src/HOL/Complex/Complex.thy
author paulson
Tue Jan 13 10:37:52 2004 +0100 (2004-01-13)
changeset 14354 988aa4648597
parent 14353 79f9fbef9106
child 14373 67a628beb981
permissions -rw-r--r--
types complex and hcomplex are now instances of class ringpower:
omitting redundant lemmas
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(*  Title:       Complex.thy
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    Author:      Jacques D. Fleuriot
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    Copyright:   2001 University of Edinburgh
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    Description: Complex numbers
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*)
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theory Complex = HLog:
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typedef complex = "{p::(real*real). True}"
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  by blast
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instance complex :: zero ..
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instance complex :: one ..
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instance complex :: plus ..
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instance complex :: times ..
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instance complex :: minus ..
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instance complex :: inverse ..
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instance complex :: power ..
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consts
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  "ii"    :: complex        ("ii")
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constdefs
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  (*--- real and Imaginary parts ---*)
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  Re :: "complex => real"
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  "Re(z) == fst(Rep_complex z)"
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  Im :: "complex => real"
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  "Im(z) == snd(Rep_complex z)"
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  (*----------- modulus ------------*)
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  cmod :: "complex => real"
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  "cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)"
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  (*----- injection from reals -----*)
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  complex_of_real :: "real => complex"
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  "complex_of_real r == Abs_complex(r,0::real)"
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  (*------- complex conjugate ------*)
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  cnj :: "complex => complex"
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  "cnj z == Abs_complex(Re z, -Im z)"
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  (*------------ Argand -------------*)
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  sgn :: "complex => complex"
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  "sgn z == z / complex_of_real(cmod z)"
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  arg :: "complex => real"
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  "arg z == @a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi"
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defs (overloaded)
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  complex_zero_def:
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  "0 == Abs_complex(0::real,0)"
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  complex_one_def:
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  "1 == Abs_complex(1,0::real)"
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  (*------ imaginary unit ----------*)
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  i_def:
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  "ii == Abs_complex(0::real,1)"
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  (*----------- negation -----------*)
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  complex_minus_def:
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  "- (z::complex) == Abs_complex(-Re z, -Im z)"
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  (*----------- inverse -----------*)
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  complex_inverse_def:
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  "inverse (z::complex) == Abs_complex(Re(z)/(Re(z) ^ 2 + Im(z) ^ 2),
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                            -Im(z)/(Re(z) ^ 2 + Im(z) ^ 2))"
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  complex_add_def:
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  "w + (z::complex) == Abs_complex(Re(w) + Re(z),Im(w) + Im(z))"
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  complex_diff_def:
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  "w - (z::complex) == w + -(z::complex)"
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  complex_mult_def:
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  "w * (z::complex) == Abs_complex(Re(w) * Re(z) - Im(w) * Im(z),
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			Re(w) * Im(z) + Im(w) * Re(z))"
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  (*----------- division ----------*)
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  complex_divide_def:
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  "w / (z::complex) == w * inverse z"
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constdefs
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  (* abbreviation for (cos a + i sin a) *)
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  cis :: "real => complex"
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  "cis a == complex_of_real(cos a) + ii * complex_of_real(sin a)"
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  (* abbreviation for r*(cos a + i sin a) *)
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  rcis :: "[real, real] => complex"
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  "rcis r a == complex_of_real r * cis a"
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  (* e ^ (x + iy) *)
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  expi :: "complex => complex"
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  "expi z == complex_of_real(exp (Re z)) * cis (Im z)"
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lemma inj_Rep_complex: "inj Rep_complex"
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apply (rule inj_on_inverseI)
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apply (rule Rep_complex_inverse)
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done
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lemma inj_Abs_complex: "inj Abs_complex"
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apply (rule inj_on_inverseI)
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apply (rule Abs_complex_inverse)
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apply (simp (no_asm) add: complex_def)
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done
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declare inj_Abs_complex [THEN injD, simp]
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lemma Abs_complex_cancel_iff: "(Abs_complex x = Abs_complex y) = (x = y)"
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by (auto dest: inj_Abs_complex [THEN injD])
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declare Abs_complex_cancel_iff [simp]
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lemma pair_mem_complex: "(x,y) : complex"
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by (unfold complex_def, auto)
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declare pair_mem_complex [simp]
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lemma Abs_complex_inverse2: "Rep_complex (Abs_complex (x,y)) = (x,y)"
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apply (simp (no_asm) add: Abs_complex_inverse)
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done
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declare Abs_complex_inverse2 [simp]
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lemma eq_Abs_complex: "(!!x y. z = Abs_complex(x,y) ==> P) ==> P"
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apply (rule_tac p = "Rep_complex z" in PairE)
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apply (drule_tac f = Abs_complex in arg_cong)
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apply (force simp add: Rep_complex_inverse)
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done
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lemma Re: "Re(Abs_complex(x,y)) = x"
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apply (unfold Re_def)
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apply (simp (no_asm))
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done
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declare Re [simp]
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lemma Im: "Im(Abs_complex(x,y)) = y"
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apply (unfold Im_def)
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apply (simp (no_asm))
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done
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declare Im [simp]
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lemma Abs_complex_cancel: "Abs_complex(Re(z),Im(z)) = z"
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apply (rule_tac z = z in eq_Abs_complex)
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apply (simp (no_asm_simp))
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done
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declare Abs_complex_cancel [simp]
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lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))"
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apply (rule_tac z = w in eq_Abs_complex)
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apply (rule_tac z = z in eq_Abs_complex)
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apply (auto dest: inj_Abs_complex [THEN injD])
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done
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lemma complex_Re_zero: "Re 0 = 0"
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apply (unfold complex_zero_def)
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apply (simp (no_asm))
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done
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lemma complex_Im_zero: "Im 0 = 0"
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apply (unfold complex_zero_def)
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apply (simp (no_asm))
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done
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declare complex_Re_zero [simp] complex_Im_zero [simp]
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lemma complex_Re_one: "Re 1 = 1"
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apply (unfold complex_one_def)
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apply (simp (no_asm))
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done
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declare complex_Re_one [simp]
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lemma complex_Im_one: "Im 1 = 0"
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apply (unfold complex_one_def)
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apply (simp (no_asm))
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done
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declare complex_Im_one [simp]
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lemma complex_Re_i: "Re(ii) = 0"
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by (unfold i_def, auto)
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declare complex_Re_i [simp]
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lemma complex_Im_i: "Im(ii) = 1"
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by (unfold i_def, auto)
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declare complex_Im_i [simp]
