converting Complex/Complex.ML to Isar

--- a/src/HOL/Complex/CLim.ML	Tue Dec 23 16:52:49 2003 +0100
+++ b/src/HOL/Complex/CLim.ML	Tue Dec 23 16:53:33 2003 +0100
@@ -1059,7 +1059,7 @@
      "x ~= 0 ==> NSCDERIV (%x. inverse(x)) x :> (- (inverse x ^ 2))";
 by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1);
 by (forward_tac [CInfinitesimal_add_not_zero] 1);
-by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_commute,two_eq_Suc_Suc]) 2); 
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_commute,numeral_2_eq_2]) 2); 
 by (auto_tac (claset(),
      simpset() addsimps [starfunC_inverse_inverse,hcomplex_diff_def] 
                delsimps [hcomplex_minus_mult_eq1 RS sym,
@@ -1116,7 +1116,7 @@
 
 
 Goal "x ~= (0::complex) \\<Longrightarrow> (x * inverse(x) ^ 2) = inverse x";
-by (auto_tac (claset(),simpset() addsimps [two_eq_Suc_Suc]));
+by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]));
 qed "lemma_complex_mult_inverse_squared";
 Addsimps [lemma_complex_mult_inverse_squared];
 

--- a/src/HOL/Complex/Complex.ML	Tue Dec 23 16:52:49 2003 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1588 +0,0 @@
-(*  Title:       Complex.ML
-    Author:      Jacques D. Fleuriot
-    Copyright:   2001  University of Edinburgh
-    Description: Complex numbers
-*)
-
-Goal "inj Rep_complex";
-by (rtac inj_inverseI 1);
-by (rtac Rep_complex_inverse 1);
-qed "inj_Rep_complex";
-
-Goal "inj Abs_complex";
-by (rtac inj_inverseI 1);
-by (rtac Abs_complex_inverse 1);
-by (simp_tac (simpset() addsimps [complex_def]) 1);
-qed "inj_Abs_complex";
-Addsimps [inj_Abs_complex RS injD];
-
-Goal "(Abs_complex x = Abs_complex y) = (x = y)";
-by (auto_tac (claset() addDs [inj_Abs_complex RS injD],simpset()));
-qed "Abs_complex_cancel_iff";
-Addsimps [Abs_complex_cancel_iff]; 
-
-Goalw [complex_def] "(x,y) : complex";
-by (Auto_tac);
-qed "pair_mem_complex";
-Addsimps [pair_mem_complex];
-
-Goal "Rep_complex (Abs_complex (x,y)) = (x,y)";
-by (simp_tac (simpset() addsimps [Abs_complex_inverse]) 1);
-qed "Abs_complex_inverse2";
-Addsimps [Abs_complex_inverse2];
-
-val [prem] = goal Complex.thy
-    "(!!x y. z = Abs_complex(x,y) ==> P) ==> P";
-by (res_inst_tac [("p","Rep_complex z")] PairE 1);
-by (dres_inst_tac [("f","Abs_complex")] arg_cong 1);
-by (res_inst_tac [("x","x"),("y","y")] prem 1);
-by (asm_full_simp_tac (simpset() addsimps [Rep_complex_inverse]) 1);
-qed "eq_Abs_complex";
-
-Goalw [Re_def] "Re(Abs_complex(x,y)) = x";
-by (Simp_tac 1);
-qed "Re";
-Addsimps [Re];
-
-Goalw [Im_def] "Im(Abs_complex(x,y)) = y";
-by (Simp_tac 1);
-qed "Im";
-Addsimps [Im];
-
-Goal "Abs_complex(Re(z),Im(z)) = z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (Asm_simp_tac 1);
-qed "Abs_complex_cancel";
-Addsimps [Abs_complex_cancel];
-
-Goal "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))";
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset() addDs [inj_Abs_complex RS injD],simpset()));
-qed "complex_Re_Im_cancel_iff";
-
-Goalw [complex_zero_def] "Re 0 = 0";
-by (Simp_tac 1);
-qed "complex_Re_zero";
-
-Goalw [complex_zero_def] "Im 0 = 0";
-by (Simp_tac 1);
-qed "complex_Im_zero";
-Addsimps [complex_Re_zero,complex_Im_zero];
-
-Goalw [complex_one_def] "Re 1 = 1";
-by (Simp_tac 1);
-qed "complex_Re_one";
-Addsimps [complex_Re_one];
-
-Goalw [complex_one_def] "Im 1 = 0";
-by (Simp_tac 1);
-qed "complex_Im_one";
-Addsimps [complex_Im_one];
-
-Goalw [i_def] "Re(ii) = 0";
-by Auto_tac;
-qed "complex_Re_i";
-Addsimps [complex_Re_i];
-
-Goalw [i_def] "Im(ii) = 1";
-by Auto_tac;
-qed "complex_Im_i";
-Addsimps [complex_Im_i];
-
-Goalw [complex_of_real_def] "Re(complex_of_real 0) = 0";
-by (Simp_tac 1);
-qed "Re_complex_of_real_zero";
-Addsimps [Re_complex_of_real_zero];
-
-Goalw [complex_of_real_def] "Im(complex_of_real 0) = 0";
-by (Simp_tac 1);
-qed "Im_complex_of_real_zero";
-Addsimps [Im_complex_of_real_zero];
-
-Goalw [complex_of_real_def] "Re(complex_of_real 1) = 1";
-by (Simp_tac 1);
-qed "Re_complex_of_real_one";
-Addsimps [Re_complex_of_real_one];
-
-Goalw [complex_of_real_def] "Im(complex_of_real 1) = 0";
-by (Simp_tac 1);
-qed "Im_complex_of_real_one";
-Addsimps [Im_complex_of_real_one];
-
-Goalw [complex_of_real_def] "Re(complex_of_real z) = z";
-by Auto_tac;
-qed "Re_complex_of_real";
-Addsimps [Re_complex_of_real];
-
-Goalw [complex_of_real_def] "Im(complex_of_real z) = 0";
-by Auto_tac;
-qed "Im_complex_of_real";
-Addsimps [Im_complex_of_real];
-
-(*** negation ***)
-
-Goalw [complex_minus_def] "- Abs_complex(x,y) = Abs_complex(-x,-y)";
-by (Simp_tac 1);
-qed "complex_minus";
-
-Goalw [Re_def] "Re (-z) = - Re z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_minus]));
-qed "complex_Re_minus";
-
-Goalw [Im_def] "Im (-z) = - Im z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_minus]));
-qed "complex_Im_minus";
-
-Goalw [complex_minus_def] "- (- z) = (z::complex)";
-by (Simp_tac 1);
-qed "complex_minus_minus";
-Addsimps [complex_minus_minus];
-
-Goal "inj(%r::complex. -r)";
-by (rtac injI 1);
-by (dres_inst_tac [("f","uminus")] arg_cong 1);
-by (Asm_full_simp_tac 1);
-qed "inj_complex_minus";
-
-Goalw [complex_zero_def] "-(0::complex) = 0";
-by (simp_tac (simpset() addsimps [complex_minus]) 1);
-qed "complex_minus_zero";
-Addsimps [complex_minus_zero];
-
-Goal "(-x = 0) = (x = (0::complex))"; 
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (auto_tac (claset() addDs [inj_Abs_complex RS injD],simpset() 
-    addsimps [complex_zero_def,complex_minus]));
-qed "complex_minus_zero_iff";
-Addsimps [complex_minus_zero_iff];
-
-Goal "(0 = -x) = (x = (0::real))"; 
-by (auto_tac (claset() addDs [sym],simpset()));
-qed "complex_minus_zero_iff2";
-Addsimps [complex_minus_zero_iff2];
-
-Goal "(-x ~= 0) = (x ~= (0::complex))"; 
-by Auto_tac;
-qed "complex_minus_not_zero_iff";
-
-(*** addition ***)
-
-Goalw [complex_add_def]
-      "Abs_complex(x1,y1) + Abs_complex(x2,y2) = Abs_complex(x1+x2,y1+y2)";
-by (Simp_tac 1);
-qed "complex_add";
-
-Goalw [Re_def] "Re(x + y) = Re(x) + Re(y)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_add]));
-qed "complex_Re_add";
-
-Goalw [Im_def] "Im(x + y) = Im(x) + Im(y)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_add]));
-qed "complex_Im_add";
-
-Goalw [complex_add_def] "(u::complex) + v = v + u";
-by (simp_tac (simpset() addsimps [real_add_commute]) 1);
-qed "complex_add_commute";
-
-Goalw [complex_add_def] "((u::complex) + v) + w = u + (v + w)";
-by (simp_tac (simpset() addsimps [real_add_assoc]) 1);
-qed "complex_add_assoc";
-
-Goalw [complex_add_def] "(x::complex) + (y + z) = y + (x + z)";
-by (simp_tac (simpset() addsimps [real_add_left_commute]) 1);
-qed "complex_add_left_commute";
-
-val complex_add_ac = [complex_add_assoc,complex_add_commute,
-                      complex_add_left_commute];
-
-Goalw [complex_add_def,complex_zero_def] "(0::complex) + z = z";
-by (Simp_tac 1);
-qed "complex_add_zero_left";
-Addsimps [complex_add_zero_left];
-
-Goalw [complex_add_def,complex_zero_def] "z + (0::complex) = z";
-by (Simp_tac 1);
-qed "complex_add_zero_right";
-Addsimps [complex_add_zero_right];
-
-Goalw [complex_add_def,complex_minus_def,complex_zero_def] 
-      "z + -z = (0::complex)";
-by (Simp_tac 1);
-qed "complex_add_minus_right_zero";
-Addsimps [complex_add_minus_right_zero];
-
-Goalw [complex_add_def,complex_minus_def,complex_zero_def] 
-      "-z + z = (0::complex)";
-by (Simp_tac 1);
-qed "complex_add_minus_left_zero";
-Addsimps [complex_add_minus_left_zero];
-
-Goal "z + (- z + w) = (w::complex)";
-by (simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
-qed "complex_add_minus_cancel";
-
-Goal "(-z) + (z + w) = (w::complex)";
-by (simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
-qed "complex_minus_add_cancel";
-
-Addsimps [complex_add_minus_cancel, complex_minus_add_cancel];
-
-Goal "x + y = (0::complex) ==> x = -y";
-by (auto_tac (claset(),simpset() addsimps [complex_Re_Im_cancel_iff,
-    complex_Re_add,complex_Im_add,complex_Re_minus,complex_Im_minus]));
-qed "complex_add_minus_eq_minus";
-
-Goal "-(x + y) = -x + -(y::complex)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add]));
-qed "complex_minus_add_distrib";
-Addsimps [complex_minus_add_distrib];
-
-Goal "((x::complex) + y = x + z) = (y = z)";
-by (Step_tac 1);
-by (dres_inst_tac [("f","%t.-x + t")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
-qed "complex_add_left_cancel";
-AddIffs [complex_add_left_cancel];
-
-Goal "(y + (x::complex)= z + x) = (y = z)";
-by (simp_tac (simpset() addsimps [complex_add_commute]) 1);
-qed "complex_add_right_cancel";
-Addsimps [complex_add_right_cancel];
-
-Goal "((x::complex) = y) = (0 = x + - y)";
-by (Step_tac 1);
-by (res_inst_tac [("x1","-y")] 
-      (complex_add_right_cancel RS iffD1) 2);
-by (Auto_tac);
-qed "complex_eq_minus_iff"; 
-
-Goal "((x::complex) = y) = (x + - y = 0)";
-by (Step_tac 1);
-by (res_inst_tac [("x1","-y")] 
-      (complex_add_right_cancel RS iffD1) 2);
-by (Auto_tac);
-qed "complex_eq_minus_iff2"; 
-
-Goal "(0::complex) - x = -x";
-by (simp_tac (simpset() addsimps [complex_diff_def]) 1);
-qed "complex_diff_0";
-
-Goal "x - (0::complex) = x";
-by (simp_tac (simpset() addsimps [complex_diff_def]) 1);
-qed "complex_diff_0_right";
-
-Goal "x - x = (0::complex)";
-by (simp_tac (simpset() addsimps [complex_diff_def]) 1);
-qed "complex_diff_self";
-
-Addsimps [complex_diff_0, complex_diff_0_right, complex_diff_self];
-
-Goalw [complex_diff_def]
-      "Abs_complex(x1,y1) - Abs_complex(x2,y2) = Abs_complex(x1-x2,y1-y2)";
-by (simp_tac (simpset() addsimps [complex_add,complex_minus]) 1);
-qed "complex_diff";
-
-Goal "((x::complex) - y = z) = (x = z + y)";
-by (auto_tac (claset(),simpset() addsimps [complex_diff_def,complex_add_assoc]));
-qed "complex_diff_eq_eq";
-
-(*** complex multiplication ***)
-
-Goalw [complex_mult_def]
-      "Abs_complex(x1,y1) * Abs_complex(x2,y2) = \
-\      Abs_complex(x1*x2 - y1*y2,x1*y2 + y1*x2)";
-by (Simp_tac 1);
-qed "complex_mult";
-
-Goalw [complex_mult_def] "(w::complex) * z = z * w";
-by (simp_tac (simpset() addsimps [real_mult_commute,real_add_commute]) 1);
-qed "complex_mult_commute";
-
-Goalw [complex_mult_def] "((u::complex) * v) * w = u * (v * w)";
-by (simp_tac (simpset() addsimps [complex_Re_Im_cancel_iff,
-    real_mult_assoc,real_diff_mult_distrib2,
-    real_add_mult_distrib2,real_diff_mult_distrib,
-    real_add_mult_distrib,real_mult_left_commute]) 1);
-qed "complex_mult_assoc";
-
-Goalw [complex_mult_def] "(x::complex) * (y * z) = y * (x * z)";
-by (simp_tac (simpset() addsimps [complex_Re_Im_cancel_iff,
-    real_mult_left_commute,real_diff_mult_distrib2,
-    real_add_mult_distrib2]) 1);
-qed "complex_mult_left_commute";
-
-val complex_mult_ac = [complex_mult_assoc,complex_mult_commute,
-                      complex_mult_left_commute];
-
-Goalw [complex_one_def] "(1::complex) * z = z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_mult]) 1);
-qed "complex_mult_one_left";
-Addsimps [complex_mult_one_left];
-
-Goal "z * (1::complex) = z";
-by (simp_tac (simpset() addsimps [complex_mult_commute]) 1);
-qed "complex_mult_one_right";
-Addsimps [complex_mult_one_right];
-
-Goalw [complex_zero_def] "(0::complex) * z = 0";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_mult]) 1);
-qed "complex_mult_zero_left";
-Addsimps [complex_mult_zero_left];
-
-Goal "z * 0 = (0::complex)";
-by (simp_tac (simpset() addsimps [complex_mult_commute]) 1);
-qed "complex_mult_zero_right";
-Addsimps [complex_mult_zero_right];
-
-Goalw [complex_divide_def] "0 / z = (0::complex)";
-by Auto_tac;
-qed "complex_divide_zero";
-Addsimps [complex_divide_zero];
-
-Goal "-(x * y) = -x * (y::complex)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_minus,
-    real_diff_def]));
-qed "complex_minus_mult_eq1";
-
-Goal "-(x * y) = x * -(y::complex)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_minus,
-    real_diff_def]));
-qed "complex_minus_mult_eq2";
-
-Addsimps [ complex_minus_mult_eq1 RS sym, complex_minus_mult_eq2 RS sym];
-
-Goal "-(1::complex) * z = -z";
-by (Simp_tac 1);
-qed "complex_mult_minus_one";
-Addsimps [complex_mult_minus_one];
-
-Goal "z * -(1::complex) = -z";
-by (stac complex_mult_commute 1);
-by (Simp_tac 1);
-qed "complex_mult_minus_one_right";
-Addsimps [complex_mult_minus_one_right];
-
-Goal "-x * -y = x * (y::complex)";
-by (Simp_tac 1);
-qed "complex_minus_mult_cancel";
-Addsimps [complex_minus_mult_cancel];
-
-Goal "-x * y = x * -(y::complex)";
-by (Simp_tac 1);
-qed "complex_minus_mult_commute";
-
-qed_goal "complex_add_assoc_cong" thy
-    "!!z. (z::complex) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
- (fn _ => [(asm_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1)]);
-
-qed_goal "complex_add_assoc_swap" thy "(z::complex) + (v + w) = v + (z + w)"
- (fn _ => [(REPEAT (ares_tac [complex_add_commute RS complex_add_assoc_cong] 1))]);
-
-Goal "((z1::complex) + z2) * w = (z1 * w) + (z2 * w)";
-by (res_inst_tac [("z","z1")] eq_Abs_complex 1);
-by (res_inst_tac [("z","z2")] eq_Abs_complex 1);
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add,
-    real_add_mult_distrib,real_diff_def] @ real_add_ac));
-qed "complex_add_mult_distrib";
-
-Goal "(w::complex) * (z1 + z2) = (w * z1) + (w * z2)";
-by (res_inst_tac [("z1","z1 + z2")] (complex_mult_commute RS ssubst) 1);
-by (simp_tac (simpset() addsimps [complex_add_mult_distrib]) 1);
-by (simp_tac (simpset() addsimps [complex_mult_commute]) 1);
