:: Little {B}ezout Theorem (Factor Theorem)
:: by Piotr Rudnicki
::
:: Received December 30, 2003
:: Copyright (c) 2003-2011 Association of Mizar Users

begin

theorem :: UPROOTS:1

canceled;

theorem Th2: :: UPROOTS:2

for f being FinSequence of NAT st ( for i being Element of NAT st i in dom f holds
f . i <> 0 ) holds
( Sum f = len f iff f = (len f) |-> 1 )

proof end;

scheme :: UPROOTS:sch 1
IndFinSeq0{ F₁() -> natural number , P₁[ set ] } :

for i being Element of NAT st 1 <= i & i <= F₁() holds
P₁[i]

provided

A1: P₁[1] and
A2: for i being Element of NAT st 1 <= i & i < F₁() & P₁[i] holds
P₁[i + 1]

proof end;

theorem Th3: :: UPROOTS:3

for L being non empty right_complementable add-associative right_zeroed addLoopStr
for r being FinSequence of L st len r >= 2 & ( for k being Element of NAT st 2 < k & k in dom r holds
r . k = 0. L ) holds
Sum r = (r /. 1) + (r /. 2)

proof end;

begin

registration

let A be finite set ;
cluster Relation-like NAT -defined A -valued Function-like bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of A;
existence
ex b₁ being FinSequence of A st b₁ is bijective

proof end;

end;

definition

let A be finite set ;
func canFS A -> FinSequence of A equals :: UPROOTS:def 1
the bijective FinSequence of A;
coherence
the bijective FinSequence of A is FinSequence of A ;

end;

:: deftheorem defines canFS UPROOTS:def 1 :
for A being finite set holds canFS A = the bijective FinSequence of A;

registration

let A be finite set ;
cluster canFS A -> bijective ;
coherence
canFS A is bijective ;

end;

theorem :: UPROOTS:4

canceled;

theorem Th5: :: UPROOTS:5

for A being finite set holds len (canFS A) = card A

proof end;

registration

let A be non empty finite set ;
cluster canFS A -> non empty ;
coherence
not canFS A is empty

proof end;

end;

theorem Th6: :: UPROOTS:6

for a being set holds canFS {a} = <*a*>

proof end;

theorem Th7: :: UPROOTS:7

for A being finite set holds (canFS A) " is Function of A,(Seg (card A))

proof end;

begin

definition

let X be set ;
let S be finite Subset of X;
let n be Element of NAT ;
func (S,n) -bag -> Element of Bags X equals :: UPROOTS:def 2
(EmptyBag X) +* (S --> n);
correctness
coherence
(EmptyBag X) +* (S --> n) is Element of Bags X;

proof end;

end;

:: deftheorem defines -bag UPROOTS:def 2 :
for X being set
for S being finite Subset of X
for n being Element of NAT holds (S,n) -bag = (EmptyBag X) +* (S --> n);

theorem Th8: :: UPROOTS:8

for X being set
for S being finite Subset of X
for n being Element of NAT
for i being set st not i in S holds
((S,n) -bag) . i = 0

proof end;

theorem Th9: :: UPROOTS:9

for X being set
for S being finite Subset of X
for n being Element of NAT
for i being set st i in S holds
((S,n) -bag) . i = n

proof end;

theorem Th10: :: UPROOTS:10

for X being set
for S being finite Subset of X
for n being Element of NAT st n <> 0 holds
support ((S,n) -bag) = S

proof end;

theorem :: UPROOTS:11

for X being set
for S being finite Subset of X
for n being Element of NAT st ( S is empty or n = 0 ) holds
(S,n) -bag = EmptyBag X

proof end;

theorem Th12: :: UPROOTS:12

for X being set
for S, T being finite Subset of X
for n being Element of NAT st S misses T holds
((S \/ T),n) -bag = ((S,n) -bag) + ((T,n) -bag)

proof end;

definition

let X be set ;
mode Rbag of X is real-valued finite-support ManySortedSet of X;

end;

definition

let A be set ;
let b be Rbag of A;
func Sum b -> real number means :Def3: :: UPROOTS:def 3
ex f being FinSequence of REAL st
( it = Sum f & f = b * (canFS (support b)) );
existence
ex b₁ being real number ex f being FinSequence of REAL st
( b₁ = Sum f & f = b * (canFS (support b)) )

proof end;

uniqueness
for b₁, b₂ being real number st ex f being FinSequence of REAL st
( b₁ = Sum f & f = b * (canFS (support b)) ) & ex f being FinSequence of REAL st
( b₂ = Sum f & f = b * (canFS (support b)) ) holds
b₁ = b₂ ;

end;

