set "baseuri" "cic:/matita/CoRN-Decl/complex/NRootCC".
+include "CoRN.ma".
+
(* $Id: NRootCC.v,v 1.9 2004/04/23 10:00:55 lcf Exp $ *)
(*#* printing sqrt_Half %\ensuremath{\sqrt{\frac12}}% *)
(*#* printing nroot_minus_I %\ensuremath{\sqrt[n]{-\imath}}% *)
-(* INCLUDE
-CComplex
-*)
-
-(* INCLUDE
-Wf_nat
-*)
-
-(* INCLUDE
-ArithRing
-*)
+include "complex/CComplex.ma".
(*#* * Roots of Complex Numbers
Section CC_ap_zero.
*)
-inline cic:/CoRN/complex/NRootCC/cc_ap_zero.con.
+inline "cic:/CoRN/complex/NRootCC/cc_ap_zero.con".
-inline cic:/CoRN/complex/NRootCC/C_cc_ap_zero.con.
+inline "cic:/CoRN/complex/NRootCC/C_cc_ap_zero.con".
(* UNEXPORTED
End CC_ap_zero.
Section Imag_to_Real.
*)
-inline cic:/CoRN/complex/NRootCC/imag_to_real.con.
+inline "cic:/CoRN/complex/NRootCC/imag_to_real.con".
(* UNEXPORTED
End Imag_to_Real.
Section NRootI.
*)
-inline cic:/CoRN/complex/NRootCC/sqrt_Half.con.
+inline "cic:/CoRN/complex/NRootCC/sqrt_Half.con".
-inline cic:/CoRN/complex/NRootCC/sqrt_I.con.
+inline "cic:/CoRN/complex/NRootCC/sqrt_I.con".
-inline cic:/CoRN/complex/NRootCC/sqrt_I_nexp.con.
+inline "cic:/CoRN/complex/NRootCC/sqrt_I_nexp.con".
-inline cic:/CoRN/complex/NRootCC/nroot_I_nexp_aux.con.
+inline "cic:/CoRN/complex/NRootCC/nroot_I_nexp_aux.con".
-inline cic:/CoRN/complex/NRootCC/nroot_I.con.
+inline "cic:/CoRN/complex/NRootCC/nroot_I.con".
-inline cic:/CoRN/complex/NRootCC/nroot_I_nexp.con.
+inline "cic:/CoRN/complex/NRootCC/nroot_I_nexp.con".
(* UNEXPORTED
Hint Resolve nroot_I_nexp: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nroot_minus_I.con.
+inline "cic:/CoRN/complex/NRootCC/nroot_minus_I.con".
-inline cic:/CoRN/complex/NRootCC/nroot_minus_I_nexp.con.
+inline "cic:/CoRN/complex/NRootCC/nroot_minus_I_nexp.con".
(* UNEXPORTED
End NRootI.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/a.var.
+inline "cic:/CoRN/complex/NRootCC/a.var".
-inline cic:/CoRN/complex/NRootCC/b.var.
+inline "cic:/CoRN/complex/NRootCC/b.var".
-inline cic:/CoRN/complex/NRootCC/b_.var.
+inline "cic:/CoRN/complex/NRootCC/b_.var".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/c2.con.
+inline "cic:/CoRN/complex/NRootCC/c2.con".
(* end hide *)
-inline cic:/CoRN/complex/NRootCC/nrCC1_c2pos.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_c2pos.con".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/c.con.
+inline "cic:/CoRN/complex/NRootCC/c.con".
-inline cic:/CoRN/complex/NRootCC/a'2.con.
+inline "cic:/CoRN/complex/NRootCC/a'2.con".
(* end hide *)
-inline cic:/CoRN/complex/NRootCC/nrCC1_a'2pos.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a'2pos.con".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/a'.con.
+inline "cic:/CoRN/complex/NRootCC/a'.con".
-inline cic:/CoRN/complex/NRootCC/b'2.con.
+inline "cic:/CoRN/complex/NRootCC/b'2.con".
(* end hide *)
-inline cic:/CoRN/complex/NRootCC/nrCC1_b'2pos.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_b'2pos.con".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/b'.con.
+inline "cic:/CoRN/complex/NRootCC/b'.con".
(* end hide *)
-inline cic:/CoRN/complex/NRootCC/nrCC1_a3.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a3.con".
-inline cic:/CoRN/complex/NRootCC/nrCC1_a4.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a4.con".
(* UNEXPORTED
Hint Resolve nrCC1_a4: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC1_a5.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a5.con".
-inline cic:/CoRN/complex/NRootCC/nrCC1_a6.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a6.con".
-inline cic:/CoRN/complex/NRootCC/nrCC1_a6'.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a6'.con".
(* UNEXPORTED
Hint Resolve nrCC1_a5: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC1_a7_upper.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a7_upper.con".
-inline cic:/CoRN/complex/NRootCC/nrCC1_a7_lower.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC1_a7_lower.con".
(* UNEXPORTED
Hint Resolve nrCC1_a3 nrCC1_a7_upper nrCC1_a7_lower: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_1_upper.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_1_upper.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_1_lower.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_1_lower.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_1_ap_real.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_1_ap_real.con".
