--- a/src/HOL/Complex/NSComplex.thy Mon Dec 22 16:22:14 2003 +0100
+++ b/src/HOL/Complex/NSComplex.thy Mon Dec 22 18:29:20 2003 +0100
@@ -4,124 +4,2232 @@
Description: Nonstandard Complex numbers
*)
-NSComplex = NSInduct +
+theory NSComplex = NSInduct:
constdefs
hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
- "hcomplexrel == {p. EX X Y. p = ((X::nat=>complex),Y) &
+ "hcomplexrel == {p. EX X Y. p = ((X::nat=>complex),Y) &
{n::nat. X(n) = Y(n)}: FreeUltrafilterNat}"
-typedef hcomplex = "{x::nat=>complex. True}//hcomplexrel" (quotient_def)
+typedef hcomplex = "{x::nat=>complex. True}//hcomplexrel"
+ by (auto simp add: quotient_def)
-instance
- hcomplex :: {zero,one,plus,times,minus,power,inverse}
-
-defs
- hcomplex_zero_def
+instance hcomplex :: zero ..
+instance hcomplex :: one ..
+instance hcomplex :: plus ..
+instance hcomplex :: times ..
+instance hcomplex :: minus ..
+instance hcomplex :: inverse ..
+instance hcomplex :: power ..
+
+defs (overloaded)
+ hcomplex_zero_def:
"0 == Abs_hcomplex(hcomplexrel `` {%n. (0::complex)})"
-
- hcomplex_one_def
+
+ hcomplex_one_def:
"1 == Abs_hcomplex(hcomplexrel `` {%n. (1::complex)})"
- hcomplex_minus_def
- "- z == Abs_hcomplex(UN X: Rep_hcomplex(z). hcomplexrel `` {%n::nat. - (X n)})"
+ hcomplex_minus_def:
+ "- z == Abs_hcomplex(UN X: Rep_hcomplex(z).
+ hcomplexrel `` {%n::nat. - (X n)})"
- hcomplex_diff_def
+ hcomplex_diff_def:
"w - z == w + -(z::hcomplex)"
-
+
constdefs
- hcomplex_of_complex :: complex => hcomplex
+ hcomplex_of_complex :: "complex => hcomplex"
"hcomplex_of_complex z == Abs_hcomplex(hcomplexrel `` {%n. z})"
-
- hcinv :: hcomplex => hcomplex
- "inverse(P) == Abs_hcomplex(UN X: Rep_hcomplex(P).
+
+ hcinv :: "hcomplex => hcomplex"
+ "inverse(P) == Abs_hcomplex(UN X: Rep_hcomplex(P).
hcomplexrel `` {%n. inverse(X n)})"
(*--- real and Imaginary parts ---*)
-
- hRe :: hcomplex => hypreal
+
+ hRe :: "hcomplex => hypreal"
"hRe(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Re (X n)})"
- hIm :: hcomplex => hypreal
- "hIm(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Im (X n)})"
+ hIm :: "hcomplex => hypreal"
+ "hIm(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Im (X n)})"
(*----------- modulus ------------*)
- hcmod :: hcomplex => hypreal
+ hcmod :: "hcomplex => hypreal"
"hcmod z == Abs_hypreal(UN X: Rep_hcomplex(z).
hyprel `` {%n. cmod (X n)})"
- (*------ imaginary unit ----------*)
-
- iii :: hcomplex
+ (*------ imaginary unit ----------*)
+
+ iii :: hcomplex
"iii == Abs_hcomplex(hcomplexrel `` {%n. ii})"
(*------- complex conjugate ------*)
- hcnj :: hcomplex => hcomplex
+ hcnj :: "hcomplex => hcomplex"
"hcnj z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. cnj (X n)})"
- (*------------ Argand -------------*)
+ (*------------ Argand -------------*)
- hsgn :: hcomplex => hcomplex
+ hsgn :: "hcomplex => hcomplex"
"hsgn z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. sgn(X n)})"
- harg :: hcomplex => hypreal
+ harg :: "hcomplex => hypreal"
"harg z == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. arg(X n)})"
(* abbreviation for (cos a + i sin a) *)
- hcis :: hypreal => hcomplex
+ hcis :: "hypreal => hcomplex"
"hcis a == Abs_hcomplex(UN X:Rep_hypreal(a). hcomplexrel `` {%n. cis (X n)})"
(* abbreviation for r*(cos a + i sin a) *)
- hrcis :: [hypreal, hypreal] => hcomplex
+ hrcis :: "[hypreal, hypreal] => hcomplex"
"hrcis r a == hcomplex_of_hypreal r * hcis a"
- (*----- injection from hyperreals -----*)
-
- hcomplex_of_hypreal :: hypreal => hcomplex
+ (*----- injection from hyperreals -----*)
+
+ hcomplex_of_hypreal :: "hypreal => hcomplex"
"hcomplex_of_hypreal r == Abs_hcomplex(UN X:Rep_hypreal(r).
hcomplexrel `` {%n. complex_of_real (X n)})"
(*------------ e ^ (x + iy) ------------*)
- hexpi :: hcomplex => hcomplex
+ hexpi :: "hcomplex => hcomplex"
"hexpi z == hcomplex_of_hypreal(( *f* exp) (hRe z)) * hcis (hIm z)"
-
+
-defs
-
+defs (overloaded)
(*----------- division ----------*)
- hcomplex_divide_def
+ hcomplex_divide_def:
"w / (z::hcomplex) == w * inverse z"
-
- hcomplex_add_def
+
+ hcomplex_add_def:
"w + z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
hcomplexrel `` {%n. X n + Y n})"
- hcomplex_mult_def
+ hcomplex_mult_def:
"w * z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
- hcomplexrel `` {%n. X n * Y n})"
+ hcomplexrel `` {%n. X n * Y n})"
primrec
- hcomplexpow_0 "z ^ 0 = 1"
- hcomplexpow_Suc "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
+ hcomplexpow_0: "z ^ 0 = 1"
+ hcomplexpow_Suc: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
consts
- "hcpow" :: [hcomplex,hypnat] => hcomplex (infixr 80)
+ "hcpow" :: "[hcomplex,hypnat] => hcomplex" (infixr 80)
defs
(* hypernatural powers of nonstandard complex numbers *)
- hcpow_def
- "(z::hcomplex) hcpow (n::hypnat)
+ hcpow_def:
+ "(z::hcomplex) hcpow (n::hypnat)
== Abs_hcomplex(UN X:Rep_hcomplex(z). UN Y: Rep_hypnat(n).
hcomplexrel `` {%n. (X n) ^ (Y n)})"
+
+lemma hcomplexrel_iff:
+ "((X,Y): hcomplexrel) = ({n. X n = Y n}: FreeUltrafilterNat)"
+apply (unfold hcomplexrel_def)
+apply fast
+done
+
+lemma hcomplexrel_refl: "(x,x): hcomplexrel"
+apply (simp add: hcomplexrel_iff)
+done
+
+lemma hcomplexrel_sym: "(x,y): hcomplexrel ==> (y,x):hcomplexrel"
+apply (auto simp add: hcomplexrel_iff eq_commute)
+done
+
+lemma hcomplexrel_trans:
+ "[|(x,y): hcomplexrel; (y,z):hcomplexrel|] ==> (x,z):hcomplexrel"
+apply (simp add: hcomplexrel_iff)
+apply ultra
+done
+
+lemma equiv_hcomplexrel: "equiv UNIV hcomplexrel"
+apply (simp add: equiv_def refl_def sym_def trans_def hcomplexrel_refl)
+apply (blast intro: hcomplexrel_sym hcomplexrel_trans)
+done
+
+lemmas equiv_hcomplexrel_iff =
+ eq_equiv_class_iff [OF equiv_hcomplexrel UNIV_I UNIV_I, simp]
+
+lemma hcomplexrel_in_hcomplex [simp]: "hcomplexrel``{x} : hcomplex"
+apply (unfold hcomplex_def hcomplexrel_def quotient_def)
+apply blast
+done
+
+lemma inj_on_Abs_hcomplex: "inj_on Abs_hcomplex hcomplex"
+apply (rule inj_on_inverseI)
+apply (erule Abs_hcomplex_inverse)
+done
+
+declare inj_on_Abs_hcomplex [THEN inj_on_iff, simp]
+ Abs_hcomplex_inverse [simp]
+
+declare equiv_hcomplexrel [THEN eq_equiv_class_iff, simp]
+
+declare hcomplexrel_iff [iff]
+
+lemma inj_Rep_hcomplex: "inj(Rep_hcomplex)"
+apply (rule inj_on_inverseI)
+apply (rule Rep_hcomplex_inverse)
+done
+
+lemma lemma_hcomplexrel_refl: "x: hcomplexrel `` {x}"
+apply (unfold hcomplexrel_def)
+apply (safe)
+apply auto
+done
+declare lemma_hcomplexrel_refl [simp]
+
+lemma hcomplex_empty_not_mem: "{} ~: hcomplex"
+apply (unfold hcomplex_def)
+apply (auto elim!: quotientE)
+done
+declare hcomplex_empty_not_mem [simp]
+
+lemma Rep_hcomplex_nonempty: "Rep_hcomplex x ~= {}"
+apply (cut_tac x = "x" in Rep_hcomplex)
+apply auto
+done
+declare Rep_hcomplex_nonempty [simp]
+
+lemma eq_Abs_hcomplex:
+ "(!!x. z = Abs_hcomplex(hcomplexrel `` {x}) ==> P) ==> P"
+apply (rule_tac x1=z in Rep_hcomplex [unfolded hcomplex_def, THEN quotientE])
+apply (drule_tac f = Abs_hcomplex in arg_cong)
+apply (force simp add: Rep_hcomplex_inverse)
+done
+
+
+subsection{*Properties of Nonstandard Real and Imaginary Parts*}
+
+lemma hRe:
+ "hRe(Abs_hcomplex (hcomplexrel `` {X})) =
+ Abs_hypreal(hyprel `` {%n. Re(X n)})"
+apply (unfold hRe_def)
+apply (rule_tac f = "Abs_hypreal" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hIm:
+ "hIm(Abs_hcomplex (hcomplexrel `` {X})) =
+ Abs_hypreal(hyprel `` {%n. Im(X n)})"
+apply (unfold hIm_def)
+apply (rule_tac f = "Abs_hypreal" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hcomplex_hRe_hIm_cancel_iff: "(w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (auto simp add: hRe hIm complex_Re_Im_cancel_iff)
+apply (ultra+)
+done
+
+lemma hcomplex_hRe_zero: "hRe 0 = 0"
+apply (unfold hcomplex_zero_def)
+apply (simp (no_asm) add: hRe hypreal_zero_num)
+done
+declare hcomplex_hRe_zero [simp]
+
+lemma hcomplex_hIm_zero: "hIm 0 = 0"
+apply (unfold hcomplex_zero_def)
+apply (simp (no_asm) add: hIm hypreal_zero_num)
+done
+declare hcomplex_hIm_zero [simp]
+
+lemma hcomplex_hRe_one: "hRe 1 = 1"
+apply (unfold hcomplex_one_def)
+apply (simp (no_asm) add: hRe hypreal_one_num)
+done
+declare hcomplex_hRe_one [simp]
+
+lemma hcomplex_hIm_one: "hIm 1 = 0"
+apply (unfold hcomplex_one_def)
+apply (simp (no_asm) add: hIm hypreal_one_def hypreal_zero_num)
+done
+declare hcomplex_hIm_one [simp]
+
+(*-----------------------------------------------------------------------*)
+(* hcomplex_of_complex: the injection from complex to hcomplex *)
+(* ----------------------------------------------------------------------*)
+
+lemma inj_hcomplex_of_complex: "inj(hcomplex_of_complex)"
+apply (rule inj_onI , rule ccontr)
+apply (auto simp add: hcomplex_of_complex_def)
+done
+
+lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex ii"
+apply (unfold iii_def hcomplex_of_complex_def)
+apply (simp (no_asm))
+done
+
+(*-----------------------------------------------------------------------*)
+(* Addition for nonstandard complex numbers: hcomplex_add *)
+(* ----------------------------------------------------------------------*)
+
+lemma hcomplex_add_congruent2:
+ "congruent2 hcomplexrel (%X Y. hcomplexrel `` {%n. X n + Y n})"
+apply (unfold congruent2_def)
+apply safe
+apply (ultra+)
+done
+
+lemma hcomplex_add:
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) + Abs_hcomplex(hcomplexrel``{%n. Y n}) =
+ Abs_hcomplex(hcomplexrel``{%n. X n + Y n})"
+apply (unfold hcomplex_add_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply auto
+apply (ultra)
+done
+
