:: The Ordering of Points on a Curve, Part { II }
:: by Adam Grabowski and Yatsuka Nakamura
::
:: Received September 10, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, EUCLID, PRE_TOPC, FUNCT_1, BORSUK_1, RELAT_1,
REAL_1, TOPS_2, CARD_1, XXREAL_0, ARYTM_3, XBOOLE_0, ORDINAL2, STRUCT_0,
RCOMP_1, TOPREAL1, TARSKI, JORDAN3, XXREAL_1, FINSEQ_1, PARTFUN1,
RLTOPSP1, ARYTM_1, TREAL_1, VALUED_1, TOPMETR, NAT_1, SUPINF_2, JORDAN5C,
FUNCT_2;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, REAL_1, XCMPLX_0,
XREAL_0, NAT_1, RCOMP_1, RELAT_1, FINSEQ_1, FUNCT_1, RELSET_1, PARTFUN1,
FUNCT_2, TOPMETR, JORDAN3, STRUCT_0, TOPREAL1, PRE_TOPC, TOPS_2,
COMPTS_1, RLVECT_1, RLTOPSP1, EUCLID, TREAL_1, XXREAL_0;
constructors REAL_1, RCOMP_1, TOPS_2, COMPTS_1, TOPREAL1, TREAL_1, JORDAN3;
registrations FUNCT_1, RELSET_1, FUNCT_2, MEMBERED, STRUCT_0, PRE_TOPC,
BORSUK_1, EUCLID, TOPREAL1, SPPOL_1, XREAL_0, NAT_1, ORDINAL1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
begin :: First and last point of a curve
theorem :: JORDAN5C:1
for P, Q being Subset of TOP-REAL 2, p1, p2, q1 being Point of
TOP-REAL 2, f being Function of I[01], (TOP-REAL 2)|P,
s1 be Real st q1 in P &
f.s1 = q1 & f is being_homeomorphism & f.0 = p1 & f.1 = p2 & 0 <= s1 & s1 <= 1
& (for t being Real st 0 <= t & t < s1 holds not f.t in Q)
for g being Function of I[01], (TOP-REAL 2)|P, s2 be Real
st g is being_homeomorphism & g.0
= p1 & g.1 = p2 & g.s2 = q1 & 0 <= s2 & s2 <= 1
for t being Real st 0 <=
t & t < s2 holds not g.t in Q;
definition
let P, Q be Subset of TOP-REAL 2, p1, p2 be Point of TOP-REAL 2;
assume that
P meets Q & P /\ Q is closed and
P is_an_arc_of p1,p2;
func First_Point(P,p1,p2,Q) -> Point of TOP-REAL 2 means
:: JORDAN5C:def 1
it in P /\ Q
& for g being Function of I[01], (TOP-REAL 2)|P, s2 being Real
st g is
being_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = it & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds not g.t in Q;
end;
theorem :: JORDAN5C:2
for P, Q being Subset of TOP-REAL 2, p, p1, p2 being Point of TOP-REAL
2 st p in P & P is_an_arc_of p1, p2 & Q = {p} holds First_Point (P,p1,p2,Q) = p
;
theorem :: JORDAN5C:3
for P, Q being Subset of TOP-REAL 2, p1, p2 being Point of
TOP-REAL 2 st p1 in Q & P /\ Q is closed & P is_an_arc_of p1, p2 holds
First_Point (P, p1, p2, Q) = p1;
theorem :: JORDAN5C:4
for P, Q being Subset of TOP-REAL 2, p1, p2, q1 being Point of
TOP-REAL 2, f being Function of I[01], (TOP-REAL 2)|P, s1 be Real
st q1 in P &
f.s1 = q1 & f is being_homeomorphism & f.0 = p1 & f.1 = p2 & 0 <= s1 & s1 <= 1
&
(for t being Real st 1 >= t & t > s1 holds not f.t in Q)
for g being
Function of I[01], (TOP-REAL 2)|P, s2 being Real
st g is being_homeomorphism &
g.0 = p1 & g.1 = p2 & g.s2 = q1 & 0 <= s2 & s2 <= 1
for t being Real st 1 >= t & t > s2 holds not g.t in Q;
definition
let P, Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2;
assume that
P meets Q & P /\ Q is closed and
P is_an_arc_of p1,p2;
func Last_Point (P,p1,p2,Q) -> Point of TOP-REAL 2 means
:: JORDAN5C:def 2
it in P /\ Q
& for g being Function of I[01], (TOP-REAL 2)|P, s2 be Real st g is
being_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = it & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds not g.t in Q;
end;
theorem :: JORDAN5C:5
for P, Q being Subset of TOP-REAL 2, p, p1,p2 being Point of TOP-REAL
2 st p in P & P is_an_arc_of p1, p2 & Q = {p} holds Last_Point (P, p1, p2, Q) =
p;
theorem :: JORDAN5C:6
for P,Q being Subset of TOP-REAL 2, p1, p2 being Point of
TOP-REAL 2 st p2 in Q & P /\ Q is closed & P is_an_arc_of p1, p2 holds
Last_Point (P, p1, p2, Q) = p2;
theorem :: JORDAN5C:7
for P, Q being Subset of TOP-REAL 2, p1, p2 being Point of
TOP-REAL 2 st P c= Q & P is closed & P is_an_arc_of p1, p2 holds First_Point (P
, p1, p2, Q) = p1 & Last_Point (P, p1, p2, Q) = p2;
begin :: The ordering of points on a curve
definition
let P be Subset of TOP-REAL 2, p1, p2, q1, q2 be Point of TOP-REAL 2;
pred LE q1, q2, P, p1, p2 means
:: JORDAN5C:def 3
q1 in P & q2 in P & for g being
Function of I[01], (TOP-REAL 2)|P, s1, s2 being Real st g is
being_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1 & 0 <= s1 & s1 <= 1 & g.
