:: On the Order on a Special Polygon
:: by Andrzej Trybulec and Yatsuka Nakamura
::
:: Received November 30, 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, XBOOLE_0, FINSEQ_1, XXREAL_0, RELAT_1,
JORDAN3, ARYTM_3, ARYTM_1, CARD_1, PARTFUN1, FUNCT_1, RCOMP_1, EUCLID,
PRE_TOPC, MCART_1, PSCOMP_1, RLTOPSP1, TOPREAL1, TARSKI, GOBOARD1,
REAL_1, SUPINF_2, ZFMISC_1, ORDINAL4, NAT_1, GOBOARD4, FINSEQ_6,
GOBOARD5, STRUCT_0, SPPOL_1, SPRECT_1, SPPOL_2, FINSEQ_5, FINSEQ_4,
TOPREAL2, SPRECT_2, SEQ_4;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0,
REAL_1, FUNCT_1, RELSET_1, PARTFUN1, FINSEQ_1, STRUCT_0, FINSEQ_4,
FINSEQ_5, ZFMISC_1, FINSEQ_6, NAT_1, NAT_D, PRE_TOPC, COMPTS_1, RLVECT_1,
RLTOPSP1, EUCLID, TOPREAL1, TOPREAL2, GOBOARD1, GOBOARD4, SPPOL_1,
SPPOL_2, GOBOARD5, PSCOMP_1, SPRECT_1, XXREAL_0;
constructors REAL_1, FINSEQ_4, NAT_D, FINSEQ_5, COMPTS_1, TOPREAL2, GOBOARD1,
GOBOARD4, SPPOL_1, GOBOARD5, PSCOMP_1, SPRECT_1, SEQ_4, RELSET_1,
CONVEX1;
registrations RELSET_1, NUMBERS, XREAL_0, FINSEQ_1, STRUCT_0, PRE_TOPC,
EUCLID, GOBOARD2, SPPOL_2, SPRECT_1, XBOOLE_0, VALUED_0, FUNCT_1,
RLTOPSP1, MEASURE6, NAT_1, ORDINAL1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
begin :: Finite sequences
reserve i,j,k,l,m,n for Nat,
D for non empty set,
f for FinSequence of D;
theorem :: SPRECT_2:1
i <= j & i in dom f & j in dom f & k in dom mid(f,i,j) implies k+
i-'1 in dom f;
theorem :: SPRECT_2:2
i > j & i in dom f & j in dom f & k in dom mid(f,i,j) implies i
-' k + 1 in dom f;
theorem :: SPRECT_2:3
i <= j & i in dom f & j in dom f & k in dom mid(f,i,j) implies (
mid(f,i,j))/.k = f/.(k+i-'1);
theorem :: SPRECT_2:4
i > j & i in dom f & j in dom f & k in dom mid(f,i,j) implies (
mid(f,i,j))/.k = f/.(i-' k +1);
theorem :: SPRECT_2:5
i in dom f & j in dom f implies len mid(f,i,j) >= 1;
theorem :: SPRECT_2:6
i in dom f & j in dom f & len mid(f,i,j) = 1 implies i = j;
theorem :: SPRECT_2:7
i in dom f & j in dom f implies mid(f,i,j) is non empty;
theorem :: SPRECT_2:8
i in dom f & j in dom f implies (mid(f,i,j))/.1 = f/.i;
theorem :: SPRECT_2:9
i in dom f & j in dom f implies (mid(f,i,j))/.len mid(f,i,j) = f /.j;
begin :: Compact sets on the plane
reserve X for compact Subset of TOP-REAL 2;
theorem :: SPRECT_2:10
for p being Point of TOP-REAL 2 st p in X & p`2 = N-bound X
holds p in N-most X;
theorem :: SPRECT_2:11
for p being Point of TOP-REAL 2 st p in X & p`2 = S-bound X
holds p in S-most X;
theorem :: SPRECT_2:12
for p being Point of TOP-REAL 2 st p in X & p`1 = W-bound X
holds p in W-most X;
theorem :: SPRECT_2:13
for p being Point of TOP-REAL 2 st p in X & p`1 = E-bound X
holds p in E-most X;
begin :: Finite sequences on the plane
theorem :: SPRECT_2:14
for f being FinSequence of TOP-REAL 2 st 1 <= i & i <= j & j <=
