:: POLYNOM2 semantic presentation
scheme :: POLYNOM2:sch 3
s3{ F
1()
-> set , F
2()
-> Element of F
1(), F
3()
-> Nat, F
4()
-> FinSequence of F
1(), F
5()
-> FinSequence of F
1(), P
1[
set ,
set ,
set ] } :
provided
E1:
for b
1 being
Nat holds
( 1
<= b
1 & b
1 <= F
3()
- 1 implies for b
2, b
3, b
4 being
Element of F
1() holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 ) )
and
E2:
(
len F
4()
= F
3() & ( F
4()
/. 1
= F
2() or F
3()
= 0 ) & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 <= F
3()
- 1 implies P
1[b
1,F
4()
/. b
1,F
4()
/. (b1 + 1)] ) ) )
and
E3:
(
len F
5()
= F
3() & ( F
5()
/. 1
= F
2() or F
3()
= 0 ) & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 <= F
3()
- 1 implies P
1[b
1,F
5()
/. b
1,F
5()
/. (b1 + 1)] ) ) )
scheme :: POLYNOM2:sch 4
s4{ F
1()
-> Nat, F
2()
-> Nat, P
1[
Nat] } :
for b
1 being
Nat holds
( F
1()
<= b
1 & b
1 <= F
2() implies P
1[b
1] )
provided
E1:
P
1[F
1()]
and
E2:
for b
1 being
Nat holds
( F
1()
<= b
1 & b
1 < F
2() & P
1[b
1] implies P
1[b
1 + 1] )
scheme :: POLYNOM2:sch 5
s5{ F
1()
-> Nat, F
2()
-> Nat, P
1[
Nat] } :
for b
1 being
Nat holds
( F
1()
<= b
1 & b
1 <= F
2() implies P
1[b
1] )
provided
E1:
P
1[F
1()]
and
E2:
for b
1 being
Nat holds
( F
1()
<= b
1 & b
1 < F
2() & ( for b
2 being
Nat holds
( F
1()
<= b
2 & b
2 <= b
1 implies P
1[b
2] ) ) implies P
1[b
1 + 1] )
Lemma1:
for b1 being set
for b2 being Subset of b1
for b3 being Order of b1 holds
( b3 is_reflexive_in b2 & b3 is_antisymmetric_in b2 & b3 is_transitive_in b2 )
Lemma2:
for b1 being set
for b2 being Subset of b1
for b3 being Order of b1 holds
( b3 is_connected_in b1 implies b3 is_connected_in b2 )
Lemma3:
for b1 being set
for b2 being Subset of b1
for b3 being Order of b1 holds
( b3 is_linear-order implies b3 linearly_orders b2 )
theorem Th1: :: POLYNOM2:1
canceled;
theorem Th2: :: POLYNOM2:2
theorem Th3: :: POLYNOM2:3
canceled;
theorem Th4: :: POLYNOM2:4
Lemma6:
for b1 being set
for b2 being finite Subset of b1
for b3 being Element of b1
for b4 being Order of b1 holds
( b4 linearly_orders {b3} \/ b2 implies b4 linearly_orders b2 )
theorem Th5: :: POLYNOM2:5
theorem Th6: :: POLYNOM2:6
theorem Th7: :: POLYNOM2:7
theorem Th8: :: POLYNOM2:8
theorem Th9: :: POLYNOM2:9
theorem Th10: :: POLYNOM2:10
Lemma12:
for b1 being set
for b2 being FinSequence of b1 holds
( dom b2 <> {} implies 1 in dom b2 )
theorem Th11: :: POLYNOM2:11
theorem Th12: :: POLYNOM2:12
theorem Th13: :: POLYNOM2:13
theorem Th14: :: POLYNOM2:14
Lemma17:
for b1 being set
for b2 being bag of b1 holds
b2 is PartFunc of b1, NAT
:: deftheorem Def1 defines empty POLYNOM2:def 1 :
theorem Th15: :: POLYNOM2:15
definition
let c
1 be
Ordinal;
let c
2 be
bag of c
1;
let c
3 be non
empty unital non
trivial doubleLoopStr ;
let c
4 be
Function of c
1,c
3;
func eval c
2,c
4 -> Element of a
3 means :
Def2:
:: POLYNOM2:def 2
ex b
1 being
FinSequence of the
carrier of a
3 st
(
len b
1 = len (SgmX (RelIncl a1),(support a2)) & a
5 = Product b
1 & ( for b
2 being
Nat holds
( 1
<= b
2 & b
2 <= len b
1 implies b
1 /. b
2 = (power a3) . ((a4 * (SgmX (RelIncl a1),(support a2))) /. b2),
