:: SCHEME1 semantic presentation
theorem Th1: :: SCHEME1:1
for b
1 being
Nat holds
not for b
2 being
Nat holds
( not b
1 = 2
* b
2 & not b
1 = (2 * b2) + 1 )
theorem Th2: :: SCHEME1:2
for b
1 being
Nat holds
not for b
2 being
Nat holds
( not b
1 = 3
* b
2 & not b
1 = (3 * b2) + 1 & not b
1 = (3 * b2) + 2 )
theorem Th3: :: SCHEME1:3
for b
1 being
Nat holds
not for b
2 being
Nat holds
( not b
1 = 4
* b
2 & not b
1 = (4 * b2) + 1 & not b
1 = (4 * b2) + 2 & not b
1 = (4 * b2) + 3 )
theorem Th4: :: SCHEME1:4
for b
1 being
Nat holds
not for b
2 being
Nat holds
( not b
1 = 5
* b
2 & not b
1 = (5 * b2) + 1 & not b
1 = (5 * b2) + 2 & not b
1 = (5 * b2) + 3 & not b
1 = (5 * b2) + 4 )
scheme :: SCHEME1:sch 9
s9{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ], P
2[
set ], P
3[
set ], F
3(
set )
-> Element of F
2(), F
4(
set )
-> Element of F
2(), F
5(
set )
-> Element of F
2() } :
ex b
1 being
PartFunc of F
1(),F
2() st
( ( for b
2 being
Element of F
1() holds
( b
2 in dom b
1 iff not ( not P
1[b
2] & not P
2[b
2] & not P
3[b
2] ) ) ) & ( for b
2 being
Element of F
1() holds
( b
2 in dom b
1 implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) ) ) ) )
provided
E4:
for b
1 being
Element of F
1() holds
( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) )
scheme :: SCHEME1:sch 10
s10{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ], P
2[
set ], P
3[
set ], F
3(
set )
-> Element of F
2(), F
4(
set )
-> Element of F
2(), F
5(
set )
-> Element of F
2() } :
ex b
1 being
PartFunc of F
1(),F
2() st
( ( for b
2 being
Element of F
1() holds
( b
2 in dom b
1 iff not ( not P
1[b
2] & not P
2[b
2] & not P
3[b
2] ) ) ) & ( for b
2 being
Element of F
1() holds
( b
2 in dom b
1 implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) ) ) ) )
provided
E4:
for b
1 being
Element of F
1() holds
( ( P
1[b
1] & P
2[b
1] implies F
3(b
1)
= F
4(b
1) ) & ( P
1[b
1] & P
3[b
1] implies F
3(b
1)
= F
5(b
1) ) & ( P
2[b
1] & P
3[b
1] implies F
4(b
1)
= F
5(b
1) ) )
scheme :: SCHEME1:sch 11
s11{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ], P
2[
set ], P
3[
set ], P
4[
set ], F
3(
set )
-> Element of F
2(), F
4(
set )
-> Element of F
2(), F
5(
set )
-> Element of F
2(), F
6(
set )
-> Element of F
2() } :
ex b
1 being
PartFunc of F
1(),F
2() st
( ( for b
2 being
Element of F
1() holds
( b
2 in dom b
1 iff not ( not P
1[b
2] & not P
2[b
2] & not P
3[b
2] & not P
4[b
2] ) ) ) & ( for b
2 being
Element of F
1() holds
( b
2 in dom b
1 implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) & ( P
4[b
2] implies b
1 . b
2 = F
6(b
2) ) ) ) ) )
provided
E4:
for b
1 being
Element of F
1() holds
( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
1[b
1] & P
4[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
4[b
1] ) & not ( P
3[b
1] & P
4[b
1] ) )
scheme :: SCHEME1:sch 12
s12{ F
1()
-> set , F
2()
-> set , P
1[
set ], P
2[
set ], F
3(
set )
-> set , F
