Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

### The abstract of the Mizar article:

### Special Polygons

**by****Czeslaw Bylinski, and****Yatsuka Nakamura**- Received January 30, 1995
- MML identifier: SPPOL_2

- [ Mizar article, MML identifier index ]

environ vocabulary EUCLID, FINSEQ_1, PRE_TOPC, ARYTM, MCART_1, TOPREAL1, FINSEQ_5, RELAT_1, FINSEQ_4, BOOLE, RFINSEQ, ARYTM_1, TARSKI, FUNCT_1, REALSET1, TOPREAL4, RCOMP_1, COMPTS_1, BORSUK_1, TOPS_2, ORDINAL2, SUBSET_1, SPPOL_2; notation TARSKI, XBOOLE_0, ORDINAL1, XREAL_0, STRUCT_0, REAL_1, NAT_1, FUNCT_1, FINSEQ_1, REALSET1, FINSEQ_4, RFINSEQ, PRE_TOPC, COMPTS_1, TOPS_2, EUCLID, BORSUK_1, TOPREAL1, TOPREAL4, FINSEQ_5; constructors TOPMETR, REAL_1, NAT_1, REALSET1, RFINSEQ, COMPTS_1, TOPS_2, TOPREAL1, TOPREAL4, FINSEQ_4, FINSEQ_5, INT_1, MEMBERED, XBOOLE_0; clusters PRE_TOPC, FINSEQ_5, RELSET_1, STRUCT_0, EUCLID, XREAL_0, FINSEQ_1, INT_1, TOPREAL1, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: Segments in TOP-REAL 2 reserve P for Subset of TOP-REAL 2, f,f1,f2,g for FinSequence of TOP-REAL 2, p,p1,p2,q,q1,q2 for Point of TOP-REAL 2, r1,r2,r1',r2' for Real, i,j,k,n for Nat; theorem :: SPPOL_2:1 for r1, r2, r1', r2' being real number st |[ r1,r2 ]| = |[ r1',r2' ]| holds r1 = r1' & r2 = r2'; theorem :: SPPOL_2:2 i+j = len f implies LSeg(f,i) = LSeg(Rev f,j); theorem :: SPPOL_2:3 i+1 <= len(f|n) implies LSeg(f|n,i) = LSeg(f,i); theorem :: SPPOL_2:4 n <= len f & 1 <= i implies LSeg(f/^n,i) = LSeg(f,n+i); theorem :: SPPOL_2:5 1 <= i & i+1 <= len f - n implies LSeg(f/^n,i) = LSeg(f,n+i); theorem :: SPPOL_2:6 i+1 <= len f implies LSeg(f^g,i) = LSeg(f,i); theorem :: SPPOL_2:7 1 <= i implies LSeg(f^g,len f + i) = LSeg(g,i); theorem :: SPPOL_2:8 f is non empty & g is non empty implies LSeg(f^g,len f) = LSeg(f/.len f,g/.1); theorem :: SPPOL_2:9 i+1 <= len(f-:p) implies LSeg(f-:p,i) = LSeg(f,i); theorem :: SPPOL_2:10 p in rng f implies LSeg(f:-p,i+1) = LSeg(f,i+p..f); theorem :: SPPOL_2:11 L~<*>(the carrier of TOP-REAL 2) = {}; theorem :: SPPOL_2:12 L~<*p*> = {}; theorem :: SPPOL_2:13 p in L~f implies ex i st 1 <= i & i+1 <= len f & p in LSeg(f,i); theorem :: SPPOL_2:14 p in L~f implies ex i st 1 <= i & i+1 <= len f & p in LSeg(f/.i,f/.(i+1)); theorem :: SPPOL_2:15 1 <= i & i+1 <= len f & p in LSeg(f/.i,f/.(i+1)) implies p in L~f; theorem :: SPPOL_2:16 1 <= i & i+1 <= len f implies LSeg(f/.i,f/.(i+1)) c= L~f; theorem :: SPPOL_2:17 p in LSeg(f,i) implies p in L~f; theorem :: SPPOL_2:18 len f >= 2 implies rng f c= L~f; theorem :: SPPOL_2:19 f is non empty implies L~(f^<*p*>) = L~f \/ LSeg(f/.len f,p); theorem :: SPPOL_2:20 f is non empty implies L~(<*p*>^f) = LSeg(p,f/.1) \/ L~f; theorem :: SPPOL_2:21 L~<*p,q*> = LSeg(p,q); theorem :: SPPOL_2:22 L~f = L~(Rev f); theorem :: SPPOL_2:23 f1 is non empty & f2 is non empty implies L~(f1^f2) = L~f1 \/ LSeg(f1/.len f1,f2/.1) \/ L~f2; canceled; theorem :: SPPOL_2:25 q in rng f implies L~f = L~(f-:q) \/ L~(f:-q); theorem :: SPPOL_2:26 p in LSeg(f,n) implies L~f = L~Ins(f,n,p); begin definition cluster being_S-Seq FinSequence of TOP-REAL 2; cluster being_S-Seq -> one-to-one unfolded s.n.c. special non trivial FinSequence of TOP-REAL 2; cluster one-to-one unfolded s.n.c. special non trivial -> being_S-Seq FinSequence of TOP-REAL 2; cluster being_S-Seq -> non empty FinSequence of TOP-REAL 2; end; definition cluster one-to-one unfolded s.n.c. special non trivial FinSequence of TOP-REAL 2; end; theorem :: SPPOL_2:27 len f <= 2 implies f is unfolded; definition let f be unfolded FinSequence of TOP-REAL 2, n; cluster f|n -> unfolded; cluster f/^n -> unfolded; end; theorem :: SPPOL_2:28 p in rng f & f is unfolded implies f:-p is unfolded; definition let f be unfolded FinSequence of TOP-REAL 