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lemma Re_complex_of_real_zero: "Re(complex_of_real 0) = 0"
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apply (unfold complex_of_real_def)
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apply (simp (no_asm))
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done
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declare Re_complex_of_real_zero [simp]
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lemma Im_complex_of_real_zero: "Im(complex_of_real 0) = 0"
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apply (unfold complex_of_real_def)
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apply (simp (no_asm))
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done
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declare Im_complex_of_real_zero [simp]
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lemma Re_complex_of_real_one: "Re(complex_of_real 1) = 1"
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apply (unfold complex_of_real_def)
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apply (simp (no_asm))
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done
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declare Re_complex_of_real_one [simp]
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lemma Im_complex_of_real_one: "Im(complex_of_real 1) = 0"
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apply (unfold complex_of_real_def)
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apply (simp (no_asm))
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done
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declare Im_complex_of_real_one [simp]
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lemma Re_complex_of_real: "Re(complex_of_real z) = z"
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by (unfold complex_of_real_def, auto)
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declare Re_complex_of_real [simp]
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lemma Im_complex_of_real: "Im(complex_of_real z) = 0"
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by (unfold complex_of_real_def, auto)
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declare Im_complex_of_real [simp]
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subsection{*Negation*}
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lemma complex_minus: "- Abs_complex(x,y) = Abs_complex(-x,-y)"
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apply (unfold complex_minus_def)
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apply (simp (no_asm))
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done
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lemma complex_Re_minus: "Re (-z) = - Re z"
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apply (unfold Re_def)
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apply (rule_tac z = z in eq_Abs_complex)
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apply (auto simp add: complex_minus)
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done
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lemma complex_Im_minus: "Im (-z) = - Im z"
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apply (unfold Im_def)
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apply (rule_tac z = z in eq_Abs_complex)
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apply (auto simp add: complex_minus)
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done
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lemma complex_minus_minus: "- (- z) = (z::complex)"
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apply (unfold complex_minus_def)
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apply (simp (no_asm))
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done
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declare complex_minus_minus [simp]
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lemma inj_complex_minus: "inj(%r::complex. -r)"
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apply (rule inj_onI)
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apply (drule_tac f = uminus in arg_cong, simp)
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done
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lemma complex_minus_zero: "-(0::complex) = 0"
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apply (unfold complex_zero_def)
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apply (simp (no_asm) add: complex_minus)
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done
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declare complex_minus_zero [simp]
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lemma complex_minus_zero_iff: "(-x = 0) = (x = (0::complex))"
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apply (rule_tac z = x in eq_Abs_complex)
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apply (auto dest: inj_Abs_complex [THEN injD]
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            simp add: complex_zero_def complex_minus)
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done
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declare complex_minus_zero_iff [simp]
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lemma complex_minus_zero_iff2: "(0 = -x) = (x = (0::real))"
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by (auto dest: sym)
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declare complex_minus_zero_iff2 [simp]
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lemma complex_minus_not_zero_iff: "(-x \<noteq> 0) = (x \<noteq> (0::complex))"
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by auto
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subsection{*Addition*}
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lemma complex_add:
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      "Abs_complex(x1,y1) + Abs_complex(x2,y2) = Abs_complex(x1+x2,y1+y2)"
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apply (unfold complex_add_def)
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apply (simp (no_asm))
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done
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lemma complex_Re_add: "Re(x + y) = Re(x) + Re(y)"
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apply (unfold Re_def)
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apply (rule_tac z = x in eq_Abs_complex)
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apply (rule_tac z = y in eq_Abs_complex)
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apply (auto simp add: complex_add)
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done
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lemma complex_Im_add: "Im(x + y) = Im(x) + Im(y)"
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apply (unfold Im_def)
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apply (rule_tac z = x in eq_Abs_complex)
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apply (rule_tac z = y in eq_Abs_complex)
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apply (auto simp add: complex_add)
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done
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lemma complex_add_commute: "(u::complex) + v = v + u"
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apply (unfold complex_add_def)
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apply (simp (no_asm) add: real_add_commute)
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done
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lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)"
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apply (unfold complex_add_def)
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apply (simp (no_asm) add: real_add_assoc)
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done
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lemma complex_add_left_commute: "(x::complex) + (y + z) = y + (x + z)"
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apply (unfold complex_add_def)
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apply (simp (no_asm) add: add_left_commute)
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done
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lemmas complex_add_ac = complex_add_assoc complex_add_commute
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                      complex_add_left_commute
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lemma complex_add_zero_left: "(0::complex) + z = z"
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apply (unfold complex_add_def complex_zero_def)
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apply (simp (no_asm))
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done
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declare complex_add_zero_left [simp]
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lemma complex_add_zero_right: "z + (0::complex) = z"
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apply (unfold complex_add_def complex_zero_def)
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apply (simp (no_asm))
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done
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declare complex_add_zero_right [simp]
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lemma complex_add_minus_right_zero:
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      "z + -z = (0::complex)"
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apply (unfold complex_add_def complex_minus_def complex_zero_def)
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apply (simp (no_asm))
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done
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declare complex_add_minus_right_zero [simp]
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lemma complex_add_minus_left:
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      "-z + z = (0::complex)"
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apply (unfold complex_add_def complex_minus_def complex_zero_def)
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apply (simp (no_asm))
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done
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lemma complex_add_minus_cancel: "z + (- z + w) = (w::complex)"
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apply (simp (no_asm) add: complex_add_assoc [symmetric])
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done
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lemma complex_minus_add_cancel: "(-z) + (z + w) = (w::complex)"
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by (simp add: complex_add_minus_left complex_add_assoc [symmetric])
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declare complex_add_minus_cancel [simp] complex_minus_add_cancel [simp]
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lemma complex_add_minus_eq_minus: "x + y = (0::complex) ==> x = -y"
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by (auto simp add: complex_Re_Im_cancel_iff complex_Re_add complex_Im_add complex_Re_minus complex_Im_minus)
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lemma complex_minus_add_distrib: "-(x + y) = -x + -(y::complex)"
paulson@14334
   360
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   361
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   362
apply (auto simp add: complex_minus complex_add)
paulson@14323
   363
done
paulson@14323
   364
declare complex_minus_add_distrib [simp]
paulson@14323
   365
paulson@14323
   366
lemma complex_add_left_cancel: "((x::complex) + y = x + z) = (y = z)"
paulson@14334
   367
apply safe
paulson@14323
   368
apply (drule_tac f = "%t.-x + t" in arg_cong)
paulson@14341
   369
apply (simp add: complex_add_minus_left complex_add_assoc [symmetric])
paulson@14323
   370
done
paulson@14323
   371
declare complex_add_left_cancel [iff]
paulson@14323
   372
paulson@14323
   373
lemma complex_add_right_cancel: "(y + (x::complex)= z + x) = (y = z)"
paulson@14323
   374
apply (simp (no_asm) add: complex_add_commute)
paulson@14323
   375
done
paulson@14323
   376
declare complex_add_right_cancel [simp]
paulson@14323
   377
paulson@14323
   378
