# HG changeset patch
# User huffman
# Date 1126050589 -7200
# Node ID 5798fbf42a6ac9bf4ea59c674ece6e87ad1c1eb3
# Parent  c6eecde058e48458fc8fa4a970f66ccfce229342
replace type hcomplex with complex star

diff -r c6eecde058e4 -r 5798fbf42a6a src/HOL/Complex/CLim.thy
--- a/src/HOL/Complex/CLim.thy	Wed Sep 07 00:48:50 2005 +0200
+++ b/src/HOL/Complex/CLim.thy	Wed Sep 07 01:49:49 2005 +0200
@@ -123,8 +123,9 @@
 lemma CLIM_NSCLIM:
       "f -- x --C> L ==> f -- x --NSC> L"
 apply (simp add: CLIM_def NSCLIM_def capprox_def, auto)
-apply (rule_tac z = xa in eq_Abs_hcomplex)
+apply (rule_tac z = xa in eq_Abs_star)
 apply (auto simp add: hcomplex_of_complex_def starfunC hcomplex_diff 
+         star_of_def star_n_def
          CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff)
 apply (rule bexI, rule_tac [2] lemma_starrel_refl, safe)
 apply (drule_tac x = u in spec, auto)
@@ -132,10 +133,10 @@
 apply (drule sym, auto)
 done
 
-lemma eq_Abs_hcomplex_ALL:
-     "(\<forall>t. P t) = (\<forall>X. P (Abs_hcomplex(hcomplexrel `` {X})))"
+lemma eq_Abs_star_ALL:
+     "(\<forall>t. P t) = (\<forall>X. P (Abs_star(starrel `` {X})))"
 apply auto
-apply (rule_tac z = t in eq_Abs_hcomplex, auto)
+apply (rule_tac z = t in eq_Abs_star, auto)
 done
 
 lemma lemma_CLIM:
@@ -168,15 +169,17 @@
      "f -- x --NSC> L ==> f -- x --C> L"
 apply (simp add: CLIM_def NSCLIM_def)
 apply (rule ccontr) 
-apply (auto simp add: eq_Abs_hcomplex_ALL starfunC 
+apply (auto simp add: eq_Abs_star_ALL starfunC 
             CInfinitesimal_capprox_minus [symmetric] hcomplex_diff
             CInfinitesimal_hcmod_iff hcomplex_of_complex_def 
+            star_of_def star_n_def
             Infinitesimal_FreeUltrafilterNat_iff hcmod)
 apply (simp add: linorder_not_less)
 apply (drule lemma_skolemize_CLIM2, safe)
 apply (drule_tac x = X in spec, auto)
 apply (drule lemma_csimp [THEN complex_seq_to_hcomplex_CInfinitesimal])
 apply (simp add: CInfinitesimal_hcmod_iff hcomplex_of_complex_def 
+            star_of_def star_n_def
             Infinitesimal_FreeUltrafilterNat_iff hcomplex_diff hcmod,  blast)
 apply (drule_tac x = r in spec, clarify)
 apply (drule FreeUltrafilterNat_all, ultra, arith)
@@ -192,7 +195,7 @@
 
 lemma CRLIM_NSCRLIM: "f -- x --CR> L ==> f -- x --NSCR> L"
 apply (simp add: CRLIM_def NSCRLIM_def capprox_def, auto)
-apply (rule_tac z = xa in eq_Abs_hcomplex)
+apply (rule_tac z = xa in eq_Abs_star)
 apply (auto simp add: hcomplex_of_complex_def starfunCR hcomplex_diff
               CInfinitesimal_hcmod_iff hcmod hypreal_diff
               Infinitesimal_FreeUltrafilterNat_iff
@@ -232,15 +235,17 @@
 lemma NSCRLIM_CRLIM: "f -- x --NSCR> L ==> f -- x --CR> L"
 apply (simp add: CRLIM_def NSCRLIM_def capprox_def)
 apply (rule ccontr) 
-apply (auto simp add: eq_Abs_hcomplex_ALL starfunCR hcomplex_diff 
+apply (auto simp add: eq_Abs_star_ALL starfunCR hcomplex_diff 
              hcomplex_of_complex_def hypreal_diff CInfinitesimal_hcmod_iff 
-             hcmod Infinitesimal_approx_minus [symmetric] 
+             hcmod Infinitesimal_approx_minus [symmetric]
+             star_of_def star_n_def 
              Infinitesimal_FreeUltrafilterNat_iff)
 apply (simp add: linorder_not_less)
 apply (drule lemma_skolemize_CRLIM2, safe)
 apply (drule_tac x = X in spec, auto)
 apply (drule lemma_crsimp [THEN complex_seq_to_hcomplex_CInfinitesimal])
 apply (simp add: CInfinitesimal_hcmod_iff hcomplex_of_complex_def 
+             star_of_def star_n_def
              Infinitesimal_FreeUltrafilterNat_iff hcomplex_diff hcmod)
 apply (auto simp add: hypreal_of_real_def star_of_def star_n_def hypreal_diff)
 apply (drule_tac x = r in spec, clarify)
@@ -408,8 +413,8 @@
    "(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)"
 apply (auto simp add: NSCLIM_def NSCRLIM_def CInfinitesimal_capprox_minus [symmetric] CInfinitesimal_hcmod_iff)
 apply (auto dest!: spec) 
-apply (rule_tac [!] z = xa in eq_Abs_hcomplex)
-apply (auto simp add: hcomplex_diff starfunC starfunCR hcomplex_of_complex_def hcmod mem_infmal_iff)
+apply (rule_tac [!] z = xa in eq_Abs_star)
+apply (auto simp add: hcomplex_diff starfunC starfunCR hcomplex_of_complex_def hcmod mem_infmal_iff star_of_def star_n_def)
 done
 
