set "baseuri" "cic:/matita/CoRN-Decl/complex/CComplex".
+include "CoRN.ma".
+
(* $Id: CComplex.v,v 1.8 2004/04/23 10:00:55 lcf Exp $ *)
(*#* printing Re %\ensuremath{\Re}% #ℜ# *)
(*#* printing CCX %\ensuremath{\mathbb C[X]}% #<b>C</b>[X]# *)
-(* INCLUDE
-NRootIR
-*)
+include "reals/NRootIR.ma".
(*#* * Complex Numbers
** Algebraic structure
*)
(* UNEXPORTED
-Section Complex_Numbers.
+Section Complex_Numbers
*)
-inline cic:/CoRN/complex/CComplex/CC_set.ind.
+inline "cic:/CoRN/complex/CComplex/CC_set.ind".
-inline cic:/CoRN/complex/CComplex/cc_ap.con.
+inline "cic:/CoRN/complex/CComplex/cc_ap.con".
-inline cic:/CoRN/complex/CComplex/cc_eq.con.
+inline "cic:/CoRN/complex/CComplex/cc_eq.con".
-inline cic:/CoRN/complex/CComplex/cc_is_CSetoid.con.
+inline "cic:/CoRN/complex/CComplex/cc_is_CSetoid.con".
-inline cic:/CoRN/complex/CComplex/cc_csetoid.con.
+inline "cic:/CoRN/complex/CComplex/cc_csetoid.con".
-inline cic:/CoRN/complex/CComplex/cc_plus.con.
+inline "cic:/CoRN/complex/CComplex/cc_plus.con".
-inline cic:/CoRN/complex/CComplex/cc_mult.con.
+inline "cic:/CoRN/complex/CComplex/cc_mult.con".
-inline cic:/CoRN/complex/CComplex/cc_zero.con.
+inline "cic:/CoRN/complex/CComplex/cc_zero.con".
-inline cic:/CoRN/complex/CComplex/cc_one.con.
+inline "cic:/CoRN/complex/CComplex/cc_one.con".
-inline cic:/CoRN/complex/CComplex/cc_i.con.
+inline "cic:/CoRN/complex/CComplex/cc_i.con".
-inline cic:/CoRN/complex/CComplex/cc_inv.con.
+inline "cic:/CoRN/complex/CComplex/cc_inv.con".
(* not needed anymore
Lemma cc_plus_op_proof : (bin_op_wd cc_csetoid cc_plus).
Qed.
*)
-inline cic:/CoRN/complex/CComplex/cc_inv_strext.con.
+inline "cic:/CoRN/complex/CComplex/cc_inv_strext.con".
-inline cic:/CoRN/complex/CComplex/cc_plus_strext.con.
+inline "cic:/CoRN/complex/CComplex/cc_plus_strext.con".
-inline cic:/CoRN/complex/CComplex/cc_mult_strext.con.
+inline "cic:/CoRN/complex/CComplex/cc_mult_strext.con".
-inline cic:/CoRN/complex/CComplex/cc_inv_op.con.
+inline "cic:/CoRN/complex/CComplex/cc_inv_op.con".
-inline cic:/CoRN/complex/CComplex/cc_plus_op.con.
+inline "cic:/CoRN/complex/CComplex/cc_plus_op.con".
-inline cic:/CoRN/complex/CComplex/cc_mult_op.con.
+inline "cic:/CoRN/complex/CComplex/cc_mult_op.con".
-inline cic:/CoRN/complex/CComplex/cc_csg_associative.con.
+inline "cic:/CoRN/complex/CComplex/cc_csg_associative.con".
-inline cic:/CoRN/complex/CComplex/cc_cr_mult_associative.con.
+inline "cic:/CoRN/complex/CComplex/cc_cr_mult_associative.con".
-inline cic:/CoRN/complex/CComplex/cc_csemi_grp.con.
+inline "cic:/CoRN/complex/CComplex/cc_csemi_grp.con".
