:: Vieta's Formula about the Sum of Roots of Polynomials
:: by Artur Korni{\l}owicz and Karol P\kak
::
:: Received May 25, 2017
:: Copyright (c) 2017 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 NUMBERS, FINSEQ_1, SUBSET_1, RELAT_1, FUNCT_1, CARD_1, CARD_3,
FINSEQ_2, ARYTM_3, ORDINAL4, TARSKI, NAT_1, XXREAL_0, ARYTM_1, XBOOLE_0,
RLVECT_1, ALGSTR_0, SUPINF_2, PARTFUN1, FINSET_1, ALGSTR_1, ZFMISC_1,
PRE_POLY, FUNCT_4, VALUED_0, BINOP_1, STRUCT_0, POLYNOM1, POLYNOM3,
AFINSQ_1, MESFUNC1, VECTSP_1, POLYNOM5, VECTSP_2, POLYNOM2, FUNCT_2,
UPROOTS, AOFA_I00, XCMPLX_0, POLYVIE1, BINOM, SGRAPH1, CLASSES1;
notations TARSKI, XBOOLE_0, XCMPLX_0, XXREAL_0, SUBSET_1, NUMBERS, RVSUM_1,
RELAT_1, FINSET_1, FUNCT_1, RELSET_1, PARTFUN1, DOMAIN_1, FUNCT_2,
CARD_1, CLASSES1, FINSEQ_1, FINSEQ_2, FUNCT_7, XXREAL_2, NAT_1, NAT_D,
STRUCT_0, VECTSP_2, ALGSTR_0, ALGSTR_1, BINOM, RLVECT_1, RLAFFIN3,
VECTSP_1, PRE_POLY, ALGSEQ_1, POLYNOM3, POLYNOM4, POLYNOM5, FVSUM_1,
GROUP_1, UPROOTS, NIVEN;
constructors NAT_D, ALGSTR_1, POLYNOM4, POLYNOM5, RECDEF_1, XXREAL_2, FUNCT_7,
FVSUM_1, ALGSEQ_1, UPROOTS, RLAFFIN3, BINOM, FINSEQ_4, NIVEN, RVSUM_1,
CLASSES1, WSIERP_1;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, FINSET_1,
XXREAL_0, XREAL_0, NAT_1, INT_1, CARD_1, MEMBERED, FINSEQ_1, STRUCT_0,
VECTSP_1, ALGSTR_1, PRE_POLY, POLYNOM3, POLYNOM4, POLYNOM5, VALUED_0,
XXREAL_2, FUNCT_2, RELSET_1, UPROOTS, NAT_2, RLVECT_1, RING_3, ALGSTR_0,
RATFUNC1, NEWTON04, FINSEQ_2;
requirements NUMERALS, BOOLE, SUBSET, ARITHM, REAL;
begin
registration
let F be FinSequence;
let f be Function of dom F,dom F;
cluster F*f -> FinSequence-like;
end;
theorem :: POLYVIE1:1
for a,b being object st a <> b holds
canFS({a,b}) = <*a,b*> or canFS({a,b}) = <*b,a*>;
theorem :: POLYVIE1:2
for X being finite set holds canFS X is Enumeration of X;
registration
let A be set;
let X be finite Subset of A;
cluster canFS X -> A-valued;
end;
theorem :: POLYVIE1:3
for L being right_zeroed non empty addLoopStr, a being Element of L
holds 2 * a = a + a;
registration
let L be almost_left_invertible multLoopStr_0;
cluster non zero -> left_invertible for Element of L;
end;
registration
let L be almost_right_invertible multLoopStr_0;
cluster non zero -> right_invertible for Element of L;
end;
registration
let L be almost_left_cancelable multLoopStr_0;
cluster non zero -> left_mult-cancelable for Element of L;
end;
registration
let L be almost_right_cancelable multLoopStr_0;
cluster non zero -> right_mult-cancelable for Element of L;
end;
theorem :: POLYVIE1:4
for L being right_unital associative non trivial doubleLoopStr
for a,b being Element of L st b is left_invertible right_mult-cancelable
& b*(/b) = (/b)*b holds a*b/b = a;
registration
let L be non degenerated ZeroOneStr;
let z0 be Element of L;
let z1 be non zero Element of L;
cluster <%z0,z1%> -> non-zero;
cluster <%z1,z0%> -> non-zero;
end;
theorem :: POLYVIE1:5
for L being non trivial ZeroStr, p being Polynomial of L st len p = 1
ex a being non zero Element of L st p = <%a%>;
theorem :: POLYVIE1:6
for L being non trivial ZeroStr, p being Polynomial of L st len p = 2
ex a being Element of L, b being non zero Element of L st p = <%a,b%>;
theorem :: POLYVIE1:7
for L being non trivial ZeroStr, p being Polynomial of L st len p = 3
ex a,b being Element of L, c being non zero Element of L st p = <%a,b,c%>;
theorem :: POLYVIE1:8
for L being add-associative right_zeroed right_complementable
associative commutative left-distributive well-unital
almost_left_invertible non empty doubleLoopStr
for a,b,x being Element of L st b <> 0.L
holds eval(<%a,b%>,-a/b) = 0.L;
theorem :: POLYVIE1:9
for L being Field
for a,x being Element of L
for b being non zero Element of L holds
x is_a_root_of <%a,b%> iff x = -a/b;
theorem :: POLYVIE1:10
for L being Field
for a being Element of L
for b being non zero Element of L holds
Roots <%a,b%> = {-a/b};
theorem :: POLYVIE1:11
for L being Field
for a being Element of L
for b being non zero Element of L holds
multiplicity(<%a,b%>,-a/b) = 1;
theorem :: POLYVIE1:12
for L being Field
for a being Element of L
for b being non zero Element of L holds
BRoots <%a,b%> = ({-a/b},1)-bag;
theorem :: POLYVIE1:13
for L being Field
for a,c being Element of L
