src/HOL/Complex/Complex.thy
author paulson
Tue Jan 06 10:40:15 2004 +0100 (2004-01-06)
changeset 14341 a09441bd4f1e
parent 14335 9c0b5e081037
child 14348 744c868ee0b7
permissions -rw-r--r--
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
This allows more theorems to be proved for semirings, but
requires a redundant axiom to be proved for rings, etc.
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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 <= 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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primrec
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     complexpow_0:   "z ^ 0       = complex_of_real 1"
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     complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"
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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 ~= 0) = (x ~= (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]
paulson@14323
   360
paulson@14323
   361
lemma complex_add_minus_eq_minus: "x + y = (0::complex) ==> x = -y"
paulson@14334
   362
by (auto simp add: complex_Re_Im_cancel_iff complex_Re_add complex_Im_add complex_Re_minus complex_Im_minus)
paulson@14323
   363
paulson@14323
   364
lemma complex_minus_add_distrib: "-(x + y) = -x + -(y::complex)"
paulson@14334
   365
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   366
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   367
apply (auto simp add: complex_minus complex_add)
paulson@14323
   368
done
paulson@14323
   369
declare complex_minus_add_distrib [simp]
paulson@14323
   370
paulson@14323
   371
lemma complex_add_left_cancel: "((x::complex) + y = x + z) = (y = z)"
paulson@14334
   372
apply safe
paulson@14323
   373
apply (drule_tac f = "%t.-x + t" in arg_cong)
paulson@14341
   374
apply (simp add: complex_add_minus_left complex_add_assoc [symmetric])
paulson@14323
   375
done
paulson@14323
   376
declare complex_add_left_cancel [iff]
paulson@14323
   377
paulson@14323
   378
lemma complex_add_right_cancel: "(y + (x::complex)= z + x) = (y = z)"
paulson@14323
   379
apply (simp (no_asm) add: complex_add_commute)
paulson@14323
   380
done
paulson@14323
   381
declare complex_add_right_cancel [simp]
paulson@14323
   382
paulson@14323
   383
lemma complex_eq_minus_iff: "((x::complex) = y) = (0 = x + - y)"
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_eq_minus_iff2: "((x::complex) = y) = (x + - y = 0)"
paulson@14334
   389
apply safe
paulson@14334
   390
apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto)
paulson@14323
   391
done
paulson@14323
   392
paulson@14323
   393
lemma complex_diff_0: "(0::complex) - x = -x"
paulson@14323
   394
apply (simp (no_asm) add: complex_diff_def)
paulson@14323
   395
done
paulson@14323
   396
paulson@14323
   397
lemma complex_diff_0_right: "x - (0::complex) = x"
paulson@14323
   398
apply (simp (no_asm) add: complex_diff_def)
paulson@14323
   399
done
paulson@14323
   400
paulson@14323
   401
lemma complex_diff_self: "x - x = (0::complex)"
paulson@14323
   402
apply (simp (no_asm) add: complex_diff_def)
paulson@14323
   403
done
paulson@14323
   404
paulson@14323
   405
declare complex_diff_0 [simp] complex_diff_0_right [simp] complex_diff_self [simp]
paulson@14323
   406
paulson@14323
   407
lemma complex_diff:
paulson@14323
   408
      "Abs_complex(x1,y1) - Abs_complex(x2,y2) = Abs_complex(x1-x2,y1-y2)"
paulson@14323
   409
apply (unfold complex_diff_def)
paulson@14323
   410
apply (simp (no_asm) add: complex_add complex_minus)
paulson@14323
   411
done
paulson@14323
   412
paulson@14323
   413
lemma complex_diff_eq_eq: "((x::complex) - y = z) = (x = z + y)"
paulson@14341
   414
by (auto simp add: complex_add_minus_left complex_diff_def complex_add_assoc)
paulson@14323
   415
paulson@14323
   416
paulson@14323
   417
subsection{*Multiplication*}
paulson@14323
   418
paulson@14323
   419
lemma complex_mult:
paulson@14323
   420
      "Abs_complex(x1,y1) * Abs_complex(x2,y2) =
paulson@14323
   421
       Abs_complex(x1*x2 - y1*y2,x1*y2 + y1*x2)"
paulson@14323
   422
apply (unfold complex_mult_def)
paulson@14323
   423
apply (simp (no_asm))
paulson@14323
   424
done
paulson@14323
   425
paulson@14323
   426
lemma complex_mult_commute: "(w::complex) * z = z * w"
paulson@14323
   427
apply (unfold complex_mult_def)
paulson@14323
   428
apply (simp (no_asm) add: real_mult_commute real_add_commute)
paulson@14323
   429
done
paulson@14323
   430
paulson@14323
   431
lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
paulson@14323
   432
apply (unfold complex_mult_def)
paulson@14334
   433
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
   434
done
paulson@14323
   435
paulson@14323
   436
lemma complex_mult_left_commute: "(x::complex) * (y * z) = y * (x * z)"
paulson@14323
   437
apply (unfold complex_mult_def)
paulson@14334
   438
apply (simp (no_asm) add: complex_Re_Im_cancel_iff mult_left_commute right_diff_distrib right_distrib)
paulson@14323
   439
done
paulson@14323
   440
paulson@14323
   441
lemmas complex_mult_ac = complex_mult_assoc complex_mult_commute
paulson@14323
   442
                      complex_mult_left_commute
paulson@14323
   443
paulson@14323
   444
lemma complex_mult_one_left: "(1::complex) * z = z"
paulson@14323
   445
apply (unfold complex_one_def)
paulson@14334
   446
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   447
apply (simp (no_asm_simp) add: complex_mult)
paulson@14323
   448
done
paulson@14323
   449
declare complex_mult_one_left [simp]
paulson@14323
   450
paulson@14323
   451
lemma complex_mult_one_right: "z * (1::complex) = z"
paulson@14323
   452
apply (simp (no_asm) add: complex_mult_commute)
paulson@14323
   453
done
paulson@14323
   454
declare complex_mult_one_right [simp]
paulson@14323
   455
paulson@14323
   456
lemma complex_mult_zero_left: "(0::complex) * z = 0"
paulson@14323
   457
apply (unfold complex_zero_def)
paulson@14334
   458
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   459
apply (simp (no_asm_simp) add: complex_mult)
paulson@14323
   460
done
paulson@14323
   461
declare complex_mult_zero_left [simp]
paulson@14323
   462
paulson@14323
   463
lemma complex_mult_zero_right: "z * 0 = (0::complex)"
paulson@14323
   464
apply (simp (no_asm) add: complex_mult_commute)
paulson@14323
   465
done
paulson@14323
   466
declare complex_mult_zero_right [simp]
paulson@14323
   467
paulson@14323
   468
lemma complex_divide_zero: "0 / z = (0::complex)"
paulson@14334
   469
by (unfold complex_divide_def, auto)
paulson@14323
   470
declare complex_divide_zero [simp]
paulson@14323
   471
paulson@14323
   472
lemma complex_minus_mult_eq1: "-(x * y) = -x * (y::complex)"
paulson@14334
   473
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   474
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   475
apply (auto simp add: complex_mult complex_minus real_diff_def)
paulson@14323
   476
done
paulson@14323
   477
paulson@14323
   478
lemma complex_minus_mult_eq2: "-(x * y) = x * -(y::complex)"
paulson@14334
   479
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   480
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   481
apply (auto simp add: complex_mult complex_minus real_diff_def)
paulson@14323
   482
done
paulson@14323
   483
paulson@14323
   484
lemma complex_add_mult_distrib: "((z1::complex) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14334
   485
apply (rule_tac z = z1 in eq_Abs_complex)
paulson@14334
   486
apply (rule_tac z = z2 in eq_Abs_complex)
paulson@14334
   487
apply (rule_tac z = w in eq_Abs_complex)
paulson@14334
   488
apply (auto simp add: complex_mult complex_add left_distrib real_diff_def add_ac)
paulson@14323
   489
done
paulson@14323
   490
paulson@14323
   491
lemma complex_add_mult_distrib2: "(w::complex) * (z1 + z2) = (w * z1) + (w * z2)"
paulson@14323
   492
apply (rule_tac z1 = "z1 + z2" in complex_mult_commute [THEN ssubst])
paulson@14323
   493
apply (simp (no_asm) add: complex_add_mult_distrib)
paulson@14323
   494
apply (simp (no_asm) add: complex_mult_commute)
paulson@14323
   495
done
paulson@14323
   496
paulson@14323
   497
lemma complex_zero_not_eq_one: "(0::complex) ~= 1"
paulson@14323
   498
apply (unfold complex_zero_def complex_one_def)
paulson@14323
   499
apply (simp (no_asm) add: complex_Re_Im_cancel_iff)
paulson@14323
   500
done
paulson@14323
   501
declare complex_zero_not_eq_one [simp]
paulson@14323
   502
declare complex_zero_not_eq_one [THEN not_sym, simp]
paulson@14323
   503
paulson@14323
   504
paulson@14323
   505
subsection{*Inverse*}
paulson@14323
   506
paulson@14323
   507
lemma complex_inverse: "inverse (Abs_complex(x,y)) =
paulson@14323
   508
     Abs_complex(x/(x ^ 2 + y ^ 2),-y/(x ^ 2 + y ^ 2))"
paulson@14323
   509
apply (unfold complex_inverse_def)
paulson@14323
   510
apply (simp (no_asm))
paulson@14323
   511
done
paulson@14323
   512
paulson@14323
   513
lemma COMPLEX_INVERSE_ZERO: "inverse 0 = (0::complex)"
paulson@14334
   514
by (unfold complex_inverse_def complex_zero_def, auto)
paulson@14323
   515
paulson@14323
   516
lemma COMPLEX_DIVISION_BY_ZERO: "a / (0::complex) = 0"
paulson@14323
   517
apply (simp (no_asm) add: complex_divide_def COMPLEX_INVERSE_ZERO)
paulson@14323
   518
done
paulson@14323
   519
paulson@14335
   520
instance complex :: division_by_zero
paulson@14335
   521
proof
paulson@14335
   522
  fix x :: complex
paulson@14335
   523
  show "inverse 0 = (0::complex)" by (rule COMPLEX_INVERSE_ZERO)
paulson@14335
   524
  show "x/0 = 0" by (rule COMPLEX_DIVISION_BY_ZERO) 
paulson@14335
   525
qed
paulson@14335
   526
paulson@14323
   527
lemma complex_mult_inv_left: "z ~= (0::complex) ==> inverse(z) * z = 1"
paulson@14334
   528
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
   529
apply (auto simp add: complex_mult complex_inverse complex_one_def 
paulson@14334
   530
       complex_zero_def add_divide_distrib [symmetric] real_power_two mult_ac)
paulson@14334
   531
apply (drule_tac y = y in real_sum_squares_not_zero)
paulson@14334
   532
apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto)
paulson@14323
   533
done
paulson@14323
   534
declare complex_mult_inv_left [simp]
paulson@14323
   535
paulson@14323
   536
lemma complex_mult_inv_right: "z ~= (0::complex) ==> z * inverse(z) = 1"
paulson@14334
   537
by (auto intro: complex_mult_commute [THEN subst])
paulson@14323
   538
declare complex_mult_inv_right [simp]
paulson@14323
   539
paulson@14335
   540
paulson@14335
   541
subsection {* The field of complex numbers *}
paulson@14335
   542
paulson@14335
   543
instance complex :: field
paulson@14335
   544
proof
paulson@14335
   545
  fix z u v w :: complex
paulson@14335
   546
  show "(u + v) + w = u + (v + w)"
paulson@14335
   547
    by (rule complex_add_assoc) 
paulson@14335
   548
  show "z + w = w + z"
paulson@14335
   549
    by (rule complex_add_commute) 
paulson@14335
   550
  show "0 + z = z"
paulson@14335
   551