:: deftheorem Def3 defines Sum UPROOTS:def 3 :
for A being set
for b being Rbag of A
for b₃ being real number holds
( b₃ = Sum b iff ex f being FinSequence of REAL st
( b₃ = Sum f & f = b * (canFS (support b)) ) );

notation

let A be set ;
let b be bag of A;
synonym degree b for Sum b;

end;

definition

let A be set ;
let b be bag of A;
:: original: Sum
redefine func degree b -> Element of NAT means :Def4: :: UPROOTS:def 4
ex f being FinSequence of NAT st
( it = Sum f & f = b * (canFS (support b)) );
coherence
Sum b is Element of NAT

proof end;

compatibility
for b₁ being Element of NAT holds
( b₁ = Sum b iff ex f being FinSequence of NAT st
( b₁ = Sum f & f = b * (canFS (support b)) ) )

proof end;

end;

:: deftheorem Def4 defines degree UPROOTS:def 4 :
for A being set
for b being bag of A
for b₃ being Element of NAT holds
( b₃ = degree b iff ex f being FinSequence of NAT st
( b₃ = Sum f & f = b * (canFS (support b)) ) );

theorem Th13: :: UPROOTS:13

for A being set
for b being Rbag of A st b = EmptyBag A holds
Sum b = 0

proof end;

theorem Th14: :: UPROOTS:14

for A being set
for b being bag of A st Sum b = 0 holds
b = EmptyBag A

proof end;

theorem Th15: :: UPROOTS:15

for A being set
for S being finite Subset of A
for b being bag of A holds
( ( S = support b & degree b = card S ) iff b = (S,1) -bag )

proof end;

theorem Th16: :: UPROOTS:16

for A being set
for S being finite Subset of A
for b being Rbag of A st support b c= S holds
ex f being FinSequence of REAL st
( f = b * (canFS S) & Sum b = Sum f )

proof end;

theorem Th17: :: UPROOTS:17

for A being set
for b, b1, b2 being Rbag of A st b = b1 + b2 holds
Sum b = (Sum b1) + (Sum b2)

proof end;

theorem Th18: :: UPROOTS:18

for L being non empty unital associative commutative multMagma
for f, g being FinSequence of L
for p being Permutation of (dom f) st g = f * p holds
Product g = Product f

proof end;

begin

definition

let L be non empty ZeroStr ;
let p be Polynomial of L;
attr p is non-zero means :Def5: :: UPROOTS:def 5
p <> 0_. L;

end;

:: deftheorem Def5 defines non-zero UPROOTS:def 5 :
for L being non empty ZeroStr
for p being Polynomial of L holds
( p is non-zero iff p <> 0_. L );

theorem Th19: :: UPROOTS:19

for L being non empty ZeroStr
for p being Polynomial of L holds
( p is non-zero iff len p > 0 )

proof end;

registration

let L be non empty non trivial ZeroStr ;
cluster Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support non-zero Element of bool [:NAT, the carrier of L:];
existence
ex b₁ being Polynomial of L st b₁ is non-zero

proof end;

end;

registration

let L be non empty non degenerated multLoopStr_0 ;
let x be Element of L;
cluster <%x,(1. L)%> -> non-zero ;
correctness
coherence
<%x,(1. L)%> is non-zero ;

proof end;

end;

theorem Th20: :: UPROOTS:20

for L being non empty ZeroStr
for p being Polynomial of L st len p > 0 holds
p . ((len p) -' 1) <> 0. L

proof end;

