--- a/src/HOL/Complex/NSComplex.thy Wed Sep 07 00:48:50 2005 +0200
+++ b/src/HOL/Complex/NSComplex.thy Wed Sep 07 01:49:49 2005 +0200
@@ -11,6 +11,8 @@
imports Complex
begin
+types hcomplex = "complex star"
+(*
constdefs
hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
"hcomplexrel == {p. \<exists>X Y. p = ((X::nat=>complex),Y) &
@@ -39,54 +41,56 @@
hcinv_def:
"inverse(P) == Abs_hcomplex(UN X: Rep_hcomplex(P).
hcomplexrel `` {%n. inverse(X n)})"
+*)
constdefs
hcomplex_of_complex :: "complex => hcomplex"
- "hcomplex_of_complex z == Abs_hcomplex(hcomplexrel `` {%n. z})"
+(* "hcomplex_of_complex z == Abs_star(starrel `` {%n. z})"*)
+ "hcomplex_of_complex z == star_of z"
(*--- real and Imaginary parts ---*)
hRe :: "hcomplex => hypreal"
- "hRe(z) == Abs_star(UN X:Rep_hcomplex(z). starrel `` {%n. Re (X n)})"
+ "hRe(z) == Abs_star(UN X:Rep_star(z). starrel `` {%n. Re (X n)})"
hIm :: "hcomplex => hypreal"
- "hIm(z) == Abs_star(UN X:Rep_hcomplex(z). starrel `` {%n. Im (X n)})"
+ "hIm(z) == Abs_star(UN X:Rep_star(z). starrel `` {%n. Im (X n)})"
(*----------- modulus ------------*)
hcmod :: "hcomplex => hypreal"
- "hcmod z == Abs_star(UN X: Rep_hcomplex(z).
+ "hcmod z == Abs_star(UN X: Rep_star(z).
starrel `` {%n. cmod (X n)})"
(*------ imaginary unit ----------*)
iii :: hcomplex
- "iii == Abs_hcomplex(hcomplexrel `` {%n. ii})"
+ "iii == Abs_star(starrel `` {%n. ii})"
(*------- complex conjugate ------*)
hcnj :: "hcomplex => hcomplex"
- "hcnj z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. cnj (X n)})"
+ "hcnj z == Abs_star(UN X:Rep_star(z). starrel `` {%n. cnj (X n)})"
(*------------ Argand -------------*)
hsgn :: "hcomplex => hcomplex"
- "hsgn z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. sgn(X n)})"
+ "hsgn z == Abs_star(UN X:Rep_star(z). starrel `` {%n. sgn(X n)})"
harg :: "hcomplex => hypreal"
- "harg z == Abs_star(UN X:Rep_hcomplex(z). starrel `` {%n. arg(X n)})"
+ "harg z == Abs_star(UN X:Rep_star(z). starrel `` {%n. arg(X n)})"
(* abbreviation for (cos a + i sin a) *)
hcis :: "hypreal => hcomplex"
- "hcis a == Abs_hcomplex(UN X:Rep_star(a). hcomplexrel `` {%n. cis (X n)})"
+ "hcis a == Abs_star(UN X:Rep_star(a). starrel `` {%n. cis (X n)})"
(*----- injection from hyperreals -----*)
hcomplex_of_hypreal :: "hypreal => hcomplex"
- "hcomplex_of_hypreal r == Abs_hcomplex(UN X:Rep_star(r).
- hcomplexrel `` {%n. complex_of_real (X n)})"
+ "hcomplex_of_hypreal r == Abs_star(UN X:Rep_star(r).
+ starrel `` {%n. complex_of_real (X n)})"
(* abbreviation for r*(cos a + i sin a) *)
hrcis :: "[hypreal, hypreal] => hcomplex"
@@ -102,22 +106,22 @@
HComplex :: "[hypreal,hypreal] => hcomplex"
"HComplex x y == hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y"
-
+(*
defs (overloaded)
-
+*)
(*----------- division ----------*)
-
+(*
hcomplex_divide_def:
"w / (z::hcomplex) == w * inverse z"
hcomplex_add_def:
- "w + z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
- hcomplexrel `` {%n. X n + Y n})"
+ "w + z == Abs_star(UN X:Rep_star(w). UN Y:Rep_star(z).
+ starrel `` {%n. X n + Y n})"
hcomplex_mult_def:
- "w * z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
- hcomplexrel `` {%n. X n * Y n})"
-
+ "w * z == Abs_star(UN X:Rep_star(w). UN Y:Rep_star(z).
+ starrel `` {%n. X n * Y n})"
+*)
consts
@@ -127,83 +131,32 @@
(* hypernatural powers of nonstandard complex numbers *)
hcpow_def:
"(z::hcomplex) hcpow (n::hypnat)
- == Abs_hcomplex(UN X:Rep_hcomplex(z). UN Y: Rep_star(n).
