tidying of the complex numbers

--- a/src/HOL/Complex/CLim.ML	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/CLim.ML	Tue Feb 03 11:06:36 2004 +0100
@@ -284,7 +284,7 @@
 (*** NSCLIM_zero, CLIM_zero, etc. ***)
 
 Goal "f -- a --NSC> l ==> (%x. f(x) - l) -- a --NSC> 0";
-by (res_inst_tac [("z1","l")] (complex_add_minus_right_zero RS subst) 1);
+by (res_inst_tac [("a1","l")] (right_minus RS subst) 1);
 by (rewtac complex_diff_def);
 by (rtac NSCLIM_add 1 THEN Auto_tac);
 qed "NSCLIM_zero";
@@ -701,7 +701,7 @@
 by (Step_tac 1);
 by (dres_inst_tac [("x","x - a")] spec 1);
 by (dres_inst_tac [("x","x + a")] spec 2);
-by (auto_tac (claset(), simpset() addsimps complex_add_ac));
+by (auto_tac (claset(), simpset() addsimps add_ac));
 qed "CDERIV_CLIM_iff";
 
 Goalw [cderiv_def] "(CDERIV f x :> D) = \
@@ -755,10 +755,10 @@
     [mem_cinfmal_iff RS sym,hcomplex_add_commute]));
 by (dres_inst_tac [("c","xa + - hcomplex_of_complex x")] capprox_mult1 1);
 by (auto_tac (claset() addIs [CInfinitesimal_subset_CFinite
-    RS subsetD],simpset() addsimps [hcomplex_mult_assoc]));
+    RS subsetD],simpset() addsimps [mult_assoc]));
 by (dres_inst_tac [("x3","D")] (CFinite_hcomplex_of_complex RSN
     (2,CInfinitesimal_CFinite_mult) RS (mem_cinfmal_iff RS iffD1)) 1);
-by (blast_tac (claset() addIs [capprox_trans,hcomplex_mult_commute RS subst,
+by (blast_tac (claset() addIs [capprox_trans,mult_commute RS subst,
     (capprox_minus_iff RS iffD2)]) 1);
 qed "NSCDERIV_isNSContc";
 
@@ -833,10 +833,9 @@
 by (dres_inst_tac [("D","Db")] lemma_nscderiv2 1);
 by (dtac (capprox_minus_iff RS iffD2 RS (bex_CInfinitesimal_iff2 RS iffD2)) 4);
 by (auto_tac (claset() addSIs [capprox_add_mono1],
-      simpset() addsimps [hcomplex_add_mult_distrib, right_distrib, 
-			  hcomplex_mult_commute, hcomplex_add_assoc]));
-by (res_inst_tac [("w1","hcomplex_of_complex Db * hcomplex_of_complex (f x)")]
-    (hcomplex_add_commute RS subst) 1);
+     simpset() addsimps [left_distrib, right_distrib, mult_commute, add_assoc]));
+by (res_inst_tac [("b1","hcomplex_of_complex Db * hcomplex_of_complex (f x)")]
+    (add_commute RS subst) 1);
 by (auto_tac (claset() addSIs [CInfinitesimal_add_capprox_self2 RS capprox_sym,
 			       CInfinitesimal_add, CInfinitesimal_mult,
 			       CInfinitesimal_hcomplex_of_complex_mult,
@@ -853,7 +852,7 @@
 Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. c * f x) x :> c*D";
 by (asm_full_simp_tac 
     (simpset() addsimps [times_divide_eq_right RS sym, NSCDERIV_NSCLIM_iff,
-                         minus_mult_right, complex_add_mult_distrib2 RS sym,
+                         minus_mult_right, right_distrib RS sym,
                          complex_diff_def] 
              delsimps [times_divide_eq_right, minus_mult_right RS sym]) 1);
 by (etac (NSCLIM_const RS NSCLIM_mult) 1);
@@ -866,9 +865,8 @@
 
 Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. -(f x)) x :> -D";
 by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,complex_diff_def]) 1);
-by (res_inst_tac [("t","f x")] (complex_minus_minus RS subst) 1);
-by (asm_simp_tac (simpset() addsimps [complex_minus_add_distrib RS sym] 
-                   delsimps [complex_minus_add_distrib, complex_minus_minus]
+by (res_inst_tac [("t","f x")] (minus_minus RS subst) 1);
+by (asm_simp_tac (simpset() addsimps [minus_add_distrib RS sym] 
                    delsimps [minus_add_distrib, minus_minus]
 
 ) 1);
@@ -1032,12 +1030,12 @@
 by (dtac (CDERIV_Id RS CDERIV_mult) 2);
 by (auto_tac (claset(), 
               simpset() addsimps [complex_of_real_add RS sym,
-                        complex_add_mult_distrib,real_of_nat_Suc] 
+                        left_distrib,real_of_nat_Suc] 
                  delsimps [complex_of_real_add]));
 by (case_tac "n" 1);
 by (auto_tac (claset(), 
-              simpset() addsimps [complex_mult_assoc, complex_add_commute]));
-by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+              simpset() addsimps [mult_assoc, add_commute]));
+by (auto_tac (claset(),simpset() addsimps [mult_commute]));
 qed "CDERIV_pow";
 Addsimps [CDERIV_pow,simplify (simpset()) CDERIV_pow];
 
@@ -1095,10 +1093,9 @@
 
 Goal "[| CDERIV f x :> d; f(x) ~= 0 |] \
 \     ==> CDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))";
-by (rtac (complex_mult_commute RS subst) 1);
-by (asm_simp_tac (simpset() addsimps [complex_minus_mult_eq1,
-    power_inverse] delsimps [complexpow_Suc, minus_mult_left RS sym,
-complex_minus_mult_eq1 RS sym]) 1);
+by (rtac (mult_commute RS subst) 1);
+by (asm_simp_tac (simpset() addsimps [minus_mult_left,
+    power_inverse] delsimps [complexpow_Suc, minus_mult_left RS sym]) 1);
 by (fold_goals_tac [o_def]);
 by (blast_tac (claset() addSIs [CDERIV_chain,CDERIV_inverse]) 1);
 qed "CDERIV_inverse_fun";
@@ -1126,8 +1123,8 @@
 by (dtac CDERIV_mult 2);
 by (REPEAT(assume_tac 1));
 by (asm_full_simp_tac
-    (simpset() addsimps [complex_divide_def, complex_add_mult_distrib2,
-                         power_inverse,complex_minus_mult_eq1] @ complex_mult_ac 
+    (simpset() addsimps [complex_divide_def, right_distrib,
+                         power_inverse,minus_mult_left] @ mult_ac 
        delsimps [complexpow_Suc, minus_mult_right RS sym, minus_mult_left RS sym]) 1);
 qed "CDERIV_quotient";
 
@@ -1161,7 +1158,7 @@
 by (Step_tac 1);
 by (res_inst_tac 
     [("x","%z. if  z = x then l else (f(z) - f(x)) / (z - x)")] exI 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult_assoc,
+by (auto_tac (claset(),simpset() addsimps [mult_assoc,
     CLAIM "z ~= x ==> z - x ~= (0::complex)"]));
 by (auto_tac (claset(),simpset() addsimps [isContc_iff,CDERIV_iff]));
 by (ALLGOALS(rtac (CLIM_equal RS iffD1)));

--- a/src/HOL/Complex/CSeries.ML	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/CSeries.ML	Tue Feb 03 11:06:36 2004 +0100
@@ -37,7 +37,7 @@
 by (induct_tac "n" 1);
 by (Auto_tac);
 by (auto_tac (claset(),simpset() addsimps 
-    [complex_add_mult_distrib2]));
+    [right_distrib]));
 qed "sumc_mult";
 
 Goal "n < p --> sumc 0 n f + sumc n p f = sumc 0 p f";
@@ -49,7 +49,7 @@
 Goal "n < p ==> sumc 0 p f + \
 \                - sumc 0 n f = sumc n p f";
 by (dres_inst_tac [("f1","f")] (sumc_split_add RS sym) 1);
-by (asm_simp_tac (simpset() addsimps complex_add_ac) 1);
+by (asm_simp_tac (simpset() addsimps add_ac) 1);
 qed "sumc_split_add_minus";
 
 Goal "cmod(sumc m n f) <= sumr m n (%i. cmod(f i))";
@@ -67,7 +67,7 @@
 Goal "sumc 0 n (%i. r) = complex_of_real (real n) * r";
 by (induct_tac "n" 1);
 by (auto_tac (claset(),
-              simpset() addsimps [complex_add_mult_distrib,
+              simpset() addsimps [left_distrib,
                                   complex_of_real_add RS sym,
                                   real_of_nat_Suc]));
 qed "sumc_const";
@@ -109,7 +109,7 @@
 by (induct_tac "na" 1);
 by (auto_tac (claset(),simpset() addsimps [left_distrib, Suc_diff_n,
                                       real_of_nat_Suc,complex_of_real_add RS sym,
-                                      complex_add_mult_distrib]));
+                                      left_distrib]));
 qed_spec_mp "sumc_interval_const";
 
 Goal "(ALL n. m <= n --> f n = r) & m <= na \
@@ -117,7 +117,7 @@
 by (induct_tac "na" 1);
 by (auto_tac (claset(),simpset() addsimps [left_distrib, Suc_diff_n,
                                       real_of_nat_Suc,complex_of_real_add RS sym,
-                                      complex_add_mult_distrib]));
+                                      left_distrib]));
 qed_spec_mp "sumc_interval_const2";
 
 (*** 

--- a/src/HOL/Complex/Complex.thy	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/Complex.thy	Tue Feb 03 11:06:36 2004 +0100
@@ -1,13 +1,15 @@
 (*  Title:       Complex.thy
     Author:      Jacques D. Fleuriot
     Copyright:   2001 University of Edinburgh
-    Description: Complex numbers
 *)
 
+header {* Complex numbers *}
+
 theory Complex = HLog:
 
-typedef complex = "{p::(real*real). True}"
-  by blast
+subsection {* Representation of complex numbers *}
+
+datatype complex = Complex real real
 
 instance complex :: zero ..
 instance complex :: one ..
@@ -18,18 +20,19 @@
 instance complex :: power ..
 
 consts
-  "ii"    :: complex        ("ii")
+  "ii"    :: complex    ("\<i>")
+
+consts Re :: "complex => real"
+primrec "Re (Complex x y) = x"
+
+consts Im :: "complex => real"
+primrec "Im (Complex x y) = y"
+
+lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
+  by (induct z) simp
 
 constdefs
 
-  (*--- real and Imaginary parts ---*)
-
-  Re :: "complex => real"
-  "Re(z) == fst(Rep_complex z)"
-
-  Im :: "complex => real"
-  "Im(z) == snd(Rep_complex z)"
-
   (*----------- modulus ------------*)
 
   cmod :: "complex => real"
@@ -38,12 +41,12 @@
   (*----- injection from reals -----*)
 
   complex_of_real :: "real => complex"
-  "complex_of_real r == Abs_complex(r,0::real)"
+  "complex_of_real r == Complex r 0"
 
   (*------- complex conjugate ------*)
 
   cnj :: "complex => complex"
-  "cnj z == Abs_complex(Re z, -Im z)"
+  "cnj z == Complex (Re z) (-Im z)"
 
   (*------------ Argand -------------*)
 
@@ -57,41 +60,29 @@
 defs (overloaded)
 
   complex_zero_def:
-  "0 == Abs_complex(0::real,0)"
+  "0 == Complex 0 0"
 
   complex_one_def:
-  "1 == Abs_complex(1,0::real)"
-
-  (*------ imaginary unit ----------*)
+  "1 == Complex 1 0"
 
-  i_def:
-  "ii == Abs_complex(0::real,1)"
-
-  (*----------- negation -----------*)
+  i_def: "ii == Complex 0 1"
 