theorem Th21: :: UPROOTS:21

for L being non empty ZeroStr
for p being AlgSequence of L st len p = 1 holds
( p = <%(p . 0)%> & p . 0 <> 0. L )

proof end;

theorem Th22: :: UPROOTS:22

for L being non empty right_complementable right-distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of L holds p *' (0_. L) = 0_. L

proof end;

registration

cluster non empty non degenerated right_complementable unital associative commutative distributive Abelian add-associative right_zeroed domRing-like algebraic-closed doubleLoopStr ;
existence
ex b₁ being non empty unital doubleLoopStr st
( b₁ is algebraic-closed & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is Abelian & b₁ is commutative & b₁ is associative & b₁ is distributive & b₁ is domRing-like & not b₁ is degenerated )

proof end;

end;

theorem Th23: :: UPROOTS:23

for L being non empty right_complementable distributive add-associative right_zeroed domRing-like doubleLoopStr
for p, q being Polynomial of L holds
( not p *' q = 0_. L or p = 0_. L or q = 0_. L )

proof end;

registration

let L be non empty right_complementable distributive add-associative right_zeroed domRing-like doubleLoopStr ;
cluster Polynom-Ring L -> domRing-like ;
correctness
coherence
Polynom-Ring L is domRing-like ;

proof end;

end;

registration

let L be domRing;
let p, q be non-zero Polynomial of L;
cluster p *' q -> non-zero ;
correctness
coherence
p *' q is non-zero ;

proof end;

end;

theorem :: UPROOTS:24

for L being non degenerated comRing
for p, q being Polynomial of L holds (Roots p) \/ (Roots q) c= Roots (p *' q)

proof end;

theorem Th25: :: UPROOTS:25

for L being domRing
for p, q being Polynomial of L holds Roots (p *' q) = (Roots p) \/ (Roots q)

proof end;

theorem Th26: :: UPROOTS:26

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of L
for pc being Element of (Polynom-Ring L) st p = pc holds
- p = - pc

proof end;

theorem Th27: :: UPROOTS:27

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of L
for pc, qc being Element of (Polynom-Ring L) st p = pc & q = qc holds
p - q = pc - qc

proof end;

theorem Th28: :: UPROOTS:28

for L being non empty right_complementable distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q, r being Polynomial of L holds (p *' q) - (p *' r) = p *' (q - r)

proof end;

theorem Th29: :: UPROOTS:29

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of L st p - q = 0_. L holds
p = q

proof end;

theorem Th30: :: UPROOTS:30

for L being non empty right_complementable distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for p, q, r being Polynomial of L st p <> 0_. L & p *' q = p *' r holds
q = r

proof end;

theorem Th31: :: UPROOTS:31

for L being domRing
for n being Element of NAT
for p being Polynomial of L st p <> 0_. L holds
p `^ n <> 0_. L

proof end;

theorem Th32: :: UPROOTS:32

for L being comRing
for i, j being Element of NAT
for p being Polynomial of L holds (p `^ i) *' (p `^ j) = p `^ (i + j)

proof end;

theorem Th33: :: UPROOTS:33

for L being non empty multLoopStr_0 holds 1_. L = <%(1. L)%>

proof end;

theorem :: UPROOTS:34

for L being non empty right_complementable right-distributive well-unital add-associative right_zeroed doubleLoopStr
for p being Polynomial of L holds p *' <%(1. L)%> = p

proof end;

theorem Th35: :: UPROOTS:35

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of L st ( len p = 0 or len q = 0 ) holds
len (p *' q) = 0

proof end;

theorem Th36: :: UPROOTS:36

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of L st p *' q is non-zero holds
( p is non-zero & q is non-zero )

proof end;

theorem :: UPROOTS:37

for L being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of L st (p . ((len p) -' 1)) * (q . ((len q) -' 1)) <> 0. L holds
0 < len (p *' q)

proof end;

theorem Th38: :: UPROOTS:38

for L being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p, q being Polynomial of L st 1 < len p & 1 < len q holds
len p < len (p *' q)

proof end;

