:: TRIANG_1 semantic presentation
scheme :: TRIANG_1:sch 1
s1{ F
1()
-> Nat, P
1[
set ] } :
for b
1 being
Nat holds
( b
1 <= F
1() implies P
1[b
1] )
provided
E1:
P
1[F
1()]
and
E2:
for b
1 being
Nat holds
( b
1 < F
1() & P
1[b
1 + 1] implies P
1[b
1] )
theorem Th1: :: TRIANG_1:1
theorem Th2: :: TRIANG_1:2
for b
1 being non
empty Posetfor b
2 being
Subset of b
1 holds
not ( b
2 is
finite & b
2 <> {} & ( for b
3, b
4 being
Element of b
1 holds
not ( b
3 in b
2 & b
4 in b
2 & not b
3 <= b
4 & not b
4 <= b
3 ) ) & ( for b
3 being
Element of b
1 holds
not ( b
3 in b
2 & ( for b
4 being
Element of b
1 holds
( b
4 in b
2 implies b
3 <= b
4 ) ) ) ) )
theorem Th3: :: TRIANG_1:3
definition
let c
1 be
set ;
let c
2 be
finite Subset of c
1;
let c
3 be
Order of c
1;
assume E2:
c
3 linearly_orders c
2
;
canceled;func SgmX c
3,c
2 -> FinSequence of a
1 means :
Def2:
:: TRIANG_1:def 2
(
rng a
4 = a
2 & ( for b
1, b
2 being
Nat holds
( b
1 in dom a
4 & b
2 in dom a
4 & b
1 < b
2 implies ( a
4 /. b
1 <> a
4 /. b
2 &
[(a4 /. b1),(a4 /. b2)] in a
3 ) ) ) );
existence
ex b1 being FinSequence of c1 st
( rng b1 = c2 & ( for b2, b3 being Nat holds
( b2 in dom b1 & b3 in dom b1 & b2 < b3 implies ( b1 /. b2 <> b1 /. b3 & [(b1 /. b2),(b1 /. b3)] in c3 ) ) ) )
uniqueness
for b1, b2 being FinSequence of c1 holds
( rng b1 = c2 & ( for b3, b4 being Nat holds
( b3 in dom b1 & b4 in dom b1 & b3 < b4 implies ( b1 /. b3 <> b1 /. b4 & [(b1 /. b3),(b1 /. b4)] in c3 ) ) ) & rng b2 = c2 & ( for b3, b4 being Nat holds
( b3 in dom b2 & b4 in dom b2 & b3 < b4 implies ( b2 /. b3 <> b2 /. b4 & [(b2 /. b3),(b2 /. b4)] in c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 TRIANG_1:def 1 :
canceled;
:: deftheorem Def2 defines SgmX TRIANG_1:def 2 :
theorem Th4: :: TRIANG_1:4
:: deftheorem Def3 defines symplexes TRIANG_1:def 3 :
theorem Th5: :: TRIANG_1:5
theorem Th6: :: TRIANG_1:6
theorem Th7: :: TRIANG_1:7
theorem Th8: :: TRIANG_1:8
theorem Th9: :: TRIANG_1:9
theorem Th10: :: TRIANG_1:10
:: deftheorem Def4 defines lower_non-empty TRIANG_1:def 4 :
:: deftheorem Def5 defines FuncsSeq TRIANG_1:def 5 :
:: deftheorem Def6 defines @ TRIANG_1:def 6 :
:: deftheorem Def7 defines NatEmbSeq TRIANG_1:def 7 :
:: deftheorem Def8 TRIANG_1:def 8 :
canceled;
:: deftheorem Def9 defines lower_non-empty TRIANG_1:def 9 :
definition
let c
1 be
lower_non-empty TriangStr ;
let c
2 be
Nat;
let c
3 be
Symplex of c
1,
(c2 + 1);
let c
4 be
Face of c
2;
assume E13:
the
SkeletonSeq of c
1 . (c2 + 1) <> {}
;
func face c
3,c
4 -> Symplex of a
1,a
2 means :
Def10:
:: TRIANG_1:def 10
for b
1, b
2 being
Function holds
( b
1 = the
FacesAssign of a
1 . a
2 & b
2 = b
1 . a
4 implies a
5 = b
2 . a
3 );
existence
ex b1 being Symplex of c1,c2 st
for b2, b3 being Function holds
( b2 = the FacesAssign of c1 . c2 & b3 = b2 . c4 implies b1 = b3 . c3 )
