:: A Test for the Stability of Networks
:: by Agnieszka Rowi\'nska-Schwarzweller and Christoph Schwarzweller
::
:: Received January 17, 2013
:: Copyright (c) 2013-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies ARYTM_3, POLYNOM2, COMPLFLD, POLYNOM1, ARYTM_1, FUNCT_1, REAL_1,
COMPLEX1, POLYNOM3, RELAT_1, RLVECT_1, FINSEQ_1, ZFMISC_1, INT_1,
VECTSP_1, ALGSEQ_1, GROUP_1, ALGSTR_1, POLYNOM5, SGRAPH1, ALGSTR_0,
FUNCT_4, RATFUNC1, MCART_1, HURWITZ, HURWITZ2, XBOOLE_0, POLYNOM4,
BINOP_1, PARTFUN1, LATTICES, XREAL_0, ABIAN, XCMPLX_0, SQUARE_1,
STRUCT_0, SUBSET_1, SUPINF_2, NAT_1, NUMBERS, CARD_1, MESFUNC1, XXREAL_0,
VALUED_0, YELLOW_8;
notations TARSKI, SUBSET_1, VALUED_0, NUMBERS, ORDINAL1, XCMPLX_0, COMPLEX1,
COMPLFLD, XREAL_0, SQUARE_1, RELAT_1, ALGSTR_0, FUNCT_7, VECTSP_1,
ARYTM_3, FUNCT_1, FUNCT_2, NAT_1, NAT_D, INT_1, GROUP_1, ABIAN, BINOP_1,
XXREAL_0, STRUCT_0, ALGSTR_1, RLVECT_1, VFUNCT_1, NORMSP_1, ALGSEQ_1,
POLYNOM3, POLYNOM4, POLYNOM5, RATFUNC1, HURWITZ;
constructors BINOP_1, REAL_1, BINARITH, ALGSTR_1, POLYNOM5, BHSP_1, RELSET_1,
FUNCT_4, ARYTM_3, ABIAN, NAT_D, SQUARE_1, POLYNOM4, VFUNCT_1, RATFUNC1,
HURWITZ, ALGSEQ_1, BINOP_2, XXREAL_2;
registrations XBOOLE_0, ORDINAL1, RELSET_1, XCMPLX_0, XXREAL_0, INT_1, NAT_1,
MEMBERED, STRUCT_0, VECTSP_1, COMPLFLD, ALGSTR_1, POLYNOM4, POLYNOM5,
XREAL_0, VALUED_0, ABIAN, FUNCT_2, RATFUNC1, POLYNOM3, VFUNCT_1,
XTUPLE_0;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
begin :: Preliminaries
theorem :: HURWITZ2:1
for x,y being Complex st Im(x) = 0 & Re(y) = 0 holds Re(x/y) = 0;
theorem :: HURWITZ2:2
for a being Complex holds a * a*' = |.a.|^2;
registration
cluster Hurwitz for Polynomial of F_Complex;
end;
registration
cluster 0 -> even;
end;
theorem :: HURWITZ2:3
for L being add-associative right_zeroed right_complementable
associative distributive non empty doubleLoopStr
for k being even Element of NAT
for x being Element of L
holds (power L).(-x,k) = (power L).(x,k);
theorem :: HURWITZ2:4
for L being add-associative right_zeroed right_complementable
associative distributive non empty doubleLoopStr
for k being odd Element of NAT
for x being Element of L
holds (power L).(-x,k) = - ((power L).(x,k));
theorem :: HURWITZ2:5
for k being even Element of NAT
for x being Element of F_Complex st Re(x) = 0
holds Im((power F_Complex).(x,k)) = 0;
theorem :: HURWITZ2:6
for k being odd Element of NAT
for x being Element of F_Complex st Re(x) = 0
holds Re((power F_Complex).(x,k)) = 0;
begin :: Even and Odd Part of Polynomials
definition
let L be non empty ZeroStr;
let p be sequence of L;
func even_part p -> sequence of L means
:: HURWITZ2:def 1
for i being even Nat holds it.i = p.i &
for i being odd Nat holds it.i = 0.L;
func odd_part p -> sequence of L means
:: HURWITZ2:def 2
for i being even Nat holds it.i = 0.L &
for i being odd Nat holds it.i = p.i;
end;
registration
let L be non empty ZeroStr;
let p be Polynomial of L;
cluster even_part p -> finite-Support;
cluster odd_part p -> finite-Support;
end;
theorem :: HURWITZ2:7
for L being non empty ZeroStr
holds even_part 0_.(L) = 0_.(L) & odd_part 0_.(L) = 0_.(L);
theorem :: HURWITZ2:8
for L being non empty multLoopStr_0
holds even_part 1_.(L) = 1_.(L) & odd_part 1_.(L) = 0_.(L);
theorem :: HURWITZ2:9
for L being left_zeroed right_zeroed non empty addLoopStr,
p being Polynomial of L
holds even_part p + odd_part p = p;
theorem :: HURWITZ2:10
for L being left_zeroed right_zeroed non empty addLoopStr,
p being Polynomial of L
holds odd_part p + even_part p = p;
theorem :: HURWITZ2:11
for L being add-associative right_zeroed right_complementable
non empty addLoopStr,
p being Polynomial of L
holds p - odd_part p = even_part p;
theorem :: HURWITZ2:12
for L being add-associative right_zeroed right_complementable
non empty addLoopStr,
p being Polynomial of L
holds p - even_part p = odd_part p;
theorem :: HURWITZ2:13
for L being add-associative right_zeroed right_complementable
Abelian non empty addLoopStr,
p being Polynomial of L
holds even_part p - p = - odd_part p;
theorem :: HURWITZ2:14
for L being add-associative right_zeroed right_complementable
Abelian non empty addLoopStr,
p being Polynomial of L
