:: Abstract Simplicial Complexes
:: by Karol P\c{a}k
::
:: Received December 18, 2009
:: Copyright (c) 2009 Association of Mizar Users
environ
vocabularies ARYTM_1, XBOOLE_0, CARD_1, CLASSES1, COHSP_1, FINSET_1, FUNCT_1,
GROUP_4, INT_1, MATROID0, ORDERS_1, PARTFUN1, PRE_TOPC, RELAT_1, RELAT_2,
NUMBERS, SETFAM_1, SGRAPH1, SUBSET_1, TARSKI, TRIANG_1, WELLORD1,
ZFMISC_1, SIMPLEX0, XXREAL_0, NAT_1, ORDINAL1, FUNCOP_1, ARYTM_3,
RCOMP_1, STRUCT_0, GLIB_000, XREAL_0;
notations TARSKI, XBOOLE_0, SUBSET_1, ZFMISC_1, RELAT_1, RELAT_2, FUNCT_1,
FUNCT_2, NUMBERS, WELLORD1, ORDINAL1, ORDERS_1, FINSET_1, CARD_1,
XREAL_0, XXREAL_0, XXREAL_3, NAT_1, INT_1, SETFAM_1, DOMAIN_1, STRUCT_0,
PRE_TOPC, TOPS_2, PENCIL_1, CLASSES1, MATROID0, RELSET_1, COHSP_1,
FUNCOP_1;
constructors SETFAM_1, XXREAL_3, REAL_1, TOPS_2, BORSUK_1, WELLORD1, BINOP_2,
CLASSES1, COH_SP, LEXBFS, MATROID0, COHSP_1, FUNCOP_1;
registrations BORSUK_3, CARD_1, COHSP_1, FIB_NUM2, FINSET_1, FUNCT_1, INT_1,
MATROID0, NAT_1, ORDINAL1, PENCIL_1, REALSET1, RELAT_1, SETFAM_1,
SUBSET_1, STRUCT_0, XBOOLE_0, XREAL_0, XXREAL_0, XXREAL_3, FUNCOP_1,
DILWORTH;
requirements REAL, SUBSET, NUMERALS, ARITHM, BOOLE;
definitions CLASSES1, COHSP_1, FUNCOP_1, MATROID0, ORDINAL1, SETFAM_1,
STRUCT_0, SUBSET_1, TARSKI, TOPS_2, VECTSP_1, XBOOLE_0, XXREAL_3;
theorems CARD_1, CARD_2, CARD_3, CLASSES1, COHSP_1, COMBGRAS, COMPL_SP,
FINSET_1, FUNCT_1, FUNCT_2, FUNCOP_1, INT_1, MATROID0, NAT_1, NUMBERS,
ORDERS_1, ORDINAL1, PENCIL_1, PRE_TOPC, RELAT_1, RELAT_2, SETFAM_1,
STIRL2_1, SUBSET_1, WELLORD1, WELLORD2, TARSKI, TOPS_2, XREAL_0,
XBOOLE_0, XBOOLE_1, XREAL_1, XXREAL_0, XXREAL_3, ZFMISC_1;
schemes CLASSES1, FRAENKEL, FUNCT_2, NAT_1;
begin :: Preliminaries
reserve x,y,X,Y,Z for set,
D for non empty set,
n,k for Nat,
i,i1,i2 for Integer;
notation let X;
antonym X is with_empty_element for X is with_non-empty_elements;
end;
registration
cluster empty -> finite-membered set;
coherence
proof
let X;
assume A1: X is empty;
let x;
thus thesis by A1;
end;
let X be finite set;
cluster {X} -> finite-membered;
coherence
proof
let x;
thus thesis by TARSKI:def 1;
end;
cluster bool X -> finite-membered;
coherence
proof
let x;
thus thesis;
end;
let Y be finite set;
cluster {X,Y} -> finite-membered;
coherence
proof
let x;
thus thesis by TARSKI:def 2;
end;
end;
registration
let X be finite-membered set;
cluster -> finite-membered Subset of X;
coherence
proof
let S be Subset of X;
let x be set;
thus thesis by MATROID0:def 5;
end;
let Y be finite-membered set;
cluster X \/ Y -> finite-membered;
coherence
proof
let x be set;
assume x in X\/Y;
then x in X or x in Y by XBOOLE_0:def 3;
hence thesis by MATROID0:def 5;
end;
end;
registration
let X be finite finite-membered set;
cluster union X -> finite;
coherence
proof
for x st x in X holds x is finite by MATROID0:def 5;
hence thesis by FINSET_1:25;
end;
end;
registration
cluster empty -> subset-closed set;
coherence
proof
let X be set;
assume X is empty;
then for x,y st x in X & y c=x holds y in X;
hence thesis by CLASSES1:def 1;
end;
cluster with_empty_element -> non empty set;
coherence by SETFAM_1:def 9;
cluster non empty subset-closed -> with_empty_element set;
coherence
proof
let X;
assume A1: X is non empty subset-closed;
then consider x such that
A2: x in X by XBOOLE_0:def 1;
{}c=x by XBOOLE_1:2;
then {} in X by A1,A2,CLASSES1:def 1;
hence thesis;
end;
end;
registration
let X;
cluster Sub_of_Fin X -> finite-membered;
coherence
proof
let x;
assume x in Sub_of_Fin X;
hence thesis by COHSP_1:def 3;
end;
end;
registration
let X be subset-closed set;
cluster Sub_of_Fin X -> subset-closed;
coherence
proof
let x,y;
assume x in Sub_of_Fin X & y c=x;
hence thesis by CLASSES1:def 1;
end;
end;
theorem
Y is subset-closed iff for X st X in Y holds bool X c= Y
proof
hereby assume A1: Y is subset-closed;
let X;
assume A2: X in Y;
thus bool X c=Y
proof
let x be set;
assume x in bool X;
hence thesis by A1,A2,CLASSES1:def 1;
end;
end;
assume A3: for X st X in Y holds bool X c=Y;
let x,y;
assume that
A4: x in Y and
A5: y c=x;
A6: y in bool x by A5;
bool x c=Y by A3,A4;
hence thesis by A6;
end;
registration
let A,B be subset-closed set;
cluster A \/ B -> subset-closed;
coherence
proof
let x,y;
assume that
A1: x in A\/B and
A2: y c=x;
x in A or x in B by A1,XBOOLE_0:def 3;
then y in A or y in B by A2,CLASSES1:def 1;
hence y in A\/B by XBOOLE_0:def 3;
end;
cluster A /\ B -> subset-closed;
coherence
proof
let x,y;
assume that
A3: x in A/\B and
A4: y c=x;
x in B by A3,XBOOLE_0:def 4;
then A5: y in B by A4,CLASSES1:def 1;
x in A by A3,XBOOLE_0:def 4;
then y in A by A4,CLASSES1:def 1;
hence y in A/\B by A5,XBOOLE_0:def 4;
end;
end;
Lm1: for X ex F be subset-closed set st F=union{bool x where x is Element of X:
x in X} & X c=F & for Y st X c=Y & Y is subset-closed holds F c=Y
proof
let X;
set B={bool x where x is Element of X:x in X};
now let a,b be set such that
A1: a in union B and
A2: b c=a;
consider y such that
A3: a in y and
A4: y in B by A1,TARSKI:def 4;
consider x be Element of X such that
A5: y=bool x and
x in X by A4;
b c=x by A2,A3,A5,XBOOLE_1:1;
hence b in union B by A4,A5,TARSKI:def 4;
end;
then reconsider U=union B as subset-closed set by CLASSES1:def 1;
take U;
thus U=union B;
thus X c=U
proof
let x such that
A6: x in X;
x c=x & bool x in B by A6;
hence thesis by TARSKI:def 4;
end;
let Y such that
A7: X c=Y & Y is subset-closed;
let x;
assume x in U;
then consider y such that
A8: x in y and
A9: y in B by TARSKI:def 4;
ex x be Element of X st y=bool x & x in X by A9;
hence x in Y by A7,A8,CLASSES1:def 1;
end;
definition
let X;
func subset-closed_closure_of X -> subset-closed set means :Def1:
X c= it & for Y st X c= Y & Y is subset-closed holds it c= Y;
existence
proof
ex F be subset-closed set st F=union{bool x where x is Element of X:x in X} &
X c=F & for Y st X c=Y & Y is subset-closed holds F c=Y by Lm1;
hence thesis;
end;
uniqueness
proof
let F1,F2 be subset-closed set;
assume X c=F1 & (for Y st X c=Y & Y is subset-closed holds F1 c=Y) & X c=F2 &
for Y st X c=Y & Y is subset-closed holds F2 c=Y;
then F1 c=F2 & F2 c=F1;
hence thesis by XBOOLE_0:def 10;
end;
end;
theorem Th2:
x in subset-closed_closure_of X iff ex y st x c= y & y in X
proof
set B={bool x1 where x1 is Element of X:x1 in X};
consider F be subset-closed set such that
A1: F=union B and
A2: X c=F & for Y st X c=Y & Y is subset-closed holds F c=Y by Lm1;
A3: F=subset-closed_closure_of X by A2,Def1;
hereby assume x in subset-closed_closure_of X;
then consider y such that
A4: x in y and
A5: y in B by A1,A3,TARSKI:def 4;
consider x1 be Element of X such that
A6: bool x1=y & x1 in X by A5;
reconsider y=x1 as set;
take y;
thus x c=y & y in X by A4,A6;
end;
given y such that
A7: x c=y and
A8: y in X;
bool y in B by A8;
hence thesis by A1,A3,A7,TARSKI:def 4;
end;
definition
let X;
let F be Subset-Family of X;
redefine func subset-closed_closure_of F -> subset-closed Subset-Family of X;
coherence
proof
set FIC=subset-closed_closure_of F;
FIC c=bool X
proof
let x be set;
assume x in FIC;
then ex y st x c=y & y in F by Th2;
then x c=X by XBOOLE_1:1;
hence thesis;
end;
hence thesis;
end;
end;
registration
cluster subset-closed_closure_of {} -> empty;
coherence
proof
assume subset-closed_closure_of{} is non empty;
then consider x be set such that
A1: x in subset-closed_closure_of{} by XBOOLE_0:def 1;
ex y st x c=y & y in {} by A1,Th2;
hence thesis;
end;
let X be non empty set;
cluster subset-closed_closure_of X -> non empty;
coherence
proof
ex x st x in X by XBOOLE_0:def 1;
hence thesis by Th2;
end;
end;
registration
let X be with_non-empty_element set;
cluster subset-closed_closure_of X -> with_non-empty_element;
coherence
proof
consider b be non empty set such that
A1: b in X by SETFAM_1:def 11;
b in subset-closed_closure_of X by A1,Th2;
hence thesis by SETFAM_1:def 11;
end;
end;
registration
let X be finite-membered set;
cluster subset-closed_closure_of X -> finite-membered;
coherence
proof
let x be set;
assume x in subset-closed_closure_of X;
then consider y such that
A1: x c=y and
A2: y in X by Th2;
y is finite by A2,MATROID0:def 5;
hence thesis by A1;
end;
end;
theorem Th3:
X c=Y & Y is subset-closed implies subset-closed_closure_of X c= Y
proof
assume A1: X c=Y & Y is subset-closed;
let x;
assume x in subset-closed_closure_of X;
then ex y st x c=y & y in X by Th2;
hence thesis by A1,CLASSES1:def 1;
end;
theorem Th4:
subset-closed_closure_of {X} = bool X
proof
set f=subset-closed_closure_of{X};
A1: X in {X} by TARSKI:def 1;
hereby let x;
assume x in f;
then consider y such that
A2: x c=y and
A3: y in {X} by Th2;
y=X by A3,TARSKI:def 1;
hence x in bool X by A2;
end;
let x;
assume x in bool X;
hence thesis by A1,Th2;
end;
theorem
subset-closed_closure_of (X\/Y) =
subset-closed_closure_of X \/ subset-closed_closure_of Y
proof
set fxy=subset-closed_closure_of(X\/Y);
set fx=subset-closed_closure_of X;
set fy=subset-closed_closure_of Y;
hereby let x;
assume x in fxy;
then consider y such that
A1: x c=y and
A2: y in X\/Y by Th2;
y in X or y in Y by A2,XBOOLE_0:def 3;
then x in fx or x in fy by A1,Th2;
hence x in fx\/fy by XBOOLE_0:def 3;
end;
let x;
assume A3: x in fx\/fy;
per cases by A3,XBOOLE_0:def 3;
suppose x in fx;
then consider y such that
A4: x c=y and
A5: y in X by Th2;
y in X\/Y by A5,XBOOLE_0:def 3;
hence thesis by A4,Th2;
end;
suppose x in fy;
then consider y such that
A6: x c=y and
A7: y in Y by Th2;
y in X\/Y by A7,XBOOLE_0:def 3;
hence thesis by A6,Th2;
end;
end;
theorem Th6:
X is_finer_than Y iff
subset-closed_closure_of X c= subset-closed_closure_of Y
proof
set fx=subset-closed_closure_of X;
set fy=subset-closed_closure_of Y;
hereby assume A1: X is_finer_than Y;
thus fx c=fy
proof
let x;
assume x in fx;
then consider y such that
A2: x c=y and
A3: y in X by Th2;
consider c be set such that
A4: c in Y and
A5: y c=c by A1,A3,SETFAM_1:def 2;
x c=c by A2,A5,XBOOLE_1:1;
hence thesis by A4,Th2;
end;
end;
assume A6: fx c=fy;
let x;
assume x in X;
then x in fx by Th2;
then ex y st x c=y & y in Y by A6,Th2;
hence thesis;
end;
theorem Th7:
X is subset-closed implies subset-closed_closure_of X = X
proof
set f=subset-closed_closure_of X;
assume A1: X is subset-closed;
thus f c=X
proof
let x;
assume x in f;
then ex y st x c=y & y in X by Th2;
hence x in X by A1,CLASSES1:def 1;
end;
thus thesis by Def1;
end;
theorem
subset-closed_closure_of X c= X implies X is subset-closed
proof
set f=subset-closed_closure_of X;
assume A1: f c=X;
let x,y;
assume x in X & y c=x;
then y in f by Th2;
hence y in X by A1;
end;
definition
let Y,X;
let n be set;
func the_subsets_with_limited_card(n,X,Y) -> Subset-Family of Y means :Def2:
for A be Subset of Y holds A in it iff A in X & card A c= card n;
existence
proof
set SS={A where A is Subset of Y:A in X & card A c=card n};
SS c=bool Y
proof
let x;
assume x in SS;
then ex A be Subset of Y st x=A & A in X & card A c=card n;
hence thesis;
end;
then reconsider SS as Subset-Family of Y;
take SS;
let A be Subset of Y;
hereby assume A in SS;
then ex B be Subset of Y st B=A & B in X & card B c=card n;
hence A in X & card A c=card n;
end;
assume A in X & card A c=card n;
hence thesis;
end;
uniqueness
proof
let S1,S2 be Subset-Family of Y;
assume that
A1: for A be Subset of Y holds A in S1 iff A in X & card A c=card n and
A2: for A be Subset of Y holds A in S2 iff A in X & card A c=card n;
thus S1 c=S2
proof
let x;
assume A3: x in S1;
then x in X & card x c=card n by A1;
hence thesis by A2,A3;
end;
let x;
assume A4: x in S2;
then x in X & card x c=card n by A2;
hence thesis by A1,A4;
end;
end;
registration
let D;
cluster finite with_non-empty_element subset-closed
finite-membered Subset-Family of D;
existence
proof
consider x be set such that
A1: x in D by XBOOLE_0:def 1;
{x}c=D by A1,ZFMISC_1:37;
then reconsider XX=bool{x} as Subset-Family of D by ZFMISC_1:79;
take XX;
{x} in bool{x} by ZFMISC_1:def 1;
hence thesis by SETFAM_1:def 11;
end;
end;
registration
let Y,X;
let n be finite set;
cluster the_subsets_with_limited_card(n,X,Y) -> finite-membered;
coherence
proof
let A be set;
assume A in the_subsets_with_limited_card(n,X,Y);
then card A c=card n by Def2;
hence thesis;
end;
end;
registration
let Y;
let X be subset-closed set;
let n be set;
cluster the_subsets_with_limited_card(n,X,Y) -> subset-closed;
coherence
proof
let x,y;
assume that
A1: x in the_subsets_with_limited_card(n,X,Y) and
A2: y c=x;
x in X by A1,Def2;
then A3: y in X by A2,CLASSES1:def 1;
card x c=card n & card y c=card x by A1,A2,Def2,CARD_1:27;
then A4: card y c=card n by XBOOLE_1:1;
y c=Y by A1,A2,XBOOLE_1:1;
hence thesis by A3,A4,Def2;
end;
end;
registration
let Y;
let X be with_empty_element set;
let n be set;
cluster the_subsets_with_limited_card(n,X,Y) -> with_empty_element;
coherence
proof
A1: {}c=Y by XBOOLE_1:2;
card{}c=card n & {} in X by SETFAM_1:def 9;
then {} in the_subsets_with_limited_card(n,X,Y) by A1,Def2;
hence thesis;
end;
end;
registration
let D;
let X be with_non-empty_element subset-closed Subset-Family of D;
let n be non empty set;
cluster the_subsets_with_limited_card(n,X,D) -> with_non-empty_element;
coherence
proof
consider x be non empty set such that
A1: x in X by SETFAM_1:def 11;
consider y be set such that
A2: y in x by XBOOLE_0:def 1;
{}c=card n;
then {} in card n by CARD_1:13;
then 1 c=card n by CARD_3:109;