-  complex_minus_def:
-  "- (z::complex) == Abs_complex(-Re z, -Im z)"
-
+  complex_minus_def: "- z == Complex (- Re z) (- Im z)"
 
-  (*----------- inverse -----------*)
   complex_inverse_def:
-  "inverse (z::complex) == Abs_complex(Re(z)/(Re(z) ^ 2 + Im(z) ^ 2),
-                            -Im(z)/(Re(z) ^ 2 + Im(z) ^ 2))"
+   "inverse z ==
+    Complex (Re z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>)) (- Im z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>))"
 
   complex_add_def:
-  "w + (z::complex) == Abs_complex(Re(w) + Re(z),Im(w) + Im(z))"
+    "z + w == Complex (Re z + Re w) (Im z + Im w)"
 
   complex_diff_def:
-  "w - (z::complex) == w + -(z::complex)"
+    "z - w == z + - (w::complex)"
 
-  complex_mult_def:
-  "w * (z::complex) == Abs_complex(Re(w) * Re(z) - Im(w) * Im(z),
-			Re(w) * Im(z) + Im(w) * Re(z))"
+  complex_mult_def: 
+    "z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)"
 
-
-  (*----------- division ----------*)
-  complex_divide_def:
-  "w / (z::complex) == w * inverse z"
+  complex_divide_def: "w / (z::complex) == w * inverse z"
 
 
 constdefs
@@ -109,429 +100,152 @@
   "expi z == complex_of_real(exp (Re z)) * cis (Im z)"
 
 
-lemma inj_Rep_complex: "inj Rep_complex"
-apply (rule inj_on_inverseI)
-apply (rule Rep_complex_inverse)
-done
-
-lemma inj_Abs_complex: "inj Abs_complex"
-apply (rule inj_on_inverseI)
-apply (rule Abs_complex_inverse)
-apply (simp (no_asm) add: complex_def)
-done
-declare inj_Abs_complex [THEN injD, simp]
-
-lemma Abs_complex_cancel_iff: "(Abs_complex x = Abs_complex y) = (x = y)"
-by (auto dest: inj_Abs_complex [THEN injD])
-declare Abs_complex_cancel_iff [simp]
+lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w"
+  by (induct z, induct w) simp
 
-lemma pair_mem_complex: "(x,y) : complex"
-by (unfold complex_def, auto)
-declare pair_mem_complex [simp]
-
-lemma Abs_complex_inverse2: "Rep_complex (Abs_complex (x,y)) = (x,y)"
-apply (simp (no_asm) add: Abs_complex_inverse)
-done
-declare Abs_complex_inverse2 [simp]
-
-lemma eq_Abs_complex: "(!!x y. z = Abs_complex(x,y) ==> P) ==> P"
-apply (rule_tac p = "Rep_complex z" in PairE)
-apply (drule_tac f = Abs_complex in arg_cong)
-apply (force simp add: Rep_complex_inverse)
-done
-
-lemma Re: "Re(Abs_complex(x,y)) = x"
-apply (unfold Re_def)
-apply (simp (no_asm))
-done
+lemma Re: "Re(Complex x y) = x"
+by simp
 declare Re [simp]
 
-lemma Im: "Im(Abs_complex(x,y)) = y"
-apply (unfold Im_def)
-apply (simp (no_asm))
-done
+lemma Im: "Im(Complex x y) = y"
+by simp
 declare Im [simp]
 
-lemma Abs_complex_cancel: "Abs_complex(Re(z),Im(z)) = z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp))
-done
-declare Abs_complex_cancel [simp]
-
 lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))"
-apply (rule_tac z = w in eq_Abs_complex)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto dest: inj_Abs_complex [THEN injD])
-done
+by (induct w, induct z, simp)
 
 lemma complex_Re_zero: "Re 0 = 0"
-apply (unfold complex_zero_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_zero_def)
 
 lemma complex_Im_zero: "Im 0 = 0"
-apply (unfold complex_zero_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_zero_def)
 declare complex_Re_zero [simp] complex_Im_zero [simp]
 
 lemma complex_Re_one: "Re 1 = 1"
-apply (unfold complex_one_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_one_def)
 declare complex_Re_one [simp]
 
 lemma complex_Im_one: "Im 1 = 0"
-apply (unfold complex_one_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_one_def)
 declare complex_Im_one [simp]
 
 lemma complex_Re_i: "Re(ii) = 0"
-by (unfold i_def, auto)
+by (simp add: i_def)
 declare complex_Re_i [simp]
 
 lemma complex_Im_i: "Im(ii) = 1"
-by (unfold i_def, auto)
+by (simp add: i_def)
 declare complex_Im_i [simp]
 
 lemma Re_complex_of_real_zero: "Re(complex_of_real 0) = 0"
-apply (unfold complex_of_real_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_of_real_def)
 declare Re_complex_of_real_zero [simp]
 
 lemma Im_complex_of_real_zero: "Im(complex_of_real 0) = 0"
-apply (unfold complex_of_real_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_of_real_def)
 declare Im_complex_of_real_zero [simp]
 
 lemma Re_complex_of_real_one: "Re(complex_of_real 1) = 1"
-apply (unfold complex_of_real_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_of_real_def)
 declare Re_complex_of_real_one [simp]
 
 lemma Im_complex_of_real_one: "Im(complex_of_real 1) = 0"
-apply (unfold complex_of_real_def)
-apply (simp (no_asm))
-done
+by (simp add: complex_of_real_def)
 declare Im_complex_of_real_one [simp]
 
 lemma Re_complex_of_real: "Re(complex_of_real z) = z"
-by (unfold complex_of_real_def, auto)
+by (simp add: complex_of_real_def)
 declare Re_complex_of_real [simp]
 
 lemma Im_complex_of_real: "Im(complex_of_real z) = 0"
-by (unfold complex_of_real_def, auto)
+by (simp add: complex_of_real_def)
 declare Im_complex_of_real [simp]
 
 
 subsection{*Negation*}
 
-lemma complex_minus: "- Abs_complex(x,y) = Abs_complex(-x,-y)"
-apply (unfold complex_minus_def)
-apply (simp (no_asm))
-done
+lemma complex_minus: "- (Complex x y) = Complex (-x) (-y)"
+by (simp add: complex_minus_def)
 
 lemma complex_Re_minus: "Re (-z) = - Re z"
-apply (unfold Re_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_minus)
-done
+by (simp add: complex_minus_def)
 
 lemma complex_Im_minus: "Im (-z) = - Im z"
-apply (unfold Im_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_minus)
-done
-
-lemma complex_minus_minus: "- (- z) = (z::complex)"
-apply (unfold complex_minus_def)
-apply (simp (no_asm))
-done
-declare complex_minus_minus [simp]
-
-lemma inj_complex_minus: "inj(%r::complex. -r)"
-apply (rule inj_onI)
-apply (drule_tac f = uminus in arg_cong, simp)
-done
+by (simp add: complex_minus_def)
 
 lemma complex_minus_zero: "-(0::complex) = 0"
-apply (unfold complex_zero_def)
-apply (simp (no_asm) add: complex_minus)
-done
+by (simp add: complex_minus_def complex_zero_def)
 declare complex_minus_zero [simp]
 
 lemma complex_minus_zero_iff: "(-x = 0) = (x = (0::complex))"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (auto dest: inj_Abs_complex [THEN injD]
-            simp add: complex_zero_def complex_minus)
-done
+by (induct x, simp add: complex_minus_def complex_zero_def)
 declare complex_minus_zero_iff [simp]
 
-lemma complex_minus_zero_iff2: "(0 = -x) = (x = (0::real))"
-by (auto dest: sym)
-declare complex_minus_zero_iff2 [simp]
-
-lemma complex_minus_not_zero_iff: "(-x \<noteq> 0) = (x \<noteq> (0::complex))"
-by auto
-
 
 subsection{*Addition*}
 
-lemma complex_add:
-      "Abs_complex(x1,y1) + Abs_complex(x2,y2) = Abs_complex(x1+x2,y1+y2)"
-apply (unfold complex_add_def)
-apply (simp (no_asm))
-done
+lemma complex_add: "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)"
+by (simp add: complex_add_def)
 
 lemma complex_Re_add: "Re(x + y) = Re(x) + Re(y)"
-apply (unfold Re_def)
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
-apply (auto simp add: complex_add)
-done
+by (simp add: complex_add_def)
 
 lemma complex_Im_add: "Im(x + y) = Im(x) + Im(y)"
-apply (unfold Im_def)
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
-apply (auto simp add: complex_add)
-done
+by (simp add: complex_add_def)
 
 lemma complex_add_commute: "(u::complex) + v = v + u"
-apply (unfold complex_add_def)
-apply (simp (no_asm) add: real_add_commute)
-done
+by (simp add: complex_add_def add_commute)
 
 lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)"
-apply (unfold complex_add_def)
-apply (simp (no_asm) add: real_add_assoc)
-done
-
-lemma complex_add_left_commute: "(x::complex) + (y + z) = y + (x + z)"
-apply (unfold complex_add_def)
-apply (simp (no_asm) add: add_left_commute)
-done
-
-lemmas complex_add_ac = complex_add_assoc complex_add_commute
-                      complex_add_left_commute
+by (simp add: complex_add_def add_assoc)
 
 lemma complex_add_zero_left: "(0::complex) + z = z"
-apply (unfold complex_add_def complex_zero_def)
-apply (simp (no_asm))
-done
-declare complex_add_zero_left [simp]
+by (simp add: complex_add_def complex_zero_def)
 
 lemma complex_add_zero_right: "z + (0::complex) = z"
-apply (unfold complex_add_def complex_zero_def)
-apply (simp (no_asm))
-done
-declare complex_add_zero_right [simp]
-
-lemma complex_add_minus_right_zero:
-      "z + -z = (0::complex)"
-apply (unfold complex_add_def complex_minus_def complex_zero_def)
-apply (simp (no_asm))
-done
-declare complex_add_minus_right_zero [simp]
-
-lemma complex_add_minus_left:
-      "-z + z = (0::complex)"
-apply (unfold complex_add_def complex_minus_def complex_zero_def)
-apply (simp (no_asm))
-done
-
-lemma complex_add_minus_cancel: "z + (- z + w) = (w::complex)"
-apply (simp (no_asm) add: complex_add_assoc [symmetric])
-done
-
-lemma complex_minus_add_cancel: "(-z) + (z + w) = (w::complex)"
-by (simp add: complex_add_minus_left complex_add_assoc [symmetric])
-
-declare complex_add_minus_cancel [simp] complex_minus_add_cancel [simp]
-
-lemma complex_add_minus_eq_minus: "x + y = (0::complex) ==> x = -y"
-by (auto simp add: complex_Re_Im_cancel_iff complex_Re_add complex_Im_add complex_Re_minus complex_Im_minus)
+by (simp add: complex_add_def complex_zero_def)
 
