:: BINOM semantic presentation
:: deftheorem Def1 BINOM:def 1 :
canceled;
:: deftheorem Def2 BINOM:def 2 :
canceled;
:: deftheorem Def3 defines add-cancelable BINOM:def 3 :
theorem Th1: :: BINOM:1
theorem Th2: :: BINOM:2
scheme :: BINOM:sch 1
s1{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> Element of F
2(), F
4()
-> Function of
[:F1(),F2():],F
2() } :
ex b
1 being
Function of
[:NAT ,F1():],F
2() st
for b
2 being
Element of F
1() holds
( b
1 . 0,b
2 = F
3() & ( for b
3 being
Nat holds b
1 . (b3 + 1),b
2 = F
4()
. b
2,
(b1 . b3,b2) ) )
scheme :: BINOM:sch 2
s2{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> Element of F
2(), F
4()
-> Function of
[:F2(),F1():],F
2() } :
ex b
1 being
Function of
[:F1(),NAT :],F
2() st
for b
2 being
Element of F
1() holds
( b
1 . b
2,0
= F
3() & ( for b
3 being
Nat holds b
1 . b
2,
(b3 + 1) = F
4()
. (b1 . b2,b3),b
2 ) )
theorem Th3: :: BINOM:3
theorem Th4: :: BINOM:4
theorem Th5: :: BINOM:5
theorem Th6: :: BINOM:6
:: deftheorem Def4 defines + BINOM:def 4 :
theorem Th7: :: BINOM:7
:: deftheorem Def5 defines |^ BINOM:def 5 :
theorem Th8: :: BINOM:8
theorem Th9: :: BINOM:9
theorem Th10: :: BINOM:10
theorem Th11: :: BINOM:11
theorem Th12: :: BINOM:12
definition
let c
1 be non
empty LoopStr ;
func Nat-mult-left c
1 -> Function of
[:NAT ,the carrier of a1:],the
carrier of a
1 means :
Def6:
:: BINOM:def 6
for b
1 being
Element of a
1 holds
( a
2 . 0,b
1 = 0. a
1 & ( for b
2 being
Nat holds a
2 . (b2 + 1),b
1 = b
1 + (a2 . b2,b1) ) );
existence
ex b1 being Function of [:NAT ,the carrier of c1:],the carrier of c1 st
for b2 being Element of c1 holds
( b1 . 0,b2 = 0. c1 & ( for b3 being Nat holds b1 . (b3 + 1),b2 = b2 + (b1 . b3,b2) ) )
uniqueness
for b1, b2 being Function of [:NAT ,the carrier of c1:],the carrier of c1 holds
( ( for b3 being Element of c1 holds
( b1 . 0,b3 = 0. c1 & ( for b4 being Nat holds b1 . (b4 + 1),b3 = b3 + (b1 . b4,b3) ) ) ) & ( for b3 being Element of c1 holds
( b2 . 0,b3 = 0. c1 & ( for b4 being Nat holds b2 . (b4 + 1),b3 = b3 + (b2 . b4,b3) ) ) ) implies b1 = b2 )
func Nat-mult-right c
1 -> Function of
[:the carrier of a1,NAT :],the
carrier of a
1 means :
Def7:
:: BINOM:def 7
for b
1 being
Element of a
1 holds
( a
2 . b
1,0
= 0. a
1 & ( for b
2 being
Nat holds a
2 . b
1,
(b2 + 1) = (a2 . b1,b2) + b
1 ) );
existence
ex b1 being Function of [:the carrier of c1,NAT :],the carrier of c1 st
for b2 being Element of c1 holds
( b1 . b2,0 = 0. c1 & ( for b3 being Nat holds b1 . b2,(b3 + 1) = (b1 . b2,b3) + b2 ) )
uniqueness
for b1, b2 being Function of [:the carrier of c1,NAT :],the carrier of c1 holds
( ( for b3 being Element of c1 holds
( b1 . b3,0 = 0. c1 & ( for b4 being Nat holds b1 . b3,(b4 + 1) = (b1 . b3,b4) + b3 ) ) ) & ( for b3 being Element of c1 holds
( b2 . b3,0 = 0. c1 & ( for b4 being Nat holds b2 . b3,(b4 + 1) = (b2 . b3,b4) + b3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines Nat-mult-left BINOM:def 6 :
:: deftheorem Def7 defines Nat-mult-right BINOM:def 7 :
:: deftheorem Def8 defines * BINOM:def 8 :
:: deftheorem Def9 defines * BINOM:def 9 :
theorem Th13: :: BINOM:13
theorem Th14: :: BINOM:14
theorem Th15: :: BINOM:15
Lemma16:
for b1 being non empty LoopStr
for b2 being Element of b1
for b3 being Nat holds (b3 + 1) * b2 = b2 + (b3 * b2)
by Def6;
Lemma17:
for b1 being non empty LoopStr
for b2 being Element of b1
for b3 being Nat holds b2 * (b3 + 1) = (b2 * b3) + b2
by Def7;
theorem Th16: :: BINOM:16
theorem Th17: :: BINOM:17
theorem Th18: :: BINOM:18
theorem Th19: :: BINOM:19
theorem Th20: :: BINOM:20
theorem Th21: :: BINOM:21
theorem Th22: :: BINOM:22
definition
let c
1 be non
empty unital doubleLoopStr ;
let c
2, c
3 be
Element of c
1;
let c
4 be
Nat;
func c
2,c
3 In_Power c
4 -> FinSequence of the
carrier of a
1 means :
Def10:
:: BINOM:def 10
(
len a
5 = a
4 + 1 & ( for b
1, b
2, b
3 being
Nat holds
( b
1 in dom a
5 & b
3 = b
1 - 1 & b
2 = a
4 - b
3 implies a
5 /. b
1 = ((a4 choose b3) * (a2 |^ b2)) * (a3 |^ b3) ) ) );
existence
ex b1 being FinSequence of the carrier of c1 st
( len b1 = c4 + 1 & ( for b2, b3, b4 being Nat holds
( b2 in dom b1 & b4 = b2 - 1 & b3 = c4 - b4 implies b1 /. b2 = ((c4 choose b4) * (c2 |^ b3)) * (c3 |^ b4) ) ) )
uniqueness
for b1, b2 being FinSequence of the carrier of c1 holds
( len b1 = c4 + 1 & ( for b3, b4, b5 being Nat holds
( b3 in dom b1 & b5 = b3 - 1 & b4 = c4 - b5 implies b1 /. b3 = ((c4 choose b5) * (c2 |^ b4)) * (c3 |^ b5) ) ) & len b2 = c4 + 1 & ( for b3, b4, b5 being Nat holds
( b3 in dom b2 & b5 = b3 - 1 & b4 = c4 - b5 implies b2 /. b3 = ((c4 choose b5) * (c2 |^ b4)) * (c3 |^ b5) ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines In_Power BINOM:def 10 :
theorem Th23: :: BINOM:23
theorem Th24: :: BINOM:24
theorem Th25: :: BINOM:25
theorem Th26: :: BINOM:26