    by (rule complex_add_zero_left) 
paulson@14335
   552
  show "-z + z = 0"
paulson@14341
   553
    by (rule complex_add_minus_left) 
paulson@14335
   554
  show "z - w = z + -w"
paulson@14335
   555
    by (simp add: complex_diff_def)
paulson@14335
   556
  show "(u * v) * w = u * (v * w)"
paulson@14335
   557
    by (rule complex_mult_assoc) 
paulson@14335
   558
  show "z * w = w * z"
paulson@14335
   559
    by (rule complex_mult_commute) 
paulson@14335
   560
  show "1 * z = z"
paulson@14335
   561
    by (rule complex_mult_one_left) 
paulson@14341
   562
  show "0 \<noteq> (1::complex)"
paulson@14335
   563
    by (rule complex_zero_not_eq_one) 
paulson@14335
   564
  show "(u + v) * w = u * w + v * w"
paulson@14335
   565
    by (rule complex_add_mult_distrib) 
paulson@14341
   566
  show "z+u = z+v ==> u=v"
paulson@14341
   567
    proof -
paulson@14341
   568
      assume eq: "z+u = z+v" 
paulson@14341
   569
      hence "(-z + z) + u = (-z + z) + v" by (simp only: eq complex_add_assoc)
paulson@14341
   570
      thus "u = v" by (simp add: complex_add_minus_left)
paulson@14341
   571
    qed
paulson@14335
   572
  assume neq: "w \<noteq> 0"
paulson@14335
   573
  thus "z / w = z * inverse w"
paulson@14335
   574
    by (simp add: complex_divide_def)
paulson@14335
   575
  show "inverse w * w = 1"
paulson@14335
   576
    by (simp add: neq complex_mult_inv_left) 
paulson@14335
   577
qed
paulson@14335
   578
paulson@14335
   579
paulson@14335
   580
lemma complex_mult_minus_one: "-(1::complex) * z = -z"
paulson@14335
   581
apply (simp (no_asm))
paulson@14335
   582
done
paulson@14335
   583
declare complex_mult_minus_one [simp]
paulson@14335
   584
paulson@14335
   585
lemma complex_mult_minus_one_right: "z * -(1::complex) = -z"
paulson@14335
   586
apply (subst complex_mult_commute)
paulson@14335
   587
apply (simp (no_asm))
paulson@14335
   588
done
paulson@14335
   589
declare complex_mult_minus_one_right [simp]
paulson@14335
   590
paulson@14335
   591
lemma complex_minus_mult_cancel: "-x * -y = x * (y::complex)"
paulson@14335
   592
apply (simp (no_asm))
paulson@14335
   593
done
paulson@14335
   594
declare complex_minus_mult_cancel [simp]
paulson@14335
   595
paulson@14335
   596
lemma complex_minus_mult_commute: "-x * y = x * -(y::complex)"
paulson@14335
   597
apply (simp (no_asm))
paulson@14335
   598
done
paulson@14335
   599
paulson@14335
   600
paulson@14323
   601
lemma complex_mult_left_cancel: "(c::complex) ~= 0 ==> (c*a=c*b) = (a=b)"
paulson@14323
   602
apply auto
paulson@14323
   603
apply (drule_tac f = "%x. x*inverse c" in arg_cong)
paulson@14323
   604
apply (simp add: complex_mult_ac)
paulson@14323
   605
done
paulson@14323
   606
paulson@14323
   607
lemma complex_mult_right_cancel: "(c::complex) ~= 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   608
apply safe
paulson@14323
   609
apply (drule_tac f = "%x. x*inverse c" in arg_cong)
paulson@14323
   610
apply (simp add: complex_mult_ac)
paulson@14323
   611
done
paulson@14323
   612
paulson@14323
   613
lemma complex_inverse_not_zero: "z ~= 0 ==> inverse(z::complex) ~= 0"
paulson@14334
   614
apply safe
paulson@14323
   615
apply (frule complex_mult_right_cancel [THEN iffD2])
paulson@14323
   616
apply (erule_tac [2] V = "inverse z = 0" in thin_rl)
paulson@14334
   617
apply (assumption, auto)
paulson@14323
   618
done
paulson@14323
   619
declare complex_inverse_not_zero [simp]
paulson@14323
   620
paulson@14323
   621
lemma complex_mult_not_zero: "!!x. [| x ~= 0; y ~= (0::complex) |] ==> x * y ~= 0"
paulson@14334
   622
apply safe
paulson@14323
   623
apply (drule_tac f = "%z. inverse x*z" in arg_cong)
paulson@14323
   624
apply (simp add: complex_mult_assoc [symmetric])
paulson@14323
   625
done
paulson@14323
   626
paulson@14323
   627
lemmas complex_mult_not_zeroE = complex_mult_not_zero [THEN notE, standard]
paulson@14323
   628
paulson@14323
   629
lemma complex_inverse_inverse: "inverse(inverse (x::complex)) = x"
paulson@14323
   630
apply (case_tac "x = 0", simp add: COMPLEX_INVERSE_ZERO)
paulson@14323
   631
apply (rule_tac c1 = "inverse x" in complex_mult_right_cancel [THEN iffD1])
paulson@14323
   632
apply (erule complex_inverse_not_zero)
paulson@14323
   633
apply (auto dest: complex_inverse_not_zero)
paulson@14323
   634
done
paulson@14323
   635
declare complex_inverse_inverse [simp]
paulson@14323
   636
paulson@14323
   637
lemma complex_inverse_one: "inverse(1::complex) = 1"
paulson@14323
   638
apply (unfold complex_one_def)
paulson@14323
   639
apply (simp (no_asm) add: complex_inverse)
paulson@14323
   640
done
paulson@14323
   641
declare complex_inverse_one [simp]
paulson@14323
   642
paulson@14323
   643
lemma complex_minus_inverse: "inverse(-x) = -inverse(x::complex)"
paulson@14323
   644
apply (case_tac "x = 0", simp add: COMPLEX_INVERSE_ZERO)
paulson@14334
   645
apply (rule_tac c1 = "-x" in complex_mult_right_cancel [THEN iffD1], force)
paulson@14334
   646
apply (subst complex_mult_inv_left, auto)
paulson@14323
   647
done
paulson@14323
   648
paulson@14323
   649
lemma complex_inverse_distrib: "inverse(x*y) = inverse x * inverse (y::complex)"
paulson@14335
   650
apply (rule inverse_mult_distrib) 
paulson@14323
   651
done
paulson@14323
   652
paulson@14323
   653
paulson@14323
   654
subsection{*Division*}
paulson@14323
   655
paulson@14323
   656
(*adding some of these theorems to simpset as for reals:
paulson@14323
   657
  not 100% convinced for some*)
paulson@14323
   658
paulson@14323
   659
lemma complex_times_divide1_eq: "(x::complex) * (y/z) = (x*y)/z"
paulson@14323
   660
apply (simp (no_asm) add: complex_divide_def complex_mult_assoc)
paulson@14323
   661
done
paulson@14323
   662
paulson@14323
   663
lemma complex_times_divide2_eq: "(y/z) * (x::complex) = (y*x)/z"
paulson@14323
   664
apply (simp (no_asm) add: complex_divide_def complex_mult_ac)
paulson@14323
   665
done
paulson@14323
   666
paulson@14323
   667
declare complex_times_divide1_eq [simp] complex_times_divide2_eq [simp]
paulson@14323
   668
paulson@14323
   669
lemma complex_divide_divide1_eq: "(x::complex) / (y/z) = (x*z)/y"
paulson@14323
   670
apply (simp (no_asm) add: complex_divide_def complex_inverse_distrib complex_mult_ac)
paulson@14323
   671
done
paulson@14323
   672
paulson@14323
   673
lemma complex_divide_divide2_eq: "((x::complex) / y) / z = x/(y*z)"
paulson@14323
   674
apply (simp (no_asm) add: complex_divide_def complex_inverse_distrib complex_mult_assoc)
paulson@14323
   675
done
paulson@14323
   676
paulson@14323
   677
declare complex_divide_divide1_eq [simp] complex_divide_divide2_eq [simp]
paulson@14323
   678
paulson@14323
   679
(** As with multiplication, pull minus signs OUT of the / operator **)
paulson@14323
   680
paulson@14323
   681
lemma complex_minus_divide_eq: "(-x) / (y::complex) = - (x/y)"
paulson@14323
   682
apply (simp (no_asm) add: complex_divide_def)
paulson@14323
   683
done
paulson@14323
   684
declare complex_minus_divide_eq [simp]
paulson@14323
   685
paulson@14323
   686
lemma complex_divide_minus_eq: "(x / -(y::complex)) = - (x/y)"
paulson@14323
   687
apply (simp (no_asm) add: complex_divide_def complex_minus_inverse)
paulson@14323
   688
done
paulson@14323
   689
declare complex_divide_minus_eq [simp]
paulson@14323
   690
paulson@14323
   691
lemma complex_add_divide_distrib: "(x+y)/(z::complex) = x/z + y/z"
paulson@14323
   692
apply (simp (no_asm) add: complex_divide_def complex_add_mult_distrib)
paulson@14323
   693
done
paulson@14323
   694
paulson@14323
   695
subsection{*Embedding Properties for @{term complex_of_real} Map*}
paulson@14323
   696
paulson@14323
   697
lemma inj_complex_of_real: "inj complex_of_real"
paulson@14323
   698
apply (rule inj_onI)
paulson@14323
   699
apply (auto dest: inj_Abs_complex [THEN injD] simp add: complex_of_real_def)
paulson@14323
   700
done
paulson@14323
   701
paulson@14323
   702
lemma complex_of_real_one:
paulson@14323
   703
      "complex_of_real 1 = 1"
paulson@14323
   704
apply (unfold complex_one_def complex_of_real_def)
paulson@14323
   705
apply (rule refl)
paulson@14323
   706
done
paulson@14323
   707
declare complex_of_real_one [simp]
paulson@14323
   708
paulson@14323
   709
lemma complex_of_real_zero:
paulson@14323
   710
      "complex_of_real 0 = 0"
paulson@14323
   711
apply (unfold complex_zero_def complex_of_real_def)
paulson@14323
   712
apply (rule refl)
paulson@14323
   713
done
paulson@14323
   714
declare complex_of_real_zero [simp]
paulson@14323
   715
paulson@14323
   716
lemma complex_of_real_eq_iff: "(complex_of_real x = complex_of_real y) = (x = y)"
paulson@14334
   717
by (auto dest: inj_complex_of_real [THEN injD])
paulson@14323
   718
declare complex_of_real_eq_iff [iff]
paulson@14323
   719
paulson@14323
   720
lemma complex_of_real_minus: "complex_of_real(-x) = - complex_of_real x"
paulson@14323
   721
apply (simp (no_asm) add: complex_of_real_def complex_minus)
paulson@14323
   722
done
paulson@14323
   723
paulson@14323
   724
lemma complex_of_real_inverse: "complex_of_real(inverse x) = inverse(complex_of_real x)"
paulson@14323
   725
apply (case_tac "x=0")
paulson@14323
   726
apply (simp add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
paulson@14334
   727
apply (simp add: complex_inverse complex_of_real_def real_divide_def 
paulson@14334
   728
                 inverse_mult_distrib real_power_two)
paulson@14323
   729
done
paulson@14323
   730
paulson@14323
   731
lemma complex_of_real_add: "complex_of_real x + complex_of_real y = complex_of_real (x + y)"
paulson@14323
   732
apply (simp (no_asm) add: complex_add complex_of_real_def)
paulson@14323
   733
done
paulson@14323
   734
paulson@14323
   735
lemma complex_of_real_diff: "complex_of_real x - complex_of_real y = complex_of_real (x - y)"
paulson@14323
   736
apply (simp (no_asm) add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add)
paulson@14323
   737
done
paulson@14323
   738
paulson@14323
   739
lemma complex_of_real_mult: "complex_of_real x * complex_of_real y = complex_of_real (x * y)"
paulson@14323
   740
apply (simp (no_asm) add: complex_mult complex_of_real_def)
paulson@14323
   741
done
paulson@14323
   742
paulson@14323
   743
lemma complex_of_real_divide:
paulson@14323
   744
      "complex_of_real x / complex_of_real y = complex_of_real(x/y)"
paulson@14323
   745
apply (unfold complex_divide_def)
paulson@14323
   746
apply (case_tac "y=0")
paulson@14323
   747
apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
paulson@14323
   748
apply (simp (no_asm_simp) add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def)
paulson@14323
   749
done
paulson@14323
   750
paulson@14323
   751
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
paulson@14323
   752
apply (induct_tac "n")
paulson@14323
   753
apply (auto simp add: complex_of_real_mult [symmetric])
paulson@14323
   754
done
paulson@14323
   755
paulson@14323
   756
lemma complex_mod: "cmod (Abs_complex(x,y)) = sqrt(x ^ 2 + y ^ 2)"