(* UNEXPORTED
End NRootCC_1_ap_real.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/a.var.
+inline "cic:/CoRN/complex/NRootCC/a.var".
-inline cic:/CoRN/complex/NRootCC/b.var.
+inline "cic:/CoRN/complex/NRootCC/b.var".
-inline cic:/CoRN/complex/NRootCC/a_.var.
+inline "cic:/CoRN/complex/NRootCC/a_.var".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/c'.con.
+inline "cic:/CoRN/complex/NRootCC/c'.con".
-inline cic:/CoRN/complex/NRootCC/a'.con.
+inline "cic:/CoRN/complex/NRootCC/a'.con".
-inline cic:/CoRN/complex/NRootCC/b'.con.
+inline "cic:/CoRN/complex/NRootCC/b'.con".
(* end hide *)
Hint Resolve sqrt_I_nexp: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_1_ap_imag.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_1_ap_imag.con".
(* UNEXPORTED
End NRootCC_1_ap_imag.
(*#* We now define the roots of arbitrary non zero complex numbers. *)
-inline cic:/CoRN/complex/NRootCC/nrootCC_1.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_1.con".
(* UNEXPORTED
End NRootCC_1.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/n.var.
+inline "cic:/CoRN/complex/NRootCC/n.var".
-inline cic:/CoRN/complex/NRootCC/c.var.
+inline "cic:/CoRN/complex/NRootCC/c.var".
-inline cic:/CoRN/complex/NRootCC/z.var.
+inline "cic:/CoRN/complex/NRootCC/z.var".
-inline cic:/CoRN/complex/NRootCC/c_.var.
+inline "cic:/CoRN/complex/NRootCC/c_.var".
-inline cic:/CoRN/complex/NRootCC/nrootCC_2'.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_2'.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_2.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_2.con".
(* UNEXPORTED
End NRootCC_2.
Section NRootCC_3.
*)
-inline cic:/CoRN/complex/NRootCC/Im_poly.con.
+inline "cic:/CoRN/complex/NRootCC/Im_poly.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a1.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a1.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a2.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a2.con".
(*#*
%\begin{convention}% Let [a,b : IR], [b_ : (b [#] Zero)] and [n : nat].
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/a.var.
+inline "cic:/CoRN/complex/NRootCC/a.var".
-inline cic:/CoRN/complex/NRootCC/b.var.
+inline "cic:/CoRN/complex/NRootCC/b.var".
-inline cic:/CoRN/complex/NRootCC/b_.var.
+inline "cic:/CoRN/complex/NRootCC/b_.var".
-inline cic:/CoRN/complex/NRootCC/n.var.
+inline "cic:/CoRN/complex/NRootCC/n.var".
-inline cic:/CoRN/complex/NRootCC/nrCC3_poly''.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_poly''.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a3.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a3.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a4.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a4.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a5.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a5.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a6.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a6.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_poly'.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_poly'.con".
(* UNEXPORTED
Hint Resolve nrCC3_a3: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC3_a7.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a7.con".
-inline cic:/CoRN/complex/NRootCC/nrCC3_a8.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a8.con".
(* UNEXPORTED
Hint Resolve nth_coeff_p_mult_c_: algebra.
Hint Resolve nrCC3_a6: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC3_a9.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC3_a9.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_3_poly.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3_poly.con".
(* UNEXPORTED
Hint Resolve nrCC3_a1 nrCC3_a7: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_3_.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3_.con".
(* UNEXPORTED
Hint Resolve nrootCC_3_: algebra.
Hint Resolve calculate_Im: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_3.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3.con".
(* UNEXPORTED
Hint Resolve nrCC3_a2: algebra.
Hint Resolve nrCC3_a9: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_3_degree.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3_degree.con".
(* UNEXPORTED
End NRootCC_3.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/c.var.
+inline "cic:/CoRN/complex/NRootCC/c.var".
-inline cic:/CoRN/complex/NRootCC/n.var.
+inline "cic:/CoRN/complex/NRootCC/n.var".
-inline cic:/CoRN/complex/NRootCC/n_.var.
+inline "cic:/CoRN/complex/NRootCC/n_.var".
-inline cic:/CoRN/complex/NRootCC/nrootCC_3'_poly.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3'_poly.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_3'.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3'.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_3'_degree.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_3'_degree.con".
(* UNEXPORTED
End NRootCC_3'.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/a.var.
+inline "cic:/CoRN/complex/NRootCC/a.var".
-inline cic:/CoRN/complex/NRootCC/b.var.
+inline "cic:/CoRN/complex/NRootCC/b.var".
-inline cic:/CoRN/complex/NRootCC/b_.var.
+inline "cic:/CoRN/complex/NRootCC/b_.var".
-inline cic:/CoRN/complex/NRootCC/n.var.
+inline "cic:/CoRN/complex/NRootCC/n.var".
-inline cic:/CoRN/complex/NRootCC/n_.var.
+inline "cic:/CoRN/complex/NRootCC/n_.var".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/c.con.