theorem Th39: :: UPROOTS:39

for L being non empty right_complementable left-distributive add-associative right_zeroed doubleLoopStr
for a, b being Element of L
for p being Polynomial of L holds
( (<%a,b%> *' p) . 0 = a * (p . 0) & ( for i being Nat holds (<%a,b%> *' p) . (i + 1) = (a * (p . (i + 1))) + (b * (p . i)) ) )

proof end;

theorem Th40: :: UPROOTS:40

for L being non empty non degenerated right_complementable associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for r being Element of L
for q being non-zero Polynomial of L holds len (<%r,(1. L)%> *' q) = (len q) + 1

proof end;

theorem Th41: :: UPROOTS:41

for L being non degenerated comRing
for x being Element of L
for i being Element of NAT holds len (<%x,(1. L)%> `^ i) = i + 1

proof end;

registration

let L be non degenerated comRing;
let x be Element of L;
let n be Element of NAT ;
cluster <%x,(1. L)%> `^ n -> non-zero ;
correctness
coherence
<%x,(1. L)%> `^ n is non-zero ;

proof end;

end;

theorem Th42: :: UPROOTS:42

for L being non degenerated comRing
for x being Element of L
for q being non-zero Polynomial of L
for i being Element of NAT holds len ((<%x,(1. L)%> `^ i) *' q) = i + (len q)

proof end;

theorem Th43: :: UPROOTS:43

for L being non empty non degenerated right_complementable associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for r being Element of L
for p, q being Polynomial of L st p = <%r,(1. L)%> *' q & p . ((len p) -' 1) = 1. L holds
q . ((len q) -' 1) = 1. L

proof end;

begin

definition

let L be non empty ZeroStr ;
let p be Polynomial of L;
let n be Nat;
func poly_shift (p,n) -> Polynomial of L means :Def6: :: UPROOTS:def 6
for i being Nat holds it . i = p . (n + i);
existence
ex b₁ being Polynomial of L st
for i being Nat holds b₁ . i = p . (n + i)

proof end;

uniqueness
for b₁, b₂ being Polynomial of L st ( for i being Nat holds b₁ . i = p . (n + i) ) & ( for i being Nat holds b₂ . i = p . (n + i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines poly_shift UPROOTS:def 6 :
for L being non empty ZeroStr
for p being Polynomial of L
for n being Nat
for b₄ being Polynomial of L holds
( b₄ = poly_shift (p,n) iff for i being Nat holds b₄ . i = p . (n + i) );

theorem Th44: :: UPROOTS:44

for L being non empty ZeroStr
for p being Polynomial of L holds poly_shift (p,0) = p

proof end;

theorem Th45: :: UPROOTS:45

for L being non empty ZeroStr
for n being Element of NAT
for p being Polynomial of L st n >= len p holds
poly_shift (p,n) = 0_. L

proof end;

theorem Th46: :: UPROOTS:46

for L being non empty non degenerated multLoopStr_0
for n being Element of NAT
for p being Polynomial of L st n <= len p holds
(len (poly_shift (p,n))) + n = len p

proof end;

theorem Th47: :: UPROOTS:47

for L being non degenerated comRing
for x being Element of L
for n being Element of NAT
for p being Polynomial of L st n < len p holds
eval ((poly_shift (p,n)),x) = (x * (eval ((poly_shift (p,(n + 1))),x))) + (p . n)

proof end;

theorem Th48: :: UPROOTS:48

for L being non degenerated comRing
for p being Polynomial of L st len p = 1 holds
Roots p = {}

proof end;

definition

let L be non degenerated comRing;
let r be Element of L;
let p be Polynomial of L;
assume A1: r is_a_root_of p ;
func poly_quotient (p,r) -> Polynomial of L means :Def7: :: UPROOTS:def 7
( (len it) + 1 = len p & ( for i being Nat holds it . i = eval ((poly_shift (p,(i + 1))),r) ) ) if len p > 0
otherwise it = 0_. L;
existence
( ( len p > 0 implies ex b₁ being Polynomial of L st
( (len b₁) + 1 = len p & ( for i being Nat holds b₁ . i = eval ((poly_shift (p,(i + 1))),r) ) ) ) & ( not len p > 0 implies ex b₁ being Polynomial of L st b₁ = 0_. L ) )

