Journal of Formalized Mathematics
Volume 10, 1998
University of Bialystok
Copyright (c) 1998 Association of Mizar Users

The abstract of the Mizar article:

Graph Theoretical Properties of Arcs in the Plane and Fashoda Meet Theorem

by
Yatsuka Nakamura

Received August 21, 1998

MML identifier: JGRAPH_1
[ Mizar article, MML identifier index ]


environ

 vocabulary GRAPH_1, FINSEQ_1, ORDERS_1, SQUARE_1, ABSVALUE, ARYTM_1, GRAPH_4,
      FUNCT_1, FINSEQ_4, BOOLE, FUNCT_3, RELAT_1, FINSET_1, CARD_1, EUCLID,
      TOPREAL1, GOBOARD5, PRE_TOPC, TARSKI, MCART_1, GOBOARD1, GOBOARD4,
      COMPLEX1, METRIC_1, RLVECT_1, RVSUM_1, PCOMPS_1, COMPTS_1, WEIERSTR,
      NAT_1, ARYTM_3, BORSUK_1, TOPS_2, ORDINAL2, RCOMP_1, TBSP_1, SUBSET_1,
      PSCOMP_1, LATTICES, JGRAPH_1, ARYTM;
 notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XREAL_0,
      REAL_1, RELAT_1, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSET_1, CARD_1,
      NAT_1, GRAPH_1, STRUCT_0, BINARITH, GRAPH_2, FINSEQ_4, GRAPH_4, TOPREAL1,
      GOBOARD5, PRE_TOPC, GOBOARD1, GOBOARD4, TOPMETR, PCOMPS_1, COMPTS_1,
      WEIERSTR, TOPS_2, METRIC_1, TBSP_1, RCOMP_1, ABSVALUE, SQUARE_1,
      PSCOMP_1, RVSUM_1, TOPRNS_1, EUCLID, FUNCT_3;
 constructors REAL_1, BINARITH, GRAPH_2, GRAPH_4, GOBOARD5, GOBOARD4, WEIERSTR,
      TOPS_2, TBSP_1, RCOMP_1, ABSVALUE, SQUARE_1, PSCOMP_1, FUNCT_3, FINSEQ_4,
      TOPRNS_1, GOBOARD1, JORDAN7;
 clusters STRUCT_0, RELSET_1, GRAPH_4, EUCLID, PRE_TOPC, BORSUK_1, XREAL_0,
      FINSEQ_1, METRIC_1, ARYTM_3, MEMBERED;
 requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;


begin :: GRAPHS BY CARTESIAN PRODUCT

reserve x,y for set;
reserve G for Graph;
reserve vs,vs' for FinSequence of the Vertices of G;
reserve IT for oriented Chain of G;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;

canceled;

theorem :: JGRAPH_1:2
sqrt(r1^2+r2^2)<=abs(r1)+abs(r2);

theorem :: JGRAPH_1:3
abs(r1)<=sqrt(r1^2+r2^2) & abs(r2)<=sqrt(r1^2+r2^2);

theorem :: JGRAPH_1:4
for vs st IT is Simple & vs is_oriented_vertex_seq_of IT
  holds for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs;

definition let X be set;
 func PGraph(X) -> MultiGraphStruct equals
:: JGRAPH_1:def 1
 MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#);
end;

theorem :: JGRAPH_1:5
for X being non empty set holds PGraph(X) is Graph;

theorem :: JGRAPH_1:6
for X being set holds the Vertices of PGraph(X)=X;

definition let f be FinSequence;
 func PairF(f) -> FinSequence means
:: JGRAPH_1:def 2
  len it=len f-'1 &
   for i being Nat st 1<=i & i<len f holds it.i=[f.i,f.(i+1)];
end;

reserve X for non empty set;

definition let X be non empty set;
 cluster PGraph(X) -> Graph-like;
end;

theorem :: JGRAPH_1:7
for f being FinSequence of X
  holds f is FinSequence of the Vertices of PGraph(X);

theorem :: JGRAPH_1:8
for f being FinSequence of X
  holds PairF(f) is FinSequence of the Edges of PGraph(X);

definition let X be non empty set,f be FinSequence of X;
 redefine func PairF(f) -> FinSequence of the Edges of PGraph(X);
end;

theorem :: JGRAPH_1:9
for n being Nat,f being FinSequence of X
  st 1 <= n & n <= len PairF(f) holds
  (PairF(f)).n in the Edges of PGraph(X);

theorem :: JGRAPH_1:10
for f being FinSequence of X holds
 PairF(f) is oriented Chain of PGraph(X);