paulson@14323
   757
apply (unfold cmod_def)
paulson@14323
   758
apply (simp (no_asm))
paulson@14323
   759
done
paulson@14323
   760
paulson@14323
   761
lemma complex_mod_zero: "cmod(0) = 0"
paulson@14323
   762
apply (unfold cmod_def)
paulson@14323
   763
apply (simp (no_asm))
paulson@14323
   764
done
paulson@14323
   765
declare complex_mod_zero [simp]
paulson@14323
   766
paulson@14323
   767
lemma complex_mod_one: "cmod(1) = 1"
paulson@14334
   768
by (unfold cmod_def, simp)
paulson@14323
   769
declare complex_mod_one [simp]
paulson@14323
   770
paulson@14323
   771
lemma complex_mod_complex_of_real: "cmod(complex_of_real x) = abs x"
paulson@14323
   772
apply (unfold complex_of_real_def)
paulson@14323
   773
apply (simp (no_asm) add: complex_mod)
paulson@14323
   774
done
paulson@14323
   775
declare complex_mod_complex_of_real [simp]
paulson@14323
   776
paulson@14323
   777
lemma complex_of_real_abs: "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
paulson@14323
   778
apply (simp (no_asm))
paulson@14323
   779
done
paulson@14323
   780
paulson@14323
   781
paulson@14323
   782
subsection{*Conjugation is an Automorphism*}
paulson@14323
   783
paulson@14323
   784
lemma complex_cnj: "cnj (Abs_complex(x,y)) = Abs_complex(x,-y)"
paulson@14323
   785
apply (unfold cnj_def)
paulson@14323
   786
apply (simp (no_asm))
paulson@14323
   787
done
paulson@14323
   788
paulson@14323
   789
lemma inj_cnj: "inj cnj"
paulson@14323
   790
apply (rule inj_onI)
paulson@14323
   791
apply (auto simp add: cnj_def Abs_complex_cancel_iff complex_Re_Im_cancel_iff)
paulson@14323
   792
done
paulson@14323
   793
paulson@14323
   794
lemma complex_cnj_cancel_iff: "(cnj x = cnj y) = (x = y)"
paulson@14334
   795
by (auto dest: inj_cnj [THEN injD])
paulson@14323
   796
declare complex_cnj_cancel_iff [simp]
paulson@14323
   797
paulson@14323
   798
lemma complex_cnj_cnj: "cnj (cnj z) = z"
paulson@14323
   799
apply (unfold cnj_def)
paulson@14323
   800
apply (simp (no_asm))
paulson@14323
   801
done
paulson@14323
   802
declare complex_cnj_cnj [simp]
paulson@14323
   803
paulson@14323
   804
lemma complex_cnj_complex_of_real: "cnj (complex_of_real x) = complex_of_real x"
paulson@14323
   805
apply (unfold complex_of_real_def)
paulson@14323
   806
apply (simp (no_asm) add: complex_cnj)
paulson@14323
   807
done
paulson@14323
   808
declare complex_cnj_complex_of_real [simp]
paulson@14323
   809
paulson@14323
   810
lemma complex_mod_cnj: "cmod (cnj z) = cmod z"
paulson@14334
   811
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   812
apply (simp (no_asm_simp) add: complex_cnj complex_mod real_power_two)
paulson@14323
   813
done
paulson@14323
   814
declare complex_mod_cnj [simp]
paulson@14323
   815
paulson@14323
   816
lemma complex_cnj_minus: "cnj (-z) = - cnj z"
paulson@14323
   817
apply (unfold cnj_def)
paulson@14323
   818
apply (simp (no_asm) add: complex_minus complex_Re_minus complex_Im_minus)
paulson@14323
   819
done
paulson@14323
   820
paulson@14323
   821
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
paulson@14334
   822
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   823
apply (simp (no_asm_simp) add: complex_cnj complex_inverse real_power_two)
paulson@14323
   824
done
paulson@14323
   825
paulson@14323
   826
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
paulson@14334
   827
apply (rule_tac z = w in eq_Abs_complex)
paulson@14334
   828
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   829
apply (simp (no_asm_simp) add: complex_cnj complex_add)
paulson@14323
   830
done
paulson@14323
   831
paulson@14323
   832
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
paulson@14323
   833
apply (unfold complex_diff_def)
paulson@14323
   834
apply (simp (no_asm) add: complex_cnj_add complex_cnj_minus)
paulson@14323
   835
done
paulson@14323
   836
paulson@14323
   837
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
paulson@14334
   838
apply (rule_tac z = w in eq_Abs_complex)
paulson@14334
   839
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   840
apply (simp (no_asm_simp) add: complex_cnj complex_mult)
paulson@14323
   841
done
paulson@14323
   842
paulson@14323
   843
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
paulson@14323
   844
apply (unfold complex_divide_def)
paulson@14323
   845
apply (simp (no_asm) add: complex_cnj_mult complex_cnj_inverse)
paulson@14323
   846
done
paulson@14323
   847
paulson@14323
   848
lemma complex_cnj_one: "cnj 1 = 1"
paulson@14323
   849
apply (unfold cnj_def complex_one_def)
paulson@14323
   850
apply (simp (no_asm))
paulson@14323
   851
done
paulson@14323
   852
declare complex_cnj_one [simp]
paulson@14323
   853
paulson@14323
   854
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
paulson@14323
   855
apply (induct_tac "n")
paulson@14323
   856
apply (auto simp add: complex_cnj_mult)
paulson@14323
   857
done
paulson@14323
   858
paulson@14323
   859
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
paulson@14334
   860
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   861
apply (simp (no_asm_simp) add: complex_add complex_cnj complex_of_real_def)
paulson@14323
   862
done
paulson@14323
   863
paulson@14323
   864
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
paulson@14334
   865
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   866
apply (simp (no_asm_simp) add: complex_add complex_cnj complex_of_real_def complex_diff_def complex_minus i_def complex_mult)
paulson@14323
   867
done
paulson@14323
   868
paulson@14323
   869
lemma complex_cnj_zero: "cnj 0 = 0"
paulson@14334
   870
by (simp add: cnj_def complex_zero_def)
paulson@14323
   871
declare complex_cnj_zero [simp]
paulson@14323
   872
paulson@14323
   873
lemma complex_cnj_zero_iff: "(cnj z = 0) = (z = 0)"
paulson@14334
   874
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   875
apply (auto simp add: complex_zero_def complex_cnj)
paulson@14323
   876
done
paulson@14323
   877
declare complex_cnj_zero_iff [iff]
paulson@14323
   878
paulson@14323
   879
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
paulson@14334
   880
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   881
apply (auto simp add: complex_cnj complex_mult complex_of_real_def real_power_two)
paulson@14323
   882
done
paulson@14323
   883
paulson@14323
   884
paulson@14323
   885
subsection{*Algebra*}
paulson@14323
   886
paulson@14323
   887
lemma complex_mult_zero_iff: "(x*y = (0::complex)) = (x = 0 | y = 0)"
paulson@14323
   888
apply auto
paulson@14323
   889
apply (auto intro: ccontr dest: complex_mult_not_zero)
paulson@14323
   890
done
paulson@14323
   891
declare complex_mult_zero_iff [iff]
paulson@14323
   892
paulson@14323
   893
lemma complex_add_left_cancel_zero: "(x + y = x) = (y = (0::complex))"
paulson@14323
   894
apply (unfold complex_zero_def)
paulson@14334
   895
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   896
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   897
apply (auto simp add: complex_add)
paulson@14323
   898
done
paulson@14323
   899
declare complex_add_left_cancel_zero [simp]
paulson@14323
   900
paulson@14323
   901
lemma complex_diff_mult_distrib:
paulson@14323
   902
      "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)"
paulson@14323
   903
apply (unfold complex_diff_def)
paulson@14323
   904
apply (simp (no_asm) add: complex_add_mult_distrib)
paulson@14323
   905
done
paulson@14323
   906
paulson@14323
   907
lemma complex_diff_mult_distrib2:
paulson@14323
   908
      "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)"
paulson@14323
   909
apply (unfold complex_diff_def)
paulson@14323
   910
apply (simp (no_asm) add: complex_add_mult_distrib2)
paulson@14323
   911
done
paulson@14323
   912
paulson@14323
   913
paulson@14323
   914
subsection{*Modulus*}
paulson@14323
   915
paulson@14323
   916
(*
paulson@14323
   917
Goal "[| sqrt(x) = 0; 0 <= x |] ==> x = 0"
paulson@14323
   918
by (auto_tac (claset() addIs [real_sqrt_eq_zero_cancel],
paulson@14323
   919
    simpset()));
paulson@14323
   920
qed "real_sqrt_eq_zero_cancel2";
paulson@14323
   921
*)
paulson@14323
   922
paulson@14323
   923
lemma complex_mod_eq_zero_cancel: "(cmod x = 0) = (x = 0)"
paulson@14334
   924
apply (rule_tac z = x in eq_Abs_complex)
paulson@14323
   925
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: complex_mod complex_zero_def real_power_two)
paulson@14323
   926
done
paulson@14323
   927
declare complex_mod_eq_zero_cancel [simp]
paulson@14323
   928
paulson@14323
   929
lemma complex_mod_complex_of_real_of_nat: "cmod (complex_of_real(real (n::nat))) = real n"
paulson@14323
   930
apply (simp (no_asm))
paulson@14323
   931
done
paulson@14323
   932
declare complex_mod_complex_of_real_of_nat [simp]
paulson@14323
   933
paulson@14323
   934
lemma complex_mod_minus: "cmod (-x) = cmod(x)"
paulson@14334
   935
apply (rule_tac z = x in eq_Abs_complex)
paulson@14323
   936
apply (simp (no_asm_simp) add: complex_mod complex_minus real_power_two)
paulson@14323
   937
done
paulson@14323
   938
declare complex_mod_minus [simp]
paulson@14323
   939
paulson@14323
   940
lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
paulson@14334
   941
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
   942
apply (simp (no_asm_simp) add: complex_mod complex_cnj complex_mult);
paulson@14323
   943
apply (simp (no_asm) add: real_power_two real_abs_def)
paulson@14323
   944
done
paulson@14323
   945
paulson@14323
   946
lemma complex_mod_squared: "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2"
paulson@14334
   947
by (unfold cmod_def, auto)
paulson@14323
   948
paulson@14323
   949
lemma complex_mod_ge_zero: "0 <= cmod x"
paulson@14323
   950
apply (unfold cmod_def)
paulson@14323
   951
apply (auto intro: real_sqrt_ge_zero)
paulson@14323
   952
done
paulson@14323
   953
declare complex_mod_ge_zero [simp]
paulson@14323
   954
paulson@14323
   955
lemma abs_cmod_cancel: "abs(cmod x) = cmod x"
paulson@14334
   956
by (auto intro: abs_eqI1)
paulson@14323
   957
declare abs_cmod_cancel [simp]
paulson@14323
   958
paulson@14323
   959
lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)"
paulson@14334
   960
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   961
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   962
apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2 [symmetric] simp del: realpow_Suc)
paulson@14334
   963
apply (rule_tac n = 1 in realpow_Suc_cancel_eq)
paulson@14323
   964
apply (auto simp add: real_power_two [symmetric] simp del: realpow_Suc)
paulson@14334
   965
apply (auto simp add: real_diff_def real_power_two right_distrib left_distrib add_ac mult_ac)
paulson@14323
   966
done
paulson@14323
   967
paulson@14323
   968
lemma complex_mod_add_squared_eq: "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
paulson@14334
   969
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   970
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   971
apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
paulson@14334
   972
apply (auto simp add: right_distrib left_distrib real_power_two mult_ac add_ac)
paulson@14323
   973
done
paulson@14323
   974
paulson@14323
   975
lemma complex_Re_mult_cnj_le_cmod: "Re(x * cnj y) <= cmod(x * cnj y)"
paulson@14334
   976
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
   977
apply (rule_tac z = y in eq_Abs_complex)
paulson@14323
   978
apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
paulson@14323
   979
done
paulson@14323
   980
declare complex_Re_mult_cnj_le_cmod [simp]
paulson@14323
   981
paulson@14323
   982
lemma complex_Re_mult_cnj_le_cmod2: "Re(x * cnj y) <= cmod(x * y)"
paulson@14334
   983
apply (cut_tac x = x and y = y in complex_Re_mult_cnj_le_cmod)
paulson@14323
   984
apply (simp add: complex_mod_mult)
paulson@14323
   985
done
paulson@14323
   986
declare complex_Re_mult_cnj_le_cmod2 [simp]
paulson@14323
   987
paulson@14323
   988
lemma real_sum_squared_expand: "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y"
paulson@14334
   989
apply (simp (no_asm) add: left_distrib right_distrib real_power_two)
paulson@14323
   990
done
paulson@14323
   991
paulson@14323
   992
lemma complex_mod_triangle_squared: "cmod (x + y) ^ 2 <= (cmod(x) + cmod(y)) ^ 2"
paulson@14323
   993
apply (simp (no_asm) add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
paulson@14323
   994
done
paulson@14323
   995
declare complex_mod_triangle_squared [simp]
paulson@14323
   996
paulson@14323
   997
lemma complex_mod_minus_le_complex_mod: "- cmod x <= cmod x"
paulson@14323
   998
apply (rule order_trans [OF _ complex_mod_ge_zero]) 
paulson@14323
   999
apply (simp (no_asm))
paulson@14323
  1000
done
paulson@14323
  1001
declare complex_mod_minus_le_complex_mod [simp]
paulson@14323
  1002
paulson@14323
  1003
lemma complex_mod_triangle_ineq: "cmod (x + y) <= cmod(x) + cmod(y)"
paulson@14334
  1004
apply (rule_tac n = 1 in realpow_increasing)
paulson@14323
  1005
apply (auto intro:  order_trans [OF _ complex_mod_ge_zero]
paulson@14323
  1006
            simp add: real_power_two [symmetric])
paulson@14323
  1007
done
paulson@14323
  1008
declare complex_mod_triangle_ineq [simp]
paulson@14323
  1009
paulson@14323
  1010
lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b <= cmod a"
paulson@14334
  1011
apply (cut_tac x1 = b and y1 = a and c = "-cmod b" 
paulson@14334
  1012
       in complex_mod_triangle_ineq [THEN add_right_mono])
paulson@14323
  1013
apply (simp (no_asm))
paulson@14323
  1014
done
paulson@14323
  1015
declare complex_mod_triangle_ineq2 [simp]
paulson@14323
  1016
paulson@14323
  1017
lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)"
paulson@14334
  1018
apply (rule_tac z = x in eq_Abs_complex)
paulson@14334
  1019
apply (rule_tac z = y in eq_Abs_complex)
paulson@14334
  1020
apply (auto simp add: complex_diff complex_mod right_diff_distrib real_power_two left_diff_distrib add_ac mult_ac)
paulson@14323
  1021
done
paulson@14323
  1022
paulson@14323
  1023
lemma complex_mod_add_less: "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
paulson@14334
  1024
by (auto intro: order_le_less_trans complex_mod_triangle_ineq)
paulson@14323
  1025
paulson@14323
  1026
lemma complex_mod_mult_less: "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
paulson@14334
  1027
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
paulson@14323
  1028
paulson@14323
  1029
lemma complex_mod_diff_ineq: "cmod(a) - cmod(b) <= cmod(a + b)"
paulson@14323
  1030
apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
paulson@14323
  1031
apply auto
paulson@14334
  1032
apply (rule order_trans [of _ 0], rule order_less_imp_le)
paulson@14334
  1033
apply (simp add: compare_rls, simp)  
paulson@14323
  1034
apply (simp add: compare_rls)
paulson@14323
  1035
apply (rule complex_mod_minus [THEN subst])
paulson@14323
  1036
apply (rule order_trans)
paulson@14323
  1037
apply (rule_tac [2] complex_mod_triangle_ineq)
paulson@14323
  1038
apply (auto simp add: complex_add_ac)
paulson@14323
  1039
done
paulson@14323
  1040
declare complex_mod_diff_ineq [simp]
paulson@14323
  1041
paulson@14323
  1042
lemma complex_Re_le_cmod: "Re z <= cmod z"
paulson@14334
  1043
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1044
apply (auto simp add: complex_mod simp del: realpow_Suc)
paulson@14323
  1045
done
paulson@14323
  1046
declare complex_Re_le_cmod [simp]
paulson@14323
  1047
paulson@14323
  1048
lemma complex_mod_gt_zero: "z ~= 0 ==> 0 < cmod z"
paulson@14334
  1049
apply (cut_tac x = z in complex_mod_ge_zero)
paulson@14334
  1050
apply (drule order_le_imp_less_or_eq, auto)
paulson@14323
  1051
done
paulson@14323
  1052
paulson@14323
  1053
paulson@14323
  1054
subsection{*A Few More Theorems*}
paulson@14323
  1055
paulson@14323
  1056
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
paulson@14323
  1057
apply (induct_tac "n")
paulson@14323
  1058
apply (auto simp add: complex_mod_mult)
paulson@14323
  1059
done
paulson@14323
  1060
paulson@14323
  1061
lemma complexpow_minus: "(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))"
paulson@14334
  1062
by (induct_tac "n", auto)
paulson@14323
  1063
paulson@14323
  1064
lemma complex_inverse_minus: "inverse (-x) = - inverse (x::complex)"
paulson@14334
  1065
apply (rule_tac z = x in eq_Abs_complex)
paulson@14323
  1066
apply (simp (no_asm_simp) add: complex_inverse complex_minus real_power_two)
paulson@14323
  1067
done
paulson@14323
  1068
paulson@14323
  1069
lemma complex_divide_one: "x / (1::complex) = x"
paulson@14323
  1070
apply (unfold complex_divide_def)
paulson@14323
  1071
apply (simp (no_asm))
paulson@14323
  1072
done
paulson@14323
  1073
declare complex_divide_one [simp]
paulson@14323
  1074
paulson@14323
  1075
lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
paulson@14323
  1076
apply (case_tac "x=0", simp add: COMPLEX_INVERSE_ZERO)
paulson@14323
  1077
apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1])
paulson@14323
  1078
apply (auto simp add: complex_mod_mult [symmetric])
paulson@14323
  1079
done
paulson@14323
  1080
paulson@14323
  1081
lemma complex_mod_divide:
paulson@14323
  1082
      "cmod(x/y) = cmod(x)/(cmod y)"
paulson@14323
  1083
apply (unfold complex_divide_def real_divide_def)
paulson@14323
  1084
apply (auto simp add: complex_mod_mult complex_mod_inverse)
paulson@14323
  1085
done
paulson@14323
  1086
paulson@14323
  1087
lemma complex_inverse_divide:
paulson@14323
  1088
      "inverse(x/y) = y/(x::complex)"
paulson@14323
  1089
apply (unfold complex_divide_def)
paulson@14323
  1090
apply (auto simp add: complex_inverse_distrib complex_mult_commute)
paulson@14323
  1091
done
paulson@14323
  1092
declare complex_inverse_divide [simp]
paulson@14323
  1093
paulson@14323
  1094
lemma complexpow_mult: "((r::complex) * s) ^ n = (r ^ n) * (s ^ n)"
paulson@14323
  1095
apply (induct_tac "n")
paulson@14323
  1096
apply (auto simp add: complex_mult_ac)
paulson@14323
  1097
done
paulson@14323
  1098
paulson@14323
  1099
paulson@14323
  1100
subsection{*More Exponentiation*}
paulson@14323
  1101
paulson@14323
  1102
lemma complexpow_zero: "(0::complex) ^ (Suc n) = 0"
paulson@14334
  1103
by auto
paulson@14323
  1104
declare complexpow_zero [simp]
paulson@14323
  1105
paulson@14323
  1106
lemma complexpow_not_zero [rule_format (no_asm)]: "r ~= (0::complex) --> r ^ n ~= 0"
paulson@14323
  1107
apply (induct_tac "n")
paulson@14323
  1108
apply (auto simp add: complex_mult_not_zero)
paulson@14323
  1109
done
paulson@14323
  1110
declare complexpow_not_zero [simp]
paulson@14323
  1111
declare complexpow_not_zero [intro]
paulson@14323
  1112
paulson@14323
  1113
lemma complexpow_zero_zero: "r ^ n = (0::complex) ==> r = 0"
paulson@14334
  1114
by (blast intro: ccontr dest: complexpow_not_zero)
paulson@14323
  1115
paulson@14323
  1116
lemma complexpow_i_squared: "ii ^ 2 = -(1::complex)"
paulson@14323
  1117
apply (unfold i_def)
paulson@14323
  1118
apply (auto simp add: complex_mult complex_one_def complex_minus numeral_2_eq_2)
paulson@14323
  1119
done
paulson@14323
  1120
declare complexpow_i_squared [simp]
paulson@14323
  1121
paulson@14323
  1122
lemma complex_i_not_zero: "ii ~= 0"
paulson@14334
  1123
by (unfold i_def complex_zero_def, auto)
paulson@14323
  1124
declare complex_i_not_zero [simp]
paulson@14323
  1125
paulson@14323
  1126
lemma complex_mult_eq_zero_cancel1: "x * y ~= (0::complex) ==> x ~= 0"
paulson@14334
  1127
by auto
paulson@14323
  1128
paulson@14323
  1129
lemma complex_mult_eq_zero_cancel2: "x * y ~= 0 ==> y ~= (0::complex)"
paulson@14334
  1130
by auto
paulson@14323
  1131
paulson@14323
  1132
lemma complex_mult_not_eq_zero_iff: "(x * y ~= 0) = (x ~= 0 & y ~= (0::complex))"
paulson@14334
  1133
by auto
paulson@14323
  1134
declare complex_mult_not_eq_zero_iff [iff]
paulson@14323
  1135
paulson@14323
  1136
lemma complexpow_inverse: "inverse ((r::complex) ^ n) = (inverse r) ^ n"
paulson@14323
  1137
apply (induct_tac "n")
paulson@14323
  1138
apply (auto simp add: complex_inverse_distrib)
paulson@14323
  1139
done
paulson@14323
  1140
paulson@14323
  1141
(*---------------------------------------------------------------------------*)
paulson@14323
  1142
(* sgn                                                                       *)
paulson@14323
  1143
(*---------------------------------------------------------------------------*)
paulson@14323
  1144
paulson@14323
  1145
lemma sgn_zero: "sgn 0 = 0"
paulson@14323
  1146
paulson@14323
  1147
apply (unfold sgn_def)
paulson@14323
  1148
apply (simp (no_asm))
paulson@14323
  1149
done
paulson@14323
  1150
declare sgn_zero [simp]
paulson@14323
  1151
paulson@14323
  1152
lemma sgn_one: "sgn 1 = 1"
paulson@14323
  1153
apply (unfold sgn_def)
paulson@14323
  1154
apply (simp (no_asm))
paulson@14323
  1155
done
paulson@14323
  1156
declare sgn_one [simp]
paulson@14323
  1157
paulson@14323
  1158
lemma sgn_minus: "sgn (-z) = - sgn(z)"
paulson@14334
  1159
by (unfold sgn_def, auto)
paulson@14323
  1160
paulson@14323
  1161
lemma sgn_eq:
paulson@14323
  1162
    "sgn z = z / complex_of_real (cmod z)"
paulson@14323
  1163
apply (unfold sgn_def)
paulson@14323
  1164
apply (simp (no_asm))
paulson@14323
  1165
done
paulson@14323
  1166
paulson@14323
  1167
lemma complex_split: "EX x y. z = complex_of_real(x) + ii * complex_of_real(y)"
paulson@14334
  1168
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1169
apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
paulson@14323
  1170
done
paulson@14323
  1171
paulson@14323
  1172
lemma Re_complex_i: "Re(complex_of_real(x) + ii * complex_of_real(y)) = x"
paulson@14334
  1173
by (auto simp add: complex_of_real_def i_def complex_mult complex_add)
paulson@14323
  1174
declare Re_complex_i [simp]
paulson@14323
  1175
paulson@14323
  1176
lemma Im_complex_i: "Im(complex_of_real(x) + ii * complex_of_real(y)) = y"
paulson@14334
  1177
by (auto simp add: complex_of_real_def i_def complex_mult complex_add)