lemma complex_eq_minus_iff: "((x::complex) = y) = (0 = x + - y)"
paulson@14334
   379
apply safe
paulson@14334
   380
apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto)
paulson@14323
   381
done
paulson@14323
   382
paulson@14323
   383
lemma complex_eq_minus_iff2: "((x::complex) = y) = (x + - y = 0)"
paulson@14334
   384
apply safe
paulson@14334
   385
apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto)
paulson@14323
   386
done
paulson@14323
   387
paulson@14323
   388
lemma complex_diff_0: "(0::complex) - x = -x"
paulson@14323
   389
apply (simp (no_asm) add: complex_diff_def)
paulson@14323
   390
done
paulson@14323
   391
paulson@14323
   392
lemma complex_diff_0_right: "x - (0::complex) = x"
paulson@14323
   393
apply (simp (no_asm) add: complex_diff_def)
paulson@14323
   394
done
paulson@14323
   395
paulson@14323
   396
lemma complex_diff_self: "x - x = (0::complex)"
paulson@14323
   397
apply (simp (no_asm) add: complex_diff_def)
paulson@14323
   398
done
paulson@14323
   399
paulson@14323
   400
declare complex_diff_0 [simp] complex_diff_0_right [simp] complex_diff_self [simp]
paulson@14323
   401
paulson@14323
   402
lemma complex_diff:
paulson@14323
   403
      "Abs_complex(x1,y1) - Abs_complex(x2,y2) = Abs_complex(x1-x2,y1-y2)"
paulson@14323
   404
apply (unfold complex_diff_def)
paulson@14323
   405
apply (simp (no_asm) add: complex_add complex_minus)
paulson@14323
   406
done
paulson@14323
   407
paulson@14323
   408
lemma complex_diff_eq_eq: "((x::complex) - y = z) = (x = z + y)"
paulson@14341
   409
by (auto simp add: complex_add_minus_left complex_diff_def complex_add_assoc)
paulson@14323
   410
paulson@14323
   411
paulson@14323
   412
subsection{*Multiplication*}
paulson@14323
   413
paulson@14323
   414
lemma complex_mult:
paulson@14323
   415
      "Abs_complex(x1,y1) * Abs_complex(x2,y2) =
paulson@14323
   416
       Abs_complex(x1*x2 - y1*y2,x1*y2 + y1*x2)"
paulson@14323
   417
apply (unfold complex_mult_def)
paulson@14323
   418
apply (simp (no_asm))
paulson@14323
   419
done
paulson@14323
   420
paulson@14323
   421
lemma complex_mult_commute: "(w::complex) * z = z * w"
paulson@14323
   422
apply (unfold complex_mult_def)
paulson@14323
   423
apply (simp (no_asm) add: real_mult_commute real_add_commute)
paulson@14323
   424
done
paulson@14323
   425
paulson@14323
   426
lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
paulson@14323
   427
apply (unfold complex_mult_def)
paulson@14334
   428
apply (simp (no_asm) add: complex_Re_Im_cancel_iff real_mult_assoc right_diff_distrib right_distrib left_diff_distrib left_distrib mult_left_commute)
paulson@14323
   429
done
paulson@14323
   430
paulson@14323
   431
lemma complex_mult_left_commute: "(x::complex) * (y * z) = y * (x * z)"
paulson@14323
   432
apply (unfold complex_mult_def)
paulson@14334
   433
apply (simp (no_asm) add: complex_Re_Im_cancel_iff mult_left_commute right_diff_distrib right_distrib)
paulson@14323
   434
done
paulson@14323
   435
paulson@14323
   436
lemmas complex_mult_ac = complex_mult_assoc complex_mult_commute
paulson@14323
   437
                      complex_mult_left_commute
paulson@14323
   438
paulson@14323
   439
lemma complex_mult_one_left: "(1::complex) * z = z"
paulson@14323
   440
apply (unfold complex_one_def)
paulson@14334
   441
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   442
apply (simp (no_asm_simp) add: complex_mult)
paulson@14323
   443
done
paulson@14323
   444
declare complex_mult_one_left [simp]
paulson@14323
   445
paulson@14323
   446
lemma complex_mult_one_right: "z * (1::complex) = z"
paulson@14323
   447
apply (simp (no_asm) add: complex_mult_commute)
paulson@14323
   448
done
paulson@14323
   449
declare complex_mult_one_right [simp]
paulson@14323
   450
paulson@14323
   451
lemma complex_mult_zero_left: "(0::complex) * z = 0"
paulson@14323
   452
apply (unfold complex_zero_def)
paulson@14334
   453
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   454
apply (simp (no_asm_simp) add: complex_mult)
paulson@14323
   455
done
paulson@14323
   456
declare complex_mult_zero_left [simp]
paulson@14323
   457
paulson@14323
   458
lemma complex_mult_zero_right: "z * 0 = (0::complex)"
paulson@14323
   459
apply (simp (no_asm) add: complex_mult_commute)
paulson@14323
   460
done
paulson@14323
   461
declare complex_mult_zero_right [simp]
paulson@14323
   462
paulson@14323
   463
lemma complex_divide_zero: "0 / z = (0::complex)"
paulson@14334
   464
by (unfold complex_divide_def, auto)
paulson@14323
   465
declare complex_divide_zero [simp]
paulson@14323
   466
paulson@14323
   467
lemma complex_minus_mult_eq1: "-(x * y) = -x * (y::complex)"
paulson@14334
   468
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   469
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   470
apply (auto simp add: complex_mult complex_minus real_diff_def)
paulson@14323
   471
done
paulson@14323
   472
paulson@14323
   473
lemma complex_minus_mult_eq2: "-(x * y) = x * -(y::complex)"
paulson@14334
   474
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   475
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   476
apply (auto simp add: complex_mult complex_minus real_diff_def)
paulson@14323
   477
done
paulson@14323
   478
paulson@14323
   479
lemma complex_add_mult_distrib: "((z1::complex) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14334
   480
apply (rule_tac z = z1 in eq_Abs_complex)
paulson@14334
   481
apply (rule_tac z = z2 in eq_Abs_complex)
paulson@14334
   482
apply (rule_tac z = w in eq_Abs_complex)
paulson@14334
   483
apply (auto simp add: complex_mult complex_add left_distrib real_diff_def add_ac)
paulson@14323
   484
done
paulson@14323
   485
paulson@14323
   486
lemma complex_add_mult_distrib2: "(w::complex) * (z1 + z2) = (w * z1) + (w * z2)"
paulson@14323
   487
apply (rule_tac z1 = "z1 + z2" in complex_mult_commute [THEN ssubst])
paulson@14323
   488
apply (simp (no_asm) add: complex_add_mult_distrib)
paulson@14323
   489
apply (simp (no_asm) add: complex_mult_commute)
paulson@14323
   490
done
paulson@14323
   491
paulson@14354
   492
lemma complex_zero_not_eq_one: "(0::complex) \<noteq> 1"
paulson@14323
   493
apply (unfold complex_zero_def complex_one_def)
paulson@14323
   494
apply (simp (no_asm) add: complex_Re_Im_cancel_iff)
paulson@14323
   495
done
paulson@14323
   496
declare complex_zero_not_eq_one [simp]
paulson@14323
   497
declare complex_zero_not_eq_one [THEN not_sym, simp]
paulson@14323
   498
paulson@14323
   499
paulson@14323
   500
subsection{*Inverse*}
paulson@14323
   501
paulson@14354
   502
lemma complex_inverse:
paulson@14354
   503
     "inverse (Abs_complex(x,y)) =
paulson@14354
   504
      Abs_complex(x/(x ^ 2 + y ^ 2),-y/(x ^ 2 + y ^ 2))"
paulson@14323
   505
apply (unfold complex_inverse_def)
paulson@14323
   506
apply (simp (no_asm))
paulson@14323
   507
done
paulson@14323
   508
paulson@14323
   509
lemma COMPLEX_INVERSE_ZERO: "inverse 0 = (0::complex)"
paulson@14334
   510
by (unfold complex_inverse_def complex_zero_def, auto)
paulson@14323
   511
paulson@14323
   512
lemma COMPLEX_DIVISION_BY_ZERO: "a / (0::complex) = 0"
paulson@14323
   513
apply (simp (no_asm) add: complex_divide_def COMPLEX_INVERSE_ZERO)
paulson@14323
   514
done
paulson@14323
   515
paulson@14335
   516
instance complex :: division_by_zero
paulson@14335
   517
proof
paulson@14335
   518
  fix x :: complex
paulson@14335
   519
  show "inverse 0 = (0::complex)" by (rule COMPLEX_INVERSE_ZERO)
paulson@14335
   520
  show "x/0 = 0" by (rule COMPLEX_DIVISION_BY_ZERO) 
paulson@14335
   521
qed
paulson@14335
   522
paulson@14354
   523
lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
paulson@14334
   524
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
   525
apply (auto simp add: complex_mult complex_inverse complex_one_def 
paulson@14353
   526
       complex_zero_def add_divide_distrib [symmetric] power2_eq_square mult_ac)
paulson@14334
   527
apply (drule_tac y = y in real_sum_squares_not_zero)
paulson@14334
   528
apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto)
paulson@14323
   529
done
paulson@14323
   530
declare complex_mult_inv_left [simp]
paulson@14323
   531
paulson@14354
   532
lemma complex_mult_inv_right: "z \<noteq> (0::complex) ==> z * inverse(z) = 1"
paulson@14334
   533
by (auto intro: complex_mult_commute [THEN subst])
paulson@14323
   534
declare complex_mult_inv_right [simp]
paulson@14323
   535
paulson@14335
   536
paulson@14335
   537
subsection {* The field of complex numbers *}
paulson@14335
   538
paulson@14335
   539
instance complex :: field
paulson@14335
   540
proof
paulson@14335
   541
  fix z u v w :: complex
paulson@14335
   542
  show "(u + v) + w = u + (v + w)"
paulson@14335
   543
    by (rule complex_add_assoc) 
paulson@14335
   544
  show "z + w = w + z"
paulson@14335
   545
    by (rule complex_add_commute) 
paulson@14335
   546
  show "0 + z = z"
paulson@14335
   547
    by (rule complex_add_zero_left) 
paulson@14335
   548
  show "-z + z = 0"
paulson@14341
   549
    by (rule complex_add_minus_left) 
paulson@14335
   550
  show "z - w = z + -w"
paulson@14335
   551
    by (simp add: complex_diff_def)
paulson@14335
   552
  show "(u * v) * w = u * (v * w)"
paulson@14335
   553
    by (rule complex_mult_assoc) 
paulson@14335
   554
  show "z * w = w * z"
paulson@14335
   555
    by (rule complex_mult_commute) 
paulson@14335
   556
  show "1 * z = z"
paulson@14335
   557
    by (rule complex_mult_one_left) 
paulson@14341
   558
  show "0 \<noteq> (1::complex)"
paulson@14335
   559
    by (rule complex_zero_not_eq_one) 
paulson@14335
   560
  show "(u + v) * w = u * w + v * w"
paulson@14335
   561
    by (rule complex_add_mult_distrib) 
paulson@14341
   562
  show "z+u = z+v ==> u=v"
paulson@14341
   563
    proof -
paulson@14341
   564
      assume eq: "z+u = z+v" 
paulson@14341
   565
      hence "(-z + z) + u = (-z + z) + v" by (simp only: eq complex_add_assoc)
paulson@14341
   566
      thus "u = v" by (simp add: complex_add_minus_left)
paulson@14341
   567
    qed
paulson@14335
   568
  assume neq: "w \<noteq> 0"
paulson@14335
   569
  thus "z / w = z * inverse w"
paulson@14335
   570
    by (simp add: complex_divide_def)
paulson@14335
   571
  show "inverse w * w = 1"
paulson@14335
   572
    by (simp add: neq complex_mult_inv_left) 
paulson@14335
   573
qed
paulson@14335
   574
paulson@14335
   575
paulson@14335
   576
lemma complex_minus_mult_commute: "-x * y = x * -(y::complex)"
paulson@14348
   577
apply (simp)
paulson@14323
   578
done
paulson@14323
   579
paulson@14323
   580
subsection{*Embedding Properties for @{term complex_of_real} Map*}
paulson@14323
   581
paulson@14323
   582
lemma inj_complex_of_real: "inj complex_of_real"
paulson@14323
   583
apply (rule inj_onI)
paulson@14323
   584
apply (auto dest: inj_Abs_complex [THEN injD] simp add: complex_of_real_def)
paulson@14323
   585
done
paulson@14323
   586
paulson@14323
   587
lemma complex_of_real_one:
paulson@14323
   588
      "complex_of_real 1 = 1"
paulson@14323
   589
apply (unfold complex_one_def complex_of_real_def)
paulson@14323
   590
apply (rule refl)
paulson@14323
   591
done
paulson@14323
   592
declare complex_of_real_one [simp]
paulson@14323
   593
paulson@14323
   594
lemma complex_of_real_zero:
paulson@14323
   595
      "complex_of_real 0 = 0"
paulson@14323
   596
apply (unfold complex_zero_def complex_of_real_def)
paulson@14323
   597
apply (rule refl)
paulson@14323
   598
done
paulson@14323
   599
declare complex_of_real_zero [simp]
paulson@14323
   600
paulson@14348
   601
lemma complex_of_real_eq_iff:
paulson@14348
   602
     "(complex_of_real x = complex_of_real y) = (x = y)"
paulson@14334
   603
by (auto dest: inj_complex_of_real [THEN injD])
paulson@14323
   604
declare complex_of_real_eq_iff [iff]
paulson@14323
   605
paulson@14323
   606
lemma complex_of_real_minus: "complex_of_real(-x) = - complex_of_real x"
paulson@14323
   607
apply (simp (no_asm) add: complex_of_real_def complex_minus)
paulson@14323
   608
done
paulson@14323
   609
paulson@14348
   610
lemma complex_of_real_inverse:
paulson@14348
   611
 "complex_of_real(inverse x) = inverse(complex_of_real x)"
paulson@14348
   612
apply (case_tac "x=0", simp)
paulson@14334
   613
apply (simp add: complex_inverse complex_of_real_def real_divide_def 
paulson@14353