 (** much, much easier standard proof **)
@@ -427,7 +432,7 @@
 apply (auto intro: NSCLIM_NSCRLIM_Re NSCLIM_NSCRLIM_Im)
 apply (auto simp add: NSCLIM_def NSCRLIM_def)
 apply (auto dest!: spec) 
-apply (rule_tac z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
 apply (simp add: capprox_approx_iff starfunC hcomplex_of_complex_def starfunCR hypreal_of_real_def star_of_def star_n_def)
 done
 
@@ -481,9 +486,9 @@
 apply (rule mem_cinfmal_iff [THEN iffD2, THEN CInfinitesimal_add_capprox_self [THEN capprox_sym]])
 apply (rule_tac [4] capprox_minus_iff2 [THEN iffD1])
  prefer 3 apply (simp add: compare_rls hcomplex_add_commute)
-apply (rule_tac [2] z = x in eq_Abs_hcomplex)
-apply (rule_tac [4] z = x in eq_Abs_hcomplex)
-apply (auto simp add: starfunC hcomplex_of_complex_def hcomplex_minus hcomplex_add)
+apply (rule_tac [2] z = x in eq_Abs_star)
+apply (rule_tac [4] z = x in eq_Abs_star)
+apply (auto simp add: starfunC hcomplex_of_complex_def hcomplex_minus hcomplex_add star_of_def star_n_def)
 done
 
 lemma NSCLIM_isContc_iff:
@@ -860,7 +865,7 @@
 apply (auto simp add: starfunC_lambda_cancel2 starfunC_o [symmetric])
 apply (case_tac "( *fc* g) (hcomplex_of_complex (x) + xa) = hcomplex_of_complex (g x) ")
 apply (drule_tac g = g in NSCDERIV_zero)
-apply (auto simp add: hcomplex_divide_def)
+apply (auto simp add: divide_inverse)
 apply (rule_tac z1 = "( *fc* g) (hcomplex_of_complex (x) + xa) - hcomplex_of_complex (g x) " and y1 = "inverse xa" in lemma_complex_chain [THEN ssubst])
 apply (simp (no_asm_simp))
 apply (rule capprox_mult_hcomplex_of_complex)
@@ -1028,7 +1033,7 @@
 val NSCLIM_NSCRLIM_Re = thm "NSCLIM_NSCRLIM_Re";
 val NSCLIM_NSCRLIM_Im = thm "NSCLIM_NSCRLIM_Im";
 val CLIM_NSCLIM = thm "CLIM_NSCLIM";
-val eq_Abs_hcomplex_ALL = thm "eq_Abs_hcomplex_ALL";
+val eq_Abs_star_ALL = thm "eq_Abs_star_ALL";
 val lemma_CLIM = thm "lemma_CLIM";
 val lemma_skolemize_CLIM2 = thm "lemma_skolemize_CLIM2";
 val lemma_csimp = thm "lemma_csimp";
diff -r c6eecde058e4 -r 5798fbf42a6a src/HOL/Complex/CStar.thy
--- a/src/HOL/Complex/CStar.thy	Wed Sep 07 00:48:50 2005 +0200
+++ b/src/HOL/Complex/CStar.thy	Wed Sep 07 01:49:49 2005 +0200
@@ -14,11 +14,11 @@
 
     (* nonstandard extension of sets *)
     starsetC :: "complex set => hcomplex set"          ("*sc* _" [80] 80)
-    "*sc* A  == {x. \<forall>X \<in> Rep_hcomplex(x). {n. X n \<in> A} \<in> FreeUltrafilterNat}"
+    "*sc* A  == {x. \<forall>X \<in> Rep_star(x). {n. X n \<in> A} \<in> FreeUltrafilterNat}"
 
     (* internal sets *)
     starsetC_n :: "(nat => complex set) => hcomplex set"  ("*scn* _" [80] 80)
-    "*scn* As  == {x. \<forall>X \<in> Rep_hcomplex(x). 
+    "*scn* As  == {x. \<forall>X \<in> Rep_star(x). 
                       {n. X n \<in> (As n)} \<in> FreeUltrafilterNat}"
 
     InternalCSets :: "hcomplex set set"
@@ -29,12 +29,12 @@
     starfunC :: "(complex => complex) => hcomplex => hcomplex"
                 ("*fc* _" [80] 80)
     "*fc* f  == 
-        (%x. Abs_hcomplex(\<Union>X \<in> Rep_hcomplex(x). hcomplexrel``{%n. f (X n)}))"
+        (%x. Abs_star(\<Union>X \<in> Rep_star(x). starrel``{%n. f (X n)}))"
 
     starfunC_n :: "(nat => (complex => complex)) => hcomplex => hcomplex"
                   ("*fcn* _" [80] 80)
     "*fcn* F  == 
-      (%x. Abs_hcomplex(\<Union>X \<in> Rep_hcomplex(x). hcomplexrel``{%n. (F n)(X n)}))"
+      (%x. Abs_star(\<Union>X \<in> Rep_star(x). starrel``{%n. (F n)(X n)}))"
 