then A3: card{y}c=card n by CARD_1:50;
{y}c=x by A2,ZFMISC_1:37;
then {y} in X by A1,CLASSES1:def 1;
then {y} in the_subsets_with_limited_card(n,X,D) by A3,Def2;
hence thesis by SETFAM_1:def 11;
end;
end;
notation let X;
let Y be Subset-Family of X;
let n be set;
synonym the_subsets_with_limited_card(n,Y)
for the_subsets_with_limited_card(n,Y,X);
end;
theorem Th9:
X is non empty finite c=-linear implies union X in X
proof
assume X is non empty finite c=-linear;
then consider U be set such that
A1: U in X and
A2: for x st x in X holds x c=U by FINSET_1:31;
A3: union X c=U
proof
let x;
assume x in union X;
then consider y such that
A4: x in y and
A5: y in X by TARSKI:def 4;
y c=U by A2,A5;
hence thesis by A4;
end;
U c=union X by A1,ZFMISC_1:92;
hence thesis by A1,A3,XBOOLE_0:def 10;
end;
theorem Th10:
for X be finite c=-linear set st X is with_non-empty_elements
holds card X c= card union X
proof
let X be finite c=-linear set;
defpred P[Nat] means
for X be finite c=-linear set st X is with_non-empty_elements & card union X
=$1 holds card X c=card union X;
defpred Q[Nat] means
for n be Nat st n<=$1 holds P[n];
assume A1: X is with_non-empty_elements;
A2: for m be Nat st Q[m] holds Q[m+1]
proof
let m be Nat such that
A3: Q[m];
let n be Nat;
assume A4: n<=m+1;
let X be finite c=-linear set such that
A5: X is with_non-empty_elements and
A6: card union X=n;
reconsider u=union X as finite set by A6;
reconsider Xu=X\{u} as finite c=-linear set;
per cases by A4,NAT_1:8;
suppose n<=m;
hence thesis by A3,A5,A6;
end;
suppose A7: n=m+1;
then X is non empty by A6,ZFMISC_1:2;
then A8: X=Xu\/{u} by Th9,ZFMISC_1:140;
then u=union Xu\/union{u} by ZFMISC_1:96
.=union Xu\/u by ZFMISC_1:31;
then A9: union Xu c=u by XBOOLE_1:11;
then reconsider uXu=union Xu as finite set;
uXu<>u
proof
assume A10: uXu=u;
then Xu is non empty by A6,A7,ZFMISC_1:2;
then u in Xu by A10,Th9;
hence thesis by ZFMISC_1:64;
end;
then A11: uXu c__ 1 ex x st x in Z & Z\{x} in Y
proof
let X be non empty set;
consider R be Relation such that
A1: R well_orders X by WELLORD2:26;
set RX=R|_2 X;
deffunc F(set)=X\(RX-Seg($1));
A2: for x st x in X holds F(x) in bool X;
consider f be Function of X,bool X such that
A3: for x st x in X holds f.x=F(x) from FUNCT_2:sch 2(A2);
take Y=rng f;
A4: dom f=X by FUNCT_2:def 1;
thus Y is with_non-empty_elements
proof
assume Y is with_empty_element;
then {} in Y by SETFAM_1:def 9;
then consider x such that
A5: x in dom f and
A6: f.x={} by FUNCT_1:def 5;
{}=F(x) by A3,A5,A6;
then X c=RX-Seg(x) by XBOOLE_1:37;
hence thesis by A4,A5,WELLORD1:def 1;
end;
thus Y is c=-linear
proof
let x,y;
assume that
A7: x in Y and
A8: y in Y;
consider a be set such that
A9: a in dom f & f.a=x by A7,FUNCT_1:def 5;
consider b be set such that
A10: b in dom f & f.b=y by A8,FUNCT_1:def 5;
RX-Seg(a),RX-Seg(b)are_c=-comparable by A1,WELLORD1:33,WELLORD2:25;
then RX-Seg(a)c=RX-Seg(b) or RX-Seg(b)c=RX-Seg(a) by XBOOLE_0:def 9;
then F(b)c=F(a) or F(a)c=F(b) by XBOOLE_1:34;
then A11: F(a),F(b)are_c=-comparable by XBOOLE_0:def 9;
x=F(a) by A3,A9;
hence thesis by A3,A10,A11;
end;
A12: field RX=X by A1,WELLORD2:25;
then consider x such that
A13: x in X and
A14: for y st y in X holds[x,y] in RX by A1,WELLORD1:11,WELLORD2:25;
A15: RX is well-ordering by A1,WELLORD2:25;
then A16: RX is well_founded by WELLORD1:def 4;
RX is antisymmetric by A15,WELLORD1:def 4;
then A17: RX is_antisymmetric_in X by A12,RELAT_2:def 12;
A18: RX-Seg(x)={}
proof
assume RX-Seg(x)<>{};
then consider y such that
A19: y in RX-Seg(x) by XBOOLE_0:def 1;
A20: y<>x by A19,WELLORD1:def 1;
A21: [y,x] in RX by A19,WELLORD1:def 1;
then A22: x in X by A12,RELAT_1:30;
A23: y in X by A12,A21,RELAT_1:30;
then [x,y] in RX by A14;
hence thesis by A17,A20,A21,A22,A23,RELAT_2:def 4;
end;
f.x=F(x) by A3,A13;
hence X in Y by A4,A13,A18,FUNCT_1:def 5;
now let x1,x2 be set;
reconsider R1=RX-Seg(x1),R2=RX-Seg(x2) as Subset of X by A12,WELLORD1:13;
assume that
A24: x1 in X & x2 in X and
A25: f.x1=f.x2;
R1`=f.x1 & f.x2=R2` by A3,A24;
then R1=R2 by A25,SUBSET_1:64;
then [x1,x2] in RX & [x2,x1] in RX by A12,A15,A24,WELLORD1:37;
hence x1=x2 by A17,A24,RELAT_2:def 4;
end;
then f is one-to-one by FUNCT_2:25;
then X,Y are_equipotent by A4,WELLORD2:def 4;
hence card X=card Y by CARD_1:21;
let Z;
assume that
A26: Z in Y and
A27: card Z<>1;
consider x such that
A28: x in dom f and
A29: f.x=Z by A26,FUNCT_1:def 5;
A30: {x}c=X by A28,ZFMISC_1:37;
take x;
A31: not x in RX-Seg(x) by WELLORD1:def 1;
A32: Z=X\(RX-Seg(x)) by A3,A28,A29;
hence x in Z by A28,A31,XBOOLE_0:def 5;
RX-Seg(x)c=X by A12,WELLORD1:13;
then reconsider Rxx=RX-Seg(x)\/{x} as Subset of X by A30,XBOOLE_1:8;
X\Rxx<>{}
proof
assume X\Rxx={};
then X c=Rxx by XBOOLE_1:37;
then Z=Rxx\RX-Seg(x) by A32,XBOOLE_0:def 10
.={x}\RX-Seg(x) by XBOOLE_1:40
.={x} by A31,ZFMISC_1:67;
hence contradiction by A27,CARD_1:50;
end;
then consider a be set such that
A33: a in Rxx` and
A34: RX-Seg(a)misses Rxx` by A12,A16,WELLORD1:def 2;
A35: not a in Rxx by A33,XBOOLE_0:def 5;
x in {x} by TARSKI:def 1;
then A36: x<>a by A35,XBOOLE_0:def 3;
RX is connected by A15,WELLORD1:def 4;
then RX is_connected_in X by A12,RELAT_2:def 14;
then A37: [x,a] in RX or[a,x] in RX by A28,A33,A36,RELAT_2:def 6;
A38: not a in RX-Seg(x) by A35,XBOOLE_0:def 3;
then x in RX-Seg(a) by A36,A37,WELLORD1:def 1;
then A39: {x}c=RX-Seg(a) by ZFMISC_1:37;
RX-Seg(a)c=X by A12,WELLORD1:13;
then A40: RX-Seg(a)c=Rxx by A34,SUBSET_1:44;
RX-Seg(x)c=RX-Seg(a) by A12,A15,A28,A33,A37,A38,WELLORD1:37,def 1;
then Rxx c=RX-Seg(a) by A39,XBOOLE_1:8;
then Rxx=RX-Seg(a) by A40,XBOOLE_0:def 10;
then f.a=X\Rxx by A3,A33
.=(X\RX-Seg(x))\{x} by XBOOLE_1:41
.=Z\{x} by A3,A28,A29;
hence thesis by A4,A33,FUNCT_1:def 5;
end;
theorem
for Y be Subset-Family of X st
Y is finite with_non-empty_elements c=-linear & X in Y
ex Y1 be Subset-Family of X st
Y c= Y1 & Y1 is with_non-empty_elements c=-linear & card X = card Y1 &
for Z st Z in Y1 & card Z <> 1 ex x st x in Z & Z\{x} in Y1
proof
let Y be Subset-Family of X such that
A1: Y is finite with_non-empty_elements c=-linear and
A2: X in Y;
defpred P[set,set] means
$1 in $2 & $2 in Y & for y st y in Y & $1 in y holds$2c=y;
A3: for x st x in X ex y st P[x,y]
proof
let x;
set U={A where A is Subset of X:x in A & A in Y};
A4: U c=Y
proof
let y;
assume y in U;
then ex A be Subset of X st y=A & x in A & A in Y;
hence thesis;
end;
then reconsider U as Subset-Family of X by XBOOLE_1:1;
assume x in X;
then X in U by A2;
then consider m be set such that
A5: m in U and
A6: for y st y in U holds m c=y by A1,A4,FINSET_1:30;
take m;
ex A be Subset of X st m=A & x in A & A in Y by A5;
hence x in m & m in Y;
let y;
assume y in Y & x in y;
then y in U;
hence thesis by A6;
end;
consider f be Function such that
A7: dom f=X & for x st x in X holds P[x,f.x] from CLASSES1:sch 1(A3);
defpred Q[set,set] means
($2 in Y or$2 is empty) & $2c<$1 & (for x st x in Y & x c<$1 holds x c=$2) &
(for x st x in Y & $2c=x & x c=$1 holds x=$1 or x=$2);
A8: for x st x in Y ex y st y in bool X & Q[x,y]
proof
let x;
assume A9: x in Y;
set U={A where A is Subset of X:A cx;
then y cx;
then y c1 ex x st x in Z & Z\{x} in rng h;
A21: for x st x in Y ex y st R[x,y]
proof
let y;
assume y in Y;
then g.y c{} by XBOOLE_1:105;
then consider Z be Subset-Family of y\(g.y) such that
A22: Z is with_non-empty_elements c=-linear & y\(g.y) in Z and
A23: card(y\g.y)=card Z and
A24: for z be set st z in Z & card z<>1 ex x st x in z & z\{x} in Z by Th12
;
y\g.y,Z are_equipotent by A23,CARD_1:21;
then consider h be Function such that
A25: h is one-to-one and
A26: dom h=y\g.y & rng h=Z by WELLORD2:def 4;
reconsider h as Function of y\g.y,bool(y\g.y) by A26,FUNCT_2:4;
take h;
thus thesis by A22,A24,A25,A26;
end;
consider h be Function such that
A27: dom h=Y & for x st x in Y holds R[x,h.x] from CLASSES1:sch 1(A21);
now let x;
assume x in dom h;
then ex H be Function of x\(g.x),bool(x\(g.x)) st h.x=H & H is one-to-one &
rng H is with_non-empty_elements c=-linear & dom H in rng H & for Z st Z in rng
H & card Z<>1 ex x st x in Z & Z\{x} in rng H by A27;
hence h.x is Function;
end;
then reconsider h as Function-yielding Function by FUNCOP_1:def 6;
deffunc Z(set)=g.(f.$1)\/(h.(f.$1)).$1;
A28: for x st x in X holds x in (f.x)\(g.(f.x))
proof
let x;
assume A29: x in X;
then A30: f.x in Y by A7;
assume A31: not x in (f.x)\(g.(f.x));
x in f.x by A7,A29;
then A32: x in g.(f.x) by A31,XBOOLE_0:def 5;
then g.(f.x) in Y by A20,A30;
then A33: f.x c=g.(f.x) by A7,A32;
g.(f.x)c1 ex x st x in Z & Z\{x} in rng H by A27;
A38: x in fx\(g.fx) by A28,A35;
dom H=fx\(g.fx) by FUNCT_2:def 1;
then H.x in rng H by A38,FUNCT_1:def 5;
then H.x c=fx by XBOOLE_1:1;
then A39: H.x c=X by A36,XBOOLE_1:1;
g.fx in Y or g.fx is empty by A20,A36;
then Z(x)c=X by A37,A39,XBOOLE_1:8;
hence thesis;
end;
consider z be Function of X,bool X such that
A40: for x st x in X holds z.x=Z(x) from FUNCT_2:sch 2(A34);
A41: dom z=X by FUNCT_2:def 1;
A42: Y c=rng z
proof
let y;
assume A43: y in Y;
then consider H be Function of y\(g.y),bool(y\(g.y)) such that
A44: h.y=H and
H is one-to-one and
rng H is with_non-empty_elements c=-linear and
A45: dom H in rng H and
for Z st Z in rng H & card Z<>1 ex x st x in Z & Z\{x} in rng H by A27;
consider x such that
A46: x in dom H and
A47: H.x=dom H by A45,FUNCT_1:def 5;
A48: dom H=y\(g.y) by FUNCT_2:def 1;
then A49: x in y by A46;
then A50: x in f.x by A7,A43;
g.y cy;
then f.x c1 ex x st x in Z & Z\{x} in rng z
proof
let Z;
assume that
A55: Z in rng z and
A56: card Z<>1;
consider x be set such that
A57: x in dom z and
A58: z.x=Z by A55,FUNCT_1:def 5;
set fx=f.x;
A59: fx in Y by A7,A57;
then consider H be Function of fx\(g.fx),bool(fx\(g.fx)) such that
A60: h.fx=H and
H is one-to-one and
rng H is with_non-empty_elements c=-linear and
dom H in rng H and
A61: for Z st Z in rng H & card Z<>1 ex x st x in Z & Z\{x} in rng H by
A27;
A62: z.x=g.fx\/H.x by A40,A57,A60;
A63: dom H=fx\(g.fx) by FUNCT_2:def 1;
x in fx\(g.fx) by A28,A57;
then A64: H.x in rng H by A63,FUNCT_1:def 5;
per cases;
suppose A65: card(H.x)=1;
then g.fx<>{} by A56,A58,A62;
then A66: g.fx in Y by A20,A59;
consider a be set such that
A67: H.x={a} by A65,CARD_2:60;
take a;
A68: a in H.x by A67,TARSKI:def 1;
then A69: not a in g.fx by A64,XBOOLE_0:def 5;
thus a in Z by A58,A62,A68,XBOOLE_0:def 3;
Z\{a}=(g.fx\/H.x)\H.x by A40,A57,A58,A60,A67
.=g.fx\H.x by XBOOLE_1:40
.=g.fx by A67,A69,ZFMISC_1:65;
hence thesis by A42,A66;
end;
suppose card(H.x)<>1;
then consider a be set such that
A70: a in H.x and
A71: H.x\{a} in rng H by A61,A64;
A72: not a in g.fx by A64,A70,XBOOLE_0:def 5;
take a;
thus a in Z by A58,A62,A70,XBOOLE_0:def 3;
consider b be set such that
A73: b in dom H and
A74: H.b=H.x\{a} by A71,FUNCT_1:def 5;
A75: b in fx by A63,A73;
then A76: b in f.b by A7,A59;
A77: f.b in Y by A7,A59,A75;
A78: f.b c=fx by A7,A59,A75;
fx=f.b
proof
assume fx<>f.b;
then f.b c1 ex x st x in Z & Z\{x} in rng H1 by A27;
A89: z.x1=g.fx1\/H1.x1 by A40,A82,A88;
dom H1=fx1\(g.fx1) by FUNCT_2:def 1;
then A90: H1.x1 in rng H1 by A86,FUNCT_1:def 5;
A91: x2 in fx2\(g.fx2) by A28,A84;
A92: fx2 in Y by A7,A84;
then consider H2 be Function of fx2\(g.fx2),bool(fx2\(g.fx2)) such that
A93: h.fx2=H2 and
H2 is one-to-one and
A94: rng H2 is with_non-empty_elements c=-linear and
dom H2 in rng H2 and
for Z st Z in rng H2 & card Z<>1 ex x st x in Z & Z\{x} in rng H2 by A27;
A95: z.x2=g.fx2\/H2.x2 by A40,A84,A93;
dom H2=fx2\(g.fx2) by FUNCT_2:def 1;
then A96: H2.x2 in rng H2 by A91,FUNCT_1:def 5;
A97: fx1,fx2 are_c=-comparable by A1,A87,A92,ORDINAL1:def 9;
per cases by A97,XBOOLE_0:def 9;
suppose A98: fx1=fx2;
then H1.x1,H1.x2 are_c=-comparable by A88,A90,A93,A94,A96,ORDINAL1:def
9;
then H1.x1 c=H1.x2 or H1.x2 c=H1.x1 by XBOOLE_0:def 9;
then z.x1 c=z.x2 or z.x2 c=z.x1 by A88,A89,A93,A95,A98,XBOOLE_1:9;
hence thesis by A83,A85,XBOOLE_0:def 9;
end;
suppose fx1 c=fx2 & fx1<>fx2;
then fx1 cfx2;
then fx2 c1 ex x st x in Z & Z\{x} in rng H by A27;
dom H=fx\(g.fx) by FUNCT_2:def 1;
then A111: H.x in rng H by A108,FUNCT_1:def 5;
z.x=g.fx\/H.x by A40,A106,A109;
hence contradiction by A107,A110,A111;
end;
take rng z;
for x1,x2 be set st x1 in dom z & x2 in dom z & z.x1=z.x2 holds x1=x2
proof
let x1,x2 be set;
set f1=f.x1,f2=f.x2;
assume that
A112: x1 in dom z and
A113: x2 in dom z and
A114: z.x1=z.x2;
A115: z.x1=Z(x1) & z.x2=Z(x2) by A40,A112,A113;
A116: f2 in Y by A7,A113;
then consider H2 be Function of f2\(g.f2),bool(f2\(g.f2)) such that
A117: h.f2=H2 and
A118: H2 is one-to-one and
A119: rng H2 is with_non-empty_elements c=-linear and
dom H2 in rng H2 and
for Z st Z in rng H2 & card Z<>1 ex x st x in Z & Z\{x} in rng H2 by A27;
dom H2=f2\(g.f2) by FUNCT_2:def 1;
then A120: x2 in dom H2 by A28,A113;
then A121: H2.x2 in rng H2 by FUNCT_1:def 5;
then A122: g.f2 misses H2.x2 by XBOOLE_1:63,79;
A123: f1 in Y by A7,A112;
then consider H1 be Function of f1\(g.f1),bool(f1\(g.f1)) such that
A124: h.f1=H1 and
H1 is one-to-one and
A125: rng H1 is with_non-empty_elements c=-linear and
dom H1 in rng H1 and
for Z st Z in rng H1 & card Z<>1 ex x st x in Z & Z\{x} in rng H1 by A27;
dom H1=f1\(g.f1) by FUNCT_2:def 1;
then A126: x1 in dom H1 by A28,A112;
then A127: H1.x1 in rng H1 by FUNCT_1:def 5;
then A128: g.f1 misses H1.x1 by XBOOLE_1:63,79;
A129: f1,f2 are_c=-comparable by A1,A116,A123,ORDINAL1:def 9;