paulson@14323
  1178
declare Im_complex_i [simp]
paulson@14323
  1179
paulson@14323
  1180
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
paulson@14323
  1181
apply (unfold i_def complex_of_real_def)
paulson@14323
  1182
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1183
done
paulson@14323
  1184
paulson@14323
  1185
lemma i_mult_eq2: "ii * ii = -(1::complex)"
paulson@14323
  1186
apply (unfold i_def complex_one_def)
paulson@14323
  1187
apply (simp (no_asm) add: complex_mult complex_minus)
paulson@14323
  1188
done
paulson@14323
  1189
declare i_mult_eq2 [simp]
paulson@14323
  1190
paulson@14323
  1191
lemma cmod_i: "cmod (complex_of_real(x) + ii * complex_of_real(y)) =
paulson@14323
  1192
      sqrt (x ^ 2 + y ^ 2)"
paulson@14323
  1193
apply (auto simp add: complex_mult complex_add i_def complex_of_real_def cmod_def)
paulson@14323
  1194
done
paulson@14323
  1195
paulson@14323
  1196
lemma complex_eq_Re_eq:
paulson@14323
  1197
     "complex_of_real xa + ii * complex_of_real ya =
paulson@14323
  1198
      complex_of_real xb + ii * complex_of_real yb
paulson@14323
  1199
       ==> xa = xb"
paulson@14323
  1200
apply (unfold complex_of_real_def i_def)
paulson@14323
  1201
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1202
done
paulson@14323
  1203
paulson@14323
  1204
lemma complex_eq_Im_eq:
paulson@14323
  1205
     "complex_of_real xa + ii * complex_of_real ya =
paulson@14323
  1206
      complex_of_real xb + ii * complex_of_real yb
paulson@14323
  1207
       ==> ya = yb"
paulson@14323
  1208
apply (unfold complex_of_real_def i_def)
paulson@14323
  1209
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1210
done
paulson@14323
  1211
paulson@14323
  1212
lemma complex_eq_cancel_iff: "(complex_of_real xa + ii * complex_of_real ya =
paulson@14323
  1213
       complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
paulson@14323
  1214
apply (auto intro: complex_eq_Im_eq complex_eq_Re_eq)
paulson@14323
  1215
done
paulson@14323
  1216
declare complex_eq_cancel_iff [iff]
paulson@14323
  1217
paulson@14323
  1218
lemma complex_eq_cancel_iffA: "(complex_of_real xa + complex_of_real ya * ii =
paulson@14323
  1219
       complex_of_real xb + complex_of_real yb * ii ) = ((xa = xb) & (ya = yb))"
paulson@14323
  1220
apply (auto simp add: complex_mult_commute)
paulson@14323
  1221
done
paulson@14323
  1222
declare complex_eq_cancel_iffA [iff]
paulson@14323
  1223
paulson@14323
  1224
lemma complex_eq_cancel_iffB: "(complex_of_real xa + complex_of_real ya * ii =
paulson@14323
  1225
       complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
paulson@14323
  1226
apply (auto simp add: complex_mult_commute)
paulson@14323
  1227
done
paulson@14323
  1228
declare complex_eq_cancel_iffB [iff]
paulson@14323
  1229
paulson@14323
  1230
lemma complex_eq_cancel_iffC: "(complex_of_real xa + ii * complex_of_real ya  =
paulson@14323
  1231
       complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))"
paulson@14323
  1232
apply (auto simp add: complex_mult_commute)
paulson@14323
  1233
done
paulson@14323
  1234
declare complex_eq_cancel_iffC [iff]
paulson@14323
  1235
paulson@14323
  1236
lemma complex_eq_cancel_iff2: "(complex_of_real x + ii * complex_of_real y =
paulson@14323
  1237
      complex_of_real xa) = (x = xa & y = 0)"
paulson@14334
  1238
apply (cut_tac xa = x and ya = y and xb = xa and yb = 0 in complex_eq_cancel_iff)
paulson@14323
  1239
apply (simp del: complex_eq_cancel_iff)
paulson@14323
  1240
done
paulson@14323
  1241
declare complex_eq_cancel_iff2 [simp]
paulson@14323
  1242
paulson@14323
  1243
lemma complex_eq_cancel_iff2a: "(complex_of_real x + complex_of_real y * ii =
paulson@14323
  1244
      complex_of_real xa) = (x = xa & y = 0)"
paulson@14323
  1245
apply (auto simp add: complex_mult_commute)
paulson@14323
  1246
done
paulson@14323
  1247
declare complex_eq_cancel_iff2a [simp]
paulson@14323
  1248
paulson@14323
  1249
lemma complex_eq_cancel_iff3: "(complex_of_real x + ii * complex_of_real y =
paulson@14323
  1250
      ii * complex_of_real ya) = (x = 0 & y = ya)"
paulson@14334
  1251
apply (cut_tac xa = x and ya = y and xb = 0 and yb = ya in complex_eq_cancel_iff)
paulson@14323
  1252
apply (simp del: complex_eq_cancel_iff)
paulson@14323
  1253
done
paulson@14323
  1254
declare complex_eq_cancel_iff3 [simp]
paulson@14323
  1255
paulson@14323
  1256
lemma complex_eq_cancel_iff3a: "(complex_of_real x + complex_of_real y * ii =
paulson@14323
  1257
      ii * complex_of_real ya) = (x = 0 & y = ya)"
paulson@14323
  1258
apply (auto simp add: complex_mult_commute)
paulson@14323
  1259
done
paulson@14323
  1260
declare complex_eq_cancel_iff3a [simp]
paulson@14323
  1261
paulson@14323
  1262
lemma complex_split_Re_zero:
paulson@14323
  1263
     "complex_of_real x + ii * complex_of_real y = 0
paulson@14323
  1264
      ==> x = 0"
paulson@14323
  1265
apply (unfold complex_of_real_def i_def complex_zero_def)
paulson@14323
  1266
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1267
done
paulson@14323
  1268
paulson@14323
  1269
lemma complex_split_Im_zero:
paulson@14323
  1270
     "complex_of_real x + ii * complex_of_real y = 0
paulson@14323
  1271
      ==> y = 0"
paulson@14323
  1272
apply (unfold complex_of_real_def i_def complex_zero_def)
paulson@14323
  1273
apply (auto simp add: complex_mult complex_add)
paulson@14323
  1274
done
paulson@14323
  1275
paulson@14323
  1276
lemma Re_sgn:
paulson@14323
  1277
      "Re(sgn z) = Re(z)/cmod z"
paulson@14323
  1278
apply (unfold sgn_def complex_divide_def)
paulson@14334
  1279
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1280
apply (auto simp add: complex_of_real_inverse [symmetric])
paulson@14323
  1281
apply (auto simp add: complex_of_real_def complex_mult real_divide_def)
paulson@14323
  1282
done
paulson@14323
  1283
declare Re_sgn [simp]
paulson@14323
  1284
paulson@14323
  1285
lemma Im_sgn:
paulson@14323
  1286
      "Im(sgn z) = Im(z)/cmod z"
paulson@14323
  1287
apply (unfold sgn_def complex_divide_def)
paulson@14334
  1288
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1289
apply (auto simp add: complex_of_real_inverse [symmetric])
paulson@14323
  1290
apply (auto simp add: complex_of_real_def complex_mult real_divide_def)
paulson@14323
  1291
done
paulson@14323
  1292
declare Im_sgn [simp]
paulson@14323
  1293
paulson@14323
  1294
lemma complex_inverse_complex_split:
paulson@14323
  1295
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
  1296
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
  1297
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
paulson@14323
  1298
apply (unfold complex_of_real_def i_def)
paulson@14323
  1299
apply (auto simp add: complex_mult complex_add complex_diff_def complex_minus complex_inverse real_divide_def)
paulson@14323
  1300
done
paulson@14323
  1301
paulson@14323
  1302
(*----------------------------------------------------------------------------*)
paulson@14323
  1303
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
  1304
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
  1305
(*----------------------------------------------------------------------------*)
paulson@14323
  1306
paulson@14323
  1307
lemma Re_mult_i_eq:
paulson@14323
  1308
    "Re (ii * complex_of_real y) = 0"
paulson@14323
  1309
apply (unfold i_def complex_of_real_def)
paulson@14323
  1310
apply (auto simp add: complex_mult)
paulson@14323
  1311
done
paulson@14323
  1312
declare Re_mult_i_eq [simp]
paulson@14323
  1313
paulson@14323
  1314
lemma Im_mult_i_eq:
paulson@14323
  1315
    "Im (ii * complex_of_real y) = y"
paulson@14323
  1316
apply (unfold i_def complex_of_real_def)
paulson@14323
  1317
apply (auto simp add: complex_mult)
paulson@14323
  1318
done
paulson@14323
  1319
declare Im_mult_i_eq [simp]
paulson@14323
  1320
paulson@14323
  1321
lemma complex_mod_mult_i:
paulson@14323
  1322
    "cmod (ii * complex_of_real y) = abs y"
paulson@14323
  1323
apply (unfold i_def complex_of_real_def)
paulson@14323
  1324
apply (auto simp add: complex_mult complex_mod real_power_two)
paulson@14323
  1325
done
paulson@14323
  1326
declare complex_mod_mult_i [simp]
paulson@14323
  1327
paulson@14323
  1328
lemma cos_arg_i_mult_zero:
paulson@14323
  1329
   "0 < y ==> cos (arg(ii * complex_of_real y)) = 0"
paulson@14323
  1330
apply (unfold arg_def)
paulson@14323
  1331
apply (auto simp add: abs_eqI2)
paulson@14334
  1332
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
  1333
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
  1334
done
paulson@14323
  1335
declare cos_arg_i_mult_zero [simp]
paulson@14323
  1336
paulson@14323
  1337
lemma cos_arg_i_mult_zero2:
paulson@14323
  1338
   "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0"
paulson@14323
  1339
apply (unfold arg_def)
paulson@14323
  1340
apply (auto simp add: abs_minus_eqI2)
paulson@14334
  1341
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
  1342
apply (rule order_trans [of _ 0], auto)
paulson@14323
  1343
done
paulson@14323
  1344
declare cos_arg_i_mult_zero2 [simp]
paulson@14323
  1345
paulson@14323
  1346
lemma complex_of_real_not_zero_iff:
paulson@14323
  1347
      "(complex_of_real y ~= 0) = (y ~= 0)"
paulson@14334
  1348
apply (unfold complex_zero_def complex_of_real_def, auto)
paulson@14323
  1349
done
paulson@14323
  1350
declare complex_of_real_not_zero_iff [simp]
paulson@14323
  1351
paulson@14323
  1352
lemma complex_of_real_zero_iff: "(complex_of_real y = 0) = (y = 0)"
paulson@14323
  1353
apply auto
paulson@14334
  1354
apply (rule ccontr, drule complex_of_real_not_zero_iff [THEN iffD2], simp)
paulson@14323
  1355
done
paulson@14323
  1356
declare complex_of_real_zero_iff [simp]
paulson@14323
  1357
paulson@14323
  1358
lemma cos_arg_i_mult_zero3: "y ~= 0 ==> cos (arg(ii * complex_of_real y)) = 0"
paulson@14334
  1359
by (cut_tac x = y and y = 0 in linorder_less_linear, auto)
paulson@14323
  1360
declare cos_arg_i_mult_zero3 [simp]
paulson@14323
  1361
paulson@14323
  1362
paulson@14323
  1363
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
  1364
paulson@14323
  1365
lemma complex_split_polar: "EX r a. z = complex_of_real r *
paulson@14323
  1366
      (complex_of_real(cos a) + ii * complex_of_real(sin a))"
paulson@14334
  1367
apply (cut_tac z = z in complex_split)
paulson@14323
  1368
apply (auto simp add: polar_Ex complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
paulson@14323
  1369
done
paulson@14323
  1370
paulson@14323
  1371
lemma rcis_Ex: "EX r a. z = rcis r a"
paulson@14323
  1372
apply (unfold rcis_def cis_def)
paulson@14323
  1373
apply (rule complex_split_polar)
paulson@14323
  1374
done
paulson@14323
  1375
paulson@14323
  1376
lemma Re_complex_polar: "Re(complex_of_real r *
paulson@14323
  1377
      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a"
paulson@14323
  1378
apply (auto simp add: complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
paulson@14323
  1379
done
paulson@14323
  1380