     InternalCFuns :: "(hcomplex => hcomplex) set"
     "InternalCFuns == {X. \<exists>F. X = *fcn* F}"
@@ -44,11 +44,11 @@
 
     starfunRC :: "(real => complex) => hypreal => hcomplex"
                  ("*fRc* _" [80] 80)
-    "*fRc* f  == (%x. Abs_hcomplex(\<Union>X \<in> Rep_star(x). hcomplexrel``{%n. f (X n)}))"
+    "*fRc* f  == (%x. Abs_star(\<Union>X \<in> Rep_star(x). starrel``{%n. f (X n)}))"
 
     starfunRC_n :: "(nat => (real => complex)) => hypreal => hcomplex"
                    ("*fRcn* _" [80] 80)
-    "*fRcn* F  == (%x. Abs_hcomplex(\<Union>X \<in> Rep_star(x). hcomplexrel``{%n. (F n)(X n)}))"
+    "*fRcn* F  == (%x. Abs_star(\<Union>X \<in> Rep_star(x). starrel``{%n. (F n)(X n)}))"
 
     InternalRCFuns :: "(hypreal => hcomplex) set"
     "InternalRCFuns == {X. \<exists>F. X = *fRcn* F}"
@@ -57,11 +57,11 @@
 
     starfunCR :: "(complex => real) => hcomplex => hypreal"
                  ("*fcR* _" [80] 80)
-    "*fcR* f  == (%x. Abs_star(\<Union>X \<in> Rep_hcomplex(x). starrel``{%n. f (X n)}))"
+    "*fcR* f  == (%x. Abs_star(\<Union>X \<in> Rep_star(x). starrel``{%n. f (X n)}))"
 
     starfunCR_n :: "(nat => (complex => real)) => hcomplex => hypreal"
                    ("*fcRn* _" [80] 80)
-    "*fcRn* F  == (%x. Abs_star(\<Union>X \<in> Rep_hcomplex(x). starrel``{%n. (F n)(X n)}))"
+    "*fcRn* F  == (%x. Abs_star(\<Union>X \<in> Rep_star(x). starrel``{%n. (F n)(X n)}))"
 
     InternalCRFuns :: "(hcomplex => hypreal) set"
     "InternalCRFuns == {X. \<exists>F. X = *fcRn* F}"
@@ -80,14 +80,14 @@
 lemma STARC_Un: "*sc* (A Un B) = *sc* A Un *sc* B"
 apply (auto simp add: starsetC_def)
 apply (drule bspec, assumption)
-apply (rule_tac z = x in eq_Abs_hcomplex, simp, ultra)
+apply (rule_tac z = x in eq_Abs_star, simp, ultra)
 apply (blast intro: FreeUltrafilterNat_subset)+
 done
 
 lemma starsetC_n_Un: "*scn* (%n. (A n) Un (B n)) = *scn* A Un *scn* B"
 apply (auto simp add: starsetC_n_def)
 apply (drule_tac x = Xa in bspec)
-apply (rule_tac [2] z = x in eq_Abs_hcomplex)
+apply (rule_tac [2] z = x in eq_Abs_star)
 apply (auto dest!: bspec, ultra+)
 done
 
@@ -112,15 +112,15 @@
 
 lemma STARC_Compl: "*sc* -A = -( *sc* A)"
 apply (auto simp add: starsetC_def)
-apply (rule_tac z = x in eq_Abs_hcomplex)
-apply (rule_tac [2] z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
+apply (rule_tac [2] z = x in eq_Abs_star)
 apply (auto dest!: bspec, ultra+)
 done
 
 lemma starsetC_n_Compl: "*scn* ((%n. - A n)) = -( *scn* A)"
 apply (auto simp add: starsetC_n_def)
-apply (rule_tac z = x in eq_Abs_hcomplex)
-apply (rule_tac [2] z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
+apply (rule_tac [2] z = x in eq_Abs_star)
 apply (auto dest!: bspec, ultra+)
 done
 
@@ -136,8 +136,8 @@
 lemma starsetC_n_diff:
       "*scn* (%n. (A n) - (B n)) = *scn* A - *scn* B"
 apply (auto simp add: starsetC_n_def)
-apply (rule_tac [2] z = x in eq_Abs_hcomplex)
-apply (rule_tac [3] z = x in eq_Abs_hcomplex)
+apply (rule_tac [2] z = x in eq_Abs_star)
+apply (rule_tac [3] z = x in eq_Abs_star)
 apply (auto dest!: bspec, ultra+)
 done
 
@@ -151,13 +151,13 @@
 done
 
 lemma STARC_mem: "a \<in> A ==> hcomplex_of_complex a \<in> *sc* A"
-apply (simp add: starsetC_def hcomplex_of_complex_def)
+apply (simp add: starsetC_def hcomplex_of_complex_def star_of_def star_n_def)
 apply (auto intro: FreeUltrafilterNat_subset)
 done
 
 lemma STARC_hcomplex_of_complex_image_subset:
     "hcomplex_of_complex ` A \<le> *sc* A"
-apply (auto simp add: starsetC_def hcomplex_of_complex_def)
+apply (auto simp add: starsetC_def hcomplex_of_complex_def star_of_def star_n_def)
 apply (blast intro: FreeUltrafilterNat_subset)
 done
 
