:: RECDEF_1 semantic presentation
Lemma1:
for b1 being Nat
for b2 being non empty set
for b3 being FinSequence of b2 holds
( 1 <= b1 & b1 <= len b3 implies b3 . b1 is Element of b2 )
scheme :: RECDEF_1:sch 1
s1{ F
1()
-> set , P
1[
set ,
set ,
set ] } :
provided
E2:
for b
1 being
Natfor b
2 being
set holds
ex b
3 being
set st P
1[b
1,b
2,b
3]
and
E3:
for b
1 being
Natfor b
2, b
3, b
4 being
set holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 )
scheme :: RECDEF_1:sch 5
s5{ F
1()
-> set , F
2()
-> Nat, P
1[
set ,
set ,
set ] } :
ex b
1 being
FinSequence st
(
len b
1 = F
2() & ( b
1 . 1
= F
1() or F
2()
= 0 ) & ( for b
2 being
Nat holds
( 1
<= b
2 & b
2 < F
2() implies P
1[b
2,b
1 . b
2,b
1 . (b2 + 1)] ) ) )
provided
E2:
for b
1 being
Nat holds
( 1
<= b
1 & b
1 < F
2() implies for b
2 being
set holds
ex b
3 being
set st P
1[b
1,b
2,b
3] )
and
E3:
for b
1 being
Nat holds
( 1
<= b
1 & b
1 < F
2() implies for b
2, b
3, b
4 being
set holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 ) )
scheme :: RECDEF_1:sch 7
s7{ F
1()
-> FinSequence, P
1[
set ,
set ,
set ] } :
provided
E2:
for b
1 being
Natfor b
2 being
set holds
not ( 1
<= b
1 & b
1 < len F
1() & ( for b
3 being
set holds
not P
1[F
1()
. (b1 + 1),b
2,b
3] ) )
and
E3:
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
set holds
( 1
<= b
1 & b
1 < len F
1() & b
5 = F
1()
. (b1 + 1) & P
1[b
5,b
2,b
3] & P
1[b
5,b
2,b
4] implies b
3 = b
4 )
scheme :: RECDEF_1:sch 9
s9{ F
1()
-> set , F
2()
-> Function, F
3()
-> Function, P
1[
set ,
set ,
set ] } :
provided
E2:
(
dom F
2()
= NAT & F
2()
. 0
= F
1() & ( for b
1 being
Nat holds P
1[b
1,F
2()
. b
1,F
2()
. (b1 + 1)] ) )
and
E3:
(
dom F
3()
= NAT & F
3()
. 0
= F
1() & ( for b
1 being
Nat holds P
1[b
1,F
3()
. b
1,F
3()
. (b1 + 1)] ) )
and
E4:
for b
1 being
Natfor b
2, b
3, b
4 being
set holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 )
scheme :: RECDEF_1:sch 10
s10{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), P
1[
set ,
set ,
set ], F
3()
-> Function of
NAT ,F
1(), F
4()
-> Function of
NAT ,F
1() } :
provided
E2:
( F
3()
. 0
= F
2() & ( for b
1 being
Nat holds P
1[b
1,F
3()
. b
1,F
3()
. (b1 + 1)] ) )
and
E3:
( F
4()
. 0
= F
2() & ( for b
1 being
Nat holds P
1[b
1,F
4()
. b
1,F
4()
. (b1 + 1)] ) )
and
E4:
for b
1 being
Natfor 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 )
scheme :: RECDEF_1:sch 14
s14{ F
1()
-> set , F
2()
-> Nat, F
3()
-> FinSequence, F
4()
-> FinSequence, P
1[
set ,
set ,
set ] } :
provided
E2:
for b
1 being
Nat holds
( 1
<= b
1 & b
1 < F
2() implies for b
2, b
3, b
4 being
set holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 ) )
and
E3:
(
len F
3()
= F
2() & ( F
3()
. 1
= F
1() or F
2()
= 0 ) & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 < F
2() implies P
1[b
1,F
3()
. b
1,F
3()
. (b1 + 1)] ) ) )
and
E4:
(
len F
4()
= F
2() & ( F
4()
. 1
= F
1() or F
2()
= 0 ) & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 < F
2() implies P
1[b
1,F
4()
. b
1,F
4()
. (b1 + 1)] ) ) )
scheme :: RECDEF_1:sch 15
s15{ F
1()
-> non
empty 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
E2:
for b
1 being
Nat holds
( 1
<= b
1 & b
1 < F
3() 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
E3:
(
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() implies P
1[b
1,F
4()
. b
1,F
4()
. (b1 + 1)] ) ) )
and
E4:
(
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() implies P
1[b
1,F
5()
. b
1,F
5()
. (b1 + 1)] ) ) )
scheme :: RECDEF_1:sch 16
s16{ F
1()
-> FinSequence, P
1[
set ,
set ,
set ], F
2()
-> set , F
3()
-> set } :
provided
E2:
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
set holds
( 1
<= b
1 & b
1 < len F
1() & b
5 = F
1()
. (b1 + 1) & P
1[b
5,b
2,b
3] & P
1[b
5,b
2,b
4] implies b
3 = b
4 )
and
E3:
ex b
1 being
FinSequence st
( F
2()
= b
1 . (len b1) &
len b
1 = len F
1() & b
1 . 1