-lemma complex_minus_add_distrib: "-(x + y) = -x + -(y::complex)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
-apply (auto simp add: complex_minus complex_add)
-done
-declare complex_minus_add_distrib [simp]
-
-lemma complex_add_left_cancel: "((x::complex) + y = x + z) = (y = z)"
-apply safe
-apply (drule_tac f = "%t.-x + t" in arg_cong)
-apply (simp add: complex_add_minus_left complex_add_assoc [symmetric])
-done
-declare complex_add_left_cancel [iff]
-
-lemma complex_add_right_cancel: "(y + (x::complex)= z + x) = (y = z)"
-apply (simp (no_asm) add: complex_add_commute)
-done
-declare complex_add_right_cancel [simp]
-
-lemma complex_eq_minus_iff: "((x::complex) = y) = (0 = x + - y)"
-apply safe
-apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto)
-done
-
-lemma complex_eq_minus_iff2: "((x::complex) = y) = (x + - y = 0)"
-apply safe
-apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto)
-done
-
-lemma complex_diff_0: "(0::complex) - x = -x"
-apply (simp (no_asm) add: complex_diff_def)
-done
-
-lemma complex_diff_0_right: "x - (0::complex) = x"
-apply (simp (no_asm) add: complex_diff_def)
-done
-
-lemma complex_diff_self: "x - x = (0::complex)"
-apply (simp (no_asm) add: complex_diff_def)
-done
-
-declare complex_diff_0 [simp] complex_diff_0_right [simp] complex_diff_self [simp]
+lemma complex_add_minus_left: "-z + z = (0::complex)"
+by (simp add: complex_add_def complex_minus_def complex_zero_def)
 
 lemma complex_diff:
-      "Abs_complex(x1,y1) - Abs_complex(x2,y2) = Abs_complex(x1-x2,y1-y2)"
-apply (unfold complex_diff_def)
-apply (simp (no_asm) add: complex_add complex_minus)
-done
-
-lemma complex_diff_eq_eq: "((x::complex) - y = z) = (x = z + y)"
-by (auto simp add: complex_add_minus_left complex_diff_def complex_add_assoc)
-
+      "Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)"
+by (simp add: complex_add_def complex_minus_def complex_diff_def)
 
 subsection{*Multiplication*}
 
 lemma complex_mult:
-      "Abs_complex(x1,y1) * Abs_complex(x2,y2) =
-       Abs_complex(x1*x2 - y1*y2,x1*y2 + y1*x2)"
-apply (unfold complex_mult_def)
-apply (simp (no_asm))
-done
+     "Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
+by (simp add: complex_mult_def)
 
 lemma complex_mult_commute: "(w::complex) * z = z * w"
-apply (unfold complex_mult_def)
-apply (simp (no_asm) add: real_mult_commute real_add_commute)
-done
+by (simp add: complex_mult_def mult_commute add_commute)
 
 lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
-apply (unfold complex_mult_def)
-apply (simp (no_asm) add: complex_Re_Im_cancel_iff real_mult_assoc right_diff_distrib right_distrib left_diff_distrib left_distrib mult_left_commute)
-done
-
-lemma complex_mult_left_commute: "(x::complex) * (y * z) = y * (x * z)"
-apply (unfold complex_mult_def)
-apply (simp (no_asm) add: complex_Re_Im_cancel_iff mult_left_commute right_diff_distrib right_distrib)
-done
-
-lemmas complex_mult_ac = complex_mult_assoc complex_mult_commute
-                      complex_mult_left_commute
+by (simp add: complex_mult_def mult_ac add_ac 
+              right_diff_distrib right_distrib left_diff_distrib left_distrib)
 
 lemma complex_mult_one_left: "(1::complex) * z = z"
-apply (unfold complex_one_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_mult)
-done
-declare complex_mult_one_left [simp]
+by (simp add: complex_mult_def complex_one_def)
 
 lemma complex_mult_one_right: "z * (1::complex) = z"
-apply (simp (no_asm) add: complex_mult_commute)
-done
-declare complex_mult_one_right [simp]
-
-lemma complex_mult_zero_left: "(0::complex) * z = 0"
-apply (unfold complex_zero_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_mult)
-done
-declare complex_mult_zero_left [simp]
-
-lemma complex_mult_zero_right: "z * 0 = (0::complex)"
-apply (simp (no_asm) add: complex_mult_commute)
-done
-declare complex_mult_zero_right [simp]
-
-lemma complex_divide_zero: "0 / z = (0::complex)"
-by (unfold complex_divide_def, auto)
-declare complex_divide_zero [simp]
-
-lemma complex_minus_mult_eq1: "-(x * y) = -x * (y::complex)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
-apply (auto simp add: complex_mult complex_minus real_diff_def)
-done
-
-lemma complex_minus_mult_eq2: "-(x * y) = x * -(y::complex)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
-apply (auto simp add: complex_mult complex_minus real_diff_def)
-done
-
-lemma complex_add_mult_distrib: "((z1::complex) + z2) * w = (z1 * w) + (z2 * w)"
-apply (rule_tac z = z1 in eq_Abs_complex)
-apply (rule_tac z = z2 in eq_Abs_complex)
-apply (rule_tac z = w in eq_Abs_complex)
-apply (auto simp add: complex_mult complex_add left_distrib real_diff_def add_ac)
-done
-
-lemma complex_add_mult_distrib2: "(w::complex) * (z1 + z2) = (w * z1) + (w * z2)"
-apply (rule_tac z1 = "z1 + z2" in complex_mult_commute [THEN ssubst])
-apply (simp (no_asm) add: complex_add_mult_distrib)
-apply (simp (no_asm) add: complex_mult_commute)
-done
-
-lemma complex_zero_not_eq_one: "(0::complex) \<noteq> 1"
-apply (unfold complex_zero_def complex_one_def)
-apply (simp (no_asm) add: complex_Re_Im_cancel_iff)
-done
-declare complex_zero_not_eq_one [simp]
-declare complex_zero_not_eq_one [THEN not_sym, simp]
+by (simp add: complex_mult_def complex_one_def)
 
 
 subsection{*Inverse*}
 
 lemma complex_inverse:
-     "inverse (Abs_complex(x,y)) =
-      Abs_complex(x/(x ^ 2 + y ^ 2),-y/(x ^ 2 + y ^ 2))"
-apply (unfold complex_inverse_def)
-apply (simp (no_asm))
-done
-
-lemma COMPLEX_INVERSE_ZERO: "inverse 0 = (0::complex)"
-by (unfold complex_inverse_def complex_zero_def, auto)
-
-lemma COMPLEX_DIVISION_BY_ZERO: "a / (0::complex) = 0"
-apply (simp (no_asm) add: complex_divide_def COMPLEX_INVERSE_ZERO)
-done
-
-instance complex :: division_by_zero
-proof
-  fix x :: complex
-  show "inverse 0 = (0::complex)" by (rule COMPLEX_INVERSE_ZERO)
-  show "x/0 = 0" by (rule COMPLEX_DIVISION_BY_ZERO) 
-qed
+     "inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
+by (simp add: complex_inverse_def)
 
 lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
-apply (rule_tac z = z in eq_Abs_complex)
+apply (induct z) 
+apply (rename_tac x y) 
 apply (auto simp add: complex_mult complex_inverse complex_one_def 
        complex_zero_def add_divide_distrib [symmetric] power2_eq_square mult_ac)
 apply (drule_tac y = y in real_sum_squares_not_zero)
 apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto)
 done
-declare complex_mult_inv_left [simp]
-
-lemma complex_mult_inv_right: "z \<noteq> (0::complex) ==> z * inverse(z) = 1"
-by (auto intro: complex_mult_commute [THEN subst])
-declare complex_mult_inv_right [simp]
 
 
 subsection {* The field of complex numbers *}
@@ -556,14 +270,14 @@
   show "1 * z = z"
     by (rule complex_mult_one_left) 
   show "0 \<noteq> (1::complex)"
-    by (rule complex_zero_not_eq_one) 
+    by (simp add: complex_zero_def complex_one_def)
   show "(u + v) * w = u * w + v * w"
-    by (rule complex_add_mult_distrib) 
+    by (simp add: complex_mult_def complex_add_def left_distrib real_diff_def add_ac)
   show "z+u = z+v ==> u=v"
     proof -
       assume eq: "z+u = z+v" 
       hence "(-z + z) + u = (-z + z) + v" by (simp only: eq complex_add_assoc)
-      thus "u = v" by (simp add: complex_add_minus_left)
+      thus "u = v" by (simp add: complex_add_minus_left complex_add_zero_left)
     qed
   assume neq: "w \<noteq> 0"
   thus "z / w = z * inverse w"
@@ -572,40 +286,33 @@
     by (simp add: neq complex_mult_inv_left) 
 qed
 
+instance complex :: division_by_zero
+proof
+  show inv: "inverse 0 = (0::complex)"
+    by (simp add: complex_inverse_def complex_zero_def)
+  fix x :: complex
+  show "x/0 = 0" 
+    by (simp add: complex_divide_def inv)
+qed
 
-lemma complex_minus_mult_commute: "-x * y = x * -(y::complex)"
-apply (simp)
-done
 
 subsection{*Embedding Properties for @{term complex_of_real} Map*}
 
-lemma inj_complex_of_real: "inj complex_of_real"
-apply (rule inj_onI)
-apply (auto dest: inj_Abs_complex [THEN injD] simp add: complex_of_real_def)
-done
-
-lemma complex_of_real_one:
-      "complex_of_real 1 = 1"
-apply (unfold complex_one_def complex_of_real_def)
-apply (rule refl)
-done
+lemma complex_of_real_one: "complex_of_real 1 = 1"
+by (simp add: complex_one_def complex_of_real_def)
 declare complex_of_real_one [simp]
 
-lemma complex_of_real_zero:
-      "complex_of_real 0 = 0"
-apply (unfold complex_zero_def complex_of_real_def)
-apply (rule refl)
-done
+lemma complex_of_real_zero: "complex_of_real 0 = 0"
+by (simp add: complex_zero_def complex_of_real_def)
 declare complex_of_real_zero [simp]
 
 lemma complex_of_real_eq_iff:
      "(complex_of_real x = complex_of_real y) = (x = y)"
-by (auto dest: inj_complex_of_real [THEN injD])
+by (simp add: complex_of_real_def) 
 declare complex_of_real_eq_iff [iff]
 
 lemma complex_of_real_minus: "complex_of_real(-x) = - complex_of_real x"
-apply (simp (no_asm) add: complex_of_real_def complex_minus)
-done
+by (simp add: complex_of_real_def complex_minus)
 
 lemma complex_of_real_inverse:
  "complex_of_real(inverse x) = inverse(complex_of_real x)"
@@ -615,133 +322,93 @@
 done
 
 lemma complex_of_real_add:
- "complex_of_real x + complex_of_real y = complex_of_real (x + y)"
-apply (simp (no_asm) add: complex_add complex_of_real_def)
-done
+     "complex_of_real x + complex_of_real y = complex_of_real (x + y)"
+by (simp add: complex_add complex_of_real_def)
 
 lemma complex_of_real_diff:
- "complex_of_real x - complex_of_real y = complex_of_real (x - y)"
-apply (simp (no_asm) add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add)
-done
+     "complex_of_real x - complex_of_real y = complex_of_real (x - y)"
+by (simp add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add)
 
 lemma complex_of_real_mult:
- "complex_of_real x * complex_of_real y = complex_of_real (x * y)"
-apply (simp (no_asm) add: complex_mult complex_of_real_def)
-done
+     "complex_of_real x * complex_of_real y = complex_of_real (x * y)"
+by (simp add: complex_mult complex_of_real_def)
 
 lemma complex_of_real_divide:
       "complex_of_real x / complex_of_real y = complex_of_real(x/y)"
-apply (unfold complex_divide_def)
-apply (case_tac "y=0")
-apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
-apply (simp (no_asm_simp) add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def)
+apply (simp add: complex_divide_def)
+apply (case_tac "y=0", simp)
+apply (simp add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def)
 done
 
-lemma complex_mod: "cmod (Abs_complex(x,y)) = sqrt(x ^ 2 + y ^ 2)"
-apply (unfold cmod_def)
-apply (simp (no_asm))
-done
+lemma complex_mod: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)"
+by (simp add: cmod_def)
 
 lemma complex_mod_zero: "cmod(0) = 0"
-apply (unfold cmod_def)
-apply (simp (no_asm))
-done
+by (simp add: cmod_def)
 declare complex_mod_zero [simp]
 
 lemma complex_mod_one [simp]: "cmod(1) = 1"
 by (simp add: cmod_def power2_eq_square)
 
 lemma complex_mod_complex_of_real: "cmod(complex_of_real x) = abs x"
-apply (simp add: complex_of_real_def power2_eq_square complex_mod)
-done
+by (simp add: complex_of_real_def power2_eq_square complex_mod)
 declare complex_mod_complex_of_real [simp]
 
 lemma complex_of_real_abs:
      "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
-by (simp)
+by simp
 