@@ -166,11 +166,11 @@
 
 lemma STARC_hcomplex_of_complex_Int:
      "*sc* X Int SComplex = hcomplex_of_complex ` X"
-apply (auto simp add: starsetC_def hcomplex_of_complex_def SComplex_def)
-apply (fold hcomplex_of_complex_def)
+apply (auto simp add: starsetC_def hcomplex_of_complex_def SComplex_def star_of_def star_n_def)
+apply (fold star_n_def star_of_def hcomplex_of_complex_def)
 apply (rule imageI, rule ccontr)
 apply (drule bspec)
-apply (rule lemma_hcomplexrel_refl)
+apply (rule lemma_starrel_refl)
 prefer 2 apply (blast intro: FreeUltrafilterNat_subset, auto)
 done
 
@@ -207,44 +207,44 @@
 by (simp add: starfunCR_n_def starfunCR_def)
 
 lemma starfunC_congruent:
-      "(%X. hcomplexrel``{%n. f (X n)}) respects hcomplexrel"
-by (auto simp add: hcomplexrel_iff congruent_def, ultra)
+      "(%X. starrel``{%n. f (X n)}) respects starrel"
+by (auto simp add: starrel_iff congruent_def, ultra)
 
 (* f::complex => complex *)
 lemma starfunC:
-      "( *fc* f) (Abs_hcomplex(hcomplexrel``{%n. X n})) =
-       Abs_hcomplex(hcomplexrel `` {%n. f (X n)})"
+      "( *fc* f) (Abs_star(starrel``{%n. X n})) =
+       Abs_star(starrel `` {%n. f (X n)})"
 apply (simp add: starfunC_def)
-apply (rule arg_cong [where f = Abs_hcomplex])
-apply (auto iff add: hcomplexrel_iff, ultra)
+apply (rule arg_cong [where f = Abs_star])
+apply (auto iff add: starrel_iff, ultra)
 done
 
 lemma cstarfun_if_eq:
      "w \<noteq> hcomplex_of_complex x
        ==> ( *fc* (\<lambda>z. if z = x then a else g z)) w = ( *fc* g) w" 
-apply (cases w) 
-apply (simp add: hcomplex_of_complex_def starfunC, ultra)
+apply (rule_tac z=w in eq_Abs_star)
+apply (simp add: hcomplex_of_complex_def star_of_def star_n_def starfunC, ultra)
 done
 
 lemma starfunRC:
       "( *fRc* f) (Abs_star(starrel``{%n. X n})) =
-       Abs_hcomplex(hcomplexrel `` {%n. f (X n)})"
+       Abs_star(starrel `` {%n. f (X n)})"
 apply (simp add: starfunRC_def)
-apply (rule arg_cong [where f = Abs_hcomplex], auto, ultra)
+apply (rule arg_cong [where f = Abs_star], auto, ultra)
 done
 
 lemma starfunCR:
-      "( *fcR* f) (Abs_hcomplex(hcomplexrel``{%n. X n})) =
+      "( *fcR* f) (Abs_star(starrel``{%n. X n})) =
        Abs_star(starrel `` {%n. f (X n)})"
 apply (simp add: starfunCR_def)
 apply (rule arg_cong [where f = Abs_star])
-apply (auto iff add: hcomplexrel_iff, ultra)
+apply (auto iff add: starrel_iff, ultra)
 done
 
 (**  multiplication: ( *f) x ( *g) = *(f x g) **)
 
 lemma starfunC_mult: "( *fc* f) z * ( *fc* g) z = ( *fc* (%x. f x * g x)) z"
-apply (rule_tac z = z in eq_Abs_hcomplex)
+apply (rule_tac z = z in eq_Abs_star)
 apply (auto simp add: starfunC hcomplex_mult)
 done
 declare starfunC_mult [symmetric, simp]
@@ -258,7 +258,7 @@
 
 lemma starfunCR_mult:
     "( *fcR* f) z * ( *fcR* g) z = ( *fcR* (%x. f x * g x)) z"
-apply (rule_tac z = z in eq_Abs_hcomplex)
+apply (rule_tac z = z in eq_Abs_star)
 apply (simp add: starfunCR hypreal_mult)
 done
 declare starfunCR_mult [symmetric, simp]
@@ -266,7 +266,7 @@
 (**  addition: ( *f) + ( *g) = *(f + g)  **)
 
 lemma starfunC_add: "( *fc* f) z + ( *fc* g) z = ( *fc* (%x. f x + g x)) z"
-apply (rule_tac z = z in eq_Abs_hcomplex)
+apply (rule_tac z = z in eq_Abs_star)
 apply (simp add: starfunC hcomplex_add)
 done
 declare starfunC_add [symmetric, simp]
@@ -278,14 +278,14 @@
 declare starfunRC_add [symmetric, simp]
 
 lemma starfunCR_add: "( *fcR* f) z + ( *fcR* g) z = ( *fcR* (%x. f x + g x)) z"
-apply (rule_tac z = z in eq_Abs_hcomplex)
+apply (rule_tac z = z in eq_Abs_star)
 apply (simp add: starfunCR hypreal_add)
 done
 declare starfunCR_add [symmetric, simp]
 
 (**  uminus **)
 lemma starfunC_minus [simp]: "( *fc* (%x. - f x)) x = - ( *fc* f) x"
-apply (rule_tac z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
 apply (simp add: starfunC hcomplex_minus)
 done
 
@@ -295,7 +295,7 @@
 done
 
 lemma starfunCR_minus [simp]: "( *fcR* (%x. - f x)) x = - ( *fcR* f) x"
-apply (rule_tac z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
 apply (simp add: starfunCR hypreal_minus)
 done
 
@@ -319,7 +319,7 @@
 
 lemma starfunC_o2: "(%x. ( *fc* f) (( *fc* g) x)) = *fc* (%x. f (g x))"
 apply (rule ext)
-apply (rule_tac z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
 apply (simp add: starfunC)
 done
 
