm"hcpow_zero2";
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|
1972 |
val hcomplexpow_not_zero = thm"hcomplexpow_not_zero";
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|
1973 |
val hcpow_not_zero = thm"hcpow_not_zero";
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|
1974 |
val hcomplexpow_zero_zero = thm"hcomplexpow_zero_zero";
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|
1975 |
val hcpow_zero_zero = thm"hcpow_zero_zero";
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|
1976 |
val hcomplex_i_mult_eq = thm"hcomplex_i_mult_eq";
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|
1977 |
val hcomplexpow_i_squared = thm"hcomplexpow_i_squared";
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|
1978 |
val hcomplex_i_not_zero = thm"hcomplex_i_not_zero";
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|
1979 |
val hcomplex_mult_eq_zero_cancel1 = thm"hcomplex_mult_eq_zero_cancel1";
|
|
1980 |
val hcomplex_mult_eq_zero_cancel2 = thm"hcomplex_mult_eq_zero_cancel2";
|
|
1981 |
val hcomplex_mult_not_eq_zero_iff = thm"hcomplex_mult_not_eq_zero_iff";
|
|
1982 |
val hcomplex_divide = thm"hcomplex_divide";
|
|
1983 |
val hsgn = thm"hsgn";
|
|
1984 |
val hsgn_zero = thm"hsgn_zero";
|
|
1985 |
val hsgn_one = thm"hsgn_one";
|
|
1986 |
val hsgn_minus = thm"hsgn_minus";
|
|
1987 |
val hsgn_eq = thm"hsgn_eq";
|
|
1988 |
val lemma_hypreal_P_EX2 = thm"lemma_hypreal_P_EX2";
|
|
1989 |
val complex_split2 = thm"complex_split2";
|
|
1990 |
val hcomplex_split = thm"hcomplex_split";
|
|
1991 |
val hRe_hcomplex_i = thm"hRe_hcomplex_i";
|
|
1992 |
val hIm_hcomplex_i = thm"hIm_hcomplex_i";
|
|
1993 |
val hcmod_i = thm"hcmod_i";
|
|
1994 |
val hcomplex_eq_hRe_eq = thm"hcomplex_eq_hRe_eq";
|
|
1995 |
val hcomplex_eq_hIm_eq = thm"hcomplex_eq_hIm_eq";
|
|
1996 |
val hcomplex_eq_cancel_iff = thm"hcomplex_eq_cancel_iff";
|
|
1997 |
val hcomplex_eq_cancel_iffA = thm"hcomplex_eq_cancel_iffA";
|
|
1998 |
val hcomplex_eq_cancel_iffB = thm"hcomplex_eq_cancel_iffB";
|
|
1999 |
val hcomplex_eq_cancel_iffC = thm"hcomplex_eq_cancel_iffC";
|
|
2000 |
val hcomplex_eq_cancel_iff2 = thm"hcomplex_eq_cancel_iff2";
|
|
2001 |
val hcomplex_eq_cancel_iff2a = thm"hcomplex_eq_cancel_iff2a";
|
|
2002 |
val hcomplex_eq_cancel_iff3 = thm"hcomplex_eq_cancel_iff3";
|
|
2003 |
val hcomplex_eq_cancel_iff3a = thm"hcomplex_eq_cancel_iff3a";
|
|
2004 |
val hcomplex_split_hRe_zero = thm"hcomplex_split_hRe_zero";
|
|
2005 |
val hcomplex_split_hIm_zero = thm"hcomplex_split_hIm_zero";
|
|
2006 |
val hRe_hsgn = thm"hRe_hsgn";
|
|
2007 |
val hIm_hsgn = thm"hIm_hsgn";
|
|
2008 |
val real_two_squares_add_zero_iff = thm"real_two_squares_add_zero_iff";
|
|
2009 |
val hcomplex_inverse_complex_split = thm"hcomplex_inverse_complex_split";
|
|
2010 |
val hRe_mult_i_eq = thm"hRe_mult_i_eq";
|
|
2011 |
val hIm_mult_i_eq = thm"hIm_mult_i_eq";
|
|
2012 |
val hcmod_mult_i = thm"hcmod_mult_i";
|
|
2013 |
val hcmod_mult_i2 = thm"hcmod_mult_i2";
|
|
2014 |
val harg = thm"harg";
|
|
2015 |
val cos_harg_i_mult_zero = thm"cos_harg_i_mult_zero";
|
|
2016 |
val cos_harg_i_mult_zero2 = thm"cos_harg_i_mult_zero2";
|
|
2017 |
val hcomplex_of_hypreal_not_zero_iff = thm"hcomplex_of_hypreal_not_zero_iff";