+lemma hcomplex_add_commute: "(z::hcomplex) + w = w + z"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (simp (no_asm_simp) add: complex_add_commute hcomplex_add)
+done
+
+lemma hcomplex_add_assoc: "((z1::hcomplex) + z2) + z3 = z1 + (z2 + z3)"
+apply (rule_tac z = "z1" in eq_Abs_hcomplex)
+apply (rule_tac z = "z2" in eq_Abs_hcomplex)
+apply (rule_tac z = "z3" in eq_Abs_hcomplex)
+apply (simp (no_asm_simp) add: hcomplex_add complex_add_assoc)
+done
+
+(*For AC rewriting*)
+lemma hcomplex_add_left_commute: "(x::hcomplex)+(y+z)=y+(x+z)"
+apply (rule hcomplex_add_commute [THEN trans])
+apply (rule hcomplex_add_assoc [THEN trans])
+apply (rule hcomplex_add_commute [THEN arg_cong])
+done
+
+(* hcomplex addition is an AC operator *)
+lemmas hcomplex_add_ac = hcomplex_add_assoc hcomplex_add_commute
+ hcomplex_add_left_commute
+
+lemma hcomplex_add_zero_left: "(0::hcomplex) + z = z"
+apply (unfold hcomplex_zero_def)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (simp add: hcomplex_add)
+done
+
+lemma hcomplex_add_zero_right: "z + (0::hcomplex) = z"
+apply (simp (no_asm) add: hcomplex_add_zero_left hcomplex_add_commute)
+done
+declare hcomplex_add_zero_left [simp] hcomplex_add_zero_right [simp]
+
+lemma hRe_add: "hRe(x + y) = hRe(x) + hRe(y)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hRe hcomplex_add hypreal_add complex_Re_add)
+done
+
+lemma hIm_add: "hIm(x + y) = hIm(x) + hIm(y)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hIm hcomplex_add hypreal_add complex_Im_add)
+done
+
+(*-----------------------------------------------------------------------*)
+(* hypreal_minus: additive inverse on nonstandard complex *)
+(* ----------------------------------------------------------------------*)
+
+lemma hcomplex_minus_congruent:
+ "congruent hcomplexrel (%X. hcomplexrel `` {%n. - (X n)})"
+
+apply (unfold congruent_def)
+apply safe
+apply (ultra+)
+done
+
+lemma hcomplex_minus:
+ "- (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. -(X n)})"
+apply (unfold hcomplex_minus_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hcomplex_minus_minus: "- (- z) = (z::hcomplex)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (simp (no_asm_simp) add: hcomplex_minus)
+done
+declare hcomplex_minus_minus [simp]
+
+lemma inj_hcomplex_minus: "inj(%z::hcomplex. -z)"
+apply (rule inj_onI)
+apply (drule_tac f = "uminus" in arg_cong)
+apply simp
+done
+
+lemma hcomplex_minus_zero: "- 0 = (0::hcomplex)"
+apply (unfold hcomplex_zero_def)
+apply (simp (no_asm) add: hcomplex_minus)
+done
+declare hcomplex_minus_zero [simp]
+
+lemma hcomplex_minus_zero_iff: "(-x = 0) = (x = (0::hcomplex))"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_zero_def hcomplex_minus)
+done
+declare hcomplex_minus_zero_iff [simp]
+
+lemma hcomplex_minus_zero_iff2: "(0 = -x) = (x = (0::hcomplex))"
+apply (auto dest: sym)
+done
+declare hcomplex_minus_zero_iff2 [simp]
+
+lemma hcomplex_minus_not_zero_iff: "(-x ~= 0) = (x ~= (0::hcomplex))"
+apply auto
+done
+
+lemma hcomplex_add_minus_right: "z + - z = (0::hcomplex)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_add hcomplex_minus hcomplex_zero_def)
+done
+declare hcomplex_add_minus_right [simp]
+
+lemma hcomplex_add_minus_left: "-z + z = (0::hcomplex)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_add hcomplex_minus hcomplex_zero_def)
+done
+declare hcomplex_add_minus_left [simp]
+
+lemma hcomplex_add_minus_cancel: "z + (- z + w) = (w::hcomplex)"
+apply (simp (no_asm) add: hcomplex_add_assoc [symmetric])
+done
+
+lemma hcomplex_minus_add_cancel: "(-z) + (z + w) = (w::hcomplex)"
+apply (simp (no_asm) add: hcomplex_add_assoc [symmetric])
+done
+
+lemma hRe_minus: "hRe(-z) = - hRe(z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hRe hcomplex_minus hypreal_minus complex_Re_minus)
+done
+
+lemma hIm_minus: "hIm(-z) = - hIm(z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hIm hcomplex_minus hypreal_minus complex_Im_minus)
+done
+
+lemma hcomplex_add_minus_eq_minus:
+ "x + y = (0::hcomplex) ==> x = -y"
+apply (unfold hcomplex_zero_def)
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_add hcomplex_minus)
+apply ultra
+done
+
+lemma hcomplex_minus_add_distrib: "-(x + y) = -x + -(y::hcomplex)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_add hcomplex_minus)
+done
+declare hcomplex_minus_add_distrib [simp]
+
+lemma hcomplex_add_left_cancel: "((x::hcomplex) + y = x + z) = (y = z)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_add)
+done
+declare hcomplex_add_left_cancel [iff]
+
+lemma hcomplex_add_right_cancel: "(y + (x::hcomplex)= z + x) = (y = z)"
+apply (simp (no_asm) add: hcomplex_add_commute)
+done
+declare hcomplex_add_right_cancel [iff]
+
+lemma hcomplex_eq_minus_iff: "((x::hcomplex) = y) = ((0::hcomplex) = x + - y)"
+apply (safe)
+apply (rule_tac [2] x1 = "-y" in hcomplex_add_right_cancel [THEN iffD1])
+apply auto
+done
+
+lemma hcomplex_eq_minus_iff2: "((x::hcomplex) = y) = (x + - y = (0::hcomplex))"
+apply (safe)
+apply (rule_tac [2] x1 = "-y" in hcomplex_add_right_cancel [THEN iffD1])
+apply auto
+done
+
+subsection{*Subraction for Nonstandard Complex Numbers*}
+
+lemma hcomplex_diff:
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) - Abs_hcomplex(hcomplexrel``{%n. Y n}) =
+ Abs_hcomplex(hcomplexrel``{%n. X n - Y n})"
+
+apply (unfold hcomplex_diff_def)
+apply (auto simp add: hcomplex_minus hcomplex_add complex_diff_def)
+done
+
+lemma hcomplex_diff_zero: "(z::hcomplex) - z = (0::hcomplex)"
+apply (unfold hcomplex_diff_def)
+apply (simp (no_asm))
+done
+declare hcomplex_diff_zero [simp]
+
+lemma hcomplex_diff_0: "(0::hcomplex) - x = -x"
+apply (simp (no_asm) add: hcomplex_diff_def)
+done
+
+lemma hcomplex_diff_0_right: "x - (0::hcomplex) = x"
+apply (simp (no_asm) add: hcomplex_diff_def)
+done
+
+lemma hcomplex_diff_self: "x - x = (0::hcomplex)"
+apply (simp (no_asm) add: hcomplex_diff_def)
+done
+
+declare hcomplex_diff_0 [simp] hcomplex_diff_0_right [simp] hcomplex_diff_self [simp]
+
+lemma hcomplex_diff_eq_eq: "((x::hcomplex) - y = z) = (x = z + y)"
+apply (auto simp add: hcomplex_diff_def hcomplex_add_assoc)
+done
+
+subsection{*Multiplication for Nonstandard Complex Numbers*}
+
+lemma hcomplex_mult:
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) * Abs_hcomplex(hcomplexrel``{%n. Y n}) =
+ Abs_hcomplex(hcomplexrel``{%n. X n * Y n})"
+
+apply (unfold hcomplex_mult_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hcomplex_mult_commute: "(w::hcomplex) * z = z * w"
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult complex_mult_commute)
+done
+
+lemma hcomplex_mult_assoc: "((u::hcomplex) * v) * w = u * (v * w)"
+apply (rule_tac z = "u" in eq_Abs_hcomplex)
+apply (rule_tac z = "v" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult complex_mult_assoc)
+done
+
+lemma hcomplex_mult_left_commute: "(x::hcomplex) * (y * z) = y * (x * z)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult complex_mult_left_commute)
+done
+
+lemmas hcomplex_mult_ac = hcomplex_mult_assoc hcomplex_mult_commute
+ hcomplex_mult_left_commute
+
+lemma hcomplex_mult_one_left: "(1::hcomplex) * z = z"
+apply (unfold hcomplex_one_def)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult)
+done
+declare hcomplex_mult_one_left [simp]
+
+lemma hcomplex_mult_one_right: "z * (1::hcomplex) = z"
+apply (simp (no_asm) add: hcomplex_mult_commute)
+done
+declare hcomplex_mult_one_right [simp]
+
+lemma hcomplex_mult_zero_left: "(0::hcomplex) * z = 0"
+apply (unfold hcomplex_zero_def)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult)
+done
+declare hcomplex_mult_zero_left [simp]
+
+lemma hcomplex_mult_zero_right: "z * (0::hcomplex) = 0"
+apply (simp (no_asm) add: hcomplex_mult_commute)
+done
+declare hcomplex_mult_zero_right [simp]
+
+lemma hcomplex_minus_mult_eq1: "-(x * y) = -x * (y::hcomplex)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult hcomplex_minus)
+done
+
+lemma hcomplex_minus_mult_eq2: "-(x * y) = x * -(y::hcomplex)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult hcomplex_minus)
+done
+
+declare hcomplex_minus_mult_eq1 [symmetric, simp] hcomplex_minus_mult_eq2 [symmetric, simp]
+
+lemma hcomplex_mult_minus_one: "- 1 * (z::hcomplex) = -z"
+apply (simp (no_asm))
+done
+declare hcomplex_mult_minus_one [simp]
+
+lemma hcomplex_mult_minus_one_right: "(z::hcomplex) * - 1 = -z"
+apply (subst hcomplex_mult_commute)
+apply (simp (no_asm))
+done
+declare hcomplex_mult_minus_one_right [simp]
+
+lemma hcomplex_minus_mult_cancel: "-x * -y = x * (y::hcomplex)"
+apply auto
+done
+declare hcomplex_minus_mult_cancel [simp]
+
+lemma hcomplex_minus_mult_commute: "-x * y = x * -(y::hcomplex)"
+apply auto
+done
+
+lemma hcomplex_add_mult_distrib: "((z1::hcomplex) + z2) * w = (z1 * w) + (z2 * w)"
+apply (rule_tac z = "z1" in eq_Abs_hcomplex)
+apply (rule_tac z = "z2" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_mult hcomplex_add complex_add_mult_distrib)
+done
+
+lemma hcomplex_add_mult_distrib2: "(w::hcomplex) * (z1 + z2) = (w * z1) + (w * z2)"
+apply (rule_tac z1 = "z1 + z2" in hcomplex_mult_commute [THEN ssubst])
+apply (simp (no_asm) add: hcomplex_add_mult_distrib)
+apply (simp (no_asm) add: hcomplex_mult_commute)
+done
+
+lemma hcomplex_zero_not_eq_one: "(0::hcomplex) ~= (1::hcomplex)"
+apply (unfold hcomplex_zero_def hcomplex_one_def)
+apply auto
+done
+declare hcomplex_zero_not_eq_one [simp]
+declare hcomplex_zero_not_eq_one [THEN not_sym, simp]
+
+
+subsection{*Inverse of Nonstandard Complex Number*}
+
+lemma hcomplex_inverse:
+ "inverse (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. inverse (X n)})"
+apply (unfold hcinv_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma HCOMPLEX_INVERSE_ZERO: "inverse (0::hcomplex) = 0"