@@ -336,7 +336,7 @@
 lemma starfun_starfunCR_o2:
     "(%x. ( *f* f) (( *fcR* g) x)) = *fcR* (%x. f (g x))"
 apply (rule ext)
-apply (rule_tac z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
 apply (simp add: starfunCR starfun)
 done
 
@@ -347,22 +347,22 @@
 by (simp add: o_def starfun_starfunCR_o2)
 
 lemma starfunC_const_fun [simp]: "( *fc* (%x. k)) z = hcomplex_of_complex k"
-apply (cases z)
-apply (simp add: starfunC hcomplex_of_complex_def)
+apply (rule_tac z=z in eq_Abs_star)
+apply (simp add: starfunC hcomplex_of_complex_def star_of_def star_n_def)
 done
 
 lemma starfunRC_const_fun [simp]: "( *fRc* (%x. k)) z = hcomplex_of_complex k"
 apply (rule_tac z=z in eq_Abs_star)
-apply (simp add: starfunRC hcomplex_of_complex_def)
+apply (simp add: starfunRC hcomplex_of_complex_def star_of_def star_n_def)
 done
 
 lemma starfunCR_const_fun [simp]: "( *fcR* (%x. k)) z = hypreal_of_real k"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
 apply (simp add: starfunCR hypreal_of_real_def star_of_def star_n_def)
 done
 
 lemma starfunC_inverse: "inverse (( *fc* f) x) = ( *fc* (%x. inverse (f x))) x"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
 apply (simp add: starfunC hcomplex_inverse)
 done
 declare starfunC_inverse [symmetric, simp]
@@ -376,14 +376,14 @@
 
 lemma starfunCR_inverse:
     "inverse (( *fcR* f) x) = ( *fcR* (%x. inverse (f x))) x"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
 apply (simp add: starfunCR hypreal_inverse)
 done
 declare starfunCR_inverse [symmetric, simp]
 
 lemma starfunC_eq [simp]:
     "( *fc* f) (hcomplex_of_complex a) = hcomplex_of_complex (f a)"
-by (simp add: starfunC hcomplex_of_complex_def)
+by (simp add: starfunC hcomplex_of_complex_def star_of_def star_n_def)
 
 lemma starfunRC_eq [simp]:
     "( *fRc* f) (hypreal_of_real a) = hcomplex_of_complex (f a)"
@@ -410,20 +410,20 @@
 *)
 
 lemma starfunC_hcpow: "( *fc* (%z. z ^ n)) Z = Z hcpow hypnat_of_nat n"
-apply (cases Z)
+apply (rule_tac z=Z in eq_Abs_star)
 apply (simp add: hcpow starfunC hypnat_of_nat_eq)
 done
 
 lemma starfunC_lambda_cancel:
     "( *fc* (%h. f (x + h))) y  = ( *fc* f) (hcomplex_of_complex  x + y)"
-apply (cases y)
-apply (simp add: starfunC hcomplex_of_complex_def hcomplex_add)
+apply (rule_tac z=y in eq_Abs_star)
+apply (simp add: starfunC hcomplex_of_complex_def hcomplex_add star_of_def star_n_def)
 done
 
 lemma starfunCR_lambda_cancel:
     "( *fcR* (%h. f (x + h))) y  = ( *fcR* f) (hcomplex_of_complex  x + y)"
-apply (cases y)
-apply (simp add: starfunCR hcomplex_of_complex_def hcomplex_add)
+apply (rule_tac z=y in eq_Abs_star)
+apply (simp add: starfunCR hcomplex_of_complex_def hcomplex_add star_of_def star_n_def)
 done
 
 lemma starfunRC_lambda_cancel:
@@ -434,14 +434,14 @@
 
 lemma starfunC_lambda_cancel2:
     "( *fc* (%h. f(g(x + h)))) y = ( *fc* (f o g)) (hcomplex_of_complex x + y)"
-apply (cases y)
-apply (simp add: starfunC hcomplex_of_complex_def hcomplex_add)
+apply (rule_tac z=y in eq_Abs_star)
+apply (simp add: starfunC hcomplex_of_complex_def hcomplex_add star_of_def star_n_def)
 done
 
 lemma starfunCR_lambda_cancel2:
     "( *fcR* (%h. f(g(x + h)))) y = ( *fcR* (f o g)) (hcomplex_of_complex x + y)"
-apply (cases y)
-apply (simp add: starfunCR hcomplex_of_complex_def hcomplex_add)
+apply (rule_tac z=y in eq_Abs_star)
+apply (simp add: starfunCR hcomplex_of_complex_def hcomplex_add star_of_def star_n_def)
 done
 
 lemma starfunRC_lambda_cancel2:
@@ -488,12 +488,12 @@
 
 lemma starfunCR_cmod: "*fcR* cmod = hcmod"
 apply (rule ext)
-apply (rule_tac z = x in eq_Abs_hcomplex)
+apply (rule_tac z = x in eq_Abs_star)
 apply (simp add: starfunCR hcmod)
 done
 
 lemma starfunC_inverse_inverse: "( *fc* inverse) x = inverse(x)"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
 apply (simp add: starfunC hcomplex_inverse)
 done
 