- hcomplexrel `` {%n. (X n) ^ (Y n)})"
-
-
-lemma hcomplexrel_refl: "(x,x): hcomplexrel"
-by (simp add: hcomplexrel_def)
-
-lemma hcomplexrel_sym: "(x,y): hcomplexrel ==> (y,x):hcomplexrel"
-by (auto simp add: hcomplexrel_def eq_commute)
-
-lemma hcomplexrel_trans:
- "[|(x,y): hcomplexrel; (y,z):hcomplexrel|] ==> (x,z):hcomplexrel"
-by (simp add: hcomplexrel_def, ultra)
-
-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"
-by (simp add: hcomplex_def hcomplexrel_def quotient_def, blast)
-
-declare Abs_hcomplex_inject [simp] Abs_hcomplex_inverse [simp]
-declare equiv_hcomplexrel [THEN eq_equiv_class_iff, simp]
-
-lemma lemma_hcomplexrel_refl [simp]: "x: hcomplexrel `` {x}"
-by (simp add: hcomplexrel_def)
-
-lemma hcomplex_empty_not_mem [simp]: "{} \<notin> hcomplex"
-apply (simp add: hcomplex_def hcomplexrel_def)
-apply (auto elim!: quotientE)
-done
-
-lemma Rep_hcomplex_nonempty [simp]: "Rep_hcomplex x \<noteq> {}"
-by (cut_tac x = x in Rep_hcomplex, auto)
-
-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 hcomplexrel_def)
-done
-
-theorem hcomplex_cases [case_names Abs_hcomplex, cases type: hcomplex]:
- "(!!x. z = Abs_hcomplex(hcomplexrel``{x}) ==> P) ==> P"
-by (rule eq_Abs_hcomplex [of z], blast)
-
-lemma hcomplexrel_iff [simp]:
- "((X,Y): hcomplexrel) = ({n. X n = Y n}: FreeUltrafilterNat)"
-by (simp add: hcomplexrel_def)
+ == Abs_star(UN X:Rep_star(z). UN Y: Rep_star(n).
+ starrel `` {%n. (X n) ^ (Y n)})"
subsection{*Properties of Nonstandard Real and Imaginary Parts*}
lemma hRe:
- "hRe(Abs_hcomplex (hcomplexrel `` {X})) =
+ "hRe(Abs_star (starrel `` {X})) =
Abs_star(starrel `` {%n. Re(X n)})"
apply (simp add: hRe_def)
apply (rule_tac f = Abs_star in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (auto iff: starrel_iff, ultra)
done
lemma hIm:
- "hIm(Abs_hcomplex (hcomplexrel `` {X})) =
+ "hIm(Abs_star (starrel `` {X})) =
Abs_star(starrel `` {%n. Im(X n)})"
apply (simp add: hIm_def)
apply (rule_tac f = Abs_star in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (auto iff: starrel_iff, ultra)
done
lemma hcomplex_hRe_hIm_cancel_iff:
"(w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
-apply (cases z, cases w)
-apply (auto simp add: hRe hIm complex_Re_Im_cancel_iff iff: hcomplexrel_iff)
+apply (rule_tac z=z in eq_Abs_star, rule_tac z=w in eq_Abs_star)
+apply (auto simp add: hRe hIm complex_Re_Im_cancel_iff iff: starrel_iff)
apply (ultra+)
done
@@ -211,121 +164,94 @@
by (simp add: hcomplex_hRe_hIm_cancel_iff)
lemma hcomplex_hRe_zero [simp]: "hRe 0 = 0"
-by (simp add: hcomplex_zero_def hRe hypreal_zero_num)
+by (simp add: hRe hypreal_zero_num)
lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
-by (simp add: hcomplex_zero_def hIm hypreal_zero_num)
+by (simp add: hIm hypreal_zero_num)
lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
-by (simp add: hcomplex_one_def hRe hypreal_one_num)
+by (simp add: hRe hypreal_one_num)
lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
-by (simp add: hcomplex_one_def hIm hypreal_one_def hypreal_zero_num)
+by (simp add: hIm hypreal_one_def hypreal_zero_num)
subsection{*Addition for Nonstandard Complex Numbers*}
-
+(*
lemma hcomplex_add_congruent2:
- "congruent2 hcomplexrel hcomplexrel (%X Y. hcomplexrel `` {%n. X n + Y n})"
-by (auto simp add: congruent2_def iff: hcomplexrel_iff, ultra)
-
+ "congruent2 starrel starrel (%X Y. starrel `` {%n. X n + Y n})"
+by (auto simp add: congruent2_def iff: starrel_iff, ultra)
+*)
lemma hcomplex_add:
- "Abs_hcomplex(hcomplexrel``{%n. X n}) +
- Abs_hcomplex(hcomplexrel``{%n. Y n}) =
- Abs_hcomplex(hcomplexrel``{%n. X n + Y n})"
-apply (simp add: hcomplex_add_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto simp add: iff: hcomplexrel_iff, ultra)
-done
+ "Abs_star(starrel``{%n. X n}) +
+ Abs_star(starrel``{%n. Y n}) =
+ Abs_star(starrel``{%n. X n + Y n})"
+by (rule hypreal_add)
lemma hcomplex_add_commute: "(z::hcomplex) + w = w + z"
-apply (cases z, cases w)
-apply (simp add: complex_add_commute hcomplex_add)
-done
+by (rule add_commute)
lemma hcomplex_add_assoc: "((z1::hcomplex) + z2) + z3 = z1 + (z2 + z3)"
-apply (cases z1, cases z2, cases z3)
-apply (simp add: hcomplex_add complex_add_assoc)
-done