 
 
 subsection{*Conjugation is an Automorphism*}
 
-lemma complex_cnj: "cnj (Abs_complex(x,y)) = Abs_complex(x,-y)"
-apply (unfold cnj_def)
-apply (simp (no_asm))
-done
-
-lemma inj_cnj: "inj cnj"
-apply (rule inj_onI)
-apply (auto simp add: cnj_def Abs_complex_cancel_iff complex_Re_Im_cancel_iff)
-done
+lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)"
+by (simp add: cnj_def)
 
 lemma complex_cnj_cancel_iff: "(cnj x = cnj y) = (x = y)"
-by (auto dest: inj_cnj [THEN injD])
+by (simp add: cnj_def complex_Re_Im_cancel_iff)
 declare complex_cnj_cancel_iff [simp]
 
 lemma complex_cnj_cnj: "cnj (cnj z) = z"
-apply (unfold cnj_def)
-apply (simp (no_asm))
-done
+by (simp add: cnj_def)
 declare complex_cnj_cnj [simp]
 
 lemma complex_cnj_complex_of_real:
- "cnj (complex_of_real x) = complex_of_real x"
-apply (unfold complex_of_real_def)
-apply (simp (no_asm) add: complex_cnj)
-done
+     "cnj (complex_of_real x) = complex_of_real x"
+by (simp add: complex_of_real_def complex_cnj)
 declare complex_cnj_complex_of_real [simp]
 
 lemma complex_mod_cnj: "cmod (cnj z) = cmod z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_cnj complex_mod power2_eq_square)
-done
+by (induct z, simp add: complex_cnj complex_mod power2_eq_square)
 declare complex_mod_cnj [simp]
 
 lemma complex_cnj_minus: "cnj (-z) = - cnj z"
-apply (unfold cnj_def)
-apply (simp (no_asm) add: complex_minus complex_Re_minus complex_Im_minus)
-done
+by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus)
 
 lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_cnj complex_inverse power2_eq_square)
-done
+by (induct z, simp add: complex_cnj complex_inverse power2_eq_square)
 
 lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
-apply (rule_tac z = w in eq_Abs_complex)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_cnj complex_add)
-done
+by (induct w, induct z, simp add: complex_cnj complex_add)
 
 lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
-apply (unfold complex_diff_def)
-apply (simp (no_asm) add: complex_cnj_add complex_cnj_minus)
-done
+by (simp add: complex_diff_def complex_cnj_add complex_cnj_minus)
 
 lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
-apply (rule_tac z = w in eq_Abs_complex)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_cnj complex_mult)
-done
+by (induct w, induct z, simp add: complex_cnj complex_mult)
 
 lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
-apply (unfold complex_divide_def)
-apply (simp (no_asm) add: complex_cnj_mult complex_cnj_inverse)
-done
+by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
 
 lemma complex_cnj_one: "cnj 1 = 1"
-apply (unfold cnj_def complex_one_def)
-apply (simp (no_asm))
-done
+by (simp add: cnj_def complex_one_def)
 declare complex_cnj_one [simp]
 
 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_add complex_cnj complex_of_real_def)
-done
+by (induct z, simp add: complex_add complex_cnj complex_of_real_def)
 
 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
-apply (rule_tac z = z in eq_Abs_complex)
+apply (induct z)
 apply (simp add: complex_add complex_cnj complex_of_real_def complex_diff_def 
                  complex_minus i_def complex_mult)
 done
@@ -750,81 +417,62 @@
 by (simp add: cnj_def complex_zero_def)
 
 lemma complex_cnj_zero_iff: "(cnj z = 0) = (z = 0)"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_zero_def complex_cnj)
-done
+by (induct z, simp add: complex_zero_def complex_cnj)
 declare complex_cnj_zero_iff [iff]
 
 lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
-done
+by (induct z, simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
 
 
 subsection{*Algebra*}
 
 lemma complex_add_left_cancel_zero: "(x + y = x) = (y = (0::complex))"
-apply (unfold complex_zero_def)
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
-apply (auto simp add: complex_add)
-done
+by (induct x, induct y, simp add: complex_zero_def complex_add)
 declare complex_add_left_cancel_zero [simp]
 
-lemma complex_diff_mult_distrib:
-      "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)"
-apply (unfold complex_diff_def)
-apply (simp (no_asm) add: complex_add_mult_distrib)
-done
+lemma complex_diff_mult_distrib: "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)"
+by (simp add: complex_diff_def left_distrib)
 
-lemma complex_diff_mult_distrib2:
-      "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)"
-apply (unfold complex_diff_def)
-apply (simp (no_asm) add: complex_add_mult_distrib2)
-done
+lemma complex_diff_mult_distrib2: "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)"
+by (simp add: complex_diff_def right_distrib)
 
 
 subsection{*Modulus*}
 
 lemma complex_mod_eq_zero_cancel: "(cmod x = 0) = (x = 0)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: complex_mod complex_zero_def power2_eq_square)
+apply (induct x)
+apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 
+            simp add: complex_mod complex_zero_def power2_eq_square)
 done
 declare complex_mod_eq_zero_cancel [simp]
 
-lemma complex_mod_complex_of_real_of_nat: "cmod (complex_of_real(real (n::nat))) = real n"
-apply (simp (no_asm))
-done
+lemma complex_mod_complex_of_real_of_nat:
+     "cmod (complex_of_real(real (n::nat))) = real n"
+by simp
 declare complex_mod_complex_of_real_of_nat [simp]
 
 lemma complex_mod_minus: "cmod (-x) = cmod(x)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_mod complex_minus power2_eq_square)
-done
+by (induct x, simp add: complex_mod complex_minus power2_eq_square)
 declare complex_mod_minus [simp]
 
 lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (simp (no_asm_simp) add: complex_mod complex_cnj complex_mult);
-apply (simp (no_asm) add: power2_eq_square real_abs_def)
+apply (induct z, simp add: complex_mod complex_cnj complex_mult)
+apply (simp add: power2_eq_square real_abs_def)
 done
 
-lemma complex_mod_squared: "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2"
-by (unfold cmod_def, auto)
+lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2"
+by (simp add: cmod_def)
 
 lemma complex_mod_ge_zero: "0 \<le> cmod x"
-apply (unfold cmod_def)
-apply (auto intro: real_sqrt_ge_zero)
-done
+by (simp add: cmod_def)
 declare complex_mod_ge_zero [simp]
 
 lemma abs_cmod_cancel: "abs(cmod x) = cmod x"
-by (auto intro: abs_eqI1)
+by (simp add: abs_if linorder_not_less) 
 declare abs_cmod_cancel [simp]
 
 lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
+apply (induct x, induct y)
 apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2 [symmetric] simp del: realpow_Suc)
 apply (rule_tac n = 1 in power_inject_base)
 apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc)
@@ -832,38 +480,30 @@
 done
 
 lemma complex_mod_add_squared_eq: "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
+apply (induct x, induct y)
 apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
 apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac)
 done
 
 lemma complex_Re_mult_cnj_le_cmod: "Re(x * cnj y) \<le> cmod(x * cnj y)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
+apply (induct x, induct y)
 apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
 done
 declare complex_Re_mult_cnj_le_cmod [simp]
 
 lemma complex_Re_mult_cnj_le_cmod2: "Re(x * cnj y) \<le> cmod(x * y)"
-apply (cut_tac x = x and y = y in complex_Re_mult_cnj_le_cmod)
-apply (simp add: complex_mod_mult)
-done
+by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult)
 declare complex_Re_mult_cnj_le_cmod2 [simp]
 
 lemma real_sum_squared_expand: "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y"
-apply (simp (no_asm) add: left_distrib right_distrib power2_eq_square)
-done
+by (simp add: left_distrib right_distrib power2_eq_square)
 
 lemma complex_mod_triangle_squared: "cmod (x + y) ^ 2 \<le> (cmod(x) + cmod(y)) ^ 2"
-apply (simp (no_asm) add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
-done
+by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
 declare complex_mod_triangle_squared [simp]
 
 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
-apply (rule order_trans [OF _ complex_mod_ge_zero]) 
-apply (simp (no_asm))
-done
+by (rule order_trans [OF _ complex_mod_ge_zero], simp)
 declare complex_mod_minus_le_complex_mod [simp]
 
 lemma complex_mod_triangle_ineq: "cmod (x + y) \<le> cmod(x) + cmod(y)"
@@ -874,15 +514,11 @@
 declare complex_mod_triangle_ineq [simp]
 
 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
-apply (cut_tac x1 = b and y1 = a and c = "-cmod b" 
-       in complex_mod_triangle_ineq [THEN add_right_mono])
-apply (simp (no_asm))
-done
+by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp)
 declare complex_mod_triangle_ineq2 [simp]
 
 lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)"
-apply (rule_tac z = x in eq_Abs_complex)
-apply (rule_tac z = y in eq_Abs_complex)
+apply (induct x, induct y)
 apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac)
 done
 
@@ -901,18 +537,16 @@
 apply (rule complex_mod_minus [THEN subst])
 apply (rule order_trans)
 apply (rule_tac [2] complex_mod_triangle_ineq)
-apply (auto simp add: complex_add_ac)
+apply (auto simp add: add_ac)
 done
 declare complex_mod_diff_ineq [simp]
 
 lemma complex_Re_le_cmod: "Re z \<le> cmod z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_mod simp del: realpow_Suc)
-done
+by (induct z, simp add: complex_mod del: realpow_Suc)
 declare complex_Re_le_cmod [simp]
 
 lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z"
-apply (cut_tac x = z in complex_mod_ge_zero)
+apply (insert complex_mod_ge_zero [of z])
 apply (drule order_le_imp_less_or_eq, auto)
 done
 
@@ -920,22 +554,16 @@
 subsection{*A Few More Theorems*}
 
 lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
-apply (case_tac "x=0", simp add: COMPLEX_INVERSE_ZERO)
+apply (case_tac "x=0", simp)
 apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1])
 apply (auto simp add: complex_mod_mult [symmetric])
 done
 
-lemma complex_mod_divide:
-      "cmod(x/y) = cmod(x)/(cmod y)"
-apply (unfold complex_divide_def real_divide_def)
-apply (auto simp add: complex_mod_mult complex_mod_inverse)
-done
+lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)"
+by (simp add: complex_divide_def real_divide_def, simp add: complex_mod_mult complex_mod_inverse)
 
-lemma complex_inverse_divide:
-      "inverse(x/y) = y/(x::complex)"
-apply (unfold complex_divide_def)
-apply (auto simp add: inverse_mult_distrib complex_mult_commute)
-done
+lemma complex_inverse_divide: "inverse(x/y) = y/(x::complex)"
+by (simp add: complex_divide_def inverse_mult_distrib mult_commute)
 declare complex_inverse_divide [simp]
 
 
@@ -977,34 +605,29 @@
 by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2)
 
 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
-by (unfold i_def complex_zero_def, auto)
+by (simp add: i_def complex_zero_def)
 
 
 subsection{*The Function @{term sgn}*}
 
 lemma sgn_zero: "sgn 0 = 0"
-apply (unfold sgn_def)
-apply (simp (no_asm))
-done
+by (simp add: sgn_def)
 declare sgn_zero [simp]
 
 lemma sgn_one: "sgn 1 = 1"
-apply (unfold sgn_def)
-apply (simp (no_asm))
-done
+by (simp add: sgn_def)
 declare sgn_one [simp]
 
 lemma sgn_minus: "sgn (-z) = - sgn(z)"
-by (unfold sgn_def, auto)
+by (simp add: sgn_def)
 
 lemma sgn_eq:
     "sgn z = z / complex_of_real (cmod z)"
-apply (unfold sgn_def)
-apply (simp (no_asm))
+apply (simp add: sgn_def)
 done
 
 lemma complex_split: "\<exists>x y. z = complex_of_real(x) + ii * complex_of_real(y)"
-apply (rule_tac z = z in eq_Abs_complex)
+apply (induct z)
 apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
 done
 