-qed "complex_add_mult_distrib2";
-
-Goalw [complex_zero_def,complex_one_def] "(0::complex) ~= 1";
-by (simp_tac (simpset() addsimps [complex_Re_Im_cancel_iff]) 1);
-qed "complex_zero_not_eq_one";
-Addsimps [complex_zero_not_eq_one];
-Addsimps [complex_zero_not_eq_one RS not_sym];
-
-(*** inverse ***)
-Goalw [complex_inverse_def] "inverse (Abs_complex(x,y)) = \
-\    Abs_complex(x/(x ^ 2 + y ^ 2),-y/(x ^ 2 + y ^ 2))";
-by (Simp_tac 1);
-qed "complex_inverse";
-
-Goalw [complex_inverse_def,complex_zero_def] "inverse 0 = (0::complex)";
-by Auto_tac;
-qed "COMPLEX_INVERSE_ZERO";
-
-Goal "a / (0::complex) = 0";
-by (simp_tac (simpset() addsimps [complex_divide_def, COMPLEX_INVERSE_ZERO]) 1);
-qed "COMPLEX_DIVISION_BY_ZERO";  (*NOT for adding to default simpset*)
-
-fun complex_div_undefined_case_tac s i =
-  case_tac s i THEN 
-  asm_simp_tac (simpset() addsimps [COMPLEX_DIVISION_BY_ZERO, COMPLEX_INVERSE_ZERO]) i;
-
-(*REMOVE?:
-lemmas:replace previous versions to accommodate new behaviour of simplification
-Goal "x ^ 2 + y ^ 2 = 0 ==> x = (0::real)";
-by (auto_tac (claset() addIs [real_sum_squares_cancel],
-    simpset() addsimps [CLAIM "2 = Suc(Suc 0)"]));
-qed "real_sum_squares_cancel";
-
-Goal "x ^ 2 + y ^ 2 = 0 ==> y = (0::real)";
-by (auto_tac (claset() addIs [real_sum_squares_cancel2],
-    simpset() addsimps [CLAIM "2 = Suc(Suc 0)"]));
-qed "real_sum_squares_cancel2";
-*)
-
-Goal "z ~= (0::complex) ==> inverse(z) * z = 1";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_inverse,
-    complex_one_def,complex_zero_def,real_add_divide_distrib RS sym,
-    realpow_two_eq_mult] @ real_mult_ac));
-by (dres_inst_tac [("y","y")] real_sum_squares_not_zero 1);
-by (dres_inst_tac [("x","x")] real_sum_squares_not_zero2 2);
-by Auto_tac;
-qed "complex_mult_inv_left";
-Addsimps [complex_mult_inv_left];
-
-Goal "z ~= (0::complex) ==> z * inverse(z) = 1";
-by (auto_tac (claset() addIs [complex_mult_commute RS subst],simpset()));
-qed "complex_mult_inv_right";
-Addsimps [complex_mult_inv_right];
-
-Goal "(c::complex) ~= 0 ==> (c*a=c*b) = (a=b)";
-by Auto_tac;
-by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps complex_mult_ac)  1);
-qed "complex_mult_left_cancel";
-    
-Goal "(c::complex) ~= 0 ==> (a*c=b*c) = (a=b)";
-by (Step_tac 1);
-by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps complex_mult_ac)  1);
-qed "complex_mult_right_cancel";
-
-Goal "z ~= 0 ==> inverse(z::complex) ~= 0";
-by (Step_tac 1);
-by (ftac (complex_mult_right_cancel RS iffD2) 1);
-by (thin_tac "inverse z = 0" 2);
-by (assume_tac 1 THEN Auto_tac);
-qed "complex_inverse_not_zero";
-Addsimps [complex_inverse_not_zero];
-
-Goal "!!x. [| x ~= 0; y ~= (0::complex) |] ==> x * y ~= 0";
-by (Step_tac 1);
-by (dres_inst_tac [("f","%z. inverse x*z")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
-qed "complex_mult_not_zero";
-
-bind_thm ("complex_mult_not_zeroE",complex_mult_not_zero RS notE);
-
-Goal "inverse(inverse (x::complex)) = x";
-by (complex_div_undefined_case_tac "x = 0" 1);
-by (res_inst_tac [("c1","inverse x")] (complex_mult_right_cancel RS iffD1) 1);
-by (etac complex_inverse_not_zero 1);
-by (auto_tac (claset() addDs [complex_inverse_not_zero],simpset()));
-qed "complex_inverse_inverse";
-Addsimps [complex_inverse_inverse];
-
-Goalw [complex_one_def] "inverse(1::complex) = 1";
-by (simp_tac (simpset() addsimps [complex_inverse,realpow_num_two]) 1);
-qed "complex_inverse_one";
-Addsimps [complex_inverse_one];
-
-Goal "inverse(-x) = -inverse(x::complex)";
-by (complex_div_undefined_case_tac "x = 0" 1);
-by (res_inst_tac [("c1","-x")] (complex_mult_right_cancel RS iffD1) 1);
-by (stac complex_mult_inv_left 2);
-by Auto_tac;
-qed "complex_minus_inverse";
-
-Goal "inverse(x*y) = inverse x * inverse (y::complex)";
-by (complex_div_undefined_case_tac "x = 0" 1);
-by (complex_div_undefined_case_tac "y = 0" 1);
-by (res_inst_tac [("c1","x*y")] (complex_mult_left_cancel RS iffD1) 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult_not_zero]
-    @ complex_mult_ac));
-by (auto_tac (claset(),simpset() addsimps [complex_mult_not_zero,
-    complex_mult_assoc RS sym]));
-qed "complex_inverse_distrib";
-
-
-(*** division ***)
-
-(*adding some of these theorems to simpset as for reals: 
-  not 100% convinced for some*)
-
-Goal "(x::complex) * (y/z) = (x*y)/z";
-by (simp_tac (simpset() addsimps [complex_divide_def, complex_mult_assoc]) 1); 
-qed "complex_times_divide1_eq";
-
-Goal "(y/z) * (x::complex) = (y*x)/z";
-by (simp_tac (simpset() addsimps [complex_divide_def]@complex_mult_ac) 1); 
-qed "complex_times_divide2_eq";
-
-Addsimps [complex_times_divide1_eq, complex_times_divide2_eq];
-
-Goal "(x::complex) / (y/z) = (x*z)/y";
-by (simp_tac (simpset() addsimps [complex_divide_def, complex_inverse_distrib]@
-                                  complex_mult_ac) 1); 
-qed "complex_divide_divide1_eq";
-
-Goal "((x::complex) / y) / z = x/(y*z)";
-by (simp_tac (simpset() addsimps [complex_divide_def, complex_inverse_distrib, 
-                                  complex_mult_assoc]) 1); 
-qed "complex_divide_divide2_eq";
-
-Addsimps [complex_divide_divide1_eq, complex_divide_divide2_eq];
-
-(** As with multiplication, pull minus signs OUT of the / operator **)
-
-Goal "(-x) / (y::complex) = - (x/y)";
-by (simp_tac (simpset() addsimps [complex_divide_def]) 1); 
-qed "complex_minus_divide_eq";
-Addsimps [complex_minus_divide_eq];
-
-Goal "(x / -(y::complex)) = - (x/y)";
-by (simp_tac (simpset() addsimps [complex_divide_def, complex_minus_inverse]) 1); 
-qed "complex_divide_minus_eq";
-Addsimps [complex_divide_minus_eq];
-
-Goal "(x+y)/(z::complex) = x/z + y/z";
-by (simp_tac (simpset() addsimps [complex_divide_def, complex_add_mult_distrib]) 1); 
-qed "complex_add_divide_distrib";
-
-(*---------------------------------------------------------------------------*)
-(*          Embedding properties for complex_of_real map                     *)
-(*---------------------------------------------------------------------------*)
-
-Goal "inj complex_of_real";
-by (rtac injI 1);
-by (auto_tac (claset() addDs [inj_Abs_complex RS injD],
-    simpset() addsimps [complex_of_real_def]));
-qed "inj_complex_of_real";
-
-Goalw [complex_one_def,complex_of_real_def]
-      "complex_of_real 1 = 1";
-by (rtac refl 1);
-qed "complex_of_real_one";
-Addsimps [complex_of_real_one];
-
-Goalw [complex_zero_def,complex_of_real_def]
-      "complex_of_real 0 = 0";
-by (rtac refl 1);
-qed "complex_of_real_zero";
-Addsimps [complex_of_real_zero];
-
-Goal "(complex_of_real x = complex_of_real y) = (x = y)";
-by (auto_tac (claset() addDs [inj_complex_of_real RS injD],simpset()));
-qed "complex_of_real_eq_iff";
-AddIffs [complex_of_real_eq_iff];
-
-Goal "complex_of_real(-x) = - complex_of_real x";
-by (simp_tac (simpset() addsimps [complex_of_real_def,complex_minus]) 1);
-qed "complex_of_real_minus";
-
-Goal "complex_of_real(inverse x) = inverse(complex_of_real x)";
-by (case_tac "x=0" 1);
-by (simp_tac (simpset() addsimps [DIVISION_BY_ZERO,COMPLEX_INVERSE_ZERO]) 1);
-by (auto_tac (claset(),simpset() addsimps [complex_inverse,
-    complex_of_real_def,realpow_num_two,real_divide_def,
-    real_inverse_distrib]));
-qed "complex_of_real_inverse";
-
-Goal "complex_of_real x + complex_of_real y = complex_of_real (x + y)";
-by (simp_tac (simpset() addsimps [complex_add,complex_of_real_def]) 1);
-qed "complex_of_real_add";
-
-Goal "complex_of_real x - complex_of_real y = complex_of_real (x - y)";
-by (simp_tac (simpset() addsimps [complex_of_real_minus RS sym,
-    complex_diff_def,complex_of_real_add]) 1);
-qed "complex_of_real_diff";
-
-Goal "complex_of_real x * complex_of_real y = complex_of_real (x * y)";
-by (simp_tac (simpset() addsimps [complex_mult,complex_of_real_def]) 1);
-qed "complex_of_real_mult";
-
-Goalw [complex_divide_def] 
-      "complex_of_real x / complex_of_real y = complex_of_real(x/y)";
-by (case_tac "y=0" 1);
-by (asm_simp_tac (simpset() addsimps [rename_numerals DIVISION_BY_ZERO,
-    COMPLEX_INVERSE_ZERO]) 1);
-by (asm_simp_tac (simpset() addsimps [complex_of_real_mult RS sym,
-    complex_of_real_inverse,real_divide_def]) 1);
-qed "complex_of_real_divide";
-
-Goal "complex_of_real (x ^ n) = (complex_of_real x) ^ n";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_mult RS sym]));
-qed "complex_of_real_pow";
-
-Goalw [cmod_def] "cmod (Abs_complex(x,y)) = sqrt(x ^ 2 + y ^ 2)";
-by (Simp_tac 1);
-qed "complex_mod";
-
-Goalw [cmod_def] "cmod(0) = 0";
-by (Simp_tac 1);
-qed "complex_mod_zero";
-Addsimps [complex_mod_zero];
-
-Goalw [cmod_def] "cmod(1) = 1";
-by (simp_tac (simpset() addsimps [realpow_num_two]) 1);
-qed "complex_mod_one";
-Addsimps [complex_mod_one];
-
-Goalw [complex_of_real_def] "cmod(complex_of_real x) = abs x";
-by (simp_tac (simpset() addsimps [complex_mod,realpow_num_two]) 1);
-qed "complex_mod_complex_of_real";
-Addsimps [complex_mod_complex_of_real];
-
-Goal "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))";
-by (Simp_tac 1);
-qed "complex_of_real_abs";
-
-(*---------------------------------------------------------------------------*)
-(*                   conjugation is an automorphism                          *)
-(*---------------------------------------------------------------------------*)
-
-Goalw [cnj_def] "cnj (Abs_complex(x,y)) = Abs_complex(x,-y)";
-by (Simp_tac 1);
-qed "complex_cnj";
-
-Goal "inj cnj";
-by (rtac injI 1);
-by (auto_tac (claset(),simpset() addsimps [cnj_def,
-    Abs_complex_cancel_iff,complex_Re_Im_cancel_iff]));
-qed "inj_cnj";
-
-Goal "(cnj x = cnj y) = (x = y)";
-by (auto_tac (claset() addDs [inj_cnj RS injD],simpset()));
-qed "complex_cnj_cancel_iff";
-Addsimps [complex_cnj_cancel_iff];
-
-Goalw [cnj_def] "cnj (cnj z) = z";
-by (Simp_tac 1);
-qed "complex_cnj_cnj";
-Addsimps [complex_cnj_cnj];
-
-Goalw [complex_of_real_def] "cnj (complex_of_real x) = complex_of_real x";
-by (simp_tac (simpset() addsimps [complex_cnj]) 1);
-qed "complex_cnj_complex_of_real";
-Addsimps [complex_cnj_complex_of_real];
-
-Goal "cmod (cnj z) = cmod z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_mod,realpow_num_two]) 1);
-qed "complex_mod_cnj";
-Addsimps [complex_mod_cnj];
-
-Goalw [cnj_def] "cnj (-z) = - cnj z";
-by (simp_tac (simpset() addsimps [complex_minus,
-    complex_Re_minus,complex_Im_minus]) 1);
-qed "complex_cnj_minus";
-
-Goal "cnj(inverse z) = inverse(cnj z)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_inverse,
-    realpow_num_two]) 1);
-qed "complex_cnj_inverse";
-
-Goal "cnj(w + z) = cnj(w) + cnj(z)";
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_add]) 1);
-qed "complex_cnj_add";
-
-Goalw [complex_diff_def] "cnj(w - z) = cnj(w) - cnj(z)";
-by (simp_tac (simpset() addsimps [complex_cnj_add,complex_cnj_minus]) 1);
-qed "complex_cnj_diff";
-
-Goal "cnj(w * z) = cnj(w) * cnj(z)";
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_mult]) 1);
-qed "complex_cnj_mult";
-
-Goalw [complex_divide_def] "cnj(w / z) = (cnj w)/(cnj z)";
-by (simp_tac (simpset() addsimps [complex_cnj_mult,complex_cnj_inverse]) 1);
-qed "complex_cnj_divide";
-
-Goalw [cnj_def,complex_one_def] "cnj 1 = 1";
-by (Simp_tac 1);
-qed "complex_cnj_one";
-Addsimps [complex_cnj_one];
-
-Goal "cnj(z ^ n) = cnj(z) ^ n";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [complex_cnj_mult]));
-qed "complex_cnj_pow";
-
-Goal "z + cnj z = complex_of_real (2 * Re(z))";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_add,complex_cnj,
-    complex_of_real_def]) 1);
-qed "complex_add_cnj";
-
-Goal "z - cnj z = complex_of_real (2 * Im(z)) * ii";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_add,complex_cnj,
-    complex_of_real_def,complex_diff_def,complex_minus,
-    i_def,complex_mult]) 1);
-qed "complex_diff_cnj";
-
-goalw Complex.thy  [cnj_def,complex_zero_def] 
-      "cnj 0 = 0";
-by Auto_tac;
-qed "complex_cnj_zero";
-Addsimps [complex_cnj_zero];
-
-goal Complex.thy "(cnj z = 0) = (z = 0)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_zero_def,
-    complex_cnj]));
-qed "complex_cnj_zero_iff";
-AddIffs [complex_cnj_zero_iff];
-
-Goal "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_cnj,complex_mult,
-    complex_of_real_def,realpow_num_two]));
-qed "complex_mult_cnj";
-
-(*---------------------------------------------------------------------------*)
-(*                              algebra                                      *)
-(*---------------------------------------------------------------------------*)
-
-Goal "(x*y = (0::complex)) = (x = 0 | y = 0)";
-by Auto_tac;
-by (auto_tac (claset() addIs [ccontr] addDs 
-    [complex_mult_not_zero],simpset()));
-qed "complex_mult_zero_iff";
-AddIffs [complex_mult_zero_iff];
-
-Goalw [complex_zero_def] "(x + y = x) = (y = (0::complex))";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_add]));
-qed "complex_add_left_cancel_zero";
-Addsimps [complex_add_left_cancel_zero];
-
-Goalw [complex_diff_def] 
-      "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)";
-by (simp_tac (simpset() addsimps [complex_add_mult_distrib]) 1);
-qed "complex_diff_mult_distrib";
-
-Goalw [complex_diff_def]
-      "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)";
-by (simp_tac (simpset() addsimps [complex_add_mult_distrib2]) 1);
-qed "complex_diff_mult_distrib2";
-
-(*---------------------------------------------------------------------------*)