proof end;

uniqueness
for b₁, b₂ being Polynomial of L holds
( ( len p > 0 & (len b₁) + 1 = len p & ( for i being Nat holds b₁ . i = eval ((poly_shift (p,(i + 1))),r) ) & (len b₂) + 1 = len p & ( for i being Nat holds b₂ . i = eval ((poly_shift (p,(i + 1))),r) ) implies b₁ = b₂ ) & ( not len p > 0 & b₁ = 0_. L & b₂ = 0_. L implies b₁ = b₂ ) )

proof end;

consistency
for b₁ being Polynomial of L holds verum ;

end;

:: deftheorem Def7 defines poly_quotient UPROOTS:def 7 :
for L being non degenerated comRing
for r being Element of L
for p being Polynomial of L st r is_a_root_of p holds
for b₄ being Polynomial of L holds
( ( len p > 0 implies ( b₄ = poly_quotient (p,r) iff ( (len b₄) + 1 = len p & ( for i being Nat holds b₄ . i = eval ((poly_shift (p,(i + 1))),r) ) ) ) ) & ( not len p > 0 implies ( b₄ = poly_quotient (p,r) iff b₄ = 0_. L ) ) );

theorem Th49: :: UPROOTS:49

for L being non degenerated comRing
for r being Element of L
for p being non-zero Polynomial of L st r is_a_root_of p holds
len (poly_quotient (p,r)) > 0

proof end;

theorem Th50: :: UPROOTS:50

for L being non empty right_complementable left-distributive well-unital add-associative right_zeroed doubleLoopStr
for x being Element of L holds Roots <%(- x),(1. L)%> = {x}

proof end;

theorem Th51: :: UPROOTS:51

for L being non trivial comRing
for x being Element of L
for p, q being Polynomial of L st p = <%(- x),(1. L)%> *' q holds
x is_a_root_of p

proof end;

theorem Th52: :: UPROOTS:52

for L being non degenerated comRing
for r being Element of L
for p being Polynomial of L st r is_a_root_of p holds
p = <%(- r),(1. L)%> *' (poly_quotient (p,r))

proof end;

theorem :: UPROOTS:53

for L being non degenerated comRing
for r being Element of L
for p, q being Polynomial of L st p = <%(- r),(1. L)%> *' q holds
r is_a_root_of p

proof end;

Lm1: now

let L be domRing; :: thesis: for p being non-zero Polynomial of L holds
( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) )
let p be non-zero Polynomial of L; :: thesis: ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) )
defpred S₁[ Nat] means for p being Polynomial of L st len p = $1 holds
( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) );
p <> 0_. L by Def5;
then len p <> 0 by POLYNOM4:8;
then A1: len p >= 0 + 1 by NAT_1:13;
A2: for n being Nat st n >= 1 & S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( n >= 1 & S₁[n] implies S₁[n + 1] )
assume that
n >= 1 and
A3: S₁[n] ; :: thesis: S₁[n + 1]
let p be Polynomial of L; :: thesis: ( len p = n + 1 implies ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) ) )
assume A4: len p = n + 1 ; :: thesis: ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) )

per cases ( p is with_roots or not p is with_roots ) ;

suppose p is with_roots ; :: thesis: ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) )

then consider x being Element of L such that
A5: x is_a_root_of p by POLYNOM5:def 7;
set r = <%(- x),(1. L)%>;
set pq = poly_quotient (p,x);
A6: p = <%(- x),(1. L)%> *' (poly_quotient (p,x)) by A5, Th52;
then A7: Roots p = (Roots <%(- x),(1. L)%>) \/ (Roots (poly_quotient (p,x))) by Th25;
A8: Roots <%(- x),(1. L)%> = {x} by Th50;
then reconsider Rr = Roots <%(- x),(1. L)%> as finite set ;
A9: (len (poly_quotient (p,x))) + 1 = len p by A4, A5, Def7;
then consider k being Element of NAT such that
A10: k = card (Roots (poly_quotient (p,x))) and
A11: k < len (poly_quotient (p,x)) by A3, A4;
reconsider Rpq = Roots (poly_quotient (p,x)) as finite set by A3, A4, A9;
reconsider Rp = Rpq \/ Rr as finite set ;
card Rr = 1 by A8, CARD_1:50;
then A12: card Rp <= k + 1 by A10, CARD_2:62;
Roots (poly_quotient (p,x)) is finite by A3, A4, A9;
hence Roots p is finite by A8, A7; :: thesis: ex n being Element of NAT st
( n = card (Roots p) & n < len p )
take m = card Rp; :: thesis: ( m = card (Roots p) & m < len p )
thus m = card (Roots p) by A6, Th25; :: thesis: m < len p
k + 1 < n + 1 by A4, A9, A11, XREAL_1:10;
hence m < len p by A4, A12, XXREAL_0:2; :: thesis: verum