|
|
2018 |
val hcomplex_of_hypreal_zero_iff = thm"hcomplex_of_hypreal_zero_iff";
|
|
2019 |
val cos_harg_i_mult_zero3 = thm"cos_harg_i_mult_zero3";
|
|
2020 |
val complex_split_polar2 = thm"complex_split_polar2";
|
|
2021 |
val hcomplex_split_polar = thm"hcomplex_split_polar";
|
|
2022 |
val hcis = thm"hcis";
|
|
2023 |
val hcis_eq = thm"hcis_eq";
|
|
2024 |
val hrcis = thm"hrcis";
|
|
2025 |
val hrcis_Ex = thm"hrcis_Ex";
|
|
2026 |
val hRe_hcomplex_polar = thm"hRe_hcomplex_polar";
|
|
2027 |
val hRe_hrcis = thm"hRe_hrcis";
|
|
2028 |
val hIm_hcomplex_polar = thm"hIm_hcomplex_polar";
|
|
2029 |
val hIm_hrcis = thm"hIm_hrcis";
|
|
2030 |
val hcmod_complex_polar = thm"hcmod_complex_polar";
|
|
2031 |
val hcmod_hrcis = thm"hcmod_hrcis";
|
|
2032 |
val hcis_hrcis_eq = thm"hcis_hrcis_eq";
|
|
2033 |
val hrcis_mult = thm"hrcis_mult";
|
|
2034 |
val hcis_mult = thm"hcis_mult";
|
|
2035 |
val hcis_zero = thm"hcis_zero";
|
|
2036 |
val hrcis_zero_mod = thm"hrcis_zero_mod";
|
|
2037 |
val hrcis_zero_arg = thm"hrcis_zero_arg";
|
|
2038 |
val hcomplex_i_mult_minus = thm"hcomplex_i_mult_minus";
|
|
2039 |
val hcomplex_i_mult_minus2 = thm"hcomplex_i_mult_minus2";
|
|
2040 |
val hypreal_of_nat = thm"hypreal_of_nat";
|
|
2041 |
val hcis_hypreal_of_nat_Suc_mult = thm"hcis_hypreal_of_nat_Suc_mult";
|
|
2042 |
val NSDeMoivre = thm"NSDeMoivre";
|
|
2043 |
val hcis_hypreal_of_hypnat_Suc_mult = thm"hcis_hypreal_of_hypnat_Suc_mult";
|
|
2044 |
val NSDeMoivre_ext = thm"NSDeMoivre_ext";
|
|
2045 |
val DeMoivre2 = thm"DeMoivre2";
|
|
2046 |
val DeMoivre2_ext = thm"DeMoivre2_ext";
|
|
2047 |
val hcis_inverse = thm"hcis_inverse";
|
|
2048 |
val hrcis_inverse = thm"hrcis_inverse";
|
|
2049 |
val hRe_hcis = thm"hRe_hcis";
|
|
2050 |
val hIm_hcis = thm"hIm_hcis";
|
|
2051 |
val cos_n_hRe_hcis_pow_n = thm"cos_n_hRe_hcis_pow_n";
|
|
2052 |
val sin_n_hIm_hcis_pow_n = thm"sin_n_hIm_hcis_pow_n";
|
|
2053 |
val cos_n_hRe_hcis_hcpow_n = thm"cos_n_hRe_hcis_hcpow_n";
|
|
2054 |
val sin_n_hIm_hcis_hcpow_n = thm"sin_n_hIm_hcis_hcpow_n";
|
|
2055 |
val hexpi_add = thm"hexpi_add";
|
|
2056 |
val hcomplex_of_complex_add = thm"hcomplex_of_complex_add";
|
|
2057 |
val hcomplex_of_complex_mult = thm"hcomplex_of_complex_mult";
|
|
2058 |
val hcomplex_of_complex_eq_iff = thm"hcomplex_of_complex_eq_iff";
|
|
2059 |
val hcomplex_of_complex_minus = thm"hcomplex_of_complex_minus";
|
|
2060 |
val hcomplex_of_complex_one = thm"hcomplex_of_complex_one";
|
|
2061 |
val hcomplex_of_complex_zero = thm"hcomplex_of_complex_zero";
|
|
2062 |
val hcomplex_of_complex_zero_iff = thm"hcomplex_of_complex_zero_iff";
|
|
2063 |
val hcomplex_of_complex_inverse = thm"hcomplex_of_complex_inverse";
|
|
2064 |
val hcomplex_of_complex_divide = thm"hcomplex_of_complex_divide";
|
|
2065 |
val hRe_hcomplex_of_complex = thm"hRe_hcomplex_of_complex";
|
|
2066 |
val hIm_hcomplex_of_complex = thm"hIm_hcomplex_of_complex";
|
|
2067 |
val hcmod_hcomplex_of_complex = thm"hcmod_hcomplex_of_complex";
|
|
2068 |
|
|
2069 |
val hcomplex_add_ac = thms"hcomplex_add_ac";
|
|
2070 |
val hcomplex_mult_ac = thms"hcomplex_mult_ac";
|
|
2071 |
*}
|
|
2072 |
|
13957
|
2073 |
end
|