-(*                               modulus                                     *)
-(*---------------------------------------------------------------------------*)
-
-(*
-Goal "[| sqrt(x) = 0; 0 <= x |] ==> x = 0";
-by (auto_tac (claset() addIs [real_sqrt_eq_zero_cancel],
-    simpset()));
-qed "real_sqrt_eq_zero_cancel2";
-*)
-
-Goal "(cmod x = 0) = (x = 0)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (auto_tac (claset() addIs
-    [real_sum_squares_cancel,real_sum_squares_cancel2],
-    simpset() addsimps [complex_mod,complex_zero_def,
-    realpow_num_two]));
-qed "complex_mod_eq_zero_cancel";
-Addsimps [complex_mod_eq_zero_cancel];
-
-Goal "cmod (complex_of_real(real (n::nat))) = real n";
-by (Simp_tac 1);
-qed "complex_mod_complex_of_real_of_nat";
-Addsimps [complex_mod_complex_of_real_of_nat];
-
-Goal "cmod (-x) = cmod(x)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_mod,complex_minus,
-    realpow_num_two]) 1);
-qed "complex_mod_minus";
-Addsimps [complex_mod_minus];
-
-Goal "cmod(z * cnj(z)) = cmod(z) ^ 2";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_mod,complex_cnj,
-    complex_mult, CLAIM "0 ^ 2 = (0::real)"]) 1);
-by (simp_tac (simpset() addsimps [realpow_two_eq_mult, real_abs_def]) 1);
-qed "complex_mod_mult_cnj";
-
-Goalw [cmod_def] "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2";
-by Auto_tac;
-qed "complex_mod_squared";
-
-Goalw [cmod_def]  "0 <= cmod x";
-by (auto_tac (claset() addIs [real_sqrt_ge_zero],simpset()));
-qed "complex_mod_ge_zero";
-Addsimps [complex_mod_ge_zero];
-
-Goal "abs(cmod x) = cmod x";
-by (auto_tac (claset() addIs [abs_eqI1],simpset()));
-qed "abs_cmod_cancel";
-Addsimps [abs_cmod_cancel];
-
-Goal "cmod(x*y) = cmod(x) * cmod(y)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult,
-    complex_mod,real_sqrt_mult_distrib2 RS sym] delsimps [realpow_Suc]));
-by (res_inst_tac [("n","1")] realpow_Suc_cancel_eq 1);
-by (auto_tac (claset(),simpset() addsimps [realpow_num_two RS sym] 
-    delsimps [realpow_Suc]));
-by (auto_tac (claset(),simpset() addsimps [real_diff_def,realpow_num_two,
-    real_add_mult_distrib2,real_add_mult_distrib] @ real_add_ac @ 
-    real_mult_ac));
-qed "complex_mod_mult";
-
-Goal "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_add,
-    complex_mod_squared,complex_mult,complex_cnj,real_diff_def] 
-    delsimps [realpow_Suc]));
-by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2,
-    real_add_mult_distrib,realpow_num_two] @ real_mult_ac @ real_add_ac));
-qed "complex_mod_add_squared_eq";
-
-Goal "Re(x * cnj y) <= cmod(x * cnj y)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mod,
-    complex_mult,complex_cnj,real_diff_def] delsimps [realpow_Suc]));
-qed "complex_Re_mult_cnj_le_cmod";
-Addsimps [complex_Re_mult_cnj_le_cmod];
-
-Goal "Re(x * cnj y) <= cmod(x * y)";
-by (cut_inst_tac [("x","x"),("y","y")] complex_Re_mult_cnj_le_cmod 1);
-by (asm_full_simp_tac (simpset() addsimps [complex_mod_mult]) 1);
-qed "complex_Re_mult_cnj_le_cmod2";
-Addsimps [complex_Re_mult_cnj_le_cmod2];
-
-Goal "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y";
-by (simp_tac (simpset() addsimps [real_add_mult_distrib,
-    real_add_mult_distrib2,realpow_num_two]) 1);
-qed "real_sum_squared_expand";
-
-Goal "cmod (x + y) ^ 2 <= (cmod(x) + cmod(y)) ^ 2";
-by (simp_tac (simpset() addsimps [real_sum_squared_expand,
-    complex_mod_add_squared_eq,real_mult_assoc,complex_mod_mult RS sym]) 1);
-qed "complex_mod_triangle_squared";
-Addsimps [complex_mod_triangle_squared];
-
-Goal "- cmod x <= cmod x";
-by (rtac (complex_mod_ge_zero RSN (2,real_le_trans)) 1);
-by (Simp_tac 1);
-qed "complex_mod_minus_le_complex_mod";
-Addsimps [complex_mod_minus_le_complex_mod];
-
-Goal "cmod (x + y) <= cmod(x) + cmod(y)";
-by (res_inst_tac [("n","1")] realpow_increasing 1);
-by (auto_tac (claset() addIs [(complex_mod_ge_zero RSN (2,real_le_trans))],
-    simpset() addsimps [realpow_num_two RS sym]));
-qed "complex_mod_triangle_ineq";
-Addsimps [complex_mod_triangle_ineq];
-
-Goal "cmod(b + a) - cmod b <= cmod a";
-by (cut_inst_tac [("x1","b"),("y1","a"),("z","-cmod b")]
-   (complex_mod_triangle_ineq RS real_add_le_mono1) 1);
-by (Simp_tac 1);
-qed "complex_mod_triangle_ineq2";
-Addsimps [complex_mod_triangle_ineq2];
-
-Goal "cmod (x - y) = cmod (y - x)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (res_inst_tac [("z","y")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_diff,
-    complex_mod,real_diff_mult_distrib2,realpow_num_two,
-    real_diff_mult_distrib] @ real_add_ac @ real_mult_ac));
-qed "complex_mod_diff_commute";
-
-Goal "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s";
-by (auto_tac (claset() addIs [order_le_less_trans,
-    complex_mod_triangle_ineq],simpset()));
-qed "complex_mod_add_less";
-
-Goal "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s";
-by (auto_tac (claset() addIs [real_mult_less_mono'],simpset()     
-    addsimps [complex_mod_mult]));
-qed "complex_mod_mult_less";
-
-goal Complex.thy "cmod(a) - cmod(b) <= cmod(a + b)";
-by (res_inst_tac [("R1.0","cmod(a)"),("R2.0","cmod(b)")]
-    real_linear_less2 1);
-by Auto_tac;
-by (dtac (ARITH_PROVE "a < b ==> a - (b::real) < 0") 1);
-by (rtac real_le_trans 1 THEN rtac order_less_imp_le 1);
-by Auto_tac;
-by (dtac (ARITH_PROVE "a < b ==> 0 < (b::real) - a") 1);
-by (rtac (ARITH_PROVE "a  <= b + c ==> a - c <= (b::real)") 1);
-by (rtac (complex_mod_minus RS subst) 1);
-by (rtac real_le_trans 1);
-by (rtac complex_mod_triangle_ineq 2);
-by (auto_tac (claset(),simpset() addsimps complex_add_ac));
-qed "complex_mod_diff_ineq";
-Addsimps [complex_mod_diff_ineq];
-
-Goal "Re z <= cmod z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mod]
-    delsimps [realpow_Suc]));
-qed "complex_Re_le_cmod";
-Addsimps [complex_Re_le_cmod];
-
-Goal "z ~= 0 ==> 0 < cmod z";
-by (cut_inst_tac [("x","z")] complex_mod_ge_zero 1);
-by (dtac order_le_imp_less_or_eq 1);
-by Auto_tac;
-qed "complex_mod_gt_zero";
-
-
-(*---------------------------------------------------------------------------*)
-(*                       a few more theorems                                 *)
-(*---------------------------------------------------------------------------*)
-
-Goal "cmod(x ^ n) = cmod(x) ^ n";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mod_mult]));
-qed "complex_mod_complexpow";
-
-Goal "(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed "complexpow_minus";
-
-Goal "inverse (-x) = - inverse (x::complex)";
-by (res_inst_tac [("z","x")] eq_Abs_complex 1);
-by (asm_simp_tac (simpset() addsimps [complex_inverse,complex_minus,
-    realpow_num_two]) 1);
-qed "complex_inverse_minus";
-
-Goalw [complex_divide_def] "x / (1::complex) = x";
-by (Simp_tac 1);
-qed "complex_divide_one";
-Addsimps [complex_divide_one];
-
-Goal "cmod(inverse x) = inverse(cmod x)";
-by (complex_div_undefined_case_tac "x=0" 1);
-by (res_inst_tac [("c1","cmod x")] (real_mult_left_cancel RS iffD1) 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mod_mult RS sym]));
-qed "complex_mod_inverse";
-
-Goalw [complex_divide_def,real_divide_def]
-      "cmod(x/y) = cmod(x)/(cmod y)";
-by (auto_tac (claset(),simpset() addsimps [complex_mod_mult,
-    complex_mod_inverse]));
-qed "complex_mod_divide";
-
-Goalw [complex_divide_def]  
-      "inverse(x/y) = y/(x::complex)";
-by (auto_tac (claset(),simpset() addsimps [complex_inverse_distrib,
-    complex_mult_commute]));
-qed "complex_inverse_divide";
-Addsimps [complex_inverse_divide];
-
-Goal "((r::complex) * s) ^ n = (r ^ n) * (s ^ n)";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps complex_mult_ac));
-qed "complexpow_mult";
-
-(*---------------------------------------------------------------------------*)
-(*                       More exponentiation                                 *)
-(*---------------------------------------------------------------------------*)
-
-Goal "(0::complex) ^ (Suc n) = 0";
-by (Auto_tac);
-qed "complexpow_zero";
-Addsimps [complexpow_zero];
-
-Goal "r ~= (0::complex) --> r ^ n ~= 0";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult_not_zero]));
-qed_spec_mp "complexpow_not_zero";
-Addsimps [complexpow_not_zero];
-AddIs [complexpow_not_zero];
-
-Goal "r ^ n = (0::complex) ==> r = 0";
-by (blast_tac (claset() addIs [ccontr] 
-    addDs [complexpow_not_zero]) 1);
-qed "complexpow_zero_zero";
-
-Goalw [i_def] "ii ^ 2 = -(1::complex)";
-by (auto_tac (claset(),simpset() addsimps 
-    [complex_mult,complex_one_def,complex_minus,realpow_num_two]));
-qed "complexpow_i_squared";
-Addsimps [complexpow_i_squared];
-
-Goalw [i_def,complex_zero_def] "ii ~= 0";
-by Auto_tac;
-qed "complex_i_not_zero";
-Addsimps [complex_i_not_zero];
-
-Goal "x * y ~= (0::complex) ==> x ~= 0";
-by Auto_tac;
-qed "complex_mult_eq_zero_cancel1";
-
-Goal "x * y ~= 0 ==> y ~= (0::complex)";
-by Auto_tac;
-qed "complex_mult_eq_zero_cancel2";
-
-Goal "(x * y ~= 0) = (x ~= 0 & y ~= (0::complex))";
-by Auto_tac;
-qed "complex_mult_not_eq_zero_iff";
-AddIffs [complex_mult_not_eq_zero_iff];
-
-Goal "inverse ((r::complex) ^ n) = (inverse r) ^ n";
-by (induct_tac "n" 1);
-by (auto_tac (claset(), simpset() addsimps [complex_inverse_distrib]));
-qed "complexpow_inverse";
-
-(*---------------------------------------------------------------------------*)
-(* sgn                                                                       *)
-(*---------------------------------------------------------------------------*)
-
-Goalw [sgn_def] "sgn 0 = 0";
-by (Simp_tac 1);
-qed "sgn_zero";
-Addsimps[sgn_zero];
-
-Goalw [sgn_def] "sgn 1 = 1";
-by (Simp_tac 1);
-qed "sgn_one";
-Addsimps [sgn_one];
-
-Goalw [sgn_def] "sgn (-z) = - sgn(z)";
-by Auto_tac;
-qed "sgn_minus";
-
-Goalw [sgn_def] 
-    "sgn z = z / complex_of_real (cmod z)";
-by (Simp_tac 1);
-qed "sgn_eq";
-
-Goal "EX x y. z = complex_of_real(x) + ii * complex_of_real(y)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
-    i_def,complex_mult,complex_add]));
-qed "complex_split";
-
-Goal "Re(complex_of_real(x) + ii * complex_of_real(y)) = x";
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
-    i_def,complex_mult,complex_add]));
-qed "Re_complex_i";
-Addsimps [Re_complex_i];
-
-Goal "Im(complex_of_real(x) + ii * complex_of_real(y)) = y";
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
-    i_def,complex_mult,complex_add]));
-qed "Im_complex_i";
-Addsimps [Im_complex_i];
-
-Goalw [i_def,complex_of_real_def] "ii * ii = complex_of_real (-1)";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
-qed "i_mult_eq";
-
-Goalw [i_def,complex_one_def] "ii * ii = -(1::complex)";
-by (simp_tac (simpset() addsimps [complex_mult,complex_minus]) 1);
-qed "i_mult_eq2";
-Addsimps [i_mult_eq2];
-
-Goal "cmod (complex_of_real(x) + ii * complex_of_real(y)) = \
-\     sqrt (x ^ 2 + y ^ 2)";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add,
-    i_def,complex_of_real_def,cmod_def]));
-qed "cmod_i";
-
-Goalw [complex_of_real_def,i_def] 
-     "complex_of_real xa + ii * complex_of_real ya = \
-\     complex_of_real xb + ii * complex_of_real yb \
-\      ==> xa = xb";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
-qed "complex_eq_Re_eq";
-
-Goalw [complex_of_real_def,i_def] 
-     "complex_of_real xa + ii * complex_of_real ya = \
-\     complex_of_real xb + ii * complex_of_real yb \
-\      ==> ya = yb";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
-qed "complex_eq_Im_eq";
-
-Goal "(complex_of_real xa + ii * complex_of_real ya = \
-\      complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))";
-by (auto_tac (claset() addIs [complex_eq_Im_eq,complex_eq_Re_eq],simpset()));
-qed "complex_eq_cancel_iff";
-AddIffs [complex_eq_cancel_iff];
-
-Goal "(complex_of_real xa + complex_of_real ya * ii = \
-\      complex_of_real xb + complex_of_real yb * ii ) = ((xa = xb) & (ya = yb))";
-by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
-qed "complex_eq_cancel_iffA";
-AddIffs [complex_eq_cancel_iffA];
-
-Goal "(complex_of_real xa + complex_of_real ya * ii = \
-\      complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))";
-by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
-qed "complex_eq_cancel_iffB";
-AddIffs [complex_eq_cancel_iffB];
-
-Goal "(complex_of_real xa + ii * complex_of_real ya  = \
-\      complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))";
-by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
-qed "complex_eq_cancel_iffC";
-AddIffs [complex_eq_cancel_iffC];
-
-Goal"(complex_of_real x + ii * complex_of_real y = \
-\     complex_of_real xa) = (x = xa & y = 0)";
-by (cut_inst_tac [("xa","x"),("ya","y"),("xb","xa"),("yb","0")]  
-    complex_eq_cancel_iff 1);
-by (asm_full_simp_tac (simpset() delsimps [complex_eq_cancel_iff]) 1);
-qed "complex_eq_cancel_iff2";
-Addsimps [complex_eq_cancel_iff2];
-
-Goal"(complex_of_real x + complex_of_real y * ii = \
-\     complex_of_real xa) = (x = xa & y = 0)";
-by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
-qed "complex_eq_cancel_iff2a";
-Addsimps [complex_eq_cancel_iff2a];
-
-Goal "(complex_of_real x + ii * complex_of_real y = \
-\     ii * complex_of_real ya) = (x = 0 & y = ya)";
-by (cut_inst_tac [("xa","x"),("ya","y"),("xb","0"),("yb","ya")]  
-    complex_eq_cancel_iff 1);
-by (asm_full_simp_tac (simpset() delsimps [complex_eq_cancel_iff]) 1);
-qed "complex_eq_cancel_iff3";
-Addsimps [complex_eq_cancel_iff3];
-
-Goal "(complex_of_real x + complex_of_real y * ii = \