+by (rule add_assoc)
lemma hcomplex_add_zero_left: "(0::hcomplex) + z = z"
-apply (cases z)
-apply (simp add: hcomplex_zero_def hcomplex_add)
-done
+by simp
lemma hcomplex_add_zero_right: "z + (0::hcomplex) = z"
-by (simp add: hcomplex_add_zero_left hcomplex_add_commute)
+by simp
lemma hRe_add: "hRe(x + y) = hRe(x) + hRe(y)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hRe hcomplex_add hypreal_add complex_Re_add)
done
lemma hIm_add: "hIm(x + y) = hIm(x) + hIm(y)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hIm hcomplex_add hypreal_add complex_Im_add)
done
subsection{*Additive Inverse on Nonstandard Complex Numbers*}
-
+(*
lemma hcomplex_minus_congruent:
- "(%X. hcomplexrel `` {%n. - (X n)}) respects hcomplexrel"
+ "(%X. starrel `` {%n. - (X n)}) respects starrel"
by (simp add: congruent_def)
-
+*)
lemma hcomplex_minus:
- "- (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
- Abs_hcomplex(hcomplexrel `` {%n. -(X n)})"
-apply (simp add: hcomplex_minus_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
-done
+ "- (Abs_star(starrel `` {%n. X n})) =
+ Abs_star(starrel `` {%n. -(X n)})"
+by (rule hypreal_minus)
lemma hcomplex_add_minus_left: "-z + z = (0::hcomplex)"
-apply (cases z)
-apply (simp add: hcomplex_add hcomplex_minus hcomplex_zero_def)
-done
+by simp
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 (simp add: hcomplex_mult_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
-done
+ "Abs_star(starrel``{%n. X n}) *
+ Abs_star(starrel``{%n. Y n}) =
+ Abs_star(starrel``{%n. X n * Y n})"
+by (rule hypreal_mult)
lemma hcomplex_mult_commute: "(w::hcomplex) * z = z * w"
-apply (cases w, cases z)
-apply (simp add: hcomplex_mult complex_mult_commute)
-done
+by (rule mult_commute)
lemma hcomplex_mult_assoc: "((u::hcomplex) * v) * w = u * (v * w)"
-apply (cases u, cases v, cases w)
-apply (simp add: hcomplex_mult complex_mult_assoc)
-done
+by (rule mult_assoc)
lemma hcomplex_mult_one_left: "(1::hcomplex) * z = z"
-apply (cases z)
-apply (simp add: hcomplex_one_def hcomplex_mult)
-done
+by (rule mult_1_left)
lemma hcomplex_mult_zero_left: "(0::hcomplex) * z = 0"
-apply (cases z)
-apply (simp add: hcomplex_zero_def hcomplex_mult)
-done
+by (rule mult_zero_left)
lemma hcomplex_add_mult_distrib:
"((z1::hcomplex) + z2) * w = (z1 * w) + (z2 * w)"
-apply (cases z1, cases z2, cases w)
-apply (simp add: hcomplex_mult hcomplex_add left_distrib)
-done
+by (rule left_distrib)
lemma hcomplex_zero_not_eq_one: "(0::hcomplex) \<noteq> (1::hcomplex)"
-by (simp add: hcomplex_zero_def hcomplex_one_def)
+by (rule zero_neq_one)
declare hcomplex_zero_not_eq_one [THEN not_sym, simp]
@@ -333,23 +259,22 @@
subsection{*Inverse of Nonstandard Complex Number*}
lemma hcomplex_inverse:
- "inverse (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
- Abs_hcomplex(hcomplexrel `` {%n. inverse (X n)})"
-apply (simp add: hcinv_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+ "inverse (Abs_star(starrel `` {%n. X n})) =
+ Abs_star(starrel `` {%n. inverse (X n)})"
+apply (fold star_n_def)
+apply (simp add: star_inverse_def Ifun_of_def star_of_def Ifun_star_n)
done
lemma hcomplex_mult_inv_left:
"z \<noteq> (0::hcomplex) ==> inverse(z) * z = (1::hcomplex)"
-apply (cases z)
-apply (simp add: hcomplex_zero_def hcomplex_one_def hcomplex_inverse hcomplex_mult, ultra)
+apply (rule_tac z=z in eq_Abs_star)
+apply (simp add: hypreal_zero_def hypreal_one_def hcomplex_inverse hcomplex_mult, ultra)
apply (rule ccontr)
apply (drule left_inverse, auto)
done
subsection {* The Field of Nonstandard Complex Numbers *}
-
+(*
instance hcomplex :: field
proof
fix z u v w :: hcomplex
@@ -385,17 +310,17 @@
show "inverse 0 = (0::hcomplex)"
by (simp add: hcomplex_inverse hcomplex_zero_def)
qed
-
+*)
subsection{*More Minus Laws*}
lemma hRe_minus: "hRe(-z) = - hRe(z)"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hRe hcomplex_minus hypreal_minus complex_Re_minus)
done
lemma hIm_minus: "hIm(-z) = - hIm(z)"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hIm hcomplex_minus hypreal_minus complex_Im_minus)
done
@@ -406,13 +331,13 @@
done
lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
-by (simp add: iii_def hcomplex_mult hcomplex_one_def hcomplex_minus)