@@ -515,22 +515,22 @@
 subsection{*Internal Functions - Some Redundancy With *Fc* Now*}
 
 lemma starfunC_n_congruent:
-      "(%X. hcomplexrel``{%n. f n (X n)}) respects hcomplexrel"
-by (auto simp add: congruent_def hcomplexrel_iff, ultra)
+      "(%X. starrel``{%n. f n (X n)}) respects starrel"
+by (auto simp add: congruent_def starrel_iff, ultra)
 
 lemma starfunC_n:
-     "( *fcn* f) (Abs_hcomplex(hcomplexrel``{%n. X n})) =
-      Abs_hcomplex(hcomplexrel `` {%n. f n (X n)})"
+     "( *fcn* f) (Abs_star(starrel``{%n. X n})) =
+      Abs_star(starrel `` {%n. f n (X n)})"
 apply (simp add: starfunC_n_def)
-apply (rule arg_cong [where f = Abs_hcomplex])
-apply (auto iff add: hcomplexrel_iff, ultra)
+apply (rule arg_cong [where f = Abs_star])
+apply (auto iff add: starrel_iff, ultra)
 done
 
 (**  multiplication: ( *fn) x ( *gn) = *(fn x gn) **)
 
 lemma starfunC_n_mult:
     "( *fcn* f) z * ( *fcn* g) z = ( *fcn* (% i x. f i x * g i x)) z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
 apply (simp add: starfunC_n hcomplex_mult)
 done
 
@@ -538,14 +538,14 @@
 
 lemma starfunC_n_add:
     "( *fcn* f) z + ( *fcn* g) z = ( *fcn* (%i x. f i x + g i x)) z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
 apply (simp add: starfunC_n hcomplex_add)
 done
 
 (** uminus **)
 
 lemma starfunC_n_minus: "- ( *fcn* g) z = ( *fcn* (%i x. - g i x)) z"
-apply (cases z)
+apply (rule_tac z=z in eq_Abs_star)
 apply (simp add: starfunC_n hcomplex_minus)
 done
 
@@ -559,19 +559,19 @@
 
 lemma starfunC_n_const_fun [simp]:
      "( *fcn* (%i x. k)) z = hcomplex_of_complex  k"
-apply (cases z)
-apply (simp add: starfunC_n hcomplex_of_complex_def)
+apply (rule_tac z=z in eq_Abs_star)
+apply (simp add: starfunC_n hcomplex_of_complex_def star_of_def star_n_def)
 done
 
 lemma starfunC_n_eq [simp]:
-    "( *fcn* f) (hcomplex_of_complex n) = Abs_hcomplex(hcomplexrel `` {%i. f i n})"
-by (simp add: starfunC_n hcomplex_of_complex_def)
+    "( *fcn* f) (hcomplex_of_complex n) = Abs_star(starrel `` {%i. f i n})"
+by (simp add: starfunC_n hcomplex_of_complex_def star_of_def star_n_def)
 
 lemma starfunC_eq_iff: "(( *fc* f) = ( *fc* g)) = (f = g)"
 apply auto
 apply (rule ext, rule ccontr)
 apply (drule_tac x = "hcomplex_of_complex (x) " in fun_cong)
-apply (simp add: starfunC hcomplex_of_complex_def)
+apply (simp add: starfunC hcomplex_of_complex_def star_of_def star_n_def)
 done
 
 lemma starfunRC_eq_iff: "(( *fRc* f) = ( *fRc* g)) = (f = g)"
@@ -591,25 +591,25 @@
 lemma starfunC_eq_Re_Im_iff:
     "(( *fc* f) x = z) = ((( *fcR* (%x. Re(f x))) x = hRe (z)) &
                           (( *fcR* (%x. Im(f x))) x = hIm (z)))"
-apply (cases x, cases z)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=z in eq_Abs_star)
 apply (auto simp add: starfunCR starfunC hIm hRe complex_Re_Im_cancel_iff, ultra+)
 done
 
 lemma starfunC_approx_Re_Im_iff:
     "(( *fc* f) x @c= z) = ((( *fcR* (%x. Re(f x))) x @= hRe (z)) &
                             (( *fcR* (%x. Im(f x))) x @= hIm (z)))"
-apply (cases x, cases z)
+apply (rule_tac z=x in eq_Abs_star, rule_tac z=z in eq_Abs_star)
 apply (simp add: starfunCR starfunC hIm hRe capprox_approx_iff)
 done
 
 lemma starfunC_Idfun_capprox:
     "x @c= hcomplex_of_complex a ==> ( *fc* (%x. x)) x @c= hcomplex_of_complex  a"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
 apply (simp add: starfunC)
 done
 
 lemma starfunC_Id [simp]: "( *fc* (%x. x)) x = x"
-apply (cases x)
+apply (rule_tac z=x in eq_Abs_star)
 apply (simp add: starfunC)
 done
 
diff -r c6eecde058e4 -r 5798fbf42a6a src/HOL/Complex/NSCA.thy
--- a/src/HOL/Complex/NSCA.thy	Wed Sep 07 00:48:50 2005 +0200
+++ b/src/HOL/Complex/NSCA.thy	Wed Sep 07 01:49:49 2005 +0200
@@ -134,7 +134,7 @@
 lemma SComplex_hcmod_SReal: 
       "z \<in> SComplex ==> hcmod z \<in> Reals"
 apply (simp add: SComplex_def SReal_def)
-apply (rule_tac z = z in eq_Abs_hcomplex)
+apply (rule_tac z = z in eq_Abs_star)
 apply (auto simp add: hcmod hypreal_of_real_def star_of_def star_n_def hcomplex_of_complex_def cmod_def)
 apply (rule_tac x = "cmod r" in exI)
 apply (simp add: cmod_def, ultra)
@@ -144,7 +144,7 @@
 by (simp add: SComplex_def hcomplex_of_complex_zero_iff)
 
 lemma SComplex_one [simp]: "1 \<in> SComplex"
-by (simp add: SComplex_def hcomplex_of_complex_def hcomplex_one_def)
+by (simp add: SComplex_def hcomplex_of_complex_def star_of_def star_n_def hypreal_one_def)
 