@@ -1017,59 +640,46 @@
 declare Im_complex_i [simp]
 
 lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
-apply (unfold i_def complex_of_real_def)
-apply (auto simp add: complex_mult complex_add)
-done
+by (simp add: i_def complex_of_real_def complex_mult complex_add)
 
 lemma i_mult_eq2: "ii * ii = -(1::complex)"
-apply (unfold i_def complex_one_def)
-apply (simp (no_asm) add: complex_mult complex_minus)
-done
+by (simp add: i_def complex_one_def complex_mult complex_minus)
 declare i_mult_eq2 [simp]
 
 lemma cmod_i: "cmod (complex_of_real(x) + ii * complex_of_real(y)) =
       sqrt (x ^ 2 + y ^ 2)"
-apply (auto simp add: complex_mult complex_add i_def complex_of_real_def cmod_def)
-done
+by (simp add: complex_mult complex_add i_def complex_of_real_def cmod_def)
 
 lemma complex_eq_Re_eq:
      "complex_of_real xa + ii * complex_of_real ya =
       complex_of_real xb + ii * complex_of_real yb
        ==> xa = xb"
-apply (unfold complex_of_real_def i_def)
-apply (auto simp add: complex_mult complex_add)
-done
+by (simp add: complex_of_real_def i_def complex_mult complex_add)
 
 lemma complex_eq_Im_eq:
      "complex_of_real xa + ii * complex_of_real ya =
       complex_of_real xb + ii * complex_of_real yb
        ==> ya = yb"
-apply (unfold complex_of_real_def i_def)
-apply (auto simp add: complex_mult complex_add)
-done
+by (simp add: complex_of_real_def i_def complex_mult complex_add)
 
 lemma complex_eq_cancel_iff: "(complex_of_real xa + ii * complex_of_real ya =
        complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
-apply (auto intro: complex_eq_Im_eq complex_eq_Re_eq)
-done
+by (auto intro: complex_eq_Im_eq complex_eq_Re_eq)
 declare complex_eq_cancel_iff [iff]
 
 lemma complex_eq_cancel_iffA: "(complex_of_real xa + complex_of_real ya * ii =
-       complex_of_real xb + complex_of_real yb * ii ) = ((xa = xb) & (ya = yb))"
-apply (auto simp add: complex_mult_commute)
-done
+       complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))"
+by (simp add: mult_commute)
 declare complex_eq_cancel_iffA [iff]
 
 lemma complex_eq_cancel_iffB: "(complex_of_real xa + complex_of_real ya * ii =
        complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
-apply (auto simp add: complex_mult_commute)
-done
+by (auto simp add: mult_commute)
 declare complex_eq_cancel_iffB [iff]
 
 lemma complex_eq_cancel_iffC: "(complex_of_real xa + ii * complex_of_real ya  =
        complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))"
-apply (auto simp add: complex_mult_commute)
-done
+by (auto simp add: mult_commute)
 declare complex_eq_cancel_iffC [iff]
 
 lemma complex_eq_cancel_iff2: "(complex_of_real x + ii * complex_of_real y =
@@ -1081,8 +691,7 @@
 
 lemma complex_eq_cancel_iff2a: "(complex_of_real x + complex_of_real y * ii =
       complex_of_real xa) = (x = xa & y = 0)"
-apply (auto simp add: complex_mult_commute)
-done
+by (auto simp add: mult_commute)
 declare complex_eq_cancel_iff2a [simp]
 
 lemma complex_eq_cancel_iff3: "(complex_of_real x + ii * complex_of_real y =
@@ -1094,39 +703,31 @@
 
 lemma complex_eq_cancel_iff3a: "(complex_of_real x + complex_of_real y * ii =
       ii * complex_of_real ya) = (x = 0 & y = ya)"
-apply (auto simp add: complex_mult_commute)
-done
+by (auto simp add: mult_commute)
 declare complex_eq_cancel_iff3a [simp]
 
 lemma complex_split_Re_zero:
      "complex_of_real x + ii * complex_of_real y = 0
       ==> x = 0"
-apply (unfold complex_of_real_def i_def complex_zero_def)
-apply (auto simp add: complex_mult complex_add)
-done
+by (simp add: complex_of_real_def i_def complex_zero_def complex_mult complex_add)
 
 lemma complex_split_Im_zero:
      "complex_of_real x + ii * complex_of_real y = 0
       ==> y = 0"
-apply (unfold complex_of_real_def i_def complex_zero_def)
-apply (auto simp add: complex_mult complex_add)
-done
+by (simp add: complex_of_real_def i_def complex_zero_def complex_mult complex_add)
 
-lemma Re_sgn:
-      "Re(sgn z) = Re(z)/cmod z"
-apply (unfold sgn_def complex_divide_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_of_real_inverse [symmetric])
-apply (auto simp add: complex_of_real_def complex_mult real_divide_def)
+lemma Re_sgn: "Re(sgn z) = Re(z)/cmod z"
+apply (induct z)
+apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric])
+apply (simp add: complex_of_real_def complex_mult real_divide_def)
 done
 declare Re_sgn [simp]
 
 lemma Im_sgn:
       "Im(sgn z) = Im(z)/cmod z"
-apply (unfold sgn_def complex_divide_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_of_real_inverse [symmetric])
-apply (auto simp add: complex_of_real_def complex_mult real_divide_def)
+apply (induct z)
+apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric])
+apply (simp add: complex_of_real_def complex_mult real_divide_def)
 done
 declare Im_sgn [simp]
 
@@ -1134,9 +735,8 @@
      "inverse(complex_of_real x + ii * complex_of_real y) =
       complex_of_real(x/(x ^ 2 + y ^ 2)) -
       ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
-apply (unfold complex_of_real_def i_def)
-apply (auto simp add: complex_mult complex_add complex_diff_def complex_minus complex_inverse real_divide_def)
-done
+by (simp add: complex_of_real_def i_def complex_mult complex_add 
+         complex_diff_def complex_minus complex_inverse real_divide_def)
 
 (*----------------------------------------------------------------------------*)
 (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
@@ -1146,47 +746,37 @@
 lemma complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)"
 by (auto simp add: complex_zero_def complex_of_real_def)
 
-lemma Re_mult_i_eq:
-    "Re (ii * complex_of_real y) = 0"
-apply (unfold i_def complex_of_real_def)
-apply (auto simp add: complex_mult)
-done
+lemma Re_mult_i_eq: "Re (ii * complex_of_real y) = 0"
+by (simp add: i_def complex_of_real_def complex_mult)
 declare Re_mult_i_eq [simp]
 
-lemma Im_mult_i_eq:
-    "Im (ii * complex_of_real y) = y"
-apply (unfold i_def complex_of_real_def)
-apply (auto simp add: complex_mult)
-done
+lemma Im_mult_i_eq: "Im (ii * complex_of_real y) = y"
+by (simp add: i_def complex_of_real_def complex_mult)
 declare Im_mult_i_eq [simp]
 
-lemma complex_mod_mult_i:
-    "cmod (ii * complex_of_real y) = abs y"
-apply (unfold i_def complex_of_real_def)
-apply (auto simp add: complex_mult complex_mod power2_eq_square)
-done
+lemma complex_mod_mult_i: "cmod (ii * complex_of_real y) = abs y"
+by (simp add: i_def complex_of_real_def complex_mult complex_mod power2_eq_square)
 declare complex_mod_mult_i [simp]
 
 lemma cos_arg_i_mult_zero_pos:
    "0 < y ==> cos (arg(ii * complex_of_real y)) = 0"
-apply (unfold arg_def)
-apply (auto simp add: abs_eqI2)
+apply (simp add: arg_def abs_if)
 apply (rule_tac a = "pi/2" in someI2, auto)
 apply (rule order_less_trans [of _ 0], auto)
 done
 
 lemma cos_arg_i_mult_zero_neg:
    "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0"
-apply (unfold arg_def)
-apply (auto simp add: abs_minus_eqI2)
+apply (simp add: arg_def abs_if)
 apply (rule_tac a = "- pi/2" in someI2, auto)
 apply (rule order_trans [of _ 0], auto)
 done
 
 lemma cos_arg_i_mult_zero [simp]
     : "y \<noteq> 0 ==> cos (arg(ii * complex_of_real y)) = 0"
-by (cut_tac x = y and y = 0 in linorder_less_linear, 
-    auto simp add: cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
+apply (insert linorder_less_linear [of y 0]) 
+apply (auto simp add: cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
+done
 
 
 subsection{*Finally! Polar Form for Complex Numbers*}
@@ -1198,97 +788,75 @@
 done
 
 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
-apply (unfold rcis_def cis_def)
+apply (simp add: rcis_def cis_def)
 apply (rule complex_split_polar)
 done
 
 lemma Re_complex_polar: "Re(complex_of_real r *
       (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a"
-apply (auto simp add: complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
-done
+by (auto simp add: right_distrib complex_of_real_mult mult_ac)
 declare Re_complex_polar [simp]
 
 lemma Re_rcis: "Re(rcis r a) = r * cos a"
-by (unfold rcis_def cis_def, auto)
+by (simp add: rcis_def cis_def)
 declare Re_rcis [simp]
 
 lemma Im_complex_polar [simp]:
      "Im(complex_of_real r * 
          (complex_of_real(cos a) + ii * complex_of_real(sin a))) = 
       r * sin a"
-by (auto simp add: complex_add_mult_distrib2 complex_of_real_mult mult_ac)
+by (auto simp add: right_distrib complex_of_real_mult mult_ac)
 
 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
-by (unfold rcis_def cis_def, auto)
+by (simp add: rcis_def cis_def)
 
 lemma complex_mod_complex_polar [simp]:
      "cmod (complex_of_real r * 
             (complex_of_real(cos a) + ii * complex_of_real(sin a))) = 
       abs r"
-by (auto simp add: complex_add_mult_distrib2 cmod_i complex_of_real_mult
+by (auto simp add: right_distrib cmod_i complex_of_real_mult
                       right_distrib [symmetric] power_mult_distrib mult_ac 
          simp del: realpow_Suc)
 
 lemma complex_mod_rcis: "cmod(rcis r a) = abs r"
-by (unfold rcis_def cis_def, auto)
+by (simp add: rcis_def cis_def)
 declare complex_mod_rcis [simp]
 
 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
-apply (unfold cmod_def)
+apply (simp add: cmod_def)
 apply (rule real_sqrt_eq_iff [THEN iffD2])
 apply (auto simp add: complex_mult_cnj)
 done
 
 lemma complex_Re_cnj: "Re(cnj z) = Re z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_cnj)
-done
+by (induct z, simp add: complex_cnj)
 declare complex_Re_cnj [simp]
 
 lemma complex_Im_cnj: "Im(cnj z) = - Im z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_cnj)
-done
+by (induct z, simp add: complex_cnj)
 declare complex_Im_cnj [simp]
 
 lemma complex_In_mult_cnj_zero: "Im (z * cnj z) = 0"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_cnj complex_mult)
-done
+by (induct z, simp add: complex_cnj complex_mult)
 declare complex_In_mult_cnj_zero [simp]
 
 lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (rule_tac z = w in eq_Abs_complex)
-apply (auto simp add: complex_mult)
-done
+by (induct z, induct w, simp add: complex_mult)
 
 lemma complex_Re_mult_complex_of_real: "Re (z * complex_of_real c) = Re(z) * c"
-apply (unfold complex_of_real_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_mult)
-done
+by (induct z, simp add: complex_of_real_def complex_mult)
 declare complex_Re_mult_complex_of_real [simp]
 
 lemma complex_Im_mult_complex_of_real: "Im (z * complex_of_real c) = Im(z) * c"
-apply (unfold complex_of_real_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_mult)
-done
+by (induct z, simp add: complex_of_real_def complex_mult)
 declare complex_Im_mult_complex_of_real [simp]
 
 lemma complex_Re_mult_complex_of_real2: "Re (complex_of_real c * z) = c * Re(z)"
-apply (unfold complex_of_real_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_mult)
-done
+by (induct z, simp add: complex_of_real_def complex_mult)
 declare complex_Re_mult_complex_of_real2 [simp]
 
 lemma complex_Im_mult_complex_of_real2: "Im (complex_of_real c * z) = c * Im(z)"
-apply (unfold complex_of_real_def)
-apply (rule_tac z = z in eq_Abs_complex)
-apply (auto simp add: complex_mult)
-done
+by (induct z, simp add: complex_of_real_def complex_mult)
 declare complex_Im_mult_complex_of_real2 [simp]
 