((a2 * (SgmX (RelIncl a1),(support a2))) /. b2) ) ) );
existence
ex b1 being Element of c3ex b2 being FinSequence of the carrier of c3 st
( len b2 = len (SgmX (RelIncl c1),(support c2)) & b1 = Product b2 & ( for b3 being Nat holds
( 1 <= b3 & b3 <= len b2 implies b2 /. b3 = (power c3) . ((c4 * (SgmX (RelIncl c1),(support c2))) /. b3),((c2 * (SgmX (RelIncl c1),(support c2))) /. b3) ) ) )
uniqueness
for b1, b2 being Element of c3 holds
( ex b3 being FinSequence of the carrier of c3 st
( len b3 = len (SgmX (RelIncl c1),(support c2)) & b1 = Product b3 & ( for b4 being Nat holds
( 1 <= b4 & b4 <= len b3 implies b3 /. b4 = (power c3) . ((c4 * (SgmX (RelIncl c1),(support c2))) /. b4),((c2 * (SgmX (RelIncl c1),(support c2))) /. b4) ) ) ) & ex b3 being FinSequence of the carrier of c3 st
( len b3 = len (SgmX (RelIncl c1),(support c2)) & b2 = Product b3 & ( for b4 being Nat holds
( 1 <= b4 & b4 <= len b3 implies b3 /. b4 = (power c3) . ((c4 * (SgmX (RelIncl c1),(support c2))) /. b4),((c2 * (SgmX (RelIncl c1),(support c2))) /. b4) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines eval POLYNOM2:def 2 :
Lemma21:
for b1 being set holds support (EmptyBag b1) = {}
theorem Th16: :: POLYNOM2:16
theorem Th17: :: POLYNOM2:17
Lemma24:
for b1 being Ordinal
for b2 being non empty add-associative right_zeroed right_complementable unital associative commutative distributive non trivial doubleLoopStr
for b3, b4 being bag of b1
for b5 being set holds
( not b5 in support b3 & support b4 = (support b3) \/ {b5} & ( for b6 being set holds
( b6 <> b5 implies b4 . b6 = b3 . b6 ) ) implies for b6 being Function of b1,b2
for b7 being Element of b2 holds
( b7 = (power b2) . (b6 . b5),(b4 . b5) implies eval b4,b6 = b7 * (eval b3,b6) ) )
Lemma25:
for b1 being Ordinal
for b2 being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non trivial doubleLoopStr
for b3 being bag of b1 holds
( ex b4 being set st support b3 = {b4} implies for b4 being bag of b1
for b5 being Function of b1,b2 holds eval (b3 + b4),b5 = (eval b3,b5) * (eval b4,b5) )
theorem Th18: :: POLYNOM2:18
theorem Th19: :: POLYNOM2:19
theorem Th20: :: POLYNOM2:20
:: deftheorem Def3 defines @ POLYNOM2:def 3 :
definition
let c
1 be
Ordinal;
let c
2 be non
empty add-associative right_zeroed right_complementable unital distributive non
trivial doubleLoopStr ;
let c
3 be
Polynomial of c
1,c
2;
let c
4 be
Function of c
1,c
2;
func eval c
3,c
4 -> Element of a
2 means :
Def4:
:: POLYNOM2:def 4
ex b
1 being
FinSequence of the
carrier of a
2 st
(
len b
1 = len (SgmX (BagOrder a1),(Support a3)) & a
5 = Sum b
1 & ( for b
2 being
Nat holds
( 1
<= b
2 & b
2 <= len b
1 implies b
1 /. b
2 = ((a3 * (SgmX (BagOrder a1),(Support a3))) /. b2) * (eval (((SgmX (BagOrder a1),(Support a3)) /. b2) @ ),a4) ) ) );
existence
ex b1 being Element of c2ex b2 being FinSequence of the carrier of c2 st
( len b2 = len (SgmX (BagOrder c1),(Support c3)) & b1 = Sum b2 & ( for b3 being Nat holds
( 1 <= b3 & b3 <= len b2 implies b2 /. b3 = ((c3 * (SgmX (BagOrder c1),(Support c3))) /. b3) * (eval (((SgmX (BagOrder c1),(Support c3)) /. b3) @ ),c4) ) ) )
uniqueness
for b1, b2 being Element of c2 holds