 (*
 Goalw [SComplex_def,SReal_def] "hcmod z \<in> Reals ==> z \<in> SComplex"
@@ -659,7 +659,7 @@
 by (blast intro: SComplex_capprox_iff [THEN iffD1] capprox_trans2)
 
 lemma hcomplex_capproxD1:
-     "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})  
+     "Abs_star(starrel ``{%n. X n}) @c= Abs_star(starrel``{%n. Y n})  
       ==> Abs_star(starrel `` {%n. Re(X n)}) @=  
           Abs_star(starrel `` {%n. Re(Y n)})"
 apply (auto simp add: approx_FreeUltrafilterNat_iff)
@@ -678,7 +678,7 @@
 
 (* same proof *)
 lemma hcomplex_capproxD2:
-     "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})  
+     "Abs_star(starrel ``{%n. X n}) @c= Abs_star(starrel``{%n. Y n})  
       ==> Abs_star(starrel `` {%n. Im(X n)}) @=  
           Abs_star(starrel `` {%n. Im(Y n)})"
 apply (auto simp add: approx_FreeUltrafilterNat_iff)
@@ -699,7 +699,7 @@
          Abs_star(starrel `` {%n. Re(Y n)});  
          Abs_star(starrel `` {%n. Im(X n)}) @=  
          Abs_star(starrel `` {%n. Im(Y n)})  
-      |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})"
+      |] ==> Abs_star(starrel ``{%n. X n}) @c= Abs_star(starrel``{%n. Y n})"
 apply (drule approx_minus_iff [THEN iffD1])
 apply (drule approx_minus_iff [THEN iffD1])
 apply (rule capprox_minus_iff [THEN iffD2])
@@ -718,7 +718,7 @@
 done
 
 lemma capprox_approx_iff:
-     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})) = 
+     "(Abs_star(starrel ``{%n. X n}) @c= Abs_star(starrel``{%n. Y n})) = 
        (Abs_star(starrel `` {%n. Re(X n)}) @= Abs_star(starrel `` {%n. Re(Y n)}) &  
         Abs_star(starrel `` {%n. Im(X n)}) @= Abs_star(starrel `` {%n. Im(Y n)}))"
 apply (blast intro: hcomplex_capproxI hcomplex_capproxD1 hcomplex_capproxD2)
@@ -731,7 +731,7 @@
 done
 
 lemma CFinite_HFinite_Re:
-     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite  
+     "Abs_star(starrel ``{%n. X n}) \<in> CFinite  
       ==> Abs_star(starrel `` {%n. Re(X n)}) \<in> HFinite"
 apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff)
 apply (rule bexI, rule_tac [2] lemma_starrel_refl)
@@ -745,7 +745,7 @@
 done
 
 lemma CFinite_HFinite_Im:
-     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite  
+     "Abs_star(starrel ``{%n. X n}) \<in> CFinite  
       ==> Abs_star(starrel `` {%n. Im(X n)}) \<in> HFinite"
 apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff)
 apply (rule bexI, rule_tac [2] lemma_starrel_refl)
@@ -760,7 +760,7 @@
 lemma HFinite_Re_Im_CFinite:
      "[| Abs_star(starrel `` {%n. Re(X n)}) \<in> HFinite;  
          Abs_star(starrel `` {%n. Im(X n)}) \<in> HFinite  
-      |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite"
+      |] ==> Abs_star(starrel ``{%n. X n}) \<in> CFinite"
 apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff)
 apply (rename_tac Y Z u v)
 apply (rule bexI, rule_tac [2] lemma_starrel_refl)
@@ -778,20 +778,20 @@
 done
 
 lemma CFinite_HFinite_iff:
-     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite) =  
+     "(Abs_star(starrel ``{%n. X n}) \<in> CFinite) =  
       (Abs_star(starrel `` {%n. Re(X n)}) \<in> HFinite &  
        Abs_star(starrel `` {%n. Im(X n)}) \<in> HFinite)"
 by (blast intro: HFinite_Re_Im_CFinite CFinite_HFinite_Im CFinite_HFinite_Re)
 
 lemma SComplex_Re_SReal:
-     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex  
+     "Abs_star(starrel ``{%n. X n}) \<in> SComplex  
       ==> Abs_star(starrel `` {%n. Re(X n)}) \<in> Reals"
 apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def star_of_def star_n_def)
 apply (rule_tac x = "Re r" in exI, ultra)
 done
 
 lemma SComplex_Im_SReal:
-     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex  
+     "Abs_star(starrel ``{%n. X n}) \<in> SComplex  
       ==> Abs_star(starrel `` {%n. Im(X n)}) \<in> Reals"
 apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def star_of_def star_n_def)
 apply (rule_tac x = "Im r" in exI, ultra)
@@ -800,40 +800,40 @@
 lemma Reals_Re_Im_SComplex:
      "[| Abs_star(starrel `` {%n. Re(X n)}) \<in> Reals;  
          Abs_star(starrel `` {%n. Im(X n)}) \<in> Reals  
-      |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex"
+      |] ==> Abs_star(starrel ``{%n. X n}) \<in> SComplex"
 apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def star_of_def star_n_def)
 apply (rule_tac x = "Complex r ra" in exI, ultra)
 done
 