uniqueness
for b1, b2 being Symplex of c1,c2 holds
( ( for b3, b4 being Function holds
( b3 = the FacesAssign of c1 . c2 & b4 = b3 . c4 implies b1 = b4 . c3 ) ) & ( for b3, b4 being Function holds
( b3 = the FacesAssign of c1 . c2 & b4 = b3 . c4 implies b2 = b4 . c3 ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines face TRIANG_1:def 10 :
definition
let c
1 be non
empty Poset;
func Triang c
1 -> strict lower_non-empty TriangStr means :: TRIANG_1:def 11
( the
SkeletonSeq of a
2 . 0
= {{} } & ( for b
1 being
Nat holds
( b
1 > 0 implies the
SkeletonSeq of a
2 . b
1 = { (SgmX the InternalRel of a1,b2) where B is non empty Element of symplexes a1 : Card b2 = b1 } ) ) & ( for b
1 being
Natfor b
2 being
Face of b
1for b
3 being
Element of the
SkeletonSeq of a
2 . (b1 + 1) holds
( b
3 in the
SkeletonSeq of a
2 . (b1 + 1) implies for b
4 being non
empty Element of
symplexes a
1 holds
(
SgmX the
InternalRel of a
1,b
4 = b
3 implies
face b
3,b
2 = (SgmX the InternalRel of a1,b4) * b
2 ) ) ) );
existence
ex b1 being strict lower_non-empty TriangStr st
( the SkeletonSeq of b1 . 0 = {{} } & ( for b2 being Nat holds
( b2 > 0 implies the SkeletonSeq of b1 . b2 = { (SgmX the InternalRel of c1,b3) where B is non empty Element of symplexes c1 : Card b3 = b2 } ) ) & ( for b2 being Nat
for b3 being Face of b2
for b4 being Element of the SkeletonSeq of b1 . (b2 + 1) holds
( b4 in the SkeletonSeq of b1 . (b2 + 1) implies for b5 being non empty Element of symplexes c1 holds
( SgmX the InternalRel of c1,b5 = b4 implies face b4,b3 = (SgmX the InternalRel of c1,b5) * b3 ) ) ) )
uniqueness
for b1, b2 being strict lower_non-empty TriangStr holds
( the SkeletonSeq of b1 . 0 = {{} } & ( for b3 being Nat holds
( b3 > 0 implies the SkeletonSeq of b1 . b3 = { (SgmX the InternalRel of c1,b4) where B is non empty Element of symplexes c1 : Card b4 = b3 } ) ) & ( for b3 being Nat
for b4 being Face of b3
for b5 being Element of the SkeletonSeq of b1 . (b3 + 1) holds
( b5 in the SkeletonSeq of b1 . (b3 + 1) implies for b6 being non empty Element of symplexes c1 holds
( SgmX the InternalRel of c1,b6 = b5 implies face b5,b4 = (SgmX the InternalRel of c1,b6) * b4 ) ) ) & the SkeletonSeq of b2 . 0 = {{} } & ( for b3 being Nat holds
( b3 > 0 implies the SkeletonSeq of b2 . b3 = { (SgmX the InternalRel of c1,b4) where B is non empty Element of symplexes c1 : Card b4 = b3 } ) ) & ( for b3 being Nat
for b4 being Face of b3
for b5 being Element of the SkeletonSeq of b2 . (b3 + 1) holds
( b5 in the SkeletonSeq of b2 . (b3 + 1) implies for b6 being non empty Element of symplexes c1 holds
( SgmX the InternalRel of c1,b6 = b5 implies face b5,b4 = (SgmX the InternalRel of c1,b6) * b4 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines Triang TRIANG_1:def 11 :