   614
                 inverse_mult_distrib power2_eq_square)
paulson@14323
   615
done
paulson@14323
   616
paulson@14348
   617
lemma complex_of_real_add:
paulson@14348
   618
 "complex_of_real x + complex_of_real y = complex_of_real (x + y)"
paulson@14323
   619
apply (simp (no_asm) add: complex_add complex_of_real_def)
paulson@14323
   620
done
paulson@14323
   621
paulson@14348
   622
lemma complex_of_real_diff:
paulson@14348
   623
 "complex_of_real x - complex_of_real y = complex_of_real (x - y)"
paulson@14323
   624
apply (simp (no_asm) add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add)
paulson@14323
   625
done
paulson@14323
   626
paulson@14348
   627
lemma complex_of_real_mult:
paulson@14348
   628
 "complex_of_real x * complex_of_real y = complex_of_real (x * y)"
paulson@14323
   629
apply (simp (no_asm) add: complex_mult complex_of_real_def)
paulson@14323
   630
done
paulson@14323
   631
paulson@14323
   632
lemma complex_of_real_divide:
paulson@14323
   633
      "complex_of_real x / complex_of_real y = complex_of_real(x/y)"
paulson@14323
   634
apply (unfold complex_divide_def)
paulson@14323
   635
apply (case_tac "y=0")
paulson@14323
   636
apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
paulson@14323
   637
apply (simp (no_asm_simp) add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def)
paulson@14323
   638
done
paulson@14323
   639
paulson@14323
   640
lemma complex_mod: "cmod (Abs_complex(x,y)) = sqrt(x ^ 2 + y ^ 2)"
paulson@14323
   641
apply (unfold cmod_def)
paulson@14323
   642
apply (simp (no_asm))
paulson@14323
   643
done
paulson@14323
   644
paulson@14323
   645
lemma complex_mod_zero: "cmod(0) = 0"
paulson@14323
   646
apply (unfold cmod_def)
paulson@14323
   647
apply (simp (no_asm))
paulson@14323
   648
done
paulson@14323
   649
declare complex_mod_zero [simp]
paulson@14323
   650
paulson@14348
   651
lemma complex_mod_one [simp]: "cmod(1) = 1"
paulson@14353
   652
by (simp add: cmod_def power2_eq_square)
paulson@14323
   653
paulson@14323
   654
lemma complex_mod_complex_of_real: "cmod(complex_of_real x) = abs x"
paulson@14353
   655
apply (simp add: complex_of_real_def power2_eq_square complex_mod)
paulson@14323
   656
done
paulson@14323
   657
declare complex_mod_complex_of_real [simp]
paulson@14323
   658
paulson@14348
   659
lemma complex_of_real_abs:
paulson@14348
   660
     "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
paulson@14348
   661
by (simp)
paulson@14348
   662
paulson@14323
   663
paulson@14323
   664
paulson@14323
   665
subsection{*Conjugation is an Automorphism*}
paulson@14323
   666
paulson@14323
   667
lemma complex_cnj: "cnj (Abs_complex(x,y)) = Abs_complex(x,-y)"
paulson@14323
   668
apply (unfold cnj_def)
paulson@14323
   669
apply (simp (no_asm))
paulson@14323
   670
done
paulson@14323
   671
paulson@14323
   672
lemma inj_cnj: "inj cnj"
paulson@14323
   673
apply (rule inj_onI)
paulson@14323
   674
apply (auto simp add: cnj_def Abs_complex_cancel_iff complex_Re_Im_cancel_iff)
paulson@14323
   675
done
paulson@14323
   676
paulson@14323
   677
lemma complex_cnj_cancel_iff: "(cnj x = cnj y) = (x = y)"
paulson@14334
   678
by (auto dest: inj_cnj [THEN injD])
paulson@14323
   679
declare complex_cnj_cancel_iff [simp]
paulson@14323
   680
paulson@14323
   681
lemma complex_cnj_cnj: "cnj (cnj z) = z"
paulson@14323
   682
apply (unfold cnj_def)
paulson@14323
   683
apply (simp (no_asm))
paulson@14323
   684
done
paulson@14323
   685
declare complex_cnj_cnj [simp]
paulson@14323
   686
paulson@14348
   687
lemma complex_cnj_complex_of_real:
paulson@14348
   688
 "cnj (complex_of_real x) = complex_of_real x"
paulson@14323
   689
apply (unfold complex_of_real_def)
paulson@14323
   690
apply (simp (no_asm) add: complex_cnj)
paulson@14323
   691
done
paulson@14323
   692
declare complex_cnj_complex_of_real [simp]
paulson@14323
   693
paulson@14323
   694
lemma complex_mod_cnj: "cmod (cnj z) = cmod z"
paulson@14334
   695
apply (rule_tac z = z in eq_Abs_complex)
paulson@14353
   696
apply (simp (no_asm_simp) add: complex_cnj complex_mod power2_eq_square)
paulson@14323
   697
done
paulson@14323
   698
declare complex_mod_cnj [simp]
paulson@14323
   699
paulson@14323
   700
lemma complex_cnj_minus: "cnj (-z) = - cnj z"
paulson@14323
   701
apply (unfold cnj_def)
paulson@14323
   702
apply (simp (no_asm) add: complex_minus complex_Re_minus complex_Im_minus)
paulson@14323
   703
done
paulson@14323
   704
paulson@14323
   705
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
paulson@14334
   706
apply (rule_tac z = z in eq_Abs_complex)
paulson@14353
   707
apply (simp (no_asm_simp) add: complex_cnj complex_inverse power2_eq_square)
paulson@14323
   708
done
paulson@14323
   709
paulson@14323
   710
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
paulson@14334
   711
apply (rule_tac z = w in eq_Abs_complex)
paulson@14334
   712
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   713
apply (simp (no_asm_simp) add: complex_cnj complex_add)
paulson@14323
   714
done
paulson@14323
   715
paulson@14323
   716
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
paulson@14323
   717
apply (unfold complex_diff_def)
paulson@14323
   718
apply (simp (no_asm) add: complex_cnj_add complex_cnj_minus)
paulson@14323
   719
done
paulson@14323
   720
paulson@14323
   721
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
paulson@14334
   722
apply (rule_tac z = w in eq_Abs_complex)
paulson@14334
   723
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   724
apply (simp (no_asm_simp) add: complex_cnj complex_mult)
paulson@14323
   725
done
paulson@14323
   726
paulson@14323
   727
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
paulson@14323
   728
apply (unfold complex_divide_def)
paulson@14323
   729
apply (simp (no_asm) add: complex_cnj_mult complex_cnj_inverse)
paulson@14323
   730
done
paulson@14323
   731
paulson@14323
   732
lemma complex_cnj_one: "cnj 1 = 1"
paulson@14323
   733
apply (unfold cnj_def complex_one_def)
paulson@14323
   734
apply (simp (no_asm))
paulson@14323
   735
done
paulson@14323
   736
declare complex_cnj_one [simp]
paulson@14323
   737
paulson@14323
   738
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
paulson@14334
   739
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   740
apply (simp (no_asm_simp) add: complex_add complex_cnj complex_of_real_def)
paulson@14323
   741
done
paulson@14323
   742
paulson@14323
   743
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
paulson@14334
   744
apply (rule_tac z = z in eq_Abs_complex)
paulson@14354
   745
apply (simp add: complex_add complex_cnj complex_of_real_def complex_diff_def 
paulson@14354
   746
                 complex_minus i_def complex_mult)
paulson@14323
   747
done
paulson@14323
   748
paulson@14354
   749
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
paulson@14334
   750
by (simp add: cnj_def complex_zero_def)
paulson@14323
   751
paulson@14323
   752
lemma complex_cnj_zero_iff: "(cnj z = 0) = (z = 0)"
paulson@14334
   753
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   754
apply (auto simp add: complex_zero_def complex_cnj)
paulson@14323
   755
done
paulson@14323
   756
declare complex_cnj_zero_iff [iff]
paulson@14323
   757
paulson@14323
   758
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
paulson@14334
   759
apply (rule_tac z = z in eq_Abs_complex)
paulson@14353
   760
apply (auto simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
paulson@14323
   761
done
paulson@14323
   762
paulson@14323
   763
paulson@14323
   764
subsection{*Algebra*}
paulson@14323
   765
paulson@14323
   766
lemma complex_add_left_cancel_zero: "(x + y = x) = (y = (0::complex))"
paulson@14323
   767
apply (unfold complex_zero_def)
paulson@14334
   768
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   769
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   770
apply (auto simp add: complex_add)
paulson@14323
   771
done
paulson@14323
   772
declare complex_add_left_cancel_zero [simp]
paulson@14323
   773
paulson@14323
   774
lemma complex_diff_mult_distrib:
paulson@14323
   775
      "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)"
paulson@14323
   776
apply (unfold complex_diff_def)
paulson@14323
   777
apply (simp (no_asm) add: complex_add_mult_distrib)
paulson@14323
   778
done
paulson@14323
   779
paulson@14323
   780
lemma complex_diff_mult_distrib2:
paulson@14323
   781
      "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)"
paulson@14323
   782
apply (unfold complex_diff_def)
paulson@14323
   783
apply (simp (no_asm) add: complex_add_mult_distrib2)
paulson@14323
   784
done
paulson@14323
   785
paulson@14323
   786
paulson@14323
   787
subsection{*Modulus*}
paulson@14323
   788
paulson@14323
   789
lemma complex_mod_eq_zero_cancel: "(cmod x = 0) = (x = 0)"
paulson@14334
   790
apply (rule_tac z = x in eq_Abs_complex)
paulson@14353
   791
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: complex_mod complex_zero_def power2_eq_square)
paulson@14323
   792
done
paulson@14323
   793
declare complex_mod_eq_zero_cancel [simp]
paulson@14323
   794
paulson@14323
   795
lemma complex_mod_complex_of_real_of_nat: "cmod (complex_of_real(real (n::nat))) = real n"
paulson@14323
   796
apply (simp (no_asm))
paulson@14323
   797
done
paulson@14323
   798
declare complex_mod_complex_of_real_of_nat [simp]
paulson@14323
   799
paulson@14323
   800
lemma complex_mod_minus: "cmod (-x) = cmod(x)"
paulson@14334
   801
apply (rule_tac z = x in eq_Abs_complex)
paulson@14353
   802
apply (simp (no_asm_simp) add: complex_mod complex_minus power2_eq_square)
paulson@14323
   803
done
paulson@14323
   804
declare complex_mod_minus [simp]
paulson@14323
   805
paulson@14323
   806
lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
paulson@14334
   807
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   808
apply (simp (no_asm_simp) add: complex_mod complex_cnj complex_mult);
paulson@14353
   809
apply (simp (no_asm) add: power2_eq_square real_abs_def)
paulson@14323
   810
done
paulson@14323
   811
paulson@14323
   812
lemma complex_mod_squared: "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2"
paulson@14334
   813
by (unfold cmod_def, auto)
paulson@14323
   814
paulson@14354
   815
lemma complex_mod_ge_zero: "0 \<le> cmod x"
paulson@14323
   816
apply (unfold cmod_def)
paulson@14323
   817
apply (auto intro: real_sqrt_ge_zero)
paulson@14323
   818
done
paulson@14323
   819
declare complex_mod_ge_zero [simp]
paulson@14323
   820
paulson@14323
   821
lemma abs_cmod_cancel: "abs(cmod x) = cmod x"
paulson@14334
   822
by (auto intro: abs_eqI1)
paulson@14323
   823
declare abs_cmod_cancel [simp]
paulson@14323
   824
paulson@14323
   825
lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)"
paulson@14334
   826
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   827
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   828
apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2 [symmetric] simp del: realpow_Suc)
paulson@14348
   829
apply (rule_tac n = 1 in power_inject_base)
paulson@14353
   830
apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc)
paulson@14353
   831
apply (auto simp add: real_diff_def power2_eq_square right_distrib left_distrib add_ac mult_ac)
paulson@14323
   832
done
paulson@14323
   833
paulson@14323
   834
lemma complex_mod_add_squared_eq: "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
paulson@14334
   835
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   836
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   837
apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
paulson@14353
   838
apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac)
paulson@14323
   839
done
paulson@14323
   840
paulson@14354
   841
lemma complex_Re_mult_cnj_le_cmod: "Re(x * cnj y) \<le> cmod(x * cnj y)"
paulson@14334
   842
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   843
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   844
apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
paulson@14323
   845
done
paulson@14323
   846
declare complex_Re_mult_cnj_le_cmod [simp]
paulson@14323
   847
paulson@14354
   848
lemma complex_Re_mult_cnj_le_cmod2: "Re(x * cnj y) \<le> cmod(x * y)"
paulson@14334
   849
apply (cut_tac x = x and y = y in complex_Re_mult_cnj_le_cmod)