per cases by A129,XBOOLE_0:def 9;
suppose A130: f1=f2;
then H1.x1=H1.x2 by A114,A115,A117,A122,A124,A128,XBOOLE_1:71;
hence thesis by A117,A118,A120,A124,A126,A130,FUNCT_1:def 8;
end;
suppose A131: f1 c=f2 & f1<>f2;
g.f1 cf2;
g.f2 c Subset of K means :Def4:
for v be Element of K holds v in it iff v is vertex-like;
existence
proof
set B={v where v is Element of K:v is vertex-like};
B c=[#]K
proof
let x;
assume x in B;
then consider v be Element of K such that
A1: x=v and
A2: v is vertex-like;
ex S be Subset of K st S is simplex-like & v in S by A2,Def3;
hence thesis by A1;
end;
then reconsider B as Subset of K;
take B;
let v be Element of K;
hereby assume v in B;
then ex w be Element of K st v=w & w is vertex-like;
hence v is vertex-like;
end;
assume v is vertex-like;
hence thesis;
end;
uniqueness
proof
let V1,V2 be Subset of K such that
A3: for v be Element of K holds v in V1 iff v is vertex-like and
A4: for v be Element of K holds v in V2 iff v is vertex-like;
hereby let x;
assume A5: x in V1;
then reconsider v=x as Element of K;
v is vertex-like by A3,A5;
hence x in V2 by A4;
end;
let x;
assume A6: x in V2;
then reconsider v=x as Element of K;
v is vertex-like by A4,A6;
hence thesis by A3;
end;
end;
definition
let K be SimplicialComplexStr;
mode Vertex of K is Element of Vertices K;
end;
definition
let K be SimplicialComplexStr;
attr K is finite-vertices means :Def5:
Vertices K is finite;
end;
definition
let K;
attr K is locally-finite means :Def6:
for v be Vertex of K holds
{S where S is Subset of K : S is simplex-like & v in S} is finite;
end;
definition
let K;
attr K is empty-membered means :Def7:
the topology of K is empty-membered;
attr K is with_non-empty_elements means :Def8:
the topology of K is with_non-empty_elements;
end;
notation let K;
antonym K is with_non-empty_element for K is empty-membered;
antonym K is with_empty_element for K is with_non-empty_elements;
end;
definition
let X;
mode SimplicialComplexStr of X -> SimplicialComplexStr means :Def9:
[#]it c= X;
existence
proof
take S=the empty TopStruct;
thus thesis by XBOOLE_1:2;
end;
end;
definition
let X;
let KX be SimplicialComplexStr of X;
attr KX is total means :Def10:
[#]KX = X;
end;
Lm2: Vertices K is empty iff K is empty-membered
proof
hereby assume A1: Vertices K is empty;
assume K is with_non-empty_element;
then the topology of K is with_non-empty_element by Def7;
then consider D such that
A2: D in the topology of K by SETFAM_1:def 11;
reconsider S=D as Subset of K by A2;
the Element of D in S;
then reconsider v=the Element of D as Element of K;
S is simplex-like by A2,PRE_TOPC:def 5;
then v is vertex-like by Def3;
hence contradiction by A1,Def4;
end;
assume A3: K is empty-membered;
assume Vertices K is non empty;
then consider x such that
A4: x in Vertices K by XBOOLE_0:def 1;
reconsider x as Element of K by A4;
x is vertex-like by A4,Def4;
then consider S be Subset of K such that
A5: S is simplex-like and
A6: x in S by Def3;
S in the topology of K by A5,PRE_TOPC:def 5;
then the topology of K is with_non-empty_element by A6,SETFAM_1:def 11;
hence contradiction by A3,Def7;
end;
Lm3: for S be Subset of K st S is simplex-like & x in S holds x in Vertices K
proof
let S be Subset of K such that
A1: S is simplex-like and
A2: x in S;
reconsider v=x as Element of K by A2;
v is vertex-like by A1,A2,Def3;
hence thesis by Def4;
end;
Lm4: for A be Subset of K st A is simplex-like holds A c=Vertices K
proof
let A be Subset of K;
assume A1: A is simplex-like;
let x;
thus thesis by A1,Lm3;
end;
Lm5: Vertices K=union the topology of K
proof
hereby let x such that
A1: x in Vertices(K);
reconsider v=x as Element of K by A1;
v is vertex-like by A1,Def4;
then consider S be Subset of K such that
A2: S is simplex-like and
A3: v in S by Def3;
S in the topology of K by A2,PRE_TOPC:def 5;
hence x in union the topology of K by A3,TARSKI:def 4;
end;
let x;
assume x in union the topology of K;
then consider y such that
A4: x in y and
A5: y in the topology of K by TARSKI:def 4;
reconsider y as Subset of K by A5;
y is simplex-like by A5,PRE_TOPC:def 5;
hence thesis by A4,Lm3;
end;
Lm6: K is finite-vertices implies the topology of K is finite
proof
Vertices K=union the topology of K by Lm5;
then A1: the topology of K c=bool Vertices K by ZFMISC_1:100;
assume K is finite-vertices;
then Vertices K is finite by Def5;
hence thesis by A1;
end;
registration
cluster with_empty_element -> non void SimplicialComplexStr;
coherence
proof
let K;
assume K is with_empty_element;
then the topology of K is with_empty_element by Def8;
hence thesis by PENCIL_1:def 4;
end;
cluster with_non-empty_element -> non void SimplicialComplexStr;
coherence
proof
let K;
assume K is with_non-empty_element;
then the topology of K is with_non-empty_element by Def7;
hence thesis by PENCIL_1:def 4;
end;
cluster non void empty-membered -> with_empty_element SimplicialComplexStr;
coherence
proof
let K;
assume K is non void empty-membered;
then the topology of K is with_empty_element by Def7;
hence thesis by Def8;
end;
cluster non void subset-closed -> with_empty_element SimplicialComplexStr;
coherence
proof
let K;
assume A1: K is non void subset-closed;
then ex x be set st x in the_family_of K by XBOOLE_0:def 1;
hence thesis by A1,Def8;
end;
cluster empty-membered-> subset-closed finite-vertices SimplicialComplexStr;
coherence
proof
let K;
assume A2: K is empty-membered;
then A3: the topology of K is empty-membered by Def7;
thus K is subset-closed
proof
let x,y such that
A4: x in the_family_of K and
A5: y c=x;
x is empty by A3,A4,SETFAM_1:def 11;
hence thesis by A4,A5;
end;
Vertices K is empty by A2,Lm2;
hence thesis by Def5;
end;
cluster finite-vertices-> locally-finite finite-degree SimplicialComplexStr;
coherence
proof
let K;
assume A6: K is finite-vertices;
then reconsider V=Vertices(K) as finite Subset of K by Def5;
thus K is locally-finite
proof
let v be Vertex of K;
A7: {S where S is Subset of K:S is simplex-like & v in S}c=the topology of K
proof
let x;
assume x in {S where S is Subset of K:S is simplex-like & v in S};
then ex S be Subset of K st x=S & S is simplex-like & v in S;
hence thesis by PRE_TOPC:def 5;
end;
the topology of K is finite by A6,Lm6;
hence thesis by A7;
end;
thus K is finite-membered
proof
let x;
assume A8: x in the_family_of K;
then reconsider X=x as Subset of K;
X is simplex-like by A8,PRE_TOPC:def 5;
then X c=V by Lm4;
hence thesis;
end;
take card V;
let A be finite Subset of K;
assume A is open;
hence thesis by Lm4,NAT_1:44;
end;
cluster locally-finite subset-closed -> finite-membered SimplicialComplexStr;
coherence
proof
let K be SimplicialComplexStr;
assume A9: K is locally-finite subset-closed;
the_family_of K is finite-membered
proof
let x;
assume A10: x in the_family_of K;
then reconsider A=x as Subset of K;
A11: K is non void by A10,PENCIL_1:def 4;
A12: A is simplex-like by A10,PRE_TOPC:def 5;
per cases;
suppose x is empty;
hence thesis;
end;
suppose x is non empty;
then consider a be set such that
A13: a in A by XBOOLE_0:def 1;
reconsider a as Element of K by A13;
a is vertex-like by A12,A13,Def3;
then reconsider a as Vertex of K by Def4;
set Aa=A\{a};
set X={S where S is Subset of K:a in S & S c=A};
defpred P[set,set] means
$1\{a}=$2;
set Z={y where y is Element of bool Aa:ex w being Subset of K st P[w,y] & w
in X};
A14: bool Aa c=Z
proof
let y;
assume A15: y in bool Aa;
then reconsider B=y as Subset of K by XBOOLE_1:1;
not a in B by A15,ZFMISC_1:64;
then A16: (B\/{a})\{a}=B by ZFMISC_1:141;
A17: {a}c=A by A13,ZFMISC_1:37;
Aa c=A by XBOOLE_1:36;
then y c=A by A15,XBOOLE_1:1;
then A18: B\/{a}c=A by A17,XBOOLE_1:8;
then A19: B\/{a} is Subset of K by XBOOLE_1:1;
a in {a} by TARSKI:def 1;
then a in B\/{a} by XBOOLE_0:def 3;
then B\/{a} in X by A18,A19;
hence thesis by A15,A16,A19;
end;
set Y={S where S is Subset of K:S is simplex-like & a in S};
A20: X c=Y
proof
let y;
assume y in X;
then consider S be Subset of K such that
A21: y=S & a in S and
A22: S c=A;
S is simplex-like by A9,A11,A12,A22,MATROID0:1;
hence thesis by A21;
end;
Y is finite by A9,Def6;
then A23: X is finite by A20;
A24: for w be Subset of K,x,y be Element of bool Aa st P[w,x] & P[w,y]
holds x=y;
Z is finite from FRAENKEL:sch 28(A23,A24);
then A25: Aa is finite by A14;
A=Aa\/{a} by A13,ZFMISC_1:140;
hence x is finite by A25;
end;
end;
hence thesis by MATROID0:def 6;
end;
end;
registration
let X;
cluster empty void empty-membered strict SimplicialComplexStr of X;
existence
proof
[#]TopStruct(# {},{}bool{} #)c=X by XBOOLE_1:2;
then reconsider T=TopStruct(# {},{}bool{} #) as SimplicialComplexStr of X by
Def9;
take T;
thus thesis;
end;
end;
registration
let D;
cluster non empty non void total empty-membered
strict SimplicialComplexStr of D;
existence
proof
reconsider B=bool 0 as Subset-Family of D by XBOOLE_1:2,ZFMISC_1:79;
[#]TopStruct(# D,B #)c=D;
then reconsider T=TopStruct(# D,B #) as SimplicialComplexStr of D by Def9;
take T;
for x be non empty set holds not x in B;
then [#]T=D & B is empty-membered by SETFAM_1:def 11;
hence thesis by Def7,Def10,PENCIL_1:def 4;
end;
cluster non empty total with_empty_element with_non-empty_element
finite-vertices subset-closed strict SimplicialComplexStr of D;
existence
proof
set E=the Element of D;
reconsider B=bool{E} as Subset-Family of D by ZFMISC_1:79;
[#]TopStruct(# D,B #)c=[#]D;
then reconsider T=TopStruct(# D,B #) as SimplicialComplexStr of D by Def9;
take T;
A1: Vertices(T)c={E}
proof
let x such that
A2: x in Vertices(T);
reconsider v=x as Element of T by A2;
v is vertex-like by A2,Def4;
then consider S be Subset of T such that
A3: S is simplex-like and
A4: v in S by Def3;
S in B by A3,PRE_TOPC:def 5;
hence thesis by A4;
end;
{E} in B by ZFMISC_1:def 1;
then B is with_non-empty_element by SETFAM_1:def 11;
then A5: [#]T=D & T is with_non-empty_element by Def7;
B=the_family_of T;
then T is subset-closed by MATROID0:def 3;
hence thesis by A1,A5,Def5,Def10;
end;
end;
registration
cluster non empty with_empty_element with_non-empty_element finite-vertices
subset-closed strict SimplicialComplexStr;
existence
proof
take T=the non empty with_empty_element with_non-empty_element
finite-vertices subset-closed strict SimplicialComplexStr of(the non empty set
);
thus thesis;
end;
end;
registration
let K be with_non-empty_element SimplicialComplexStr;
cluster Vertices K -> non empty;
coherence by Lm2;
end;
registration
let K be finite-vertices SimplicialComplexStr;
cluster simplex-like -> finite Subset-Family of K;
coherence
proof
let S be Subset-Family of K;
assume S is simplex-like;
then A1: S c=the topology of K by Th14;
Vertices K=union the topology of K by Lm5;
then A2: the topology of K c=bool Vertices K by ZFMISC_1:100;
Vertices K is finite by Def5;
hence thesis by A1,A2;
end;
end;
registration
let K be finite-membered SimplicialComplexStr;
cluster simplex-like -> finite-membered Subset-Family of K;
coherence
proof
let S be Subset-Family of K;
assume A1: S is simplex-like;
let x;
assume A2: x in S;
then reconsider X=x as Subset of K;
X is simplex-like by A1,A2,TOPS_2:def 1;
then A3: X in the topology of K by PRE_TOPC:def 5;
the_family_of K is finite-membered by MATROID0:def 6;
hence thesis by A3,MATROID0:def 5;
end;
end;
theorem
Vertices K is empty iff K is empty-membered by Lm2;
theorem
Vertices K = union the topology of K by Lm5;
theorem
for S be Subset of K st S is simplex-like holds S c= Vertices K by Lm4;
theorem
K is finite-vertices implies the topology of K is finite by Lm6;
theorem Th19:
the topology of K is finite & K is non finite-vertices implies
K is non finite-membered
proof
defpred P[set,set] means
$1 in $2;
set V=Vertices K;
assume that
A1: the topology of K is finite and
A2: K is non finite-vertices;
A3: V is infinite by A2,Def5;
A4: V=union the topology of K by Lm5;
A5: for x st x in V ex y st y in the topology of K & P[x,y]
proof
let x;
assume x in V;
then ex y st x in y & y in the topology of K by A4,TARSKI:def 4;
hence thesis;
end;
consider f be Function of V,the topology of K such that
A6: for x st x in V holds P[x,f.x] from FUNCT_2:sch 1(A5);
the topology of K is non empty by A2,A4,Def5;
then dom f=V by FUNCT_2:def 1;
then consider x such that
A7: x in rng f and
A8: f"{x} is infinite by A1,A3,COMPL_SP:24;
x in the topology of K by A7;
then reconsider x as Subset of K;
f"{x}c=x
proof
let y;
assume y in f"{x};
then f.y in {x} & y in f.y by A6,FUNCT_1:def 13;
hence thesis by TARSKI:def 1;
end;
then x is non finite by A8;
then the_family_of K is non finite-membered by A7,MATROID0:def 5;
hence thesis by MATROID0:def 6;
end;
theorem Th20:
K is subset-closed & the topology of K is finite implies K is finite-vertices
proof
assume that
A1: K is subset-closed and
A2: the topology of K is finite;
assume K is non finite-vertices;
then K is non finite-membered by A2,Th19;
then the_family_of K is non finite-membered by MATROID0:def 6;
then consider x such that
A3: x in the_family_of K and
A4: x is non finite by MATROID0:def 5;
{x}c=the_family_of K by A3,ZFMISC_1:37;
then subset-closed_closure_of{x}c=the_family_of K by A1,Th3;
then bool x c=the_family_of K by Th4;
hence contradiction by A2,A4;
end;
begin :: The Simplicial Complex Generated on the Set
definition
let X;
let Y be Subset-Family of X;
func Complex_of Y -> strict SimplicialComplexStr of X equals
TopStruct(# X,subset-closed_closure_of Y #);