-inline cic:/CoRN/complex/CComplex/cc_cm_proof.con.
+inline "cic:/CoRN/complex/CComplex/cc_cm_proof.con".
-inline cic:/CoRN/complex/CComplex/cc_cmonoid.con.
+inline "cic:/CoRN/complex/CComplex/cc_cmonoid.con".
-inline cic:/CoRN/complex/CComplex/cc_cg_proof.con.
+inline "cic:/CoRN/complex/CComplex/cc_cg_proof.con".
-inline cic:/CoRN/complex/CComplex/cc_cr_dist.con.
+inline "cic:/CoRN/complex/CComplex/cc_cr_dist.con".
-inline cic:/CoRN/complex/CComplex/cc_cr_non_triv.con.
+inline "cic:/CoRN/complex/CComplex/cc_cr_non_triv.con".
-inline cic:/CoRN/complex/CComplex/cc_cgroup.con.
+inline "cic:/CoRN/complex/CComplex/cc_cgroup.con".
-inline cic:/CoRN/complex/CComplex/cc_cabgroup.con.
+inline "cic:/CoRN/complex/CComplex/cc_cabgroup.con".
-inline cic:/CoRN/complex/CComplex/cc_cr_mult_mon.con.
+inline "cic:/CoRN/complex/CComplex/cc_cr_mult_mon.con".
-inline cic:/CoRN/complex/CComplex/cc_mult_commutes.con.
+inline "cic:/CoRN/complex/CComplex/cc_mult_commutes.con".
-inline cic:/CoRN/complex/CComplex/cc_isCRing.con.
+inline "cic:/CoRN/complex/CComplex/cc_isCRing.con".
-inline cic:/CoRN/complex/CComplex/cc_cring.con.
+inline "cic:/CoRN/complex/CComplex/cc_cring.con".
-inline cic:/CoRN/complex/CComplex/cc_ap_zero.con.
+inline "cic:/CoRN/complex/CComplex/cc_ap_zero.con".
-inline cic:/CoRN/complex/CComplex/cc_inv_aid.con.
+inline "cic:/CoRN/complex/CComplex/cc_inv_aid.con".
(*#*
If [x [~=] Zero] or [y [~=] Zero], then [x [/] x[^]2 [+] y[^]2 [~=] Zero] or
[[--]y[/]x[^]2[+]y[^]2 [~=] Zero].
*)
-inline cic:/CoRN/complex/CComplex/cc_inv_aid2.con.
+inline "cic:/CoRN/complex/CComplex/cc_inv_aid2.con".
(*
REMARK KEPT FOR SENTIMENTAL REASONS...
actual function.
*)
-inline cic:/CoRN/complex/CComplex/cc_recip.con.
+inline "cic:/CoRN/complex/CComplex/cc_recip.con".
-inline cic:/CoRN/complex/CComplex/cc_cfield_proof.con.
+inline "cic:/CoRN/complex/CComplex/cc_cfield_proof.con".
-inline cic:/CoRN/complex/CComplex/cc_Recip_proof.con.
+inline "cic:/CoRN/complex/CComplex/cc_Recip_proof.con".
(* UNEXPORTED
Opaque cc_recip.
Opaque cc_inv.
*)
-inline cic:/CoRN/complex/CComplex/cc_cfield.con.
+inline "cic:/CoRN/complex/CComplex/cc_cfield.con".
-inline cic:/CoRN/complex/CComplex/CC.con.
+inline "cic:/CoRN/complex/CComplex/CC.con".
(*#*
Maps from reals to complex and vice-versa are defined, as well as conjugate,
absolute value and the imaginary unit [I] *)
-inline cic:/CoRN/complex/CComplex/cc_set_CC.con.
+inline "cic:/CoRN/complex/CComplex/cc_set_CC.con".
-inline cic:/CoRN/complex/CComplex/cc_IR.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR.con".
-inline cic:/CoRN/complex/CComplex/CC_conj.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj.con".