definition let X be non empty set,f be FinSequence of X;
 redefine func PairF(f) -> oriented Chain of PGraph(X);
end;

theorem :: JGRAPH_1:11
for f being FinSequence of X,
  f1 being FinSequence of the Vertices of PGraph(X) st len f>=1 & f=f1 holds
  f1 is_oriented_vertex_seq_of PairF(f);

begin ::  SHORTCUTS OF FINITE SEQUENCES IN PLANE

definition let X be non empty set,f,g be FinSequence of X;
pred g is_Shortcut_of f means
:: JGRAPH_1:def 3
 f.1=g.1 & f.len f=g.len g &
 ex fc being FinSubsequence of PairF(f),fvs being FinSubsequence of f,
 sc being oriented simple Chain of PGraph(X),
 g1 being FinSequence of the Vertices of PGraph(X)
 st Seq fc = sc & Seq fvs = g & g1=g
 & g1 is_oriented_vertex_seq_of sc;
end;

theorem :: JGRAPH_1:12
for f,g being FinSequence of X st g is_Shortcut_of f holds
 1<=len g & len g <= len f;

theorem :: JGRAPH_1:13
for f being FinSequence of X st len f>=1 holds
 ex g being FinSequence of X st g is_Shortcut_of f;

theorem :: JGRAPH_1:14
for f,g being FinSequence of X st
 g is_Shortcut_of f holds rng PairF(g) c= rng PairF(f);

theorem :: JGRAPH_1:15
for f,g being FinSequence of X st f.1<>f.len f &
g is_Shortcut_of f holds g is one-to-one & rng PairF(g) c= rng PairF(f)
& g.1=f.1 & g.len g=f.len f;

definition let n; let IT be FinSequence of TOP-REAL n;
   attr IT is nodic means
:: JGRAPH_1:def 4
    for i,j st LSeg(IT,i) meets LSeg(IT,j)
         holds LSeg(IT,i) /\ LSeg(IT,j)={IT.i}
               &(IT.i=IT.j or IT.i=IT.(j+1)) or
               LSeg(IT,i) /\ LSeg(IT,j)={IT.(i+1)}
               &(IT.(i+1)=IT.j or IT.(i+1)=IT.(j+1)) or
               LSeg(IT,i)=LSeg(IT,j);
end;

theorem :: JGRAPH_1:16
for f being FinSequence of TOP-REAL 2 st f is s.n.c.
  holds f is s.c.c.;

theorem :: JGRAPH_1:17
for f being FinSequence of TOP-REAL 2 st f is s.c.c. &
 LSeg(f,1) misses LSeg(f,len f -'1) holds f is s.n.c.;

theorem :: JGRAPH_1:18
for f being FinSequence of TOP-REAL 2 st
  f is nodic & PairF(f) is Simple holds f is s.c.c.;

theorem :: JGRAPH_1:19
for f being FinSequence of TOP-REAL 2 st
  f is nodic & PairF(f) is Simple & f.1<>f.len f holds f is s.n.c.;

theorem :: JGRAPH_1:20
for p1,p2,p3 being Point of TOP-REAL n st
 (ex x being set st x<>p2 & x in LSeg(p1,p2)/\ LSeg(p2,p3)) holds
 p1 in LSeg(p2,p3) or p3 in LSeg(p1,p2);

theorem :: JGRAPH_1:21
for f being FinSequence of TOP-REAL 2 st f is s.n.c. &
LSeg(f,1) /\ LSeg(f,1+1) c= {f/.(1+1)} &
LSeg(f,len f-'2) /\ LSeg(f,len f-'1) c= {f/.(len f-'1)} holds
f is unfolded;

theorem :: JGRAPH_1:22
for f being FinSequence of X holds
  PairF(f) is Simple & f.1<>f.len f implies f is one-to-one & len f<>1;

theorem :: JGRAPH_1:23
for f being FinSequence of X holds
  f is one-to-one & len f>1 implies PairF(f) is Simple & f.1<>f.len f;

theorem :: JGRAPH_1:24
for f being FinSequence of TOP-REAL 2 st
  f is nodic & PairF(f) is Simple & f.1<>f.len f holds f is unfolded;

theorem :: JGRAPH_1:25
for f,g being FinSequence of TOP-REAL 2,i st
 g is_Shortcut_of f & 1 <= i & i+1 <= len g holds
 ex k1 being Nat st 1<=k1 & k1+1<=len f & f/.k1=g/.i &
   f/.(k1+1)=g/.(i+1)
    & f.k1=g.i & f.(k1+1)=g.(i+1);