declare Re_complex_polar [simp]
paulson@14323
  1381
paulson@14323
  1382
lemma Re_rcis: "Re(rcis r a) = r * cos a"
paulson@14334
  1383
by (unfold rcis_def cis_def, auto)
paulson@14323
  1384
declare Re_rcis [simp]
paulson@14323
  1385
paulson@14323
  1386
lemma Im_complex_polar: "Im(complex_of_real r *
paulson@14323
  1387
      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * sin a"
paulson@14323
  1388
apply (auto simp add: complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac)
paulson@14323
  1389
done
paulson@14323
  1390
declare Im_complex_polar [simp]
paulson@14323
  1391
paulson@14323
  1392
lemma Im_rcis: "Im(rcis r a) = r * sin a"
paulson@14334
  1393
by (unfold rcis_def cis_def, auto)
paulson@14323
  1394
declare Im_rcis [simp]
paulson@14323
  1395
paulson@14323
  1396
lemma complex_mod_complex_polar: "cmod (complex_of_real r *
paulson@14323
  1397
      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = abs r"
paulson@14334
  1398
apply (auto simp add: complex_add_mult_distrib2 cmod_i complex_of_real_mult right_distrib [symmetric] realpow_mult complex_mult_ac mult_ac simp del: realpow_Suc)
paulson@14323
  1399
done
paulson@14323
  1400
declare complex_mod_complex_polar [simp]
paulson@14323
  1401
paulson@14323
  1402
lemma complex_mod_rcis: "cmod(rcis r a) = abs r"
paulson@14334
  1403
by (unfold rcis_def cis_def, auto)
paulson@14323
  1404
declare complex_mod_rcis [simp]
paulson@14323
  1405
paulson@14323
  1406
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
paulson@14323
  1407
apply (unfold cmod_def)
paulson@14323
  1408
apply (rule real_sqrt_eq_iff [THEN iffD2])
paulson@14323
  1409
apply (auto simp add: complex_mult_cnj)
paulson@14323
  1410
done
paulson@14323
  1411
paulson@14323
  1412
lemma complex_Re_cnj: "Re(cnj z) = Re z"
paulson@14334
  1413
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1414
apply (auto simp add: complex_cnj)
paulson@14323
  1415
done
paulson@14323
  1416
declare complex_Re_cnj [simp]
paulson@14323
  1417
paulson@14323
  1418
lemma complex_Im_cnj: "Im(cnj z) = - Im z"
paulson@14334
  1419
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1420
apply (auto simp add: complex_cnj)
paulson@14323
  1421
done
paulson@14323
  1422
declare complex_Im_cnj [simp]
paulson@14323
  1423
paulson@14323
  1424
lemma complex_In_mult_cnj_zero: "Im (z * cnj z) = 0"
paulson@14334
  1425
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1426
apply (auto simp add: complex_cnj complex_mult)
paulson@14323
  1427
done
paulson@14323
  1428
declare complex_In_mult_cnj_zero [simp]
paulson@14323
  1429
paulson@14323
  1430
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
paulson@14334
  1431
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
  1432
apply (rule_tac z = w in eq_Abs_complex)
paulson@14323
  1433
apply (auto simp add: complex_mult)
paulson@14323
  1434
done
paulson@14323
  1435
paulson@14323
  1436
lemma complex_Re_mult_complex_of_real: "Re (z * complex_of_real c) = Re(z) * c"
paulson@14323
  1437
apply (unfold complex_of_real_def)
paulson@14334
  1438
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1439
apply (auto simp add: complex_mult)
paulson@14323
  1440
done
paulson@14323
  1441
declare complex_Re_mult_complex_of_real [simp]
paulson@14323
  1442
paulson@14323
  1443
lemma complex_Im_mult_complex_of_real: "Im (z * complex_of_real c) = Im(z) * c"
paulson@14323
  1444
apply (unfold complex_of_real_def)
paulson@14334
  1445
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1446
apply (auto simp add: complex_mult)
paulson@14323
  1447
done
paulson@14323
  1448
declare complex_Im_mult_complex_of_real [simp]
paulson@14323
  1449
paulson@14323
  1450
lemma complex_Re_mult_complex_of_real2: "Re (complex_of_real c * z) = c * Re(z)"
paulson@14323
  1451
apply (unfold complex_of_real_def)
paulson@14334
  1452
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1453
apply (auto simp add: complex_mult)
paulson@14323
  1454
done
paulson@14323
  1455
declare complex_Re_mult_complex_of_real2 [simp]
paulson@14323
  1456
paulson@14323
  1457
lemma complex_Im_mult_complex_of_real2: "Im (complex_of_real c * z) = c * Im(z)"
paulson@14323
  1458
apply (unfold complex_of_real_def)
paulson@14334
  1459
apply (rule_tac z = z in eq_Abs_complex)
paulson@14323
  1460
apply (auto simp add: complex_mult)
paulson@14323
  1461
done
paulson@14323
  1462
declare complex_Im_mult_complex_of_real2 [simp]
paulson@14323
  1463
paulson@14323
  1464
(*---------------------------------------------------------------------------*)
paulson@14323
  1465
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
paulson@14323
  1466
(*---------------------------------------------------------------------------*)
paulson@14323
  1467
paulson@14323
  1468
lemma cis_rcis_eq: "cis a = rcis 1 a"
paulson@14323
  1469
apply (unfold rcis_def)
paulson@14323
  1470
apply (simp (no_asm))
paulson@14323
  1471
done
paulson@14323
  1472
paulson@14323
  1473
lemma rcis_mult:
paulson@14323
  1474
  "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
paulson@14323
  1475
apply (unfold rcis_def cis_def)
paulson@14323
  1476
apply (auto simp add: cos_add sin_add complex_add_mult_distrib2 complex_add_mult_distrib complex_mult_ac complex_add_ac)
paulson@14323
  1477
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
  1478
apply (auto simp add: complex_add_ac)
paulson@14334
  1479
apply (auto simp add: complex_add_assoc [symmetric] complex_of_real_add right_distrib real_diff_def mult_ac add_ac)
paulson@14323
  1480
done
paulson@14323
  1481
paulson@14323
  1482
lemma cis_mult: "cis a * cis b = cis (a + b)"
paulson@14323
  1483
apply (simp (no_asm) add: cis_rcis_eq rcis_mult)
paulson@14323
  1484
done
paulson@14323
  1485
paulson@14323
  1486
lemma cis_zero: "cis 0 = 1"
paulson@14334
  1487
by (unfold cis_def, auto)
paulson@14323
  1488
declare cis_zero [simp]
paulson@14323
  1489
paulson@14323
  1490
lemma cis_zero2: "cis 0 = complex_of_real 1"
paulson@14334
  1491
by (unfold cis_def, auto)
paulson@14323
  1492
declare cis_zero2 [simp]
paulson@14323
  1493
paulson@14323
  1494
lemma rcis_zero_mod: "rcis 0 a = 0"
paulson@14323
  1495
apply (unfold rcis_def)
paulson@14323
  1496
apply (simp (no_asm))
paulson@14323
  1497
done
paulson@14323
  1498
declare rcis_zero_mod [simp]
paulson@14323
  1499
paulson@14323
  1500
lemma rcis_zero_arg: "rcis r 0 = complex_of_real r"
paulson@14323
  1501
apply (unfold rcis_def)
paulson@14323
  1502
apply (simp (no_asm))
paulson@14323
  1503
done
paulson@14323
  1504
declare rcis_zero_arg [simp]
paulson@14323
  1505
paulson@14323
  1506
lemma complex_of_real_minus_one:
paulson@14323
  1507
   "complex_of_real (-(1::real)) = -(1::complex)"
paulson@14323
  1508
apply (unfold complex_of_real_def complex_one_def)
paulson@14323
  1509
apply (simp (no_asm) add: complex_minus)
paulson@14323
  1510
done
paulson@14323
  1511
paulson@14323
  1512
lemma complex_i_mult_minus: "ii * (ii * x) = - x"
paulson@14323
  1513
apply (simp (no_asm) add: complex_mult_assoc [symmetric])
paulson@14323
  1514
done
paulson@14323
  1515
declare complex_i_mult_minus [simp]
paulson@14323
  1516
paulson@14323
  1517
lemma complex_i_mult_minus2: "ii * ii * x = - x"
paulson@14323
  1518
apply (simp (no_asm))
paulson@14323
  1519
done
paulson@14323
  1520
declare complex_i_mult_minus2 [simp]
paulson@14323
  1521
paulson@14323
  1522
lemma cis_real_of_nat_Suc_mult:
paulson@14323
  1523
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
paulson@14323
  1524
apply (unfold cis_def)
paulson@14334
  1525
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
  1526
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
  1527
done
paulson@14323
  1528
paulson@14323
  1529
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
paulson@14323
  1530
apply (induct_tac "n")
paulson@14323
  1531
apply (auto simp add: cis_real_of_nat_Suc_mult)
paulson@14323
  1532
done
paulson@14323
  1533
paulson@14323
  1534
lemma DeMoivre2:
paulson@14323
  1535
   "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
paulson@14323
  1536
apply (unfold rcis_def)
paulson@14323
  1537
apply (auto simp add: complexpow_mult DeMoivre complex_of_real_pow)
paulson@14323
  1538
done
paulson@14323
  1539
paulson@14323
  1540
lemma cis_inverse: "inverse(cis a) = cis (-a)"
paulson@14323
  1541
apply (unfold cis_def)
paulson@14323
  1542
apply (auto simp add: complex_inverse_complex_split complex_of_real_minus complex_diff_def)
paulson@14323
  1543
done
paulson@14323
  1544
declare cis_inverse [simp]
paulson@14323
  1545
paulson@14323
  1546
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
paulson@14323
  1547
apply (case_tac "r=0")
paulson@14323
  1548
apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
paulson@14334
  1549
apply (auto simp add: complex_inverse_complex_split complex_add_mult_distrib2 complex_of_real_mult rcis_def cis_def real_power_two complex_mult_ac mult_ac)
paulson@14334
  1550
apply (auto simp add: right_distrib [symmetric] complex_of_real_minus complex_diff_def)
paulson@14323
  1551
done
paulson@14323
  1552
paulson@14323
  1553
lemma cis_divide: "cis a / cis b = cis (a - b)"
paulson@14323
  1554
apply (unfold complex_divide_def)
paulson@14323
  1555
apply (auto simp add: cis_mult real_diff_def)
paulson@14323
  1556
done
paulson@14323
  1557
paulson@14323
  1558
lemma rcis_divide:
paulson@14323
  1559
 "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
paulson@14323
  1560
apply (unfold complex_divide_def)
paulson@14323
  1561
apply (case_tac "r2=0")
paulson@14323
  1562
apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO)
paulson@14323
  1563
apply (auto simp add: rcis_inverse rcis_mult real_diff_def)
paulson@14323
  1564
done
paulson@14323
  1565
paulson@14323
  1566
lemma Re_cis: "Re(cis a) = cos a"
paulson@14334
  1567
by (unfold cis_def, auto)
paulson@14323
  1568
declare Re_cis [simp]
paulson@14323
  1569
paulson@14323
  1570
lemma Im_cis: "Im(cis a) = sin a"
paulson@14334
  1571
by (unfold cis_def, auto)
paulson@14323
  1572
declare Im_cis [simp]
paulson@14323
  1573
paulson@14323
  1574
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
paulson@14334
  1575
by (auto simp add: DeMoivre)
paulson@14323
  1576
paulson@14323
  1577
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
paulson@14334
  1578
by (auto simp add: DeMoivre)
paulson@14323
  1579
paulson@14323
  1580
lemma expi_Im_split:
paulson@14323
  1581
    "expi (ii * complex_of_real y) =
paulson@14323
  1582
     complex_of_real (cos y) + ii * complex_of_real (sin y)"
paulson@14334
  1583
apply (unfold expi_def cis_def, auto)
paulson@14323
  1584
done
paulson@14323
  1585
paulson@14323
  1586
lemma expi_Im_cis:
paulson@14323
  1587
    "expi (ii * complex_of_real y) = cis y"
paulson@14334
  1588
apply (unfold expi_def, auto)
paulson@14323
  1589
done
paulson@14323
  1590
paulson@14323
  1591
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
paulson@14323
  1592
apply (unfold expi_def)
paulson@14323
  1593