 lemma SComplex_SReal_iff:
-     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex) =  
+     "(Abs_star(starrel ``{%n. X n}) \<in> SComplex) =  
       (Abs_star(starrel `` {%n. Re(X n)}) \<in> Reals &  
        Abs_star(starrel `` {%n. Im(X n)}) \<in> Reals)"
 by (blast intro: SComplex_Re_SReal SComplex_Im_SReal Reals_Re_Im_SComplex)
 
 lemma CInfinitesimal_Infinitesimal_iff:
-     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CInfinitesimal) =  
+     "(Abs_star(starrel ``{%n. X n}) \<in> CInfinitesimal) =  
       (Abs_star(starrel `` {%n. Re(X n)}) \<in> Infinitesimal &  
        Abs_star(starrel `` {%n. Im(X n)}) \<in> Infinitesimal)"
 by (simp add: mem_cinfmal_iff mem_infmal_iff hcomplex_zero_num hypreal_zero_num capprox_approx_iff)
 
-lemma eq_Abs_hcomplex_EX:
-     "(\<exists>t. P t) = (\<exists>X. P (Abs_hcomplex(hcomplexrel `` {X})))"
+lemma eq_Abs_star_EX:
+     "(\<exists>t. P t) = (\<exists>X. P (Abs_star(starrel `` {X})))"
 apply auto
-apply (rule_tac z = t in eq_Abs_hcomplex, auto)
+apply (rule_tac z = t in eq_Abs_star, auto)
 done
 
-lemma eq_Abs_hcomplex_Bex:
-     "(\<exists>t \<in> A. P t) = (\<exists>X. (Abs_hcomplex(hcomplexrel `` {X})) \<in> A &  
-                         P (Abs_hcomplex(hcomplexrel `` {X})))"
+lemma eq_Abs_star_Bex:
+     "(\<exists>t \<in> A. P t) = (\<exists>X. (Abs_star(starrel `` {X})) \<in> A &  
+                         P (Abs_star(starrel `` {X})))"
 apply auto
-apply (rule_tac z = t in eq_Abs_hcomplex, auto)
+apply (rule_tac z = t in eq_Abs_star, auto)
 done
 
 (* Here we go - easy proof now!! *)
 lemma stc_part_Ex: "x:CFinite ==> \<exists>t \<in> SComplex. x @c= t"
-apply (rule_tac z = x in eq_Abs_hcomplex)
-apply (auto simp add: CFinite_HFinite_iff eq_Abs_hcomplex_Bex SComplex_SReal_iff capprox_approx_iff)
+apply (rule_tac z = x in eq_Abs_star)
+apply (auto simp add: CFinite_HFinite_iff eq_Abs_star_Bex SComplex_SReal_iff capprox_approx_iff)
 apply (drule st_part_Ex, safe)+
 apply (rule_tac z = t in eq_Abs_star)
 apply (rule_tac z = ta in eq_Abs_star, auto)
@@ -1171,7 +1171,7 @@
 by (simp add: CInfinitesimal_hcmod_iff)
 
 lemma CInfinite_HInfinite_iff:
-     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CInfinite) =  
+     "(Abs_star(starrel ``{%n. X n}) \<in> CInfinite) =  
       (Abs_star(starrel `` {%n. Re(X n)}) \<in> HInfinite |  
        Abs_star(starrel `` {%n. Im(X n)}) \<in> HInfinite)"
 by (simp add: CInfinite_CFinite_iff HInfinite_HFinite_iff CFinite_HFinite_iff)
@@ -1216,8 +1216,8 @@
 
 lemma complex_seq_to_hcomplex_CInfinitesimal:
      "\<forall>n. cmod (X n - x) < inverse (real (Suc n)) ==>  
-      Abs_hcomplex(hcomplexrel``{X}) - hcomplex_of_complex x \<in> CInfinitesimal"
-apply (simp add: hcomplex_diff CInfinitesimal_hcmod_iff hcomplex_of_complex_def Infinitesimal_FreeUltrafilterNat_iff hcmod)
+      Abs_star(starrel``{X}) - hcomplex_of_complex x \<in> CInfinitesimal"
+apply (simp add: hcomplex_diff CInfinitesimal_hcmod_iff hcomplex_of_complex_def star_of_def star_n_def Infinitesimal_FreeUltrafilterNat_iff hcmod)
 apply (rule bexI, auto)
 apply (auto dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset)
 done
@@ -1384,7 +1384,7 @@
 val Reals_Re_Im_SComplex = thm "Reals_Re_Im_SComplex";
 val SComplex_SReal_iff = thm "SComplex_SReal_iff";
 val CInfinitesimal_Infinitesimal_iff = thm "CInfinitesimal_Infinitesimal_iff";
-val eq_Abs_hcomplex_Bex = thm "eq_Abs_hcomplex_Bex";
+val eq_Abs_star_Bex = thm "eq_Abs_star_Bex";
 val stc_part_Ex = thm "stc_part_Ex";
 val stc_part_Ex1 = thm "stc_part_Ex1";
 val CFinite_Int_CInfinite_empty = thm "CFinite_Int_CInfinite_empty";
diff -r c6eecde058e4 -r 5798fbf42a6a src/HOL/Complex/NSComplex.thy
--- 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";