coherence
proof
[#]TopStruct(# X,subset-closed_closure_of Y #)=X;
hence thesis by Def9;
end;
end;
registration
let X;
let Y be Subset-Family of X;
cluster Complex_of Y -> total subset-closed;
coherence
proof
[#](Complex_of Y)=X & subset-closed_closure_of Y=the_family_of TopStruct(# X,
subset-closed_closure_of Y #);
hence thesis by Def10,MATROID0:def 3;
end;
end;
registration
let X;
let Y be non empty Subset-Family of X;
cluster Complex_of Y -> with_empty_element;
coherence by Def8;
end;
registration
let X;
let Y be finite-membered Subset-Family of X;
cluster Complex_of Y -> finite-membered;
coherence
proof
subset-closed_closure_of Y=the_family_of Complex_of Y;
hence thesis by MATROID0:def 6;
end;
end;
registration
let X;
let Y be finite finite-membered Subset-Family of X;
cluster Complex_of Y -> finite-vertices;
coherence
proof
set C=Complex_of Y;
reconsider Y1=Y as Subset-Family of union Y by ZFMISC_1:100;
subset-closed_closure_of Y1 c=bool union Y;
then A1: union the topology of C c=union bool union Y by ZFMISC_1:95;
union bool union Y=union Y by ZFMISC_1:99;
then Vertices C c=union Y by A1,Lm5;
hence thesis by Def5;
end;
end;
theorem
K is subset-closed implies the TopStruct of K = Complex_of the topology of K
proof
assume K is subset-closed;
then the_family_of K is subset-closed;
hence thesis by Th7;
end;
definition
let X;
mode SimplicialComplex of X is finite-membered subset-closed
SimplicialComplexStr of X;
end;
definition
let K be non void SimplicialComplexStr;
mode Simplex of K is simplex-like Subset of K;
end;
registration
let K be with_empty_element SimplicialComplexStr;
cluster empty -> simplex-like Subset of K;
coherence
proof
let S be Subset of K;
assume A1: S is empty;
the topology of K is with_empty_element by Def8;
then S in the topology of K by A1,SETFAM_1:def 9;
hence thesis by PRE_TOPC:def 5;
end;
cluster empty Simplex of K;
existence
proof
{}K is simplex-like;
hence thesis;
end;
end;
registration
let K be non void finite-membered SimplicialComplexStr;
cluster finite Simplex of K;
existence
proof
consider S be set such that
A1: S in the topology of K by XBOOLE_0:def 1;
reconsider S as Simplex of K by A1,PRE_TOPC:def 5;
S is finite;
hence thesis;
end;
end;
begin :: The Degree of Simplicial Complexes
definition
let K;
func degree K -> ext-real number means :Def12:
(for S be finite Subset of K st S is simplex-like holds card S <= it+1) &
ex S be Subset of K st S is simplex-like & card S = it+1
if K is non void finite-degree,
it = -1 if K is void
otherwise
it = +infty;
existence
proof
now defpred P[Nat] means
for S be finite Subset of K st S is simplex-like holds card S<=$1;
assume A1: K is non void finite-degree;
then A2: the_family_of K is finite-membered by MATROID0:def 6;
A3: ex n be Nat st P[n] by A1,MATROID0:def 7;
consider k be Nat such that
A4: P[k] and
A5: for n be Nat st P[n] holds k<=n from NAT_1:sch 5(A3);
take D=k-1;
thus for S be finite Subset of K st S is simplex-like holds card S<=D+1 by
A4;
assume A6: for S be Subset of K st S is simplex-like holds card S<>D+1;
per cases;
suppose A7: k=0;
consider S be set such that
A8: S in the topology of K by A1,XBOOLE_0:def 1;
reconsider S as finite Subset of K by A2,A8,MATROID0:def 5;
A9: S is simplex-like by A8,PRE_TOPC:def 5;
then card S=0 by A4,A7;
hence contradiction by A6,A7,A9;
end;
suppose k>0;
then reconsider k1=k-1 as Element of NAT by NAT_1:20;
P[k1]
proof
let S be finite Subset of K;
assume A10: S is simplex-like;
then card S<=k1+1 by A4;
then card S natural;
coherence
proof
per cases;
suppose K is void;
then degree K+1=-1+1 by Def12
.=0;
hence thesis;
end;
suppose K is non void;
then consider S be Subset of K such that
A1: S is simplex-like and
A2: card S=degree K+1 by Def12;
the_family_of K is finite-membered & S in the topology of K by A1,MATROID0:
def 6,PRE_TOPC:def 5;
then S is finite by MATROID0:def 5;
hence thesis by A2;
end;
end;
cluster degree K -> integer;
coherence
proof
degree K=(degree K+1)-1 by XXREAL_3:24;
hence thesis;
end;
end;
theorem Th22:
degree K = -1 iff K is empty-membered
proof
per cases;
suppose K is void;
hence thesis by Def12;
end;
suppose A1: K is non void;
hereby assume A2: degree K=-1;
then A3: K is finite-degree by A1,Def12;
assume K is with_non-empty_element;
then the topology of K is with_non-empty_element by Def7;
then consider S be non empty set such that
A4: S in the topology of K by SETFAM_1:def 11;
reconsider S as Subset of K by A4;
A5: S is simplex-like by A4,PRE_TOPC:def 5;
then reconsider S as finite Subset of K by A1,A3;
card S<=-1+1 by A1,A2,A3,A5,Def12;
then card S=0;
hence contradiction;
end;
assume A6: K is empty-membered;
then consider S be Subset of K such that
A7: S is simplex-like and
A8: card S=degree K+1 by A1,Def12;
A9: S in the topology of K by A7,PRE_TOPC:def 5;
assume degree K<>-1;
then card S<>-1+1 by A8,XXREAL_3:11;
then A10: S is non empty;
the topology of K is empty-membered by A6,Def7;
hence thesis by A9,A10,SETFAM_1:def 11;
end;
end;
theorem Th23:
-1 <= degree K
proof
per cases;
suppose K is void;
hence thesis by Def12;
end;
suppose A1: K is non void finite-degree;
then reconsider d=degree K as Integer;
0=-1+1 & d+1>=0 by A1;
hence thesis by XREAL_1:8;
end;
suppose K is non void non finite-degree;
hence thesis by Def12;
end;
end;
theorem Th24:
for S be finite Subset of K st S is simplex-like holds card S <= degree K+1
proof
let S be finite Subset of K such that
A1: S is simplex-like;
S in the topology of K by A1,PRE_TOPC:def 5;
then A2: K is non void by PENCIL_1:def 4;
per cases;
suppose K is finite-degree;
hence thesis by A1,A2,Def12;
end;
suppose K is non finite-degree;
then degree K=+infty by A2,Def12;
then degree K+1=+infty by XXREAL_3:def 2;
hence thesis by XXREAL_0:3;
end;
end;
theorem Th25:
K is non void or i >= -1 implies (degree K <= i iff K is finite-membered &
for S be finite Subset of K st S is simplex-like holds card S <= i+1)
proof
assume A1: K is non void or i>=-1;
per cases;
suppose A2: K is void;
A3: now let S be finite Subset of K;
assume S is simplex-like;
then S in the topology of K by PRE_TOPC:def 5;
hence card S<=i+1 by A2,PENCIL_1:def 4;
end;
K is empty-membered by A2;
hence thesis by A1,A2,A3,Th22;
end;
suppose A4: K is non void;
hereby assume A5: degree K<=i;
then A6: degree K+1<=i+1 by XXREAL_3:36;
i in REAL by XREAL_0:def 1;
then A7: degree K<>+infty by A5,XXREAL_0:9;
then K is finite-degree or K is empty-membered by Def12;
hence K is finite-membered;
let S be finite Subset of K;
assume A8: S is simplex-like;
K is non void finite-degree or K is void by A7,Def12;
then card S<=degree K+1 by A4,A8,Def12;
hence card S<=i+1 by A6,XXREAL_0:2;
end;
assume that
A9: K is finite-membered and
A10: for S be finite Subset of K st S is simplex-like holds card S<=i+1;
consider S be set such that
A11: S in the topology of K by A4,XBOOLE_0:def 1;
reconsider S as Subset of K by A11;
A12: S is simplex-like by A11,PRE_TOPC:def 5;
then reconsider S as finite Subset of K by A4,A9;
card S<=i+1 by A10,A12;
then reconsider i1=i+1 as Element of NAT by INT_1:16;
for A be finite Subset of K st A is open holds card A<=i1 by A10;
then A13: K is finite-degree by A9,MATROID0:def 7;
then reconsider d=degree K as Integer;
ex S1 be Subset of K st S1 is simplex-like & card S1=degree K+1 by
A4,A13,
Def12;
then d+1<=i+1 by A4,A9,A10;
hence thesis by XREAL_1:8;
end;
end;
theorem
for A be finite Subset of X holds degree Complex_of {A} = card A - 1
proof
let A be finite Subset of X;
set C=Complex_of{A};
A1: the topology of C=bool A by Th4;
now let S be finite Subset of C;
assume S is simplex-like;
then S in the topology of C by PRE_TOPC:def 5;
hence card S<=(card A-1)+1 by A1,NAT_1:44;
end;
then A2: degree C<=card A-1 by Th25;
A c=A;
then reconsider A1=A as finite Simplex of C by A1,PRE_TOPC:def 5;
card A=card A-1+1 & card A1<=degree C+1 by Def12;
then card A-1<=degree C by XREAL_1:10;
hence card A-1=degree C by A2,XXREAL_0:1;
end;
begin :: Subcomplexes
definition
let X;
let KX be SimplicialComplexStr of X;
mode SubSimplicialComplex of KX -> SimplicialComplex of X means :Def13:
[#]it c= [#]KX & the topology of it c= the topology of KX;
existence
proof
set T=TopStruct(# 0,{}bool 0 #);
the_family_of T is empty & [#]T c=X by XBOOLE_1:2;
then reconsider T=TopStruct(# 0,{}bool 0 #) as SimplicialComplex of X by Def9
,MATROID0:def 3,def 6;
take T;
thus thesis by XBOOLE_1:2;
end;
end;
reserve KX for SimplicialComplexStr of X,
SX for SubSimplicialComplex of KX;
registration
let X,KX;
cluster empty void strict SubSimplicialComplex of KX;
existence
proof
set C=Complex_of{}bool{};
the topology of C={};
then A1: [#]C c=[#]KX & the topology of C c=the topology of KX by XBOOLE_1:2;
[#]C c=X by XBOOLE_1:2;
then C is SimplicialComplexStr of X by Def9;
then reconsider C as SubSimplicialComplex of KX by A1,Def13;
take C;
thus thesis;
end;
end;
registration
let X;
let KX be void SimplicialComplexStr of X;
cluster -> void SubSimplicialComplex of KX;
coherence
proof
let S be SubSimplicialComplex of KX;
the topology of S c=the topology of KX & the topology of KX is empty by Def13
,PENCIL_1:def 4;
hence thesis;
end;
end;
registration
let D;
let KD be non void subset-closed SimplicialComplexStr of D;
cluster non void SubSimplicialComplex of KD;
existence
proof
A1: the_family_of KD is subset-closed;
consider x such that
A2: x in the topology of KD by XBOOLE_0:def 1;
per cases;
suppose x is empty;
then reconsider x as empty Subset of KD by A2;
set S=Complex_of{x};
A3: [#]S=[#]KD;
[#]KD c=D by Def9;
then reconsider S as SimplicialComplex of D by A3,Def9;
take S;
{x}c=the topology of KD by A2,ZFMISC_1:37;
then subset-closed_closure_of{x}c=the topology of KD by A1,Th3;
hence thesis by A3,Def13;
end;
suppose A4: x is non empty;
set d=the Element of x;
d in x by A4;
then reconsider dd={d} as Subset of KD by A2,ZFMISC_1:37;
set S=Complex_of{dd};
A5: [#]S=[#]KD;
[#]KD c=D by Def9;
then reconsider S as SimplicialComplex of D by A5,Def9;
take S;
dd c=x by A4,ZFMISC_1:37;
then dd in the topology of KD by A1,A2,CLASSES1:def 1;
then {dd}c=the topology of KD by ZFMISC_1:37;
then the topology of S c=the topology of KD by A1,Th3;
hence thesis by A5,Def13;
end;
end;
end;
registration
let X;
let KX be finite-vertices SimplicialComplexStr of X;
cluster -> finite-vertices SubSimplicialComplex of KX;
coherence
proof
let SX be SubSimplicialComplex of KX;
A1: Vertices KX is finite by Def5;
the topology of SX c=the topology of KX by Def13;
then A2: union the topology of SX c=union the topology of KX by ZFMISC_1:95;
union the topology of SX=Vertices SX & union the topology of KX=Vertices KX
by Lm5;
hence thesis by A2,A1,Def5;
end;
end;
registration
let X;
let KX be finite-degree SimplicialComplexStr of X;
cluster -> finite-degree SubSimplicialComplex of KX;
coherence
proof
let S be SubSimplicialComplex of KX;
consider n be Nat such that
A1: for A be finite Subset of KX st A is simplex-like holds card A<=n by
MATROID0:def 7;
thus S is finite-membered;
take n;
let A be finite Subset of S;
assume A is simplex-like;
then A2: A in the topology of S by PRE_TOPC:def 5;
[#]S c=[#]KX by Def13;
then reconsider A1=A as finite Subset of KX by XBOOLE_1:1;
the topology of S c=the topology of KX by Def13;
then A1 is simplex-like by A2,PRE_TOPC:def 5;
hence thesis by A1;
end;
end;
theorem Th27:
for S1 be SubSimplicialComplex of SX holds S1 is SubSimplicialComplex of KX
proof
let S1 be SubSimplicialComplex of SX;
[#]SX c=[#]KX & [#]S1 c=[#]SX by Def13;
then A1: [#]S1 c=[#]KX by XBOOLE_1:1;
the topology of SX c=the topology of KX & the topology of S1 c=the topology
of SX by Def13;
then the topology of S1 c=the topology of KX by XBOOLE_1:1;
hence thesis by A1,Def13;
end;
theorem Th28:
for A be Subset of KX for S be finite-membered Subset-Family of A st
subset-closed_closure_of S c=the topology of KX
holds Complex_of S is strict SubSimplicialComplex of KX
proof
let A be Subset of KX;
let S be finite-membered Subset-Family of A such that
A1: subset-closed_closure_of S c=the topology of KX;
set C=Complex_of S;
[#]KX c=X by Def9;
then [#]C c=X by XBOOLE_1:1;
then [#]C c=[#]KX & C is strict SimplicialComplexStr of X by Def9;
hence thesis by A1,Def13;
end;
theorem
for KX be subset-closed SimplicialComplexStr of X
for A be Subset of KX for S be finite-membered Subset-Family of A st
S c= the topology of KX
holds Complex_of S is strict SubSimplicialComplex of KX
proof
let KX be subset-closed SimplicialComplexStr of X;
let A be Subset of KX;
let S be finite-membered Subset-Family of A;
A1: the_family_of KX is subset-closed;
assume S c=the topology of KX;
hence thesis by A1,Th3,Th28;
end;
theorem
for Y1,Y2 be Subset-Family of X st
Y1 is finite-membered & Y1 is_finer_than Y2
holds Complex_of Y1 is SubSimplicialComplex of Complex_of Y2
proof
let Y1,Y2 be Subset-Family of X such that
A1: Y1 is finite-membered and
A2: Y1 is_finer_than Y2;
set C1=Complex_of Y1,C2=Complex_of Y2;
A3: [#]C1=X & [#]C2=X;
subset-closed_closure_of Y1 c=subset-closed_closure_of Y2 by A2,Th6;
hence thesis by A1,A3,Def13;
end;
theorem
Vertices SX c= Vertices KX
proof
let x;
assume A1: x in Vertices SX;
then reconsider v=x as Element of SX;
v is vertex-like by A1,Def4;
then consider S be Subset of SX such that
A2: S is simplex-like and
A3: v in S by Def3;
A4: the topology of SX c=the topology of KX by Def13;