apply (auto simp add: complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult complex_mult_ac)
paulson@14323
  1594
done
paulson@14323
  1595
paulson@14323
  1596
lemma expi_complex_split:
paulson@14323
  1597
     "expi(complex_of_real x + ii * complex_of_real y) =
paulson@14323
  1598
      complex_of_real (exp(x)) * cis y"
paulson@14334
  1599
apply (unfold expi_def, auto)
paulson@14323
  1600
done
paulson@14323
  1601
paulson@14323
  1602
lemma expi_zero: "expi (0::complex) = 1"
paulson@14334
  1603
by (unfold expi_def, auto)
paulson@14323
  1604
declare expi_zero [simp]
paulson@14323
  1605
paulson@14323
  1606
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
paulson@14334
  1607
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
  1608
apply (rule_tac z = w in eq_Abs_complex)
paulson@14323
  1609
apply (auto simp add: complex_mult)
paulson@14323
  1610
done
paulson@14323
  1611
paulson@14323
  1612
lemma complex_Im_mult_eq:
paulson@14323
  1613
     "Im (w * z) = Re w * Im z + Im w * Re z"
paulson@14334
  1614
apply (rule_tac z = z in eq_Abs_complex)
paulson@14334
  1615
apply (rule_tac z = w in eq_Abs_complex)
paulson@14323
  1616
apply (auto simp add: complex_mult)
paulson@14323
  1617
done
paulson@14323
  1618
paulson@14323
  1619
lemma complex_expi_Ex: 
paulson@14323
  1620
   "EX a r. z = complex_of_real r * expi a"
paulson@14334
  1621
apply (cut_tac z = z in rcis_Ex)
paulson@14323
  1622
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult)
paulson@14334
  1623
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
  1624
done
paulson@14323
  1625
paulson@14323
  1626
paulson@14323
  1627
(****
paulson@14323
  1628
Goal "[| - pi < a; a <= pi |] ==> (-pi < a & a <= 0) | (0 <= a & a <= pi)"
paulson@14334
  1629
by Auto_tac
paulson@14323
  1630
qed "lemma_split_interval";
paulson@14323
  1631
paulson@14323
  1632
Goalw [arg_def]
paulson@14323
  1633
  "[| r ~= 0; - pi < a; a <= pi |] \
paulson@14323
  1634
\  ==> arg(complex_of_real r * \
paulson@14323
  1635
\      (complex_of_real(cos a) + ii * complex_of_real(sin a))) = a";
paulson@14334
  1636
by Auto_tac
paulson@14323
  1637
by (cut_inst_tac [("x","0"),("y","r")] linorder_less_linear 1);
paulson@14323
  1638
by (auto_tac (claset(),simpset() addsimps (map (full_rename_numerals thy)
paulson@14323
  1639
    [rabs_eqI2,rabs_minus_eqI2,real_minus_rinv]) [real_divide_def,
paulson@14334
  1640
    minus_mult_right RS sym] mult_ac));
paulson@14323
  1641
by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
paulson@14334
  1642
by (dtac lemma_split_interval 1 THEN safe)
paulson@14323
  1643
****)
paulson@14323
  1644
paulson@14323
  1645
paulson@14323
  1646
ML
paulson@14323
  1647
{*
paulson@14323
  1648
val complex_zero_def = thm"complex_zero_def";
paulson@14323
  1649
val complex_one_def = thm"complex_one_def";
paulson@14323
  1650
val complex_minus_def = thm"complex_minus_def";
paulson@14323
  1651
val complex_diff_def = thm"complex_diff_def";
paulson@14323
  1652
val complex_divide_def = thm"complex_divide_def";
paulson@14323
  1653
val complex_mult_def = thm"complex_mult_def";
paulson@14323
  1654
val complex_add_def = thm"complex_add_def";
paulson@14323
  1655
val complex_of_real_def = thm"complex_of_real_def";
paulson@14323
  1656
val i_def = thm"i_def";
paulson@14323
  1657
val expi_def = thm"expi_def";
paulson@14323
  1658
val cis_def = thm"cis_def";
paulson@14323
  1659
val rcis_def = thm"rcis_def";
paulson@14323
  1660
val cmod_def = thm"cmod_def";
paulson@14323
  1661
val cnj_def = thm"cnj_def";
paulson@14323
  1662
val sgn_def = thm"sgn_def";
paulson@14323
  1663
val arg_def = thm"arg_def";
paulson@14323
  1664
val complexpow_0 = thm"complexpow_0";
paulson@14323
  1665
val complexpow_Suc = thm"complexpow_Suc";
paulson@14323
  1666
paulson@14323
  1667
val inj_Rep_complex = thm"inj_Rep_complex";
paulson@14323
  1668
val inj_Abs_complex = thm"inj_Abs_complex";
paulson@14323
  1669
val Abs_complex_cancel_iff = thm"Abs_complex_cancel_iff";
paulson@14323
  1670
val pair_mem_complex = thm"pair_mem_complex";
paulson@14323
  1671
val Abs_complex_inverse2 = thm"Abs_complex_inverse2";
paulson@14323
  1672
val eq_Abs_complex = thm"eq_Abs_complex";
paulson@14323
  1673
val Re = thm"Re";
paulson@14323
  1674
val Im = thm"Im";
paulson@14323
  1675
val Abs_complex_cancel = thm"Abs_complex_cancel";
paulson@14323
  1676
val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff";
paulson@14323
  1677
val complex_Re_zero = thm"complex_Re_zero";
paulson@14323
  1678
val complex_Im_zero = thm"complex_Im_zero";
paulson@14323
  1679
val complex_Re_one = thm"complex_Re_one";
paulson@14323
  1680
val complex_Im_one = thm"complex_Im_one";
paulson@14323
  1681
val complex_Re_i = thm"complex_Re_i";
paulson@14323
  1682
val complex_Im_i = thm"complex_Im_i";
paulson@14323
  1683
val Re_complex_of_real_zero = thm"Re_complex_of_real_zero";
paulson@14323
  1684
val Im_complex_of_real_zero = thm"Im_complex_of_real_zero";
paulson@14323
  1685
val Re_complex_of_real_one = thm"Re_complex_of_real_one";
paulson@14323
  1686
val Im_complex_of_real_one = thm"Im_complex_of_real_one";
paulson@14323
  1687
val Re_complex_of_real = thm"Re_complex_of_real";
paulson@14323
  1688
val Im_complex_of_real = thm"Im_complex_of_real";
paulson@14323
  1689
val complex_minus = thm"complex_minus";
paulson@14323
  1690
val complex_Re_minus = thm"complex_Re_minus";
paulson@14323
  1691
val complex_Im_minus = thm"complex_Im_minus";
paulson@14323
  1692
val complex_minus_minus = thm"complex_minus_minus";
paulson@14323
  1693
val inj_complex_minus = thm"inj_complex_minus";
paulson@14323
  1694
val complex_minus_zero = thm"complex_minus_zero";
paulson@14323
  1695
val complex_minus_zero_iff = thm"complex_minus_zero_iff";
paulson@14323
  1696
val complex_minus_zero_iff2 = thm"complex_minus_zero_iff2";
paulson@14323
  1697
val complex_minus_not_zero_iff = thm"complex_minus_not_zero_iff";
paulson@14323
  1698
val complex_add = thm"complex_add";
paulson@14323
  1699
val complex_Re_add = thm"complex_Re_add";
paulson@14323
  1700
val complex_Im_add = thm"complex_Im_add";
paulson@14323
  1701
val complex_add_commute = thm"complex_add_commute";
paulson@14323
  1702
val complex_add_assoc = thm"complex_add_assoc";
paulson@14323
  1703
val complex_add_left_commute = thm"complex_add_left_commute";
paulson@14323
  1704
val complex_add_zero_left = thm"complex_add_zero_left";
paulson@14323
  1705
val complex_add_zero_right = thm"complex_add_zero_right";
paulson@14323
  1706
val complex_add_minus_right_zero = thm"complex_add_minus_right_zero";
paulson@14323
  1707
val complex_add_minus_cancel = thm"complex_add_minus_cancel";
paulson@14323
  1708
val complex_minus_add_cancel = thm"complex_minus_add_cancel";
paulson@14323
  1709
val complex_add_minus_eq_minus = thm"complex_add_minus_eq_minus";
paulson@14323
  1710
val complex_minus_add_distrib = thm"complex_minus_add_distrib";
paulson@14323
  1711
val complex_add_left_cancel = thm"complex_add_left_cancel";
paulson@14323
  1712
val complex_add_right_cancel = thm"complex_add_right_cancel";
paulson@14323
  1713
val complex_eq_minus_iff = thm"complex_eq_minus_iff";
paulson@14323
  1714
val complex_eq_minus_iff2 = thm"complex_eq_minus_iff2";
paulson@14323
  1715
val complex_diff_0 = thm"complex_diff_0";
paulson@14323
  1716
val complex_diff_0_right = thm"complex_diff_0_right";
paulson@14323
  1717
val complex_diff_self = thm"complex_diff_self";
paulson@14323
  1718
val complex_diff = thm"complex_diff";
paulson@14323
  1719
val complex_diff_eq_eq = thm"complex_diff_eq_eq";
paulson@14323
  1720
val complex_mult = thm"complex_mult";
paulson@14323
  1721
val complex_mult_commute = thm"complex_mult_commute";
paulson@14323
  1722
val complex_mult_assoc = thm"complex_mult_assoc";
paulson@14323
  1723
val complex_mult_left_commute = thm"complex_mult_left_commute";
paulson@14323
  1724
val complex_mult_one_left = thm"complex_mult_one_left";
paulson@14323
  1725
val complex_mult_one_right = thm"complex_mult_one_right";
paulson@14323
  1726
val complex_mult_zero_left = thm"complex_mult_zero_left";
paulson@14323
  1727
val complex_mult_zero_right = thm"complex_mult_zero_right";
paulson@14323
  1728
val complex_divide_zero = thm"complex_divide_zero";
paulson@14323
  1729
val complex_minus_mult_eq1 = thm"complex_minus_mult_eq1";
paulson@14323
  1730
val complex_minus_mult_eq2 = thm"complex_minus_mult_eq2";
paulson@14323
  1731
val complex_mult_minus_one = thm"complex_mult_minus_one";
paulson@14323
  1732
val complex_mult_minus_one_right = thm"complex_mult_minus_one_right";
paulson@14323
  1733
val complex_minus_mult_cancel = thm"complex_minus_mult_cancel";
paulson@14323
  1734
val complex_minus_mult_commute = thm"complex_minus_mult_commute";
paulson@14323
  1735
val complex_add_mult_distrib = thm"complex_add_mult_distrib";
paulson@14323
  1736
val complex_add_mult_distrib2 = thm"complex_add_mult_distrib2";
paulson@14323
  1737
val complex_zero_not_eq_one = thm"complex_zero_not_eq_one";
paulson@14323
  1738
val complex_inverse = thm"complex_inverse";
paulson@14323
  1739
val COMPLEX_INVERSE_ZERO = thm"COMPLEX_INVERSE_ZERO";
paulson@14323
  1740
val COMPLEX_DIVISION_BY_ZERO = thm"COMPLEX_DIVISION_BY_ZERO";
paulson@14323
  1741
val complex_mult_inv_left = thm"complex_mult_inv_left";
paulson@14323
  1742
val complex_mult_inv_right = thm"complex_mult_inv_right";
paulson@14323
  1743
val complex_mult_left_cancel = thm"complex_mult_left_cancel";
paulson@14323
  1744
val complex_mult_right_cancel = thm"complex_mult_right_cancel";
paulson@14323
  1745
val complex_inverse_not_zero = thm"complex_inverse_not_zero";
paulson@14323
  1746
val complex_mult_not_zero = thm"complex_mult_not_zero";
paulson@14323
  1747
val complex_inverse_inverse = thm"complex_inverse_inverse";
paulson@14323
  1748
val complex_inverse_one = thm"complex_inverse_one";
paulson@14323
  1749
val complex_minus_inverse = thm"complex_minus_inverse";
paulson@14323
  1750
val complex_inverse_distrib = thm"complex_inverse_distrib";
paulson@14323
  1751
val complex_times_divide1_eq = thm"complex_times_divide1_eq";
paulson@14323
  1752
val complex_times_divide2_eq = thm"complex_times_divide2_eq";
paulson@14323
  1753
val complex_divide_divide1_eq = thm"complex_divide_divide1_eq";
paulson@14323
  1754
val complex_divide_divide2_eq = thm"complex_divide_divide2_eq";
paulson@14323
  1755
val complex_minus_divide_eq = thm"complex_minus_divide_eq";
paulson@14323
  1756
val complex_divide_minus_eq = thm"complex_divide_minus_eq";
paulson@14323
  1757
val complex_add_divide_distrib = thm"complex_add_divide_distrib";
paulson@14323
  1758
val inj_complex_of_real = thm"inj_complex_of_real";
paulson@14323
  1759
val complex_of_real_one = thm"complex_of_real_one";
paulson@14323
  1760
val complex_of_real_zero = thm"complex_of_real_zero";
paulson@14323
  1761
val complex_of_real_eq_iff = thm"complex_of_real_eq_iff";
paulson@14323
  1762
val complex_of_real_minus = thm"complex_of_real_minus";
paulson@14323
  1763