+by (simp add: iii_def hcomplex_mult hypreal_one_def hcomplex_minus)
lemma hcomplex_i_mult_left [simp]: "iii * (iii * z) = -z"
by (simp add: mult_assoc [symmetric])
lemma hcomplex_i_not_zero [simp]: "iii \<noteq> 0"
-by (simp add: iii_def hcomplex_zero_def)
+by (simp add: iii_def hypreal_zero_def)
subsection{*More Multiplication Laws*}
@@ -438,9 +363,9 @@
subsection{*Subraction and Division*}
lemma hcomplex_diff:
- "Abs_hcomplex(hcomplexrel``{%n. X n}) - Abs_hcomplex(hcomplexrel``{%n. Y n}) =
- Abs_hcomplex(hcomplexrel``{%n. X n - Y n})"
-by (simp add: hcomplex_diff_def hcomplex_minus hcomplex_add complex_diff_def)
+ "Abs_star(starrel``{%n. X n}) - Abs_star(starrel``{%n. Y n}) =
+ Abs_star(starrel``{%n. X n - Y n})"
+by (rule hypreal_diff)
lemma hcomplex_diff_eq_eq [simp]: "((x::hcomplex) - y = z) = (x = z + y)"
by (rule OrderedGroup.diff_eq_eq)
@@ -453,9 +378,9 @@
lemma hcomplex_of_hypreal:
"hcomplex_of_hypreal (Abs_star(starrel `` {%n. X n})) =
- Abs_hcomplex(hcomplexrel `` {%n. complex_of_real (X n)})"
+ Abs_star(starrel `` {%n. complex_of_real (X n)})"
apply (simp add: hcomplex_of_hypreal_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong, auto iff: hcomplexrel_iff, ultra)
+apply (rule_tac f = Abs_star in arg_cong, auto iff: starrel_iff, ultra)
done
lemma hcomplex_of_hypreal_cancel_iff [iff]:
@@ -465,10 +390,10 @@
done
lemma hcomplex_of_hypreal_one [simp]: "hcomplex_of_hypreal 1 = 1"
-by (simp add: hcomplex_one_def hcomplex_of_hypreal hypreal_one_num)
+by (simp add: hypreal_one_def hcomplex_of_hypreal hypreal_one_num)
lemma hcomplex_of_hypreal_zero [simp]: "hcomplex_of_hypreal 0 = 0"
-by (simp add: hcomplex_zero_def hypreal_zero_def hcomplex_of_hypreal)
+by (simp add: hypreal_zero_def hypreal_zero_def hcomplex_of_hypreal)
lemma hcomplex_of_hypreal_minus [simp]:
"hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x"
@@ -491,7 +416,7 @@
lemma hcomplex_of_hypreal_diff [simp]:
"hcomplex_of_hypreal (x - y) =
hcomplex_of_hypreal x - hcomplex_of_hypreal y "
-by (simp add: hcomplex_diff_def hypreal_diff_def)
+by (simp add: hypreal_diff_def)
lemma hcomplex_of_hypreal_mult [simp]:
"hcomplex_of_hypreal (x * y) = hcomplex_of_hypreal x * hcomplex_of_hypreal y"
@@ -501,10 +426,7 @@
lemma hcomplex_of_hypreal_divide [simp]:
"hcomplex_of_hypreal(x/y) = hcomplex_of_hypreal x / hcomplex_of_hypreal y"
-apply (simp add: hcomplex_divide_def)
-apply (case_tac "y=0", simp)
-apply (simp add: hypreal_divide_def)
-done
+by (simp add: divide_inverse)
lemma hRe_hcomplex_of_hypreal [simp]: "hRe(hcomplex_of_hypreal z) = z"
apply (rule_tac z=z in eq_Abs_star)
@@ -518,7 +440,7 @@
lemma hcomplex_of_hypreal_epsilon_not_zero [simp]:
"hcomplex_of_hypreal epsilon \<noteq> 0"
-by (auto simp add: hcomplex_of_hypreal epsilon_def star_n_def hcomplex_zero_def)
+by (auto simp add: hcomplex_of_hypreal epsilon_def star_n_def hypreal_zero_def)
subsection{*HComplex theorems*}
@@ -534,15 +456,15 @@
done
text{*Relates the two nonstandard constructions*}
-lemma HComplex_eq_Abs_hcomplex_Complex:
+lemma HComplex_eq_Abs_star_Complex:
"HComplex (Abs_star (starrel `` {X})) (Abs_star (starrel `` {Y})) =
- Abs_hcomplex(hcomplexrel `` {%n::nat. Complex (X n) (Y n)})";
+ Abs_star(starrel `` {%n::nat. Complex (X n) (Y n)})";
by (simp add: hcomplex_hRe_hIm_cancel_iff hRe hIm)
lemma hcomplex_surj [simp]: "HComplex (hRe z) (hIm z) = z"
by (simp add: hcomplex_equality)
-lemma hcomplex_induct [case_names rect, induct type: hcomplex]:
+lemma hcomplex_induct [case_names rect(*, induct type: hcomplex*)]:
"(\<And>x y. P (HComplex x y)) ==> P z"
by (rule hcomplex_surj [THEN subst], blast)
@@ -550,19 +472,19 @@
subsection{*Modulus (Absolute Value) of Nonstandard Complex Number*}
lemma hcmod:
- "hcmod (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ "hcmod (Abs_star(starrel `` {%n. X n})) =
Abs_star(starrel `` {%n. cmod (X n)})"
apply (simp add: hcmod_def)
apply (rule_tac f = Abs_star in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (auto iff: starrel_iff, ultra)
done
lemma hcmod_zero [simp]: "hcmod(0) = 0"
-by (simp add: hcomplex_zero_def hypreal_zero_def hcmod)
+by (simp add: hypreal_zero_def hypreal_zero_def hcmod)