+apply (unfold hcomplex_zero_def)
+apply (auto simp add: hcomplex_inverse)
+done
+
+lemma HCOMPLEX_DIVISION_BY_ZERO: "a / (0::hcomplex) = 0"
+apply (simp (no_asm) add: hcomplex_divide_def HCOMPLEX_INVERSE_ZERO)
+done
+
+lemma hcomplex_mult_inv_left:
+ "z ~= (0::hcomplex) ==> inverse(z) * z = (1::hcomplex)"
+apply (unfold hcomplex_zero_def hcomplex_one_def)
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_inverse hcomplex_mult)
+apply (ultra)
+apply (rule ccontr)
+apply (drule complex_mult_inv_left)
+apply auto
+done
+declare hcomplex_mult_inv_left [simp]
+
+lemma hcomplex_mult_inv_right: "z ~= (0::hcomplex) ==> z * inverse(z) = (1::hcomplex)"
+apply (auto intro: hcomplex_mult_commute [THEN subst])
+done
+declare hcomplex_mult_inv_right [simp]
+
+lemma hcomplex_mult_left_cancel: "(c::hcomplex) ~= (0::hcomplex) ==> (c*a=c*b) = (a=b)"
+apply auto
+apply (drule_tac f = "%x. x*inverse c" in arg_cong)
+apply (simp add: hcomplex_mult_ac)
+done
+
+lemma hcomplex_mult_right_cancel: "(c::hcomplex) ~= (0::hcomplex) ==> (a*c=b*c) = (a=b)"
+apply (safe)
+apply (drule_tac f = "%x. x*inverse c" in arg_cong)
+apply (simp add: hcomplex_mult_ac)
+done
+
+lemma hcomplex_inverse_not_zero: "z ~= (0::hcomplex) ==> inverse(z) ~= 0"
+apply (safe)
+apply (frule hcomplex_mult_right_cancel [THEN iffD2])
+apply (erule_tac [2] V = "inverse z = 0" in thin_rl)
+apply (assumption , auto)
+done
+declare hcomplex_inverse_not_zero [simp]
+
+lemma hcomplex_mult_not_zero: "[| x ~= (0::hcomplex); y ~= 0 |] ==> x * y ~= 0"
+apply (safe)
+apply (drule_tac f = "%z. inverse x*z" in arg_cong)
+apply (simp add: hcomplex_mult_assoc [symmetric])
+done
+
+lemmas hcomplex_mult_not_zeroE = hcomplex_mult_not_zero [THEN notE, standard]
+
+lemma hcomplex_inverse_inverse: "inverse(inverse x) = (x::hcomplex)"
+apply (case_tac "x = 0", simp add: HCOMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "inverse x" in hcomplex_mult_right_cancel [THEN iffD1])
+apply (erule hcomplex_inverse_not_zero)
+apply (auto dest: hcomplex_inverse_not_zero)
+done
+declare hcomplex_inverse_inverse [simp]
+
+lemma hcomplex_inverse_one: "inverse((1::hcomplex)) = 1"
+apply (unfold hcomplex_one_def)
+apply (simp (no_asm) add: hcomplex_inverse)
+done
+declare hcomplex_inverse_one [simp]
+
+lemma hcomplex_minus_inverse: "inverse(-x) = -inverse(x::hcomplex)"
+apply (case_tac "x = 0", simp add: HCOMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "-x" in hcomplex_mult_right_cancel [THEN iffD1])
+apply (force );
+apply (subst hcomplex_mult_inv_left)
+apply auto
+done
+
+lemma hcomplex_inverse_distrib: "inverse(x*y) = inverse x * inverse (y::hcomplex)"
+apply (case_tac "x = 0", simp add: HCOMPLEX_INVERSE_ZERO)
+apply (case_tac "y = 0", simp add: HCOMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "x*y" in hcomplex_mult_left_cancel [THEN iffD1])
+apply (auto simp add: hcomplex_mult_not_zero hcomplex_mult_ac)
+apply (auto simp add: hcomplex_mult_not_zero hcomplex_mult_assoc [symmetric])
+done
+
+subsection{*Division*}
+
+lemma hcomplex_times_divide1_eq: "(x::hcomplex) * (y/z) = (x*y)/z"
+apply (simp (no_asm) add: hcomplex_divide_def hcomplex_mult_assoc)
+done
+
+lemma hcomplex_times_divide2_eq: "(y/z) * (x::hcomplex) = (y*x)/z"
+apply (simp (no_asm) add: hcomplex_divide_def hcomplex_mult_ac)
+done
+
+declare hcomplex_times_divide1_eq [simp] hcomplex_times_divide2_eq [simp]
+
+lemma hcomplex_divide_divide1_eq: "(x::hcomplex) / (y/z) = (x*z)/y"
+apply (simp (no_asm) add: hcomplex_divide_def hcomplex_inverse_distrib hcomplex_mult_ac)
+done
+
+lemma hcomplex_divide_divide2_eq: "((x::hcomplex) / y) / z = x/(y*z)"
+apply (simp (no_asm) add: hcomplex_divide_def hcomplex_inverse_distrib hcomplex_mult_assoc)
+done
+
+declare hcomplex_divide_divide1_eq [simp] hcomplex_divide_divide2_eq [simp]
+
+(** As with multiplication, pull minus signs OUT of the / operator **)
+
+lemma hcomplex_minus_divide_eq: "(-x) / (y::hcomplex) = - (x/y)"
+apply (simp (no_asm) add: hcomplex_divide_def)
+done
+declare hcomplex_minus_divide_eq [simp]
+
+lemma hcomplex_divide_minus_eq: "(x / -(y::hcomplex)) = - (x/y)"
+apply (simp (no_asm) add: hcomplex_divide_def hcomplex_minus_inverse)
+done
+declare hcomplex_divide_minus_eq [simp]
+
+lemma hcomplex_add_divide_distrib: "(x+y)/(z::hcomplex) = x/z + y/z"
+apply (simp (no_asm) add: hcomplex_divide_def hcomplex_add_mult_distrib)
+done
+
+
+subsection{*Embedding Properties for @{term hcomplex_of_hypreal} Map*}
+
+lemma hcomplex_of_hypreal:
+ "hcomplex_of_hypreal (Abs_hypreal(hyprel `` {%n. X n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. complex_of_real (X n)})"
+apply (unfold hcomplex_of_hypreal_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply auto
+apply (ultra)
+done
+
+lemma inj_hcomplex_of_hypreal: "inj hcomplex_of_hypreal"
+apply (rule inj_onI)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal)
+done
+
+lemma hcomplex_of_hypreal_cancel_iff: "(hcomplex_of_hypreal x = hcomplex_of_hypreal y) = (x = y)"
+apply (auto dest: inj_hcomplex_of_hypreal [THEN injD])
+done
+declare hcomplex_of_hypreal_cancel_iff [iff]
+
+lemma hcomplex_of_hypreal_minus: "hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_minus hypreal_minus complex_of_real_minus)
+done
+
+lemma hcomplex_of_hypreal_inverse: "hcomplex_of_hypreal(inverse x) = inverse(hcomplex_of_hypreal x)"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hypreal_inverse hcomplex_inverse complex_of_real_inverse)
+done
+
+lemma hcomplex_of_hypreal_add: "hcomplex_of_hypreal x + hcomplex_of_hypreal y =
+ hcomplex_of_hypreal (x + y)"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hypreal_add hcomplex_add complex_of_real_add)
+done
+
+lemma hcomplex_of_hypreal_diff:
+ "hcomplex_of_hypreal x - hcomplex_of_hypreal y =
+ hcomplex_of_hypreal (x - y)"
+apply (unfold hcomplex_diff_def)
+apply (auto simp add: hcomplex_of_hypreal_minus [symmetric] hcomplex_of_hypreal_add hypreal_diff_def)
+done
+
+lemma hcomplex_of_hypreal_mult: "hcomplex_of_hypreal x * hcomplex_of_hypreal y =
+ hcomplex_of_hypreal (x * y)"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hypreal_mult hcomplex_mult complex_of_real_mult)
+done
+
+lemma hcomplex_of_hypreal_divide:
+ "hcomplex_of_hypreal x / hcomplex_of_hypreal y = hcomplex_of_hypreal(x/y)"
+apply (unfold hcomplex_divide_def)
+apply (case_tac "y=0")
+apply (simp (no_asm_simp) add: HYPREAL_DIVISION_BY_ZERO HYPREAL_INVERSE_ZERO HCOMPLEX_INVERSE_ZERO)
+apply (auto simp add: hcomplex_of_hypreal_mult hcomplex_of_hypreal_inverse [symmetric])
+apply (simp (no_asm) add: hypreal_divide_def)
+done
+
+lemma hcomplex_of_hypreal_one [simp]:
+ "hcomplex_of_hypreal 1 = 1"
+apply (unfold hcomplex_one_def)
+apply (auto simp add: hcomplex_of_hypreal hypreal_one_num)
+done
+
+lemma hcomplex_of_hypreal_zero [simp]:
+ "hcomplex_of_hypreal 0 = 0"
+apply (unfold hcomplex_zero_def hypreal_zero_def)
+apply (auto simp add: hcomplex_of_hypreal)
+done
+
+lemma hcomplex_of_hypreal_pow:
+ "hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: hcomplex_of_hypreal_mult [symmetric])
+done
+
+lemma hRe_hcomplex_of_hypreal: "hRe(hcomplex_of_hypreal z) = z"
+apply (rule_tac z = "z" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hRe)
+done
+declare hRe_hcomplex_of_hypreal [simp]
+
+lemma hIm_hcomplex_of_hypreal: "hIm(hcomplex_of_hypreal z) = 0"
+apply (rule_tac z = "z" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hIm hypreal_zero_num)
+done
+declare hIm_hcomplex_of_hypreal [simp]
+
+lemma hcomplex_of_hypreal_epsilon_not_zero: "hcomplex_of_hypreal epsilon ~= 0"
+apply (auto simp add: hcomplex_of_hypreal epsilon_def hcomplex_zero_def)
+done
+declare hcomplex_of_hypreal_epsilon_not_zero [simp]
+
+(*---------------------------------------------------------------------------*)
+(* Modulus (absolute value) of nonstandard complex number *)
+(*---------------------------------------------------------------------------*)
+
+lemma hcmod:
+ "hcmod (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ Abs_hypreal(hyprel `` {%n. cmod (X n)})"
+
+apply (unfold hcmod_def)
+apply (rule_tac f = "Abs_hypreal" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hcmod_zero [simp]:
+ "hcmod(0) = 0"
+apply (unfold hcomplex_zero_def hypreal_zero_def)
+apply (auto simp add: hcmod)
+done
+
+lemma hcmod_one:
+ "hcmod(1) = 1"
+apply (unfold hcomplex_one_def)
+apply (auto simp add: hcmod hypreal_one_num)
+done
+declare hcmod_one [simp]
+
+lemma hcmod_hcomplex_of_hypreal: "hcmod(hcomplex_of_hypreal x) = abs x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: hcmod hcomplex_of_hypreal hypreal_hrabs)
+done
+declare hcmod_hcomplex_of_hypreal [simp]
+
+lemma hcomplex_of_hypreal_abs: "hcomplex_of_hypreal (abs x) =
+ hcomplex_of_hypreal(hcmod(hcomplex_of_hypreal x))"
+apply (simp (no_asm))
+done
+
+
+subsection{*Conjugation*}
+
+lemma hcnj:
+ "hcnj (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. cnj(X n)})"
+
+apply (unfold hcnj_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma inj_hcnj: "inj hcnj"
+apply (rule inj_onI)
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj)
+done
+
+lemma hcomplex_hcnj_cancel_iff: "(hcnj x = hcnj y) = (x = y)"
+apply (auto dest: inj_hcnj [THEN injD])
+done
+declare hcomplex_hcnj_cancel_iff [simp]
+
+lemma hcomplex_hcnj_hcnj: "hcnj (hcnj z) = z"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj)
+done
+declare hcomplex_hcnj_hcnj [simp]
+
+lemma hcomplex_hcnj_hcomplex_of_hypreal: "hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: hcnj hcomplex_of_hypreal)
+done
+declare hcomplex_hcnj_hcomplex_of_hypreal [simp]