A5: S in the topology of SX by A2,PRE_TOPC:def 5;
then S in the topology of KX by A4;
then reconsider S as Subset of KX;
v in S by A3;
then reconsider v as Element of KX;
S is simplex-like by A4,A5,PRE_TOPC:def 5;
then v is vertex-like by A3,Def3;
hence thesis by Def4;
end;
theorem Th32:
degree SX <= degree KX
proof
per cases;
suppose A1: SX is non finite-degree;
SX is non void
proof
assume SX is void;
then SX is empty-membered;
hence contradiction by A1;
end;
then KX is non finite-degree non void & degree SX=+infty by A1,Def12;
hence thesis by Def12;
end;
suppose SX is void;
then degree SX=-1 by Def12;
hence thesis by Th23;
end;
suppose A2: SX is non void finite-degree;
then A3: KX is non void;
per cases;
suppose KX is non finite-degree;
then degree KX=+infty by A3,Def12;
hence thesis by XXREAL_0:3;
end;
suppose A4: KX is finite-degree;
A5: the topology of SX c=the topology of KX by Def13;
consider S be Subset of SX such that
A6: S is simplex-like and
A7: card S=degree SX+1 by A2,Def12;
A8: S in the topology of SX by A6,PRE_TOPC:def 5;
then S in the topology of KX by A5;
then reconsider S1=S as finite Subset of KX by A4,A7;
S1 is simplex-like by A5,A8,PRE_TOPC:def 5;
then degree SX+1<=degree KX+1 by A3,A4,A7,Def12;
then A9: degree SX+1-1<=degree KX+1-1 by XXREAL_3:39;
degree SX+1-1=degree SX by XXREAL_3:24;
hence thesis by A9,XXREAL_3:24;
end;
end;
end;
definition
let X,KX,SX;
attr SX is maximal means :Def14:
for A be Subset of SX st A in the topology of KX holds A is simplex-like;
end;
theorem Th33:
SX is maximal iff bool[#]SX /\ the topology of KX c= the topology of SX
proof
hereby assume A1: SX is maximal;
thus bool[#]SX/\the topology of KX c=the topology of SX
proof
let x;
assume A2: x in bool[#]SX/\the topology of KX;
then reconsider A=x as Subset of SX by XBOOLE_0:def 4;
A in the topology of KX by A2,XBOOLE_0:def 4;
then A is simplex-like by A1,Def14;
hence thesis by PRE_TOPC:def 5;
end;
end;
assume A3: bool[#]SX/\the topology of KX c=the topology of SX;
let A be Subset of SX;
assume A in the topology of KX;
then A in bool[#]SX/\the topology of KX by XBOOLE_0:def 4;
hence thesis by A3,PRE_TOPC:def 5;
end;
registration
let X,KX;
cluster maximal strict SubSimplicialComplex of KX;
existence
proof
set B=bool{}/\the topology of KX;
reconsider B as finite-membered subset-closed Subset-Family of{}KX by
XBOOLE_1:17,ZFMISC_1:1,39;
subset-closed_closure_of B=B by Th7;
then reconsider C=Complex_of B as strict SubSimplicialComplex of KX by Th28,
XBOOLE_1:17;
take C;
C=TopStruct(# {}KX,B #) & bool[#]C=bool{} by Th7;
hence thesis by Th33;
end;
end;
theorem Th34:
for S1 be SubSimplicialComplex of SX st SX is maximal & S1 is maximal
holds S1 is maximal SubSimplicialComplex of KX
proof
let S1 be SubSimplicialComplex of SX;
reconsider s1=S1 as SubSimplicialComplex of KX by Th27;
assume that
A1: SX is maximal and
A2: S1 is maximal;
A3: [#]S1 c=[#]SX by Def13;
now let A be Subset of s1;
reconsider a=A as Subset of SX by A3,XBOOLE_1:1;
assume A in the topology of KX;
then a is simplex-like by A1,Def14;
then a in the topology of SX by PRE_TOPC:def 5;
hence A is simplex-like by A2,Def14;
end;
hence S1 is maximal SubSimplicialComplex of KX by Def14;
end;
theorem
for S1 be SubSimplicialComplex of SX st
S1 is maximal SubSimplicialComplex of KX
holds S1 is maximal
proof
let S1 be SubSimplicialComplex of SX;
assume S1 is maximal SubSimplicialComplex of KX;
then reconsider s1=S1 as maximal SubSimplicialComplex of KX;
the topology of SX c=the topology of KX by Def13;
then A1: bool[#]s1/\the topology of SX c=bool[#]s1/\the topology of KX by
XBOOLE_1:26;
bool[#]s1/\the topology of KX c=the topology of s1 by Th33;
then bool[#]S1/\the topology of SX c=the topology of S1 by A1,XBOOLE_1:1;
hence thesis by Th33;
end;
theorem Th36:
for K1,K2 be maximal SubSimplicialComplex of KX st [#]K1 = [#]K2
holds the TopStruct of K1 = the TopStruct of K2
proof
let K1,K2 be maximal SubSimplicialComplex of KX;
assume A1: [#]K1=[#]K2;
set TOP1=the topology of K1,TOP2=the topology of K2,TOP=the topology of KX;
A2: TOP/\bool[#]K2 c=TOP2 by Th33;
TOP1 c=TOP by Def13;
then A3: TOP1 c=TOP/\bool[#]K1 by XBOOLE_1:19;
TOP2 c=TOP by Def13;
then A4: TOP2 c=TOP/\bool[#]K2 by XBOOLE_1:19;
TOP/\bool[#]K1 c=TOP1 by Th33;
then TOP1=TOP/\bool[#]K1 by A3,XBOOLE_0:def 10
.=TOP2 by A1,A2,A4,XBOOLE_0:def 10;
hence thesis by A1;
end;
definition
let X;
let KX be subset-closed SimplicialComplexStr of X;
let A be Subset of KX such that
A1: bool A /\ the topology of KX is finite-membered;
func KX|A -> maximal strict SubSimplicialComplex of KX means :Def15:
[#]it = A;
existence
proof
the_family_of KX is subset-closed;
then reconsider BA=bool A/\the topology of KX as finite-membered
subset-closed Subset-Family of A by A1,XBOOLE_1:17;
set KA=TopStruct(# A,BA #);
A2: [#]KA c=[#]KX & the topology of KA c=the topology of KX by XBOOLE_1:17;
BA=the_family_of KA;
then A3: KA is subset-closed finite-membered by MATROID0:def 3,def 6;
[#]KX c=X by Def9;
then [#]KA c=X by XBOOLE_1:1;
then KA is SimplicialComplexStr of X by Def9;
then reconsider KA as maximal strict SubSimplicialComplex of KX by A2,A3,
Def13,Th33;
take KA;
thus thesis;
end;
uniqueness by Th36;
end;
reserve SC for SimplicialComplex of X;
definition
let X,SC;
let A be Subset of SC;
redefine func SC|A means :Def16:
[#]it = A;
compatibility
proof
let KA be maximal strict SubSimplicialComplex of SC;
the_family_of SC is finite-membered & bool A/\the topology of SC c=the
topology of SC by MATROID0:def 6,XBOOLE_1:17;
hence thesis by Def15;
end;
end;
theorem Th37:
for A be Subset of SC holds
the topology of SC|A = bool A /\ the topology of SC
proof
let A be Subset of SC;
A1: [#](SC|A)=A by Def16;
then A2: bool A/\the topology of SC c=the topology of SC|A by Th33;
the topology of SC|A c=the topology of SC by Def13;
then the topology of SC|A c=bool A/\the topology of SC by A1,XBOOLE_1:19;
hence thesis by A2,XBOOLE_0:def 10;
end;
theorem
for A,B be Subset of SC for B1 be Subset of SC|A st B1 = B
holds SC|A|B1 = SC|B
proof
let A,B be Subset of SC;
let B1 be Subset of SC|A;
reconsider SCAB=SC|A|B1 as maximal strict SubSimplicialComplex of SC by
Th34;
assume A1: B1=B;
[#]SCAB=B1 by Def16;
hence thesis by A1,Def16;
end;
theorem
SC|[#]SC = the TopStruct of SC
proof
set T=the TopStruct of SC;
A1: [#]T c=[#]SC & bool[#]T/\the topology of SC=the topology of SC by
XBOOLE_1:28;
the_family_of SC=the_family_of T & the_family_of SC is finite-membered
subset-closed by MATROID0:def 6;
then A2: T is finite-membered subset-closed by MATROID0:def 3,def 6;
[#]SC c=X by Def9;
then [#]T c=X;
then T is SimplicialComplexStr of X by Def9;
then reconsider T as maximal SubSimplicialComplex of SC by A1,A2,Def13,Th33;
[#]T=[#]SC;
hence thesis by Def16;
end;
theorem
for A,B be Subset of SC st A c= B holds SC|A is SubSimplicialComplex of SC|B
proof
let A,B be Subset of SC;
A1: bool A/\the topology of SC=the topology of SC|A & bool B/\the topology of
SC=the topology of SC|B by Th37;
assume A2: A c=B;
then bool A c=bool B by ZFMISC_1:79;
then A3: bool A/\the topology of SC c=bool B/\the topology of SC by XBOOLE_1:
27;
[#](SC|A)=A & [#](SC|B)=B by Def16;
hence thesis by A2,A3,A1,Def13;
end;
registration
cluster -> finite Integer;
coherence
proof
let i be Integer;
i in NAT\/[:{0},NAT:]\{[0,0]} by INT_1:def 2,NUMBERS:def 4;
then i in NAT or i in [:{0},NAT:] by XBOOLE_0:def 3;
then i in NAT or ex x,y be set st x in {0} & y in NAT & i=[x,y] by ZFMISC_1:
def 2;
hence thesis;
end;
end;
begin :: The Skelton of a Simplicial Complex
definition
let X,KX;
let i be real number;
func Skeleton_of(KX,i) -> SimplicialComplexStr of X equals
Complex_of the_subsets_with_limited_card(i + 1,the topology of KX);
coherence
proof
set C=Complex_of the_subsets_with_limited_card(i+1,the topology of KX);
[#]KX c=X by Def9;
then [#]C c=X;
hence thesis by Def9;
end;
end;
registration
let X,KX;
cluster Skeleton_of(KX,-1) -> empty-membered;
coherence
proof
assume Skeleton_of(KX,-1) is with_non-empty_element;
then the topology of Skeleton_of(KX,-1) is with_non-empty_element by Def7;
then consider x be non empty set such that
A1: x in the topology of Skeleton_of(KX,-1) by SETFAM_1:def 11;
consider b be set such that
A2: x c=b and
A3: b in the_subsets_with_limited_card(-1+1,the topology of KX) by A1,Th2;
card b c=card(-1+1) by A3,Def2;
then card b c={};
then b is empty;
hence thesis by A2;
end;
let i;
cluster Skeleton_of(KX,i) -> finite-degree;
coherence
proof
reconsider i1=i+1 as Integer;
set SUB=the_subsets_with_limited_card(i+1,the topology of KX);
set C=Complex_of SUB;
now let A be finite Subset of C;
assume A is open;
then A in subset-closed_closure_of SUB by PRE_TOPC:def 5;
then consider B be set such that
A4: A c=B & B in SUB by Th2;
card A c=card B & card B c=card i1 by A4,Def2,CARD_1:27;
then card A c=card i1 by XBOOLE_1:1;
hence card A<=card i1 by NAT_1:40;
end;
hence thesis by MATROID0:def 7;
end;
end;
registration
let X;
let KX be empty-membered SimplicialComplexStr of X;
let i;
cluster Skeleton_of(KX,i) -> empty-membered;
coherence
proof
assume Skeleton_of(KX,i) is with_non-empty_element;
then the topology of Skeleton_of(KX,i) is with_non-empty_element by Def7;
then consider x be non empty set such that
A1: x in the topology of Skeleton_of(KX,i) by SETFAM_1:def 11;
A2: the topology of KX is empty-membered by Def7;
consider y such that
A3: x c=y and
A4: y in the_subsets_with_limited_card(i+1,the topology of KX) by A1,Th2;
y in the topology of KX by A4,Def2;
then y is empty by A2,SETFAM_1:def 11;
hence thesis by A3;
end;
end;
registration
let D;
let KD be non void subset-closed SimplicialComplexStr of D;
let i;
cluster Skeleton_of(KD,i) -> non void;
coherence
proof
consider x such that
A1: x in the topology of KD by XBOOLE_0:def 1;
reconsider x as Simplex of KD by A1,PRE_TOPC:def 5;
reconsider E={} as Subset of KD by XBOOLE_1:2;
E in the topology of KD & card E c=card(i+1) by PRE_TOPC:def 5;
then E in the_subsets_with_limited_card(i+1,the topology of KD) by Def2;
hence thesis;
end;
end;
theorem
-1 <= i1 & i1 <= i2 implies
Skeleton_of(KX,i1) is SubSimplicialComplex of Skeleton_of(KX,i2)
proof
assume that
A1: -1<=i1 and
A2: i1<=i2;
-1<=i2 by A1,A2,XXREAL_0:2;
then A3: -1+1<=i2+1 by XREAL_1:8;
-1+1<=i1+1 by A1,XREAL_1:8;
then reconsider I1=i1+1,I2=i2+1 as Element of NAT by A3,INT_1:16;
the_subsets_with_limited_card(i1+1,the topology of KX)c=
the_subsets_with_limited_card(i2+1,the topology of KX)
proof
let x;
I1<=I2 by A2,XREAL_1:8;
then A4: card I1 c=card I2 by NAT_1:41;
assume A5: x in the_subsets_with_limited_card(i1+1,the topology of KX);
then card x c=card I1 by Def2;
then A6: card x c=card I2 by A4,XBOOLE_1:1;
x in the topology of KX by A5,Def2;
hence thesis by A6,Def2;
end;
then the_subsets_with_limited_card(i1+1,the topology of KX)is_finer_than
the_subsets_with_limited_card(i2+1,the topology of KX) by SETFAM_1:17;
then A7: the topology of Skeleton_of(KX,i1)c=the topology of Skeleton_of(KX,
i2) by Th6;
[#]Skeleton_of(KX,i1)=[#]Skeleton_of(KX,i2);
hence thesis by A7,Def13;
end;
definition
let X;
let KX be subset-closed SimplicialComplexStr of X;
let i;
redefine func Skeleton_of(KX,i) -> SubSimplicialComplex of KX;
coherence
proof
A1: the_subsets_with_limited_card(i+1,the topology of KX)c=the topology of KX
proof
let x;
assume x in the_subsets_with_limited_card(i+1,the topology of KX);
hence thesis by Def2;
end;
the_family_of KX is subset-closed;
then [#]Skeleton_of(KX,i)=[#]KX & subset-closed_closure_of
the_subsets_with_limited_card(i+1,the topology of KX)c=the topology of KX by A1
,Th3;
hence thesis by Def13;
end;
end;
theorem
KX is subset-closed & Skeleton_of(KX,i) is empty-membered implies
KX is empty-membered or i = -1
proof
assume KX is subset-closed;
then A1: the_family_of KX is subset-closed;
assume A2: Skeleton_of(KX,i) is empty-membered;
assume KX is with_non-empty_element;
then the topology of KX is with_non-empty_element by Def7;
then consider x be non empty set such that
A3: x in the topology of KX by SETFAM_1:def 11;
consider y such that
A4: y in x by XBOOLE_0:def 1;
assume i<>-1;
then {}c=card(i+1) & i+1 is non empty;
then {} in card(i+1) by CARD_1:13;
then 1 c=card(i+1) by CARD_3:109;
then A5: card{y}c=card(i+1) by CARD_1:50;
{y}c=x by A4,ZFMISC_1:37;
then {y} in the topology of KX by A1,A3,CLASSES1:def 1;
then {y} in the_subsets_with_limited_card(i+1,the topology of KX) by A5,Def2
;
then A6: {y} in the topology of Skeleton_of(KX,i) by Th2;
the topology of Skeleton_of(KX,i) is empty-membered by A2,Def7;
hence thesis by A6,SETFAM_1:def 11;
end;
theorem
degree Skeleton_of(KX,i) <= degree KX
proof
set S=Skeleton_of(KX,i);
per cases;
suppose KX is void or S is void;
then KX is empty-membered or S is empty-membered;
then degree Skeleton_of(KX,i)=-1 by Th22;
hence thesis by Th23;
end;
suppose A1: KX is non void finite-degree & S is non void;
then reconsider d=degree KX as Integer;
now let s be finite Subset of S;
assume s is simplex-like;
then s in subset-closed_closure_of the_subsets_with_limited_card(i+1,the
topology of KX) by PRE_TOPC:def 5;
then consider a be set such that
A2: s c=a and
A3: a in the_subsets_with_limited_card(i+1,the topology of KX) by Th2;
A4: card a c=card(i+1) by A3,Def2;
A5: a in the topology of KX by A3,Def2;
reconsider a as finite Subset of KX by A3,A4;
card s c=card a by A2,CARD_1:27;