paulson@14323
   850
apply (simp add: complex_mod_mult)
paulson@14323
   851
done
paulson@14323
   852
declare complex_Re_mult_cnj_le_cmod2 [simp]
paulson@14323
   853
paulson@14323
   854
lemma real_sum_squared_expand: "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y"
paulson@14353
   855
apply (simp (no_asm) add: left_distrib right_distrib power2_eq_square)
paulson@14323
   856
done
paulson@14323
   857
paulson@14354
   858
lemma complex_mod_triangle_squared: "cmod (x + y) ^ 2 \<le> (cmod(x) + cmod(y)) ^ 2"
paulson@14323
   859
apply (simp (no_asm) add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
paulson@14323
   860
done
paulson@14323
   861
declare complex_mod_triangle_squared [simp]
paulson@14323
   862
paulson@14354
   863
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
paulson@14323
   864
apply (rule order_trans [OF _ complex_mod_ge_zero]) 
paulson@14323
   865
apply (simp (no_asm))
paulson@14323
   866
done
paulson@14323
   867
declare complex_mod_minus_le_complex_mod [simp]
paulson@14323
   868
paulson@14354
   869
lemma complex_mod_triangle_ineq: "cmod (x + y) \<le> cmod(x) + cmod(y)"
paulson@14334
   870
apply (rule_tac n = 1 in realpow_increasing)
paulson@14323
   871
apply (auto intro:  order_trans [OF _ complex_mod_ge_zero]
paulson@14353
   872
            simp add: power2_eq_square [symmetric])
paulson@14323
   873
done
paulson@14323
   874
declare complex_mod_triangle_ineq [simp]
paulson@14323
   875
paulson@14354
   876
lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
paulson@14334
   877
apply (cut_tac x1 = b and y1 = a and c = "-cmod b" 
paulson@14334
   878
       in complex_mod_triangle_ineq [THEN add_right_mono])
paulson@14323
   879
apply (simp (no_asm))
paulson@14323
   880
done
paulson@14323
   881
declare complex_mod_triangle_ineq2 [simp]
paulson@14323
   882
paulson@14323
   883
lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)"
paulson@14334
   884
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   885
apply (rule_tac z = y in eq_Abs_complex)
paulson@14353
   886
apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac)
paulson@14323
   887
done
paulson@14323
   888
paulson@14323
   889
lemma complex_mod_add_less: "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
paulson@14334
   890
by (auto intro: order_le_less_trans complex_mod_triangle_ineq)
paulson@14323
   891
paulson@14323
   892
lemma complex_mod_mult_less: "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
paulson@14334
   893
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
paulson@14323
   894
paulson@14354
   895
lemma complex_mod_diff_ineq: "cmod(a) - cmod(b) \<le> cmod(a + b)"
paulson@14323
   896
apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
paulson@14323
   897
apply auto
paulson@14334
   898
apply (rule order_trans [of _ 0], rule order_less_imp_le)
paulson@14334
   899
apply (simp add: compare_rls, simp)  
paulson@14323
   900
apply (simp add: compare_rls)
paulson@14323
   901
apply (rule complex_mod_minus [THEN subst])
paulson@14323
   902
apply (rule order_trans)
paulson@14323
   903
apply (rule_tac [2] complex_mod_triangle_ineq)
paulson@14323
   904
apply (auto simp add: complex_add_ac)
paulson@14323
   905
done
paulson@14323
   906
declare complex_mod_diff_ineq [simp]
paulson@14323
   907
paulson@14354
   908
lemma complex_Re_le_cmod: "Re z \<le> cmod z"
paulson@14334
   909
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   910
apply (auto simp add: complex_mod simp del: realpow_Suc)
paulson@14323
   911
done
paulson@14323
   912
declare complex_Re_le_cmod [simp]
paulson@14323
   913
paulson@14354
   914
lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z"
paulson@14334
   915
apply (cut_tac x = z in complex_mod_ge_zero)
paulson@14334
   916
apply (drule order_le_imp_less_or_eq, auto)
paulson@14323
   917
done
paulson@14323
   918
paulson@14323
   919
paulson@14323
   920
subsection{*A Few More Theorems*}
paulson@14323
   921
paulson@14323
   922
lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
paulson@14323
   923
apply (case_tac "x=0", simp add: COMPLEX_INVERSE_ZERO)
paulson@14323
   924
apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1])
paulson@14323
   925
apply (auto simp add: complex_mod_mult [symmetric])
paulson@14323
   926
done
paulson@14323
   927
paulson@14323
   928
lemma complex_mod_divide:
paulson@14323
   929
      "cmod(x/y) = cmod(x)/(cmod y)"
paulson@14323
   930
apply (unfold complex_divide_def real_divide_def)
paulson@14323
   931
apply (auto simp add: complex_mod_mult complex_mod_inverse)
paulson@14323
   932
done
paulson@14323
   933
paulson@14323
   934
lemma complex_inverse_divide:
paulson@14323
   935
      "inverse(x/y) = y/(x::complex)"
paulson@14323
   936
apply (unfold complex_divide_def)
paulson@14348
   937
apply (auto simp add: inverse_mult_distrib complex_mult_commute)
paulson@14323
   938
done
paulson@14323
   939
declare complex_inverse_divide [simp]
paulson@14323
   940
paulson@14354
   941
paulson@14354
   942
subsection{*Exponentiation*}
paulson@14354
   943
paulson@14354
   944
primrec
paulson@14354
   945
     complexpow_0:   "z ^ 0       = 1"
paulson@14354
   946
     complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"
paulson@14354
   947
paulson@14354
   948
paulson@14354
   949
instance complex :: ringpower
paulson@14354
   950
proof
paulson@14354
   951
  fix z :: complex
paulson@14354
   952
  fix n :: nat
paulson@14354
   953
  show "z^0 = 1" by simp
paulson@14354
   954
  show "z^(Suc n) = z * (z^n)" by simp
paulson@14354
   955
qed
paulson@14323
   956
paulson@14323
   957
paulson@14354
   958
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
paulson@14323
   959
apply (induct_tac "n")
paulson@14354
   960
apply (auto simp add: complex_of_real_mult [symmetric])
paulson@14323
   961
done
paulson@14323
   962
paulson@14354
   963
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
paulson@14323
   964
apply (induct_tac "n")
paulson@14354
   965
apply (auto simp add: complex_cnj_mult)
paulson@14323
   966
done
paulson@14323
   967
paulson@14354
   968
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
paulson@14354
   969
apply (induct_tac "n")
paulson@14354
   970
apply (auto simp add: complex_mod_mult)
paulson@14354
   971
done
paulson@14354
   972
paulson@14354
   973
lemma complexpow_minus: "(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))"
paulson@14354
   974
by (induct_tac "n", auto)
paulson@14354
   975
paulson@14354
   976
lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)"
paulson@14354
   977
by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2)
paulson@14354
   978
paulson@14354
   979
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
paulson@14354
   980
by (unfold i_def complex_zero_def, auto)
paulson@14354
   981
paulson@14354
   982
paulson@14354
   983
subsection{*The Function @{term sgn}*}
paulson@14323
   984
paulson@14323
   985
lemma sgn_zero: "sgn 0 = 0"
paulson@14323
   986
apply (unfold sgn_def)
paulson@14323
   987
apply (simp (no_asm))
paulson@14323
   988
done
paulson@14323
   989
declare sgn_zero [simp]
paulson@14323
   990
paulson@14323
   991
lemma sgn_one: "sgn 1 = 1"
paulson@14323
   992
apply (unfold sgn_def)
paulson@14323
   993
apply (simp (no_asm))
paulson@14323
   994
done
paulson@14323
   995
declare sgn_one [simp]
paulson@14323
   996
paulson@14323
   997
lemma sgn_minus: "sgn (-z) = - sgn(z)"
paulson@14334
   998
by (unfold sgn_def, auto)
paulson@14323
   999
paulson@14323
  1000
lemma sgn_eq:
paulson@14323
  1001
    "sgn z = z / complex_of_real (cmod z)"
paulson@14323
  1002
apply (unfold sgn_def)
paulson@14323
  1003
apply (simp (no_asm))
paulson@14323
  1004
done
paulson@14323
  1005
paulson@14354
  1006
lemma complex_split: "\<exists>x y. z = complex_of_real(x) + ii * complex_of_real(y)"
paulson@14334
  1007
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1008
apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
paulson@14323
  1009
done
paulson@14323
  1010
paulson@14323
  1011
lemma Re_complex_i: "Re(complex_of_real(x) + ii * complex_of_real(y)) = x"
paulson@14334
  1012
by (auto simp add: complex_of_real_def i_def complex_mult complex_add)
paulson@14323
  1013
declare Re_complex_i [simp]
paulson@14323
  1014
paulson@14323
  1015
lemma Im_complex_i: "Im(complex_of_real(x) + ii * complex_of_real(y)) = y"
paulson@14334
  1016
by (auto simp add: complex_of_real_def i_def complex_mult complex_add)
paulson@14323
  1017
declare Im_complex_i [simp]
paulson@14323
  1018
paulson@14323
  1019
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
paulson@14323
  1020
apply (unfold i_def complex_of_real_def)
paulson@14323
  1021
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1022
done
paulson@14323
  1023
paulson@14323
  1024
lemma i_mult_eq2: "ii * ii = -(1::complex)"
paulson@14323
  1025
apply (unfold i_def complex_one_def)
paulson@14323
  1026
apply (simp (no_asm) add: complex_mult complex_minus)
paulson@14323
  1027
done
paulson@14323
  1028
declare i_mult_eq2 [simp]
paulson@14323
  1029
paulson@14323
  1030
lemma cmod_i: "cmod (complex_of_real(x) + ii * complex_of_real(y)) =
paulson@14323
  1031
      sqrt (x ^ 2 + y ^ 2)"
paulson@14323
  1032
apply (auto simp add: complex_mult complex_add i_def complex_of_real_def cmod_def)
paulson@14323
  1033
done
paulson@14323
  1034
paulson@14323
  1035
lemma complex_eq_Re_eq:
paulson@14323
  1036
     "complex_of_real xa + ii * complex_of_real ya =
paulson@14323
  1037
      complex_of_real xb + ii * complex_of_real yb
paulson@14323
  1038
       ==> xa = xb"
paulson@14323
  1039
apply (unfold complex_of_real_def i_def)
paulson@14323
  1040
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1041
done
paulson@14323
  1042
paulson@14323
  1043
lemma complex_eq_Im_eq:
paulson@14323
  1044
     "complex_of_real xa + ii * complex_of_real ya =
paulson@14323
  1045
      complex_of_real xb + ii * complex_of_real yb
paulson@14323
  1046
       ==> ya = yb"
paulson@14323
  1047
apply (unfold complex_of_real_def i_def)
paulson@14323
  1048
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1049
done
paulson@14323
  1050
paulson@14323
  1051
lemma complex_eq_cancel_iff: "(complex_of_real xa + ii * complex_of_real ya =
paulson@14323
  1052
       complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
paulson@14323
  1053
apply (auto intro: complex_eq_Im_eq complex_eq_Re_eq)
paulson@14323
  1054
done
paulson@14323
  1055
declare complex_eq_cancel_iff [iff]
paulson@14323
  1056
paulson@14323
  1057
lemma complex_eq_cancel_iffA: "(complex_of_real xa + complex_of_real ya * ii =
paulson@14323
  1058
       complex_of_real xb + complex_of_real yb * ii ) = ((xa = xb) & (ya = yb))"
paulson@14323
  1059
apply (auto simp add: complex_mult_commute)
paulson@14323
  1060
done
paulson@14323
  1061
declare complex_eq_cancel_iffA [iff]
paulson@14323
  1062
paulson@14323
  1063
lemma complex_eq_cancel_iffB: "(complex_of_real xa + complex_of_real ya * ii =
paulson@14323
  1064
       complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
paulson@14323
  1065
apply (auto simp add: complex_mult_commute)
paulson@14323
  1066
done
paulson@14323
  1067
declare complex_eq_cancel_iffB [iff]
paulson@14323
  1068
paulson@14323
  1069
lemma complex_eq_cancel_iffC: "(complex_of_real xa + ii * complex_of_real ya  =
paulson@14323
  1070
       complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))"
paulson@14323
  1071
apply (auto simp add: complex_mult_commute)
paulson@14323
  1072
done
paulson@14323
  1073
declare complex_eq_cancel_iffC [iff]
paulson@14323
  1074
paulson@14323
  1075
lemma complex_eq_cancel_iff2: "(complex_of_real x + ii * complex_of_real y =
paulson@14323
  1076
      complex_of_real xa) = (x = xa & y = 0)"
paulson@14334
  1077
apply (cut_tac xa = x and ya = y and xb = xa and yb = 0 in complex_eq_cancel_iff)
paulson@14323
  1078
apply (simp del: complex_eq_cancel_iff)
paulson@14323
  1079
done
paulson@14323
  1080
declare complex_eq_cancel_iff2 [simp]
paulson@14323
  1081
paulson@14323
  1082
lemma complex_eq_cancel_iff2a: "(complex_of_real x + complex_of_real y * ii =
paulson@14323
  1083
      complex_of_real xa) = (x = xa & y = 0)"
paulson@14323
  1084
apply (auto simp add: complex_mult_commute)
paulson@14323
  1085
done
paulson@14323
  1086