+
+lemma hcomplex_hmod_hcnj: "hcmod (hcnj z) = hcmod z"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcmod)
+done
+declare hcomplex_hmod_hcnj [simp]
+
+lemma hcomplex_hcnj_minus: "hcnj (-z) = - hcnj z"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcomplex_minus complex_cnj_minus)
+done
+
+lemma hcomplex_hcnj_inverse: "hcnj(inverse z) = inverse(hcnj z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcomplex_inverse complex_cnj_inverse)
+done
+
+lemma hcomplex_hcnj_add: "hcnj(w + z) = hcnj(w) + hcnj(z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcomplex_add complex_cnj_add)
+done
+
+lemma hcomplex_hcnj_diff: "hcnj(w - z) = hcnj(w) - hcnj(z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcomplex_diff complex_cnj_diff)
+done
+
+lemma hcomplex_hcnj_mult: "hcnj(w * z) = hcnj(w) * hcnj(z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (rule_tac z = "w" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcomplex_mult complex_cnj_mult)
+done
+
+lemma hcomplex_hcnj_divide: "hcnj(w / z) = (hcnj w)/(hcnj z)"
+apply (unfold hcomplex_divide_def)
+apply (simp (no_asm) add: hcomplex_hcnj_mult hcomplex_hcnj_inverse)
+done
+
+lemma hcnj_one: "hcnj 1 = 1"
+apply (unfold hcomplex_one_def)
+apply (simp (no_asm) add: hcnj)
+done
+declare hcnj_one [simp]
+
+lemma hcomplex_hcnj_pow: "hcnj(z ^ n) = hcnj(z) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: hcomplex_hcnj_mult)
+done
+
+(* MOVE to NSComplexBin
+Goal "z + hcnj z =
+ hcomplex_of_hypreal (2 * hRe(z))"
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
+ hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
+qed "hcomplex_add_hcnj";
+
+Goal "z - hcnj z = \
+\ hcomplex_of_hypreal (hypreal_of_real 2 * hIm(z)) * iii";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
+ hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
+ complex_diff_cnj,iii_def,hcomplex_mult]));
+qed "hcomplex_diff_hcnj";
+*)
+
+lemma hcomplex_hcnj_zero:
+ "hcnj 0 = 0"
+apply (unfold hcomplex_zero_def)
+apply (auto simp add: hcnj)
+done
+declare hcomplex_hcnj_zero [simp]
+
+lemma hcomplex_hcnj_zero_iff: "(hcnj z = 0) = (z = 0)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_zero_def hcnj)
+done
+declare hcomplex_hcnj_zero_iff [iff]
+
+lemma hcomplex_mult_hcnj: "z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add hypreal_mult complex_mult_cnj two_eq_Suc_Suc)
+done
+
+
+(*---------------------------------------------------------------------------*)
+(* some algebra etc. *)
+(*---------------------------------------------------------------------------*)
+
+lemma hcomplex_mult_zero_iff: "(x*y = (0::hcomplex)) = (x = 0 | y = 0)"
+apply auto
+apply (auto intro: ccontr dest: hcomplex_mult_not_zero)
+done
+declare hcomplex_mult_zero_iff [simp]
+
+lemma hcomplex_add_left_cancel_zero: "(x + y = x) = (y = (0::hcomplex))"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_add hcomplex_zero_def)
+done
+declare hcomplex_add_left_cancel_zero [simp]
+
+lemma hcomplex_diff_mult_distrib:
+ "((z1::hcomplex) - z2) * w = (z1 * w) - (z2 * w)"
+apply (unfold hcomplex_diff_def)
+apply (simp (no_asm) add: hcomplex_add_mult_distrib)
+done
+
+lemma hcomplex_diff_mult_distrib2:
+ "(w::hcomplex) * (z1 - z2) = (w * z1) - (w * z2)"
+apply (unfold hcomplex_diff_def)
+apply (simp (no_asm) add: hcomplex_add_mult_distrib2)
+done
+
+(*---------------------------------------------------------------------------*)
+(* More theorems about hcmod *)
+(*---------------------------------------------------------------------------*)
+
+lemma hcomplex_hcmod_eq_zero_cancel: "(hcmod x = 0) = (x = 0)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_zero_def hypreal_zero_num)
+done
+declare hcomplex_hcmod_eq_zero_cancel [simp]
+
+(* not proved already? strange! *)
+lemma hypreal_of_nat_le_iff:
+ "(hypreal_of_nat n <= hypreal_of_nat m) = (n <= m)"
+apply (unfold hypreal_le_def)
+apply auto
+done
+declare hypreal_of_nat_le_iff [simp]
+
+lemma hypreal_of_nat_ge_zero: "0 <= hypreal_of_nat n"
+apply (simp (no_asm) add: hypreal_of_nat_zero [symmetric]
+ del: hypreal_of_nat_zero)
+done
+declare hypreal_of_nat_ge_zero [simp]
+
+declare hypreal_of_nat_ge_zero [THEN hrabs_eqI1, simp]
+
+lemma hypreal_of_hypnat_ge_zero: "0 <= hypreal_of_hypnat n"
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (simp (no_asm_simp) add: hypreal_of_hypnat hypreal_zero_num hypreal_le)
+done
+declare hypreal_of_hypnat_ge_zero [simp]
+
+declare hypreal_of_hypnat_ge_zero [THEN hrabs_eqI1, simp]
+
+lemma hcmod_hcomplex_of_hypreal_of_nat: "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n"
+apply auto
+done
+declare hcmod_hcomplex_of_hypreal_of_nat [simp]
+
+lemma hcmod_hcomplex_of_hypreal_of_hypnat: "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
+apply auto
+done
+declare hcmod_hcomplex_of_hypreal_of_hypnat [simp]
+
+lemma hcmod_minus: "hcmod (-x) = hcmod(x)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_minus)
+done
+declare hcmod_minus [simp]
+
+lemma hcmod_mult_hcnj: "hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj two_eq_Suc_Suc)
+done
+
+lemma hcmod_ge_zero: "(0::hypreal) <= hcmod x"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hypreal_zero_num hypreal_le)
+done
+declare hcmod_ge_zero [simp]
+
+lemma hrabs_hcmod_cancel: "abs(hcmod x) = hcmod x"
+apply (auto intro: hrabs_eqI1)
+done
+declare hrabs_hcmod_cancel [simp]
+
+lemma hcmod_mult: "hcmod(x*y) = hcmod(x) * hcmod(y)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_mult hypreal_mult complex_mod_mult)
+done
+
+lemma hcmod_add_squared_eq:
+ "hcmod(x + y) ^ 2 = hcmod(x) ^ 2 + hcmod(y) ^ 2 + 2 * hRe(x * hcnj y)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_add hypreal_mult hRe hcnj hcomplex_mult
+ two_eq_Suc_Suc realpow_two [symmetric]
+ simp del: realpow_Suc)
+apply (auto simp add: two_eq_Suc_Suc [symmetric] complex_mod_add_squared_eq
+ hypreal_add [symmetric] hypreal_mult [symmetric]
+ hypreal_of_real_def [symmetric])
+done
+
+lemma hcomplex_hRe_mult_hcnj_le_hcmod: "hRe(x * hcnj y) <= hcmod(x * hcnj y)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcnj hcomplex_mult hRe hypreal_le)
+done
+declare hcomplex_hRe_mult_hcnj_le_hcmod [simp]
+
+lemma hcomplex_hRe_mult_hcnj_le_hcmod2: "hRe(x * hcnj y) <= hcmod(x * y)"
+apply (cut_tac x = "x" and y = "y" in hcomplex_hRe_mult_hcnj_le_hcmod)
+apply (simp add: hcmod_mult)
+done
+declare hcomplex_hRe_mult_hcnj_le_hcmod2 [simp]
+
+lemma hcmod_triangle_squared: "hcmod (x + y) ^ 2 <= (hcmod(x) + hcmod(y)) ^ 2"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcnj hcomplex_add hypreal_mult hypreal_add
+ hypreal_le realpow_two [symmetric] two_eq_Suc_Suc
+ simp del: realpow_Suc)
+apply (simp (no_asm) add: two_eq_Suc_Suc [symmetric])
+done
+declare hcmod_triangle_squared [simp]
+
+lemma hcmod_triangle_ineq: "hcmod (x + y) <= hcmod(x) + hcmod(y)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_add hypreal_add hypreal_le)
+done
+declare hcmod_triangle_ineq [simp]
+
+lemma hcmod_triangle_ineq2: "hcmod(b + a) - hcmod b <= hcmod a"
+apply (cut_tac x1 = "b" and y1 = "a" and c = "-hcmod b" in hcmod_triangle_ineq [THEN add_right_mono])
+apply (simp add: hypreal_add_ac)
+done
+declare hcmod_triangle_ineq2 [simp]
+
+lemma hcmod_diff_commute: "hcmod (x - y) = hcmod (y - x)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_diff complex_mod_diff_commute)
+done
+
+lemma hcmod_add_less: "[| hcmod x < r; hcmod y < s |] ==> hcmod (x + y) < r + s"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (rule_tac z = "r" in eq_Abs_hypreal)
+apply (rule_tac z = "s" in eq_Abs_hypreal)
+apply (auto simp add: hcmod hcomplex_add hypreal_add hypreal_less)
+apply ultra
+apply (auto intro: complex_mod_add_less)
+done
+
+lemma hcmod_mult_less: "[| hcmod x < r; hcmod y < s |] ==> hcmod (x * y) < r * s"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "y" in eq_Abs_hcomplex)
+apply (rule_tac z = "r" in eq_Abs_hypreal)
+apply (rule_tac z = "s" in eq_Abs_hypreal)
+apply (auto simp add: hcmod hypreal_mult hypreal_less hcomplex_mult)
+apply ultra
+apply (auto intro: complex_mod_mult_less)
+done
+
+lemma hcmod_diff_ineq: "hcmod(a) - hcmod(b) <= hcmod(a + b)"
+apply (rule_tac z = "a" in eq_Abs_hcomplex)
+apply (rule_tac z = "b" in eq_Abs_hcomplex)
+apply (auto simp add: hcmod hcomplex_add hypreal_diff hypreal_le)
+done
+declare hcmod_diff_ineq [simp]
+
+
+
+subsection{*A Few Nonlinear Theorems*}
+
+lemma hcpow:
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) hcpow
+ Abs_hypnat(hypnatrel``{%n. Y n}) =
+ Abs_hcomplex(hcomplexrel``{%n. X n ^ Y n})"
+apply (unfold hcpow_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hcomplex_of_hypreal_hyperpow: "hcomplex_of_hypreal (x pow n) = (hcomplex_of_hypreal x) hcpow n"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcomplex_of_hypreal hyperpow hcpow complex_of_real_pow)
+done
+
+lemma hcmod_hcomplexpow: "hcmod(x ^ n) = hcmod(x) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: hcmod_mult)
+done
+
+lemma hcmod_hcpow: "hcmod(x hcpow n) = hcmod(x) pow n"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcpow hyperpow hcmod complex_mod_complexpow)
+done
+
+lemma hcomplexpow_minus: "(-x::hcomplex) ^ n = (if even n then (x ^ n) else -(x ^ n))"
+apply (induct_tac "n")
+apply auto
+done
+
+lemma hcpow_minus: "(-x::hcomplex) hcpow n =
+ (if ( *pNat* even) n then (x hcpow n) else -(x hcpow n))"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcpow hyperpow starPNat hcomplex_minus)
+apply ultra
+apply (auto simp add: complexpow_minus)
+apply ultra
+done
+
+lemma hccomplex_inverse_minus: "inverse(-x) = - inverse (x::hcomplex)"
+apply (rule_tac z = "x" in eq_Abs_hcomplex)