-\     ii * complex_of_real ya) = (x = 0 & y = ya)";
-by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
-qed "complex_eq_cancel_iff3a";
-Addsimps [complex_eq_cancel_iff3a];
-
-Goalw [complex_of_real_def,i_def,complex_zero_def] 
-     "complex_of_real x + ii * complex_of_real y = 0 \
-\     ==> x = 0";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
-qed "complex_split_Re_zero";
-
-Goalw [complex_of_real_def,i_def,complex_zero_def] 
-     "complex_of_real x + ii * complex_of_real y = 0 \
-\     ==> y = 0";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
-qed "complex_split_Im_zero";
-
-Goalw [sgn_def,complex_divide_def] 
-      "Re(sgn z) = Re(z)/cmod z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_inverse RS sym]));
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
-    complex_mult,real_divide_def]));
-qed "Re_sgn";
-Addsimps [Re_sgn];
-
-Goalw [sgn_def,complex_divide_def] 
-      "Im(sgn z) = Im(z)/cmod z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_inverse RS sym]));
-by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
-    complex_mult,real_divide_def]));
-qed "Im_sgn";
-Addsimps [Im_sgn];
-
-Goalw [complex_of_real_def,i_def] 
-     "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))";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add,
-    complex_diff_def,complex_minus,complex_inverse,real_divide_def]));
-qed "complex_inverse_complex_split";
-
-(*----------------------------------------------------------------------------*)
-(* Many of the theorems below need to be moved elsewhere e.g. Transc.ML. Also *)
-(* many of the theorems are not used - so should they be kept?                *)
-(*----------------------------------------------------------------------------*)
-
-Goalw [i_def,complex_of_real_def]
-    "Re (ii * complex_of_real y) = 0";
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "Re_mult_i_eq";
-Addsimps [Re_mult_i_eq];
-
-Goalw [i_def,complex_of_real_def]
-    "Im (ii * complex_of_real y) = y";
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "Im_mult_i_eq";
-Addsimps [Im_mult_i_eq];
-
-Goalw [i_def,complex_of_real_def]
-    "cmod (ii * complex_of_real y) = abs y";
-by (auto_tac (claset(),simpset() addsimps [complex_mult,
-    complex_mod,realpow_num_two]));
-qed "complex_mod_mult_i";
-Addsimps [complex_mod_mult_i];
-
-Goalw [arg_def] 
-   "0 < y ==> cos (arg(ii * complex_of_real y)) = 0";
-by (auto_tac (claset(),simpset() addsimps [abs_eqI2]));
-by (res_inst_tac [("a","pi/2")] someI2 1);
-by Auto_tac;
-by (res_inst_tac [("R2.0","0")] real_less_trans 1);
-by Auto_tac;
-qed "cos_arg_i_mult_zero";
-Addsimps [cos_arg_i_mult_zero];
-
-Goalw [arg_def] 
-   "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0";
-by (auto_tac (claset(),simpset() addsimps [abs_minus_eqI2]));
-by (res_inst_tac [("a","- pi/2")] someI2 1);
-by Auto_tac;
-by (res_inst_tac [("j","0")] real_le_trans 1);
-by Auto_tac;
-qed "cos_arg_i_mult_zero2";
-Addsimps [cos_arg_i_mult_zero2];
-
-Goalw [complex_zero_def,complex_of_real_def] 
-      "(complex_of_real y ~= 0) = (y ~= 0)";
-by Auto_tac;
-qed "complex_of_real_not_zero_iff";
-Addsimps [complex_of_real_not_zero_iff];
-
-Goal "(complex_of_real y = 0) = (y = 0)";
-by Auto_tac;
-by (rtac ccontr 1 THEN dtac (complex_of_real_not_zero_iff RS iffD2) 1);
-by (Asm_full_simp_tac 1);
-qed "complex_of_real_zero_iff";
-Addsimps [complex_of_real_zero_iff];
-
-Goal "y ~= 0 ==> cos (arg(ii * complex_of_real y)) = 0";
-by (cut_inst_tac [("x","y"),("y","0")] linorder_less_linear 1);
-by Auto_tac;
-qed "cos_arg_i_mult_zero3";
-Addsimps [cos_arg_i_mult_zero3];
-
-(*---------------------------------------------------------------------------*)
-(* Finally! Polar form for complex numbers                                   *) 
-(*---------------------------------------------------------------------------*)
-
-Goal "EX r a. z = complex_of_real r * \
-\     (complex_of_real(cos a) + ii * complex_of_real(sin a))";
-by (cut_inst_tac [("z","z")] complex_split 1);
-by (auto_tac (claset(),simpset() addsimps [polar_Ex,
-    complex_add_mult_distrib2,complex_of_real_mult] @ complex_mult_ac));
-qed "complex_split_polar";
-
-Goalw [rcis_def,cis_def] "EX r a. z = rcis r a";
-by (rtac complex_split_polar 1);
-qed "rcis_Ex";
-
-Goal "Re(complex_of_real r * \
-\     (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a";
-by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2,
-    complex_of_real_mult] @ complex_mult_ac));
-qed "Re_complex_polar";
-Addsimps [Re_complex_polar];
-
-Goalw [rcis_def,cis_def] "Re(rcis r a) = r * cos a";
-by Auto_tac;
-qed "Re_rcis";
-Addsimps [Re_rcis];
-
-Goal "Im(complex_of_real r * \
-\     (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * sin a";
-by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2,
-    complex_of_real_mult] @ complex_mult_ac));
-qed "Im_complex_polar";
-Addsimps [Im_complex_polar];
-
-Goalw [rcis_def,cis_def] "Im(rcis r a) = r * sin a";
-by Auto_tac;
-qed "Im_rcis";
-Addsimps [Im_rcis];
-
-Goal "cmod (complex_of_real r * \
-\     (complex_of_real(cos a) + ii * complex_of_real(sin a))) = abs r";
-by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2,
-    cmod_i,complex_of_real_mult,real_add_mult_distrib2
-    RS sym,realpow_mult]  @ complex_mult_ac@ real_mult_ac 
-    delsimps [realpow_Suc]));
-qed "complex_mod_complex_polar";
-Addsimps [complex_mod_complex_polar];
-
-Goalw [rcis_def,cis_def] "cmod(rcis r a) = abs r";
-by Auto_tac;
-qed "complex_mod_rcis";
-Addsimps [complex_mod_rcis];
-
-Goalw [cmod_def] "cmod z = sqrt (Re (z * cnj z))";
-by (rtac (real_sqrt_eq_iff RS iffD2) 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult_cnj]));
-qed "complex_mod_sqrt_Re_mult_cnj";
-
-Goal "Re(cnj z) = Re z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_cnj]));
-qed "complex_Re_cnj";
-Addsimps [complex_Re_cnj];
-
-Goal "Im(cnj z) = - Im z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_cnj]));
-qed "complex_Im_cnj";
-Addsimps [complex_Im_cnj];
-
-Goal "Im (z * cnj z) = 0";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_cnj,complex_mult]));
-qed "complex_In_mult_cnj_zero";
-Addsimps [complex_In_mult_cnj_zero];
-
-Goal "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Re_mult";
-
-Goalw [complex_of_real_def] "Re (z * complex_of_real c) = Re(z) * c";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Re_mult_complex_of_real";
-Addsimps [complex_Re_mult_complex_of_real];
-
-Goalw [complex_of_real_def] "Im (z * complex_of_real c) = Im(z) * c";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Im_mult_complex_of_real";
-Addsimps [complex_Im_mult_complex_of_real];
-
-Goalw [complex_of_real_def] "Re (complex_of_real c * z) = c * Re(z)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Re_mult_complex_of_real2";
-Addsimps [complex_Re_mult_complex_of_real2];
-
-Goalw [complex_of_real_def] "Im (complex_of_real c * z) = c * Im(z)";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Im_mult_complex_of_real2";
-Addsimps [complex_Im_mult_complex_of_real2];
-
-(*---------------------------------------------------------------------------*)
-(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *) 
-(*---------------------------------------------------------------------------*)
-
-Goalw [rcis_def] "cis a = rcis 1 a";
-by (Simp_tac 1);
-qed "cis_rcis_eq";
-
-Goalw [rcis_def,cis_def] 
-  "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)";
-by (auto_tac (claset(),simpset() addsimps [cos_add,sin_add,
-    complex_add_mult_distrib2,complex_add_mult_distrib]
-     @ complex_mult_ac @ complex_add_ac));
-by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2 RS sym,
-    complex_mult_assoc RS sym,complex_of_real_mult,complex_of_real_add,
-    complex_add_assoc RS sym,i_mult_eq] delsimps [i_mult_eq2]));
-by (auto_tac (claset(),simpset() addsimps complex_add_ac));
-by (auto_tac (claset(),simpset() addsimps [complex_add_assoc RS sym,
-    complex_of_real_add,real_add_mult_distrib2,
-    real_diff_def] @ real_mult_ac @ real_add_ac));
-qed "rcis_mult";
-
-Goal "cis a * cis b = cis (a + b)";
-by (simp_tac (simpset() addsimps [cis_rcis_eq,rcis_mult]) 1);
-qed "cis_mult";
-
-Goalw [cis_def] "cis 0 = 1";
-by Auto_tac;
-qed "cis_zero";
-Addsimps [cis_zero];
-
-Goalw [cis_def] "cis 0 = complex_of_real 1";
-by Auto_tac;
-qed "cis_zero2";
-Addsimps [cis_zero2];
-
-Goalw [rcis_def] "rcis 0 a = 0";
-by (Simp_tac 1);
-qed "rcis_zero_mod";
-Addsimps [rcis_zero_mod];
-
-Goalw [rcis_def] "rcis r 0 = complex_of_real r";
-by (Simp_tac 1);
-qed "rcis_zero_arg";
-Addsimps [rcis_zero_arg];
-
-Goalw [complex_of_real_def,complex_one_def] 
-   "complex_of_real (-(1::real)) = -(1::complex)";
-by (simp_tac (simpset() addsimps [complex_minus]) 1);
-qed "complex_of_real_minus_one";
-
-Goal "ii * (ii * x) = - x";
-by (simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
-qed "complex_i_mult_minus";
-Addsimps [complex_i_mult_minus];
-
-Goal "ii * ii * x = - x";
-by (Simp_tac 1);
-qed "complex_i_mult_minus2";
-Addsimps [complex_i_mult_minus2];
-
-Goalw [cis_def] 
-   "cis (real (Suc n) * a) = cis a * cis (real n * a)";
-by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,
-    real_add_mult_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));
-by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2 RS sym,
-    complex_mult_assoc RS sym,i_mult_eq,complex_of_real_mult,
-    complex_of_real_add,complex_add_assoc RS sym,complex_of_real_minus
-    RS sym,real_diff_def] @ real_mult_ac delsimps [i_mult_eq2]));
-qed "cis_real_of_nat_Suc_mult";
-
-Goal "(cis a) ^ n = cis (real n * a)";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [cis_real_of_nat_Suc_mult]));
-qed "DeMoivre";
-
-Goalw [rcis_def] 
-   "(rcis r a) ^ n = rcis (r ^ n) (real n * a)";
-by (auto_tac (claset(),simpset() addsimps [complexpow_mult,
-    DeMoivre,complex_of_real_pow]));
-qed "DeMoivre2";
-
-Goalw [cis_def] "inverse(cis a) = cis (-a)";
-by (auto_tac (claset(),simpset() addsimps [complex_inverse_complex_split,
-    complex_of_real_minus,complex_diff_def]));
-qed "cis_inverse";
-Addsimps [cis_inverse];
-
-Goal "inverse(rcis r a) = rcis (1/r) (-a)";
-by (case_tac "r=0" 1);
-by (asm_simp_tac (simpset() addsimps [rename_numerals DIVISION_BY_ZERO,
-    COMPLEX_INVERSE_ZERO]) 1);
-by (auto_tac (claset(),simpset() addsimps [complex_inverse_complex_split,
-    complex_add_mult_distrib2,complex_of_real_mult,rcis_def,cis_def,
-    realpow_num_two] @ complex_mult_ac @ real_mult_ac));
-by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2 RS sym,
-    complex_of_real_minus,complex_diff_def]));
-qed "rcis_inverse";
-
-Goalw [complex_divide_def] "cis a / cis b = cis (a - b)";
-by (auto_tac (claset(),simpset() addsimps [cis_mult,real_diff_def]));
-qed "cis_divide";
-
-Goalw [complex_divide_def]
- "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)";
-by (case_tac "r2=0" 1);
-by (asm_simp_tac (simpset() addsimps [rename_numerals DIVISION_BY_ZERO,
-    COMPLEX_INVERSE_ZERO]) 1);
-by (auto_tac (claset(),simpset() addsimps [rcis_inverse,rcis_mult,
-    real_diff_def]));
-qed "rcis_divide";
-
-Goalw [cis_def] "Re(cis a) = cos a";
-by Auto_tac;
-qed "Re_cis";
-Addsimps [Re_cis];
-
-Goalw [cis_def] "Im(cis a) = sin a";
-by Auto_tac;
-qed "Im_cis";
-Addsimps [Im_cis];
-
-Goal "cos (real n * a) = Re(cis a ^ n)";
-by (auto_tac (claset(),simpset() addsimps [DeMoivre]));
-qed "cos_n_Re_cis_pow_n";
- 
-Goal "sin (real n * a) = Im(cis a ^ n)";
-by (auto_tac (claset(),simpset() addsimps [DeMoivre]));
-qed "sin_n_Im_cis_pow_n";
-
-Goalw [expi_def,cis_def]
-    "expi (ii * complex_of_real y) = \
-\    complex_of_real (cos y) + ii * complex_of_real (sin y)";
-by Auto_tac;
-qed "expi_Im_split";
-
-Goalw [expi_def]
-    "expi (ii * complex_of_real y) = cis y";
-by Auto_tac;
-qed "expi_Im_cis";
-
-Goalw [expi_def] "expi(a + b) = expi(a) * expi(b)";
-by (auto_tac (claset(),simpset() addsimps [complex_Re_add,exp_add,
-    complex_Im_add,cis_mult RS sym,complex_of_real_mult] @
-    complex_mult_ac));
-qed "expi_add";
-
-Goalw [expi_def] 
-     "expi(complex_of_real x + ii * complex_of_real y) = \
-\     complex_of_real (exp(x)) * cis y";
-by Auto_tac;
-qed "expi_complex_split";
-
-Goalw [expi_def] "expi (0::complex) = 1";
-by Auto_tac;
-qed "expi_zero";
-Addsimps [expi_zero];
-
-goal Complex.thy 
-     "Re (w * z) = Re w * Re z - Im w * Im z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Re_mult_eq";
-
-goal Complex.thy 
-     "Im (w * z) = Re w * Im z + Im w * Re z";
-by (res_inst_tac [("z","z")] eq_Abs_complex 1);
-by (res_inst_tac [("z","w")] eq_Abs_complex 1);
-by (auto_tac (claset(),simpset() addsimps [complex_mult]));
-qed "complex_Im_mult_eq";
-
-goal Complex.thy 
-   "EX a r. z = complex_of_real r * expi a";
-by (cut_inst_tac [("z","z")] rcis_Ex 1);
-by (auto_tac (claset(),simpset() addsimps [expi_def,rcis_def,
-    complex_mult_assoc RS sym,complex_of_real_mult]));
-by (res_inst_tac [("x","ii * complex_of_real a")] exI 1);
-by Auto_tac;
-qed "complex_expi_Ex";
-
-
-(****
-Goal "[| - pi < a; a <= pi |] ==> (-pi < a & a <= 0) | (0 <= a & a <= pi)";
-by Auto_tac;
-qed "lemma_split_interval";
-
-Goalw [arg_def] 
-  "[| r ~= 0; - pi < a; a <= 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,
-    real_minus_mult_eq2 RS sym] @ real_mult_ac));
-by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
-by (dtac lemma_split_interval 1 THEN Step_tac 1);
-****)
-