+inline "cic:/CoRN/complex/NRootCC/c.con".
(* end hide *)
Hint Resolve nrootCC_3: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC4_a1.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a1.con".
(*#*
%\begin{convention}% Let [r2',c2 : IR] and [r2'_ : (r2' [#] Zero)].
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/r2'.var.
+inline "cic:/CoRN/complex/NRootCC/r2'.var".
-inline cic:/CoRN/complex/NRootCC/c2.var.
+inline "cic:/CoRN/complex/NRootCC/c2.var".
-inline cic:/CoRN/complex/NRootCC/r2'_.var.
+inline "cic:/CoRN/complex/NRootCC/r2'_.var".
(* UNEXPORTED
Hint Resolve nrootCC_3': algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC4_a1'.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a1'.con".
(* UNEXPORTED
End NRootCC_4_solutions.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/r.var.
+inline "cic:/CoRN/complex/NRootCC/r.var".
-inline cic:/CoRN/complex/NRootCC/r_property.var.
+inline "cic:/CoRN/complex/NRootCC/r_property.var".
-inline cic:/CoRN/complex/NRootCC/y2.var.
+inline "cic:/CoRN/complex/NRootCC/y2.var".
-inline cic:/CoRN/complex/NRootCC/y2_property.var.
+inline "cic:/CoRN/complex/NRootCC/y2_property.var".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a2.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a2.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a3.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a3.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a4.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a4.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_y.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_y.con".
-inline cic:/CoRN/complex/NRootCC/y.con.
+inline "cic:/CoRN/complex/NRootCC/y.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_x.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_x.con".
-inline cic:/CoRN/complex/NRootCC/x.con.
+inline "cic:/CoRN/complex/NRootCC/x.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a5.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a5.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a6.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a6.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_z.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_z.con".
-inline cic:/CoRN/complex/NRootCC/z.con.
+inline "cic:/CoRN/complex/NRootCC/z.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a7.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a7.con".
(* UNEXPORTED
Hint Resolve nrCC4_a6: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC4_a8.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a8.con".
-inline cic:/CoRN/complex/NRootCC/nrCC4_a9.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a9.con".
(* UNEXPORTED
End NRootCC_4_equations.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC4_a10.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC4_a10.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_4_ap_real.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_4_ap_real.con".
(* UNEXPORTED
End NRootCC_4_ap_real.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/a.var.
+inline "cic:/CoRN/complex/NRootCC/a.var".
-inline cic:/CoRN/complex/NRootCC/b.var.
+inline "cic:/CoRN/complex/NRootCC/b.var".
-inline cic:/CoRN/complex/NRootCC/a_.var.
+inline "cic:/CoRN/complex/NRootCC/a_.var".
-inline cic:/CoRN/complex/NRootCC/n.var.
+inline "cic:/CoRN/complex/NRootCC/n.var".
-inline cic:/CoRN/complex/NRootCC/n_.var.
+inline "cic:/CoRN/complex/NRootCC/n_.var".
(* begin hide *)
-inline cic:/CoRN/complex/NRootCC/c'.con.
+inline "cic:/CoRN/complex/NRootCC/c'.con".
-inline cic:/CoRN/complex/NRootCC/a'.con.
+inline "cic:/CoRN/complex/NRootCC/a'.con".
-inline cic:/CoRN/complex/NRootCC/b'.con.
+inline "cic:/CoRN/complex/NRootCC/b'.con".
(* end hide *)
-inline cic:/CoRN/complex/NRootCC/nrootCC_4_ap_real'.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_4_ap_real'.con".
(* UNEXPORTED
Hint Resolve nroot_minus_I_nexp: algebra.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_4_ap_imag.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_4_ap_imag.con".
(* UNEXPORTED
End NRootCC_4_ap_imag.
*)
-inline cic:/CoRN/complex/NRootCC/nrootCC_4.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_4.con".
(* UNEXPORTED
End NRootCC_4.
Section NRootCC_5.
*)
-inline cic:/CoRN/complex/NRootCC/nrCC_5a2.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC_5a2.con".
-inline cic:/CoRN/complex/NRootCC/nrCC_5a3.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC_5a3.con".
(* UNEXPORTED
Hint Resolve nrCC_5a3: algebra.
%\end{convention}%
*)
-inline cic:/CoRN/complex/NRootCC/c.var.
+inline "cic:/CoRN/complex/NRootCC/c.var".
-inline cic:/CoRN/complex/NRootCC/c_.var.
+inline "cic:/CoRN/complex/NRootCC/c_.var".
-inline cic:/CoRN/complex/NRootCC/nrCC_5a4.con.
+inline "cic:/CoRN/complex/NRootCC/nrCC_5a4.con".
-inline cic:/CoRN/complex/NRootCC/nrootCC_5.con.
+inline "cic:/CoRN/complex/NRootCC/nrootCC_5.con".
(* UNEXPORTED
End NRootCC_5.
(*#* Final definition *)
-inline cic:/CoRN/complex/NRootCC/CnrootCC.con.
+inline "cic:/CoRN/complex/NRootCC/CnrootCC.con".