s2 = q2 & 0 <= s2 & s2 <= 1 holds s1 <= s2;
end;
theorem :: JORDAN5C:8
for P being Subset of TOP-REAL 2, p1, p2, q1, q2 being Point of
TOP-REAL 2, g being Function of I[01], (TOP-REAL 2)|P,
s1, s2 being Real st P
is_an_arc_of p1,p2 & g is being_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1
& 0 <= s1 & s1 <= 1 & g.s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds LE q1,q2,P
,p1,p2;
theorem :: JORDAN5C:9
for P being Subset of TOP-REAL 2, p1,p2,q1 being Point of
TOP-REAL 2 st q1 in P holds LE q1,q1,P,p1,p2;
theorem :: JORDAN5C:10
for P being Subset of TOP-REAL 2, p1,p2,q1 being Point of
TOP-REAL 2 st P is_an_arc_of p1,p2 & q1 in P holds LE p1,q1,P,p1,p2 & LE q1,p2,
P,p1,p2;
theorem :: JORDAN5C:11
for P being Subset of TOP-REAL 2, p1,p2 being Point of TOP-REAL 2 st P
is_an_arc_of p1,p2 holds LE p1,p2,P,p1,p2;
theorem :: JORDAN5C:12
for P being Subset of TOP-REAL 2, p1, p2, q1, q2 being Point of
TOP-REAL 2 st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q1,P,p1,p2 holds
q1=q2;
theorem :: JORDAN5C:13
for P being Subset of TOP-REAL 2, p1,p2,q1,q2,q3 being Point of
TOP-REAL 2 st LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds LE q1,q3,P,p1,p2;
theorem :: JORDAN5C:14
for P being Subset of TOP-REAL 2, p1, p2, q1, q2 being Point of
TOP-REAL 2 st P is_an_arc_of p1, p2 & q1 in P & q2 in P & q1 <> q2 holds LE q1,
q2,P,p1,p2 & not LE q2,q1,P,p1,p2 or LE q2,q1,P,p1,p2 & not LE q1,q2,P,p1,p2;
begin :: Some properties of the ordering of points on a curve
theorem :: JORDAN5C:15
for f being FinSequence of TOP-REAL 2, Q being Subset of
TOP-REAL 2, q being Point of TOP-REAL 2 st f is being_S-Seq & L~f /\ Q is
closed & q in L~f & q in Q holds LE First_Point(L~f,f/.1,f/.len f,Q), q, L~f, f
/.1, f/.len f;
theorem :: JORDAN5C:16
for f being FinSequence of TOP-REAL 2, Q being Subset of
TOP-REAL 2, q being Point of TOP-REAL 2 st f is being_S-Seq & L~f /\ Q is
closed & q in L~f & q in Q holds LE q, Last_Point(L~f,f/.1,f/.len f,Q), L~f, f
/.1, f/.len f;
theorem :: JORDAN5C:17
for q1, q2, p1, p2 being Point of TOP-REAL 2 st p1 <> p2 holds
LE q1, q2, LSeg(p1, p2), p1, p2 implies LE q1, q2, p1, p2;
theorem :: JORDAN5C:18
for P, Q being Subset of TOP-REAL 2, p1, p2 being Point of TOP-REAL 2
st P is_an_arc_of p1,p2 & P meets Q & P /\ Q is closed holds First_Point(P,p1,
p2,Q) = Last_Point(P,p2,p1,Q) & Last_Point(P,p1,p2,Q) = First_Point(P,p2,p1,Q);
theorem :: JORDAN5C:19
for f being FinSequence of TOP-REAL 2, Q being Subset of
TOP-REAL 2, i being Nat st L~f meets Q & Q is closed & f is
being_S-Seq & 1 <= i & i+1 <= len f & First_Point (L~f, f/.1, f/.len f, Q) in
LSeg (f, i) holds First_Point (L~f, f/.1, f/.len f, Q) = First_Point (LSeg (f,
i), f/.i, f/.(i+1), Q);
theorem :: JORDAN5C:20
for f being FinSequence of TOP-REAL 2, Q being Subset of
TOP-REAL 2, i being Nat st L~f meets Q & Q is closed & f is