for b,d being non zero Element of L holds
Roots(<%a,b%>*'<%c,d%>) = {-a/b,-c/d};
theorem :: POLYVIE1:14
for L being Field
for a,x being Element of L
for b being non zero Element of L st x <> -a/b holds
multiplicity(<%a,b%>,x) = 0;
theorem :: POLYVIE1:15
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L st not -a/b in Roots(p)
holds card Roots(<%a,b%>*'p) = 1 + card Roots(p);
theorem :: POLYVIE1:16
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L st not -a/b in Roots(p)
holds (canFS Roots(p))^<*-a/b*> is Enumeration of Roots(<%a,b%>*'p);
theorem :: POLYVIE1:17
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L
for E being Enumeration of Roots(<%a,b%>*'p) st E = (canFS Roots(p))^<*-a/b*>
holds len E = 1 + card Roots(p) &
E.(1+card Roots(p)) = -a/b &
for n being Nat st 1 <= n <= card Roots(p) holds E.n = (canFS Roots(p)).n;
definition
let L be non empty doubleLoopStr;
let B be bag of the carrier of L;
let E be (the carrier of L)-valued FinSequence;
func B(++)E -> FinSequence of L means
:: POLYVIE1:def 1
len it = len E &
for n being Nat st 1 <= n <= len it holds it.n = (B*E).n * (E/.n);
end;
theorem :: POLYVIE1:18
for L being domRing
for p being non-zero Polynomial of L
for B being bag of the carrier of L
for E being Enumeration of Roots p st Roots p = {}
holds B(++)E = {};
theorem :: POLYVIE1:19
for L being left_zeroed add-associative non empty doubleLoopStr
for B1,B2 being bag of the carrier of L
for E being (the carrier of L)-valued FinSequence
holds (B1+B2)(++)E = (B1(++)E) + (B2(++)E);
theorem :: POLYVIE1:20
for L being left_zeroed add-associative non empty doubleLoopStr
for B being bag of the carrier of L
for E,F being (the carrier of L)-valued FinSequence
holds B(++)(E^F) = (B(++)E) ^ (B(++)F);
theorem :: POLYVIE1:21
for L being left_zeroed add-associative non empty doubleLoopStr
for B1,B2 being bag of the carrier of L
for E,F being (the carrier of L)-valued FinSequence
holds (B1+B2)(++)(E^F) = (B1(++)E)^(B1(++)F) + (B2(++)E)^(B2(++)F);
theorem :: POLYVIE1:22
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L
for E being Enumeration of Roots(<%a,b%>*'p)
for P being Permutation of dom E
holds (BRoots(<%a,b%>*'p)(++)E)*P = BRoots(<%a,b%>*'p)(++)(E*P);
theorem :: POLYVIE1:23
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L st not -a/b in Roots(p)
for E being Enumeration of Roots(<%a,b%>*'p) st E = (canFS Roots(p))^<*-a/b*>
holds (canFS Roots(<%a,b%>*'p))" * E is Permutation of dom E;
theorem :: POLYVIE1:24
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L st not -a/b in Roots(p)
for E being Enumeration of Roots(<%a,b%>*'p) st E = (canFS Roots(p))^<*-a/b*>
holds
Sum(BRoots(<%a,b%>*'p)(++)E) =
Sum(BRoots(<%a,b%>*'p)(++)canFS Roots(<%a,b%>*'p));
theorem :: POLYVIE1:25
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L
for E being Enumeration of Roots(<%a,b%>*'p)
holds Sum(BRoots(<%a,b%>)(++)E) = -a/b;
definition
let L be domRing;
let p be non-zero Polynomial of L;
func SumRoots(p) -> Element of L equals
:: POLYVIE1:def 2
Sum (BRoots(p)(++)canFS Roots p);
end;
theorem :: POLYVIE1:26
for L being domRing
for p being non-zero Polynomial of L st Roots p = {} holds
SumRoots p = 0.L;
theorem :: POLYVIE1:27
for L being Field
for a being Element of L
for b being non zero Element of L holds
SumRoots <%a,b%> = -a/b;
theorem :: POLYVIE1:28
for L being Field
for p being non-zero Polynomial of L
for a being Element of L
for b being non zero Element of L
holds SumRoots(<%a,b%>*'p) = -a/b + SumRoots(p);
theorem :: POLYVIE1:29
for L being Field
for a,c being Element of L
for b,d being non zero Element of L
holds SumRoots(<%a,b%>*'<%c,d%>) = (-a/b) + (-c/d);
theorem :: POLYVIE1:30
for L being algebraic-closed Field
for p,q being non-zero Polynomial of L st
len p >= 2 holds SumRoots(p*'q) = SumRoots(p) + SumRoots(q);
theorem :: POLYVIE1:31
for L being algebraic-closed domRing, p being non-zero Polynomial of L
for r being FinSequence of L st r is one-to-one & len r = len p-'1 &
Roots p = rng r holds Sum r = SumRoots p;
::$N Vieta's formula about the sum of roots
theorem :: POLYVIE1:32
for L being algebraic-closed Field, p being non-zero Polynomial of L holds
len p >= 2 implies SumRoots p = - p.(len p-'2) / p.(len p-'1);