lemma hcmod_one [simp]: "hcmod(1) = 1"
-by (simp add: hcomplex_one_def hcmod hypreal_one_num)
+by (simp add: hypreal_one_def hcmod hypreal_one_num)
lemma hcmod_hcomplex_of_hypreal [simp]: "hcmod(hcomplex_of_hypreal x) = abs x"
apply (rule_tac z=x in eq_Abs_star)
@@ -633,20 +555,20 @@
subsection{*Conjugation*}
lemma hcnj:
- "hcnj (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
- Abs_hcomplex(hcomplexrel `` {%n. cnj(X n)})"
+ "hcnj (Abs_star(starrel `` {%n. X n})) =
+ Abs_star(starrel `` {%n. cnj(X n)})"
apply (simp add: hcnj_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (rule_tac f = Abs_star in arg_cong)
+apply (auto iff: starrel_iff, ultra)
done
lemma hcomplex_hcnj_cancel_iff [iff]: "(hcnj x = hcnj y) = (x = y)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hcnj)
done
lemma hcomplex_hcnj_hcnj [simp]: "hcnj (hcnj z) = z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hcnj)
done
@@ -657,52 +579,52 @@
done
lemma hcomplex_hmod_hcnj [simp]: "hcmod (hcnj z) = hcmod z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hcnj hcmod)
done
lemma hcomplex_hcnj_minus: "hcnj (-z) = - hcnj z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hcnj hcomplex_minus complex_cnj_minus)
done
lemma hcomplex_hcnj_inverse: "hcnj(inverse z) = inverse(hcnj z)"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hcnj hcomplex_inverse complex_cnj_inverse)
done
lemma hcomplex_hcnj_add: "hcnj(w + z) = hcnj(w) + hcnj(z)"
-apply (cases z, cases w)
+apply (rule_tac z=z in eq_Abs_star, rule_tac z=w in eq_Abs_star)
apply (simp add: hcnj hcomplex_add complex_cnj_add)
done
lemma hcomplex_hcnj_diff: "hcnj(w - z) = hcnj(w) - hcnj(z)"
-apply (cases z, cases w)
+apply (rule_tac z=z in eq_Abs_star, rule_tac z=w in eq_Abs_star)
apply (simp add: hcnj hcomplex_diff complex_cnj_diff)
done
lemma hcomplex_hcnj_mult: "hcnj(w * z) = hcnj(w) * hcnj(z)"
-apply (cases z, cases w)
+apply (rule_tac z=z in eq_Abs_star, rule_tac z=w in eq_Abs_star)
apply (simp add: hcnj hcomplex_mult complex_cnj_mult)
done
lemma hcomplex_hcnj_divide: "hcnj(w / z) = (hcnj w)/(hcnj z)"
-by (simp add: hcomplex_divide_def hcomplex_hcnj_mult hcomplex_hcnj_inverse)
+by (simp add: divide_inverse hcomplex_hcnj_mult hcomplex_hcnj_inverse)
lemma hcnj_one [simp]: "hcnj 1 = 1"
-by (simp add: hcomplex_one_def hcnj)
+by (simp add: hypreal_one_def hcnj)
lemma hcomplex_hcnj_zero [simp]: "hcnj 0 = 0"
-by (simp add: hcomplex_zero_def hcnj)
+by (simp add: hypreal_zero_def hcnj)
lemma hcomplex_hcnj_zero_iff [iff]: "(hcnj z = 0) = (z = 0)"
-apply (cases z)
-apply (simp add: hcomplex_zero_def hcnj)
+apply (rule_tac z=z in eq_Abs_star)
+apply (simp add: hypreal_zero_def hcnj)
done
lemma hcomplex_mult_hcnj:
"z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add
hypreal_mult complex_mult_cnj numeral_2_eq_2)
done
@@ -711,8 +633,8 @@
subsection{*More Theorems about the Function @{term hcmod}*}
lemma hcomplex_hcmod_eq_zero_cancel [simp]: "(hcmod x = 0) = (x = 0)"
-apply (cases x)
-apply (simp add: hcmod hcomplex_zero_def hypreal_zero_num)
+apply (rule_tac z=x in eq_Abs_star)
+apply (simp add: hcmod hypreal_zero_def hypreal_zero_num)
done
lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
@@ -726,17 +648,17 @@
done
lemma hcmod_minus [simp]: "hcmod (-x) = hcmod(x)"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
apply (simp add: hcmod hcomplex_minus)
done
lemma hcmod_mult_hcnj: "hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj numeral_2_eq_2)
done
lemma hcmod_ge_zero [simp]: "(0::hypreal) \<le> hcmod x"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
apply (simp add: hcmod hypreal_zero_num hypreal_le)
done
@@ -744,13 +666,13 @@
by (simp add: abs_if linorder_not_less)
lemma hcmod_mult: "hcmod(x*y) = hcmod(x) * hcmod(y)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (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 (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hcmod hcomplex_add hypreal_mult hRe hcnj hcomplex_mult
numeral_2_eq_2 realpow_two [symmetric]
del: realpow_Suc)
@@ -761,7 +683,7 @@
done
lemma hcomplex_hRe_mult_hcnj_le_hcmod [simp]: "hRe(x * hcnj y) \<le> hcmod(x * hcnj y)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hcmod hcnj hcomplex_mult hRe hypreal_le)