declare complex_eq_cancel_iff2a [simp]
paulson@14323
  1087
paulson@14323
  1088
lemma complex_eq_cancel_iff3: "(complex_of_real x + ii * complex_of_real y =
paulson@14323
  1089
      ii * complex_of_real ya) = (x = 0 & y = ya)"
paulson@14334
  1090
apply (cut_tac xa = x and ya = y and xb = 0 and yb = ya in complex_eq_cancel_iff)
paulson@14323
  1091
apply (simp del: complex_eq_cancel_iff)
paulson@14323
  1092
done
paulson@14323
  1093
declare complex_eq_cancel_iff3 [simp]
paulson@14323
  1094
paulson@14323
  1095
lemma complex_eq_cancel_iff3a: "(complex_of_real x + complex_of_real y * ii =
paulson@14323
  1096
      ii * complex_of_real ya) = (x = 0 & y = ya)"
paulson@14323
  1097
apply (auto simp add: complex_mult_commute)
paulson@14323
  1098
done
paulson@14323
  1099
declare complex_eq_cancel_iff3a [simp]
paulson@14323
  1100
paulson@14323
  1101
lemma complex_split_Re_zero:
paulson@14323
  1102
     "complex_of_real x + ii * complex_of_real y = 0
paulson@14323
  1103
      ==> x = 0"
paulson@14323
  1104
apply (unfold complex_of_real_def i_def complex_zero_def)
paulson@14323
  1105
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1106
done
paulson@14323
  1107
paulson@14323
  1108
lemma complex_split_Im_zero:
paulson@14323
  1109
     "complex_of_real x + ii * complex_of_real y = 0
paulson@14323
  1110
      ==> y = 0"
paulson@14323
  1111
apply (unfold complex_of_real_def i_def complex_zero_def)
paulson@14323
  1112
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1113
done
paulson@14323
  1114
paulson@14323
  1115
lemma Re_sgn:
paulson@14323
  1116
      "Re(sgn z) = Re(z)/cmod z"
paulson@14323
  1117
apply (unfold sgn_def complex_divide_def)
paulson@14334
  1118
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1119
apply (auto simp add: complex_of_real_inverse [symmetric])
paulson@14323
  1120
apply (auto simp add: complex_of_real_def complex_mult real_divide_def)
paulson@14323
  1121
done
paulson@14323
  1122
declare Re_sgn [simp]
paulson@14323
  1123
paulson@14323
  1124
lemma Im_sgn:
paulson@14323
  1125
      "Im(sgn z) = Im(z)/cmod z"
paulson@14323
  1126
apply (unfold sgn_def complex_divide_def)
paulson@14334
  1127
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1128
apply (auto simp add: complex_of_real_inverse [symmetric])
paulson@14323
  1129
apply (auto simp add: complex_of_real_def complex_mult real_divide_def)
paulson@14323
  1130
done
paulson@14323
  1131
declare Im_sgn [simp]
paulson@14323
  1132
paulson@14323
  1133
lemma complex_inverse_complex_split:
paulson@14323
  1134
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
  1135
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
  1136
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
paulson@14323
  1137
apply (unfold complex_of_real_def i_def)
paulson@14323
  1138
apply (auto simp add: complex_mult complex_add complex_diff_def complex_minus complex_inverse real_divide_def)
paulson@14323
  1139
done
paulson@14323
  1140
paulson@14323
  1141
(*----------------------------------------------------------------------------*)
paulson@14323
  1142
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
  1143
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
  1144
(*----------------------------------------------------------------------------*)
paulson@14323
  1145
paulson@14354
  1146
lemma complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)"
paulson@14354
  1147
by (auto simp add: complex_zero_def complex_of_real_def)
paulson@14354
  1148
paulson@14323
  1149
lemma Re_mult_i_eq:
paulson@14323
  1150
    "Re (ii * complex_of_real y) = 0"
paulson@14323
  1151
apply (unfold i_def complex_of_real_def)
paulson@14323
  1152
apply (auto simp add: complex_mult)
paulson@14323
  1153
done
paulson@14323
  1154
declare Re_mult_i_eq [simp]
paulson@14323
  1155
paulson@14323
  1156
lemma Im_mult_i_eq:
paulson@14323
  1157
    "Im (ii * complex_of_real y) = y"
paulson@14323
  1158
apply (unfold i_def complex_of_real_def)
paulson@14323
  1159
apply (auto simp add: complex_mult)
paulson@14323
  1160
done
paulson@14323
  1161
declare Im_mult_i_eq [simp]
paulson@14323
  1162
paulson@14323
  1163
lemma complex_mod_mult_i:
paulson@14323
  1164
    "cmod (ii * complex_of_real y) = abs y"
paulson@14323
  1165
apply (unfold i_def complex_of_real_def)
paulson@14353
  1166
apply (auto simp add: complex_mult complex_mod power2_eq_square)
paulson@14323
  1167
done
paulson@14323
  1168
declare complex_mod_mult_i [simp]
paulson@14323
  1169
paulson@14354
  1170
lemma cos_arg_i_mult_zero_pos:
paulson@14323
  1171
   "0 < y ==> cos (arg(ii * complex_of_real y)) = 0"
paulson@14323
  1172
apply (unfold arg_def)
paulson@14323
  1173
apply (auto simp add: abs_eqI2)
paulson@14334
  1174
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
  1175
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
  1176
done
paulson@14323
  1177
paulson@14354
  1178
lemma cos_arg_i_mult_zero_neg:
paulson@14323
  1179
   "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0"
paulson@14323
  1180
apply (unfold arg_def)
paulson@14323
  1181
apply (auto simp add: abs_minus_eqI2)
paulson@14334
  1182
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
  1183
apply (rule order_trans [of _ 0], auto)
paulson@14323
  1184
done
paulson@14323
  1185
paulson@14354
  1186
lemma cos_arg_i_mult_zero [simp]
paulson@14354
  1187
    : "y \<noteq> 0 ==> cos (arg(ii * complex_of_real y)) = 0"
paulson@14354
  1188
by (cut_tac x = y and y = 0 in linorder_less_linear, 
paulson@14354
  1189
    auto simp add: cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
paulson@14323
  1190
paulson@14323
  1191
paulson@14323
  1192
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
  1193
paulson@14354
  1194
lemma complex_split_polar: "\<exists>r a. z = complex_of_real r *
paulson@14323
  1195
      (complex_of_real(cos a) + ii * complex_of_real(sin a))"
paulson@14334
  1196
apply (cut_tac z = z in complex_split)
paulson@14354
  1197
apply (auto simp add: polar_Ex right_distrib complex_of_real_mult mult_ac)
paulson@14323
  1198
done
paulson@14323
  1199
paulson@14354
  1200
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
paulson@14323
  1201
apply (unfold rcis_def cis_def)
paulson@14323
  1202
apply (rule complex_split_polar)
paulson@14323
  1203
done
paulson@14323
  1204
paulson@14323
  1205
lemma Re_complex_polar: "Re(complex_of_real r *
paulson@14323
  1206
      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a"
paulson@14323
  1207
apply (auto simp add: complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
paulson@14323
  1208
done
paulson@14323
  1209
declare Re_complex_polar [simp]
paulson@14323
  1210
paulson@14323
  1211
lemma Re_rcis: "Re(rcis r a) = r * cos a"
paulson@14334
  1212
by (unfold rcis_def cis_def, auto)
paulson@14323
  1213
declare Re_rcis [simp]
paulson@14323
  1214
paulson@14348
  1215
lemma Im_complex_polar [simp]:
paulson@14348
  1216
     "Im(complex_of_real r * 
paulson@14348
  1217
         (complex_of_real(cos a) + ii * complex_of_real(sin a))) = 
paulson@14348
  1218
      r * sin a"
paulson@14348
  1219
by (auto simp add: complex_add_mult_distrib2 complex_of_real_mult mult_ac)
paulson@14323
  1220
paulson@14348
  1221
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
paulson@14334
  1222
by (unfold rcis_def cis_def, auto)
paulson@14323
  1223
paulson@14348
  1224
lemma complex_mod_complex_polar [simp]:
paulson@14348
  1225
     "cmod (complex_of_real r * 
paulson@14348
  1226
            (complex_of_real(cos a) + ii * complex_of_real(sin a))) = 
paulson@14348
  1227
      abs r"
paulson@14348
  1228
by (auto simp add: complex_add_mult_distrib2 cmod_i complex_of_real_mult
paulson@14348
  1229
                      right_distrib [symmetric] power_mult_distrib mult_ac 
paulson@14348
  1230
         simp del: realpow_Suc)
paulson@14323
  1231
paulson@14323
  1232
lemma complex_mod_rcis: "cmod(rcis r a) = abs r"
paulson@14334
  1233
by (unfold rcis_def cis_def, auto)
paulson@14323
  1234
declare complex_mod_rcis [simp]
paulson@14323
  1235
paulson@14323
  1236
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
paulson@14323
  1237
apply (unfold cmod_def)
paulson@14323
  1238
apply (rule real_sqrt_eq_iff [THEN iffD2])
paulson@14323
  1239
apply (auto simp add: complex_mult_cnj)
paulson@14323
  1240
done
paulson@14323
  1241
paulson@14323
  1242
lemma complex_Re_cnj: "Re(cnj z) = Re z"
paulson@14334
  1243
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1244
apply (auto simp add: complex_cnj)
paulson@14323
  1245
done
paulson@14323
  1246
declare complex_Re_cnj [simp]
paulson@14323
  1247
paulson@14323
  1248
lemma complex_Im_cnj: "Im(cnj z) = - Im z"
paulson@14334
  1249
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1250
apply (auto simp add: complex_cnj)
paulson@14323
  1251
done
paulson@14323
  1252
declare complex_Im_cnj [simp]
paulson@14323
  1253
paulson@14323
  1254
lemma complex_In_mult_cnj_zero: "Im (z * cnj z) = 0"
paulson@14334
  1255
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1256
apply (auto simp add: complex_cnj complex_mult)
paulson@14323
  1257
done
paulson@14323
  1258
declare complex_In_mult_cnj_zero [simp]
paulson@14323
  1259
paulson@14323
  1260
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
paulson@14334
  1261
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
  1262
apply (rule_tac z = w in eq_Abs_complex)
paulson@14323
  1263
apply (auto simp add: complex_mult)
paulson@14323
  1264
done
paulson@14323
  1265
paulson@14323
  1266
lemma complex_Re_mult_complex_of_real: "Re (z * complex_of_real c) = Re(z) * c"
paulson@14323
  1267
apply (unfold complex_of_real_def)
paulson@14334
  1268
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1269
apply (auto simp add: complex_mult)
paulson@14323
  1270
done
paulson@14323
  1271
declare complex_Re_mult_complex_of_real [simp]
paulson@14323
  1272
paulson@14323
  1273
lemma complex_Im_mult_complex_of_real: "Im (z * complex_of_real c) = Im(z) * c"
paulson@14323
  1274
apply (unfold complex_of_real_def)
paulson@14334
  1275
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1276
apply (auto simp add: complex_mult)
paulson@14323
  1277
done
paulson@14323
  1278
declare complex_Im_mult_complex_of_real [simp]
paulson@14323
  1279
paulson@14323
  1280
lemma complex_Re_mult_complex_of_real2: "Re (complex_of_real c * z) = c * Re(z)"
paulson@14323
  1281
apply (unfold complex_of_real_def)
paulson@14334
  1282
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1283
apply (auto simp add: complex_mult)
paulson@14323
  1284
done
paulson@14323
  1285
declare complex_Re_mult_complex_of_real2 [simp]
paulson@14323
  1286
paulson@14323
  1287
lemma complex_Im_mult_complex_of_real2: "Im (complex_of_real c * z) = c * Im(z)"
paulson@14323
  1288
apply (unfold complex_of_real_def)
paulson@14334
  1289
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1290
apply (auto simp add: complex_mult)
paulson@14323
  1291
done
paulson@14323
  1292
declare complex_Im_mult_complex_of_real2 [simp]
paulson@14323
  1293
paulson@14323
  1294
(*---------------------------------------------------------------------------*)
paulson@14323
  1295
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
paulson@14323
  1296
(*---------------------------------------------------------------------------*)
paulson@14323
  1297
paulson@14323
  1298
lemma cis_rcis_eq: "cis a = rcis 1 a"
paulson@14323
  1299
apply (unfold rcis_def)
paulson@14323
  1300
apply (simp (no_asm))
paulson@14323
  1301
done
paulson@14323
  1302
paulson@14323
  1303
lemma rcis_mult:
paulson@14323
  1304
  "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
paulson@14323
  1305
apply (unfold rcis_def cis_def)
paulson@14323
  1306
apply (auto simp add: cos_add sin_add complex_add_mult_distrib2 complex_add_mult_distrib complex_mult_ac complex_add_ac)
paulson@14323
  1307
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)
paulson@14323
  1308
apply (auto simp add: complex_add_ac)
paulson@14334
  1309
apply (auto simp add: complex_add_assoc [symmetric] complex_of_real_add right_distrib real_diff_def mult_ac add_ac)
paulson@14323
  1310
done
paulson@14323
  1311
paulson@14323
  1312
lemma cis_mult: "cis a * cis b = cis (a + b)"
paulson@14323
  1313
apply (simp (no_asm) add: cis_rcis_eq rcis_mult)
paulson@14323
  1314
done
