:: YELLOW15 semantic presentation
scheme :: YELLOW15:sch 1
s1{ F
1()
-> Nat, F
2()
-> non
empty set , P
1[
set ], F
3(
set )
-> set , F
4(
set )
-> set } :
ex b
1 being
FinSequence of F
2() st
(
len b
1 = F
1() & ( for b
2 being
Nat holds
( b
2 in Seg F
1() implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( not P
1[b
2] implies b
1 . b
2 = F
4(b
2) ) ) ) ) )
provided
E1:
for b
1 being
Nat holds
( b
1 in Seg F
1() implies ( ( P
1[b
1] implies F
3(b
1)
in F
2() ) & ( not P
1[b
1] implies F
4(b
1)
in F
2() ) ) )
theorem Th1: :: YELLOW15:1
canceled;
theorem Th2: :: YELLOW15:2
theorem Th3: :: YELLOW15:3
theorem Th4: :: YELLOW15:4
definition
let c
1 be
set ;
let c
2 be
FinSequence of
bool c
1;
let c
3 be
FinSequence of
BOOLEAN ;
func MergeSequence c
2,c
3 -> FinSequence of
bool a
1 means :
Def1:
:: YELLOW15:def 1
(
len a
4 = len a
2 & ( for b
1 being
Nat holds
( b
1 in dom a
2 implies a
4 . b
1 = IFEQ (a3 . b1),
TRUE ,
(a2 . b1),
(a1 \ (a2 . b1)) ) ) );
existence
ex b1 being FinSequence of bool c1 st
( len b1 = len c2 & ( for b2 being Nat holds
( b2 in dom c2 implies b1 . b2 = IFEQ (c3 . b2),TRUE ,(c2 . b2),(c1 \ (c2 . b2)) ) ) )
uniqueness
for b1, b2 being FinSequence of bool c1 holds
( len b1 = len c2 & ( for b3 being Nat holds
( b3 in dom c2 implies b1 . b3 = IFEQ (c3 . b3),TRUE ,(c2 . b3),(c1 \ (c2 . b3)) ) ) & len b2 = len c2 & ( for b3 being Nat holds
( b3 in dom c2 implies b2 . b3 = IFEQ (c3 . b3),TRUE ,(c2 . b3),(c1 \ (c2 . b3)) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines MergeSequence YELLOW15:def 1 :
theorem Th5: :: YELLOW15:5
theorem Th6: :: YELLOW15:6
theorem Th7: :: YELLOW15:7
theorem Th8: :: YELLOW15:8
theorem Th9: :: YELLOW15:9
theorem Th10: :: YELLOW15:10
theorem Th11: :: YELLOW15:11
theorem Th12: :: YELLOW15:12
theorem Th13: :: YELLOW15:13
theorem Th14: :: YELLOW15:14
theorem Th15: :: YELLOW15:15
for b
1 being
set for b
2, b
3, b
4 being
Subset of b
1for b
5 being
FinSequence of
BOOLEAN holds
( ( b
5 . 1
= TRUE implies
(MergeSequence <*b2,b3,b4*>,b5) . 1
= b
2 ) & ( b
5 . 1
= FALSE implies
(MergeSequence <*b2,b3,b4*>,b5) . 1
= b
1 \ b
2 ) & ( b
5 . 2
= TRUE implies
(MergeSequence <*b2,b3,b4*>,b5) . 2
= b
3 ) & ( b
5 . 2
= FALSE implies
(MergeSequence <*b2,b3,b4*>,b5) . 2
= b
1 \ b
3 ) & ( b
5 . 3
= TRUE implies
(MergeSequence <*b2,b3,b4*>,b5) . 3
= b
4 ) & ( b
5 . 3
= FALSE implies
(MergeSequence <*b2,b3,b4*>,b5) . 3
= b
1 \ b
4 ) )
theorem Th16: :: YELLOW15:16
:: deftheorem Def2 defines Components YELLOW15:def 2 :
theorem Th17: :: YELLOW15:17
theorem Th18: :: YELLOW15:18
theorem Th19: :: YELLOW15:19
theorem Th20: :: YELLOW15:20
:: deftheorem Def3 defines in_general_position YELLOW15:def 3 :
theorem Th21: :: YELLOW15:21
theorem Th22: :: YELLOW15:22
theorem Th23: :: YELLOW15:23
theorem Th24: :: YELLOW15:24
theorem Th25: :: YELLOW15:25
theorem Th26: :: YELLOW15:26
theorem Th27: :: YELLOW15:27
theorem Th28: :: YELLOW15:28
theorem Th29: :: YELLOW15:29
theorem Th30: :: YELLOW15:30
theorem Th31: :: YELLOW15:31
theorem Th32: :: YELLOW15:32