end;

suppose A13: not p is with_roots ; :: thesis: ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) )

A14: now

assume Roots p <> {} ; :: thesis: contradiction
then consider x being set such that
A15: x in Roots p by XBOOLE_0:def 1;
reconsider x = x as Element of L by A15;
x is_a_root_of p by A15, POLYNOM5:def 9;
hence contradiction by A13, POLYNOM5:def 7; :: thesis: verum

end;

hence Roots p is finite ; :: thesis: ex n being Element of NAT st
( n = card (Roots p) & n < len p )
take 0 ; :: thesis: ( 0 = card (Roots p) & 0 < len p )
thus 0 = card (Roots p) by A14; :: thesis: 0 < len p
thus 0 < len p by A4; :: thesis: verum

end;

end;

end;

A16: S₁[1]

proof

let p be Polynomial of L; :: thesis: ( len p = 1 implies ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) ) )
assume A17: len p = 1 ; :: thesis: ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) )
hence Roots p is finite by Th48; :: thesis: ex n being Element of NAT st
( n = card (Roots p) & n < len p )
take 0 ; :: thesis: ( 0 = card (Roots p) & 0 < len p )
thus 0 = card (Roots p) by A17, Th48, CARD_1:47; :: thesis: 0 < len p
thus 0 < len p by A17; :: thesis: verum

end;

for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A16, A2);
hence ( Roots p is finite & ex n being Element of NAT st
( n = card (Roots p) & n < len p ) ) by A1; :: thesis: verum

end;

begin

registration

let L be domRing;
let p be non-zero Polynomial of L;
cluster Roots p -> finite ;
correctness
coherence
Roots p is finite ;
by Lm1;

end;

definition

let L be non degenerated comRing;
let x be Element of L;
let p be non-zero Polynomial of L;
func multiplicity (p,x) -> Element of NAT means :Def8: :: UPROOTS:def 8
ex F being non empty finite Subset of NAT st
( F = { k where k is Element of NAT : ex q being Polynomial of L st p = (<%(- x),(1. L)%> `^ k) *' q } & it = max F );
existence
ex b<sub>1</sub> being Element of NAT ex F being non empty finite Subset of NAT st
( F = { k where k is Element of NAT : ex q being Polynomial of L st p = (<%(- x),(1. L)%> `^ k) *' q } & b₁ = max F )

proof end;

uniqueness
for b₁, b₂ being Element of NAT st ex F being non empty finite Subset of NAT st
( F = { k where k is Element of NAT : ex q being Polynomial of L st p = (<%(- x),(1. L)%> `^ k) *' q } & b<sub>1</sub> = max F ) & ex F being non empty finite Subset of NAT st
( F = { k where k is Element of NAT : ex q being Polynomial of L st p = (<%(- x),(1. L)%> `^ k) *' q } & b₂ = max F ) holds
b₁ = b₂ ;

end;