+apply (auto simp add: hcomplex_inverse hcomplex_minus complex_inverse_minus)
+done
+
+lemma hcomplex_div_one: "x / (1::hcomplex) = x"
+apply (unfold hcomplex_divide_def)
+apply (simp (no_asm))
+done
+declare hcomplex_div_one [simp]
+
+lemma hcmod_hcomplex_inverse: "hcmod(inverse x) = inverse(hcmod x)"
+apply (case_tac "x = 0", simp add: HCOMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "hcmod x" in hypreal_mult_left_cancel [THEN iffD1])
+apply (auto simp add: hcmod_mult [symmetric])
+done
+
+lemma hcmod_divide:
+ "hcmod(x/y) = hcmod(x)/(hcmod y)"
+apply (unfold hcomplex_divide_def hypreal_divide_def)
+apply (auto simp add: hcmod_mult hcmod_hcomplex_inverse)
+done
+
+lemma hcomplex_inverse_divide:
+ "inverse(x/y) = y/(x::hcomplex)"
+apply (unfold hcomplex_divide_def)
+apply (auto simp add: hcomplex_inverse_distrib hcomplex_mult_commute)
+done
+declare hcomplex_inverse_divide [simp]
+
+lemma hcomplexpow_mult: "((r::hcomplex) * s) ^ n = (r ^ n) * (s ^ n)"
+apply (induct_tac "n")
+apply (auto simp add: hcomplex_mult_ac)
+done
+
+lemma hcpow_mult: "((r::hcomplex) * s) hcpow n = (r hcpow n) * (s hcpow n)"
+apply (rule_tac z = "r" in eq_Abs_hcomplex)
+apply (rule_tac z = "s" in eq_Abs_hcomplex)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcpow hypreal_mult hcomplex_mult complexpow_mult)
+done
+
+lemma hcomplexpow_zero: "(0::hcomplex) ^ (Suc n) = 0"
+apply auto
+done
+declare hcomplexpow_zero [simp]
+
+lemma hcpow_zero:
+ "0 hcpow (n + 1) = 0"
+apply (unfold hcomplex_zero_def hypnat_one_def)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcpow hypnat_add)
+done
+declare hcpow_zero [simp]
+
+lemma hcpow_zero2:
+ "0 hcpow (hSuc n) = 0"
+apply (unfold hSuc_def)
+apply (simp (no_asm))
+done
+declare hcpow_zero2 [simp]
+
+lemma hcomplexpow_not_zero [rule_format (no_asm)]: "r ~= (0::hcomplex) --> r ^ n ~= 0"
+apply (induct_tac "n")
+apply (auto simp add: hcomplex_mult_not_zero)
+done
+declare hcomplexpow_not_zero [simp]
+declare hcomplexpow_not_zero [intro]
+
+lemma hcpow_not_zero: "r ~= 0 ==> r hcpow n ~= (0::hcomplex)"
+apply (rule_tac z = "r" in eq_Abs_hcomplex)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcpow hcomplex_zero_def)
+apply ultra
+apply (auto dest: complexpow_zero_zero)
+done
+declare hcpow_not_zero [simp]
+declare hcpow_not_zero [intro]
+
+lemma hcomplexpow_zero_zero: "r ^ n = (0::hcomplex) ==> r = 0"
+apply (blast intro: ccontr dest: hcomplexpow_not_zero)
+done
+
+lemma hcpow_zero_zero: "r hcpow n = (0::hcomplex) ==> r = 0"
+apply (blast intro: ccontr dest: hcpow_not_zero)
+done
+
+lemma hcomplex_i_mult_eq: "iii * iii = - 1"
+apply (unfold iii_def)
+apply (auto simp add: hcomplex_mult hcomplex_one_def hcomplex_minus)
+done
+declare hcomplex_i_mult_eq [simp]
+
+lemma hcomplexpow_i_squared: "iii ^ 2 = - 1"
+apply (simp (no_asm) add: two_eq_Suc_Suc)
+done
+declare hcomplexpow_i_squared [simp]
+
+lemma hcomplex_i_not_zero: "iii ~= 0"
+apply (unfold iii_def hcomplex_zero_def)
+apply auto
+done
+declare hcomplex_i_not_zero [simp]
+
+lemma hcomplex_mult_eq_zero_cancel1: "x * y ~= (0::hcomplex) ==> x ~= 0"
+apply auto
+done
+
+lemma hcomplex_mult_eq_zero_cancel2: "x * y ~= (0::hcomplex) ==> y ~= 0"
+apply auto
+done
+
+lemma hcomplex_mult_not_eq_zero_iff: "(x * y ~= (0::hcomplex)) = (x ~= 0 & y ~= 0)"
+apply auto
+done
+declare hcomplex_mult_not_eq_zero_iff [iff]
+
+lemma hcomplex_divide:
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) / Abs_hcomplex(hcomplexrel``{%n. Y n}) =
+ Abs_hcomplex(hcomplexrel``{%n. X n / Y n})"
+apply (unfold hcomplex_divide_def complex_divide_def)
+apply (auto simp add: hcomplex_inverse hcomplex_mult)
+done
+
+
+subsection{*The Function @{term hsgn}*}
+
+lemma hsgn:
+ "hsgn (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. sgn (X n)})"
+apply (unfold hsgn_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma hsgn_zero: "hsgn 0 = 0"
+apply (unfold hcomplex_zero_def)
+apply (simp (no_asm) add: hsgn)
+done
+declare hsgn_zero [simp]
+
+
+lemma hsgn_one: "hsgn 1 = 1"
+apply (unfold hcomplex_one_def)
+apply (simp (no_asm) add: hsgn)
+done
+declare hsgn_one [simp]
+
+lemma hsgn_minus: "hsgn (-z) = - hsgn(z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hsgn hcomplex_minus sgn_minus)
+done
+
+lemma hsgn_eq: "hsgn z = z / hcomplex_of_hypreal (hcmod z)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hsgn hcomplex_divide hcomplex_of_hypreal hcmod sgn_eq)
+done
+
+lemma lemma_hypreal_P_EX2: "(EX (x::hypreal) y. P x y) =
+ (EX f g. P (Abs_hypreal(hyprel `` {f})) (Abs_hypreal(hyprel `` {g})))"
+apply auto
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply auto
+done
+
+lemma complex_split2: "ALL (n::nat). EX x y. (z n) = complex_of_real(x) + ii * complex_of_real(y)"
+apply (blast intro: complex_split)
+done
+
+(* Interesting proof! *)
+lemma hcomplex_split: "EX x y. z = hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def hcomplex_add hcomplex_mult)
+apply (cut_tac z = "x" in complex_split2)
+apply (drule choice , safe)+
+apply (rule_tac x = "f" in exI)
+apply (rule_tac x = "fa" in exI)
+apply auto
+done
+
+lemma hRe_hcomplex_i: "hRe(hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hRe iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
+done
+declare hRe_hcomplex_i [simp]
+
+lemma hIm_hcomplex_i: "hIm(hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = y"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hIm iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
+done
+declare hIm_hcomplex_i [simp]
+
+lemma hcmod_i: "hcmod (hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) =
+ ( *f* sqrt) (x ^ 2 + y ^ 2)"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+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)
+done
+
+lemma hcomplex_eq_hRe_eq:
+ "hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
+ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb
+ ==> xa = xb"
+apply (unfold iii_def)
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (rule_tac z = "ya" in eq_Abs_hypreal)
+apply (rule_tac z = "xb" in eq_Abs_hypreal)
+apply (rule_tac z = "yb" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_mult hcomplex_add hcomplex_of_hypreal)
+apply (ultra)
+done
+
+lemma hcomplex_eq_hIm_eq:
+ "hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
+ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb
+ ==> ya = yb"
+apply (unfold iii_def)
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (rule_tac z = "ya" in eq_Abs_hypreal)
+apply (rule_tac z = "xb" in eq_Abs_hypreal)
+apply (rule_tac z = "yb" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_mult hcomplex_add hcomplex_of_hypreal)
+apply (ultra)
+done
+
+lemma hcomplex_eq_cancel_iff: "(hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
+ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb) =
+ ((xa = xb) & (ya = yb))"
+apply (auto intro: hcomplex_eq_hIm_eq hcomplex_eq_hRe_eq)
+done
+declare hcomplex_eq_cancel_iff [simp]
+
+lemma hcomplex_eq_cancel_iffA: "(hcomplex_of_hypreal xa + hcomplex_of_hypreal ya * iii =
+ hcomplex_of_hypreal xb + hcomplex_of_hypreal yb * iii ) = ((xa = xb) & (ya = yb))"
+apply (auto simp add: hcomplex_mult_commute)
+done
+declare hcomplex_eq_cancel_iffA [iff]
+
+lemma hcomplex_eq_cancel_iffB: "(hcomplex_of_hypreal xa + hcomplex_of_hypreal ya * iii =
+ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb) = ((xa = xb) & (ya = yb))"
+apply (auto simp add: hcomplex_mult_commute)
+done
+declare hcomplex_eq_cancel_iffB [iff]
+
+lemma hcomplex_eq_cancel_iffC: "(hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya =
+ hcomplex_of_hypreal xb + hcomplex_of_hypreal yb * iii) = ((xa = xb) & (ya = yb))"
+apply (auto simp add: hcomplex_mult_commute)
+done
+declare hcomplex_eq_cancel_iffC [iff]
+
+lemma hcomplex_eq_cancel_iff2: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y =
+ hcomplex_of_hypreal xa) = (x = xa & y = 0)"
+apply (cut_tac xa = "x" and ya = "y" and xb = "xa" and yb = "0" in hcomplex_eq_cancel_iff)
+apply (simp del: hcomplex_eq_cancel_iff)
+done
+declare hcomplex_eq_cancel_iff2 [simp]
+
+lemma hcomplex_eq_cancel_iff2a: "(hcomplex_of_hypreal x + hcomplex_of_hypreal y * iii =
+ hcomplex_of_hypreal xa) = (x = xa & y = 0)"
+apply (auto simp add: hcomplex_mult_commute)
+done
+declare hcomplex_eq_cancel_iff2a [simp]
+
+lemma hcomplex_eq_cancel_iff3: "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y =
+ iii * hcomplex_of_hypreal ya) = (x = 0 & y = ya)"
+apply (cut_tac xa = "x" and ya = "y" and xb = "0" and yb = "ya" in hcomplex_eq_cancel_iff)
+apply (simp del: hcomplex_eq_cancel_iff)
+done
+declare hcomplex_eq_cancel_iff3 [simp]
+
+lemma hcomplex_eq_cancel_iff3a: "(hcomplex_of_hypreal x + hcomplex_of_hypreal y * iii =
+ iii * hcomplex_of_hypreal ya) = (x = 0 & y = ya)"
+apply (auto simp add: hcomplex_mult_commute)
+done
+declare hcomplex_eq_cancel_iff3a [simp]
+
+lemma hcomplex_split_hRe_zero:
+ "hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = 0
+ ==> x = 0"
+apply (unfold iii_def)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_add hcomplex_mult hcomplex_zero_def hypreal_zero_num)
+apply ultra
+apply (auto simp add: complex_split_Re_zero)
+done
+
+lemma hcomplex_split_hIm_zero:
+ "hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = 0
+ ==> y = 0"
+apply (unfold iii_def)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_add hcomplex_mult hcomplex_zero_def hypreal_zero_num)
+apply ultra
+apply (auto simp add: complex_split_Im_zero)
+done
+
+lemma hRe_hsgn: "hRe(hsgn z) = hRe(z)/hcmod z"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hsgn hcmod hRe hypreal_divide)