(* old def
Definition CC_conj' : CC->CC := [z:CC_set] (CC_set_rec [_:CC_set]CC_set [Re0,Im0:IR] (Build_CC_set Re0 [--]Im0) z).
*)
-inline cic:/CoRN/complex/CComplex/AbsCC.con.
+inline "cic:/CoRN/complex/CComplex/AbsCC.con".
-inline cic:/CoRN/complex/CComplex/TwoCC_ap_zero.con.
+inline "cic:/CoRN/complex/CComplex/TwoCC_ap_zero.con".
(* UNEXPORTED
-End Complex_Numbers.
+End Complex_Numbers
*)
(* begin hide *)
+(* NOTATION
+Notation CCX := (cpoly_cring CC).
+*)
+
(* end hide *)
-inline cic:/CoRN/complex/CComplex/II.con.
+inline "cic:/CoRN/complex/CComplex/II.con".
+
+(* NOTATION
+Infix "[+I*]" := cc_set_CC (at level 48, no associativity).
+*)
(*#* ** Properties of [II] *)
(* UNEXPORTED
-Section I_properties.
+Section I_properties
*)
-inline cic:/CoRN/complex/CComplex/I_square.con.
+inline "cic:/CoRN/complex/CComplex/I_square.con".
(* UNEXPORTED
Hint Resolve I_square: algebra.
*)
-inline cic:/CoRN/complex/CComplex/I_square'.con.
+inline "cic:/CoRN/complex/CComplex/I_square'.con".
-inline cic:/CoRN/complex/CComplex/I_recip_lft.con.
+inline "cic:/CoRN/complex/CComplex/I_recip_lft.con".
-inline cic:/CoRN/complex/CComplex/I_recip_rht.con.
+inline "cic:/CoRN/complex/CComplex/I_recip_rht.con".
-inline cic:/CoRN/complex/CComplex/mult_I.con.
+inline "cic:/CoRN/complex/CComplex/mult_I.con".
-inline cic:/CoRN/complex/CComplex/I_wd.con.
+inline "cic:/CoRN/complex/CComplex/I_wd.con".
(*#* ** Properties of [Re] and [Im] *)
-inline cic:/CoRN/complex/CComplex/calculate_norm.con.
+inline "cic:/CoRN/complex/CComplex/calculate_norm.con".
-inline cic:/CoRN/complex/CComplex/calculate_Re.con.
+inline "cic:/CoRN/complex/CComplex/calculate_Re.con".
-inline cic:/CoRN/complex/CComplex/calculate_Im.con.
+inline "cic:/CoRN/complex/CComplex/calculate_Im.con".
-inline cic:/CoRN/complex/CComplex/Re_wd.con.
+inline "cic:/CoRN/complex/CComplex/Re_wd.con".
-inline cic:/CoRN/complex/CComplex/Im_wd.con.
+inline "cic:/CoRN/complex/CComplex/Im_wd.con".
-inline cic:/CoRN/complex/CComplex/Re_resp_plus.con.
+inline "cic:/CoRN/complex/CComplex/Re_resp_plus.con".
-inline cic:/CoRN/complex/CComplex/Re_resp_inv.con.
+inline "cic:/CoRN/complex/CComplex/Re_resp_inv.con".
-inline cic:/CoRN/complex/CComplex/Im_resp_plus.con.
+inline "cic:/CoRN/complex/CComplex/Im_resp_plus.con".
-inline cic:/CoRN/complex/CComplex/Im_resp_inv.con.
+inline "cic:/CoRN/complex/CComplex/Im_resp_inv.con".
-inline cic:/CoRN/complex/CComplex/cc_calculate_square.con.
+inline "cic:/CoRN/complex/CComplex/cc_calculate_square.con".
(* UNEXPORTED
-End I_properties.
+End I_properties
*)
(* UNEXPORTED
(*#* ** Properties of conjugation *)
(* UNEXPORTED
-Section Conj_properties.