 (*---------------------------------------------------------------------------*)
@@ -1296,64 +864,51 @@
 (*---------------------------------------------------------------------------*)
 
 lemma cis_rcis_eq: "cis a = rcis 1 a"
-apply (unfold rcis_def)
-apply (simp (no_asm))
-done
+by (simp add: rcis_def)
 
 lemma rcis_mult:
   "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
-apply (unfold rcis_def cis_def)
-apply (auto simp add: cos_add sin_add complex_add_mult_distrib2 complex_add_mult_distrib complex_mult_ac complex_add_ac)
-apply (auto simp add: complex_add_mult_distrib2 [symmetric] complex_mult_assoc [symmetric] complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] i_mult_eq simp del: i_mult_eq2)
-apply (auto simp add: complex_add_ac)
+apply (simp add: rcis_def cis_def cos_add sin_add right_distrib left_distrib 
+                 mult_ac add_ac)
+apply (auto simp add: right_distrib [symmetric] complex_mult_assoc [symmetric] complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] i_mult_eq simp del: i_mult_eq2)
+apply (auto simp add: add_ac)
 apply (auto simp add: complex_add_assoc [symmetric] complex_of_real_add right_distrib real_diff_def mult_ac add_ac)
 done
 
 lemma cis_mult: "cis a * cis b = cis (a + b)"
-apply (simp (no_asm) add: cis_rcis_eq rcis_mult)
-done
+by (simp add: cis_rcis_eq rcis_mult)
 
 lemma cis_zero: "cis 0 = 1"
-by (unfold cis_def, auto)
+by (simp add: cis_def)
 declare cis_zero [simp]
 
 lemma cis_zero2: "cis 0 = complex_of_real 1"
-by (unfold cis_def, auto)
+by (simp add: cis_def)
 declare cis_zero2 [simp]
 
 lemma rcis_zero_mod: "rcis 0 a = 0"
-apply (unfold rcis_def)
-apply (simp (no_asm))
-done
+by (simp add: rcis_def)
 declare rcis_zero_mod [simp]
 
 lemma rcis_zero_arg: "rcis r 0 = complex_of_real r"
-apply (unfold rcis_def)
-apply (simp (no_asm))
-done
+by (simp add: rcis_def)
 declare rcis_zero_arg [simp]
 
 lemma complex_of_real_minus_one:
    "complex_of_real (-(1::real)) = -(1::complex)"
-apply (unfold complex_of_real_def complex_one_def)
-apply (simp (no_asm) add: complex_minus)
+apply (simp add: complex_of_real_def complex_one_def complex_minus)
 done
 
 lemma complex_i_mult_minus: "ii * (ii * x) = - x"
-apply (simp (no_asm) add: complex_mult_assoc [symmetric])
-done
+by (simp add: complex_mult_assoc [symmetric])
 declare complex_i_mult_minus [simp]
 
-lemma complex_i_mult_minus2: "ii * ii * x = - x"
-apply (simp (no_asm))
-done
-declare complex_i_mult_minus2 [simp]
 
 lemma cis_real_of_nat_Suc_mult:
    "cis (real (Suc n) * a) = cis a * cis (real n * a)"
-apply (unfold cis_def)
-apply (auto simp add: real_of_nat_Suc left_distrib cos_add sin_add complex_add_mult_distrib complex_add_mult_distrib2 complex_of_real_add complex_of_real_mult complex_mult_ac complex_add_ac)
-apply (auto simp add: complex_add_mult_distrib2 [symmetric] complex_mult_assoc [symmetric] i_mult_eq complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] complex_of_real_minus [symmetric] real_diff_def mult_ac simp del: i_mult_eq2)
+apply (simp add: cis_def)
+apply (auto simp add: real_of_nat_Suc left_distrib cos_add sin_add left_distrib right_distrib complex_of_real_add complex_of_real_mult mult_ac add_ac)
+apply (auto simp add: right_distrib [symmetric] complex_mult_assoc [symmetric] i_mult_eq complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] complex_of_real_minus [symmetric] real_diff_def mult_ac simp del: i_mult_eq2)
 done
 
 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
@@ -1363,14 +918,11 @@
 
 lemma DeMoivre2:
    "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
-apply (unfold rcis_def)
-apply (auto simp add: power_mult_distrib DeMoivre complex_of_real_pow)
+apply (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow)
 done
 
 lemma cis_inverse: "inverse(cis a) = cis (-a)"
-apply (unfold cis_def)
-apply (auto simp add: complex_inverse_complex_split complex_of_real_minus complex_diff_def)
-done
+by (simp add: cis_def complex_inverse_complex_split complex_of_real_minus complex_diff_def)
 declare cis_inverse [simp]
 
 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
@@ -1381,23 +933,20 @@
 done
 
 lemma cis_divide: "cis a / cis b = cis (a - b)"
-apply (unfold complex_divide_def)
-apply (auto simp add: cis_mult real_diff_def)
-done
+by (simp add: complex_divide_def cis_mult real_diff_def)
 
 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
-apply (unfold complex_divide_def)
-apply (case_tac "r2=0")
-apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
-apply (auto simp add: rcis_inverse rcis_mult real_diff_def)
+apply (simp add: complex_divide_def)
+apply (case_tac "r2=0", simp)
+apply (simp add: rcis_inverse rcis_mult real_diff_def)
 done
 
 lemma Re_cis: "Re(cis a) = cos a"
-by (unfold cis_def, auto)
+by (simp add: cis_def)
 declare Re_cis [simp]
 
 lemma Im_cis: "Im(cis a) = sin a"
-by (unfold cis_def, auto)
+by (simp add: cis_def)
 declare Im_cis [simp]
 
 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
@@ -1409,65 +958,40 @@
 lemma expi_Im_split:
     "expi (ii * complex_of_real y) =
      complex_of_real (cos y) + ii * complex_of_real (sin y)"
-by (unfold expi_def cis_def, auto)
+by (simp add: expi_def cis_def)
 
 lemma expi_Im_cis:
     "expi (ii * complex_of_real y) = cis y"
-by (unfold expi_def, auto)
+by (simp add: expi_def)
 
 lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
-apply (unfold expi_def)
-apply (auto simp add: complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult complex_mult_ac)
-done
+by (simp add: expi_def complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult mult_ac)
 
 lemma expi_complex_split:
      "expi(complex_of_real x + ii * complex_of_real y) =
       complex_of_real (exp(x)) * cis y"
-by (unfold expi_def, auto)
+by (simp add: expi_def)
 
 lemma expi_zero: "expi (0::complex) = 1"
-by (unfold expi_def, auto)
+by (simp add: expi_def)
 declare expi_zero [simp]
 
 lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (rule_tac z = w in eq_Abs_complex)
-apply (auto simp add: complex_mult)
-done
+by (induct z, induct w, simp add: complex_mult)
 
 lemma complex_Im_mult_eq:
      "Im (w * z) = Re w * Im z + Im w * Re z"
-apply (rule_tac z = z in eq_Abs_complex)
-apply (rule_tac z = w in eq_Abs_complex)
-apply (auto simp add: complex_mult)
+apply (induct z, induct w, simp add: complex_mult)
 done
 
 lemma complex_expi_Ex: 
    "\<exists>a r. z = complex_of_real r * expi a"
-apply (cut_tac z = z in rcis_Ex)
+apply (insert rcis_Ex [of z])
 apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult)
 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
 done
 
 
-(****
-Goal "[| - pi < a; a \<le> pi |] ==> (-pi < a & a \<le> 0) | (0 \<le> a & a \<le> pi)"
-by Auto_tac
-qed "lemma_split_interval";
-
-Goalw [arg_def]
-  "[| r \<noteq> 0; - pi < a; a \<le> pi |] \
-\  ==> arg(complex_of_real r * \
-\      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = a";
-by Auto_tac
-by (cut_inst_tac [("x","0"),("y","r")] linorder_less_linear 1);
-by (auto_tac (claset(),simpset() addsimps (map (full_rename_numerals thy)
-    [rabs_eqI2,rabs_minus_eqI2,real_minus_rinv]) [real_divide_def,
-    minus_mult_right RS sym] mult_ac));
-by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
-by (dtac lemma_split_interval 1 THEN safe)
-****)
-
 
 ML
 {*
@@ -1490,15 +1014,8 @@
 val complexpow_0 = thm"complexpow_0";
 val complexpow_Suc = thm"complexpow_Suc";
 