theorem :: JGRAPH_1:26
for f,g being FinSequence of TOP-REAL 2 st
 g is_Shortcut_of f holds rng g c= rng f;

theorem :: JGRAPH_1:27
for f,g being FinSequence of TOP-REAL 2 st
 g is_Shortcut_of f holds L~g c= L~f;

theorem :: JGRAPH_1:28
for f,g being FinSequence of TOP-REAL 2 st f is special
 & g is_Shortcut_of f holds g is special;

theorem :: JGRAPH_1:29
for f being FinSequence of TOP-REAL 2 st f is special & 2<=len f
 & f.1 <> f.len f holds ex g being FinSequence of TOP-REAL 2 st
 2<=len g & g is special & g is one-to-one & L~g c= L~f & f.1=g.1
 & f.len f=g.len g & rng g c= rng f;

::  Goboard Theorem for general special sequences
theorem :: JGRAPH_1:30
for f1,f2 being FinSequence of TOP-REAL 2 st
 f1 is special & f2 is special & 2<=len f1 & 2<=len f2 & f1.1<>f1.len f1
 & f2.1<>f2.len f2 &
 X_axis(f1) lies_between (X_axis(f1)).1, (X_axis(f1)).(len f1) &
 X_axis(f2) lies_between (X_axis(f1)).1, (X_axis(f1)).(len f1) &
 Y_axis(f1) lies_between (Y_axis(f2)).1, (Y_axis(f2)).(len f2) &
 Y_axis(f2) lies_between (Y_axis(f2)).1, (Y_axis(f2)).(len f2) holds
  L~f1 meets L~f2;

begin  :: NORM OF POINTS IN TOPREAL n

theorem :: JGRAPH_1:31
for a,b,r1,r2 being Real
  st a<=r1 & r1<=b & a<=r2 & r2<=b holds
  abs(r1-r2)<=b-a;

definition let n;let p be Point of TOP-REAL n;
 redefine func |. p .| means
:: JGRAPH_1:def 5
 for w being Element of REAL n st p=w holds it=|.w.|;
end;

reserve p,p1,p2 for Point of TOP-REAL n;

canceled 13;

theorem :: JGRAPH_1:45
for x1,x2 being Point of Euclid n st x1=p1 & x2=p2 holds
 |.p1 - p2.| = dist(x1,x2);

theorem :: JGRAPH_1:46
for p being Point of TOP-REAL 2 holds
 (|.p.|)^2 = (p`1)^2+(p`2)^2;

theorem :: JGRAPH_1:47
for p being Point of TOP-REAL 2 holds
 |.p.| = sqrt((p`1)^2+(p`2)^2);

theorem :: JGRAPH_1:48
for p being Point of TOP-REAL 2 holds |.p.|<=abs(p`1)+abs(p`2);

theorem :: JGRAPH_1:49
for p1,p2 being Point of TOP-REAL 2 holds
       |.p1-p2.|<=abs(p1`1-p2`1)+abs(p1`2-p2`2);

theorem :: JGRAPH_1:50
for p being Point of TOP-REAL 2 holds
      abs(p`1)<=|.p.| & abs(p`2)<=|.p.|;

theorem :: JGRAPH_1:51
for p1,p2 being Point of TOP-REAL 2 holds
      abs(p1`1-p2`1)<=|.p1-p2.| & abs(p1`2-p2`2)<=|.p1-p2.|;

theorem :: JGRAPH_1:52
p in LSeg(p1,p2) implies ex r st
  0<=r & r<=1 & p=(1-r)*p1+r*p2;

theorem :: JGRAPH_1:53
p in LSeg(p1,p2) implies
 |.p-p1.|<=|.p1-p2.| & |.p-p2.|<=|.p1-p2.|;

begin ::  EXTENDED GOBOARD THEOREM AND FASHODA MEET THEOREM

reserve M for non empty MetrSpace;

theorem :: JGRAPH_1:54
for P,Q being Subset of TopSpaceMetr(M) st
   P <> {} & P is compact & Q <> {} & Q is compact holds
   min_dist_min(P,Q)>=0;

theorem :: JGRAPH_1:55
for P,Q being Subset of TopSpaceMetr(M) st
   P <> {} & P is compact & Q <> {} & Q is compact holds
   P misses Q iff min_dist_min(P,Q)>0;