+Section Conj_properties
*)
-inline cic:/CoRN/complex/CComplex/CC_conj_plus.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_plus.con".
-inline cic:/CoRN/complex/CComplex/CC_conj_mult.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_mult.con".
(* UNEXPORTED
Hint Resolve CC_conj_mult: algebra.
*)
-inline cic:/CoRN/complex/CComplex/CC_conj_strext.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_strext.con".
-inline cic:/CoRN/complex/CComplex/CC_conj_conj.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_conj.con".
-inline cic:/CoRN/complex/CComplex/CC_conj_zero.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_zero.con".
-inline cic:/CoRN/complex/CComplex/CC_conj_one.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_one.con".
(* UNEXPORTED
Hint Resolve CC_conj_one: algebra.
*)
-inline cic:/CoRN/complex/CComplex/CC_conj_nexp.con.
+inline "cic:/CoRN/complex/CComplex/CC_conj_nexp.con".
(* UNEXPORTED
-End Conj_properties.
+End Conj_properties
*)
(* UNEXPORTED
(*#* ** Properties of the real axis *)
(* UNEXPORTED
-Section cc_IR_properties.
+Section cc_IR_properties
*)
-inline cic:/CoRN/complex/CComplex/Re_cc_IR.con.
+inline "cic:/CoRN/complex/CComplex/Re_cc_IR.con".
-inline cic:/CoRN/complex/CComplex/Im_cc_IR.con.
+inline "cic:/CoRN/complex/CComplex/Im_cc_IR.con".
-inline cic:/CoRN/complex/CComplex/cc_IR_wd.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_wd.con".
(* UNEXPORTED
Hint Resolve cc_IR_wd: algebra_c.
*)
-inline cic:/CoRN/complex/CComplex/cc_IR_resp_ap.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_resp_ap.con".
-inline cic:/CoRN/complex/CComplex/cc_IR_mult.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_mult.con".
(* UNEXPORTED
Hint Resolve cc_IR_mult: algebra.
*)
-inline cic:/CoRN/complex/CComplex/cc_IR_mult_lft.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_mult_lft.con".
-inline cic:/CoRN/complex/CComplex/cc_IR_mult_rht.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_mult_rht.con".
-inline cic:/CoRN/complex/CComplex/cc_IR_plus.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_plus.con".
(* UNEXPORTED
Hint Resolve cc_IR_plus: algebra.
*)
-inline cic:/CoRN/complex/CComplex/cc_IR_minus.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_minus.con".
-inline cic:/CoRN/complex/CComplex/cc_IR_zero.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_zero.con".
(* UNEXPORTED
Hint Resolve cc_IR_zero: algebra.
*)
-inline cic:/CoRN/complex/CComplex/cc_IR_one.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_one.con".
(* UNEXPORTED
Hint Resolve cc_IR_one: algebra.
*)
-inline cic:/CoRN/complex/CComplex/cc_IR_nring.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_nring.con".
-inline cic:/CoRN/complex/CComplex/cc_IR_nexp.con.
+inline "cic:/CoRN/complex/CComplex/cc_IR_nexp.con".
(* UNEXPORTED
-End cc_IR_properties.
+End cc_IR_properties
*)
(* UNEXPORTED
(*#* ** [CC] has characteristic zero *)
-(* INCLUDE
-Transparent_algebra
-*)
+include "tactics/Transparent_algebra.ma".
-inline cic:/CoRN/complex/CComplex/char0_CC.con.
+inline "cic:/CoRN/complex/CComplex/char0_CC.con".
-(* INCLUDE
-Opaque_algebra
-*)
+include "tactics/Opaque_algebra.ma".
-inline cic:/CoRN/complex/CComplex/poly_apzero_CC.con.
+inline "cic:/CoRN/complex/CComplex/poly_apzero_CC.con".
-inline cic:/CoRN/complex/CComplex/poly_CC_extensional.con.
+inline "cic:/CoRN/complex/CComplex/poly_CC_extensional.con".