paulson@14323
  1315
paulson@14323
  1316
lemma cis_zero: "cis 0 = 1"
paulson@14334
  1317
by (unfold cis_def, auto)
paulson@14323
  1318
declare cis_zero [simp]
paulson@14323
  1319
paulson@14323
  1320
lemma cis_zero2: "cis 0 = complex_of_real 1"
paulson@14334
  1321
by (unfold cis_def, auto)
paulson@14323
  1322
declare cis_zero2 [simp]
paulson@14323
  1323
paulson@14323
  1324
lemma rcis_zero_mod: "rcis 0 a = 0"
paulson@14323
  1325
apply (unfold rcis_def)
paulson@14323
  1326
apply (simp (no_asm))
paulson@14323
  1327
done
paulson@14323
  1328
declare rcis_zero_mod [simp]
paulson@14323
  1329
paulson@14323
  1330
lemma rcis_zero_arg: "rcis r 0 = complex_of_real r"
paulson@14323
  1331
apply (unfold rcis_def)
paulson@14323
  1332
apply (simp (no_asm))
paulson@14323
  1333
done
paulson@14323
  1334
declare rcis_zero_arg [simp]
paulson@14323
  1335
paulson@14323
  1336
lemma complex_of_real_minus_one:
paulson@14323
  1337
   "complex_of_real (-(1::real)) = -(1::complex)"
paulson@14323
  1338
apply (unfold complex_of_real_def complex_one_def)
paulson@14323
  1339
apply (simp (no_asm) add: complex_minus)
paulson@14323
  1340
done
paulson@14323
  1341
paulson@14323
  1342
lemma complex_i_mult_minus: "ii * (ii * x) = - x"
paulson@14323
  1343
apply (simp (no_asm) add: complex_mult_assoc [symmetric])
paulson@14323
  1344
done
paulson@14323
  1345
declare complex_i_mult_minus [simp]
paulson@14323
  1346
paulson@14323
  1347
lemma complex_i_mult_minus2: "ii * ii * x = - x"
paulson@14323
  1348
apply (simp (no_asm))
paulson@14323
  1349
done
paulson@14323
  1350
declare complex_i_mult_minus2 [simp]
paulson@14323
  1351
paulson@14323
  1352
lemma cis_real_of_nat_Suc_mult:
paulson@14323
  1353
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
paulson@14323
  1354
apply (unfold cis_def)
paulson@14334
  1355
apply (auto simp add: real_of_nat_Suc left_distrib cos_add sin_add complex_add_mult_distrib complex_add_mult_distrib2 complex_of_real_add complex_of_real_mult complex_mult_ac complex_add_ac)
paulson@14323
  1356
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)
paulson@14323
  1357
done
paulson@14323
  1358
paulson@14323
  1359
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
paulson@14323
  1360
apply (induct_tac "n")
paulson@14323
  1361
apply (auto simp add: cis_real_of_nat_Suc_mult)
paulson@14323
  1362
done
paulson@14323
  1363
paulson@14323
  1364
lemma DeMoivre2:
paulson@14323
  1365
   "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
paulson@14323
  1366
apply (unfold rcis_def)
paulson@14354
  1367
apply (auto simp add: power_mult_distrib DeMoivre complex_of_real_pow)
paulson@14323
  1368
done
paulson@14323
  1369
paulson@14323
  1370
lemma cis_inverse: "inverse(cis a) = cis (-a)"
paulson@14323
  1371
apply (unfold cis_def)
paulson@14323
  1372
apply (auto simp add: complex_inverse_complex_split complex_of_real_minus complex_diff_def)
paulson@14323
  1373
done
paulson@14323
  1374
declare cis_inverse [simp]
paulson@14323
  1375
paulson@14323
  1376
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
paulson@14354
  1377
apply (case_tac "r=0", simp)
paulson@14354
  1378
apply (auto simp add: complex_inverse_complex_split right_distrib 
paulson@14354
  1379
            complex_of_real_mult rcis_def cis_def power2_eq_square mult_ac)
paulson@14334
  1380
apply (auto simp add: right_distrib [symmetric] complex_of_real_minus complex_diff_def)
paulson@14323
  1381
done
paulson@14323
  1382
paulson@14323
  1383
lemma cis_divide: "cis a / cis b = cis (a - b)"
paulson@14323
  1384
apply (unfold complex_divide_def)
paulson@14323
  1385
apply (auto simp add: cis_mult real_diff_def)
paulson@14323
  1386
done
paulson@14323
  1387
paulson@14354
  1388
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
paulson@14323
  1389
apply (unfold complex_divide_def)
paulson@14323
  1390
apply (case_tac "r2=0")
paulson@14323
  1391
apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
paulson@14323
  1392
apply (auto simp add: rcis_inverse rcis_mult real_diff_def)
paulson@14323
  1393
done
paulson@14323
  1394
paulson@14323
  1395
lemma Re_cis: "Re(cis a) = cos a"
paulson@14334
  1396
by (unfold cis_def, auto)
paulson@14323
  1397
declare Re_cis [simp]
paulson@14323
  1398
paulson@14323
  1399
lemma Im_cis: "Im(cis a) = sin a"
paulson@14334
  1400
by (unfold cis_def, auto)
paulson@14323
  1401
declare Im_cis [simp]
paulson@14323
  1402
paulson@14323
  1403
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
paulson@14334
  1404
by (auto simp add: DeMoivre)
paulson@14323
  1405
paulson@14323
  1406
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
paulson@14334
  1407
by (auto simp add: DeMoivre)
paulson@14323
  1408
paulson@14323
  1409
lemma expi_Im_split:
paulson@14323
  1410
    "expi (ii * complex_of_real y) =
paulson@14323
  1411
     complex_of_real (cos y) + ii * complex_of_real (sin y)"
paulson@14354
  1412
by (unfold expi_def cis_def, auto)
paulson@14323
  1413
paulson@14323
  1414
lemma expi_Im_cis:
paulson@14323
  1415
    "expi (ii * complex_of_real y) = cis y"
paulson@14354
  1416
by (unfold expi_def, auto)
paulson@14323
  1417
paulson@14323
  1418
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
paulson@14323
  1419
apply (unfold expi_def)
paulson@14323
  1420
apply (auto simp add: complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult complex_mult_ac)
paulson@14323
  1421
done
paulson@14323
  1422
paulson@14323
  1423
lemma expi_complex_split:
paulson@14323
  1424
     "expi(complex_of_real x + ii * complex_of_real y) =
paulson@14323
  1425
      complex_of_real (exp(x)) * cis y"
paulson@14354
  1426
by (unfold expi_def, auto)
paulson@14323
  1427
paulson@14323
  1428
lemma expi_zero: "expi (0::complex) = 1"
paulson@14334
  1429
by (unfold expi_def, auto)
paulson@14323
  1430
declare expi_zero [simp]
paulson@14323
  1431
paulson@14323
  1432
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
paulson@14334
  1433
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
  1434
apply (rule_tac z = w in eq_Abs_complex)
paulson@14323
  1435
apply (auto simp add: complex_mult)
paulson@14323
  1436
done
paulson@14323
  1437
paulson@14323
  1438
lemma complex_Im_mult_eq:
paulson@14323
  1439
     "Im (w * z) = Re w * Im z + Im w * Re z"
paulson@14334
  1440
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
  1441
apply (rule_tac z = w in eq_Abs_complex)
paulson@14323
  1442
apply (auto simp add: complex_mult)
paulson@14323
  1443
done
paulson@14323
  1444
paulson@14323
  1445
lemma complex_expi_Ex: 
paulson@14354
  1446
   "\<exists>a r. z = complex_of_real r * expi a"
paulson@14334
  1447
apply (cut_tac z = z in rcis_Ex)
paulson@14323
  1448
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult)
paulson@14334
  1449
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
  1450
done
paulson@14323
  1451
paulson@14323
  1452
paulson@14323
  1453
(****
paulson@14354
  1454
Goal "[| - pi < a; a \<le> pi |] ==> (-pi < a & a \<le> 0) | (0 \<le> a & a \<le> pi)"
paulson@14334
  1455
by Auto_tac
paulson@14323
  1456
qed "lemma_split_interval";
paulson@14323
  1457
paulson@14323
  1458
Goalw [arg_def]
paulson@14354
  1459
  "[| r \<noteq> 0; - pi < a; a \<le> pi |] \
paulson@14323
  1460
\  ==> arg(complex_of_real r * \
paulson@14323
  1461
\      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = a";
paulson@14334
  1462
by Auto_tac
paulson@14323
  1463
by (cut_inst_tac [("x","0"),("y","r")] linorder_less_linear 1);
paulson@14323
  1464
by (auto_tac (claset(),simpset() addsimps (map (full_rename_numerals thy)
paulson@14323
  1465
    [rabs_eqI2,rabs_minus_eqI2,real_minus_rinv]) [real_divide_def,
paulson@14334
  1466
    minus_mult_right RS sym] mult_ac));
paulson@14323
  1467
by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
paulson@14334
  1468
by (dtac lemma_split_interval 1 THEN safe)
paulson@14323
  1469
****)
paulson@14323
  1470
paulson@14323
  1471
paulson@14323
  1472
ML
paulson@14323
  1473
{*
paulson@14323
  1474
val complex_zero_def = thm"complex_zero_def";
paulson@14323
  1475
val complex_one_def = thm"complex_one_def";
paulson@14323
  1476
val complex_minus_def = thm"complex_minus_def";
paulson@14323
  1477
val complex_diff_def = thm"complex_diff_def";
paulson@14323
  1478
val complex_divide_def = thm"complex_divide_def";
paulson@14323
  1479
val complex_mult_def = thm"complex_mult_def";
paulson@14323
  1480
val complex_add_def = thm"complex_add_def";
paulson@14323
  1481
val complex_of_real_def = thm"complex_of_real_def";
paulson@14323
  1482
val i_def = thm"i_def";
paulson@14323
  1483
val expi_def = thm"expi_def";
paulson@14323
  1484
val cis_def = thm"cis_def";
paulson@14323
  1485
val rcis_def = thm"rcis_def";
paulson@14323
  1486
val cmod_def = thm"cmod_def";
paulson@14323
  1487
val cnj_def = thm"cnj_def";
paulson@14323
  1488
val sgn_def = thm"sgn_def";
paulson@14323
  1489
val arg_def = thm"arg_def";
paulson@14323
  1490
val complexpow_0 = thm"complexpow_0";
paulson@14323
  1491
val complexpow_Suc = thm"complexpow_Suc";
paulson@14323
  1492
paulson@14323
  1493
val inj_Rep_complex = thm"inj_Rep_complex";
paulson@14323
  1494
val inj_Abs_complex = thm"inj_Abs_complex";
paulson@14323
  1495
val Abs_complex_cancel_iff = thm"Abs_complex_cancel_iff";
paulson@14323
  1496
val pair_mem_complex = thm"pair_mem_complex";
paulson@14323
  1497
val Abs_complex_inverse2 = thm"Abs_complex_inverse2";
paulson@14323
  1498
val eq_Abs_complex = thm"eq_Abs_complex";
paulson@14323
  1499
val Re = thm"Re";
paulson@14323
  1500
val Im = thm"Im";
paulson@14323
  1501
val Abs_complex_cancel = thm"Abs_complex_cancel";
paulson@14323
  1502
val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff";
paulson@14323
  1503
val complex_Re_zero = thm"complex_Re_zero";
paulson@14323
  1504
val complex_Im_zero = thm"complex_Im_zero";
paulson@14323
  1505
val complex_Re_one = thm"complex_Re_one";
paulson@14323
  1506
val complex_Im_one = thm"complex_Im_one";
paulson@14323
  1507
val complex_Re_i = thm"complex_Re_i";
paulson@14323
  1508
val complex_Im_i = thm"complex_Im_i";
paulson@14323
  1509
val Re_complex_of_real_zero = thm"Re_complex_of_real_zero";
paulson@14323
  1510
val Im_complex_of_real_zero = thm"Im_complex_of_real_zero";
paulson@14323
  1511
val Re_complex_of_real_one = thm"Re_complex_of_real_one";
paulson@14323
  1512
val Im_complex_of_real_one = thm"Im_complex_of_real_one";
paulson@14323
  1513
val Re_complex_of_real = thm"Re_complex_of_real";
paulson@14323
  1514
val Im_complex_of_real = thm"Im_complex_of_real";
paulson@14323
  1515
val complex_minus = thm"complex_minus";
paulson@14323
  1516
val complex_Re_minus = thm"complex_Re_minus";
paulson@14323
  1517
val complex_Im_minus = thm"complex_Im_minus";
paulson@14323
  1518
val complex_minus_minus = thm"complex_minus_minus";
paulson@14323
  1519
val inj_complex_minus = thm"inj_complex_minus";
paulson@14323
  1520
val complex_minus_zero = thm"complex_minus_zero";
paulson@14323
  1521
val complex_minus_zero_iff = thm"complex_minus_zero_iff";
paulson@14323
  1522
val complex_minus_zero_iff2 = thm"complex_minus_zero_iff2";
paulson@14323
  1523
val complex_minus_not_zero_iff = thm"complex_minus_not_zero_iff";
paulson@14323
  1524
val complex_add = thm"complex_add";
paulson@14323
  1525
val complex_Re_add = thm"complex_Re_add";
paulson@14323
  1526
val complex_Im_add = thm"complex_Im_add";
paulson@14323
  1527
val complex_add_commute = thm"complex_add_commute";
paulson@14323
  1528
val complex_add_assoc = thm"complex_add_assoc";
paulson@14323
  1529
val complex_add_left_commute = thm"complex_add_left_commute";
paulson@14323
  1530
val complex_add_zero_left = thm"complex_add_zero_left";
paulson@14323
  1531
val complex_add_zero_right = thm"complex_add_zero_right";
paulson@14323
  1532
val complex_add_minus_right_zero = thm"complex_add_minus_right_zero";
paulson@14323
  1533
val complex_add_minus_cancel = thm"complex_add_minus_cancel";
paulson@14323
  1534
val complex_minus_add_cancel = thm"complex_minus_add_cancel";
paulson@14323
  1535
val complex_add_minus_eq_minus = thm"complex_add_minus_eq_minus";
paulson@14323
  1536
val complex_minus_add_distrib = thm"complex_minus_add_distrib";
paulson@14323
  1537
val complex_add_left_cancel = thm"complex_add_left_cancel";
paulson@14323
  1538
val complex_add_right_cancel = thm"complex_add_right_cancel";
