:: NECKLA_2 semantic presentation
theorem Th1: :: NECKLA_2:1
for b
1 being
Universefor b
2, b
3 being
set holds
( b
2 in b
1 & b
3 in b
1 implies for b
4 being
Relation of b
2,b
3 holds b
4 in b
1 )
theorem Th2: :: NECKLA_2:2
the
InternalRel of
(Necklace 4) = {[0,1],[1,0],[1,2],[2,1],[2,3],[3,2]}
theorem Th3: :: NECKLA_2:3
:: deftheorem Def1 defines N-free NECKLA_2:def 1 :
:: deftheorem Def2 defines union_of NECKLA_2:def 2 :
definition
let c
1, c
2 be
RelStr ;
func sum_of c
1,c
2 -> strict RelStr means :
Def3:
:: NECKLA_2:def 3
( the
carrier of a
3 = the
carrier of a
1 \/ the
carrier of a
2 & the
InternalRel of a
3 = ((the InternalRel of a1 \/ the InternalRel of a2) \/ [:the carrier of a1,the carrier of a2:]) \/ [:the carrier of a2,the carrier of a1:] );
existence
ex b1 being strict RelStr st
( the carrier of b1 = the carrier of c1 \/ the carrier of c2 & the InternalRel of b1 = ((the InternalRel of c1 \/ the InternalRel of c2) \/ [:the carrier of c1,the carrier of c2:]) \/ [:the carrier of c2,the carrier of c1:] )
uniqueness
for b1, b2 being strict RelStr holds
( the carrier of b1 = the carrier of c1 \/ the carrier of c2 & the InternalRel of b1 = ((the InternalRel of c1 \/ the InternalRel of c2) \/ [:the carrier of c1,the carrier of c2:]) \/ [:the carrier of c2,the carrier of c1:] & the carrier of b2 = the carrier of c1 \/ the carrier of c2 & the InternalRel of b2 = ((the InternalRel of c1 \/ the InternalRel of c2) \/ [:the carrier of c1,the carrier of c2:]) \/ [:the carrier of c2,the carrier of c1:] implies b1 = b2 )
;
end;
:: deftheorem Def3 defines sum_of NECKLA_2:def 3 :
:: deftheorem Def4 defines fin_RelStr NECKLA_2:def 4 :
definition
func fin_RelStr_sp -> Subset of
fin_RelStr means :
Def5:
:: NECKLA_2:def 5
( ( for b
1 being
strict RelStr holds
( not the
carrier of b
1 is
empty & the
carrier of b
1 is
trivial & the
carrier of b
1 in FinSETS implies b
1 in a
1 ) ) & ( for b
1, b
2 being
strict RelStr holds
( the
carrier of b
1 misses the
carrier of b
2 & b
1 in a
1 & b
2 in a
1 implies (
union_of b
1,b
2 in a
1 &
sum_of b
1,b
2 in a
1 ) ) ) & ( for b
1 being
Subset of
fin_RelStr holds
( ( for b
2 being
strict RelStr holds
( not the
carrier of b
2 is
empty & the
carrier of b
2 is
trivial & the
carrier of b
2 in FinSETS implies b
2 in b
1 ) ) & ( for b
2, b
3 being
strict RelStr holds
( the
carrier of b
2 misses the
carrier of b
3 & b
2 in b
1 & b
3 in b
1 implies (
union_of b
2,b
3 in b
1 &
sum_of b
2,b
3 in b
1 ) ) ) implies a
1 c= b
1 ) ) );
existence
ex b1 being Subset of fin_RelStr st
( ( for b2 being strict RelStr holds
( not the carrier of b2 is empty & the carrier of b2 is trivial & the carrier of b2 in FinSETS implies b2 in b1 ) ) & ( for b2, b3 being strict RelStr holds
( the carrier of b2 misses the carrier of b3 & b2 in b1 & b3 in b1 implies ( union_of b2,b3 in b1 & sum_of b2,b3 in b1 ) ) ) & ( for b2 being Subset of fin_RelStr holds
( ( for b3 being strict RelStr holds
( not the carrier of b3 is empty & the carrier of b3 is trivial & the carrier of b3 in FinSETS implies b3 in b2 ) ) & ( for b3, b4 being strict RelStr holds
( the carrier of b3 misses the carrier of b4 & b3 in b2 & b4 in b2 implies ( union_of b3,b4 in b2 & sum_of b3,b4 in b2 ) ) ) implies b1 c= b2 ) ) )
uniqueness
for b1, b2 being Subset of fin_RelStr holds
( ( for b3 being strict RelStr holds
( not the carrier of b3 is empty & the carrier of b3 is trivial & the carrier of b3 in FinSETS implies b3 in b1 ) ) & ( for b3, b4 being strict RelStr holds
( the carrier of b3 misses the carrier of b4 & b3 in b1 & b4 in b1 implies ( union_of b3,b4 in b1 & sum_of b3,b4 in b1 ) ) ) & ( for b3 being Subset of fin_RelStr holds
( ( for b4 being strict RelStr holds
( not the carrier of b4 is empty & the carrier of b4 is trivial & the carrier of b4 in FinSETS implies b4 in b3 ) ) & ( for b4, b5 being strict RelStr holds
( the carrier of b4 misses the carrier of b5 & b4 in b3 & b5 in b3 implies ( union_of b4,b5 in b3 & sum_of b4,b5 in b3 ) ) ) implies b1 c= b3 ) ) & ( for b3 being strict RelStr holds
( not the carrier of b3 is empty & the carrier of b3 is trivial & the carrier of b3 in FinSETS implies b3 in b2 ) ) & ( for b3, b4 being strict RelStr holds
( the carrier of b3 misses the carrier of b4 & b3 in b2 & b4 in b2 implies ( union_of b3,b4 in b2 & sum_of b3,b4 in b2 ) ) ) & ( for b3 being Subset of fin_RelStr holds
( ( for b4 being strict RelStr holds
( not the carrier of b4 is empty & the carrier of b4 is trivial & the carrier of b4 in FinSETS implies b4 in b3 ) ) & ( for b4, b5 being strict RelStr holds
( the carrier of b4 misses the carrier of b5 & b4 in b3 & b5 in b3 implies ( union_of b4,b5 in b3 & sum_of b4,b5 in b3 ) ) ) implies b2 c= b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines fin_RelStr_sp NECKLA_2:def 5 :
theorem Th4: :: NECKLA_2:4
theorem Th5: :: NECKLA_2:5
theorem Th6: :: NECKLA_2:6
Lemma11:
the carrier of (Necklace 4) = {0,1,2,3}
by NECKLACE:2, NECKLACE:21;
theorem Th7: :: NECKLA_2:7