::  Approximation of finite sequence by special finite sequence
theorem :: JGRAPH_1:56
for f being FinSequence of TOP-REAL 2,a,c,d being Real st
 1<=len f & X_axis(f) lies_between (X_axis(f)).1, (X_axis(f)).(len f) &
 Y_axis(f) lies_between c, d & a>0 &
 (for i st 1<=i & i+1<=len f holds
 |. f/.i-f/.(i+1) .|<a)
 holds
 ex g being FinSequence of TOP-REAL 2 st g is special &
 g.1=f.1 & g.len g=f.len f & len g>=len f &
 X_axis(g) lies_between (X_axis(f)).1, (X_axis(f)).(len f) &
 Y_axis(g) lies_between c, d &
 (for j st j in dom g holds ex k st k in dom f & |. g/.j - f/.k .|<a)&
 (for j st 1<=j & j+1<=len g holds |. g/.j - g/.(j+1) .|<a);

theorem :: JGRAPH_1:57
for f being FinSequence of TOP-REAL 2,a,c,d being Real st
 1<=len f &
 Y_axis(f) lies_between (Y_axis(f)).1, (Y_axis(f)).(len f) &
 X_axis(f) lies_between c, d & a>0 &
 (for i st 1<=i & i+1<=len f holds |. (f/.i)-(f/.(i+1)) .|<a)
 holds
 ex g being FinSequence of TOP-REAL 2 st g is special &
 g.1=f.1 & g.len g=f.len f & len g>=len f &
 Y_axis(g) lies_between (Y_axis(f)).1, (Y_axis(f)).(len f) &
 X_axis(g) lies_between c, d &
 (for j st j in dom g holds ex k st k in dom f & |. g/.j - f/.k .| < a)&
 (for j st 1<=j & j+1<=len g holds |. g/.j - g/.(j+1) .|<a);

canceled;

theorem :: JGRAPH_1:59
  for f being FinSequence of TOP-REAL 2 st 1 <=len f
    holds len X_axis(f)=len f &
  X_axis(f).1=(f/.1)`1 & X_axis(f).len f=(f/.len f)`1;

theorem :: JGRAPH_1:60
  for f being FinSequence of TOP-REAL 2 st 1 <=len f
    holds len Y_axis(f)=len f &
  Y_axis(f).1=(f/.1)`2 & Y_axis(f).len f=(f/.len f)`2;

theorem :: JGRAPH_1:61
  for f being FinSequence of TOP-REAL 2 st i in dom f
    holds X_axis(f).i=(f/.i)`1 & Y_axis(f).i=(f/.i)`2;

:: Goboard Theorem in continuous case
theorem :: JGRAPH_1:62
for P,Q being non empty Subset of TOP-REAL 2,
p1,p2,q1,q2 being Point of TOP-REAL 2 st
P is_an_arc_of p1,p2 & Q is_an_arc_of q1,q2 &
 (for p being Point of TOP-REAL 2 st p in P holds p1`1<=p`1 & p`1<= p2`1) &
 (for p being Point of TOP-REAL 2 st p in Q holds p1`1<=p`1 & p`1<= p2`1) &
 (for p being Point of TOP-REAL 2 st p in P holds q1`2<=p`2 & p`2<= q2`2) &
 (for p being Point of TOP-REAL 2 st p in Q holds q1`2<=p`2 & p`2<= q2`2)
 holds P meets Q;

reserve X,Y for non empty TopSpace;

theorem :: JGRAPH_1:63
for f being map of X,Y,
 P being non empty Subset of Y,
 f1 being map of X,Y|P st f=f1 & f is continuous holds
 f1 is continuous;

theorem :: JGRAPH_1:64
for f being map of X,Y,
 P being non empty Subset of Y
 st X is compact & Y is_T2 &
 f is continuous one-to-one & P=rng f holds
 ex f1 being map of X,(Y|P) st f=f1 & f1 is_homeomorphism;

::  Fashoda Meet Theorem
theorem :: JGRAPH_1:65
for f,g being map of I[01],TOP-REAL 2,a,b,c,d being real number,
  O,I being Point of I[01] st O=0 & I=1 &
  f is continuous one-to-one &
  g is continuous one-to-one &
  (f.O)`1=a & (f.I)`1=b &
  (g.O)`2=c & (g.I)`2=d &
  (for r being Point of I[01] holds
    a <=(f.r)`1 & (f.r)`1<=b & a <=(g.r)`1 & (g.r)`1<=b &
    c <=(f.r)`2 & (f.r)`2<=d & c <=(g.r)`2 & (g.r)`2<=d)
   holds rng f meets rng g;


Back to top