4(
set )
-> set } :
ex b
1 being
PartFunc of F
1(),F
2() st
( ( for b
2 being
set holds
( b
2 in dom b
1 iff ( b
2 in F
1() & ( P
1[b
2] or P
2[b
2] ) ) ) ) & ( for b
2 being
set holds
( b
2 in dom b
1 implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) ) ) ) )
provided
E4:
for b
1 being
set holds
not ( b
1 in F
1() & P
1[b
1] & P
2[b
1] )
and
E5:
for b
1 being
set holds
( b
1 in F
1() & P
1[b
1] implies F
3(b
1)
in F
2() )
and
E6:
for b
1 being
set holds
( b
1 in F
1() & P
2[b
1] implies F
4(b
1)
in F
2() )
scheme :: SCHEME1:sch 13
s13{ F
1()
-> set , F
2()
-> set , P
1[
set ], P
2[
set ], P
3[
set ], F
3(
set )
-> set , F
4(
set )
-> set , F
5(
set )
-> set } :
ex b
1 being
PartFunc of F
1(),F
2() st
( ( for b
2 being
set holds
( b
2 in dom b
1 iff ( b
2 in F
1() & not ( not P
1[b
2] & not P
2[b
2] & not P
3[b
2] ) ) ) ) & ( for b
2 being
set holds
( b
2 in dom b
1 implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) ) ) ) )
provided
E4:
for b
1 being
set holds
( b
1 in F
1() implies ( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) ) )
and
E5:
for b
1 being
set holds
( b
1 in F
1() & P
1[b
1] implies F
3(b
1)
in F
2() )
and
E6:
for b
1 being
set holds
( b
1 in F
1() & P
2[b
1] implies F
4(b
1)
in F
2() )
and
E7:
for b
1 being
set holds
( b
1 in F
1() & P
3[b
1] implies F
5(b
1)
in F
2() )
scheme :: SCHEME1:sch 14
s14{ F
1()
-> set , F
2()
-> set , P
1[
set ], P
2[
set ], P
3[
set ], P
4[
set ], F
3(
set )
-> set , F
4(
set )
-> set , F
5(
set )
-> set , F
6(
set )
-> set } :
ex b
1 being
PartFunc of F
1(),F
2() st
( ( for b
2 being
set holds
( b
2 in dom b
1 iff ( b
2 in F
1() & not ( not P
1[b
2] & not P
2[b
2] & not P
3[b
2] & not P
4[b
2] ) ) ) ) & ( for b
2 being
set holds
( b
2 in dom b
1 implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) & ( P
4[b
2] implies b
1 . b
2 = F
6(b
2) ) ) ) ) )
provided
E4:
for b
1 being
set holds
( b
1 in F
1() implies ( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
1[b
1] & P
4[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
4[b
1] ) & not ( P
3[b
1] & P
4[b
1] ) ) )
and
E5:
for b
1 being
set holds
( b
1 in F
1() & P
1[b
1] implies F
3(b
1)
in F
2() )
and
E6:
for b
1 being
set holds
( b
1 in F
1() & P
2[b
1] implies F
4(b
1)
in F
2() )
and
E7:
for b
1 being
set holds
( b
1 in F
1() & P
3[b
1] implies F
5(b
1)
in F
2() )
and
E8:
for b
1 being
set holds
( b
1 in F
1() & P
4[b
1] implies F
6(b
1)
in F
2() )
scheme :: SCHEME1:sch 15
s15{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , P
1[
set ,
set ], P
2[
set ,
set ], F
4(
set ,
set )
-> Element of F
3(), F
5(
set ,
set )
-> Element of F
3() } :
ex b
1 being
PartFunc of
[:F1(),F2():],F
3() st
( ( for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 iff ( P
1[b
2,b
3] or P
2[b
2,b
3] ) ) ) & ( for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 implies ( ( P
1[b
2,b
3] implies b
1 . [b2,b3] = F