then A6: card s<=card a by NAT_1:40;
a is simplex-like & KX is non void by A5,PENCIL_1:def 4,PRE_TOPC:def 5;
then card a<=d+1 by Th25;
hence card s<=d+1 by A6,XXREAL_0:2;
end;
hence thesis by A1,Th25;
end;
suppose A7: KX is non finite-degree;
KX is non void
proof
assume KX is void;
then KX is empty-membered;
hence thesis by A7;
end;
then degree KX=+infty by A7,Def12;
hence thesis by XXREAL_0:3;
end;
end;
theorem
-1 <= i implies degree Skeleton_of(KX,i) <= i
proof
set swlc=the_subsets_with_limited_card(i+1,the topology of KX);
set S=Skeleton_of(KX,i);
assume A1: -1<=i;
then -1+1<=i+1 by XREAL_1:8;
then reconsider i1=i+1 as Element of NAT by INT_1:16;
now let A be finite Subset of S;
assume A is simplex-like;
then A in the topology of S by PRE_TOPC:def 5;
then consider x such that
A2: A c=x & x in swlc by Th2;
card x c=card(i+1) & card A c=card x by A2,Def2,CARD_1:27;
then A3: card A c=card i1 by XBOOLE_1:1;
card card A=card A;
hence card A<=i+1 by A3,NAT_1:41;
end;
hence thesis by A1,Th25;
end;
theorem
-1 <= i & Skeleton_of(KX,i) = the TopStruct of KX implies degree KX <= i
proof
assume A1: -1<=i;
per cases;
suppose KX is empty-membered;
hence thesis by A1,Th22;
end;
suppose A2: KX is with_non-empty_element;
-1+1<=i+1 by A1,XREAL_1:8;
then reconsider i1=i+1 as Element of NAT by INT_1:16;
assume A3: Skeleton_of(KX,i)=the TopStruct of KX;
A4: now let S be finite Subset of KX;
assume S is simplex-like;
then S in subset-closed_closure_of the_subsets_with_limited_card(i1,the
topology of KX) by A3,PRE_TOPC:def 5;
then consider y such that
A5: S c=y & y in the_subsets_with_limited_card(i1,the topology of KX) by
Th2;
card S c=card y & card y c=card i1 by A5,Def2,CARD_1:27;
then A6: card S c=card i1 by XBOOLE_1:1;
card S=card card S;
hence card S<=i1 by A6,NAT_1:41;
end;
now let x;
assume x in the topology of KX;
then consider y such that
A7: x c=y & y in the_subsets_with_limited_card(i1,the topology of KX) by
A3,Th2;
card x c=card y & card y c=card i1 by A7,Def2,CARD_1:27;
hence x is finite;
end;
then the_family_of KX is finite-membered by MATROID0:def 5;
then KX is finite-membered by MATROID0:def 6;
hence thesis by A2,A4,Th25;
end;
end;
theorem
KX is subset-closed & degree KX <= i implies
Skeleton_of(KX,i) = the TopStruct of KX
proof
assume that
A1: KX is subset-closed and
A2: degree KX<=i;
set S=Skeleton_of(KX,i);
i in REAL by XREAL_0:def 1;
then degree KX<+infty by A2,XXREAL_0:2,9;
then A3: KX is finite-degree or KX is empty-membered by Def12;
then A4: the_family_of KX is finite-membered by MATROID0:def 6;
A5: the topology of KX c=the topology of S
proof
let x;
A6: degree KX+1<=i+1 by A2,XXREAL_3:38;
assume A7: x in the topology of KX;
then reconsider A=x as finite Subset of KX by A4,MATROID0:def 5;
A is simplex-like & KX is non void by A7,PENCIL_1:def 4,PRE_TOPC:def 5;
then card A<=degree KX+1 by A3,Def12;
then card A<=i+1 & i+1 in NAT by A6,INT_1:16,XXREAL_0:2;
then card card A c=card(i+1) by NAT_1:41;
then A in the_subsets_with_limited_card(i+1,the topology of KX) by A7,Def2;
hence thesis by Th2;
end;
A8: the_subsets_with_limited_card(i+1,the topology of KX)c=the topology of KX
proof
let x;
thus thesis by Def2;
end;
the_family_of KX is subset-closed by A1;
then the topology of S c=the topology of KX by A8,Th3;
hence thesis by A5,XBOOLE_0:def 10;
end;
reserve K for non void subset-closed SimplicialComplexStr;
Lm7: -1<=i & i<=degree K implies ex S be Simplex of K st card S=i+1
proof
assume that
A1: -1<=i and
A2: i<=degree K;
-1+1<=i+1 by A1,XREAL_1:8;
then reconsider i1=i+1 as Element of NAT by INT_1:16;
per cases;
suppose A3: not K is finite-degree;
per cases by A3,MATROID0:def 7;
suppose not K is finite-membered;
then not the_family_of K is finite-membered by MATROID0:def 6;
then consider A be set such that
A4: A in the_family_of K and
A5: A is infinite by MATROID0:def 5;
card i1 c=card A by A5;
then consider f be Function such that
A6: f is one-to-one & dom f=i1 and
A7: rng f c=A by CARD_1:26;
rng f in the_family_of K by A4,A7,CLASSES1:def 1;
then reconsider R=rng f as Simplex of K by PRE_TOPC:def 5;
take R;
R,i1 are_equipotent by A6,WELLORD2:def 4;
hence thesis by CARD_1:def 5;
end;
suppose for n be Nat ex A be finite Subset of K st A is open & card A>n;
then consider A be finite Subset of K such that
A8: A is simplex-like and
A9: card A>i1;
card i1 in card card A by A9,NAT_1:42;
then card i1 c=card A by CARD_1:13;
then consider f be Function such that
A10: f is one-to-one & dom f=i1 and
A11: rng f c=A by CARD_1:26;
A in the_family_of K by A8,PRE_TOPC:def 5;
then rng f in the_family_of K by A11,CLASSES1:def 1;
then reconsider R=rng f as Simplex of K by PRE_TOPC:def 5;
take R;
R,i1 are_equipotent by A10,WELLORD2:def 4;
hence thesis by CARD_1:def 5;
end;
end;
suppose A12: K is finite-degree;
then reconsider d=degree K as Integer;
consider S be Subset of K such that
A13: S is simplex-like and
A14: card S=degree K+1 by A12,Def12;
reconsider s=S as finite Subset of K by A12,A13;
i+1<=d+1 by A2,XREAL_1:8;
then i1 c=card s by A14,NAT_1:40;
then card i1 c=card card s by CARD_1:72;
then consider f be Function such that
A15: f is one-to-one & dom f=i1 and
A16: rng f c=S by CARD_1:26;
S in the_family_of K by A13,PRE_TOPC:def 5;
then rng f in the_family_of K by A16,CLASSES1:def 1;
then reconsider R=rng f as Simplex of K by PRE_TOPC:def 5;
take R;
R,i1 are_equipotent by A15,WELLORD2:def 4;
hence thesis by CARD_1:def 5;
end;
end;
definition
let K;
let i be real number such that
A1: i is integer;
mode Simplex of i,K -> finite Simplex of K means :Def18:
card it = i+1 if -1 <= i & i <= degree K otherwise it is empty;
existence
proof
A2: now reconsider S={}K as Simplex of K;
assume A3: -1>i or i>degree K;
take S;
S is empty;
hence thesis by A3;
end;
now assume A4: -1<=i & i<=degree K;
then consider S be Simplex of K such that
A5: card S=i+1 by A1,Lm7;
take S;
S is finite by A1,A5;
hence thesis by A4,A5;
end;
hence thesis by A2;
end;
correctness;
end;
registration
let K;
cluster -> empty Simplex of-1,K;
coherence
proof
let S be Simplex of-1,K;
-1<=degree K by Th23;
then card S=-1+1 by Def18
.=0;
hence thesis;
end;
end;
theorem
for S be Simplex of i,K st S is non empty holds i is natural
proof
let S be Simplex of i,K;
assume S is non empty;
then -1<=i & i<>-1 by Def18;
then -1__*=0 by INT_1:21;
then i in NAT by INT_1:16;
hence thesis;
end;
theorem
for S be finite Simplex of K holds S is Simplex of card S - 1,K
proof
let S be finite Simplex of K;
card S<=degree K+1 by Th24;
then card S-1<=degree K+1-1 by XXREAL_3:39;
then A1: card S-1<=degree K by XXREAL_3:24;
card S-1>=0-1 & card S=card S-1+1 by XREAL_1:8;
hence thesis by A1,Def18;
end;
theorem
for K be non void subset-closed SimplicialComplexStr of D
for S be non void SubSimplicialComplex of K
for i be Integer,A be Simplex of i,S st
A is non empty or i <= degree S or degree S = degree K
holds A is Simplex of i,K
proof
let K be non void subset-closed SimplicialComplexStr of D;
let S be non void SubSimplicialComplex of K;
let i be Integer,A be Simplex of i,S such that
A1: A is non empty or i<=degree S or degree S=degree K;
A in the topology of S & the topology of S c=the topology of K by Def13,
PRE_TOPC:def 5;
then A in the topology of K;
then reconsider B=A as Simplex of K by PRE_TOPC:def 5;
A2: degree S<=degree K by Th32;
per cases by A1;
suppose A3: A is non empty;
then A4: -1<=i by Def18;
A5: i<=degree S by A3,Def18;
-1<=i by A3,Def18;
then A6: card B=i+1 by A5,Def18;
i<=degree K by A2,A5,XXREAL_0:2;
hence thesis by A6,A4,Def18;
end;
suppose A7: i<=degree S;
then A8: i<=degree K by A2,XXREAL_0:2;
per cases;
suppose A9: -1<=i;
then card B=i+1 by A7,Def18;
hence thesis by A8,A9,Def18;
end;
suppose A10: -1>i;
then B is empty by Def18;
hence thesis by A10,Def18;
end;
end;
suppose A11: degree S=degree K;
per cases;
suppose A12: -1<=i & i<=degree S;
then card B=i+1 by Def18;
hence thesis by A11,A12,Def18;
end;
suppose A13: -1>i or i>degree S;
then B is empty by Def18;
hence thesis by A11,A13,Def18;
end;
end;
end;
definition
let K;
let i be real number such that
A1: i is integer and
A2: i <= degree K;
let S be Simplex of i,K;
mode Face of S -> Simplex of max(i-1,-1),K means :Def19:
it c= S;
existence
proof
per cases;
suppose i<0;
then i-1<0-1 by XREAL_1:10;
then reconsider F=the Simplex of-1,K as Simplex of max(i-1,-1),K
by XXREAL_0:def 10;
take F;
thus thesis by XBOOLE_1:2;
end;
suppose A3: i>=0;
then A4: card S=i+1 by A1,A2,Def18;
then S is non empty by A3;
then consider x be set such that
A5: x in S by XBOOLE_0:def 1;
reconsider I=i as Element of NAT by A1,A3,INT_1:16;
i-1<=degree K-0 by A2,XXREAL_3:39;
then A6: i-1<=degree K by XXREAL_3:4;
reconsider Sx=S\{x} as Simplex of K;
A7: card Sx=I-1+1 by A4,A5,STIRL2_1:65;
A8: i-1>=0-1 by A3,XREAL_1:8;
then max(i-1,-1)=i-1 by XXREAL_0:def 10;
then reconsider Sx as Simplex of max(i-1,-1),K by A6,A7,A8,Def18;
take Sx;
thus thesis by XBOOLE_1:36;
end;
end;
end;
theorem
for S be Simplex of n,K st n<=degree K holds
X is Face of S iff ex x st x in S & S\{x} = X
proof
let S be Simplex of n,K such that
A1: n<=degree K;
n-1<=n-0 by XREAL_1:8;
then A2: n-1<=degree K by A1,XXREAL_0:2;
reconsider N=n as Integer;
A3: n-1>=0-1 by XREAL_1:8;
then A4: max(n-1,-1)=n-1 by XXREAL_0:def 10;
A5: card S=N+1 by A1,Def18;
hereby assume X is Face of S;
then reconsider f=X as Face of S;
A6: f c=S by A1,Def19;
card f=n-1+1 by A2,A3,A4,Def18;
then card(S\f)=n+1-n by A5,A6,CARD_2:63
.=1;
then consider x such that
A7: S\f={x} by CARD_2:60;
take x;
x in {x} by TARSKI:def 1;
hence x in S by A7,XBOOLE_0:def 5;
thus S\{x}=f/\S by A7,XBOOLE_1:48
.=X by A6,XBOOLE_1:28;
end;
given x such that
A8: x in S and
A9: S\{x}=X;
reconsider f=X as finite Simplex of K by A9;
card f=n-1+1 by A5,A8,A9,STIRL2_1:65;
then A10: f is Simplex of n-1,K by A2,A3,Def18;
X c=S by A9,XBOOLE_1:36;
hence thesis by A1,A4,A10,Def19;
end;
begin :: The Subdivision of a Simplicial Complex
reserve P for Function;
definition
let X,KX,P;
func subdivision(P,KX) -> strict SimplicialComplexStr of X means :Def20:
[#]it = [#]KX & for A be Subset of it holds A is simplex-like iff
ex S be c=-linear finite simplex-like Subset-Family of KX st A = P.:S;
existence
proof
defpred P[set] means
ex S be c=-linear finite simplex-like Subset-Family of KX st$1=P.:S;
set SS={A where A is Subset of KX:P[A]};
SS c=bool the carrier of KX
proof
let x;
assume x in SS;
then ex A be Subset of KX st x=A & P[A];
hence thesis;
end;
then reconsider SS as Subset-Family of KX;
set PP=TopStruct(# the carrier of KX,SS #);
[#]PP=[#]KX & [#]KX c=X by Def9;
then reconsider PP as strict SimplicialComplexStr of X by Def9;
take PP;
thus[#]PP=[#]KX;
let A be Subset of PP;
hereby assume A is simplex-like;
then A in SS by PRE_TOPC:def 5;
then ex B be Subset of KX st B=A & P[B];
hence P[A];
end;
assume P[A];
then A in SS;
hence thesis by PRE_TOPC:def 5;
end;
uniqueness
proof
let P1,P2 be strict SimplicialComplexStr of X such that
A1: [#]P1=[#]KX and
A2: for A be Subset of P1 holds A is simplex-like iff ex S be c=-linear
finite simplex-like Subset-Family of KX st A=P.:S and
A3: [#]P2=[#]KX and
A4: for A be Subset of P2 holds A is simplex-like iff ex S be c=-linear
finite simplex-like Subset-Family of KX st A=P.:S;
now let x;
hereby assume A5: x in the topology of P1;
then reconsider A=x as Subset of P1;
reconsider B=A as Subset of P2 by A1,A3;
A is simplex-like by A5,PRE_TOPC:def 5;
then ex S be c=-linear finite simplex-like Subset-Family of KX st A=P.:S by
A2;
then B is simplex-like by A4;
hence x in the topology of P2 by PRE_TOPC:def 5;
end;
assume A6: x in the topology of P2;
then reconsider A=x as Subset of P2;
reconsider B=A as Subset of P1 by A1,A3;
A is simplex-like by A6,PRE_TOPC:def 5;
then ex S be c=-linear finite simplex-like Subset-Family of KX st A=P.:S by
A4;
then B is simplex-like by A2;
hence x in the topology of P1 by PRE_TOPC:def 5;
end;
hence thesis by A1,A3,TARSKI:2;
end;
end;
registration
let X,KX,P;
cluster subdivision(P,KX)
-> with_empty_element subset-closed finite-membered;
coherence
proof
set PP=subdivision(P,KX);
set S=the empty c=-linear simplex-like Subset-Family of KX;
P.:S={}KX by RELAT_1:149;
then {}PP is simplex-like by Def20;
then {} in the topology of PP by PRE_TOPC:def 5;
then the topology of PP is with_empty_element;
hence PP is with_empty_element by Def8;
thus PP is subset-closed
proof
let x,y;
assume that
A1: x in the_family_of PP and
A2: y c=x;
reconsider X=x,Y=y as Subset of PP by A1,A2,XBOOLE_1:1;
X is simplex-like by A1,PRE_TOPC:def 5;
then consider S be c=-linear finite simplex-like Subset-Family of KX such
that
A3: X=P.:S by Def20;
set S1={A where A is Subset of KX:A in S & P.A in Y};
S1 c=S
proof
let x;
assume x in S1;
then ex A be Subset of KX st A=x & A in S & P.A in Y;
hence thesis;
end;
then reconsider S1 as c=-linear finite simplex-like Subset-Family of KX
by
TOPS_2:18,XBOOLE_1:1;
A4: P.:S1 c=Y
proof
let x;
assume x in P.:S1;
then consider y such that
y in dom P and
A5: y in S1 and
A6: P.y=x by FUNCT_1:def 12;
ex B be Subset of KX st y=B & B in S & P.B in Y by A5;
hence thesis by A6;
end;
Y c=P.:S1
proof
let x;
assume A7: x in Y;
then consider y such that
A8: y in dom P and
A9: y in S and
A10: P.y=x by A2,A3,FUNCT_1:def 12;
y in S1 by A7,A9,A10;
hence thesis by A8,A10,FUNCT_1:def 12;
end;
then P.:S1=Y by A4,XBOOLE_0:def 10;