holds odd_part p - p = - even_part p;
theorem :: HURWITZ2:15
for L being add-associative right_zeroed right_complementable
Abelian non empty addLoopStr,
p,q being Polynomial of L
holds even_part(p+q) = even_part(p) + even_part(q);
theorem :: HURWITZ2:16
for L being add-associative right_zeroed right_complementable
Abelian non empty addLoopStr,
p,q being Polynomial of L
holds odd_part(p+q) = odd_part(p) + odd_part(q);
theorem :: HURWITZ2:17
for L being well-unital non empty doubleLoopStr
for p being Polynomial of L st deg p is even
holds even_part Leading-Monomial(p) = Leading-Monomial(p);
theorem :: HURWITZ2:18
for L being well-unital non empty doubleLoopStr
for p being Polynomial of L st deg p is odd
holds even_part Leading-Monomial(p) = 0_.(L);
theorem :: HURWITZ2:19
for L being well-unital non empty doubleLoopStr
for p being Polynomial of L st deg p is even
holds odd_part Leading-Monomial(p) = 0_.(L);
theorem :: HURWITZ2:20
for L being well-unital non empty doubleLoopStr
for p being Polynomial of L st deg p is odd
holds odd_part Leading-Monomial(p) = Leading-Monomial(p);
theorem :: HURWITZ2:21
for L being well-unital add-associative right_zeroed right_complementable
Abelian associative distributive non degenerated doubleLoopStr
for p being non zero Polynomial of L
holds deg even_part(p) <> deg odd_part(p);
theorem :: HURWITZ2:22
for L being well-unital add-associative right_zeroed right_complementable
associative Abelian distributive non degenerated doubleLoopStr
for p being Polynomial of L
holds deg even_part(p) <= deg p & deg odd_part(p) <= deg p;
theorem :: HURWITZ2:23
for L being well-unital add-associative right_zeroed right_complementable
associative Abelian distributive non degenerated doubleLoopStr
for p being Polynomial of L
holds deg p = max( deg even_part(p), deg odd_part(p));
begin :: Even and Odd Polynomials and Rational Functions
definition
let L be non empty addLoopStr;
let f be Function of L,L;
attr f is even means
:: HURWITZ2:def 3
for x being Element of L holds f.(-x) = f.x;
attr f is odd means
:: HURWITZ2:def 4
for x being Element of L holds f.(-x) = - f.x;
end;
definition
let L be well-unital non empty doubleLoopStr;
let p be Polynomial of L;
attr p is even means
:: HURWITZ2:def 5
Polynomial-Function(L,p) is even;
attr p is odd means
:: HURWITZ2:def 6
Polynomial-Function(L,p) is odd;
end;
definition
let L be well-unital non trivial doubleLoopStr;
let Z be rational_function of L;
attr Z is odd means
:: HURWITZ2:def 7
(Z`1 is even & Z`2 is odd) or (Z`1 is odd & Z`2 is even);
end;
notation
let L be well-unital non trivial doubleLoopStr;
let Z be rational_function of L;
antonym Z is even for Z is odd;
end;
registration
let L be well-unital non empty doubleLoopStr;
cluster even for Polynomial of L;
end;
registration
let L be add-associative right_zeroed right_complementable
well-unital non empty doubleLoopStr;
cluster odd for Polynomial of L;
end;
registration
let L be well-unital add-associative right_zeroed right_complementable
associative non degenerated doubleLoopStr;
cluster non zero even for Polynomial of L;
end;
registration
let L be add-associative right_zeroed right_complementable
Abelian well-unital non degenerated doubleLoopStr;
cluster non zero odd for Polynomial of L;
end;
theorem :: HURWITZ2:24
for L being well-unital non empty doubleLoopStr
for p being even Polynomial of L
for x being Element of L
holds eval(p,-x) = eval(p,x);
theorem :: HURWITZ2:25
for L being add-associative right_zeroed right_complementable Abelian
well-unital non degenerated doubleLoopStr
for p being odd Polynomial of L
for x being Element of L
holds eval(p,-x) = - eval(p,x);
registration
let L be well-unital non empty doubleLoopStr;
cluster 0_.(L) -> even;
end;
registration
let L be add-associative right_zeroed right_complementable
well-unital non empty doubleLoopStr;
cluster 0_.(L) -> odd;
end;
registration
let L be well-unital add-associative right_zeroed right_complementable
associative non degenerated doubleLoopStr;
cluster 1_.(L) -> even;
end;
registration
let L be Abelian add-associative right_zeroed right_complementable
well-unital left-distributive non empty doubleLoopStr;
let p,q be even Polynomial of L;