+done
+declare hRe_hsgn [simp]
+
+lemma hIm_hsgn: "hIm(hsgn z) = hIm(z)/hcmod z"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: hsgn hcmod hIm hypreal_divide)
+done
+declare hIm_hsgn [simp]
+
+lemma real_two_squares_add_zero_iff: "(x*x + y*y = 0) = ((x::real) = 0 & y = 0)"
+apply (auto intro: real_sum_squares_cancel)
+done
+declare real_two_squares_add_zero_iff [simp]
+
+lemma hcomplex_inverse_complex_split: "inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) =
+ hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) -
+ iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+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)
+done
+
+lemma hRe_mult_i_eq:
+ "hRe (iii * hcomplex_of_hypreal y) = 0"
+apply (unfold iii_def)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hRe hypreal_zero_num)
+done
+declare hRe_mult_i_eq [simp]
+
+lemma hIm_mult_i_eq:
+ "hIm (iii * hcomplex_of_hypreal y) = y"
+apply (unfold iii_def)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hIm hypreal_zero_num)
+done
+declare hIm_mult_i_eq [simp]
+
+lemma hcmod_mult_i: "hcmod (iii * hcomplex_of_hypreal y) = abs y"
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hcmod hypreal_hrabs iii_def hcomplex_mult)
+done
+declare hcmod_mult_i [simp]
+
+lemma hcmod_mult_i2: "hcmod (hcomplex_of_hypreal y * iii) = abs y"
+apply (auto simp add: hcomplex_mult_commute)
+done
+declare hcmod_mult_i2 [simp]
+
+(*---------------------------------------------------------------------------*)
+(* harg *)
+(*---------------------------------------------------------------------------*)
+
+lemma harg:
+ "harg (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ Abs_hypreal(hyprel `` {%n. arg (X n)})"
+
+apply (unfold harg_def)
+apply (rule_tac f = "Abs_hypreal" in arg_cong)
+apply (auto , ultra)
+done
+
+lemma cos_harg_i_mult_zero: "0 < y ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_mult hypreal_zero_num hypreal_less starfun harg)
+apply (ultra)
+done
+declare cos_harg_i_mult_zero [simp]
+
+lemma cos_harg_i_mult_zero2: "y < 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_mult hypreal_zero_num hypreal_less starfun harg)
+apply (ultra)
+done
+declare cos_harg_i_mult_zero2 [simp]
+
+lemma hcomplex_of_hypreal_not_zero_iff: "(hcomplex_of_hypreal y ~= 0) = (y ~= 0)"
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
+done
+declare hcomplex_of_hypreal_not_zero_iff [simp]
+
+lemma hcomplex_of_hypreal_zero_iff: "(hcomplex_of_hypreal y = 0) = (y = 0)"
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
+done
+declare hcomplex_of_hypreal_zero_iff [simp]
+
+lemma cos_harg_i_mult_zero3: "y ~= 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
+apply (cut_tac x = "y" and y = "0" in hypreal_linear)
+apply auto
+done
+declare cos_harg_i_mult_zero3 [simp]
+
+(*---------------------------------------------------------------------------*)
+(* Polar form for nonstandard complex numbers *)
+(*---------------------------------------------------------------------------*)
+
+lemma complex_split_polar2: "ALL n. EX r a. (z n) = complex_of_real r *
+ (complex_of_real(cos a) + ii * complex_of_real(sin a))"
+apply (blast intro: complex_split_polar)
+done
+
+lemma hcomplex_split_polar:
+ "EX r a. z = hcomplex_of_hypreal r *
+ (hcomplex_of_hypreal(( *f* cos) a) + iii * hcomplex_of_hypreal(( *f* sin) a))"
+apply (rule_tac z = "z" in eq_Abs_hcomplex)
+apply (auto simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def starfun hcomplex_add hcomplex_mult)
+apply (cut_tac z = "x" in complex_split_polar2)
+apply (drule choice , safe)+
+apply (rule_tac x = "f" in exI)
+apply (rule_tac x = "fa" in exI)
+apply auto
+done
+
+lemma hcis:
+ "hcis (Abs_hypreal(hyprel `` {%n. X n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. cis (X n)})"
+apply (unfold hcis_def)
+apply (rule_tac f = "Abs_hcomplex" in arg_cong)
+apply auto
+apply (ultra)
+done
+
+lemma hcis_eq:
+ "hcis a =
+ (hcomplex_of_hypreal(( *f* cos) a) +
+ iii * hcomplex_of_hypreal(( *f* sin) a))"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: starfun hcis hcomplex_of_hypreal iii_def hcomplex_mult hcomplex_add cis_def)
+done
+
+lemma hrcis:
+ "hrcis (Abs_hypreal(hyprel `` {%n. X n})) (Abs_hypreal(hyprel `` {%n. Y n})) =
+ Abs_hcomplex(hcomplexrel `` {%n. rcis (X n) (Y n)})"
+apply (unfold hrcis_def)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hcis rcis_def)
+done
+
+lemma hrcis_Ex: "EX r a. z = hrcis r a"
+apply (simp (no_asm) add: hrcis_def hcis_eq)
+apply (rule hcomplex_split_polar)
+done
+
+lemma hRe_hcomplex_polar: "hRe(hcomplex_of_hypreal r *
+ (hcomplex_of_hypreal(( *f* cos) a) +
+ iii * hcomplex_of_hypreal(( *f* sin) a))) = r * ( *f* cos) a"
+apply (auto simp add: hcomplex_add_mult_distrib2 hcomplex_of_hypreal_mult hcomplex_mult_ac)
+done
+declare hRe_hcomplex_polar [simp]
+
+lemma hRe_hrcis: "hRe(hrcis r a) = r * ( *f* cos) a"
+apply (unfold hrcis_def)
+apply (auto simp add: hcis_eq)
+done
+declare hRe_hrcis [simp]
+
+lemma hIm_hcomplex_polar: "hIm(hcomplex_of_hypreal r *
+ (hcomplex_of_hypreal(( *f* cos) a) +
+ iii * hcomplex_of_hypreal(( *f* sin) a))) = r * ( *f* sin) a"
+apply (auto simp add: hcomplex_add_mult_distrib2 hcomplex_of_hypreal_mult hcomplex_mult_ac)
+done
+declare hIm_hcomplex_polar [simp]
+
+lemma hIm_hrcis: "hIm(hrcis r a) = r * ( *f* sin) a"
+apply (unfold hrcis_def)
+apply (auto simp add: hcis_eq)
+done
+declare hIm_hrcis [simp]
+
+lemma hcmod_complex_polar: "hcmod (hcomplex_of_hypreal r *
+ (hcomplex_of_hypreal(( *f* cos) a) +
+ iii * hcomplex_of_hypreal(( *f* sin) a))) = abs r"
+apply (rule_tac z = "r" in eq_Abs_hypreal)
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: iii_def starfun hcomplex_of_hypreal hcomplex_mult hcmod hcomplex_add hypreal_hrabs)
+done
+declare hcmod_complex_polar [simp]
+
+lemma hcmod_hrcis: "hcmod(hrcis r a) = abs r"
+apply (unfold hrcis_def)
+apply (auto simp add: hcis_eq)
+done
+declare hcmod_hrcis [simp]
+
+(*---------------------------------------------------------------------------*)
+(* (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b) *)
+(*---------------------------------------------------------------------------*)
+
+lemma hcis_hrcis_eq: "hcis a = hrcis 1 a"
+
+apply (unfold hrcis_def)
+apply (simp (no_asm))
+done
+declare hcis_hrcis_eq [symmetric, simp]
+
+lemma hrcis_mult:
+ "hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)"
+apply (unfold hrcis_def)
+apply (rule_tac z = "r1" in eq_Abs_hypreal)
+apply (rule_tac z = "r2" in eq_Abs_hypreal)
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (rule_tac z = "b" in eq_Abs_hypreal)
+apply (auto simp add: hrcis hcis hypreal_add hypreal_mult hcomplex_of_hypreal
+ hcomplex_mult cis_mult [symmetric]
+ complex_of_real_mult [symmetric])
+done
+
+lemma hcis_mult: "hcis a * hcis b = hcis (a + b)"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (rule_tac z = "b" in eq_Abs_hypreal)
+apply (auto simp add: hcis hcomplex_mult hypreal_add cis_mult)
+done
+
+lemma hcis_zero:
+ "hcis 0 = 1"
+apply (unfold hcomplex_one_def)
+apply (auto simp add: hcis hypreal_zero_num)
+done
+declare hcis_zero [simp]
+
+lemma hrcis_zero_mod:
+ "hrcis 0 a = 0"
+apply (unfold hcomplex_zero_def)
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: hrcis hypreal_zero_num)
+done
+declare hrcis_zero_mod [simp]
+
+lemma hrcis_zero_arg: "hrcis r 0 = hcomplex_of_hypreal r"
+apply (rule_tac z = "r" in eq_Abs_hypreal)
+apply (auto simp add: hrcis hypreal_zero_num hcomplex_of_hypreal)
+done
+declare hrcis_zero_arg [simp]
+
+lemma hcomplex_i_mult_minus: "iii * (iii * x) = - x"
+apply (simp (no_asm) add: hcomplex_mult_assoc [symmetric])
+done
+declare hcomplex_i_mult_minus [simp]
+
+lemma hcomplex_i_mult_minus2: "iii * iii * x = - x"
+apply (simp (no_asm))
+done
+declare hcomplex_i_mult_minus2 [simp]
+
+(* Move to one of the hyperreal theories *)
+lemma hypreal_of_nat: "hypreal_of_nat m = Abs_hypreal(hyprel `` {%n. real m})"
+apply (induct_tac "m")
+apply (auto simp add: hypreal_zero_def hypreal_of_nat_Suc hypreal_zero_num hypreal_one_num hypreal_add real_of_nat_Suc)
+done
+
+lemma hcis_hypreal_of_nat_Suc_mult:
+ "hcis (hypreal_of_nat (Suc n) * a) = hcis a * hcis (hypreal_of_nat n * a)"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: hypreal_of_nat hcis hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
+done
+
+lemma NSDeMoivre: "(hcis a) ^ n = hcis (hypreal_of_nat n * a)"
+apply (induct_tac "n")
+apply (auto simp add: hcis_hypreal_of_nat_Suc_mult)
+done
+
+lemma hcis_hypreal_of_hypnat_Suc_mult: "hcis (hypreal_of_hypnat (n + 1) * a) =
+ hcis a * hcis (hypreal_of_hypnat n * a)"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcis hypreal_of_hypnat hypnat_add hypnat_one_def hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
+done
+
+lemma NSDeMoivre_ext: "(hcis a) hcpow n = hcis (hypreal_of_hypnat n * a)"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (rule_tac z = "n" in eq_Abs_hypnat)
+apply (auto simp add: hcis hypreal_of_hypnat hypreal_mult hcpow DeMoivre)
+done
+
+lemma DeMoivre2:
+ "(hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
+apply (unfold hrcis_def)
+apply (auto simp add: hcomplexpow_mult NSDeMoivre hcomplex_of_hypreal_pow)
+done
+
+lemma DeMoivre2_ext:
+ "(hrcis r a) hcpow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
+apply (unfold hrcis_def)
+apply (auto simp add: hcpow_mult NSDeMoivre_ext hcomplex_of_hypreal_hyperpow)
+done
+
+lemma hcis_inverse: "inverse(hcis a) = hcis (-a)"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_inverse hcis hypreal_minus)
+done
+declare hcis_inverse [simp]
+