= F
1()
. 1 & ( for b
2 being
Nat holds
( 1
<= b
2 & b
2 < len F
1() implies P
1[F
1()
. (b2 + 1),b
1 . b
2,b
1 . (b2 + 1)] ) ) )
and
E4:
ex b
1 being
FinSequence st
( F
3()
= b
1 . (len b1) &
len b
1 = len F
1() & b
1 . 1
= F
1()
. 1 & ( for b
2 being
Nat holds
( 1
<= b
2 & b
2 < len F
1() implies P
1[F
1()
. (b2 + 1),b
1 . b
2,b
1 . (b2 + 1)] ) ) )
scheme :: RECDEF_1:sch 18
s18{ F
1()
-> set , F
2()
-> Nat, P
1[
set ,
set ,
set ] } :
( ex b
1 being
set ex b
2 being
Function st
( b
1 = b
2 . F
2() &
dom b
2 = NAT & b
2 . 0
= F
1() & ( for b
3 being
Nat holds P
1[b
3,b
2 . b
3,b
2 . (b3 + 1)] ) ) & ( for b
1, b
2 being
set holds
( ex b
3 being
Function st
( b
1 = b
3 . F
2() &
dom b
3 = NAT & b
3 . 0
= F
1() & ( for b
4 being
Nat holds P
1[b
4,b
3 . b
4,b
3 . (b4 + 1)] ) ) & ex b
3 being
Function st
( b
2 = b
3 . F
2() &
dom b
3 = NAT & b
3 . 0
= F
1() & ( for b
4 being
Nat holds P
1[b
4,b
3 . b
4,b
3 . (b4 + 1)] ) ) implies b
1 = b
2 ) ) )
provided
E2:
for b
1 being
Natfor b
2 being
set holds
ex b
3 being
set st P
1[b
1,b
2,b
3]
and
E3:
for b
1 being
Natfor b
2, b
3, b
4 being
set holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 )
scheme :: RECDEF_1:sch 20
s20{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Nat, P
1[
set ,
set ,
set ] } :
( ex b
1 being
Element of F
1()ex b
2 being
Function of
NAT ,F
1() st
( b
1 = b
2 . F
3() & b
2 . 0
= F
2() & ( for b
3 being
Nat holds P
1[b
3,b
2 . b
3,b
2 . (b3 + 1)] ) ) & ( for b
1, b
2 being
Element of F
1() holds
( ex b
3 being
Function of
NAT ,F
1() st
( b
1 = b
3 . F
3() & b
3 . 0
= F
2() & ( for b
4 being
Nat holds P
1[b
4,b
3 . b
4,b
3 . (b4 + 1)] ) ) & ex b
3 being
Function of
NAT ,F
1() st
( b
2 = b
3 . F
3() & b
3 . 0
= F
2() & ( for b
4 being
Nat holds P
1[b
4,b
3 . b
4,b
3 . (b4 + 1)] ) ) implies b
1 = b
2 ) ) )
provided
E2:
for b
1 being
Natfor b
2 being
Element of F
1() holds
ex b
3 being
Element of F
1() st P
1[b
1,b
2,b
3]
and
E3:
for b
1 being
Natfor 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 )
scheme :: RECDEF_1:sch 21
s21{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Nat, F
4(
set ,
set )
-> Element of F
1() } :
( ex b
1 being
Element of F
1()ex b
2 being
Function of
NAT ,F
1() st
( b
1 = b
2 . F
3() & b
2 . 0
= F
2() & ( for b
3 being
Nat holds b
2 . (b3 + 1) = F
4(b
3,
(b2 . b3)) ) ) & ( for b
1, b
2 being
Element of F
1() holds
( ex b
3 being
Function of
NAT ,F
1() st
( b
1 = b
3 . F
3() & b
3 . 0
= F
2() & ( for b
4 being
Nat holds b
3 . (b4 + 1) = F
4(b
4,
(b3 . b4)) ) ) & ex b
3 being
Function of
NAT ,F
1() st
( b
2 = b
3 . F
3() & b
3 . 0
= F
2() & ( for b
4 being
Nat holds b
3 . (b4 + 1) = F
4(b
4,
(b3 . b4)) ) ) implies b
1 = b
2 ) ) )
scheme :: RECDEF_1:sch 22
s22{ F
1()
-> FinSequence, P
1[
set ,
set ,
set ] } :
( ex b
1 being
set ex b
2 being
FinSequence st
( b
1 = b
2 . (len b2) &
len b
2 = len F
1() & b
2 . 1
= F
1()
. 1 & ( for b
3 being
Nat holds
( 1
<= b
3 & b
3 < len F
1() implies P
1[F
1()
. (b3 + 1),b
2 . b
3,b
2 . (b3 + 1)] ) ) ) & ( for b
1, b
2 being
set holds
( ex b
3 being
FinSequence st
( b
1 = b
3 . (len b3) &
len b
3 = len F
1() & b
3 . 1
= F
1()
. 1 & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 < len F
1() implies P
1[F
1()
. (b4 + 1),b
3 . b
4,b
3 . (b4 + 1)] ) ) ) & ex b
3 being
FinSequence st
( b
2 = b
3 . (len b3) &
len b
3 = len F
1() & b
3 . 1
= F
1()
. 1 & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 < len F
1() implies P
1[F
1()
. (b4 + 1),b
3 . b
4,b
3 . (b4 + 1)] ) ) ) implies b
1 = b
2 ) ) )
provided
E2:
for b
1 being
Natfor b
2 being
set holds
not ( 1
<= b
1 & b
1 < len F
1() & ( for b
3 being
set holds
not P
1[F
1()
. (b1 + 1),b
2,b
3] ) )
and
E3:
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
set holds
( 1
<= b
1 & b
1 < len F
1() & b
5 = F
1()
. (b1 + 1) & P
1[b
5,b
2,b
3] & P
1[b
5,b
2,b
4] implies b
3 = b
4 )