4(b
2,b
3) ) & ( P
2[b
2,b
3] implies b
1 . [b2,b3] = F
5(b
2,b
3) ) ) ) ) )
provided
E4:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
not ( P
1[b
1,b
2] & P
2[b
1,b
2] )
scheme :: SCHEME1:sch 16
s16{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , P
1[
set ,
set ], P
2[
set ,
set ], P
3[
set ,
set ], F
4(
set ,
set )
-> Element of F
3(), F
5(
set ,
set )
-> Element of F
3(), F
6(
set ,
set )
-> Element of F
3() } :
ex b
1 being
PartFunc of
[:F1(),F2():],F
3() st
( ( for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 iff not ( not P
1[b
2,b
3] & not P
2[b
2,b
3] & not P
3[b
2,b
3] ) ) ) & ( for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 implies ( ( P
1[b
2,b
3] implies b
1 . [b2,b3] = F
4(b
2,b
3) ) & ( P
2[b
2,b
3] implies b
1 . [b2,b3] = F
5(b
2,b
3) ) & ( P
3[b
2,b
3] implies b
1 . [b2,b3] = F
6(b
2,b
3) ) ) ) ) )
provided
E4:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( not ( P
1[b
1,b
2] & P
2[b
1,b
2] ) & not ( P
1[b
1,b
2] & P
3[b
1,b
2] ) & not ( P
2[b
1,b
2] & P
3[b
1,b
2] ) )
scheme :: SCHEME1:sch 17
s17{ F
1()
-> set , F
2()
-> set , F
3()
-> set , P
1[
set ,
set ], P
2[
set ,
set ], F
4(
set ,
set )
-> set , F
5(
set ,
set )
-> set } :
ex b
1 being
PartFunc of
[:F1(),F2():],F
3() st
( ( for b
2, b
3 being
set holds
(
[b2,b3] in dom b
1 iff ( b
2 in F
1() & b
3 in F
2() & ( P
1[b
2,b
3] or P
2[b
2,b
3] ) ) ) ) & ( for b
2, b
3 being
set holds
(
[b2,b3] in dom b
1 implies ( ( P
1[b
2,b
3] implies b
1 . [b2,b3] = F
4(b
2,b
3) ) & ( P
2[b
2,b
3] implies b
1 . [b2,b3] = F
5(b
2,b
3) ) ) ) ) )
provided
E4:
for b
1, b
2 being
set holds
not ( b
1 in F
1() & b
2 in F
2() & P
1[b
1,b
2] & P
2[b
1,b
2] )
and
E5:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
1[b
1,b
2] implies F
4(b
1,b
2)
in F
3() )
and
E6:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
2[b
1,b
2] implies F
5(b
1,b
2)
in F
3() )
scheme :: SCHEME1:sch 18
s18{ F
1()
-> set , F
2()
-> set , F
3()
-> set , P
1[
set ,
set ], P
2[
set ,
set ], P
3[
set ,
set ], F
4(
set ,
set )
-> set , F
5(
set ,
set )
-> set , F
6(
set ,
set )
-> set } :
ex b
1 being
PartFunc of
[:F1(),F2():],F
3() st
( ( for b
2, b
3 being
set holds
(
[b2,b3] in dom b
1 iff ( b
2 in F
1() & b
3 in F
2() & not ( not P
1[b
2,b
3] & not P
2[b
2,b
3] & not P
3[b
2,b
3] ) ) ) ) & ( for b
2, b
3 being
set holds
(
[b2,b3] in dom b
1 implies ( ( P
1[b
2,b
3] implies b
1 . [b2,b3] = F
4(b
2,b
3) ) & ( P
2[b
2,b
3] implies b
1 . [b2,b3] = F
5(b
2,b
3) ) & ( P
3[b
2,b
3] implies b
1 . [b2,b3] = F
6(b
2,b
3) ) ) ) ) )
provided
E4:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() implies ( not ( P
1[b
1,b
2] & P
2[b
1,b
2] ) & not ( P
1[b
1,b
2] & P
3[b
1,b
2] ) & not ( P
2[b
1,b
2] & P
3[b
1,b
2] ) ) )
and
E5:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
1[b
1,b
2] implies F
4(b
1,b
2)
in F
3() )
and
E6:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