len f holds L~mid(f,i,j) = union{ LSeg(f,k): i <= k & k < j};
theorem :: SPRECT_2:15
for f being FinSequence of TOP-REAL 2 holds dom X_axis f = dom f;
theorem :: SPRECT_2:16
for f being FinSequence of TOP-REAL 2 holds dom Y_axis f = dom f;
reserve r for Real;
theorem :: SPRECT_2:17
for a,b,c being Point of TOP-REAL 2 st b in LSeg(a,c) & a`1 <= b
`1 & c`1 <= b`1 holds a = b or b = c or a`1 = b`1 & c`1 = b`1;
theorem :: SPRECT_2:18
for a,b,c being Point of TOP-REAL 2 st b in LSeg(a,c) & a`2 <= b
`2 & c`2 <= b`2 holds a = b or b = c or a`2 = b`2 & c`2 = b`2;
theorem :: SPRECT_2:19
for a,b,c being Point of TOP-REAL 2 st b in LSeg(a,c) & a`1 >= b
`1 & c`1 >= b`1 holds a = b or b = c or a`1 = b`1 & c`1 = b`1;
theorem :: SPRECT_2:20
for a,b,c being Point of TOP-REAL 2 st b in LSeg(a,c) & a`2 >= b
`2 & c`2 >= b`2 holds a = b or b = c or a`2 = b`2 & c`2 = b`2;
begin :: The area of a sequence
definition
let f, g be FinSequence of TOP-REAL 2;
pred g is_in_the_area_of f means
:: SPRECT_2:def 1
for n st n in dom g holds W-bound L~
f <= (g/.n)`1 & (g/.n)`1 <= E-bound L~f & S-bound L~f <= (g/.n)`2 & (g/.n)`2 <=
N-bound L~f;
end;
theorem :: SPRECT_2:21
for f being non trivial FinSequence of TOP-REAL 2 holds f is_in_the_area_of f
;
theorem :: SPRECT_2:22
for f, g being FinSequence of TOP-REAL 2 st g is_in_the_area_of
f for i,j st i in dom g & j in dom g holds mid(g,i,j) is_in_the_area_of f;
theorem :: SPRECT_2:23
for f being non trivial FinSequence of TOP-REAL 2 for i,j st i in dom
f & j in dom f holds mid(f,i,j) is_in_the_area_of f;
theorem :: SPRECT_2:24
for f,g,h being FinSequence of TOP-REAL 2 st g is_in_the_area_of
f & h is_in_the_area_of f holds g^h is_in_the_area_of f;
theorem :: SPRECT_2:25
for f being non trivial FinSequence of TOP-REAL 2 holds <*
NE-corner L~f*> is_in_the_area_of f;
theorem :: SPRECT_2:26
for f being non trivial FinSequence of TOP-REAL 2 holds <*
NW-corner L~f*> is_in_the_area_of f;
theorem :: SPRECT_2:27
for f being non trivial FinSequence of TOP-REAL 2 holds <*
SE-corner L~f*> is_in_the_area_of f;
theorem :: SPRECT_2:28
for f being non trivial FinSequence of TOP-REAL 2 holds <*
SW-corner L~f*> is_in_the_area_of f;
begin :: Horizontal and vertical connections
definition
let f, g be FinSequence of TOP-REAL 2;
pred g is_a_h.c._for f means
:: SPRECT_2:def 2
g is_in_the_area_of f & (g/.1)`1 = W-bound L~f & (g/.len g)`1 = E-bound L~f;
pred g is_a_v.c._for f means
:: SPRECT_2:def 3
g is_in_the_area_of f & (g/.1)`2 = S-bound L~f & (g/.len g)`2 = N-bound L~f;
end;
theorem :: SPRECT_2:29
for f being FinSequence of TOP-REAL 2, g,h being one-to-one
special FinSequence of TOP-REAL 2 st 2 <= len g & 2 <= len h & g is_a_h.c._for
f & h is_a_v.c._for f holds L~g meets L~h;
begin :: Orientation
definition