+lemma hrcis_inverse: "inverse(hrcis r a) = hrcis (inverse r) (-a)"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (rule_tac z = "r" in eq_Abs_hypreal)
+apply (auto simp add: hcomplex_inverse hrcis hypreal_minus hypreal_inverse rcis_inverse)
+apply (ultra)
+apply (unfold real_divide_def)
+apply (auto simp add: INVERSE_ZERO)
+done
+
+lemma hRe_hcis: "hRe(hcis a) = ( *f* cos) a"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: hcis starfun hRe)
+done
+declare hRe_hcis [simp]
+
+lemma hIm_hcis: "hIm(hcis a) = ( *f* sin) a"
+apply (rule_tac z = "a" in eq_Abs_hypreal)
+apply (auto simp add: hcis starfun hIm)
+done
+declare hIm_hcis [simp]
+
+lemma cos_n_hRe_hcis_pow_n: "( *f* cos) (hypreal_of_nat n * a) = hRe(hcis a ^ n)"
+apply (auto simp add: NSDeMoivre)
+done
+
+lemma sin_n_hIm_hcis_pow_n: "( *f* sin) (hypreal_of_nat n * a) = hIm(hcis a ^ n)"
+apply (auto simp add: NSDeMoivre)
+done
+
+lemma cos_n_hRe_hcis_hcpow_n: "( *f* cos) (hypreal_of_hypnat n * a) = hRe(hcis a hcpow n)"
+apply (auto simp add: NSDeMoivre_ext)
+done
+
+lemma sin_n_hIm_hcis_hcpow_n: "( *f* sin) (hypreal_of_hypnat n * a) = hIm(hcis a hcpow n)"
+apply (auto simp add: NSDeMoivre_ext)
+done
+
+lemma hexpi_add: "hexpi(a + b) = hexpi(a) * hexpi(b)"
+apply (unfold hexpi_def)
+apply (rule_tac z = "a" in eq_Abs_hcomplex)
+apply (rule_tac z = "b" in eq_Abs_hcomplex)
+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)
+done
+
+
+subsection{*@{term hcomplex_of_complex} Preserves Field Properties*}
+
+lemma hcomplex_of_complex_add:
+ "hcomplex_of_complex (z1 + z2) = hcomplex_of_complex z1 + hcomplex_of_complex z2"
+apply (unfold hcomplex_of_complex_def)
+apply (simp (no_asm) add: hcomplex_add)
+done
+declare hcomplex_of_complex_add [simp]
+
+lemma hcomplex_of_complex_mult:
+ "hcomplex_of_complex (z1 * z2) = hcomplex_of_complex z1 * hcomplex_of_complex z2"
+apply (unfold hcomplex_of_complex_def)
+apply (simp (no_asm) add: hcomplex_mult)
+done
+declare hcomplex_of_complex_mult [simp]
+
+lemma hcomplex_of_complex_eq_iff:
+ "(hcomplex_of_complex z1 = hcomplex_of_complex z2) = (z1 = z2)"
+apply (unfold hcomplex_of_complex_def)
+apply auto
+done
+declare hcomplex_of_complex_eq_iff [simp]
+
+lemma hcomplex_of_complex_minus: "hcomplex_of_complex (-r) = - hcomplex_of_complex r"
+apply (unfold hcomplex_of_complex_def)
+apply (auto simp add: hcomplex_minus)
+done
+declare hcomplex_of_complex_minus [simp]
+
+lemma hcomplex_of_complex_one:
+ "hcomplex_of_complex 1 = 1"
+apply (unfold hcomplex_of_complex_def hcomplex_one_def)
+apply auto
+done
+
+lemma hcomplex_of_complex_zero:
+ "hcomplex_of_complex 0 = 0"
+apply (unfold hcomplex_of_complex_def hcomplex_zero_def)
+apply (simp (no_asm))
+done
+
+lemma hcomplex_of_complex_zero_iff: "(hcomplex_of_complex r = 0) = (r = 0)"
+apply (auto intro: FreeUltrafilterNat_P simp add: hcomplex_of_complex_def hcomplex_zero_def)
+done
+
+lemma hcomplex_of_complex_inverse: "hcomplex_of_complex (inverse r) = inverse (hcomplex_of_complex r)"
+apply (case_tac "r=0")
+apply (simp (no_asm_simp) add: COMPLEX_INVERSE_ZERO HCOMPLEX_INVERSE_ZERO hcomplex_of_complex_zero COMPLEX_DIVIDE_ZERO)
+apply (rule_tac c1 = "hcomplex_of_complex r" in hcomplex_mult_left_cancel [THEN iffD1])
+apply (force simp add: hcomplex_of_complex_zero_iff)
+apply (subst hcomplex_of_complex_mult [symmetric])
+apply (auto simp add: hcomplex_of_complex_one hcomplex_of_complex_zero_iff);
+done
+declare hcomplex_of_complex_inverse [simp]
+
+lemma hcomplex_of_complex_divide: "hcomplex_of_complex (z1 / z2) = hcomplex_of_complex z1 / hcomplex_of_complex z2"
+apply (simp (no_asm) add: hcomplex_divide_def complex_divide_def)
+done
+declare hcomplex_of_complex_divide [simp]
+
+lemma hRe_hcomplex_of_complex:
+ "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
+apply (unfold hcomplex_of_complex_def hypreal_of_real_def)
+apply (auto simp add: hRe)
+done
+
+lemma hIm_hcomplex_of_complex:
+ "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
+apply (unfold hcomplex_of_complex_def hypreal_of_real_def)
+apply (auto simp add: hIm)
+done
+
+lemma hcmod_hcomplex_of_complex:
+ "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
+apply (unfold hypreal_of_real_def hcomplex_of_complex_def)
+apply (auto simp add: hcmod)
+done
+
+ML
+{*
+val hcomplex_zero_def = thm"hcomplex_zero_def";
+val hcomplex_one_def = thm"hcomplex_one_def";
+val hcomplex_minus_def = thm"hcomplex_minus_def";
+val hcomplex_diff_def = thm"hcomplex_diff_def";
+val hcomplex_divide_def = thm"hcomplex_divide_def";
+val hcomplex_mult_def = thm"hcomplex_mult_def";
+val hcomplex_add_def = thm"hcomplex_add_def";
+val hcomplex_of_complex_def = thm"hcomplex_of_complex_def";
+val iii_def = thm"iii_def";
+
+val hcomplexrel_iff = thm"hcomplexrel_iff";
+val hcomplexrel_refl = thm"hcomplexrel_refl";
+val hcomplexrel_sym = thm"hcomplexrel_sym";
+val hcomplexrel_trans = thm"hcomplexrel_trans";
+val equiv_hcomplexrel = thm"equiv_hcomplexrel";
+val equiv_hcomplexrel_iff = thm"equiv_hcomplexrel_iff";
+val hcomplexrel_in_hcomplex = thm"hcomplexrel_in_hcomplex";
+val inj_on_Abs_hcomplex = thm"inj_on_Abs_hcomplex";
+val inj_Rep_hcomplex = thm"inj_Rep_hcomplex";
+val lemma_hcomplexrel_refl = thm"lemma_hcomplexrel_refl";
+val hcomplex_empty_not_mem = thm"hcomplex_empty_not_mem";
+val Rep_hcomplex_nonempty = thm"Rep_hcomplex_nonempty";
+val eq_Abs_hcomplex = thm"eq_Abs_hcomplex";
+val hRe = thm"hRe";
+val hIm = thm"hIm";
+val hcomplex_hRe_hIm_cancel_iff = thm"hcomplex_hRe_hIm_cancel_iff";
+val hcomplex_hRe_zero = thm"hcomplex_hRe_zero";
+val hcomplex_hIm_zero = thm"hcomplex_hIm_zero";
+val hcomplex_hRe_one = thm"hcomplex_hRe_one";
+val hcomplex_hIm_one = thm"hcomplex_hIm_one";
+val inj_hcomplex_of_complex = thm"inj_hcomplex_of_complex";
+val hcomplex_of_complex_i = thm"hcomplex_of_complex_i";
+val hcomplex_add_congruent2 = thm"hcomplex_add_congruent2";
+val hcomplex_add = thm"hcomplex_add";
+val hcomplex_add_commute = thm"hcomplex_add_commute";
+val hcomplex_add_assoc = thm"hcomplex_add_assoc";
+val hcomplex_add_left_commute = thm"hcomplex_add_left_commute";
+val hcomplex_add_zero_left = thm"hcomplex_add_zero_left";
+val hcomplex_add_zero_right = thm"hcomplex_add_zero_right";
+val hRe_add = thm"hRe_add";
+val hIm_add = thm"hIm_add";
+val hcomplex_minus_congruent = thm"hcomplex_minus_congruent";
+val hcomplex_minus = thm"hcomplex_minus";
+val hcomplex_minus_minus = thm"hcomplex_minus_minus";
+val inj_hcomplex_minus = thm"inj_hcomplex_minus";
+val hcomplex_minus_zero = thm"hcomplex_minus_zero";
+val hcomplex_minus_zero_iff = thm"hcomplex_minus_zero_iff";
+val hcomplex_minus_zero_iff2 = thm"hcomplex_minus_zero_iff2";
+val hcomplex_minus_not_zero_iff = thm"hcomplex_minus_not_zero_iff";
+val hcomplex_add_minus_right = thm"hcomplex_add_minus_right";
+val hcomplex_add_minus_left = thm"hcomplex_add_minus_left";
+val hcomplex_add_minus_cancel = thm"hcomplex_add_minus_cancel";
+val hcomplex_minus_add_cancel = thm"hcomplex_minus_add_cancel";
+val hRe_minus = thm"hRe_minus";
+val hIm_minus = thm"hIm_minus";
+val hcomplex_add_minus_eq_minus = thm"hcomplex_add_minus_eq_minus";
+val hcomplex_minus_add_distrib = thm"hcomplex_minus_add_distrib";
+val hcomplex_add_left_cancel = thm"hcomplex_add_left_cancel";
+val hcomplex_add_right_cancel = thm"hcomplex_add_right_cancel";
+val hcomplex_eq_minus_iff = thm"hcomplex_eq_minus_iff";
+val hcomplex_eq_minus_iff2 = thm"hcomplex_eq_minus_iff2";
+val hcomplex_diff = thm"hcomplex_diff";
+val hcomplex_diff_zero = thm"hcomplex_diff_zero";
+val hcomplex_diff_0 = thm"hcomplex_diff_0";
+val hcomplex_diff_0_right = thm"hcomplex_diff_0_right";
+val hcomplex_diff_self = thm"hcomplex_diff_self";
+val hcomplex_diff_eq_eq = thm"hcomplex_diff_eq_eq";
+val hcomplex_mult = thm"hcomplex_mult";
+val hcomplex_mult_commute = thm"hcomplex_mult_commute";
+val hcomplex_mult_assoc = thm"hcomplex_mult_assoc";
+val hcomplex_mult_left_commute = thm"hcomplex_mult_left_commute";
+val hcomplex_mult_one_left = thm"hcomplex_mult_one_left";
+val hcomplex_mult_one_right = thm"hcomplex_mult_one_right";
+val hcomplex_mult_zero_left = thm"hcomplex_mult_zero_left";
+val hcomplex_mult_zero_right = thm"hcomplex_mult_zero_right";
+val hcomplex_minus_mult_eq1 = thm"hcomplex_minus_mult_eq1";
+val hcomplex_minus_mult_eq2 = thm"hcomplex_minus_mult_eq2";
+val hcomplex_mult_minus_one = thm"hcomplex_mult_minus_one";
+val hcomplex_mult_minus_one_right = thm"hcomplex_mult_minus_one_right";
+val hcomplex_minus_mult_cancel = thm"hcomplex_minus_mult_cancel";
+val hcomplex_minus_mult_commute = thm"hcomplex_minus_mult_commute";
+val hcomplex_add_mult_distrib = thm"hcomplex_add_mult_distrib";
+val hcomplex_add_mult_distrib2 = thm"hcomplex_add_mult_distrib2";
+val hcomplex_zero_not_eq_one = thm"hcomplex_zero_not_eq_one";
+val hcomplex_inverse = thm"hcomplex_inverse";
+val HCOMPLEX_INVERSE_ZERO = thm"HCOMPLEX_INVERSE_ZERO";
+val HCOMPLEX_DIVISION_BY_ZERO = thm"HCOMPLEX_DIVISION_BY_ZERO";
+val hcomplex_mult_inv_left = thm"hcomplex_mult_inv_left";
+val hcomplex_mult_inv_right = thm"hcomplex_mult_inv_right";
+val hcomplex_mult_left_cancel = thm"hcomplex_mult_left_cancel";
+val hcomplex_mult_right_cancel = thm"hcomplex_mult_right_cancel";
+val hcomplex_inverse_not_zero = thm"hcomplex_inverse_not_zero";
+val hcomplex_mult_not_zero = thm"hcomplex_mult_not_zero";
+val hcomplex_mult_not_zeroE = thm"hcomplex_mult_not_zeroE";
+val hcomplex_inverse_inverse = thm"hcomplex_inverse_inverse";
+val hcomplex_inverse_one = thm"hcomplex_inverse_one";
+val hcomplex_minus_inverse = thm"hcomplex_minus_inverse";
+val hcomplex_inverse_distrib = thm"hcomplex_inverse_distrib";