--- a/src/HOL/Complex/Complex.thy	Tue Dec 23 16:52:49 2003 +0100
+++ b/src/HOL/Complex/Complex.thy	Tue Dec 23 16:53:33 2003 +0100
@@ -4,108 +4,1958 @@
     Description: Complex numbers
 *)
 
-Complex = HLog + 
+theory Complex = HLog:
 
 typedef complex = "{p::(real*real). True}"
+  by blast
 
-instance
-  complex :: {ord,zero,one,plus,minus,times,power,inverse}
+instance complex :: zero ..
+instance complex :: one ..
+instance complex :: plus ..
+instance complex :: times ..
+instance complex :: minus ..
+instance complex :: inverse ..
+instance complex :: power ..
 
 consts
-  "ii"    :: complex        ("ii") 
+  "ii"    :: complex        ("ii")
 
 constdefs
 
   (*--- real and Imaginary parts ---*)
-  
-  Re :: complex => real
+
+  Re :: "complex => real"
   "Re(z) == fst(Rep_complex z)"
 
-  Im :: complex => real
+  Im :: "complex => real"
   "Im(z) == snd(Rep_complex z)"
 
   (*----------- modulus ------------*)
 
-  cmod :: complex => real
-  "cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)"			      
+  cmod :: "complex => real"
+  "cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)"
 
-  (*----- injection from reals -----*)			   
- 
-  complex_of_real :: real => complex
+  (*----- injection from reals -----*)
+
+  complex_of_real :: "real => complex"
   "complex_of_real r == Abs_complex(r,0::real)"
-				    
+
   (*------- complex conjugate ------*)
 
-  cnj :: complex => complex
+  cnj :: "complex => complex"
   "cnj z == Abs_complex(Re z, -Im z)"
 
-  (*------------ Argand -------------*)		       
+  (*------------ Argand -------------*)
 
-  sgn :: complex => complex
+  sgn :: "complex => complex"
   "sgn z == z / complex_of_real(cmod z)"
 
-  arg :: complex => real
+  arg :: "complex => real"
   "arg z == @a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a <= pi"
-									  
-defs
+
 
-  complex_zero_def
+defs (overloaded)
+
+  complex_zero_def:
   "0 == Abs_complex(0::real,0)"
 
-  complex_one_def
+  complex_one_def:
   "1 == Abs_complex(1,0::real)"
 
-  (*------ imaginary unit ----------*)					 
-			      
-  i_def 
+  (*------ imaginary unit ----------*)
+
+  i_def:
   "ii == Abs_complex(0::real,1)"
 
   (*----------- negation -----------*)
-				     
-  complex_minus_def
-  "- (z::complex) == Abs_complex(-Re z, -Im z)"				     
+
+  complex_minus_def:
+  "- (z::complex) == Abs_complex(-Re z, -Im z)"
 
-  
+
   (*----------- inverse -----------*)
-  complex_inverse_def
+  complex_inverse_def:
   "inverse (z::complex) == Abs_complex(Re(z)/(Re(z) ^ 2 + Im(z) ^ 2),
                             -Im(z)/(Re(z) ^ 2 + Im(z) ^ 2))"
 
-  complex_add_def
+  complex_add_def:
   "w + (z::complex) == Abs_complex(Re(w) + Re(z),Im(w) + Im(z))"
 
-  complex_diff_def
+  complex_diff_def:
   "w - (z::complex) == w + -(z::complex)"
 
-  complex_mult_def
+  complex_mult_def:
   "w * (z::complex) == Abs_complex(Re(w) * Re(z) - Im(w) * Im(z),
 			Re(w) * Im(z) + Im(w) * Re(z))"
 
 
   (*----------- division ----------*)
-  complex_divide_def
+  complex_divide_def:
   "w / (z::complex) == w * inverse z"
-  
+
 
 primrec
-     complexpow_0   "z ^ 0       = complex_of_real 1"
-     complexpow_Suc "z ^ (Suc n) = (z::complex) * (z ^ n)"
+     complexpow_0:   "z ^ 0       = complex_of_real 1"
+     complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"
 
 
 constdefs
 
   (* abbreviation for (cos a + i sin a) *)
-  cis :: real => complex
+  cis :: "real => complex"
   "cis a == complex_of_real(cos a) + ii * complex_of_real(sin a)"
 
   (* abbreviation for r*(cos a + i sin a) *)
-  rcis :: [real, real] => complex
+  rcis :: "[real, real] => complex"
   "rcis r a == complex_of_real r * cis a"
 