let f be FinSequence of TOP-REAL 2;
attr f is clockwise_oriented means
:: SPRECT_2:def 4
(Rotate(f,N-min L~f))/.2 in N-most L~f;
end;
theorem :: SPRECT_2:30
for f being non constant standard special_circular_sequence st f
/.1 = N-min L~f holds f is clockwise_oriented iff f/.2 in N-most L~f;
registration
cluster R^2-unit_square -> compact;
end;
theorem :: SPRECT_2:31
N-bound R^2-unit_square = 1;
theorem :: SPRECT_2:32
W-bound R^2-unit_square = 0;
theorem :: SPRECT_2:33
E-bound R^2-unit_square = 1;
theorem :: SPRECT_2:34
S-bound R^2-unit_square = 0;
theorem :: SPRECT_2:35
N-most R^2-unit_square = LSeg(|[0,1]|,|[1,1]|);
theorem :: SPRECT_2:36
N-min R^2-unit_square = |[0,1]|;
registration
let X be non vertical non horizontal non empty compact Subset of TOP-REAL 2;
cluster SpStSeq X -> clockwise_oriented;
end;
registration
cluster clockwise_oriented for
non constant standard special_circular_sequence;
end;
theorem :: SPRECT_2:37
for f being non constant standard special_circular_sequence, i,j
st i > j & (1 < j & i <= len f or 1 <= j & i < len f) holds mid(f,i,j) is
S-Sequence_in_R2;
theorem :: SPRECT_2:38
for f being non constant standard special_circular_sequence, i,j
st i < j & (1 < i & j <= len f or 1 <= i & j < len f) holds mid(f,i,j) is
S-Sequence_in_R2;
reserve f for non trivial FinSequence of TOP-REAL 2;
theorem :: SPRECT_2:39
N-min L~f in rng f;
theorem :: SPRECT_2:40
N-max L~f in rng f;
theorem :: SPRECT_2:41
S-min L~f in rng f;
theorem :: SPRECT_2:42
S-max L~f in rng f;
theorem :: SPRECT_2:43
W-min L~f in rng f;
theorem :: SPRECT_2:44
W-max L~f in rng f;
theorem :: SPRECT_2:45
E-min L~f in rng f;
theorem :: SPRECT_2:46
E-max L~f in rng f;
reserve f for non constant standard special_circular_sequence;
theorem :: SPRECT_2:47
1 <= i & i <= j & j < m & m <= n & n <= len f & (1 < i or n <
len f) implies L~mid(f,i,j) misses L~mid(f,m,n);
theorem :: SPRECT_2:48
1 <= i & i <= j & j < m & m <= n & n <= len f & (1 < i or n <
len f) implies L~mid(f,i,j) misses L~mid(f,n,m);
theorem :: SPRECT_2:49
1 <= i & i <= j & j < m & m <= n & n <= len f & (1 < i or n <
len f) implies L~mid(f,j,i) misses L~mid(f,n,m);
theorem :: SPRECT_2:50
1 <= i & i <= j & j < m & m <= n & n <= len f & (1 < i or n <
len f) implies L~mid(f,j,i) misses L~mid(f,m,n);
theorem :: SPRECT_2:51
(N-min L~f)`1 < (N-max L~f)`1;
theorem :: SPRECT_2:52
N-min L~f <> N-max L~f;
theorem :: SPRECT_2:53
(E-min L~f)`2 < (E-max L~f)`2;
theorem :: SPRECT_2:54
E-min L~f <> E-max L~f;
theorem :: SPRECT_2:55
(S-min L~f)`1 < (S-max L~f)`1;
theorem :: SPRECT_2:56
S-min L~f <> S-max L~f;
theorem :: SPRECT_2:57
(W-min L~f)`2 < (W-max L~f)`2;
theorem :: SPRECT_2:58
W-min L~f <> W-max L~f;
theorem :: SPRECT_2:59
LSeg(NW-corner L~f,N-min L~f) misses LSeg(N-max L~f,NE-corner L~ f);