+val hcomplex_times_divide1_eq = thm"hcomplex_times_divide1_eq";
+val hcomplex_times_divide2_eq = thm"hcomplex_times_divide2_eq";
+val hcomplex_divide_divide1_eq = thm"hcomplex_divide_divide1_eq";
+val hcomplex_divide_divide2_eq = thm"hcomplex_divide_divide2_eq";
+val hcomplex_minus_divide_eq = thm"hcomplex_minus_divide_eq";
+val hcomplex_divide_minus_eq = thm"hcomplex_divide_minus_eq";
+val hcomplex_add_divide_distrib = thm"hcomplex_add_divide_distrib";
+val hcomplex_of_hypreal = thm"hcomplex_of_hypreal";
+val inj_hcomplex_of_hypreal = thm"inj_hcomplex_of_hypreal";
+val hcomplex_of_hypreal_cancel_iff = thm"hcomplex_of_hypreal_cancel_iff";
+val hcomplex_of_hypreal_minus = thm"hcomplex_of_hypreal_minus";
+val hcomplex_of_hypreal_inverse = thm"hcomplex_of_hypreal_inverse";
+val hcomplex_of_hypreal_add = thm"hcomplex_of_hypreal_add";
+val hcomplex_of_hypreal_diff = thm"hcomplex_of_hypreal_diff";
+val hcomplex_of_hypreal_mult = thm"hcomplex_of_hypreal_mult";
+val hcomplex_of_hypreal_divide = thm"hcomplex_of_hypreal_divide";
+val hcomplex_of_hypreal_one = thm"hcomplex_of_hypreal_one";
+val hcomplex_of_hypreal_zero = thm"hcomplex_of_hypreal_zero";
+val hcomplex_of_hypreal_pow = thm"hcomplex_of_hypreal_pow";
+val hRe_hcomplex_of_hypreal = thm"hRe_hcomplex_of_hypreal";
+val hIm_hcomplex_of_hypreal = thm"hIm_hcomplex_of_hypreal";
+val hcomplex_of_hypreal_epsilon_not_zero = thm"hcomplex_of_hypreal_epsilon_not_zero";
+val hcmod = thm"hcmod";
+val hcmod_zero = thm"hcmod_zero";
+val hcmod_one = thm"hcmod_one";
+val hcmod_hcomplex_of_hypreal = thm"hcmod_hcomplex_of_hypreal";
+val hcomplex_of_hypreal_abs = thm"hcomplex_of_hypreal_abs";
+val hcnj = thm"hcnj";
+val inj_hcnj = thm"inj_hcnj";
+val hcomplex_hcnj_cancel_iff = thm"hcomplex_hcnj_cancel_iff";
+val hcomplex_hcnj_hcnj = thm"hcomplex_hcnj_hcnj";
+val hcomplex_hcnj_hcomplex_of_hypreal = thm"hcomplex_hcnj_hcomplex_of_hypreal";
+val hcomplex_hmod_hcnj = thm"hcomplex_hmod_hcnj";
+val hcomplex_hcnj_minus = thm"hcomplex_hcnj_minus";
+val hcomplex_hcnj_inverse = thm"hcomplex_hcnj_inverse";
+val hcomplex_hcnj_add = thm"hcomplex_hcnj_add";
+val hcomplex_hcnj_diff = thm"hcomplex_hcnj_diff";
+val hcomplex_hcnj_mult = thm"hcomplex_hcnj_mult";
+val hcomplex_hcnj_divide = thm"hcomplex_hcnj_divide";
+val hcnj_one = thm"hcnj_one";
+val hcomplex_hcnj_pow = thm"hcomplex_hcnj_pow";
+val hcomplex_hcnj_zero = thm"hcomplex_hcnj_zero";
+val hcomplex_hcnj_zero_iff = thm"hcomplex_hcnj_zero_iff";
+val hcomplex_mult_hcnj = thm"hcomplex_mult_hcnj";
+val hcomplex_mult_zero_iff = thm"hcomplex_mult_zero_iff";
+val hcomplex_add_left_cancel_zero = thm"hcomplex_add_left_cancel_zero";
+val hcomplex_diff_mult_distrib = thm"hcomplex_diff_mult_distrib";
+val hcomplex_diff_mult_distrib2 = thm"hcomplex_diff_mult_distrib2";
+val hcomplex_hcmod_eq_zero_cancel = thm"hcomplex_hcmod_eq_zero_cancel";
+val hypreal_of_nat_le_iff = thm"hypreal_of_nat_le_iff";
+val hypreal_of_nat_ge_zero = thm"hypreal_of_nat_ge_zero";
+val hypreal_of_hypnat_ge_zero = thm"hypreal_of_hypnat_ge_zero";
+val hcmod_hcomplex_of_hypreal_of_nat = thm"hcmod_hcomplex_of_hypreal_of_nat";
+val hcmod_hcomplex_of_hypreal_of_hypnat = thm"hcmod_hcomplex_of_hypreal_of_hypnat";
+val hcmod_minus = thm"hcmod_minus";
+val hcmod_mult_hcnj = thm"hcmod_mult_hcnj";
+val hcmod_ge_zero = thm"hcmod_ge_zero";
+val hrabs_hcmod_cancel = thm"hrabs_hcmod_cancel";
+val hcmod_mult = thm"hcmod_mult";
+val hcmod_add_squared_eq = thm"hcmod_add_squared_eq";
+val hcomplex_hRe_mult_hcnj_le_hcmod = thm"hcomplex_hRe_mult_hcnj_le_hcmod";
+val hcomplex_hRe_mult_hcnj_le_hcmod2 = thm"hcomplex_hRe_mult_hcnj_le_hcmod2";
+val hcmod_triangle_squared = thm"hcmod_triangle_squared";
+val hcmod_triangle_ineq = thm"hcmod_triangle_ineq";
+val hcmod_triangle_ineq2 = thm"hcmod_triangle_ineq2";
+val hcmod_diff_commute = thm"hcmod_diff_commute";
+val hcmod_add_less = thm"hcmod_add_less";
+val hcmod_mult_less = thm"hcmod_mult_less";
+val hcmod_diff_ineq = thm"hcmod_diff_ineq";
+val hcpow = thm"hcpow";
+val hcomplex_of_hypreal_hyperpow = thm"hcomplex_of_hypreal_hyperpow";
+val hcmod_hcomplexpow = thm"hcmod_hcomplexpow";
+val hcmod_hcpow = thm"hcmod_hcpow";
+val hcomplexpow_minus = thm"hcomplexpow_minus";
+val hcpow_minus = thm"hcpow_minus";
+val hccomplex_inverse_minus = thm"hccomplex_inverse_minus";
+val hcomplex_div_one = thm"hcomplex_div_one";
+val hcmod_hcomplex_inverse = thm"hcmod_hcomplex_inverse";
+val hcmod_divide = thm"hcmod_divide";
+val hcomplex_inverse_divide = thm"hcomplex_inverse_divide";
+val hcomplexpow_mult = thm"hcomplexpow_mult";
+val hcpow_mult = thm"hcpow_mult";
+val hcomplexpow_zero = thm"hcomplexpow_zero";
+val hcpow_zero = thm"hcpow_zero";
+val hcpow_zero2 = thm"hcpow_zero2";
+val hcomplexpow_not_zero = thm"hcomplexpow_not_zero";
+val hcpow_not_zero = thm"hcpow_not_zero";
+val hcomplexpow_zero_zero = thm"hcomplexpow_zero_zero";
+val hcpow_zero_zero = thm"hcpow_zero_zero";
+val hcomplex_i_mult_eq = thm"hcomplex_i_mult_eq";
+val hcomplexpow_i_squared = thm"hcomplexpow_i_squared";
+val hcomplex_i_not_zero = thm"hcomplex_i_not_zero";
+val hcomplex_mult_eq_zero_cancel1 = thm"hcomplex_mult_eq_zero_cancel1";
+val hcomplex_mult_eq_zero_cancel2 = thm"hcomplex_mult_eq_zero_cancel2";
+val hcomplex_mult_not_eq_zero_iff = thm"hcomplex_mult_not_eq_zero_iff";
+val hcomplex_divide = thm"hcomplex_divide";
+val hsgn = thm"hsgn";
+val hsgn_zero = thm"hsgn_zero";
+val hsgn_one = thm"hsgn_one";
+val hsgn_minus = thm"hsgn_minus";
+val hsgn_eq = thm"hsgn_eq";
+val lemma_hypreal_P_EX2 = thm"lemma_hypreal_P_EX2";
+val complex_split2 = thm"complex_split2";
+val hcomplex_split = thm"hcomplex_split";
+val hRe_hcomplex_i = thm"hRe_hcomplex_i";
+val hIm_hcomplex_i = thm"hIm_hcomplex_i";
+val hcmod_i = thm"hcmod_i";
+val hcomplex_eq_hRe_eq = thm"hcomplex_eq_hRe_eq";
+val hcomplex_eq_hIm_eq = thm"hcomplex_eq_hIm_eq";
+val hcomplex_eq_cancel_iff = thm"hcomplex_eq_cancel_iff";
+val hcomplex_eq_cancel_iffA = thm"hcomplex_eq_cancel_iffA";
+val hcomplex_eq_cancel_iffB = thm"hcomplex_eq_cancel_iffB";
+val hcomplex_eq_cancel_iffC = thm"hcomplex_eq_cancel_iffC";
+val hcomplex_eq_cancel_iff2 = thm"hcomplex_eq_cancel_iff2";
+val hcomplex_eq_cancel_iff2a = thm"hcomplex_eq_cancel_iff2a";
+val hcomplex_eq_cancel_iff3 = thm"hcomplex_eq_cancel_iff3";
+val hcomplex_eq_cancel_iff3a = thm"hcomplex_eq_cancel_iff3a";
+val hcomplex_split_hRe_zero = thm"hcomplex_split_hRe_zero";
+val hcomplex_split_hIm_zero = thm"hcomplex_split_hIm_zero";
+val hRe_hsgn = thm"hRe_hsgn";
+val hIm_hsgn = thm"hIm_hsgn";
+val real_two_squares_add_zero_iff = thm"real_two_squares_add_zero_iff";
+val hcomplex_inverse_complex_split = thm"hcomplex_inverse_complex_split";
+val hRe_mult_i_eq = thm"hRe_mult_i_eq";
+val hIm_mult_i_eq = thm"hIm_mult_i_eq";
+val hcmod_mult_i = thm"hcmod_mult_i";
+val hcmod_mult_i2 = thm"hcmod_mult_i2";
+val harg = thm"harg";
+val cos_harg_i_mult_zero = thm"cos_harg_i_mult_zero";
+val cos_harg_i_mult_zero2 = thm"cos_harg_i_mult_zero2";
+val hcomplex_of_hypreal_not_zero_iff = thm"hcomplex_of_hypreal_not_zero_iff";
+val hcomplex_of_hypreal_zero_iff = thm"hcomplex_of_hypreal_zero_iff";
+val cos_harg_i_mult_zero3 = thm"cos_harg_i_mult_zero3";
+val complex_split_polar2 = thm"complex_split_polar2";
+val hcomplex_split_polar = thm"hcomplex_split_polar";
+val hcis = thm"hcis";
+val hcis_eq = thm"hcis_eq";
+val hrcis = thm"hrcis";
+val hrcis_Ex = thm"hrcis_Ex";
+val hRe_hcomplex_polar = thm"hRe_hcomplex_polar";
+val hRe_hrcis = thm"hRe_hrcis";
+val hIm_hcomplex_polar = thm"hIm_hcomplex_polar";
+val hIm_hrcis = thm"hIm_hrcis";
+val hcmod_complex_polar = thm"hcmod_complex_polar";
+val hcmod_hrcis = thm"hcmod_hrcis";
+val hcis_hrcis_eq = thm"hcis_hrcis_eq";
+val hrcis_mult = thm"hrcis_mult";
+val hcis_mult = thm"hcis_mult";
+val hcis_zero = thm"hcis_zero";
+val hrcis_zero_mod = thm"hrcis_zero_mod";
+val hrcis_zero_arg = thm"hrcis_zero_arg";
+val hcomplex_i_mult_minus = thm"hcomplex_i_mult_minus";
+val hcomplex_i_mult_minus2 = thm"hcomplex_i_mult_minus2";
+val hypreal_of_nat = thm"hypreal_of_nat";
+val hcis_hypreal_of_nat_Suc_mult = thm"hcis_hypreal_of_nat_Suc_mult";
+val NSDeMoivre = thm"NSDeMoivre";
+val hcis_hypreal_of_hypnat_Suc_mult = thm"hcis_hypreal_of_hypnat_Suc_mult";
+val NSDeMoivre_ext = thm"NSDeMoivre_ext";
+val DeMoivre2 = thm"DeMoivre2";
+val DeMoivre2_ext = thm"DeMoivre2_ext";
+val hcis_inverse = thm"hcis_inverse";
+val hrcis_inverse = thm"hrcis_inverse";
+val hRe_hcis = thm"hRe_hcis";
+val hIm_hcis = thm"hIm_hcis";
+val cos_n_hRe_hcis_pow_n = thm"cos_n_hRe_hcis_pow_n";
+val sin_n_hIm_hcis_pow_n = thm"sin_n_hIm_hcis_pow_n";
+val cos_n_hRe_hcis_hcpow_n = thm"cos_n_hRe_hcis_hcpow_n";
+val sin_n_hIm_hcis_hcpow_n = thm"sin_n_hIm_hcis_hcpow_n";
+val hexpi_add = thm"hexpi_add";
+val hcomplex_of_complex_add = thm"hcomplex_of_complex_add";
+val hcomplex_of_complex_mult = thm"hcomplex_of_complex_mult";
+val hcomplex_of_complex_eq_iff = thm"hcomplex_of_complex_eq_iff";
+val hcomplex_of_complex_minus = thm"hcomplex_of_complex_minus";
+val hcomplex_of_complex_one = thm"hcomplex_of_complex_one";
+val hcomplex_of_complex_zero = thm"hcomplex_of_complex_zero";
+val hcomplex_of_complex_zero_iff = thm"hcomplex_of_complex_zero_iff";
+val hcomplex_of_complex_inverse = thm"hcomplex_of_complex_inverse";
+val hcomplex_of_complex_divide = thm"hcomplex_of_complex_divide";
+val hRe_hcomplex_of_complex = thm"hRe_hcomplex_of_complex";
+val hIm_hcomplex_of_complex = thm"hIm_hcomplex_of_complex";
+val hcmod_hcomplex_of_complex = thm"hcmod_hcomplex_of_complex";
+
+val hcomplex_add_ac = thms"hcomplex_add_ac";
+val hcomplex_mult_ac = thms"hcomplex_mult_ac";
+*}
+
end