   (* e ^ (x + iy) *)
-  expi :: complex => complex
+  expi :: "complex => complex"
   "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)"
+apply (auto dest: inj_Abs_complex [THEN injD])
+done
+declare Abs_complex_cancel_iff [simp]
+
+lemma pair_mem_complex: "(x,y) : complex"
+apply (unfold complex_def)
+apply auto
+done
+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
+declare Re [simp]
+
+lemma Im: "Im(Abs_complex(x,y)) = y"
+apply (unfold Im_def)
+apply (simp (no_asm))
+done
+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
+
+lemma complex_Re_zero: "Re 0 = 0"
+apply (unfold complex_zero_def)
+apply (simp (no_asm))
+done
+
+lemma complex_Im_zero: "Im 0 = 0"
+apply (unfold complex_zero_def)
+apply (simp (no_asm))
+done
+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
+declare complex_Re_one [simp]
+
+lemma complex_Im_one: "Im 1 = 0"
+apply (unfold complex_one_def)
+apply (simp (no_asm))
+done
+declare complex_Im_one [simp]
+
+lemma complex_Re_i: "Re(ii) = 0"
+apply (unfold i_def)
+apply auto
+done
+declare complex_Re_i [simp]
+
+lemma complex_Im_i: "Im(ii) = 1"
+apply (unfold i_def)
+apply auto
+done
+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
+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
+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
+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
+declare Im_complex_of_real_one [simp]
+
+lemma Re_complex_of_real: "Re(complex_of_real z) = z"
+apply (unfold complex_of_real_def)
+apply auto
+done
+declare Re_complex_of_real [simp]
+
+lemma Im_complex_of_real: "Im(complex_of_real z) = 0"
+apply (unfold complex_of_real_def)
+apply auto
+done
+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_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
+
+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)
+apply simp
+done
+
+lemma complex_minus_zero: "-(0::complex) = 0"
+apply (unfold complex_zero_def)
+apply (simp (no_asm) add: complex_minus)
+done
+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
+declare complex_minus_zero_iff [simp]
+
+lemma complex_minus_zero_iff2: "(0 = -x) = (x = (0::real))"
+apply (auto dest: sym)
+done
+declare complex_minus_zero_iff2 [simp]
+
+lemma complex_minus_not_zero_iff: "(-x ~= 0) = (x ~= (0::complex))"
+apply auto
+done
+
+
+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_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
+
+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
+
+lemma complex_add_commute: "(u::complex) + v = v + u"
+apply (unfold complex_add_def)
+apply (simp (no_asm) add: real_add_commute)
+done
+
+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: real_add_left_commute)
+done
+
+lemmas complex_add_ac = complex_add_assoc complex_add_commute
+                      complex_add_left_commute
+
+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]
+
+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_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_left_zero [simp]
+
+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)"
+apply (simp (no_asm) add: complex_add_assoc [symmetric])
+done
+
+declare complex_add_minus_cancel [simp] complex_minus_add_cancel [simp]
+
+lemma complex_add_minus_eq_minus: "x + y = (0::complex) ==> x = -y"
+apply (auto simp add: complex_Re_Im_cancel_iff complex_Re_add complex_Im_add complex_Re_minus complex_Im_minus)
+done
+
+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_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])
+apply 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])
+apply 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_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)"
+apply (auto simp add: complex_diff_def complex_add_assoc)
+done
+
+
+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
+
+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
+
+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 real_diff_mult_distrib2 real_add_mult_distrib2 real_diff_mult_distrib real_add_mult_distrib real_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 real_mult_left_commute real_diff_mult_distrib2 real_add_mult_distrib2)
+done
+
+lemmas complex_mult_ac = complex_mult_assoc complex_mult_commute
+                      complex_mult_left_commute
+
+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]
+
+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)"
+apply (unfold complex_divide_def)
+apply auto
+done
+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
+
+declare complex_minus_mult_eq1 [symmetric, simp] complex_minus_mult_eq2 [symmetric, simp]
+
+lemma complex_mult_minus_one: "-(1::complex) * z = -z"
+apply (simp (no_asm))
+done
+declare complex_mult_minus_one [simp]
+
+lemma complex_mult_minus_one_right: "z * -(1::complex) = -z"
+apply (subst complex_mult_commute)
+apply (simp (no_asm))
+done
+declare complex_mult_minus_one_right [simp]
+
+lemma complex_minus_mult_cancel: "-x * -y = x * (y::complex)"
+apply (simp (no_asm))
+done
+declare complex_minus_mult_cancel [simp]
+
+lemma complex_minus_mult_commute: "-x * y = x * -(y::complex)"
+apply (simp (no_asm))
+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 real_add_mult_distrib real_diff_def real_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) ~= 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]
+
+
+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)"
+apply (unfold complex_inverse_def complex_zero_def)
+apply auto
+done
+
+lemma COMPLEX_DIVISION_BY_ZERO: "a / (0::complex) = 0"
+apply (simp (no_asm) add: complex_divide_def COMPLEX_INVERSE_ZERO)
+done
+
+lemma complex_mult_inv_left: "z ~= (0::complex) ==> inverse(z) * z = 1"
+apply (rule_tac z = "z" in eq_Abs_complex)
+apply (auto simp add: complex_mult complex_inverse complex_one_def complex_zero_def real_add_divide_distrib [symmetric] real_power_two 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)
+apply auto
+done
+declare complex_mult_inv_left [simp]
+
+lemma complex_mult_inv_right: "z ~= (0::complex) ==> z * inverse(z) = 1"
+apply (auto intro: complex_mult_commute [THEN subst])
+done
+declare complex_mult_inv_right [simp]
+
+lemma complex_mult_left_cancel: "(c::complex) ~= 0 ==> (c*a=c*b) = (a=b)"
+apply auto
+apply (drule_tac f = "%x. x*inverse c" in arg_cong)
+apply (simp add: complex_mult_ac)
+done
+
+lemma complex_mult_right_cancel: "(c::complex) ~= 0 ==> (a*c=b*c) = (a=b)"
+apply (safe)
+apply (drule_tac f = "%x. x*inverse c" in arg_cong)
+apply (simp add: complex_mult_ac)
+done
+
+lemma complex_inverse_not_zero: "z ~= 0 ==> inverse(z::complex) ~= 0"
+apply (safe)
+apply (frule complex_mult_right_cancel [THEN iffD2])
+apply (erule_tac [2] V = "inverse z = 0" in thin_rl)
+apply (assumption , auto)
+done
+declare complex_inverse_not_zero [simp]
+
+lemma complex_mult_not_zero: "!!x. [| x ~= 0; y ~= (0::complex) |] ==> x * y ~= 0"
+apply (safe)
+apply (drule_tac f = "%z. inverse x*z" in arg_cong)
+apply (simp add: complex_mult_assoc [symmetric])
+done
+
+lemmas complex_mult_not_zeroE = complex_mult_not_zero [THEN notE, standard]
+
+lemma complex_inverse_inverse: "inverse(inverse (x::complex)) = x"
+apply (case_tac "x = 0", simp add: COMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "inverse x" in complex_mult_right_cancel [THEN iffD1])
+apply (erule complex_inverse_not_zero)
+apply (auto dest: complex_inverse_not_zero)
+done
+declare complex_inverse_inverse [simp]
+
+lemma complex_inverse_one: "inverse(1::complex) = 1"
+apply (unfold complex_one_def)
+apply (simp (no_asm) add: complex_inverse)
+done
+declare complex_inverse_one [simp]
+
+lemma complex_minus_inverse: "inverse(-x) = -inverse(x::complex)"
+apply (case_tac "x = 0", simp add: COMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "-x" in complex_mult_right_cancel [THEN iffD1])
+apply force
+apply (subst complex_mult_inv_left)
+apply auto
+done
+
+lemma complex_inverse_distrib: "inverse(x*y) = inverse x * inverse (y::complex)"
+apply (case_tac "x = 0", simp add: COMPLEX_INVERSE_ZERO)
+apply (case_tac "y = 0", simp add: COMPLEX_INVERSE_ZERO)
+apply (rule_tac c1 = "x*y" in complex_mult_left_cancel [THEN iffD1])
+apply (auto simp add: complex_mult_not_zero complex_mult_ac)
+apply (auto simp add: complex_mult_not_zero complex_mult_assoc [symmetric])
+done
+
+
+subsection{*Division*}
+
+(*adding some of these theorems to simpset as for reals:
+  not 100% convinced for some*)
+
+lemma complex_times_divide1_eq: "(x::complex) * (y/z) = (x*y)/z"
+apply (simp (no_asm) add: complex_divide_def complex_mult_assoc)
+done
+
+lemma complex_times_divide2_eq: "(y/z) * (x::complex) = (y*x)/z"
+apply (simp (no_asm) add: complex_divide_def complex_mult_ac)
+done
+
+declare complex_times_divide1_eq [simp] complex_times_divide2_eq [simp]
+
+lemma complex_divide_divide1_eq: "(x::complex) / (y/z) = (x*z)/y"
+apply (simp (no_asm) add: complex_divide_def complex_inverse_distrib complex_mult_ac)
+done
+
+lemma complex_divide_divide2_eq: "((x::complex) / y) / z = x/(y*z)"
+apply (simp (no_asm) add: complex_divide_def complex_inverse_distrib complex_mult_assoc)
+done
+
+declare complex_divide_divide1_eq [simp] complex_divide_divide2_eq [simp]
+
+(** As with multiplication, pull minus signs OUT of the / operator **)
+
+lemma complex_minus_divide_eq: "(-x) / (y::complex) = - (x/y)"
+apply (simp (no_asm) add: complex_divide_def)
+done
+declare complex_minus_divide_eq [simp]
+
+lemma complex_divide_minus_eq: "(x / -(y::complex)) = - (x/y)"
+apply (simp (no_asm) add: complex_divide_def complex_minus_inverse)
+done
+declare complex_divide_minus_eq [simp]
+
+lemma complex_add_divide_distrib: "(x+y)/(z::complex) = x/z + y/z"
+apply (simp (no_asm) add: complex_divide_def complex_add_mult_distrib)
+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
+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
+declare complex_of_real_zero [simp]
+
+lemma complex_of_real_eq_iff: "(complex_of_real x = complex_of_real y) = (x = y)"
+apply (auto dest: inj_complex_of_real [THEN injD])
+done
+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
+
+lemma complex_of_real_inverse: "complex_of_real(inverse x) = inverse(complex_of_real x)"
+apply (case_tac "x=0")
+apply (simp add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
+apply (simp add: complex_inverse complex_of_real_def real_divide_def real_inverse_distrib real_power_two)
+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
+
+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
+
+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
+
+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)
+done
+
+lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: complex_of_real_mult [symmetric])
+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_zero: "cmod(0) = 0"
+apply (unfold cmod_def)
+apply (simp (no_asm))
+done
+declare complex_mod_zero [simp]
+
+lemma complex_mod_one: "cmod(1) = 1"
+apply (unfold cmod_def)
+apply (simp add: );  
+done
+declare complex_mod_one [simp]
+
+lemma complex_mod_complex_of_real: "cmod(complex_of_real x) = abs x"
+apply (unfold complex_of_real_def)
+apply (simp (no_asm) add: complex_mod)
+done
+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))"
+apply (simp (no_asm))
+done
+
+
+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_cancel_iff: "(cnj x = cnj y) = (x = y)"
+apply (auto dest: inj_cnj [THEN injD])
+done
+declare complex_cnj_cancel_iff [simp]
+
+lemma complex_cnj_cnj: "cnj (cnj z) = z"
+apply (unfold cnj_def)
+apply (simp (no_asm))
+done
+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
+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 real_power_two)
+done
+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
+
+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 real_power_two)
+done
+
+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
+
+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
+
+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
+
+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
+
+lemma complex_cnj_one: "cnj 1 = 1"
+apply (unfold cnj_def complex_one_def)
+apply (simp (no_asm))
+done
+declare complex_cnj_one [simp]
+
+lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: complex_cnj_mult)
+done
+
+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
+
+lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
+apply (rule_tac z = "z" in eq_Abs_complex)
+apply (simp (no_asm_simp) add: complex_add complex_cnj complex_of_real_def complex_diff_def complex_minus i_def complex_mult)
+done
+
+lemma complex_cnj_zero: "cnj 0 = 0"
+apply (simp add: cnj_def complex_zero_def) 
+done
+declare complex_cnj_zero [simp]
+
+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
+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 real_power_two)
+done
+
+
+subsection{*Algebra*}
+
+lemma complex_mult_zero_iff: "(x*y = (0::complex)) = (x = 0 | y = 0)"
+apply auto
+apply (auto intro: ccontr dest: complex_mult_not_zero)
+done
+declare complex_mult_zero_iff [iff]
+
+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
+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_distrib2:
+      "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)"
+apply (unfold complex_diff_def)
+apply (simp (no_asm) add: complex_add_mult_distrib2)
+done
+
+
+subsection{*Modulus*}
+
+(*
+Goal "[| sqrt(x) = 0; 0 <= x |] ==> x = 0"
+by (auto_tac (claset() addIs [real_sqrt_eq_zero_cancel],
+    simpset()));
+qed "real_sqrt_eq_zero_cancel2";
+*)
+
+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 real_power_two)
+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
+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 real_power_two)
+done
+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: real_power_two real_abs_def)
+done
+
+lemma complex_mod_squared: "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2"
+apply (unfold cmod_def)
+apply auto
+done
+
+lemma complex_mod_ge_zero: "0 <= cmod x"
+apply (unfold cmod_def)
+apply (auto intro: real_sqrt_ge_zero)
+done
+declare complex_mod_ge_zero [simp]
+
+lemma abs_cmod_cancel: "abs(cmod x) = cmod x"
+apply (auto intro: abs_eqI1)
+done
+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 (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2 [symmetric] simp del: realpow_Suc)
+apply (rule_tac n = "1" in realpow_Suc_cancel_eq)
+apply (auto simp add: real_power_two [symmetric] simp del: realpow_Suc)
+apply (auto simp add: real_diff_def real_power_two real_add_mult_distrib2 real_add_mult_distrib real_add_ac real_mult_ac)
+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 (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
+apply (auto simp add: real_add_mult_distrib2 real_add_mult_distrib real_power_two real_mult_ac real_add_ac)
+done
+
+lemma complex_Re_mult_cnj_le_cmod: "Re(x * cnj y) <= cmod(x * cnj y)"
+apply (rule_tac z = "x" in eq_Abs_complex)
+apply (rule_tac z = "y" in eq_Abs_complex)
+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) <= 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
+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: real_add_mult_distrib real_add_mult_distrib2 real_power_two)
+done
+
+lemma complex_mod_triangle_squared: "cmod (x + y) ^ 2 <= (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
+declare complex_mod_triangle_squared [simp]
+
+lemma complex_mod_minus_le_complex_mod: "- cmod x <= cmod x"
+apply (rule order_trans [OF _ complex_mod_ge_zero]) 
+apply (simp (no_asm))
+done
+declare complex_mod_minus_le_complex_mod [simp]
+
+lemma complex_mod_triangle_ineq: "cmod (x + y) <= cmod(x) + cmod(y)"
+apply (rule_tac n = "1" in realpow_increasing)
+apply (auto intro:  order_trans [OF _ complex_mod_ge_zero]
+            simp add: real_power_two [symmetric])
+done
+declare complex_mod_triangle_ineq [simp]
+
+lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b <= cmod a"
+apply (cut_tac x1 = "b" and y1 = "a" and z = "-cmod b" in complex_mod_triangle_ineq [THEN real_add_le_mono1])
+apply (simp (no_asm))
+done
+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 (auto simp add: complex_diff complex_mod real_diff_mult_distrib2 real_power_two real_diff_mult_distrib real_add_ac real_mult_ac)
+done
+
+lemma complex_mod_add_less: "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
+apply (auto intro: order_le_less_trans complex_mod_triangle_ineq)
+done
+
+lemma complex_mod_mult_less: "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
+apply (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
+done
+
+lemma complex_mod_diff_ineq: "cmod(a) - cmod(b) <= cmod(a + b)"
+apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
+apply auto
+apply (rule order_trans [of _ 0] , rule order_less_imp_le)
+apply (simp add: compare_rls)
+apply (simp add: );  
+apply (simp add: compare_rls)
+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)
+done
+declare complex_mod_diff_ineq [simp]
+
+lemma complex_Re_le_cmod: "Re z <= cmod z"
+apply (rule_tac z = "z" in eq_Abs_complex)
+apply (auto simp add: complex_mod simp del: realpow_Suc)
+done
+declare complex_Re_le_cmod [simp]
+
+lemma complex_mod_gt_zero: "z ~= 0 ==> 0 < cmod z"
+apply (cut_tac x = "z" in complex_mod_ge_zero)
+apply (drule order_le_imp_less_or_eq)
+apply auto
+done
+
+
+subsection{*A Few More Theorems*}
+
+lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: complex_mod_mult)
+done
+
+lemma complexpow_minus: "(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))"
+apply (induct_tac "n")
+apply auto
+done
+
+lemma complex_inverse_minus: "inverse (-x) = - inverse (x::complex)"
+apply (rule_tac z = "x" in eq_Abs_complex)
+apply (simp (no_asm_simp) add: complex_inverse complex_minus real_power_two)
+done
+
+lemma complex_divide_one: "x / (1::complex) = x"
+apply (unfold complex_divide_def)
+apply (simp (no_asm))
+done
+declare complex_divide_one [simp]
+
+lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
+apply (case_tac "x=0", simp add: COMPLEX_INVERSE_ZERO)
+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_inverse_divide:
+      "inverse(x/y) = y/(x::complex)"
+apply (unfold complex_divide_def)
+apply (auto simp add: complex_inverse_distrib complex_mult_commute)
+done
+declare complex_inverse_divide [simp]
+
+lemma complexpow_mult: "((r::complex) * s) ^ n = (r ^ n) * (s ^ n)"
+apply (induct_tac "n")
+apply (auto simp add: complex_mult_ac)
+done
+
+
+subsection{*More Exponentiation*}
+
+lemma complexpow_zero: "(0::complex) ^ (Suc n) = 0"
+apply auto
+done
+declare complexpow_zero [simp]
+
+lemma complexpow_not_zero [rule_format (no_asm)]: "r ~= (0::complex) --> r ^ n ~= 0"
+apply (induct_tac "n")
+apply (auto simp add: complex_mult_not_zero)
+done
+declare complexpow_not_zero [simp]
+declare complexpow_not_zero [intro]
+
+lemma complexpow_zero_zero: "r ^ n = (0::complex) ==> r = 0"
+apply (blast intro: ccontr dest: complexpow_not_zero)
+done
+
+lemma complexpow_i_squared: "ii ^ 2 = -(1::complex)"
+apply (unfold i_def)
+apply (auto simp add: complex_mult complex_one_def complex_minus numeral_2_eq_2)
+done
+declare complexpow_i_squared [simp]
+
+lemma complex_i_not_zero: "ii ~= 0"
+apply (unfold i_def complex_zero_def)
+apply auto
+done
+declare complex_i_not_zero [simp]
+
+lemma complex_mult_eq_zero_cancel1: "x * y ~= (0::complex) ==> x ~= 0"
+apply auto
+done
+
+lemma complex_mult_eq_zero_cancel2: "x * y ~= 0 ==> y ~= (0::complex)"
+apply auto
+done
+
+lemma complex_mult_not_eq_zero_iff: "(x * y ~= 0) = (x ~= 0 & y ~= (0::complex))"
+apply auto
+done
+declare complex_mult_not_eq_zero_iff [iff]
+
+lemma complexpow_inverse: "inverse ((r::complex) ^ n) = (inverse r) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: complex_inverse_distrib)
+done
+
+(*---------------------------------------------------------------------------*)
+(* sgn                                                                       *)
+(*---------------------------------------------------------------------------*)
+
+lemma sgn_zero: "sgn 0 = 0"
+
+apply (unfold sgn_def)
+apply (simp (no_asm))
+done
+declare sgn_zero [simp]
+
+lemma sgn_one: "sgn 1 = 1"
+apply (unfold sgn_def)
+apply (simp (no_asm))
+done
+declare sgn_one [simp]
+
+lemma sgn_minus: "sgn (-z) = - sgn(z)"
+apply (unfold sgn_def)
+apply auto
+done
+
+lemma sgn_eq:
+    "sgn z = z / complex_of_real (cmod z)"
+apply (unfold sgn_def)
+apply (simp (no_asm))
+done
+
+lemma complex_split: "EX x y. z = complex_of_real(x) + ii * complex_of_real(y)"
+apply (rule_tac z = "z" in eq_Abs_complex)
+apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
+done
+