theorem :: SPRECT_2:60
for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
2 st f is being_S-Seq & p <> f/.1 & (p`1 = (f/.1)`1 or p`2 = (f/.1)`2) & LSeg(p
,f/.1) /\ L~f = {f/.1} holds <*p*>^f is S-Sequence_in_R2;
theorem :: SPRECT_2:61
for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
2 st f is being_S-Seq & p <> f/.len f & (p`1 = (f/.len f)`1 or p`2 = (f/.len f)
`2) & LSeg(p,f/.len f) /\ L~f = {f/.len f} holds f^<*p*> is S-Sequence_in_R2;
begin :: Appending corners
theorem :: SPRECT_2:62
for i,j st i in dom f & j in dom f & mid(f,i,j) is
S-Sequence_in_R2 & f/.j = N-max L~f & N-max L~f <> NE-corner L~f holds mid(f,i,
j)^<*NE-corner L~f*> is S-Sequence_in_R2;
theorem :: SPRECT_2:63
for i,j st i in dom f & j in dom f & mid(f,i,j) is S-Sequence_in_R2 &
f/.j = E-max L~f & E-max L~f <> NE-corner L~f holds mid(f,i,j)^<*NE-corner L~f
*> is S-Sequence_in_R2;
theorem :: SPRECT_2:64
for i,j st i in dom f & j in dom f & mid(f,i,j) is
S-Sequence_in_R2 & f/.j = S-max L~f & S-max L~f <> SE-corner L~f holds mid(f,i,
j)^<*SE-corner L~f*> is S-Sequence_in_R2;
theorem :: SPRECT_2:65
for i,j st i in dom f & j in dom f & mid(f,i,j) is
S-Sequence_in_R2 & f/.j = E-max L~f & E-max L~f <> NE-corner L~f holds mid(f,i,
j)^<*NE-corner L~f*> is S-Sequence_in_R2;
theorem :: SPRECT_2:66
for i,j st i in dom f & j in dom f & mid(f,i,j) is
S-Sequence_in_R2 & f/.i = N-min L~f & N-min L~f <> NW-corner L~f holds <*
NW-corner L~f*>^mid(f,i,j) is S-Sequence_in_R2;
theorem :: SPRECT_2:67
for i,j st i in dom f & j in dom f & mid(f,i,j) is
S-Sequence_in_R2 & f/.i = W-min L~f & W-min L~f <> SW-corner L~f holds <*
SW-corner L~f*>^mid(f,i,j) is S-Sequence_in_R2;
registration
let f be non constant standard special_circular_sequence;
cluster L~f -> being_simple_closed_curve;
end;
begin :: The order
theorem :: SPRECT_2:68
f/.1 = N-min L~f implies (N-min L~f)..f < (N-max L~f)..f;
theorem :: SPRECT_2:69
f/.1 = N-min L~f implies (N-max L~f)..f > 1;
reserve z for clockwise_oriented non constant standard
special_circular_sequence;
theorem :: SPRECT_2:70
z/.1 = N-min L~z & N-max L~z <> E-max L~z implies (N-max L~z)..z < (
E-max L~z)..z;
theorem :: SPRECT_2:71
z/.1 = N-min L~z implies (E-max L~z)..z < (E-min L~z)..z;
theorem :: SPRECT_2:72
z/.1 = N-min L~z & E-min L~z <> S-max L~z implies (E-min L~z)..z
< (S-max L~z)..z;
theorem :: SPRECT_2:73
z/.1 = N-min L~z implies (S-max L~z)..z < (S-min L~z)..z;
theorem :: SPRECT_2:74
z/.1 = N-min L~z & S-min L~z <> W-min L~z implies (S-min L~z)..z < (
W-min L~z)..z;
theorem :: SPRECT_2:75
z/.1 = N-min L~z & N-min L~z <> W-max L~z implies (W-min L~z)..z
< (W-max L~z)..z;
theorem :: SPRECT_2:76
z/.1 = N-min L~z implies (W-min L~z)..z < len z;
theorem :: SPRECT_2:77
f/.1 = N-min L~f implies (W-max L~f)..f < len f;