Volume 9, 1997

University of Bialystok

Copyright (c) 1997 Association of Mizar Users

### The abstract of the Mizar article:

### On the Order on a Special Polygon

**by****Andrzej Trybulec, and****Yatsuka Nakamura**- Received November 30, 1997
- MML identifier: SPRECT_2

- [ Mizar article, MML identifier index ]

environ vocabulary BOOLE, TARSKI, FINSEQ_1, RELAT_1, JORDAN3, ARYTM_1, FINSEQ_4, FUNCT_1, COMPTS_1, EUCLID, PRE_TOPC, MCART_1, PSCOMP_1, TOPREAL1, GOBOARD1, REALSET1, GOBOARD4, FINSEQ_6, SEQM_3, GOBOARD5, FUNCT_5, SUBSET_1, ORDINAL2, SPPOL_1, SPRECT_1, SPPOL_2, FINSEQ_5, TOPREAL2, SPRECT_2, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XREAL_0, REAL_1, FUNCT_1, FUNCT_2, FINSEQ_1, STRUCT_0, FINSEQ_4, FINSEQ_5, REALSET1, FINSEQ_6, NAT_1, BINARITH, PRE_TOPC, COMPTS_1, EUCLID, TOPREAL1, TOPREAL2, GOBOARD1, GOBOARD4, SPPOL_1, SPPOL_2, GOBOARD5, JORDAN3, PSCOMP_1, SPRECT_1; constructors PSCOMP_1, GOBOARD5, JORDAN3, GOBOARD4, BINARITH, REALSET1, SPPOL_1, FINSEQ_5, REAL_1, SEQM_3, TOPREAL4, SPRECT_1, TOPREAL2, COMPTS_1, FINSEQ_4, GOBOARD1; clusters STRUCT_0, EUCLID, RELSET_1, GOBOARD5, SPPOL_2, GOBOARD2, GOBOARD4, PSCOMP_1, SPRECT_1, XREAL_0, FINSEQ_1, ARYTM_3, MEMBERED; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: Preliminaries theorem :: SPRECT_2:1 for A,B,C,p being set st A /\ B c= {p} & p in C & C misses B holds A \/ C misses B; theorem :: SPRECT_2:2 for A,B,C,p being set st A /\ C = {p} & p in B & B c= C holds A /\ B = {p}; canceled; theorem :: SPRECT_2:4 for A,B being set st for x,y being set st x in A & y in B holds x misses y holds union A misses union B; begin :: Finite sequences reserve i,j,k,l,m,n for Nat, D for non empty set, f for FinSequence of D; theorem :: SPRECT_2:5 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_2:6 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_2:7 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_2:8 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_2:9 i in dom f & j in dom f implies len mid(f,i,j) >= 1; theorem :: SPRECT_2:10 i in dom f & j in dom f & len mid(f,i,j) = 1 implies i = j; theorem :: SPRECT_2:11 i in dom f & j in dom f implies mid(f,i,j) is non empty; theorem :: SPRECT_2:12 i in dom f & j in dom f implies (mid(f,i,j))/.1 = f/.i; theorem :: SPRECT_2:13 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_2:14 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_2:15 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_2:16 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_2:17 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_2:18 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_2:19 for f being FinSequence of TOP-REAL 2 holds dom X_axis f = dom f; theorem :: SPRECT_2:20 for f being FinSequence of TOP-REAL 2 holds dom Y_axis f = dom f; reserve p,q,r for Real; theorem :: SPRECT_2:21 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_2:22 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_2:23 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_2:24 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_2:25 for f being non trivial FinSequence of TOP-REAL 2 holds f is_in_the_area_of f; theorem :: SPRECT_2:26 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_2:27 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_2:28 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_2:29 for f being non trivial FinSequence of TOP-REAL 2 holds <*NE-corner L~f*> is_in_the_area_of f; theorem :: SPRECT_2:30 for f being non trivial FinSequence of TOP-REAL 2 holds <*NW-corner L~f*> is_in_the_area_of f; theorem :: SPRECT_2:31 for f being non trivial FinSequence of TOP-REAL 2 holds <*SE-corner L~f*> is_in_the_area_of f; theorem :: SPRECT_2:32 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_2:33 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_2:34 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; definition cluster R^2-unit_square -> compact; end; theorem :: SPRECT_2:35 N-bound R^2-unit_square = 1; theorem :: SPRECT_2:36 W-bound R^2-unit_square = 0; theorem :: SPRECT_2:37 E-bound R^2-unit_square = 1; theorem :: SPRECT_2:38 S-bound R^2-unit_square = 0; theorem :: SPRECT_2:39 N-most R^2-unit_square = LSeg(|[0,1]|,|[1,1]|); theorem :: SPRECT_2:40 N-min R^2-unit_square = |[0,1]|; definition let X be non vertical non horizontal non empty compact Subset of TOP-REAL 2; cluster SpStSeq X -> clockwise_oriented; end; definition cluster clockwise_oriented (non constant standard special_circular_sequence); end; theorem :: SPRECT_2:41 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_2:42 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_2:43 N-min L~f in rng f; theorem :: SPRECT_2:44 N-max L~f in rng f; theorem :: SPRECT_2:45 S-min L~f in rng f; theorem :: SPRECT_2:46 S-max L~f in rng f; theorem :: SPRECT_2:47 W-min L~f in rng f; theorem :: SPRECT_2:48 W-max L~f in rng f; theorem :: SPRECT_2:49 E-min L~f in rng f; theorem :: SPRECT_2:50 E-max L~f in rng f; reserve f for non constant standard special_circular_sequence; theorem :: SPRECT_2:51 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_2:52 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_2:53 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_2:54 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_2:55 (N-min L~f)`1 < (N-max L~f)`1; theorem :: SPRECT_2:56 N-min L~f <> N-max L~f; theorem :: SPRECT_2:57 (E-min L~f)`2 < (E-max L~f)`2; theorem :: SPRECT_2:58 E-min L~f <> E-max L~f; theorem :: SPRECT_2:59 (S-min L~f)`1 < (S-max L~f)`1; theorem :: SPRECT_2:60 S-min L~f <> S-max L~f; theorem :: SPRECT_2:61 (W-min L~f)`2 < (W-max L~f)`2; theorem :: SPRECT_2:62 W-min L~f <> W-max L~f; theorem :: SPRECT_2:63 LSeg(NW-corner L~f,N-min L~f) misses LSeg(N-max L~f,NE-corner L~f); theorem :: SPRECT_2:64 for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL 2 st f is_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_2:65 for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL 2 st f is_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_2:66 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_2:67 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_2:68 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_2:69 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_2:70 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_2:71 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; definition let f be non constant standard special_circular_sequence; cluster L~f -> being_simple_closed_curve; end; begin :: The order theorem :: SPRECT_2:72 f/.1 = N-min L~f implies (N-min L~f)..f < (N-max L~f)..f; theorem :: SPRECT_2:73 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_2:74 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_2:75 z/.1 = N-min L~z implies (E-max L~z)..z < (E-min L~z)..z; theorem :: SPRECT_2:76 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_2:77 z/.1 = N-min L~z implies (S-max L~z)..z < (S-min L~z)..z; theorem :: SPRECT_2:78 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_2:79 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_2:80 z/.1 = N-min L~z implies (W-min L~z)..z < len z; theorem :: SPRECT_2:81 f/.1 = N-min L~f implies (W-max L~f)..f < len f;

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