+lemma Re_complex_i: "Re(complex_of_real(x) + ii * complex_of_real(y)) = x"
+apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
+done
+declare Re_complex_i [simp]
+
+lemma Im_complex_i: "Im(complex_of_real(x) + ii * complex_of_real(y)) = y"
+apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
+done
+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
+
+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
+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
+
+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
+
+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
+
+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
+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
+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
+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
+declare complex_eq_cancel_iffC [iff]
+
+lemma complex_eq_cancel_iff2: "(complex_of_real x + ii * complex_of_real y =
+      complex_of_real xa) = (x = xa & y = 0)"
+apply (cut_tac xa = "x" and ya = "y" and xb = "xa" and yb = "0" in complex_eq_cancel_iff)
+apply (simp del: complex_eq_cancel_iff)
+done
+declare complex_eq_cancel_iff2 [simp]
+
+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
+declare complex_eq_cancel_iff2a [simp]
+
+lemma complex_eq_cancel_iff3: "(complex_of_real x + ii * complex_of_real y =
+      ii * complex_of_real ya) = (x = 0 & y = ya)"
+apply (cut_tac xa = "x" and ya = "y" and xb = "0" and yb = "ya" in complex_eq_cancel_iff)
+apply (simp del: complex_eq_cancel_iff)
+done
+declare complex_eq_cancel_iff3 [simp]
+
+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
+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
+
+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
+
+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)
+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)
+done
+declare Im_sgn [simp]
+
+lemma complex_inverse_complex_split:
+     "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
+
+(*----------------------------------------------------------------------------*)
+(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
+(* many of the theorems are not used - so should they be kept?                *)
+(*----------------------------------------------------------------------------*)
+
+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
+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
+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 real_power_two)
+done
+declare complex_mod_mult_i [simp]
+
+lemma cos_arg_i_mult_zero:
+   "0 < y ==> cos (arg(ii * complex_of_real y)) = 0"
+apply (unfold arg_def)
+apply (auto simp add: abs_eqI2)
+apply (rule_tac a = "pi/2" in someI2)
+apply auto
+apply (rule order_less_trans [of _ 0])
+apply auto
+done
+declare cos_arg_i_mult_zero [simp]
+
+lemma cos_arg_i_mult_zero2:
+   "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0"
+apply (unfold arg_def)
+apply (auto simp add: abs_minus_eqI2)
+apply (rule_tac a = "- pi/2" in someI2)
+apply auto
+apply (rule order_trans [of _ 0])
+apply auto
+done
+declare cos_arg_i_mult_zero2 [simp]
+
+lemma complex_of_real_not_zero_iff:
+      "(complex_of_real y ~= 0) = (y ~= 0)"
+apply (unfold complex_zero_def complex_of_real_def)
+apply auto
+done
+declare complex_of_real_not_zero_iff [simp]
+
+lemma complex_of_real_zero_iff: "(complex_of_real y = 0) = (y = 0)"
+apply auto
+apply (rule ccontr , drule complex_of_real_not_zero_iff [THEN iffD2])
+apply simp
+done
+declare complex_of_real_zero_iff [simp]
+
+lemma cos_arg_i_mult_zero3: "y ~= 0 ==> cos (arg(ii * complex_of_real y)) = 0"
+apply (cut_tac x = "y" and y = "0" in linorder_less_linear)
+apply auto
+done
+declare cos_arg_i_mult_zero3 [simp]
+
+
+subsection{*Finally! Polar Form for Complex Numbers*}
+
+lemma complex_split_polar: "EX r a. z = complex_of_real r *
+      (complex_of_real(cos a) + ii * complex_of_real(sin a))"
+apply (cut_tac z = "z" in complex_split)
+apply (auto simp add: polar_Ex complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
+done
+
+lemma rcis_Ex: "EX r a. z = rcis r a"
+apply (unfold 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
+declare Re_complex_polar [simp]
+
+lemma Re_rcis: "Re(rcis r a) = r * cos a"
+apply (unfold rcis_def cis_def)
+apply auto
+done
+declare Re_rcis [simp]
+
+lemma Im_complex_polar: "Im(complex_of_real r *
+      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * sin a"
+apply (auto simp add: complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
+done
+declare Im_complex_polar [simp]
+
+lemma Im_rcis: "Im(rcis r a) = r * sin a"
+apply (unfold rcis_def cis_def)
+apply auto
+done
+declare Im_rcis [simp]
+
+lemma complex_mod_complex_polar: "cmod (complex_of_real r *
+      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = abs r"
+apply (auto simp add: complex_add_mult_distrib2 cmod_i complex_of_real_mult real_add_mult_distrib2 [symmetric] realpow_mult complex_mult_ac real_mult_ac simp del: realpow_Suc)
+done
+declare complex_mod_complex_polar [simp]
+
+lemma complex_mod_rcis: "cmod(rcis r a) = abs r"
+apply (unfold rcis_def cis_def)
+apply auto
+done
+declare complex_mod_rcis [simp]
+
+lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
+apply (unfold 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
+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
+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
+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
+
+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
+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
+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
+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
+declare complex_Im_mult_complex_of_real2 [simp]
+
+(*---------------------------------------------------------------------------*)
+(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
+(*---------------------------------------------------------------------------*)
+
+lemma cis_rcis_eq: "cis a = rcis 1 a"
+apply (unfold rcis_def)
+apply (simp (no_asm))
+done
+
+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 (auto simp add: complex_add_assoc [symmetric] complex_of_real_add real_add_mult_distrib2 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
+
+lemma cis_zero: "cis 0 = 1"
+apply (unfold cis_def)
+apply auto
+done
+declare cis_zero [simp]
+
+lemma cis_zero2: "cis 0 = complex_of_real 1"
+apply (unfold cis_def)
+apply auto
+done
+declare cis_zero2 [simp]
+
+lemma rcis_zero_mod: "rcis 0 a = 0"
+apply (unfold rcis_def)
+apply (simp (no_asm))
+done
+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
+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)
+done
+
+lemma complex_i_mult_minus: "ii * (ii * x) = - x"
+apply (simp (no_asm) add: complex_mult_assoc [symmetric])
+done
+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 real_add_mult_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)
+done
+
+lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
+apply (induct_tac "n")
+apply (auto simp add: cis_real_of_nat_Suc_mult)
+done
+
+lemma DeMoivre2:
+   "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
+apply (unfold rcis_def)
+apply (auto simp add: complexpow_mult 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
+declare cis_inverse [simp]
+
+lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
+apply (case_tac "r=0")
+apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
+apply (auto simp add: complex_inverse_complex_split complex_add_mult_distrib2 complex_of_real_mult rcis_def cis_def real_power_two complex_mult_ac real_mult_ac)
+apply (auto simp add: real_add_mult_distrib2 [symmetric] complex_of_real_minus complex_diff_def)
+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
+
+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)
+done
+
+lemma Re_cis: "Re(cis a) = cos a"
+apply (unfold cis_def)
+apply auto
+done
+declare Re_cis [simp]
+
+lemma Im_cis: "Im(cis a) = sin a"
+apply (unfold cis_def)
+apply auto
+done
+declare Im_cis [simp]
+
+lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
+apply (auto simp add: DeMoivre)
+done
+
+lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
+apply (auto simp add: DeMoivre)
+done
+
+lemma expi_Im_split:
+    "expi (ii * complex_of_real y) =
+     complex_of_real (cos y) + ii * complex_of_real (sin y)"
+apply (unfold expi_def cis_def)
+apply auto
+done
+
+lemma expi_Im_cis:
+    "expi (ii * complex_of_real y) = cis y"
+apply (unfold expi_def)
+apply auto
+done
+
+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
+
+lemma expi_complex_split:
+     "expi(complex_of_real x + ii * complex_of_real y) =
+      complex_of_real (exp(x)) * cis y"
+apply (unfold expi_def)
+apply auto
+done
+
+lemma expi_zero: "expi (0::complex) = 1"
+apply (unfold expi_def)
+apply auto
+done
+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
+
+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)
+done
+
+lemma complex_expi_Ex: 
+   "EX a r. z = complex_of_real r * expi a"
+apply (cut_tac z = "z" in rcis_Ex)
+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)
+apply auto
+done
+
+
+(****
+Goal "[| - pi < a; a <= pi |] ==> (-pi < a & a <= 0) | (0 <= a & a <= pi)"
+by Auto_tac;
+qed "lemma_split_interval";
+
+Goalw [arg_def]
+  "[| r ~= 0; - pi < a; a <= 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,
+    real_minus_mult_eq2 RS sym] real_mult_ac));
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
+by (dtac lemma_split_interval 1 THEN safe);
+****)
+
+
+ML
+{*
+val complex_zero_def = thm"complex_zero_def";
+val complex_one_def = thm"complex_one_def";
+val complex_minus_def = thm"complex_minus_def";
+val complex_diff_def = thm"complex_diff_def";
+val complex_divide_def = thm"complex_divide_def";
+val complex_mult_def = thm"complex_mult_def";
+val complex_add_def = thm"complex_add_def";
+val complex_of_real_def = thm"complex_of_real_def";
+val i_def = thm"i_def";
+val expi_def = thm"expi_def";
+val cis_def = thm"cis_def";
+val rcis_def = thm"rcis_def";
+val cmod_def = thm"cmod_def";
+val cnj_def = thm"cnj_def";
+val sgn_def = thm"sgn_def";
+val arg_def = thm"arg_def";
+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";
+val complex_Re_one = thm"complex_Re_one";
+val complex_Im_one = thm"complex_Im_one";
+val complex_Re_i = thm"complex_Re_i";
+val complex_Im_i = thm"complex_Im_i";
+val Re_complex_of_real_zero = thm"Re_complex_of_real_zero";
+val Im_complex_of_real_zero = thm"Im_complex_of_real_zero";
+val Re_complex_of_real_one = thm"Re_complex_of_real_one";
+val Im_complex_of_real_one = thm"Im_complex_of_real_one";
+val Re_complex_of_real = thm"Re_complex_of_real";
+val Im_complex_of_real = thm"Im_complex_of_real";
+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_left_zero = thm"complex_add_minus_left_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_mult_minus_one = thm"complex_mult_minus_one";
+val complex_mult_minus_one_right = thm"complex_mult_minus_one_right";
+val complex_minus_mult_cancel = thm"complex_minus_mult_cancel";
+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 complex_mult_left_cancel = thm"complex_mult_left_cancel";
+val complex_mult_right_cancel = thm"complex_mult_right_cancel";
+val complex_inverse_not_zero = thm"complex_inverse_not_zero";
+val complex_mult_not_zero = thm"complex_mult_not_zero";
+val complex_inverse_inverse = thm"complex_inverse_inverse";
+val complex_inverse_one = thm"complex_inverse_one";
+val complex_minus_inverse = thm"complex_minus_inverse";
+val complex_inverse_distrib = thm"complex_inverse_distrib";
+val complex_times_divide1_eq = thm"complex_times_divide1_eq";
+val complex_times_divide2_eq = thm"complex_times_divide2_eq";
+val complex_divide_divide1_eq = thm"complex_divide_divide1_eq";
+val complex_divide_divide2_eq = thm"complex_divide_divide2_eq";
+val complex_minus_divide_eq = thm"complex_minus_divide_eq";
+val complex_divide_minus_eq = thm"complex_divide_minus_eq";
+val complex_add_divide_distrib = thm"complex_add_divide_distrib";
+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";
+val complex_of_real_minus = thm"complex_of_real_minus";
+val complex_of_real_inverse = thm"complex_of_real_inverse";
+val complex_of_real_add = thm"complex_of_real_add";
+val complex_of_real_diff = thm"complex_of_real_diff";
+val complex_of_real_mult = thm"complex_of_real_mult";
+val complex_of_real_divide = thm"complex_of_real_divide";
+val complex_of_real_pow = thm"complex_of_real_pow";
+val complex_mod = thm"complex_mod";
+val complex_mod_zero = thm"complex_mod_zero";
+val complex_mod_one = thm"complex_mod_one";
+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";
+val complex_mod_cnj = thm"complex_mod_cnj";
+val complex_cnj_minus = thm"complex_cnj_minus";
+val complex_cnj_inverse = thm"complex_cnj_inverse";
+val complex_cnj_add = thm"complex_cnj_add";
+val complex_cnj_diff = thm"complex_cnj_diff";
+val complex_cnj_mult = thm"complex_cnj_mult";
+val complex_cnj_divide = thm"complex_cnj_divide";
+val complex_cnj_one = thm"complex_cnj_one";
+val complex_cnj_pow = thm"complex_cnj_pow";
+val complex_add_cnj = thm"complex_add_cnj";
+val complex_diff_cnj = thm"complex_diff_cnj";
+val complex_cnj_zero = thm"complex_cnj_zero";
+val complex_cnj_zero_iff = thm"complex_cnj_zero_iff";
+val complex_mult_cnj = thm"complex_mult_cnj";
+val complex_mult_zero_iff = thm"complex_mult_zero_iff";
+val complex_add_left_cancel_zero = thm"complex_add_left_cancel_zero";
+val complex_diff_mult_distrib = thm"complex_diff_mult_distrib";
+val complex_diff_mult_distrib2 = thm"complex_diff_mult_distrib2";
+val complex_mod_eq_zero_cancel = thm"complex_mod_eq_zero_cancel";
+val complex_mod_complex_of_real_of_nat = thm"complex_mod_complex_of_real_of_nat";
+val complex_mod_minus = thm"complex_mod_minus";
+val complex_mod_mult_cnj = thm"complex_mod_mult_cnj";
+val complex_mod_squared = thm"complex_mod_squared";
+val complex_mod_ge_zero = thm"complex_mod_ge_zero";
+val abs_cmod_cancel = thm"abs_cmod_cancel";
+val complex_mod_mult = thm"complex_mod_mult";
+val complex_mod_add_squared_eq = thm"complex_mod_add_squared_eq";
+val complex_Re_mult_cnj_le_cmod = thm"complex_Re_mult_cnj_le_cmod";
+val complex_Re_mult_cnj_le_cmod2 = thm"complex_Re_mult_cnj_le_cmod2";
+val real_sum_squared_expand = thm"real_sum_squared_expand";
+val complex_mod_triangle_squared = thm"complex_mod_triangle_squared";
+val complex_mod_minus_le_complex_mod = thm"complex_mod_minus_le_complex_mod";
+val complex_mod_triangle_ineq = thm"complex_mod_triangle_ineq";
+val complex_mod_triangle_ineq2 = thm"complex_mod_triangle_ineq2";
+val complex_mod_diff_commute = thm"complex_mod_diff_commute";
+val complex_mod_add_less = thm"complex_mod_add_less";
+val complex_mod_mult_less = thm"complex_mod_mult_less";
+val complex_mod_diff_ineq = thm"complex_mod_diff_ineq";
+val complex_Re_le_cmod = thm"complex_Re_le_cmod";
+val complex_mod_gt_zero = thm"complex_mod_gt_zero";
+val complex_mod_complexpow = thm"complex_mod_complexpow";
+val complexpow_minus = thm"complexpow_minus";
+val complex_inverse_minus = thm"complex_inverse_minus";
+val complex_divide_one = thm"complex_divide_one";
+val complex_mod_inverse = thm"complex_mod_inverse";
+val complex_mod_divide = thm"complex_mod_divide";
+val complex_inverse_divide = thm"complex_inverse_divide";
+val complexpow_mult = thm"complexpow_mult";
+val complexpow_zero = thm"complexpow_zero";
+val complexpow_not_zero = thm"complexpow_not_zero";
+val complexpow_zero_zero = thm"complexpow_zero_zero";
+val complexpow_i_squared = thm"complexpow_i_squared";
+val complex_i_not_zero = thm"complex_i_not_zero";
+val complex_mult_eq_zero_cancel1 = thm"complex_mult_eq_zero_cancel1";
+val complex_mult_eq_zero_cancel2 = thm"complex_mult_eq_zero_cancel2";
+val complex_mult_not_eq_zero_iff = thm"complex_mult_not_eq_zero_iff";
+val complexpow_inverse = thm"complexpow_inverse";
+val sgn_zero = thm"sgn_zero";
+val sgn_one = thm"sgn_one";
+val sgn_minus = thm"sgn_minus";
+val sgn_eq = thm"sgn_eq";
+val complex_split = thm"complex_split";
+val Re_complex_i = thm"Re_complex_i";
+val Im_complex_i = thm"Im_complex_i";
+val i_mult_eq = thm"i_mult_eq";
+val i_mult_eq2 = thm"i_mult_eq2";
+val cmod_i = thm"cmod_i";
+val complex_eq_Re_eq = thm"complex_eq_Re_eq";
+val complex_eq_Im_eq = thm"complex_eq_Im_eq";
+val complex_eq_cancel_iff = thm"complex_eq_cancel_iff";
+val complex_eq_cancel_iffA = thm"complex_eq_cancel_iffA";
+val complex_eq_cancel_iffB = thm"complex_eq_cancel_iffB";
+val complex_eq_cancel_iffC = thm"complex_eq_cancel_iffC";
+val complex_eq_cancel_iff2 = thm"complex_eq_cancel_iff2";
+val complex_eq_cancel_iff2a = thm"complex_eq_cancel_iff2a";
+val complex_eq_cancel_iff3 = thm"complex_eq_cancel_iff3";
+val complex_eq_cancel_iff3a = thm"complex_eq_cancel_iff3a";
+val complex_split_Re_zero = thm"complex_split_Re_zero";
+val complex_split_Im_zero = thm"complex_split_Im_zero";
+val Re_sgn = thm"Re_sgn";
+val Im_sgn = thm"Im_sgn";
+val complex_inverse_complex_split = thm"complex_inverse_complex_split";
+val Re_mult_i_eq = thm"Re_mult_i_eq";
+val Im_mult_i_eq = thm"Im_mult_i_eq";
+val complex_mod_mult_i = thm"complex_mod_mult_i";
+val cos_arg_i_mult_zero = thm"cos_arg_i_mult_zero";
+val cos_arg_i_mult_zero2 = thm"cos_arg_i_mult_zero2";
+val complex_of_real_not_zero_iff = thm"complex_of_real_not_zero_iff";
+val complex_of_real_zero_iff = thm"complex_of_real_zero_iff";
+val cos_arg_i_mult_zero3 = thm"cos_arg_i_mult_zero3";
+val complex_split_polar = thm"complex_split_polar";
+val rcis_Ex = thm"rcis_Ex";
+val Re_complex_polar = thm"Re_complex_polar";
+val Re_rcis = thm"Re_rcis";
+val Im_complex_polar = thm"Im_complex_polar";
+val Im_rcis = thm"Im_rcis";
+val complex_mod_complex_polar = thm"complex_mod_complex_polar";
+val complex_mod_rcis = thm"complex_mod_rcis";
+val complex_mod_sqrt_Re_mult_cnj = thm"complex_mod_sqrt_Re_mult_cnj";
+val complex_Re_cnj = thm"complex_Re_cnj";
+val complex_Im_cnj = thm"complex_Im_cnj";
+val complex_In_mult_cnj_zero = thm"complex_In_mult_cnj_zero";
+val complex_Re_mult = thm"complex_Re_mult";
+val complex_Re_mult_complex_of_real = thm"complex_Re_mult_complex_of_real";
+val complex_Im_mult_complex_of_real = thm"complex_Im_mult_complex_of_real";
+val complex_Re_mult_complex_of_real2 = thm"complex_Re_mult_complex_of_real2";
+val complex_Im_mult_complex_of_real2 = thm"complex_Im_mult_complex_of_real2";
+val cis_rcis_eq = thm"cis_rcis_eq";
+val rcis_mult = thm"rcis_mult";
+val cis_mult = thm"cis_mult";
+val cis_zero = thm"cis_zero";
+val cis_zero2 = thm"cis_zero2";
+val rcis_zero_mod = thm"rcis_zero_mod";
+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";
+val cis_inverse = thm"cis_inverse";
+val rcis_inverse = thm"rcis_inverse";
+val cis_divide = thm"cis_divide";
+val rcis_divide = thm"rcis_divide";
+val Re_cis = thm"Re_cis";
+val Im_cis = thm"Im_cis";
+val cos_n_Re_cis_pow_n = thm"cos_n_Re_cis_pow_n";
+val sin_n_Im_cis_pow_n = thm"sin_n_Im_cis_pow_n";
+val expi_Im_split = thm"expi_Im_split";
+val expi_Im_cis = thm"expi_Im_cis";
+val expi_add = thm"expi_add";
+val expi_complex_split = thm"expi_complex_split";
+val expi_zero = thm"expi_zero";
+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/NSCA.ML	Tue Dec 23 16:52:49 2003 +0100
+++ b/src/HOL/Complex/NSCA.ML	Tue Dec 23 16:53:33 2003 +0100
@@ -838,11 +838,10 @@
 by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
 by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
 by (auto_tac (claset(),HOL_ss addsimps [complex_minus,complex_add,
-    complex_mod,pair_mem_complex RS Abs_complex_inverse,snd_conv,
-    fst_conv,two_eq_Suc_Suc]));
+    complex_mod, snd_conv, fst_conv,numeral_2_eq_2]));
 by (rtac (realpow_two_abs RS subst) 1);
 by (res_inst_tac [("x1","xa + - xb")] (realpow_two_abs RS subst) 1);
-by (simp_tac (simpset() addsimps [two_eq_Suc_Suc RS sym]) 1);
+by (simp_tac (simpset() addsimps [numeral_2_eq_2 RS sym]) 1);
 by (rtac lemma_sqrt_hcomplex_capprox 1);
 by Auto_tac;
 qed "hcomplex_capproxI";
@@ -869,11 +868,11 @@
 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 (auto_tac (claset(),simpset() addsimps [complex_mod,two_eq_Suc_Suc] delsimps [realpow_Suc]));
+by (auto_tac (claset(),simpset() addsimps [complex_mod,numeral_2_eq_2] delsimps [realpow_Suc]));
 by (rtac ccontr 1 THEN dtac real_leI 1);
 by (dtac order_less_le_trans 1 THEN assume_tac 1);
 by (dtac (real_sqrt_ge_abs1 RSN (2,order_less_le_trans)) 1);
-by (auto_tac (claset(),simpset() addsimps [two_eq_Suc_Suc RS sym]));
+by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2 RS sym]));
 qed "CFinite_HFinite_Re";
 
 Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) : CFinite \
@@ -901,12 +900,12 @@
 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 (auto_tac (claset(),simpset() addsimps [complex_mod,two_eq_Suc_Suc] delsimps [realpow_Suc]));
+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);
 by (rtac (realpow_two_abs RS subst) 1);
 by (res_inst_tac [("x1","Y x")] (realpow_two_abs RS subst) 1);
-by (simp_tac (simpset() addsimps [two_eq_Suc_Suc RS sym]) 1);
+by (simp_tac (simpset() addsimps [numeral_2_eq_2 RS sym]) 1);
 by (rtac lemma_sqrt_hcomplex_capprox 1);
 by Auto_tac;
 qed "HFinite_Re_Im_CFinite";

--- a/src/HOL/Complex/NSComplex.thy	Tue Dec 23 16:52:49 2003 +0100
+++ b/src/HOL/Complex/NSComplex.thy	Tue Dec 23 16:53:33 2003 +0100
@@ -866,7 +866,7 @@
 
 lemma hcomplex_mult_hcnj: "z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
 apply (rule_tac z = "z" in eq_Abs_hcomplex)
-apply (auto simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add hypreal_mult complex_mult_cnj two_eq_Suc_Suc)
+apply (auto simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add hypreal_mult complex_mult_cnj numeral_2_eq_2)
 done
 
 
@@ -951,7 +951,7 @@
 
 lemma hcmod_mult_hcnj: "hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
 apply (rule_tac z = "z" in eq_Abs_hcomplex)
-apply (auto simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj two_eq_Suc_Suc)
+apply (auto simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj numeral_2_eq_2)
 done
 
 lemma hcmod_ge_zero: "(0::hypreal) <= hcmod x"
@@ -976,9 +976,9 @@
 apply (rule_tac z = "x" in eq_Abs_hcomplex)
 apply (rule_tac z = "y" in eq_Abs_hcomplex)
 apply (auto simp add: hcmod hcomplex_add hypreal_mult hRe hcnj hcomplex_mult
-                      two_eq_Suc_Suc realpow_two [symmetric] 
+                      numeral_2_eq_2 realpow_two [symmetric] 
                  simp del: realpow_Suc)
-apply (auto simp add: two_eq_Suc_Suc [symmetric] complex_mod_add_squared_eq
+apply (auto simp add: numeral_2_eq_2 [symmetric] complex_mod_add_squared_eq
                  hypreal_add [symmetric] hypreal_mult [symmetric] 
                  hypreal_of_real_def [symmetric])
 done
@@ -1000,9 +1000,9 @@
 apply (rule_tac z = "x" in eq_Abs_hcomplex)
 apply (rule_tac z = "y" in eq_Abs_hcomplex)
 apply (auto simp add: hcmod hcnj hcomplex_add hypreal_mult hypreal_add
-                      hypreal_le realpow_two [symmetric] two_eq_Suc_Suc
+                      hypreal_le realpow_two [symmetric] numeral_2_eq_2
             simp del: realpow_Suc)
-apply (simp (no_asm) add: two_eq_Suc_Suc [symmetric])
+apply (simp (no_asm) add: numeral_2_eq_2 [symmetric])
 done
 declare hcmod_triangle_squared [simp]
 
@@ -1191,7 +1191,7 @@
 declare hcomplex_i_mult_eq [simp]
 
 lemma hcomplexpow_i_squared: "iii ^ 2 = - 1"
-apply (simp (no_asm) add: two_eq_Suc_Suc)
+apply (simp (no_asm) add: numeral_2_eq_2)
 done
 declare hcomplexpow_i_squared [simp]
 
@@ -1296,7 +1296,7 @@
       ( *f* sqrt) (x ^ 2 + y ^ 2)"
 apply (rule_tac z = "x" in eq_Abs_hypreal)
 apply (rule_tac z = "y" in eq_Abs_hypreal)
-apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_add hcomplex_mult starfun hypreal_mult hypreal_add hcmod cmod_i two_eq_Suc_Suc)
+apply (auto simp add: hcomplex_of_hypreal iii_def hcomplex_add hcomplex_mult starfun hypreal_mult hypreal_add hcmod cmod_i numeral_2_eq_2)
 done
 
 lemma hcomplex_eq_hRe_eq:
@@ -1420,7 +1420,7 @@
       iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
 apply (rule_tac z = "x" in eq_Abs_hypreal)
 apply (rule_tac z = "y" in eq_Abs_hypreal)
-apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hcomplex_add iii_def starfun hypreal_mult hypreal_add hcomplex_inverse hypreal_divide hcomplex_diff complex_inverse_complex_split two_eq_Suc_Suc)
+apply (auto simp add: hcomplex_of_hypreal hcomplex_mult hcomplex_add iii_def starfun hypreal_mult hypreal_add hcomplex_inverse hypreal_divide hcomplex_diff complex_inverse_complex_split numeral_2_eq_2)
 done
 
 lemma hRe_mult_i_eq:

--- a/src/HOL/IsaMakefile	Tue Dec 23 16:52:49 2003 +0100
+++ b/src/HOL/IsaMakefile	Tue Dec 23 16:53:33 2003 +0100
@@ -168,8 +168,7 @@
   Complex/Complex_Main.thy\
   Complex/CLim.ML Complex/CLim.thy\
   Complex/CSeries.ML Complex/CSeries.thy\
-  Complex/CStar.ML Complex/CStar.thy\
-  Complex/Complex.ML Complex/Complex.thy\
+  Complex/CStar.ML Complex/CStar.thy Complex/Complex.thy\
   Complex/ComplexArith0.ML Complex/ComplexArith0.thy\
   Complex/ComplexBin.ML Complex/ComplexBin.thy\
   Complex/NSCA.ML Complex/NSCA.thy\