JGRAPH_8: The {F}ashoda Meet Theorem for Continuous Mappings

:: The {F}ashoda Meet Theorem for Continuous Mappings
:: by Yatsuka Nakamura , Andrzej Trybulec and Artur Korni{\l}owicz
::
:: Received September 14, 2005
:: Copyright (c) 2005-2016 Association of Mizar Users

environ

vocabularies HIDDEN, NUMBERS, SUBSET_1, PRE_TOPC, EUCLID, BORSUK_1, PCOMPS_1, TOPMETR, CARD_1, FINSEQ_1, XXREAL_0, ORDINAL4, FUNCT_1, ARYTM_3, RELAT_1, PARTFUN1, ORDINAL2, ARYTM_1, JGRAPH_6, CONVEX1, MCART_1, RLTOPSP1, REAL_1, TOPREALA, TARSKI, STRUCT_0, XXREAL_1, MEASURE5, XBOOLE_0, FCONT_2, METRIC_1, NAT_1, INT_1, COMPLEX1, XXREAL_2, RELAT_2, RCOMP_1, CONNSP_1, GRAPH_1, WEIERSTR, PSCOMP_1, GOBOARD1, GOBOARD4, TOPREAL1, BORSUK_2;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XXREAL_0, XCMPLX_0, XREAL_0, COMPLEX1, REAL_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, FINSEQ_1, NAT_1, NAT_D, SEQM_3, STRUCT_0, TOPREAL1, PRE_TOPC, GOBOARD1, GOBOARD4, TOPMETR, PCOMPS_1, COMPTS_1, WEIERSTR, METRIC_1, TBSP_1, RCOMP_1, UNIFORM1, INT_1, CONNSP_1, PSCOMP_1, RLVECT_1, RLTOPSP1, EUCLID, BORSUK_1, BORSUK_2, JGRAPH_6, TOPREALA, TOPALG_2;
definitions TARSKI, XBOOLE_0, JORDAN1, TOPALG_2;
theorems TARSKI, FUNCT_1, FINSEQ_1, NAT_1, FINSEQ_4, RELAT_1, FINSEQ_3, TOPREAL1, GOBOARD4, TOPREAL3, EUCLID, UNIFORM1, WEIERSTR, PRE_TOPC, TBSP_1, HEINE, COMPTS_1, TOPMETR, PSCOMP_1, BORSUK_1, RCOMP_1, FUNCT_2, XBOOLE_0, XBOOLE_1, TOPRNS_1, XCMPLX_1, JGRAPH_1, INT_1, XREAL_0, CONNSP_1, XREAL_1, BORSUK_2, TSEP_1, JGRAPH_6, JORDAN5B, FINSEQ_2, XXREAL_0, TOPS_2, PARTFUN1, XXREAL_1, XXREAL_2, SEQM_3, RLTOPSP1, RELSET_1, ORDINAL1;
schemes FINSEQ_1, NAT_1;
registrations RELAT_1, XREAL_0, INT_1, FINSEQ_1, STRUCT_0, PRE_TOPC, BORSUK_1, EUCLID, TOPREAL1, BORSUK_2, JORDAN5A, TOPALG_2, MEMBERED, VALUED_0, FUNCT_2, XXREAL_2, RELSET_1, SPPOL_1, TOPMETR, FUNCT_1, NAT_1, ORDINAL1;
constructors HIDDEN, REAL_1, RCOMP_1, CONNSP_1, COMPTS_1, TBSP_1, TOPREAL1, GOBOARD1, GOBOARD4, PSCOMP_1, WEIERSTR, UNIFORM1, JGRAPH_6, TOPALG_2, TOPREALA, BORSUK_2, NAT_D, XXREAL_2, FUNCSDOM, CONVEX1, BINOP_2, NUMBERS;
requirements HIDDEN, REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
equalities XBOOLE_0, RLTOPSP1, EUCLID, TOPREALA, STRUCT_0;
expansions TARSKI, XBOOLE_0, TOPALG_2;

Lm1: I[01] = TopSpaceMetr (Closed-Interval-MSpace (0,1))
by TOPMETR:20, TOPMETR:def 7;

Lm2: for x being set
for f being FinSequence st 1 <= len f holds
( (f ^ <*x*>) . 1 = f . 1 & (<*x*> ^ f) . ((len f) + 1) = f . (len f) )

proof end;

Lm3: for f being FinSequence of REAL st ( for k being Nat st 1 <= k & k < len f holds
f /. k < f /. (k + 1) ) holds
f is increasing

proof end;

registration

let a, b, c, d be Real;

cluster closed_inside_of_rectangle (a,b,c,d) -> convex ;

proof end;

end;

registration

let a, b, c, d be Real;

cluster Trectangle (a,b,c,d) -> convex ;

coherence by PRE_TOPC:8;

end;

theorem Th1: :: JGRAPH_8:1

for n being Nat
for e being positive Real
for g being continuous Function of I[01],(TOP-REAL n) ex h being FinSequence of REAL st
( h . 1 = 0 & h . (len h) = 1 & 5 <= len h & rng h c= the carrier of I[01] & h is increasing & ( for i being Nat
for Q being Subset of I[01]
for W being Subset of (Euclid n) st 1 <= i & i < len h & Q = [.(h /. i),(h /. (i + 1)).] & W = g .: Q holds
diameter W < e ) )

proof end;

theorem Th2: :: JGRAPH_8:2

for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st P c= LSeg (p1,p2) & p1 in P & p2 in P & P is connected holds
P = LSeg (p1,p2)

proof end;

theorem Th3: :: JGRAPH_8:3

for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for g being Path of p1,p2 st rng g c= LSeg (p1,p2) holds
rng g = LSeg (p1,p2)

proof end;

:: Goboard Theorem in continuous case

theorem Th4: :: JGRAPH_8:4

for P, Q being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for f being Path of p1,p2
for g being Path of q1,q2 st rng f = P & rng g = Q & ( 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

proof end;

:: Fashoda Meet Theorem

theorem Th5: :: JGRAPH_8:5

for a, b, c, d being Real
for f, g being continuous Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & (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

proof end;

theorem :: JGRAPH_8:6

for a, b, c, d being Real
for ar, br, cr, dr being Point of (Trectangle (a,b,c,d))
for h being Path of ar,br
for v being Path of dr,cr
for Ar, Br, Cr, Dr being Point of (TOP-REAL 2) st Ar `1 = a & Br `1 = b & Cr `2 = c & Dr `2 = d & ar = Ar & br = Br & cr = Cr & dr = Dr holds
ex s, t being Point of I[01] st h . s = v . t

proof end;