being_S-Seq & 1 <= i & i+1 <= len f & Last_Point (L~f, f/.1, f/.len f, Q) in
LSeg (f, i) holds Last_Point (L~f, f/.1, f/.len f, Q) = Last_Point (LSeg (f, i)
, f/.i, f/.(i+1), Q);
theorem :: JORDAN5C:21
for f being FinSequence of TOP-REAL 2, i be Nat st 1 <= i &
i+1 <= len f & f is being_S-Seq & First_Point (L~f, f/.1, f/.len f, LSeg (f,i)
) in LSeg (f,i) holds First_Point (L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.i;
theorem :: JORDAN5C:22
for f being FinSequence of TOP-REAL 2, i be Nat st 1 <= i &
i+1 <= len f & f is being_S-Seq & Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i)
) in LSeg (f,i) holds Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.(i+1)
;
theorem :: JORDAN5C:23
for f be FinSequence of TOP-REAL 2, i be Nat st f is
being_S-Seq & 1 <= i & i+1 <= len f holds LE f/.i, f/.(i+1), L~f, f/.1, f/.len
f;
theorem :: JORDAN5C:24
for f be FinSequence of TOP-REAL 2, i, j be Nat st f is being_S-Seq &
1 <= i & i <= j & j <= len f holds LE f/.i, f/.j, L~f, f/.1, f/.len f;
theorem :: JORDAN5C:25
for f being FinSequence of TOP-REAL 2, q being Point of TOP-REAL
2, i being Nat st f is being_S-Seq & 1 <= i & i+1 <= len f & q in
LSeg(f,i) holds LE f/.i, q, L~f, f/.1, f/.len f;
theorem :: JORDAN5C:26
for f being FinSequence of TOP-REAL 2, q being Point of TOP-REAL
2, i being Nat st f is being_S-Seq & 1<=i & i+1<=len f & q in LSeg(f
,i) holds LE q, f/.(i+1), L~f, f/.1, f/.len f;
theorem :: JORDAN5C:27
for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q
being Point of TOP-REAL 2, i, j being Nat st L~f meets Q & f is
being_S-Seq & Q is closed & First_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1
<=i & i+1<=len f & q in LSeg(f,j) & 1 <= j & j + 1 <= len f & q in Q &
First_Point (L~f,f/.1,f/.len f,Q) <> q holds i <= j & (i=j implies LE
First_Point(L~f,f/.1,f/.len f,Q), q, f/.i, f/.(i+1));
theorem :: JORDAN5C:28
for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q
being Point of TOP-REAL 2, i, j being Nat st L~f meets Q & f is
being_S-Seq & Q is closed & Last_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=
i & i+1<=len f & q in LSeg(f,j) & 1 <= j & j + 1 <= len f & q in Q & Last_Point
(L~f,f/.1,f/.len f,Q) <> q holds i >= j & (i=j implies LE q, Last_Point(L~f,f/.
1,f/.len f,Q), f/.i, f/.(i+1));
theorem :: JORDAN5C:29
for f being FinSequence of TOP-REAL 2, q1, q2 being Point of
TOP-REAL 2, i being Nat st q1 in LSeg(f,i) & q2 in LSeg(f,i) & f is
being_S-Seq & 1 <= i & i + 1 <= len f holds LE q1, q2, L~f, f/.1, f/.len f
implies LE q1, q2, LSeg (f,i), f/.i, f/.(i+1);
theorem :: JORDAN5C:30
for f being FinSequence of TOP-REAL 2, q1, q2 being Point of TOP-REAL
2 st q1 in L~f & q2 in L~f & f is being_S-Seq & q1 <> q2 holds LE q1, q2, L~f,
f/.1, f/.len f iff for i, j being Nat st q1 in LSeg(f,i) & q2 in
LSeg(f,j) & 1 <= i & i+1 <= len f & 1 <= j & j+1 <= len f holds i <= j & (i = j
implies LE q1,q2,f/.i,f/.(i+1));