val complex_of_real_inverse = thm"complex_of_real_inverse";
paulson@14323
  1764
val complex_of_real_add = thm"complex_of_real_add";
paulson@14323
  1765
val complex_of_real_diff = thm"complex_of_real_diff";
paulson@14323
  1766
val complex_of_real_mult = thm"complex_of_real_mult";
paulson@14323
  1767
val complex_of_real_divide = thm"complex_of_real_divide";
paulson@14323
  1768
val complex_of_real_pow = thm"complex_of_real_pow";
paulson@14323
  1769
val complex_mod = thm"complex_mod";
paulson@14323
  1770
val complex_mod_zero = thm"complex_mod_zero";
paulson@14323
  1771
val complex_mod_one = thm"complex_mod_one";
paulson@14323
  1772
val complex_mod_complex_of_real = thm"complex_mod_complex_of_real";
paulson@14323
  1773
val complex_of_real_abs = thm"complex_of_real_abs";
paulson@14323
  1774
val complex_cnj = thm"complex_cnj";
paulson@14323
  1775
val inj_cnj = thm"inj_cnj";
paulson@14323
  1776
val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff";
paulson@14323
  1777
val complex_cnj_cnj = thm"complex_cnj_cnj";
paulson@14323
  1778
val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real";
paulson@14323
  1779
val complex_mod_cnj = thm"complex_mod_cnj";
paulson@14323
  1780
val complex_cnj_minus = thm"complex_cnj_minus";
paulson@14323
  1781
val complex_cnj_inverse = thm"complex_cnj_inverse";
paulson@14323
  1782
val complex_cnj_add = thm"complex_cnj_add";
paulson@14323
  1783
val complex_cnj_diff = thm"complex_cnj_diff";
paulson@14323
  1784
val complex_cnj_mult = thm"complex_cnj_mult";
paulson@14323
  1785
val complex_cnj_divide = thm"complex_cnj_divide";
paulson@14323
  1786
val complex_cnj_one = thm"complex_cnj_one";
paulson@14323
  1787
val complex_cnj_pow = thm"complex_cnj_pow";
paulson@14323
  1788
val complex_add_cnj = thm"complex_add_cnj";
paulson@14323
  1789
val complex_diff_cnj = thm"complex_diff_cnj";
paulson@14323
  1790
val complex_cnj_zero = thm"complex_cnj_zero";
paulson@14323
  1791
val complex_cnj_zero_iff = thm"complex_cnj_zero_iff";
paulson@14323
  1792
val complex_mult_cnj = thm"complex_mult_cnj";
paulson@14323
  1793
val complex_mult_zero_iff = thm"complex_mult_zero_iff";
paulson@14323
  1794
val complex_add_left_cancel_zero = thm"complex_add_left_cancel_zero";
paulson@14323
  1795
val complex_diff_mult_distrib = thm"complex_diff_mult_distrib";
paulson@14323
  1796
val complex_diff_mult_distrib2 = thm"complex_diff_mult_distrib2";
paulson@14323
  1797
val complex_mod_eq_zero_cancel = thm"complex_mod_eq_zero_cancel";
paulson@14323
  1798
val complex_mod_complex_of_real_of_nat = thm"complex_mod_complex_of_real_of_nat";
paulson@14323
  1799
val complex_mod_minus = thm"complex_mod_minus";
paulson@14323
  1800
val complex_mod_mult_cnj = thm"complex_mod_mult_cnj";
paulson@14323
  1801
val complex_mod_squared = thm"complex_mod_squared";
paulson@14323
  1802
val complex_mod_ge_zero = thm"complex_mod_ge_zero";
paulson@14323
  1803
val abs_cmod_cancel = thm"abs_cmod_cancel";
paulson@14323
  1804
val complex_mod_mult = thm"complex_mod_mult";
paulson@14323
  1805
val complex_mod_add_squared_eq = thm"complex_mod_add_squared_eq";
paulson@14323
  1806
val complex_Re_mult_cnj_le_cmod = thm"complex_Re_mult_cnj_le_cmod";
paulson@14323
  1807
val complex_Re_mult_cnj_le_cmod2 = thm"complex_Re_mult_cnj_le_cmod2";
paulson@14323
  1808
val real_sum_squared_expand = thm"real_sum_squared_expand";
paulson@14323
  1809
val complex_mod_triangle_squared = thm"complex_mod_triangle_squared";
paulson@14323
  1810
val complex_mod_minus_le_complex_mod = thm"complex_mod_minus_le_complex_mod";
paulson@14323
  1811
val complex_mod_triangle_ineq = thm"complex_mod_triangle_ineq";
paulson@14323
  1812
val complex_mod_triangle_ineq2 = thm"complex_mod_triangle_ineq2";
paulson@14323
  1813
val complex_mod_diff_commute = thm"complex_mod_diff_commute";
paulson@14323
  1814
val complex_mod_add_less = thm"complex_mod_add_less";
paulson@14323
  1815
val complex_mod_mult_less = thm"complex_mod_mult_less";
paulson@14323
  1816
val complex_mod_diff_ineq = thm"complex_mod_diff_ineq";
paulson@14323
  1817
val complex_Re_le_cmod = thm"complex_Re_le_cmod";
paulson@14323
  1818
val complex_mod_gt_zero = thm"complex_mod_gt_zero";
paulson@14323
  1819
val complex_mod_complexpow = thm"complex_mod_complexpow";
paulson@14323
  1820
val complexpow_minus = thm"complexpow_minus";
paulson@14323
  1821
val complex_inverse_minus = thm"complex_inverse_minus";
paulson@14323
  1822
val complex_divide_one = thm"complex_divide_one";
paulson@14323
  1823
val complex_mod_inverse = thm"complex_mod_inverse";
paulson@14323
  1824
val complex_mod_divide = thm"complex_mod_divide";
paulson@14323
  1825
val complex_inverse_divide = thm"complex_inverse_divide";
paulson@14323
  1826
val complexpow_mult = thm"complexpow_mult";
paulson@14323
  1827
val complexpow_zero = thm"complexpow_zero";
paulson@14323
  1828
val complexpow_not_zero = thm"complexpow_not_zero";
paulson@14323
  1829
val complexpow_zero_zero = thm"complexpow_zero_zero";
paulson@14323
  1830
val complexpow_i_squared = thm"complexpow_i_squared";
paulson@14323
  1831
val complex_i_not_zero = thm"complex_i_not_zero";
paulson@14323
  1832
val complex_mult_eq_zero_cancel1 = thm"complex_mult_eq_zero_cancel1";
paulson@14323
  1833
val complex_mult_eq_zero_cancel2 = thm"complex_mult_eq_zero_cancel2";
paulson@14323
  1834
val complex_mult_not_eq_zero_iff = thm"complex_mult_not_eq_zero_iff";
paulson@14323
  1835
val complexpow_inverse = thm"complexpow_inverse";
paulson@14323
  1836
val sgn_zero = thm"sgn_zero";
paulson@14323
  1837
val sgn_one = thm"sgn_one";
paulson@14323
  1838
val sgn_minus = thm"sgn_minus";
paulson@14323
  1839
val sgn_eq = thm"sgn_eq";
paulson@14323
  1840
val complex_split = thm"complex_split";
paulson@14323
  1841
val Re_complex_i = thm"Re_complex_i";
paulson@14323
  1842
val Im_complex_i = thm"Im_complex_i";
paulson@14323
  1843
val i_mult_eq = thm"i_mult_eq";
paulson@14323
  1844
val i_mult_eq2 = thm"i_mult_eq2";
paulson@14323
  1845
val cmod_i = thm"cmod_i";
paulson@14323
  1846
val complex_eq_Re_eq = thm"complex_eq_Re_eq";
paulson@14323
  1847
val complex_eq_Im_eq = thm"complex_eq_Im_eq";
paulson@14323
  1848
val complex_eq_cancel_iff = thm"complex_eq_cancel_iff";
paulson@14323
  1849
val complex_eq_cancel_iffA = thm"complex_eq_cancel_iffA";
paulson@14323
  1850
val complex_eq_cancel_iffB = thm"complex_eq_cancel_iffB";
paulson@14323
  1851
val complex_eq_cancel_iffC = thm"complex_eq_cancel_iffC";
paulson@14323
  1852
val complex_eq_cancel_iff2 = thm"complex_eq_cancel_iff2";
paulson@14323
  1853
val complex_eq_cancel_iff2a = thm"complex_eq_cancel_iff2a";
paulson@14323
  1854
val complex_eq_cancel_iff3 = thm"complex_eq_cancel_iff3";
paulson@14323
  1855
val complex_eq_cancel_iff3a = thm"complex_eq_cancel_iff3a";
paulson@14323
  1856
val complex_split_Re_zero = thm"complex_split_Re_zero";
paulson@14323
  1857
val complex_split_Im_zero = thm"complex_split_Im_zero";
paulson@14323
  1858
val Re_sgn = thm"Re_sgn";
paulson@14323
  1859
val Im_sgn = thm"Im_sgn";
paulson@14323
  1860
val complex_inverse_complex_split = thm"complex_inverse_complex_split";
paulson@14323
  1861
val Re_mult_i_eq = thm"Re_mult_i_eq";
paulson@14323
  1862
val Im_mult_i_eq = thm"Im_mult_i_eq";
paulson@14323
  1863
val complex_mod_mult_i = thm"complex_mod_mult_i";
paulson@14323
  1864
val cos_arg_i_mult_zero = thm"cos_arg_i_mult_zero";
paulson@14323
  1865
val cos_arg_i_mult_zero2 = thm"cos_arg_i_mult_zero2";
paulson@14323
  1866
val complex_of_real_not_zero_iff = thm"complex_of_real_not_zero_iff";
paulson@14323
  1867
val complex_of_real_zero_iff = thm"complex_of_real_zero_iff";
paulson@14323
  1868
val cos_arg_i_mult_zero3 = thm"cos_arg_i_mult_zero3";
paulson@14323
  1869
val complex_split_polar = thm"complex_split_polar";
paulson@14323
  1870
val rcis_Ex = thm"rcis_Ex";
paulson@14323
  1871
val Re_complex_polar = thm"Re_complex_polar";
paulson@14323
  1872
val Re_rcis = thm"Re_rcis";
paulson@14323
  1873
val Im_complex_polar = thm"Im_complex_polar";
paulson@14323
  1874
val Im_rcis = thm"Im_rcis";
paulson@14323
  1875
val complex_mod_complex_polar = thm"complex_mod_complex_polar";
paulson@14323
  1876
val complex_mod_rcis = thm"complex_mod_rcis";
paulson@14323
  1877
val complex_mod_sqrt_Re_mult_cnj = thm"complex_mod_sqrt_Re_mult_cnj";
paulson@14323
  1878
val complex_Re_cnj = thm"complex_Re_cnj";
paulson@14323
  1879
val complex_Im_cnj = thm"complex_Im_cnj";
paulson@14323
  1880
val complex_In_mult_cnj_zero = thm"complex_In_mult_cnj_zero";
paulson@14323
  1881
val complex_Re_mult = thm"complex_Re_mult";
paulson@14323
  1882
val complex_Re_mult_complex_of_real = thm"complex_Re_mult_complex_of_real";
paulson@14323
  1883
val complex_Im_mult_complex_of_real = thm"complex_Im_mult_complex_of_real";
paulson@14323
  1884
val complex_Re_mult_complex_of_real2 = thm"complex_Re_mult_complex_of_real2";
paulson@14323
  1885
val complex_Im_mult_complex_of_real2 = thm"complex_Im_mult_complex_of_real2";
paulson@14323
  1886
val cis_rcis_eq = thm"cis_rcis_eq";
paulson@14323
  1887
val rcis_mult = thm"rcis_mult";
paulson@14323
  1888
val cis_mult = thm"cis_mult";
paulson@14323
  1889
val cis_zero = thm"cis_zero";
paulson@14323
  1890
val cis_zero2 = thm"cis_zero2";
paulson@14323
  1891
val rcis_zero_mod = thm"rcis_zero_mod";
paulson@14323
  1892
val rcis_zero_arg = thm"rcis_zero_arg";
paulson@14323
  1893
val complex_of_real_minus_one = thm"complex_of_real_minus_one";
paulson@14323
  1894
val complex_i_mult_minus = thm"complex_i_mult_minus";
paulson@14323
  1895
val complex_i_mult_minus2 = thm"complex_i_mult_minus2";
paulson@14323
  1896
val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult";
paulson@14323
  1897
val DeMoivre = thm"DeMoivre";
paulson@14323
  1898
val DeMoivre2 = thm"DeMoivre2";
paulson@14323
  1899
val cis_inverse = thm"cis_inverse";
paulson@14323
  1900
val rcis_inverse = thm"rcis_inverse";
paulson@14323
  1901
val cis_divide = thm"cis_divide";
paulson@14323
  1902
val rcis_divide = thm"rcis_divide";
paulson@14323
  1903
val Re_cis = thm"Re_cis";
paulson@14323
  1904
val Im_cis = thm"Im_cis";
paulson@14323
  1905
val cos_n_Re_cis_pow_n = thm"cos_n_Re_cis_pow_n";
paulson@14323
  1906
val sin_n_Im_cis_pow_n = thm"sin_n_Im_cis_pow_n";
paulson@14323
  1907
val expi_Im_split = thm"expi_Im_split";
paulson@14323
  1908
val expi_Im_cis = thm"expi_Im_cis";
paulson@14323
  1909
val expi_add = thm"expi_add";
paulson@14323
  1910
val expi_complex_split = thm"expi_complex_split";
paulson@14323
  1911
val expi_zero = thm"expi_zero";
paulson@14323
  1912
val complex_Re_mult_eq = thm"complex_Re_mult_eq";
paulson@14323
  1913
val complex_Im_mult_eq = thm"complex_Im_mult_eq";
paulson@14323
  1914
val complex_expi_Ex = thm"complex_expi_Ex";
paulson@14323
  1915
paulson@14323
  1916
val complex_add_ac = thms"complex_add_ac";
paulson@14323
  1917
val complex_mult_ac = thms"complex_mult_ac";
paulson@14323
  1918
*}
paulson@14323
  1919
paulson@13957
  1920
end
paulson@13957
  1921
paulson@13957
  1922