done
@@ -771,7 +693,7 @@
done
lemma hcmod_triangle_squared [simp]: "hcmod (x + y) ^ 2 \<le> (hcmod(x) + hcmod(y)) ^ 2"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hcmod hcnj hcomplex_add hypreal_mult hypreal_add
hypreal_le realpow_two [symmetric] numeral_2_eq_2
del: realpow_Suc)
@@ -779,7 +701,7 @@
done
lemma hcmod_triangle_ineq [simp]: "hcmod (x + y) \<le> hcmod(x) + hcmod(y)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (simp add: hcmod hcomplex_add hypreal_add hypreal_le)
done
@@ -789,26 +711,28 @@
done
lemma hcmod_diff_commute: "hcmod (x - y) = hcmod (y - x)"
-apply (cases x, cases y)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
apply (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 (cases x, cases y, rule_tac z=r in eq_Abs_star, rule_tac z=s in eq_Abs_star)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
+apply (rule_tac z=r in eq_Abs_star, rule_tac z=s in eq_Abs_star)
apply (simp add: hcmod hcomplex_add hypreal_add hypreal_less, 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 (cases x, cases y, rule_tac z=r in eq_Abs_star, rule_tac z=s in eq_Abs_star)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
+apply (rule_tac z=r in eq_Abs_star, rule_tac z=s in eq_Abs_star)
apply (simp add: hcmod hypreal_mult hypreal_less hcomplex_mult, ultra)
apply (auto intro: complex_mod_mult_less)
done
lemma hcmod_diff_ineq [simp]: "hcmod(a) - hcmod(b) \<le> hcmod(a + b)"
-apply (cases a, cases b)
+apply (rule_tac z=a in eq_Abs_star, rule_tac z=b in eq_Abs_star)
apply (simp add: hcmod hcomplex_add hypreal_diff hypreal_le)
done
@@ -816,12 +740,12 @@
subsection{*A Few Nonlinear Theorems*}
lemma hcpow:
- "Abs_hcomplex(hcomplexrel``{%n. X n}) hcpow
+ "Abs_star(starrel``{%n. X n}) hcpow
Abs_star(starrel``{%n. Y n}) =
- Abs_hcomplex(hcomplexrel``{%n. X n ^ Y n})"
+ Abs_star(starrel``{%n. X n ^ Y n})"
apply (simp add: hcpow_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (rule_tac f = Abs_star in arg_cong)
+apply (auto iff: starrel_iff, ultra)
done
lemma hcomplex_of_hypreal_hyperpow:
@@ -831,7 +755,7 @@
done
lemma hcmod_hcpow: "hcmod(x hcpow n) = hcmod(x) pow n"
-apply (cases x, rule_tac z=n in eq_Abs_star)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=n in eq_Abs_star)
apply (simp add: hcpow hyperpow hcmod complex_mod_complexpow)
done
@@ -842,15 +766,18 @@
done
lemma hcmod_divide: "hcmod(x/y) = hcmod(x)/(hcmod y)"
-by (simp add: hcomplex_divide_def hypreal_divide_def hcmod_mult hcmod_hcomplex_inverse)
+by (simp add: divide_inverse hcmod_mult hcmod_hcomplex_inverse)
subsection{*Exponentiation*}
-primrec
- hcomplexpow_0: "z ^ 0 = 1"
- hcomplexpow_Suc: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
+lemma hcomplexpow_0 [simp]: "z ^ 0 = (1::hcomplex)"
+by (rule power_0)
+lemma hcomplexpow_Suc [simp]: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
+by (rule power_Suc)
+
+(*
instance hcomplex :: recpower
proof
fix z :: hcomplex
@@ -858,7 +785,7 @@
show "z^0 = 1" by simp
show "z^(Suc n) = z * (z^n)" by simp
qed
-
+*)
lemma hcomplexpow_i_squared [simp]: "iii ^ 2 = - 1"
by (simp add: power2_eq_square)
@@ -882,18 +809,19 @@
lemma hcpow_minus:
"(-x::hcomplex) hcpow n =
(if ( *pNat* even) n then (x hcpow n) else -(x hcpow n))"
-apply (cases x, rule_tac z=n in eq_Abs_star)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=n in eq_Abs_star)
apply (auto simp add: hcpow hyperpow starPNat hcomplex_minus, ultra)
apply (auto simp add: neg_power_if, ultra)
done
lemma hcpow_mult: "((r::hcomplex) * s) hcpow n = (r hcpow n) * (s hcpow n)"
-apply (cases r, cases s, rule_tac z=n in eq_Abs_star)
+apply (rule_tac z=r in eq_Abs_star, rule_tac z=s in eq_Abs_star)
+apply (rule_tac z=n in eq_Abs_star)
apply (simp add: hcpow hypreal_mult hcomplex_mult power_mult_distrib)
done
lemma hcpow_zero [simp]: "0 hcpow (n + 1) = 0"
-apply (simp add: hcomplex_zero_def hypnat_one_def, rule_tac z=n in eq_Abs_star)
+apply (simp add: hypreal_zero_def hypnat_one_def, rule_tac z=n in eq_Abs_star)
apply (simp add: hcpow hypnat_add)
done
@@ -901,17 +829,17 @@
by (simp add: hSuc_def)
lemma hcpow_not_zero [simp,intro]: "r \<noteq> 0 ==> r hcpow n \<noteq> (0::hcomplex)"
-apply (cases r, rule_tac z=n in eq_Abs_star)