then Y is simplex-like by Def20;
hence thesis by PRE_TOPC:def 5;
end;
let x;
assume A11: x in the_family_of PP;
then reconsider A=x as Subset of PP;
A is simplex-like by A11,PRE_TOPC:def 5;
then ex S be c=-linear finite simplex-like Subset-Family of KX st A=P.:S by
Def20;
hence thesis;
end;
end;
registration
let X;
let KX be void SimplicialComplexStr of X;
let P;
cluster subdivision(P,KX) -> empty-membered;
coherence
proof
set PP=subdivision(P,KX);
assume PP is with_non-empty_element;
then the topology of PP is with_non-empty_element by Def7;
then consider x be non empty set such that
A1: x in the topology of PP by SETFAM_1:def 11;
reconsider A=x as Simplex of PP by A1,PRE_TOPC:def 5;
consider S be c=-linear finite simplex-like Subset-Family of KX such that
A2: A=P.:S by Def20;
A=P.:(S/\dom P) by A2,RELAT_1:145;
then S/\dom P is non empty by RELAT_1:149;
then consider y such that
A3: y in S/\dom P by XBOOLE_0:def 1;
A4: y in S by A3,XBOOLE_0:def 4;
reconsider y as Subset of KX by A3;
y is simplex-like by A4,TOPS_2:def 1;
then y in the topology of KX by PRE_TOPC:def 5;
hence thesis by PENCIL_1:def 4;
end;
end;
theorem Th51:
degree subdivision(P,KX) <= degree KX + 1
proof
set PP=subdivision(P,KX);
per cases;
suppose KX is void;
then KX is empty-membered & degree PP=-1 by Th22;
hence thesis;
end;
suppose A1: KX is non void non finite-degree;
A2: degree PP<=+infty & degree KX+0<=degree KX+1 by XXREAL_0:3,XXREAL_3:38;
degree KX+0=degree KX & degree KX=+infty by A1,Def12,XXREAL_3:4;
hence thesis by A2,XXREAL_0:2;
end;
suppose A3: KX is non void finite-degree;
then reconsider d=degree KX as Integer;
reconsider d1=d+1 as Nat by A3;
for A be finite Subset of PP st A is simplex-like holds card A<=(d+1)+1
proof
let A be finite Subset of PP;
assume A is simplex-like;
then consider S be c=-linear finite simplex-like Subset-Family of KX such
that
A4: A=P.:S by Def20;
set Sd=S/\dom P;
A=P.:(S/\dom P) by A4,RELAT_1:145;
then card A c=card Sd by CARD_2:3;
then A5: card A<=card Sd by NAT_1:40;
Sd\{{}}c=S by XBOOLE_1:18,36;
then reconsider SP=Sd\{{}} as c=-linear finite simplex-like Subset-Family
of KX by TOPS_2:18;
SP\/{{}}=Sd\/{{}} by XBOOLE_1:39;
then A6: card{{}}=1 & card Sd<=card(SP\/{{}}) by CARD_1:50,NAT_1:44,
XBOOLE_1:7;
card(SP\/{{}})<=card SP+card{{}} by CARD_2:62;
then A7: card Sd<=card SP+1 by A6,XXREAL_0:2;
per cases;
suppose A8: SP is empty;
0+1<=d1+1 by XREAL_1:8;
then card Sd<=d1+1 by A6,A8,XXREAL_0:2;
hence thesis by A5,XXREAL_0:2;
end;
suppose A9: SP is non empty;
reconsider uSP=union SP as Subset of KX;
union SP in SP by A9,Th9;
then A10: uSP is simplex-like by TOPS_2:def 1;
not{} in SP by ZFMISC_1:64;
then SP is with_non-empty_elements by SETFAM_1:def 9;
then A11: card SP c=card union SP by Th10;
reconsider uSP as finite Subset of KX by A3;
card uSP<=d1 by A3,A10,Th25;
then card uSP c=d1 by NAT_1:40;
then card SP c=d1 by A11,XBOOLE_1:1;
then card SP<=d1 by NAT_1:40;
then card SP+1<=d1+1 by XREAL_1:8;
then card Sd<=d1+1 by A7,XXREAL_0:2;
hence thesis by A5,XXREAL_0:2;
end;
end;
hence thesis by Th25;
end;
end;
theorem Th52:
dom P is with_non-empty_elements implies
degree subdivision(P,KX) <= degree KX
proof
assume A1: dom P is with_non-empty_elements;
set PP=subdivision(P,KX);
per cases;
suppose A2: KX is non finite-degree;
KX is non void
proof
assume KX is void;
then KX is empty-membered;
hence thesis by A2;
end;
then degree KX=+infty by A2,Def12;
hence thesis by XXREAL_0:3;
end;
suppose A3: KX is finite-degree;
then reconsider d=degree KX as Integer;
reconsider d1=d+1 as Nat by A3;
for A be finite Subset of PP st A is simplex-like holds card A<=d+1
proof
let A be finite Subset of PP;
assume A is simplex-like;
then consider S be c=-linear finite simplex-like Subset-Family of KX such
that
A4: A=P.:S by Def20;
A5: A=P.:(S/\dom P) by A4,RELAT_1:145;
per cases;
suppose S/\dom P is empty;
then 0<=d1 & A={} by A5,RELAT_1:149;
hence thesis;
end;
suppose A6: S/\dom P is non empty;
reconsider uSP=union(S/\dom P) as Subset of KX;
A7: S/\dom P c=S by XBOOLE_1:17;
then union(S/\dom P) in S/\dom P by A6,Th9;
then union(S/\dom P) in S by XBOOLE_0:def 4;
then A8: uSP is simplex-like by TOPS_2:def 1;
then A9: uSP in the topology of KX by PRE_TOPC:def 5;
the_family_of KX is finite-membered by A3,MATROID0:def 6;
then reconsider uSP as finite Subset of KX by A9,MATROID0:def 5;
KX is non void by A9,PENCIL_1:def 4;
then card uSP<=d+1 by A8,Th25;
then A10: card uSP c=d1 by NAT_1:40;
card(S/\dom P)c=card union(S/\dom P) by A1,A7,Th10;
then A11: card(S/\dom P)c=d1 by A10,XBOOLE_1:1;
card A c=card(S/\dom P) by A5,CARD_2:3;
then card A c=d1 by A11,XBOOLE_1:1;
hence thesis by NAT_1:40;
end;
end;
hence thesis by Th25;
end;
end;
registration
let X;
let KX be finite-degree SimplicialComplexStr of X;
let P;
cluster subdivision(P,KX) -> finite-degree;
coherence
proof
assume subdivision(P,KX) is non finite-degree;
then degree KX+1 in REAL & degree subdivision(P,KX)=+infty
by Def12,XREAL_0:def 1;
hence thesis by Th51,XXREAL_0:9;
end;
end;
registration
let X;
let KX be finite-vertices SimplicialComplexStr of X;
let P;
cluster subdivision(P,KX) -> finite-vertices;
coherence
proof
reconsider V=Vertices KX as finite Subset of KX by Def5;
set PP=subdivision(P,KX);
set TOP=the topology of PP;
defpred F[set,set] means
P.:$2=$1;
bool V=bool union the topology of KX by Lm5;
then A1: the topology of KX c=bool V by ZFMISC_1:100;
A2: for x st x in TOP ex y st y in bool bool V & F[x,y]
proof
let x such that
A3: x in TOP;
reconsider X=x as Subset of PP by A3;
X is simplex-like by A3,PRE_TOPC:def 5;
then consider S be c=-linear finite simplex-like Subset-Family of KX such
that
A4: X=P.:S by Def20;
take S;
S c=the topology of KX
proof
let y such that
A5: y in S;
reconsider Y=y as Subset of KX by A5;
Y is simplex-like by A5,TOPS_2:def 1;
hence thesis by PRE_TOPC:def 5;
end;
then S c=bool V by A1,XBOOLE_1:1;
hence thesis by A4;
end;
consider f be Function of TOP,bool bool V such that
A6: for x st x in TOP holds F[x,f.x] from FUNCT_2:sch 1(A2);
now let x1,x2 be set such that
A7: x1 in dom f and
A8: x2 in dom f & f.x1=f.x2;
thus x1=P.:(f.x1) by A6,A7
.=x2 by A6,A8;
end;
then A9: f is one-to-one by FUNCT_1:def 8;
dom f=TOP & rng f c=bool bool V by FUNCT_2:def 1;
then card TOP c=card bool bool V by A9,CARD_1:26;
then TOP is finite;
hence thesis by Th20;
end;
end;
theorem
for KX be subset-closed SimplicialComplexStr of X
for P st dom P is with_non-empty_elements &
for n st n <= degree KX ex S be Subset of KX st
S is simplex-like & card S = n+1 & BOOL S c=dom P &
P.:BOOL S is Subset of KX & P|BOOL S is one-to-one
holds degree subdivision(P,KX) = degree KX
proof
let K be subset-closed SimplicialComplexStr of X;
let P be Function such that
A1: dom P is with_non-empty_elements and
A2: for n st n<=degree K ex S be Subset of K st S is simplex-like & card S=n
+1 & BOOL S c=dom P & P.:BOOL S is Subset of K & P|BOOL S is one-to-one;
set PP=subdivision(P,K);
A3: degree PP<=degree K by A1,Th52;
A4: for n st n<=degree K ex S be Simplex of PP st card S=n+1
proof
let n;
A5: [#]K=[#]PP by Def20;
assume n<=degree K;
then consider A be Subset of K such that
A6: A is simplex-like and
A7: card A=n+1 and
A8: BOOL A c=dom P and
A9: P.:BOOL A is Subset of K and
A10: P|BOOL A is one-to-one by A2;
A11: dom(P|BOOL A)=BOOL A by A8,RELAT_1:91;
A is non empty by A7;
then consider S be Subset-Family of A such that
A12: S is with_non-empty_elements c=-linear and
A in S and
A13: card A=card S and
for Z st Z in S & card Z<>1 ex x st x in Z & Z\{x} in S by Th12;
bool A c=bool the carrier of K by ZFMISC_1:79;
then reconsider SS=S as Subset-Family of K by XBOOLE_1:1;
A14: S c=BOOL A
proof
let x;
assume x in S;
then x in bool A\{{}} by A12,ZFMISC_1:64;
hence thesis by ORDERS_1:def 2;
end;
then P.:S c=P.:BOOL A by RELAT_1:156;
then reconsider PS=P.:SS as Subset of PP by A5,A9,XBOOLE_1:1;
A15: A in the_family_of K by A6,PRE_TOPC:def 5;
SS is simplex-like
proof
let B be Subset of K;
assume B in SS;
then B in the_family_of K by A15,CLASSES1:def 1;
hence thesis by PRE_TOPC:def 5;
end;
then SS is c=-linear finite simplex-like Subset-Family of K by A7,A12,A13;
then reconsider PS as Simplex of PP by Def20;
P.:S=(P|BOOL A).:S by A14,RELAT_1:162;
then card PS=n+1 by A7,A10,A11,A13,A14,COMBGRAS:4;
hence thesis;
end;
per cases;
suppose A16: K is empty-membered;
A17: degree PP>=-1 by Th23;
degree K=-1 by A16,Th22;
hence thesis by A3,A17,XXREAL_0:1;
end;
suppose A18: K is with_non-empty_element finite-degree;
then reconsider d=degree K,dPP=degree PP as Integer;
A19: -1<=d by Th23;
d<>-1 by A18,Th22;
then -1=d+1 by A18,Def12;
then dPP>=d by XREAL_1:8;
hence thesis by A3,XXREAL_0:1;
end;
suppose K is non void non finite-degree;
then A20: degree K=+infty by Def12;
PP is non finite-degree
proof
assume A21: PP is finite-degree;
then reconsider d=degree PP+1 as Nat;
consider S be Subset of PP such that
A22: S is simplex-like and
A23: card S=d by A21,Def12;
reconsider S as finite Subset of PP by A22;
ex S1 be Simplex of PP st card S1=card S+1 by A4,A20,XXREAL_0:3;
then d+1<=d by A21,A23,Def12;
hence contradiction by NAT_1:13;
end;
hence thesis by A20,Def12;
end;
end;
theorem Th54:
Y c=Z implies
subdivision(P|Y,KX) is SubSimplicialComplex of subdivision(P|Z,KX)
proof
set PY=subdivision(P|Y,KX);
set PZ=subdivision(P|Z,KX);
A1: dom(P|Z)=Z/\dom P by RELAT_1:90;
assume A2: Y c=Z;
then Y/\Z=Y by XBOOLE_1:28;
then A3: dom(P|Y)=(Z/\Y)/\dom P by RELAT_1:90
.=Y/\dom(P|Z) by A1,XBOOLE_1:16;
A4: [#]PY=[#]KX by Def20;
hence [#]PY c=[#]PZ by Def20;
let x;
assume x in the topology of PY;
then reconsider A=x as Simplex of PY by PRE_TOPC:def 5;
[#]PZ=[#]KX by Def20;
then reconsider B=A as Subset of PZ by A4;
consider S be c=-linear finite simplex-like Subset-Family of KX such that
A5: A=(P|Y).:S by Def20;
S/\Y c=S by XBOOLE_1:17;
then reconsider SY=S/\Y as c=-linear finite simplex-like Subset-Family of KX
by TOPS_2:18;
A6: dom(P|Y)c=Y & dom(P|Y)/\S c=dom(P|Y) by RELAT_1:87,XBOOLE_1:17;
then A7: dom(P|Y)/\S c=Y by XBOOLE_1:1;
B=(P|Y).:(dom(P|Y)/\S) by A5,RELAT_1:145
.=P.:(dom(P|Y)/\S) by A6,RELAT_1:162,XBOOLE_1:1
.=(P|Z).:(dom(P|Y)/\S) by A2,A7,RELAT_1:162,XBOOLE_1:1
.=(P|Z).:(dom(P|Z)/\SY) by A3,XBOOLE_1:16
.=(P|Z).:SY by RELAT_1:145;
then B is simplex-like by Def20;
hence thesis by PRE_TOPC:def 5;
end;
theorem
dom P/\the topology of KX c= Y implies
subdivision(P|Y,KX) = subdivision(P,KX)
proof
set PX=subdivision(P|Y,KX);
set PP=subdivision(P,KX);
A1: (P|Y)|dom(P|Y)=P|(dom(P|Y)) by RELAT_1:87,103;
A2: [#]PX=[#]KX & [#]PP=[#]KX by Def20;
assume A3: dom P/\the topology of KX c=Y;
A4: the topology of PP c=the topology of PX
proof
let x;
assume x in the topology of PP;
then reconsider A=x as Simplex of PP by PRE_TOPC:def 5;
reconsider B=A as Subset of PX by A2;
consider S be c=-linear finite simplex-like Subset-Family of KX such that
A5: A=P.:S by Def20;
A6: S/\dom P c=Y
proof
let x;
assume A7: x in S/\dom P;
then reconsider A=x as Subset of KX;
x in S by A7,XBOOLE_0:def 4;
then A is simplex-like by TOPS_2:def 1;
then A8: A in the topology of KX by PRE_TOPC:def 5;
x in dom P by A7,XBOOLE_0:def 4;
then A in (the topology of KX)/\dom P by A8,XBOOLE_0:def 4;
hence thesis by A3;
end;
then A9: S/\dom P/\Y=S/\dom P by XBOOLE_1:28;
B=P.:(S/\dom P) by A5,RELAT_1:145
.=(P|Y).:(S/\dom P/\Y) by A6,A9,RELAT_1:162
.=(P|Y).:(S/\(dom P/\Y)) by XBOOLE_1:16
.=(P|Y).:(S/\dom(P|Y)) by RELAT_1:90
.=(P|Y).:S by RELAT_1:145;
then B is simplex-like by Def20;
hence thesis by PRE_TOPC:def 5;
end;
P|dom P=P & P|Y=(P|Y)|dom(P|Y) by RELAT_1:98;
then PX is SubSimplicialComplex of PP by A1,Th54,RELAT_1:89;
then the topology of PX c=the topology of PP by Def13;
hence thesis by A2,A4,XBOOLE_0:def 10;
end;
theorem Th56:
Y c= Z implies
subdivision(Y|P,KX) is SubSimplicialComplex of subdivision(Z|P,KX)
proof
assume A1: Y c=Z;
set PZ=subdivision(Z|P,KX);
set PY=subdivision(Y|P,KX);
A2: [#]PY=[#]KX by Def20;
hence [#]PY c=[#]PZ by Def20;
let x;
assume x in the topology of PY;
then reconsider A=x as Simplex of PY by PRE_TOPC:def 5;
consider S be c=-linear finite simplex-like Subset-Family of KX such that
A3: A=Y|P.:S by Def20;
S/\dom(Y|P)c=S by XBOOLE_1:17;
then reconsider Sd=S/\dom(Y|P) as c=-linear finite simplex-like Subset-Family
of KX by TOPS_2:18;
Y|(Z|P)=Y|P by A1,RELAT_1:130;
then A4: (Y|P).:Sd c=(Z|P).:Sd by RELAT_1:117,157;
[#]PZ=[#]KX by Def20;
then reconsider A as Subset of PZ by A2;
A5: Z|P.:Sd c=Y|P.:Sd
proof
let y;
assume y in (Z|P).:Sd;
then consider x such that
A6: x in dom(Z|P) and
A7: x in Sd and
A8: (Z|P).x=y by FUNCT_1:def 12;
A9: x in dom(Y|P) by A7,XBOOLE_0:def 4;
P.x=y by A6,A8,FUNCT_1:85;
then (Y|P).x=y by A9,FUNCT_1:85;
hence thesis by A7,A9,FUNCT_1:def 12;
end;
A=Y|P.:Sd by A3,RELAT_1:145;
then A=Z|P.:Sd by A4,A5,XBOOLE_0:def 10;
then A is simplex-like by Def20;
hence thesis by PRE_TOPC:def 5;
end;
theorem
P.:(the topology of KX) c= Y implies subdivision(Y|P,KX) = subdivision(P,KX)
proof
set PK=subdivision(P,KX);
P=rng P|P & Y|(rng P|P)=(Y/\rng P)|P by RELAT_1:126,127;
then reconsider PY=subdivision(Y|P,KX) as SubSimplicialComplex of PK by Th56,
XBOOLE_1:17;
A1: [#]PY=[#]KX & [#]PK=[#]KX by Def20;
assume A2: P.:(the topology of KX)c=Y;
A3: the topology of PK c=the topology of PY
proof
let x;
assume x in the topology of PK;
then reconsider A=x as Simplex of PK by PRE_TOPC:def 5;