-val inj_Rep_complex = thm"inj_Rep_complex";
-val inj_Abs_complex = thm"inj_Abs_complex";
-val Abs_complex_cancel_iff = thm"Abs_complex_cancel_iff";
-val pair_mem_complex = thm"pair_mem_complex";
-val Abs_complex_inverse2 = thm"Abs_complex_inverse2";
-val eq_Abs_complex = thm"eq_Abs_complex";
 val Re = thm"Re";
 val Im = thm"Im";
-val Abs_complex_cancel = thm"Abs_complex_cancel";
 val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff";
 val complex_Re_zero = thm"complex_Re_zero";
 val complex_Im_zero = thm"complex_Im_zero";
@@ -1515,55 +1032,20 @@
 val complex_minus = thm"complex_minus";
 val complex_Re_minus = thm"complex_Re_minus";
 val complex_Im_minus = thm"complex_Im_minus";
-val complex_minus_minus = thm"complex_minus_minus";
-val inj_complex_minus = thm"inj_complex_minus";
 val complex_minus_zero = thm"complex_minus_zero";
 val complex_minus_zero_iff = thm"complex_minus_zero_iff";
-val complex_minus_zero_iff2 = thm"complex_minus_zero_iff2";
-val complex_minus_not_zero_iff = thm"complex_minus_not_zero_iff";
 val complex_add = thm"complex_add";
 val complex_Re_add = thm"complex_Re_add";
 val complex_Im_add = thm"complex_Im_add";
 val complex_add_commute = thm"complex_add_commute";
 val complex_add_assoc = thm"complex_add_assoc";
-val complex_add_left_commute = thm"complex_add_left_commute";
 val complex_add_zero_left = thm"complex_add_zero_left";
 val complex_add_zero_right = thm"complex_add_zero_right";
-val complex_add_minus_right_zero = thm"complex_add_minus_right_zero";
-val complex_add_minus_cancel = thm"complex_add_minus_cancel";
-val complex_minus_add_cancel = thm"complex_minus_add_cancel";
-val complex_add_minus_eq_minus = thm"complex_add_minus_eq_minus";
-val complex_minus_add_distrib = thm"complex_minus_add_distrib";
-val complex_add_left_cancel = thm"complex_add_left_cancel";
-val complex_add_right_cancel = thm"complex_add_right_cancel";
-val complex_eq_minus_iff = thm"complex_eq_minus_iff";
-val complex_eq_minus_iff2 = thm"complex_eq_minus_iff2";
-val complex_diff_0 = thm"complex_diff_0";
-val complex_diff_0_right = thm"complex_diff_0_right";
-val complex_diff_self = thm"complex_diff_self";
 val complex_diff = thm"complex_diff";
-val complex_diff_eq_eq = thm"complex_diff_eq_eq";
 val complex_mult = thm"complex_mult";
-val complex_mult_commute = thm"complex_mult_commute";
-val complex_mult_assoc = thm"complex_mult_assoc";
-val complex_mult_left_commute = thm"complex_mult_left_commute";
 val complex_mult_one_left = thm"complex_mult_one_left";
 val complex_mult_one_right = thm"complex_mult_one_right";
-val complex_mult_zero_left = thm"complex_mult_zero_left";
-val complex_mult_zero_right = thm"complex_mult_zero_right";
-val complex_divide_zero = thm"complex_divide_zero";
-val complex_minus_mult_eq1 = thm"complex_minus_mult_eq1";
-val complex_minus_mult_eq2 = thm"complex_minus_mult_eq2";
-val complex_minus_mult_commute = thm"complex_minus_mult_commute";
-val complex_add_mult_distrib = thm"complex_add_mult_distrib";
-val complex_add_mult_distrib2 = thm"complex_add_mult_distrib2";
-val complex_zero_not_eq_one = thm"complex_zero_not_eq_one";
 val complex_inverse = thm"complex_inverse";
-val COMPLEX_INVERSE_ZERO = thm"COMPLEX_INVERSE_ZERO";
-val COMPLEX_DIVISION_BY_ZERO = thm"COMPLEX_DIVISION_BY_ZERO";
-val complex_mult_inv_left = thm"complex_mult_inv_left";
-val complex_mult_inv_right = thm"complex_mult_inv_right";
-val inj_complex_of_real = thm"inj_complex_of_real";
 val complex_of_real_one = thm"complex_of_real_one";
 val complex_of_real_zero = thm"complex_of_real_zero";
 val complex_of_real_eq_iff = thm"complex_of_real_eq_iff";
@@ -1580,7 +1062,6 @@
 val complex_mod_complex_of_real = thm"complex_mod_complex_of_real";
 val complex_of_real_abs = thm"complex_of_real_abs";
 val complex_cnj = thm"complex_cnj";
-val inj_cnj = thm"inj_cnj";
 val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff";
 val complex_cnj_cnj = thm"complex_cnj_cnj";
 val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real";
@@ -1686,7 +1167,6 @@
 val rcis_zero_arg = thm"rcis_zero_arg";
 val complex_of_real_minus_one = thm"complex_of_real_minus_one";
 val complex_i_mult_minus = thm"complex_i_mult_minus";
-val complex_i_mult_minus2 = thm"complex_i_mult_minus2";
 val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult";
 val DeMoivre = thm"DeMoivre";
 val DeMoivre2 = thm"DeMoivre2";
@@ -1706,9 +1186,6 @@
 val complex_Re_mult_eq = thm"complex_Re_mult_eq";
 val complex_Im_mult_eq = thm"complex_Im_mult_eq";
 val complex_expi_Ex = thm"complex_expi_Ex";
-
-val complex_add_ac = thms"complex_add_ac";
-val complex_mult_ac = thms"complex_mult_ac";
 *}
 
 end

--- a/src/HOL/Complex/ComplexArith0.ML	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/ComplexArith0.ML	Tue Feb 03 11:06:36 2004 +0100
@@ -19,7 +19,7 @@
   val trans_tac         = Real_Numeral_Simprocs.trans_tac
   val norm_tac =  ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps @ mult_1s)) 
                   THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@complex_mult_minus_simps))
-                  THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_mult_ac))
+                  THEN ALLGOALS (simp_tac (HOL_ss addsimps mult_ac))
   val numeral_simp_tac	=  ALLGOALS (simp_tac (HOL_ss addsimps rel_complex_number_of@bin_simps))
   val simplify_meta_eq  = simplify_meta_eq
   end
@@ -105,7 +105,7 @@
   val dest_coeff	= dest_coeff
   val find_first	= find_first []
   val trans_tac         = Real_Numeral_Simprocs.trans_tac
-  val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@complex_mult_ac))
+  val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@mult_ac))
   end;
 
 
@@ -173,13 +173,13 @@
 Addsimps [complex_minus1_divide];
 
 Goal "(x + - a = (0::complex)) = (x=a)";
-by (simp_tac (simpset() addsimps [complex_diff_eq_eq,symmetric complex_diff_def]) 1);
+by (simp_tac (simpset() addsimps [diff_eq_eq,symmetric complex_diff_def]) 1);
 qed "complex_add_minus_iff";
 Addsimps [complex_add_minus_iff];
 
 Goal "(x+y = (0::complex)) = (y = -x)";
 by Auto_tac;
-by (dtac (sym RS (complex_diff_eq_eq RS iffD2)) 1);
+by (dtac (sym RS (diff_eq_eq RS iffD2)) 1);
 by Auto_tac;  
 qed "complex_add_eq_0_iff";
 AddIffs [complex_add_eq_0_iff];

--- a/src/HOL/Complex/ComplexBin.ML	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/ComplexBin.ML	Tue Feb 03 11:06:36 2004 +0100
@@ -60,11 +60,11 @@
 
 (*For specialist use: NOT as default simprules*)
 Goal "2 * z = (z+z::complex)";
-by (simp_tac (simpset () addsimps [lemma, complex_add_mult_distrib]) 1);
+by (simp_tac (simpset () addsimps [lemma, left_distrib]) 1);
 qed "complex_mult_2";
 
 Goal "z * 2 = (z+z::complex)";
-by (stac complex_mult_commute 1 THEN rtac complex_mult_2 1);
+by (stac mult_commute 1 THEN rtac complex_mult_2 1);
 qed "complex_mult_2_right";
 
 (** Equals (=) **)
@@ -88,7 +88,7 @@
 qed "complex_mult_minus1";
 
 Goal "z * -1 = -(z::complex)";
-by (stac complex_mult_commute 1 THEN rtac complex_mult_minus1 1);
+by (stac mult_commute 1 THEN rtac complex_mult_minus1 1);
 qed "complex_mult_minus1_right";
 
 Addsimps [complex_mult_minus1,complex_mult_minus1_right];
@@ -111,7 +111,7 @@
 qed "complex_add_number_of_left";
 
 Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::complex)";
-by (simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
+by (simp_tac (simpset() addsimps [mult_assoc RS sym]) 1);
 qed "complex_mult_number_of_left";
 
 Goalw [complex_diff_def]
@@ -121,7 +121,7 @@
 
 Goal "number_of v + (c - number_of w) = \
 \     number_of (bin_add v (bin_minus w)) + (c::complex)";
-by (auto_tac (claset(),simpset() addsimps [complex_diff_def]@ complex_add_ac));
+by (auto_tac (claset(),simpset() addsimps [complex_diff_def]@ add_ac));
 qed "complex_add_number_of_diff2";
 
 Addsimps [complex_add_number_of_left, complex_mult_number_of_left,
@@ -133,40 +133,10 @@
 (** Combining of literal coefficients in sums of products **)
 
 Goal "(x = y) = (x-y = (0::complex))";
-by (simp_tac (simpset() addsimps [complex_diff_eq_eq]) 1);   
+by (simp_tac (simpset() addsimps [diff_eq_eq]) 1);   
 qed "complex_eq_iff_diff_eq_0";
 
-(** For combine_numerals **)
 
-Goal "i*u + (j*u + k) = (i+j)*u + (k::complex)";
-by (asm_simp_tac (simpset() addsimps [complex_add_mult_distrib]
-    @ complex_add_ac) 1);
-qed "left_complex_add_mult_distrib";
-
-(** For cancel_numerals **)
-
-Goal "((x::complex) = u + v) = (x - (u + v) = 0)";
-by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq]));
-qed "complex_eq_add_diff_eq_0";
-
-Goal "((x::complex) = n) = (x - n = 0)";
-by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq]));
-qed "complex_eq_diff_eq_0";
-
-val complex_rel_iff_rel_0_rls = [complex_eq_diff_eq_0,complex_eq_add_diff_eq_0];
-
-Goal "!!i::complex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
-by (auto_tac (claset(), simpset() addsimps [complex_add_mult_distrib,
-    complex_diff_def] @ complex_add_ac));
-by (asm_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
-by (simp_tac (simpset() addsimps [complex_add_assoc]) 1);
-qed "complex_eq_add_iff1";
-
-Goal "!!i::complex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
-by (simp_tac (simpset() addsimps [ complex_eq_add_iff1]) 1);
-by (auto_tac (claset(), simpset() addsimps [complex_diff_def, 
-    complex_add_mult_distrib]@ complex_add_ac));
-qed "complex_eq_add_iff2";
 
 structure Complex_Numeral_Simprocs =
 struct
@@ -276,29 +246,26 @@
 		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
 
 (*To let us treat subtraction as addition*)
-val diff_simps = [complex_diff_def, complex_minus_add_distrib, 
-                  complex_minus_minus];
+val diff_simps = [complex_diff_def, minus_add_distrib, minus_minus];
 
 (* push the unary minus down: - x * y = x * - y *)
 val complex_minus_mult_eq_1_to_2 = 
-    [complex_minus_mult_eq1 RS sym, complex_minus_mult_eq2] MRS trans 
+    [minus_mult_left RS sym, minus_mult_right] MRS trans 
     |> standard;
 
 (*to extract again any uncancelled minuses*)
 val complex_minus_from_mult_simps = 
-    [complex_minus_minus, complex_minus_mult_eq1 RS sym, 
-     complex_minus_mult_eq2 RS sym];
+    [minus_minus, minus_mult_left RS sym, minus_mult_right RS sym];
 
 (*combine unary minus with numeric literals, however nested within a product*)
 val complex_mult_minus_simps =
-    [complex_mult_assoc, complex_minus_mult_eq1, complex_minus_mult_eq_1_to_2];
+    [mult_assoc, minus_mult_left, complex_minus_mult_eq_1_to_2];
 
 (*Final simplification: cancel + and *  *)
 val simplify_meta_eq = 
     Int_Numeral_Simprocs.simplify_meta_eq
-         [complex_add_zero_left, complex_add_zero_right,
- 	  complex_mult_zero_left, complex_mult_zero_right, complex_mult_one_left, 
-          complex_mult_one_right];
+         [add_zero_left, add_zero_right,
+ 	  mult_zero_left, mult_zero_right, mult_1, mult_1_right];
 
 val prep_simproc = Real_Numeral_Simprocs.prep_simproc;
 
@@ -313,11 +280,11 @@
   val trans_tac         = Real_Numeral_Simprocs.trans_tac
   val norm_tac = 
      ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
-                                         complex_minus_simps@complex_add_ac))
+                                         complex_minus_simps@add_ac))
      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@complex_mult_minus_simps))
      THEN ALLGOALS
               (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@
-                                         complex_add_ac@complex_mult_ac))
+                                         add_ac@mult_ac))
   val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   val simplify_meta_eq  = simplify_meta_eq
   end;
@@ -328,8 +295,8 @@
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_eq
   val dest_bal = HOLogic.dest_bin "op =" complexT
-  val bal_add1 = complex_eq_add_iff1 RS trans
-  val bal_add2 = complex_eq_add_iff2 RS trans
+  val bal_add1 = eq_add_iff1 RS trans
+  val bal_add2 = eq_add_iff2 RS trans
 );
 
 
@@ -348,15 +315,15 @@
   val dest_sum		= dest_sum
   val mk_coeff		= mk_coeff
   val dest_coeff	= dest_coeff 1
-  val left_distrib	= left_complex_add_mult_distrib RS trans
+  val left_distrib	= combine_common_factor RS trans
   val prove_conv	= Bin_Simprocs.prove_conv_nohyps
   val trans_tac         = Real_Numeral_Simprocs.trans_tac
   val norm_tac = 
      ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
-                                         complex_minus_simps@complex_add_ac))
+                                         complex_minus_simps@add_ac))
      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@complex_mult_minus_simps))
      THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@
-                                              complex_add_ac@complex_mult_ac))
+                                              add_ac@mult_ac))
   val numeral_simp_tac	= ALLGOALS 
                     (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   val simplify_meta_eq  = simplify_meta_eq
@@ -470,7 +437,7 @@
   val sg_ref    = Sign.self_ref (Theory.sign_of (the_context ()))
   val T	     = Complex_Numeral_Simprocs.complexT
   val plus   = Const ("op *", [T,T] ---> T)
-  val add_ac = complex_mult_ac
+  val add_ac = mult_ac
 end;
 
 structure Complex_Times_Assoc = Assoc_Fold (Complex_Times_Assoc_Data);
@@ -479,9 +446,6 @@
 
