Article POLYNOM4, MML version 4.99.1005
:: POLYNOM4:th 3
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
st 1 <= b3 & b3 <= len b2
holds b2 = ((b2 | (b3 -' 1)) ^ <*b2 . b3*>) ^ (b2 /^ b3);
:: POLYNOM4:condreg 1
registration
cluster non empty right_add-cancelable almost_left_invertible left_zeroed associative commutative right-distributive well-unital -> domRing-like (doubleLoopStr);
end;
:: POLYNOM4:exreg 1
registration
cluster non empty non degenerated non trivial right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative distributive domRing-like doubleLoopStr;
end;
:: POLYNOM4:th 5
theorem
for b1 being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(0_. b1) *' b2 = 0_. b1;
:: POLYNOM4:th 6
theorem
for b1 being non empty ZeroStr holds
len 0_. b1 = 0;
:: POLYNOM4:th 7
theorem
for b1 being non empty non degenerated multLoopStr_0 holds
len 1_. b1 = 1;
:: POLYNOM4:th 8
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
st len b2 = 0
holds b2 = 0_. b1;
:: POLYNOM4:th 9
theorem
for b1 being non empty right_zeroed addLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of NAT
st len b2 <= b4 & len b3 <= b4
holds len (b2 + b3) <= b4;
:: POLYNOM4:th 10
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
st len b2 <> len b3
holds len (b2 + b3) = max(len b2,len b3);
:: POLYNOM4:th 11
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
len - b2 = len b2;
:: POLYNOM4:th 12
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of NAT
st len b2 <= b4 & len b3 <= b4
holds len (b2 - b3) <= b4;
:: POLYNOM4:th 13
theorem
for b1 being non empty right_complementable add-associative right_zeroed associative commutative distributive left_unital doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
st (b2 . ((len b2) -' 1)) * (b3 . ((len b3) -' 1)) <> 0. b1
holds len (b2 *' b3) = ((len b2) + len b3) - 1;
:: POLYNOM4:funcnot 1 => POLYNOM4:func 1
definition
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
func Leading-Monomial A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
it . ((len a2) -' 1) = a2 . ((len a2) -' 1) &
(for b1 being Element of NAT
st b1 <> (len a2) -' 1
holds it . b1 = 0. a1);
end;
:: POLYNOM4:def 1
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 = Leading-Monomial b2
iff
b3 . ((len b2) -' 1) = b2 . ((len b2) -' 1) &
(for b4 being Element of NAT
st b4 <> (len b2) -' 1
holds b3 . b4 = 0. b1);
:: POLYNOM4:th 14
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
Leading-Monomial b2 = (0_. b1) +*((len b2) -' 1,b2 . ((len b2) -' 1));
:: POLYNOM4:funcreg 1
registration
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
cluster Leading-Monomial a2 -> Function-like quasi_total finite-Support;
end;
:: POLYNOM4:th 15
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
st len b2 = 0
holds Leading-Monomial b2 = 0_. b1;
:: POLYNOM4:th 16
theorem
for b1 being non empty ZeroStr holds
Leading-Monomial 0_. b1 = 0_. b1;
:: POLYNOM4:th 17
theorem
for b1 being non empty non degenerated multLoopStr_0 holds
Leading-Monomial 1_. b1 = 1_. b1;
:: POLYNOM4:th 18
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
len Leading-Monomial b2 = len b2;
:: POLYNOM4:th 19
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
st len b2 <> 0
holds ex b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
len b3 < len b2 &
b2 = b3 + Leading-Monomial b2 &
(for b4 being Element of NAT
st b4 < (len b2) - 1
holds b3 . b4 = b2 . b4);
:: POLYNOM4:funcnot 2 => POLYNOM4:func 2
definition
let a1 be non empty unital doubleLoopStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
func eval(A2,A3) -> Element of the carrier of a1 means
ex b1 being FinSequence of the carrier of a1 st
it = Sum b1 &
len b1 = len a2 &
(for b2 being Element of NAT
st b2 in dom b1
holds b1 . b2 = (a2 . (b2 -' 1)) * ((power a1) .(a3,b2 -' 1)));
end;
:: POLYNOM4:def 2
theorem
for b1 being non empty unital doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1 holds
b4 = eval(b2,b3)
iff
ex b5 being FinSequence of the carrier of b1 st
b4 = Sum b5 &
len b5 = len b2 &
(for b6 being Element of NAT
st b6 in dom b5
holds b5 . b6 = (b2 . (b6 -' 1)) * ((power b1) .(b3,b6 -' 1)));
:: POLYNOM4:th 20
theorem
for b1 being non empty unital doubleLoopStr
for b2 being Element of the carrier of b1 holds
eval(0_. b1,b2) = 0. b1;
:: POLYNOM4:th 21
theorem
for b1 being non empty non degenerated right_complementable add-associative right_zeroed associative well-unital doubleLoopStr
for b2 being Element of the carrier of b1 holds
eval(1_. b1,b2) = 1. b1;
:: POLYNOM4:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed unital left-distributive doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1 holds
eval(b2 + b3,b4) = (eval(b2,b4)) + eval(b3,b4);
:: POLYNOM4:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed unital distributive doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
eval(- b2,b3) = - eval(b2,b3);
:: POLYNOM4:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed unital distributive doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1 holds
eval(b2 - b3,b4) = (eval(b2,b4)) - eval(b3,b4);
:: POLYNOM4:th 25
theorem
for b1 being non empty right_complementable add-associative right_zeroed unital distributive doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
eval(Leading-Monomial b2,b3) = (b2 . ((len b2) -' 1)) * ((power b1) .(b3,(len b2) -' 1));
:: POLYNOM4:th 26
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1 holds
eval((Leading-Monomial b2) *' b3,b4) = (eval(Leading-Monomial b2,b4)) * eval(b3,b4);
:: POLYNOM4:th 27
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1 holds
eval(b2 *' b3,b4) = (eval(b2,b4)) * eval(b3,b4);
:: POLYNOM4:funcnot 3 => POLYNOM4:func 3
definition
let a1 be non empty right_complementable add-associative right_zeroed unital distributive doubleLoopStr;
let a2 be Element of the carrier of a1;
func Polynom-Evaluation(A1,A2) -> Function-like quasi_total Relation of the carrier of Polynom-Ring a1,the carrier of a1 means
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 holds
it . b1 = eval(b1,a2);
end;
:: POLYNOM4:def 3
theorem
for b1 being non empty right_complementable add-associative right_zeroed unital distributive doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of Polynom-Ring b1,the carrier of b1 holds
b3 = Polynom-Evaluation(b1,b2)
iff
for b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
b3 . b4 = eval(b4,b2);
:: POLYNOM4:funcreg 2
registration
let a1 be non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be Element of the carrier of a1;
cluster Polynom-Evaluation(a1,a2) -> Function-like quasi_total unity-preserving;
end;
:: POLYNOM4:funcreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed unital distributive doubleLoopStr;
let a2 be Element of the carrier of a1;
cluster Polynom-Evaluation(a1,a2) -> Function-like quasi_total additive;
end;
:: POLYNOM4:funcreg 4
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital doubleLoopStr;
let a2 be Element of the carrier of a1;
cluster Polynom-Evaluation(a1,a2) -> Function-like quasi_total multiplicative;
end;
:: POLYNOM4:funcreg 5
registration
let a1 be non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be Element of the carrier of a1;
cluster Polynom-Evaluation(a1,a2) -> Function-like quasi_total RingHomomorphism;
end;