:: deftheorem Def8 defines multiplicity UPROOTS:def 8 :
for L being non degenerated comRing
for x being Element of L
for p being non-zero Polynomial of L
for b₄ being Element of NAT holds
( b₄ = multiplicity (p,x) iff ex F being non empty finite Subset of NAT st
( F = { k where k is Element of NAT : ex q being Polynomial of L st p = (<%(- x),(1. L)%> `^ k) *' q } & b₄ = max F ) );

theorem Th54: :: UPROOTS:54

for L being non degenerated comRing
for p being non-zero Polynomial of L
for x being Element of L holds
( x is_a_root_of p iff multiplicity (p,x) >= 1 )

proof end;

theorem Th55: :: UPROOTS:55

for L being non degenerated comRing
for x being Element of L holds multiplicity (<%(- x),(1. L)%>,x) = 1

proof end;

definition

let L be domRing;
let p be non-zero Polynomial of L;
func BRoots p -> bag of the carrier of L means :Def9: :: UPROOTS:def 9
( support it = Roots p & ( for x being Element of L holds it . x = multiplicity (p,x) ) );
existence
ex b₁ being bag of the carrier of L st
( support b₁ = Roots p & ( for x being Element of L holds b₁ . x = multiplicity (p,x) ) )

proof end;

uniqueness
for b₁, b₂ being bag of the carrier of L st support b₁ = Roots p & ( for x being Element of L holds b₁ . x = multiplicity (p,x) ) & support b₂ = Roots p & ( for x being Element of L holds b₂ . x = multiplicity (p,x) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines BRoots UPROOTS:def 9 :
for L being domRing
for p being non-zero Polynomial of L
for b₃ being bag of the carrier of L holds
( b₃ = BRoots p iff ( support b₃ = Roots p & ( for x being Element of L holds b₃ . x = multiplicity (p,x) ) ) );

theorem Th56: :: UPROOTS:56

for L being domRing
for x being Element of L holds BRoots <%(- x),(1. L)%> = ({x},1) -bag

proof end;

theorem Th57: :: UPROOTS:57

for L being domRing
for x being Element of L
for p, q being non-zero Polynomial of L holds multiplicity ((p *' q),x) = (multiplicity (p,x)) + (multiplicity (q,x))

proof end;

theorem Th58: :: UPROOTS:58

for L being domRing
for p, q being non-zero Polynomial of L holds BRoots (p *' q) = (BRoots p) + (BRoots q)

proof end;

Lm2: now

let L be domRing; :: thesis: for p, q being non-zero Polynomial of L holds degree (BRoots (p *' q)) = (degree (BRoots p)) + (degree (BRoots q))
let p, q be non-zero Polynomial of L; :: thesis: degree (BRoots (p *' q)) = (degree (BRoots p)) + (degree (BRoots q))
BRoots (p *' q) = (BRoots p) + (BRoots q) by Th58;
hence degree (BRoots (p *' q)) = (degree (BRoots p)) + (degree (BRoots q)) by Th17; :: thesis: verum

end;

theorem Th59: :: UPROOTS:59

for L being domRing
for p being non-zero Polynomial of L st len p = 1 holds
degree (BRoots p) = 0

proof end;

theorem Th60: :: UPROOTS:60

for L being domRing
for x being Element of L
for n being Element of NAT holds degree (BRoots (<%(- x),(1. L)%> `^ n)) = n

proof end;

theorem :: UPROOTS:61

for L being algebraic-closed domRing
for p being non-zero Polynomial of L holds degree (BRoots p) = (len p) -' 1

proof end;

definition

let L be non empty right_complementable distributive add-associative right_zeroed doubleLoopStr ;
let c be Element of L;
let n be Element of NAT ;
func fpoly_mult_root (c,n) -> FinSequence of (Polynom-Ring L) means :Def10: :: UPROOTS:def 10
( len it = n & ( for i being Element of NAT st i in dom it holds
it . i = <%(- c),(1. L)%> ) );
existence
ex b₁ being FinSequence of (Polynom-Ring L) st
( len b₁ = n & ( for i being Element of NAT st i in dom b₁ holds
b₁ . i = <%(- c),(1. L)%> ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of (Polynom-Ring L) st len b₁ = n & ( for i being Element of NAT st i in dom b₁ holds
b₁ . i = <%(- c),(1. L)%> ) & len b₂ = n & ( for i being Element of NAT st i in dom b₂ holds
b₂ . i = <%(- c),(1. L)%> ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines fpoly_mult_root UPROOTS:def 10 :
for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for c being Element of L
for n being Element of NAT
for b₄ being FinSequence of (Polynom-Ring L) holds
( b₄ = fpoly_mult_root (c,n) iff ( len b₄ = n & ( for i being Element of NAT st i in dom b₄ holds
b₄ . i = <%(- c),(1. L)%> ) ) );