-apply (auto simp add: hcpow hcomplex_zero_def, ultra)
+apply (rule_tac z=r in eq_Abs_star, rule_tac z=n in eq_Abs_star)
+apply (auto simp add: hcpow hypreal_zero_def, ultra)
done
lemma hcpow_zero_zero: "r hcpow n = (0::hcomplex) ==> r = 0"
by (blast intro: ccontr dest: hcpow_not_zero)
lemma hcomplex_divide:
- "Abs_hcomplex(hcomplexrel``{%n. X n}) / Abs_hcomplex(hcomplexrel``{%n. Y n}) =
- Abs_hcomplex(hcomplexrel``{%n. X n / Y n})"
-by (simp add: hcomplex_divide_def complex_divide_def hcomplex_inverse hcomplex_mult)
+ "Abs_star(starrel``{%n. X n::complex}) / Abs_star(starrel``{%n. Y n}) =
+ Abs_star(starrel``{%n. X n / Y n})"
+by (simp add: divide_inverse complex_divide_def hcomplex_inverse hcomplex_mult)
@@ -919,33 +847,33 @@
subsection{*The Function @{term hsgn}*}
lemma hsgn:
- "hsgn (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
- Abs_hcomplex(hcomplexrel `` {%n. sgn (X n)})"
+ "hsgn (Abs_star(starrel `` {%n. X n})) =
+ Abs_star(starrel `` {%n. sgn (X n)})"
apply (simp add: hsgn_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (rule_tac f = Abs_star in arg_cong)
+apply (auto iff: starrel_iff, ultra)
done
lemma hsgn_zero [simp]: "hsgn 0 = 0"
-by (simp add: hcomplex_zero_def hsgn)
+by (simp add: hypreal_zero_def hsgn)
lemma hsgn_one [simp]: "hsgn 1 = 1"
-by (simp add: hcomplex_one_def hsgn)
+by (simp add: hypreal_one_def hsgn)
lemma hsgn_minus: "hsgn (-z) = - hsgn(z)"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hsgn hcomplex_minus sgn_minus)
done
lemma hsgn_eq: "hsgn z = z / hcomplex_of_hypreal (hcmod z)"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hsgn hcomplex_divide hcomplex_of_hypreal hcmod sgn_eq)
done
lemma hcmod_i: "hcmod (HComplex x y) = ( *f* sqrt) (x ^ 2 + y ^ 2)"
apply (rule_tac z=x in eq_Abs_star, rule_tac z=y in eq_Abs_star)
-apply (simp add: HComplex_eq_Abs_hcomplex_Complex starfun
+apply (simp add: HComplex_eq_Abs_star_Complex starfun
hypreal_mult hypreal_add hcmod numeral_2_eq_2)
done
@@ -970,12 +898,12 @@
by (simp add: i_eq_HComplex_0_1)
lemma hRe_hsgn [simp]: "hRe(hsgn z) = hRe(z)/hcmod z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hsgn hcmod hRe hypreal_divide)
done
lemma hIm_hsgn [simp]: "hIm(hsgn z) = hIm(z)/hcmod z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: hsgn hcmod hIm hypreal_divide)
done
@@ -1026,11 +954,11 @@
(*---------------------------------------------------------------------------*)
lemma harg:
- "harg (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
+ "harg (Abs_star(starrel `` {%n. X n})) =
Abs_star(starrel `` {%n. arg (X n)})"
apply (simp add: harg_def)
apply (rule_tac f = Abs_star in arg_cong)
-apply (auto iff: hcomplexrel_iff, ultra)
+apply (auto iff: starrel_iff, ultra)
done
lemma cos_harg_i_mult_zero_pos:
@@ -1055,7 +983,7 @@
lemma hcomplex_of_hypreal_zero_iff [simp]:
"(hcomplex_of_hypreal y = 0) = (y = 0)"
apply (rule_tac z=y in eq_Abs_star)
-apply (simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
+apply (simp add: hcomplex_of_hypreal hypreal_zero_num hypreal_zero_def)
done
@@ -1075,7 +1003,7 @@
lemma hcomplex_split_polar:
"\<exists>r a. z = hcomplex_of_hypreal r * (HComplex(( *f* cos) a)(( *f* sin) a))"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
apply (simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def starfun hcomplex_add hcomplex_mult HComplex_def)
apply (cut_tac z = x in complex_split_polar2)
apply (drule choice, safe)+
@@ -1085,9 +1013,9 @@
lemma hcis:
"hcis (Abs_star(starrel `` {%n. X n})) =
- Abs_hcomplex(hcomplexrel `` {%n. cis (X n)})"
+ Abs_star(starrel `` {%n. cis (X n)})"
apply (simp add: hcis_def)
-apply (rule_tac f = Abs_hcomplex in arg_cong, auto iff: hcomplexrel_iff, ultra)
+apply (rule_tac f = Abs_star in arg_cong, auto iff: starrel_iff, ultra)
done
lemma hcis_eq:
@@ -1100,7 +1028,7 @@
lemma hrcis:
"hrcis (Abs_star(starrel `` {%n. X n})) (Abs_star(starrel `` {%n. Y n})) =
- Abs_hcomplex(hcomplexrel `` {%n. rcis (X n) (Y n)})"
+ Abs_star(starrel `` {%n. rcis (X n) (Y n)})"
by (simp add: hrcis_def hcomplex_of_hypreal hcomplex_mult hcis rcis_def)
lemma hrcis_Ex: "\<exists>r a. z = hrcis r a"
@@ -1162,10 +1090,10 @@
done
lemma hcis_zero [simp]: "hcis 0 = 1"