 Addsimps [complex_of_real_zero_iff];
 
-(*Simplification of  x-y = 0 *)
-
-AddIffs [complex_eq_iff_diff_eq_0 RS sym];
 
 (*** Real and imaginary stuff ***)
 

--- a/src/HOL/Complex/NSCA.ML	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/NSCA.ML	Tue Feb 03 11:06:36 2004 +0100
@@ -4,6 +4,9 @@
     Description : Infinite, infinitesimal complex number etc! 
 *)
 
+val complex_induct = thm"complex.induct";
+
+
 (*--------------------------------------------------------------------------------------*)
 (* Closure laws for members of (embedded) set standard complex SComplex                 *)
 (* -------------------------------------------------------------------------------------*)
@@ -785,8 +788,8 @@
 by (dres_inst_tac [("x","m")] spec 1);
 by (Ultra_tac 1);
 by (rename_tac "Z x" 1);
-by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
+by (case_tac "X x" 1);
+by (case_tac "Y x" 1);
 by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add,
     complex_mod] delsimps [realpow_Suc]));
 by (rtac order_le_less_trans 1 THEN assume_tac 2);
@@ -806,8 +809,8 @@
 by (dres_inst_tac [("x","m")] spec 1);
 by (Ultra_tac 1);
 by (rename_tac "Z x" 1);
-by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
+by (case_tac "X x" 1);
+by (case_tac "Y x" 1);
 by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add,
     complex_mod] delsimps [realpow_Suc]));
 by (rtac order_le_less_trans 1 THEN assume_tac 2);
@@ -836,12 +839,13 @@
 by (TRYALL(Force_tac));
 by (ultra_tac (claset(),HOL_ss) 1);
 by (dtac sym 1 THEN dtac sym 1); 
-by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
+by (case_tac "X x" 1);
+by (case_tac "Y x" 1);
 by (auto_tac (claset(),
     HOL_ss addsimps [complex_minus,complex_add,
     complex_mod, snd_conv, fst_conv,numeral_2_eq_2]));
-by (subgoal_tac "sqrt (abs(xa + - xb) ^ 2 + abs(y + - ya) ^ 2) < u" 1);
+by (rename_tac "a b c d" 1);
+by (subgoal_tac "sqrt (abs(a + - c) ^ 2 + abs(b + - d) ^ 2) < u" 1);
 by (rtac lemma_sqrt_hcomplex_capprox 2);
 by Auto_tac;
 by (asm_full_simp_tac (simpset() addsimps [power2_eq_square]) 1); 
@@ -868,7 +872,7 @@
 by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
 by (res_inst_tac [("x","u")] exI 1 THEN Auto_tac);
 by (Ultra_tac 1);
-by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (dtac sym 1 THEN case_tac "X x" 1);
 by (auto_tac (claset(),
     simpset() addsimps [complex_mod,numeral_2_eq_2] delsimps [realpow_Suc]));
 by (rtac ccontr 1 THEN dtac (linorder_not_less RS iffD1) 1);
@@ -884,7 +888,7 @@
 by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
 by (res_inst_tac [("x","u")] exI 1 THEN Auto_tac);
 by (Ultra_tac 1);
-by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (dtac sym 1 THEN case_tac "X x" 1);
 by (auto_tac (claset(),simpset() addsimps [complex_mod] delsimps [realpow_Suc]));
 by (rtac ccontr 1 THEN dtac (linorder_not_less RS iffD1) 1);
 by (dtac order_less_le_trans 1 THEN assume_tac 1);
@@ -901,7 +905,7 @@
 by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
 by (res_inst_tac [("x","2*(u + v)")] exI 1);
 by (Ultra_tac 1);
-by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (dtac sym 1 THEN case_tac "X x" 1);
 by (auto_tac (claset(),simpset() addsimps [complex_mod,numeral_2_eq_2] delsimps [realpow_Suc]));
 by (subgoal_tac "0 < u" 1 THEN arith_tac 2);
 by (subgoal_tac "0 < v" 1 THEN arith_tac 2);
@@ -941,7 +945,7 @@
     hcomplex_of_complex_def,SReal_def,hypreal_of_real_def]));
 by (res_inst_tac [("x","complex_of_real r + ii  * complex_of_real ra")] exI 1);
 by (Ultra_tac 1);
-by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (case_tac "X x" 1);
 by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,i_def,
     complex_add,complex_mult]));
 qed "Reals_Re_Im_SComplex";

--- a/src/HOL/Complex/NSComplex.thy	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/NSComplex.thy	Tue Feb 03 11:06:36 2004 +0100
@@ -372,7 +372,7 @@
 apply (rule_tac z = "z1" in eq_Abs_hcomplex)
 apply (rule_tac z = "z2" in eq_Abs_hcomplex)
 apply (rule_tac z = "w" in eq_Abs_hcomplex)
-apply (auto simp add: hcomplex_mult hcomplex_add complex_add_mult_distrib)
+apply (auto simp add: hcomplex_mult hcomplex_add left_distrib)
 done
 
 lemma hcomplex_zero_not_eq_one: "(0::hcomplex) \<noteq> (1::hcomplex)"
@@ -400,7 +400,7 @@
 apply (auto simp add: hcomplex_inverse hcomplex_mult)
 apply (ultra)
 apply (rule ccontr)
-apply (drule complex_mult_inv_left)
+apply (drule left_inverse)
 apply auto
 done
 
@@ -744,24 +744,6 @@
 done
 declare hcnj_one [simp]
 
-
-(* MOVE to NSComplexBin
-Goal "z + hcnj z =
-      hcomplex_of_hypreal (2 * hRe(z))"
-by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
-by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
-    hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
-qed "hcomplex_add_hcnj";
-
-Goal "z - hcnj z = \
-\     hcomplex_of_hypreal (hypreal_of_real 2 * hIm(z)) * iii";
-by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
-by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
-    hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
-    complex_diff_cnj,iii_def,hcomplex_mult]));
-qed "hcomplex_diff_hcnj";
-*)
-
 lemma hcomplex_hcnj_zero:
       "hcnj 0 = 0"
 apply (unfold hcomplex_zero_def)

--- a/src/HOL/Complex/NSComplexBin.ML	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/NSComplexBin.ML	Tue Feb 03 11:06:36 2004 +0100
@@ -161,49 +161,6 @@
 
 (**** Simprocs for numeric literals ****)
 
-(** Combining of literal coefficients in sums of products **)
-
-Goal "(x = y) = (x-y = (0::hcomplex))";
-by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq]) 1);   
-qed "hcomplex_eq_iff_diff_eq_0";
-
-(** For combine_numerals **)
-
-Goal "i*u + (j*u + k) = (i+j)*u + (k::hcomplex)";
-by (asm_simp_tac (simpset() addsimps [hcomplex_add_mult_distrib]
-    @ add_ac) 1);
-qed "left_hcomplex_add_mult_distrib";
-
-(** For cancel_numerals **)
-
-Goal "((x::hcomplex) = u + v) = (x - (u + v) = 0)";
-by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
-qed "hcomplex_eq_add_diff_eq_0";
-
-Goal "((x::hcomplex) = n) = (x - n = 0)";
-by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
-qed "hcomplex_eq_diff_eq_0";
-
-val hcomplex_rel_iff_rel_0_rls = [hcomplex_eq_diff_eq_0,hcomplex_eq_add_diff_eq_0];
-
-Goal "!!i::hcomplex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
-by (auto_tac (claset(), simpset() addsimps [hcomplex_add_mult_distrib,
-    hcomplex_diff_def] @ add_ac));
-by (asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1);
-by (simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1);
-qed "hcomplex_eq_add_iff1";
-
-Goal "!!i::hcomplex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
-by (res_inst_tac [("z","i")] eq_Abs_hcomplex 1);
-by (res_inst_tac [("z","j")] eq_Abs_hcomplex 1);
-by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1);
-by (res_inst_tac [("z","m")] eq_Abs_hcomplex 1);
-by (res_inst_tac [("z","n")] eq_Abs_hcomplex 1);
-by (auto_tac (claset(), simpset() addsimps [hcomplex_diff,hcomplex_add,
-    hcomplex_mult,complex_eq_add_iff2]));
-qed "hcomplex_eq_add_iff2";
-
-
 structure HComplex_Numeral_Simprocs =
 struct
 
@@ -358,8 +315,8 @@
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_eq
   val dest_bal = HOLogic.dest_bin "op =" hcomplexT
-  val bal_add1 = hcomplex_eq_add_iff1 RS trans
-  val bal_add2 = hcomplex_eq_add_iff2 RS trans
+  val bal_add1 = eq_add_iff1 RS trans
+  val bal_add2 = eq_add_iff2 RS trans
 );
 
 
@@ -378,7 +335,7 @@
   val dest_sum		= dest_sum
   val mk_coeff		= mk_coeff
   val dest_coeff	= dest_coeff 1
-  val left_distrib	= left_hcomplex_add_mult_distrib RS trans
+  val left_distrib	= combine_common_factor RS trans
   val prove_conv	= Bin_Simprocs.prove_conv_nohyps
   val trans_tac         = Real_Numeral_Simprocs.trans_tac
   val norm_tac = 
@@ -507,9 +464,6 @@
 
 Addsimps [hcomplex_of_complex_zero_iff];
 
-(*Simplification of  x-y = 0 *)
-
-AddIffs [hcomplex_eq_iff_diff_eq_0 RS sym];
 
 (** extra thms **)
 

--- a/src/HOL/Complex/ex/BinEx.thy	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/Complex/ex/BinEx.thy	Tue Feb 03 11:06:36 2004 +0100
@@ -381,8 +381,7 @@
 text{*Multiplication requires distributive laws.  Perhaps versions instantiated
 to literal constants should be added to the simpset.*}
 
-lemmas distrib = complex_add_mult_distrib complex_add_mult_distrib2
-                 complex_diff_mult_distrib complex_diff_mult_distrib2
+lemmas distrib = left_distrib right_distrib left_diff_distrib right_diff_distrib
 
 lemma "(1 + ii) * (1 - ii) = 2"
 by (simp add: distrib)
@@ -393,10 +392,8 @@
 lemma "(-84 + 255*ii) + (51 * 255*ii) = -84 + 13260 * ii"
 by (simp add: distrib)
 
-text{*No inequalities: we have no ordering on the complex numbers.*}
+text{*No inequalities or linear arithmetic: the complex numbers are unordered!*}
 
 text{*No powers (not supported yet)*}
 
-text{*No linear arithmetic*}
-
 end

--- a/src/HOL/IsaMakefile	Tue Feb 03 10:19:21 2004 +0100
+++ b/src/HOL/IsaMakefile	Tue Feb 03 11:06:36 2004 +0100
@@ -138,7 +138,7 @@
 
 $(OUT)/HOL-Complex: $(OUT)/HOL Complex/ROOT.ML\
   Library/Zorn.thy\
-  Real/Complex_Numbers.thy Real/Lubs.thy Real/rat_arith.ML Real/RatArith.thy\
+  Real/Lubs.thy Real/rat_arith.ML Real/RatArith.thy\
   Real/Rational.thy Real/PReal.thy Real/RComplete.thy \
   Real/ROOT.ML Real/Real.thy \
   Real/RealArith.thy Real/real_arith.ML Real/RealDef.thy \