definition

let L be non empty right_complementable distributive add-associative right_zeroed doubleLoopStr ;
let b be bag of the carrier of L;
func poly_with_roots b -> Polynomial of L means :Def11: :: UPROOTS:def 11
ex f being FinSequence of the carrier of (Polynom-Ring L) * ex s being FinSequence of L st
( len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) & it = Product (FlattenSeq f) );
existence
ex b₁ being Polynomial of L ex f being FinSequence of the carrier of (Polynom-Ring L) * ex s being FinSequence of L st
( len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) & b₁ = Product (FlattenSeq f) )

proof end;

uniqueness
for b₁, b₂ being Polynomial of L st ex f being FinSequence of the carrier of (Polynom-Ring L) * ex s being FinSequence of L st
( len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) & b₁ = Product (FlattenSeq f) ) & ex f being FinSequence of the carrier of (Polynom-Ring L) * ex s being FinSequence of L st
( len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) & b₂ = Product (FlattenSeq f) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines poly_with_roots UPROOTS:def 11 :
for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b being bag of the carrier of L
for b₃ being Polynomial of L holds
( b₃ = poly_with_roots b iff ex f being FinSequence of the carrier of (Polynom-Ring L) * ex s being FinSequence of L st
( len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) & b₃ = Product (FlattenSeq f) ) );

theorem Th62: :: UPROOTS:62

for L being non empty right_complementable commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds poly_with_roots (EmptyBag the carrier of L) = <%(1. L)%>

proof end;

theorem Th63: :: UPROOTS:63

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for c being Element of L holds poly_with_roots (({c},1) -bag) = <%(- c),(1. L)%>

proof end;

theorem Th64: :: UPROOTS:64

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b being bag of the carrier of L
for f being FinSequence of the carrier of (Polynom-Ring L) *
for s being FinSequence of L st len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) holds
len (FlattenSeq f) = degree b

proof end;

theorem Th65: :: UPROOTS:65

for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b being bag of the carrier of L
for f being FinSequence of the carrier of (Polynom-Ring L) *
for s being FinSequence of L
for c being Element of L st len f = card (support b) & s = canFS (support b) & ( for i being Element of NAT st i in dom f holds
f . i = fpoly_mult_root ((s /. i),(b . (s /. i))) ) holds
( ( c in support b implies card ((FlattenSeq f) " {<%(- c),(1. L)%>}) = b . c ) & ( not c in support b implies card ((FlattenSeq f) " {<%(- c),(1. L)%>}) = 0 ) )

proof end;

theorem Th66: :: UPROOTS:66

for L being comRing
for b1, b2 being bag of the carrier of L holds poly_with_roots (b1 + b2) = (poly_with_roots b1) *' (poly_with_roots b2)

proof end;

theorem :: UPROOTS:67

for L being algebraic-closed domRing
for p being non-zero Polynomial of L st p . ((len p) -' 1) = 1. L holds
p = poly_with_roots (BRoots p)

proof end;

theorem :: UPROOTS:68

for L being comRing
for s being non empty finite Subset of L
for f being FinSequence of (Polynom-Ring L) st len f = card s & ( for i being Element of NAT
for c being Element of L st i in dom f & c = (canFS s) . i holds
f . i = <%(- c),(1. L)%> ) holds
poly_with_roots ((s,1) -bag) = Product f

proof end;

theorem :: UPROOTS:69

for L being non trivial comRing
for s being non empty finite Subset of L
for x being Element of L
for f being FinSequence of L st len f = card s & ( for i being Element of NAT
for c being Element of L st i in dom f & c = (canFS s) . i holds
f . i = eval (<%(- c),(1. L)%>,x) ) holds
eval ((poly_with_roots ((s,1) -bag)),x) = Product f

proof end;