consider S be c=-linear finite simplex-like Subset-Family of KX such that
A4: A=P.:S by Def20;
reconsider A as Subset of PY by A1;
S c=the topology of KX
proof
let y such that
A5: y in S;
reconsider y as Subset of KX by A5;
y is simplex-like by A5,TOPS_2:def 1;
hence thesis by PRE_TOPC:def 5;
end;
then A c=P.:(the topology of KX) by A4,RELAT_1:156;
then A/\Y=A by A2,XBOOLE_1:1,28;
then A=Y|P.:S by A4,FUNCT_1:126;
then A is simplex-like by Def20;
hence thesis by PRE_TOPC:def 5;
end;
the topology of PY c=the topology of PK by Def13;
hence thesis by A1,A3,XBOOLE_0:def 10;
end;
theorem Th58:
subdivision(P,SX) is SubSimplicialComplex of subdivision(P,KX)
proof
set PS=subdivision(P,SX);
set PK=subdivision(P,KX);
A1: [#]SX c=[#]KX by Def13;
A2: [#]PK=[#]KX by Def20;
hence [#]PS c=[#]PK by A1,Def20;
let x;
assume x in the topology of PS;
then reconsider A=x as Simplex of PS by PRE_TOPC:def 5;
[#]PS=[#]SX by Def20;
then reconsider B=A as Subset of PK by A1,A2,XBOOLE_1:1;
consider SS be c=-linear finite simplex-like Subset-Family of SX such that
A3: A=P.:SS by Def20;
bool[#]SX c=bool[#]KX by A1,ZFMISC_1:79;
then reconsider SK=SS as c=-linear finite Subset-Family of KX by XBOOLE_1:1;
SK is simplex-like
proof
let C be Subset of KX;
assume A4: C in SK;
then reconsider c =C as Subset of SX;
c is simplex-like by A4,TOPS_2:def 1;
then A5: c in the topology of SX by PRE_TOPC:def 5;
the topology of SX c=the topology of KX by Def13;
hence thesis by A5,PRE_TOPC:def 5;
end;
then B is simplex-like by A3,Def20;
hence thesis by PRE_TOPC:def 5;
end;
theorem
for A be Subset of subdivision(P,KX) st
dom P c=the topology of SX & A = [#]SX
holds subdivision(P,SX)=subdivision(P,KX)|A
proof
set PK=subdivision(P,KX);
reconsider PS=subdivision(P,SX) as strict SubSimplicialComplex of PK by Th58;
let A be Subset of subdivision(P,KX) such that
A1: dom P c=the topology of SX and
A2: A=[#]SX;
now let a be Subset of PS;
assume a in the topology of PK;
then reconsider b=a as Simplex of PK by PRE_TOPC:def 5;
consider SS be c=-linear finite simplex-like Subset-Family of KX such that
A3: b=P.:SS by Def20;
SS/\dom P c=dom P by XBOOLE_1:17;
then A4: SS/\dom P c=the topology of SX by A1,XBOOLE_1:1;
SS/\dom P c=SS by XBOOLE_1:17;
then reconsider Sd=SS/\dom P as c=-linear finite Subset-Family of SX by A4,
XBOOLE_1:1;
A5: Sd is simplex-like
proof
let B be Subset of SX;
assume B in Sd;
then B in dom P by XBOOLE_0:def 4;
hence thesis by A1,PRE_TOPC:def 5;
end;
P.:SS=P.:Sd by RELAT_1:145;
hence a is simplex-like by A3,A5,Def20;
end;
then A6: PS is maximal by Def14;
[#]PS=[#]SX by Def20;
hence thesis by A2,A6,Def16;
end;
theorem Th60:
for K1,K2 be SimplicialComplexStr of X st
the TopStruct of K1 = the TopStruct of K2
holds subdivision(P,K1) = subdivision(P,K2)
proof
let K1,K2 be SimplicialComplexStr of X such that
A1: the TopStruct of K1=the TopStruct of K2;
set P1=subdivision(P,K1),P2=subdivision(P,K2);
A2: [#]K1=[#]P1 & [#]K2=[#]P2 by Def20;
A3: the topology of P1 c=the topology of P2
proof
let x;
assume A4: x in the topology of P1;
then reconsider A=x as Subset of P1;
reconsider A1=A as Subset of P2 by A1,A2;
A is simplex-like by A4,PRE_TOPC:def 5;
then consider S be c=-linear finite simplex-like Subset-Family of K1 such
that
A5: A=P.:S by Def20;
reconsider S1=S as Subset-Family of K2 by A1;
S c=the topology of K1 by Th14;
then S1 is simplex-like by A1,Th14;
then A1 is simplex-like by A5,Def20;
hence thesis by PRE_TOPC:def 5;
end;
the topology of P2 c=the topology of P1
proof
let x;
assume A6: x in the topology of P2;
then reconsider A=x as Subset of P2;
reconsider A1=A as Subset of P1 by A1,A2;
A is simplex-like by A6,PRE_TOPC:def 5;
then consider S be c=-linear finite simplex-like Subset-Family of K2 such
that
A7: A=P.:S by Def20;
reconsider S1=S as Subset-Family of K1 by A1;
S c=the topology of K2 by Th14;
then S1 is simplex-like by A1,Th14;
then A1 is simplex-like by A7,Def20;
hence thesis by PRE_TOPC:def 5;
end;
hence thesis by A1,A2,A3,XBOOLE_0:def 10;
end;
definition
let X,KX,P,n;
func subdivision(n,P,KX) -> SimplicialComplexStr of X means :Def21:
ex F be Function st F.0 = KX & F.n = it & dom F = NAT &
for k for KX1 be SimplicialComplexStr of X st KX1 = F.k
holds F.(k+1) = subdivision(P,KX1);
existence
proof
defpred F[set,set] means
for SX be SimplicialComplexStr of X st the topology of SX=$1 & [#]SX=[#]KX
holds the topology of subdivision(P,SX)=$2;
set BBK=bool bool[#]KX;
A1: for x st x in BBK ex y st y in BBK & F[x,y]
proof
let x;
assume A2: x in BBK;
per cases;
suppose ex SX be SimplicialComplexStr of X st the topology of SX=x & [#]SX=
[#]KX;
then consider SX be SimplicialComplexStr of X such that
A3: the topology of SX=x and
A4: [#]SX=[#]KX;
take T=the topology of subdivision(P,SX);
[#]subdivision(P,SX)=[#]SX by Def20;
hence T in BBK by A4;
let SX1 be SimplicialComplexStr of X such that
A5: the topology of SX1=x & [#]SX1=[#]KX;
the TopStruct of SX=the TopStruct of SX1 by A3,A4,A5;
hence thesis by Th60;
end;
suppose A6: for SX be SimplicialComplexStr of X holds the topology of SX<>x
or[#]SX<>[#]KX;
take x;
thus thesis by A2,A6;
end;
end;
consider f being Function of BBK,BBK such that
A7: for x st x in BBK holds F[x,f.x] from FUNCT_2:sch 1(A1);
deffunc G(set,set)=f.$2;
consider g being Function such that
A8: dom g=NAT & g.0=the topology of KX & for n being Nat holds g.(n+1)=G(n,
g.n) from NAT_1:sch 11;
defpred P[Nat] means
(ex SX be SimplicialComplexStr of X st the topology of SX=g.$1 & [#]SX=[#]KX
& ($1>0 implies SX is strict)) & for SX be SimplicialComplexStr of X st the
topology of SX=g.$1 & [#]SX=[#]KX holds g.($1+1)=
the topology of subdivision(P,SX);
g.(0+1)=f.(the topology of KX) by A8;
then A9: g.1=the topology of subdivision(P,KX) by A7;
A10: P[0 qua Nat]
proof
thus ex SX be SimplicialComplexStr of X st the topology of SX=g.0 & [#]SX=
[#]KX & (0>0 implies SX is strict) by A8;
let SX be SimplicialComplexStr of X;
assume the topology of SX=g.0 & [#]SX=[#]KX;
then the TopStruct of SX=the TopStruct of KX by A8;
hence thesis by A9,Th60;
end;
defpred H[set,set] means
for k be Nat st k=$1 holds(k=0 implies $2=KX) & (k>0 implies for SF be
Subset-Family of KX st SF=g.k holds$2=TopStruct(# the carrier of KX,SF #));
A11: for k be Nat st P[k] holds P[k+1]
proof
let k be Nat;
set k1=k+1;
given SX be SimplicialComplexStr of X such that
A12: the topology of SX=g.k and
A13: [#]SX=[#]KX and
k>0 implies SX is strict;
assume for SX be SimplicialComplexStr of X st the topology of SX=g.k & [#]SX
=[#]KX holds g.k1=the topology of subdivision(P,SX);
then A14: g.k1=the topology of subdivision(P,SX) by A12,A13;
[#]subdivision(P,SX)=[#]KX by A13,Def20;
hence ex SX be SimplicialComplexStr of X st the topology of SX=g.k1 &
[#]SX=[#]KX & (k1>0 implies SX is strict) by A14;
let S1 be SimplicialComplexStr of X;
assume A15: the topology of S1=g.k1 & [#]S1=[#]KX;
g.(k1+1)=f.(g.k1) by A8;
hence thesis by A7,A15;
end;
A16: for k be Nat holds P[k] from NAT_1:sch 2(A10,A11);
A17: for x st x in NAT ex y st H[x,y]
proof
let x;
assume x in NAT;
then reconsider m=x as Nat;
per cases;
suppose A18: m=0;
take KX;
thus thesis by A18;
end;
suppose A19: m>0;
then reconsider m1=m-1 as Nat by NAT_1:20;
consider K1 be SimplicialComplexStr of X such that
A20: the topology of K1=g.m and
A21: [#]K1=[#]KX and
m>0 implies K1 is strict by A16;
reconsider TOP=the topology of K1 as Subset-Family of KX by A21;
take T=TopStruct(# the carrier of KX,TOP #);
thus thesis by A19,A20;
end;
end;
consider h being Function such that
A22: dom h=NAT & for x st x in NAT holds H[x,h.x] from CLASSES1:sch 1(A17);
A23: 0 in NAT by ORDINAL1:def 13;
then A24: h.0=KX by A22;
A25: for k be Nat for K1 be SimplicialComplexStr of X st h.k=K1 holds h.(k+1)
=subdivision(P,K1)
proof
let k be Nat;
let KK be SimplicialComplexStr of X such that
A26: h.k=KK;
P[k+1] by A16;
then consider K1 be SimplicialComplexStr of X such that
A27: the topology of K1=g.(k+1) and
A28: [#]K1=[#]KX;
reconsider TOP1=the topology of K1 as Subset-Family of KX by A28;
A29: k in NAT by ORDINAL1:def 13;
P[k] by A16;
then consider K2 be SimplicialComplexStr of X such that
A30: the topology of K2=g.k and
A31: [#]K2=[#]KX;
reconsider TOP2=the topology of K2 as Subset-Family of KX by A31;
A32: [#]subdivision(P,KK)=[#]KK by Def20;
per cases;
suppose A33: k=0;
then TOP1=the topology of subdivision(P,KK) by A8,A10,A24,A26,A27;
hence thesis by A22,A24,A26,A27,A32,A33;
end;
suppose k>0;
then A34: KK=TopStruct(# the carrier of KX,TOP2 #) by A22,A26,A29,A30;
then [#]KK=[#]KX;
then g.(k+1)=the topology of subdivision(P,KK) by A16,A30,A34;
hence thesis by A22,A32,A34;
end;
end;
per cases;
suppose A35: n=0;
take KX;
thus thesis by A22,A24,A25,A35;
end;
suppose A36: n>0;
consider K1 be SimplicialComplexStr of X such that
A37: the topology of K1=g.n and
A38: [#]K1=[#]KX and
A39: n>0 implies K1 is strict by A16;
reconsider K1 as strict SimplicialComplexStr of X by A36,A39;
reconsider TOP=the topology of K1 as Subset-Family of KX by A38;
take K1,h;
thus h.0=KX by A22,A23;
n in NAT by ORDINAL1:def 13;
hence thesis by A22,A25,A36,A37,A38;
end;
end;
uniqueness
proof
let P1,P2 be SimplicialComplexStr of X;
given F1 be Function such that
A40: F1.0=KX and
A41: F1.n=P1 and
dom F1=NAT and
A42: for k for KX1 be SimplicialComplexStr of X st KX1=F1.k holds F1.(k+1)=
subdivision(P,KX1);
given F2 be Function such that
A43: F2.0=KX and
A44: F2.n=P2 and
dom F2=NAT and
A45: for k for KX1 be SimplicialComplexStr of X st KX1=F2.k holds F2.(k+1)=
subdivision(P,KX1);
defpred P[Nat] means
F1.$1=F2.$1 & ex KX1 be SimplicialComplexStr of X st KX1=F1.$1;
A46: for k st P[k] holds P[k+1]
proof
let k such that
A47: P[k];
consider KX1 be SimplicialComplexStr of X such that
A48: KX1=F1.k by A47;
F1.(k+1)=subdivision(P,KX1) by A42,A48;
hence thesis by A45,A47,A48;
end;
A49: P[0 qua Nat] by A40,A43;
for k holds P[k] from NAT_1:sch 2(A49,A46);
hence thesis by A41,A44;
end;
end;
theorem Th61:
subdivision(0,P,KX) = KX
proof
ex F be Function st F.0=KX & F.0=subdivision(0,P,KX) & dom F=NAT & for k for
KX1 be SimplicialComplexStr of X st KX1=F.k holds
F.(k+1)=subdivision(P,KX1) by
Def21;
hence thesis;
end;
theorem Th62:
subdivision(1,P,KX) = subdivision(P,KX)
proof
consider F be Function such that
A1: F.0=KX and
A2: F.1=subdivision(1,P,KX) and
dom F=NAT and
A3: for k for KX1 be SimplicialComplexStr of X st KX1=F.k holds F.(k+1)=
subdivision(P,KX1) by Def21;
F.(0+1)=subdivision(P,KX) by A1,A3;
hence thesis by A2;
end;
theorem Th63:
for n,k be Element of NAT holds
subdivision(n+k,P,KX) = subdivision(n,P,subdivision(k,P,KX))
proof
let n,k be Element of NAT;
consider Fn be Function such that
A1: Fn.0=subdivision(k,P,KX) and
A2: Fn.n=subdivision(n,P,subdivision(k,P,KX)) and
dom Fn=NAT and
A3: for m be Nat for KX1 be SimplicialComplexStr of X st KX1=Fn.m holds Fn.(
m+1)=subdivision(P,KX1) by Def21;
consider Fnm be Function such that
A4: Fnm.0=KX and
A5: Fnm.(n+k)=subdivision(n+k,P,KX) and
dom Fnm=NAT and
A6: for m be Nat for KX1 be SimplicialComplexStr of X st KX1=Fnm.m holds Fnm
.(m+1)=subdivision(P,KX1) by Def21;
defpred R[Nat] means
$1<=n implies Fn.$1=Fnm.($1+k) & ex SX be SimplicialComplexStr of X st Fn.$1
=SX;
A7: for m be Nat st R[m] holds R[m+1]
proof
let m be Nat such that
A8: R[m];
assume A9: m+1<=n;
then consider SX be SimplicialComplexStr of X such that
A10: Fn.m=SX by A8,NAT_1:13;
subdivision(P,SX)=Fnm.(m+k+1) by A6,A8,A9,A10,NAT_1:13;
hence thesis by A3,A10;
end;
consider Fm be Function such that
A11: Fm.0=KX and
A12: Fm.k=subdivision(k,P,KX) and
dom Fm=NAT and
A13: for m be Nat for KX1 be SimplicialComplexStr of X st KX1=Fm.m holds Fm.
(m+1)=subdivision(P,KX1) by Def21;
defpred P[Nat] means
$1<=k implies Fm.$1=Fnm.$1 & ex SX be SimplicialComplexStr of X st Fm.$1=SX;
A14: for m be Nat st P[m] holds P[m+1]
proof
let m be Nat such that
A15: P[m];
assume A16: m+1<=k;
then consider SX be SimplicialComplexStr of X such that
A17: Fm.m=SX by A15,NAT_1:13;
subdivision(P,SX)=Fnm.(m+1) by A6,A15,A16,A17,NAT_1:13;
hence thesis by A13,A17;
end;
A18: P[0 qua Nat] by A4,A11;
for k holds P[k] from NAT_1:sch 2(A18,A14);
then A19: R[0 qua Nat] by A1,A12;
for k holds R[k] from NAT_1:sch 2(A19,A7);
hence thesis by A2,A5;
end;
theorem
[#]subdivision(n,P,KX) = [#]KX
proof
defpred P[Nat] means
[#]subdivision($1,P,KX)=[#]KX;
A1: for n st P[n] holds P[n+1]
proof
let n;
n in NAT by ORDINAL1:def 13;
then A2: subdivision(n+1,P,KX)=subdivision(1,P,subdivision(n,P,KX)) by Th63
.=subdivision(P,subdivision(n,P,KX)) by Th62;
assume[#]subdivision(n,P,KX)=[#]KX;
hence thesis by A2,Def20;
end;
A3: P[0 qua Nat] by Th61;
for n holds P[n] from NAT_1:sch 2(A3,A1);
hence thesis;
end;
theorem
subdivision(n,P,SX) is SubSimplicialComplex of subdivision(n,P,KX)
proof
defpred P[Nat] means
subdivision($1,P,SX) is SubSimplicialComplex of subdivision($1,P,KX);
A1: for n st P[n] holds P[n+1]
proof
let n;
assume P[n];
then reconsider Pn=subdivision(n,P,SX) as
SubSimplicialComplex of subdivision(n,P,KX);
A2: n in NAT by ORDINAL1:def 13;
then A3: subdivision(n+1,P,SX)=subdivision(1,P,Pn) by Th63
.=subdivision(P,Pn) by Th62;
subdivision(n+1,P,KX)=subdivision(1,P,subdivision(n,P,KX)) by A2,Th63
.=subdivision(P,subdivision(n,P,KX)) by Th62;
hence thesis by A3,Th58;
end;
subdivision(0,P,SX)=SX by Th61;
then A4: P[0] by Th61;
for n holds P[n] from NAT_1:sch 2(A4,A1);
hence thesis;
end;
*