-by (simp add: hcomplex_one_def hcis hypreal_zero_num)
+by (simp add: hypreal_one_def hcis hypreal_zero_num)
lemma hrcis_zero_mod [simp]: "hrcis 0 a = 0"
-apply (simp add: hcomplex_zero_def, rule_tac z=a in eq_Abs_star)
+apply (simp add: hypreal_zero_def, rule_tac z=a in eq_Abs_star)
apply (simp add: hrcis hypreal_zero_num)
done
@@ -1247,7 +1175,7 @@
by (simp add: NSDeMoivre_ext)
lemma hexpi_add: "hexpi(a + b) = hexpi(a) * hexpi(b)"
-apply (simp add: hexpi_def, cases a, cases b)
+apply (simp add: hexpi_def, rule_tac z=a in eq_Abs_star, rule_tac z=b in eq_Abs_star)
apply (simp add: hcis hRe hIm hcomplex_add hcomplex_mult hypreal_mult starfun hcomplex_of_hypreal cis_mult [symmetric] complex_Im_add complex_Re_add exp_add complex_of_real_mult)
done
@@ -1261,7 +1189,7 @@
done
lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex ii"
-by (simp add: iii_def hcomplex_of_complex_def)
+by (simp add: iii_def hcomplex_of_complex_def star_of_def star_n_def)
lemma hcomplex_of_complex_add [simp]:
"hcomplex_of_complex (z1 + z2) = hcomplex_of_complex z1 + hcomplex_of_complex z2"
@@ -1281,15 +1209,15 @@
by (simp add: hcomplex_of_complex_def hcomplex_minus)
lemma hcomplex_of_complex_one [simp]: "hcomplex_of_complex 1 = 1"
-by (simp add: hcomplex_of_complex_def hcomplex_one_def)
+by (simp add: hcomplex_of_complex_def hypreal_one_def)
lemma hcomplex_of_complex_zero [simp]: "hcomplex_of_complex 0 = 0"
-by (simp add: hcomplex_of_complex_def hcomplex_zero_def)
+by (simp add: hcomplex_of_complex_def hypreal_zero_def)
lemma hcomplex_of_complex_zero_iff [simp]:
"(hcomplex_of_complex r = 0) = (r = 0)"
by (auto intro: FreeUltrafilterNat_P
- simp add: hcomplex_of_complex_def hcomplex_zero_def)
+ simp add: hcomplex_of_complex_def star_of_def star_n_def hypreal_zero_def)
lemma hcomplex_of_complex_inverse [simp]:
"hcomplex_of_complex (inverse r) = inverse (hcomplex_of_complex r)"
@@ -1315,7 +1243,7 @@
lemma hcomplex_of_complex_divide [simp]:
"hcomplex_of_complex (z1 / z2) =
hcomplex_of_complex z1 / hcomplex_of_complex z2"
-by (simp add: hcomplex_divide_def complex_divide_def)
+by (simp add: divide_inverse)
lemma hRe_hcomplex_of_complex:
"hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
@@ -1332,6 +1260,7 @@
subsection{*Numerals and Arithmetic*}
+(*
instance hcomplex :: number ..
defs (overloaded)
@@ -1340,11 +1269,18 @@
instance hcomplex :: number_ring
by (intro_classes, simp add: hcomplex_number_of_def)
+*)
+lemma hcomplex_number_of_def: "(number_of w :: hcomplex) == of_int (Rep_Bin w)"
+apply (rule eq_reflection)
+apply (unfold star_number_def star_of_int_def)
+apply (rule star_of_inject [THEN iffD2])
+apply (rule number_of_eq)
+done
lemma hcomplex_of_complex_of_nat [simp]:
"hcomplex_of_complex (of_nat n) = of_nat n"
-by (induct n, simp_all)
+by (simp add: hcomplex_of_complex_def)
lemma hcomplex_of_complex_of_int [simp]:
"hcomplex_of_complex (of_int z) = of_int z"
@@ -1390,14 +1326,14 @@
(*
Goal "z + hcnj z =
hcomplex_of_hypreal (2 * hRe(z))"
-by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z")] eq_Abs_star 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 (res_inst_tac [("z","z")] eq_Abs_star 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]));
@@ -1410,12 +1346,12 @@
done
declare hcomplex_hcnj_num_zero_iff [simp]
-lemma hcomplex_zero_num: "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})"
-apply (simp add: hcomplex_zero_def)
+lemma hcomplex_zero_num: "0 = Abs_star (starrel `` {%n. 0})"
+apply (simp add: hypreal_zero_def)
done
-lemma hcomplex_one_num: "1 = Abs_hcomplex (hcomplexrel `` {%n. 1})"
-apply (simp add: hcomplex_one_def)
+lemma hcomplex_one_num: "1 = Abs_star (starrel `` {%n. 1})"
+apply (simp add: hypreal_one_def)
done
(*** Real and imaginary stuff ***)
@@ -1525,27 +1461,16 @@
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_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 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";
@@ -1562,7 +1487,7 @@
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_congruent = thm"hcomplex_minus_congruent"; *)
val hcomplex_minus = thm"hcomplex_minus";
val hcomplex_add_minus_left = thm"hcomplex_add_minus_left";
val hRe_minus = thm"hRe_minus";