( ex b3 being FinSequence of the carrier of c2 st
( len b3 = len (SgmX (BagOrder c1),(Support c3)) & b1 = Sum b3 & ( for b4 being Nat holds
( 1 <= b4 & b4 <= len b3 implies b3 /. b4 = ((c3 * (SgmX (BagOrder c1),(Support c3))) /. b4) * (eval (((SgmX (BagOrder c1),(Support c3)) /. b4) @ ),c4) ) ) ) & ex b3 being FinSequence of the carrier of c2 st
( len b3 = len (SgmX (BagOrder c1),(Support c3)) & b2 = Sum b3 & ( for b4 being Nat holds
( 1 <= b4 & b4 <= len b3 implies b3 /. b4 = ((c3 * (SgmX (BagOrder c1),(Support c3))) /. b4) * (eval (((SgmX (BagOrder c1),(Support c3)) /. b4) @ ),c4) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines eval POLYNOM2:def 4 :
theorem Th21: :: POLYNOM2:21
theorem Th22: :: POLYNOM2:22
theorem Th23: :: POLYNOM2:23
theorem Th24: :: POLYNOM2:24
Lemma34:
for b1 being Ordinal
for b2 being non empty Abelian add-associative right_zeroed right_complementable unital distributive non trivial doubleLoopStr
for b3, b4 being Polynomial of b1,b2
for b5 being Function of b1,b2
for b6 being bag of b1 holds
( not b6 in Support b3 & Support b4 = (Support b3) \/ {b6} & ( for b7 being bag of b1 holds
( b7 <> b6 implies b4 . b7 = b3 . b7 ) ) implies eval b4,b5 = (eval b3,b5) + ((b4 . b6) * (eval b6,b5)) )
Lemma35:
for b1 being Ordinal
for b2 being non empty Abelian add-associative right_zeroed right_complementable unital distributive non trivial doubleLoopStr
for b3 being Polynomial of b1,b2 holds
( ex b4 being bag of b1 st Support b3 = {b4} implies for b4 being Polynomial of b1,b2
for b5 being Function of b1,b2 holds eval (b3 + b4),b5 = (eval b3,b5) + (eval b4,b5) )
theorem Th25: :: POLYNOM2:25
theorem Th26: :: POLYNOM2:26
Lemma37:
for b1 being Ordinal
for b2 being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non trivial doubleLoopStr
for b3, b4 being Polynomial of b1,b2
for b5, b6 being bag of b1 holds
( Support b3 = {b5} & Support b4 = {b6} implies for b7 being Function of b1,b2 holds eval (b3 *' b4),b7 = (eval b3,b7) * (eval b4,b7) )
Lemma38:
for b1 being Ordinal
for b2 being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non trivial doubleLoopStr
for b3 being Polynomial of b1,b2 holds
( ex b4 being bag of b1 st Support b3 = {b4} implies for b4 being Polynomial of b1,b2
for b5 being Function of b1,b2 holds eval (b4 *' b3),b5 = (eval b4,b5) * (eval b3,b5) )
theorem Th27: :: POLYNOM2:27
definition
let c
1 be
Ordinal;
let c
2 be non
empty add-associative right_zeroed right_complementable unital distributive non
trivial doubleLoopStr ;
let c
3 be
Function of c
1,c
2;
func Polynom-Evaluation c
1,c
2,c
3 -> Function of
(Polynom-Ring a1,a2),a
2 means :
Def5:
:: POLYNOM2:def 5
for b
1 being
Polynomial of a
1,a
2 holds a
4 . b
1 = eval b
1,a
3;
existence
ex b1 being Function of (Polynom-Ring c1,c2),c2 st
for b2 being Polynomial of c1,c2 holds b1 . b2 = eval b2,c3
uniqueness
for b1, b2 being Function of (Polynom-Ring c1,c2),c2 holds
( ( for b3 being Polynomial of c1,c2 holds b1 . b3 = eval b3,c3 ) & ( for b3 being Polynomial of c1,c2 holds b2 . b3 = eval b3,c3 ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Polynom-Evaluation POLYNOM2:def 5 :