cluster p + q -> even;
end;
registration
let L be Abelian add-associative right_zeroed right_complementable
well-unital left-distributive non degenerated doubleLoopStr;
let p,q be odd Polynomial of L;
cluster p + q -> odd;
end;
registration
let L be Abelian add-associative right_zeroed right_complementable
associative well-unital distributive non degenerated doubleLoopStr;
let p be Polynomial of L;
cluster even_part(p) -> even;
cluster odd_part(p) -> odd;
end;
theorem :: HURWITZ2:26
for L being Abelian add-associative right_zeroed right_complementable
well-unital distributive non degenerated doubleLoopStr
for p being even Polynomial of L
for q being odd Polynomial of L
for x being Element of L
st x is_a_common_root_of p,q holds -x is_a_root_of p+q;
theorem :: HURWITZ2:27
for p being Hurwitz Polynomial of F_Complex
holds even_part(p), odd_part(p) have_no_common_roots;
begin :: Real Positive Polynomials and Rational Functions
definition
let p be Polynomial of F_Complex;
attr p is real means
:: HURWITZ2:def 8
for i being Nat holds p.i is Real;
attr p is positive means
:: HURWITZ2:def 9
for x being Element of F_Complex holds Re(x) > 0 implies Re(eval(p,x)) > 0;
end;
registration
cluster 0_.(F_Complex) -> real non positive;
cluster 1_.(F_Complex) -> real positive;
end;
registration
cluster non zero real positive for Polynomial of F_Complex;
end;
registration
cluster real -> real-valued for Polynomial of F_Complex;
end;
registration
let p be real Polynomial of F_Complex;
cluster even_part(p) -> real;
cluster odd_part(p) -> real;
end;
definition
let L be non empty addLoopStr;
let p be Polynomial of L;
attr p is with_all_coefficients means
:: HURWITZ2:def 10
for i being Nat st i <= deg p holds p.i <> 0;
end;
definition
let p be real Polynomial of F_Complex;
attr p is with_positive_coefficients means
:: HURWITZ2:def 11
for i being Nat st i <= deg p holds p.i > 0;
attr p is with_negative_coefficients means
:: HURWITZ2:def 12
for i being Nat st i <= deg p holds p.i < 0;
end;
registration
cluster with_positive_coefficients -> with_all_coefficients
for real Polynomial of F_Complex;
cluster with_negative_coefficients -> with_all_coefficients
for real Polynomial of F_Complex;
end;
registration
cluster non constant with_positive_coefficients
for real Polynomial of F_Complex;
end;
registration
let p be non zero with_all_coefficients real Polynomial of F_Complex;
cluster even_part p -> non zero;
end;
registration
let p be non constant with_all_coefficients real Polynomial of F_Complex;
cluster odd_part p -> non zero;
end;
definition
let Z be rational_function of F_Complex;
attr Z is real means
:: HURWITZ2:def 13
for i being Nat holds (Z`1).i is Real & (Z`2).i is Real;
attr Z is positive means
:: HURWITZ2:def 14
for x being Element of F_Complex
st Re(x) > 0 & eval(Z`2,x) <> 0 holds Re(eval(Z,x)) > 0;
end;
registration
cluster non zero odd real positive for rational_function of F_Complex;
end;
registration
let p1 be real Polynomial of F_Complex;
let p2 be non zero real Polynomial of F_Complex;
cluster [p1,p2] -> real for rational_function of F_Complex;
end;
begin :: The Theorem
definition
mode one_port_function is real positive rational_function of F_Complex;
mode reactance_one_port_function is
odd real positive rational_function of F_Complex;
end;
theorem :: HURWITZ2:28
for p being real Polynomial of F_Complex
for x being Element of F_Complex st Re(x) = 0
holds Im(eval(even_part(p),x)) = 0;
theorem :: HURWITZ2:29
for p being real Polynomial of F_Complex
for x being Element of F_Complex st Re(x) = 0
holds Re(eval(odd_part(p),x)) = 0;
theorem :: HURWITZ2:30
for p be non constant with_positive_coefficients real Polynomial of F_Complex
st [even_part(p),odd_part(p)] is positive &
even_part(p),odd_part(p) have_no_common_roots
holds (for x being Element of F_Complex
st Re(x) = 0 & eval(odd_part(p),x) <> 0
holds Re(eval([even_part(p),odd_part(p)],x)) >= 0) &
even_part(p) + odd_part(p) is Hurwitz;
theorem :: HURWITZ2:31
for p be non constant with_positive_coefficients real Polynomial of F_Complex
st [even_part(p),odd_part(p)] is one_port_function &
degree([even_part(p),odd_part(p)]) = degree p
holds p is Hurwitz;