paulson@14323
  1539
val complex_eq_minus_iff = thm"complex_eq_minus_iff";
paulson@14323
  1540
val complex_eq_minus_iff2 = thm"complex_eq_minus_iff2";
paulson@14323
  1541
val complex_diff_0 = thm"complex_diff_0";
paulson@14323
  1542
val complex_diff_0_right = thm"complex_diff_0_right";
paulson@14323
  1543
val complex_diff_self = thm"complex_diff_self";
paulson@14323
  1544
val complex_diff = thm"complex_diff";
paulson@14323
  1545
val complex_diff_eq_eq = thm"complex_diff_eq_eq";
paulson@14323
  1546
val complex_mult = thm"complex_mult";
paulson@14323
  1547
val complex_mult_commute = thm"complex_mult_commute";
paulson@14323
  1548
val complex_mult_assoc = thm"complex_mult_assoc";
paulson@14323
  1549
val complex_mult_left_commute = thm"complex_mult_left_commute";
paulson@14323
  1550
val complex_mult_one_left = thm"complex_mult_one_left";
paulson@14323
  1551
val complex_mult_one_right = thm"complex_mult_one_right";
paulson@14323
  1552
val complex_mult_zero_left = thm"complex_mult_zero_left";
paulson@14323
  1553
val complex_mult_zero_right = thm"complex_mult_zero_right";
paulson@14323
  1554
val complex_divide_zero = thm"complex_divide_zero";
paulson@14323
  1555
val complex_minus_mult_eq1 = thm"complex_minus_mult_eq1";
paulson@14323
  1556
val complex_minus_mult_eq2 = thm"complex_minus_mult_eq2";
paulson@14323
  1557
val complex_minus_mult_commute = thm"complex_minus_mult_commute";
paulson@14323
  1558
val complex_add_mult_distrib = thm"complex_add_mult_distrib";
paulson@14323
  1559
val complex_add_mult_distrib2 = thm"complex_add_mult_distrib2";
paulson@14323
  1560
val complex_zero_not_eq_one = thm"complex_zero_not_eq_one";
paulson@14323
  1561
val complex_inverse = thm"complex_inverse";
paulson@14323
  1562
val COMPLEX_INVERSE_ZERO = thm"COMPLEX_INVERSE_ZERO";
paulson@14323
  1563
val COMPLEX_DIVISION_BY_ZERO = thm"COMPLEX_DIVISION_BY_ZERO";
paulson@14323
  1564
val complex_mult_inv_left = thm"complex_mult_inv_left";
paulson@14323
  1565
val complex_mult_inv_right = thm"complex_mult_inv_right";
paulson@14323
  1566
val inj_complex_of_real = thm"inj_complex_of_real";
paulson@14323
  1567
val complex_of_real_one = thm"complex_of_real_one";
paulson@14323
  1568
val complex_of_real_zero = thm"complex_of_real_zero";
paulson@14323
  1569
val complex_of_real_eq_iff = thm"complex_of_real_eq_iff";
paulson@14323
  1570
val complex_of_real_minus = thm"complex_of_real_minus";
paulson@14323
  1571
val complex_of_real_inverse = thm"complex_of_real_inverse";
paulson@14323
  1572
val complex_of_real_add = thm"complex_of_real_add";
paulson@14323
  1573
val complex_of_real_diff = thm"complex_of_real_diff";
paulson@14323
  1574
val complex_of_real_mult = thm"complex_of_real_mult";
paulson@14323
  1575
val complex_of_real_divide = thm"complex_of_real_divide";
paulson@14323
  1576
val complex_of_real_pow = thm"complex_of_real_pow";
paulson@14323
  1577
val complex_mod = thm"complex_mod";
paulson@14323
  1578
val complex_mod_zero = thm"complex_mod_zero";
paulson@14323
  1579
val complex_mod_one = thm"complex_mod_one";
paulson@14323
  1580
val complex_mod_complex_of_real = thm"complex_mod_complex_of_real";
paulson@14323
  1581
val complex_of_real_abs = thm"complex_of_real_abs";
paulson@14323
  1582
val complex_cnj = thm"complex_cnj";
paulson@14323
  1583
val inj_cnj = thm"inj_cnj";
paulson@14323
  1584
val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff";
paulson@14323
  1585
val complex_cnj_cnj = thm"complex_cnj_cnj";
paulson@14323
  1586
val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real";
paulson@14323
  1587
val complex_mod_cnj = thm"complex_mod_cnj";
paulson@14323
  1588
val complex_cnj_minus = thm"complex_cnj_minus";
paulson@14323
  1589
val complex_cnj_inverse = thm"complex_cnj_inverse";
paulson@14323
  1590
val complex_cnj_add = thm"complex_cnj_add";
paulson@14323
  1591
val complex_cnj_diff = thm"complex_cnj_diff";
paulson@14323
  1592
val complex_cnj_mult = thm"complex_cnj_mult";
paulson@14323
  1593
val complex_cnj_divide = thm"complex_cnj_divide";
paulson@14323
  1594
val complex_cnj_one = thm"complex_cnj_one";
paulson@14323
  1595
val complex_cnj_pow = thm"complex_cnj_pow";
paulson@14323
  1596
val complex_add_cnj = thm"complex_add_cnj";
paulson@14323
  1597
val complex_diff_cnj = thm"complex_diff_cnj";
paulson@14323
  1598
val complex_cnj_zero = thm"complex_cnj_zero";
paulson@14323
  1599
val complex_cnj_zero_iff = thm"complex_cnj_zero_iff";
paulson@14323
  1600
val complex_mult_cnj = thm"complex_mult_cnj";
paulson@14323
  1601
val complex_add_left_cancel_zero = thm"complex_add_left_cancel_zero";
paulson@14323
  1602
val complex_diff_mult_distrib = thm"complex_diff_mult_distrib";
paulson@14323
  1603
val complex_diff_mult_distrib2 = thm"complex_diff_mult_distrib2";
paulson@14323
  1604
val complex_mod_eq_zero_cancel = thm"complex_mod_eq_zero_cancel";
paulson@14323
  1605
val complex_mod_complex_of_real_of_nat = thm"complex_mod_complex_of_real_of_nat";
paulson@14323
  1606
val complex_mod_minus = thm"complex_mod_minus";
paulson@14323
  1607
val complex_mod_mult_cnj = thm"complex_mod_mult_cnj";
paulson@14323
  1608
val complex_mod_squared = thm"complex_mod_squared";
paulson@14323
  1609
val complex_mod_ge_zero = thm"complex_mod_ge_zero";
paulson@14323
  1610
val abs_cmod_cancel = thm"abs_cmod_cancel";
paulson@14323
  1611
val complex_mod_mult = thm"complex_mod_mult";
paulson@14323
  1612
val complex_mod_add_squared_eq = thm"complex_mod_add_squared_eq";
paulson@14323
  1613
val complex_Re_mult_cnj_le_cmod = thm"complex_Re_mult_cnj_le_cmod";
paulson@14323
  1614
val complex_Re_mult_cnj_le_cmod2 = thm"complex_Re_mult_cnj_le_cmod2";
paulson@14323
  1615
val real_sum_squared_expand = thm"real_sum_squared_expand";
paulson@14323
  1616
val complex_mod_triangle_squared = thm"complex_mod_triangle_squared";
paulson@14323
  1617
val complex_mod_minus_le_complex_mod = thm"complex_mod_minus_le_complex_mod";
paulson@14323
  1618
val complex_mod_triangle_ineq = thm"complex_mod_triangle_ineq";
paulson@14323
  1619
val complex_mod_triangle_ineq2 = thm"complex_mod_triangle_ineq2";
paulson@14323
  1620
val complex_mod_diff_commute = thm"complex_mod_diff_commute";
paulson@14323
  1621
val complex_mod_add_less = thm"complex_mod_add_less";
paulson@14323
  1622
val complex_mod_mult_less = thm"complex_mod_mult_less";
paulson@14323
  1623
val complex_mod_diff_ineq = thm"complex_mod_diff_ineq";
paulson@14323
  1624
val complex_Re_le_cmod = thm"complex_Re_le_cmod";
paulson@14323
  1625
val complex_mod_gt_zero = thm"complex_mod_gt_zero";
paulson@14323
  1626
val complex_mod_complexpow = thm"complex_mod_complexpow";
paulson@14323
  1627
val complexpow_minus = thm"complexpow_minus";
paulson@14323
  1628
val complex_mod_inverse = thm"complex_mod_inverse";
paulson@14323
  1629
val complex_mod_divide = thm"complex_mod_divide";
paulson@14323
  1630
val complex_inverse_divide = thm"complex_inverse_divide";
paulson@14323
  1631
val complexpow_i_squared = thm"complexpow_i_squared";
paulson@14323
  1632
val complex_i_not_zero = thm"complex_i_not_zero";
paulson@14323
  1633
val sgn_zero = thm"sgn_zero";
paulson@14323
  1634
val sgn_one = thm"sgn_one";
paulson@14323
  1635
val sgn_minus = thm"sgn_minus";
paulson@14323
  1636
val sgn_eq = thm"sgn_eq";
paulson@14323
  1637
val complex_split = thm"complex_split";
paulson@14323
  1638
val Re_complex_i = thm"Re_complex_i";
paulson@14323
  1639
val Im_complex_i = thm"Im_complex_i";
paulson@14323
  1640
val i_mult_eq = thm"i_mult_eq";
paulson@14323
  1641
val i_mult_eq2 = thm"i_mult_eq2";
paulson@14323
  1642
val cmod_i = thm"cmod_i";
paulson@14323
  1643
val complex_eq_Re_eq = thm"complex_eq_Re_eq";
paulson@14323
  1644
val complex_eq_Im_eq = thm"complex_eq_Im_eq";
paulson@14323
  1645
val complex_eq_cancel_iff = thm"complex_eq_cancel_iff";
paulson@14323
  1646
val complex_eq_cancel_iffA = thm"complex_eq_cancel_iffA";
paulson@14323
  1647
val complex_eq_cancel_iffB = thm"complex_eq_cancel_iffB";
paulson@14323
  1648
val complex_eq_cancel_iffC = thm"complex_eq_cancel_iffC";
paulson@14323
  1649
val complex_eq_cancel_iff2 = thm"complex_eq_cancel_iff2";
paulson@14323
  1650
val complex_eq_cancel_iff2a = thm"complex_eq_cancel_iff2a";
paulson@14323
  1651
val complex_eq_cancel_iff3 = thm"complex_eq_cancel_iff3";
paulson@14323
  1652
val complex_eq_cancel_iff3a = thm"complex_eq_cancel_iff3a";
paulson@14323
  1653
val complex_split_Re_zero = thm"complex_split_Re_zero";
paulson@14323
  1654
val complex_split_Im_zero = thm"complex_split_Im_zero";
paulson@14323
  1655
val Re_sgn = thm"Re_sgn";
paulson@14323
  1656
val Im_sgn = thm"Im_sgn";
paulson@14323
  1657
val complex_inverse_complex_split = thm"complex_inverse_complex_split";
paulson@14323
  1658
val Re_mult_i_eq = thm"Re_mult_i_eq";
paulson@14323
  1659
val Im_mult_i_eq = thm"Im_mult_i_eq";
paulson@14323
  1660
val complex_mod_mult_i = thm"complex_mod_mult_i";
paulson@14323
  1661
val cos_arg_i_mult_zero = thm"cos_arg_i_mult_zero";
paulson@14323
  1662
val complex_of_real_zero_iff = thm"complex_of_real_zero_iff";
paulson@14323
  1663
val complex_split_polar = thm"complex_split_polar";
paulson@14323
  1664
val rcis_Ex = thm"rcis_Ex";
paulson@14323
  1665
val Re_complex_polar = thm"Re_complex_polar";
paulson@14323
  1666
val Re_rcis = thm"Re_rcis";
paulson@14323
  1667
val Im_complex_polar = thm"Im_complex_polar";
paulson@14323
  1668
val Im_rcis = thm"Im_rcis";
paulson@14323
  1669
val complex_mod_complex_polar = thm"complex_mod_complex_polar";
paulson@14323
  1670
val complex_mod_rcis = thm"complex_mod_rcis";
paulson@14323
  1671
val complex_mod_sqrt_Re_mult_cnj = thm"complex_mod_sqrt_Re_mult_cnj";
paulson@14323
  1672
val complex_Re_cnj = thm"complex_Re_cnj";
paulson@14323
  1673
val complex_Im_cnj = thm"complex_Im_cnj";
paulson@14323
  1674
val complex_In_mult_cnj_zero = thm"complex_In_mult_cnj_zero";
paulson@14323
  1675
val complex_Re_mult = thm"complex_Re_mult";
paulson@14323
  1676
val complex_Re_mult_complex_of_real = thm"complex_Re_mult_complex_of_real";
paulson@14323
  1677
val complex_Im_mult_complex_of_real = thm"complex_Im_mult_complex_of_real";
paulson@14323
  1678
val complex_Re_mult_complex_of_real2 = thm"complex_Re_mult_complex_of_real2";
paulson@14323
  1679
val complex_Im_mult_complex_of_real2 = thm"complex_Im_mult_complex_of_real2";
paulson@14323
  1680
val cis_rcis_eq = thm"cis_rcis_eq";
paulson@14323
  1681
val rcis_mult = thm"rcis_mult";
paulson@14323
  1682
val cis_mult = thm"cis_mult";
paulson@14323
  1683
val cis_zero = thm"cis_zero";
paulson@14323
  1684
val cis_zero2 = thm"cis_zero2";
paulson@14323
  1685
val rcis_zero_mod = thm"rcis_zero_mod";
paulson@14323
  1686
val rcis_zero_arg = thm"rcis_zero_arg";
paulson@14323
  1687
val complex_of_real_minus_one = thm"complex_of_real_minus_one";
paulson@14323
  1688
val complex_i_mult_minus = thm"complex_i_mult_minus";
paulson@14323
  1689
val complex_i_mult_minus2 = thm"complex_i_mult_minus2";
paulson@14323
  1690
val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult";
paulson@14323
  1691
val DeMoivre = thm"DeMoivre";
paulson@14323
  1692
val DeMoivre2 = thm"DeMoivre2";
paulson@14323
  1693
val cis_inverse = thm"cis_inverse";
paulson@14323
  1694
val rcis_inverse = thm"rcis_inverse";
paulson@14323
  1695
val cis_divide = thm"cis_divide";
paulson@14323
  1696
val rcis_divide = thm"rcis_divide";
paulson@14323
  1697
val Re_cis = thm"Re_cis";
paulson@14323
  1698
val Im_cis = thm"Im_cis";
paulson@14323
  1699
val cos_n_Re_cis_pow_n = thm"cos_n_Re_cis_pow_n";
paulson@14323
  1700
val sin_n_Im_cis_pow_n = thm"sin_n_Im_cis_pow_n";
paulson@14323
  1701
val expi_Im_split = thm"expi_Im_split";
paulson@14323
  1702
val expi_Im_cis = thm"expi_Im_cis";
paulson@14323
  1703
val expi_add = thm"expi_add";
paulson@14323
  1704
val expi_complex_split = thm"expi_complex_split";
paulson@14323
  1705
val expi_zero = thm"expi_zero";
paulson@14323
  1706
val complex_Re_mult_eq = thm"complex_Re_mult_eq";
paulson@14323
  1707
val complex_Im_mult_eq = thm"complex_Im_mult_eq";
paulson@14323
  1708
val complex_expi_Ex = thm"complex_expi_Ex";
paulson@14323
  1709
paulson@14323
  1710
val complex_add_ac = thms"complex_add_ac";
paulson@14323
  1711
val complex_mult_ac = thms"complex_mult_ac";
paulson@14323
  1712
*}
paulson@14323
  1713
paulson@13957
  1714
end
paulson@13957
  1715
paulson@13957
  1716