2[b
1,b
2] implies F
5(b
1,b
2)
in F
3() )
and
E7:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
3[b
1,b
2] implies F
6(b
1,b
2)
in F
3() )
scheme :: SCHEME1:sch 19
s19{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ], P
2[
set ], P
3[
set ], F
3(
set )
-> Element of F
2(), F
4(
set )
-> Element of F
2(), F
5(
set )
-> Element of F
2() } :
ex b
1 being
Function of F
1(),F
2() st
for b
2 being
Element of F
1() holds
( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) )
provided
E4:
for b
1 being
Element of F
1() holds
( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) )
and
E5:
for b
1 being
Element of F
1() holds
not ( not P
1[b
1] & not P
2[b
1] & not P
3[b
1] )
scheme :: SCHEME1:sch 20
s20{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ], P
2[
set ], P
3[
set ], P
4[
set ], F
3(
set )
-> Element of F
2(), F
4(
set )
-> Element of F
2(), F
5(
set )
-> Element of F
2(), F
6(
set )
-> Element of F
2() } :
ex b
1 being
Function of F
1(),F
2() st
for b
2 being
Element of F
1() holds
( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
5(b
2) ) & ( P
4[b
2] implies b
1 . b
2 = F
6(b
2) ) )
provided
E4:
for b
1 being
Element of F
1() holds
( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
1[b
1] & P
4[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
4[b
1] ) & not ( P
3[b
1] & P
4[b
1] ) )
and
E5:
for b
1 being
Element of F
1() holds
not ( not P
1[b
1] & not P
2[b
1] & not P
3[b
1] & not P
4[b
1] )
scheme :: SCHEME1:sch 21
s21{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , P
1[
set ,
set ], F
4(
set ,
set )
-> Element of F
3(), F
5(
set ,
set )
-> Element of F
3() } :
ex b
1 being
Function of
[:F1(),F2():],F
3() st
for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 implies ( ( P
1[b
2,b
3] implies b
1 . [b2,b3] = F
4(b
2,b
3) ) & ( not P
1[b
2,b
3] implies b
1 . [b2,b3] = F
5(b
2,b
3) ) ) )
scheme :: SCHEME1:sch 22
s22{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , P
1[
set ,
set ], P
2[
set ,
set ], P
3[
set ,
set ], F
4(
set ,
set )
-> Element of F
3(), F
5(
set ,
set )
-> Element of F
3(), F
6(
set ,
set )
-> Element of F
3() } :
ex b
1 being
Function of
[:F1(),F2():],F
3() st
( ( for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 iff not ( not P
1[b
2,b
3] & not P
2[b
2,b
3] & not P
3[b
2,b
3] ) ) ) & ( for b
2 being
Element of F
1()
for b
3 being
Element of F
2() holds
(
[b2,b3] in dom b
1 implies ( ( P
1[b
2,b
3] implies b
1 . [b2,b3] = F
4(b
2,b
3) ) & ( P
2[b
2,b
3] implies b
1 . [b2,b3] = F
5(b
2,b
3) ) & ( P
3[b
2,b
3] implies b
1 . [b2,b3] = F
6(b
2,b
3) ) ) ) ) )
provided
E4:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( not ( P
1[b
1,b
2] & P
2[b
1,b
2] ) & not ( P
1[b
1,b
2] & P
3[b
1,b
2] ) & not ( P
2[b
1,b
2] & P
3[b
1,b
2] ) )
and
E5:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
not ( not P
1[b
1,b
2] & not P
2[b
1,b
2] & not P
3[b
1,b
2] )