2, p; cluster f-:p -> unfolded; end; theorem :: SPPOL_2:29 f is unfolded implies Rev f is unfolded; theorem :: SPPOL_2:30 g is unfolded & LSeg(p,g/.1) /\ LSeg(g,1) = {g/.1} implies <*p*>^g is unfolded; theorem :: SPPOL_2:31 f is unfolded & k+1 = len f & LSeg(f,k) /\ LSeg(f/.len f,p) = {f/.len f} implies f^<*p*> is unfolded; theorem :: SPPOL_2:32 f is unfolded & g is unfolded & k+1 = len f & LSeg(f,k) /\ LSeg(f/.len f,g/.1) = {f/.len f} & LSeg(f/.len f,g/.1) /\ LSeg(g,1) = {g/.1} implies f^g is unfolded; theorem :: SPPOL_2:33 f is unfolded & p in LSeg(f,n) implies Ins(f,n,p) is unfolded; theorem :: SPPOL_2:34 len f <= 2 implies f is s.n.c.; definition let f be s.n.c. FinSequence of TOP-REAL 2, n; cluster f|n -> s.n.c.; cluster f/^n -> s.n.c.; end; definition let f be s.n.c. FinSequence of TOP-REAL 2, p; cluster f-:p -> s.n.c.; end; theorem :: SPPOL_2:35 p in rng f & f is s.n.c. implies f:-p is s.n.c.; theorem :: SPPOL_2:36 f is s.n.c. implies Rev f is s.n.c.; theorem :: SPPOL_2:37 f is s.n.c. & g is s.n.c. & L~f misses L~g & (for i st 1<=i & i+2 <= len f holds LSeg(f,i) misses LSeg(f/.len f,g/.1)) & (for i st 2<=i & i+1 <= len g holds LSeg(g,i) misses LSeg(f/.len f,g/.1)) implies f^g is s.n.c.; theorem :: SPPOL_2:38 f is unfolded s.n.c. & p in LSeg(f,n) & not p in rng f implies Ins(f,n,p) is s.n.c.; definition cluster <*>(the carrier of TOP-REAL 2) -> special; end; theorem :: SPPOL_2:39 <*p*> is special; theorem :: SPPOL_2:40 p`1 = q`1 or p`2 = q`2 implies <*p,q*> is special; definition let f be special FinSequence of TOP-REAL 2, n; cluster f|n -> special; cluster f/^n -> special; end; theorem :: SPPOL_2:41 p in rng f & f is special implies f:-p is special; definition let f be special FinSequence of TOP-REAL 2, p; cluster f-:p -> special; end; theorem :: SPPOL_2:42 f is special implies Rev f is special; canceled; theorem :: SPPOL_2:44 f is special & p in LSeg(f,n) implies Ins(f,n,p) is special; theorem :: SPPOL_2:45 q in rng f & 1<>q..f & q..f<>len f & f is unfolded s.n.c. implies L~(f-:q) /\ L~(f:-q) = {q}; theorem :: SPPOL_2:46 p <> q & (p`1 = q`1 or p`2 = q`2) implies <*p,q*> is being_S-Seq; definition mode S-Sequence_in_R2 is being_S-Seq FinSequence of TOP-REAL 2; end; theorem :: SPPOL_2:47 for f being S-Sequence_in_R2 holds Rev f is being_S-Seq; theorem :: SPPOL_2:48 for f being S-Sequence_in_R2 st i in dom f holds f/.i in L~f; theorem :: SPPOL_2:49 p <> q & (p`1 = q`1 or p`2 = q`2) implies LSeg(p,q) is being_S-P_arc; theorem :: SPPOL_2:50 for f being S-Sequence_in_R2 st p in rng f & p..f <> 1 holds f-:p is being_S-Seq; theorem :: SPPOL_2:51 for f being S-Sequence_in_R2 st p in rng f & p..f <> len f holds f:-p is being_S-Seq; theorem :: SPPOL_2:52 for f being S-Sequence_in_R2 st p in LSeg(f,i) & not p in rng f holds Ins(f,i,p) is being_S-Seq; begin :: Special Polygons in TOP-REAL 2 definition cluster being_S-P_arc Subset of TOP-REAL 2; end; theorem :: SPPOL_2:53 P is_S-P_arc_joining p1,p2 implies P is_S-P_arc_joining p2,p1; definition let p1,p2; let P be Subset of TOP-REAL 2; pred p1,p2 split P means :: SPPOL_2:def 1 p1 <> p2 & ex f1,f2 being S-Sequence_in_R2 st p1 = f1/.1 & p1 = f2/.1 & p2 = f1/.len f1 & p2 = f2/.len f2 & L~f1 /\ L~f2 = {p1,p2} & P = L~f1 \/ L~f2; end; theorem :: SPPOL_2:54 p1,p2 split P implies p2,p1 split P; theorem :: SPPOL_2:55 p1,p2 split P & q in P & q <> p1 implies p1,q split P; theorem :: SPPOL_2:56 p1,p2 split P & q in P & q <> p2 implies q,p2 split P; theorem :: SPPOL_2:57 p1,p2 split P & q1 in P & q2 in P & q1 <> q2 implies q1,q2 split P; definition let P be Subset of TOP-REAL 2; redefine attr P is being_special_polygon means :: SPPOL_2:def 2 ex p1,p2 st p1,p2 split P; synonym P is special_polygonal; end; definition let r1,r2,r1',r2'; func [.r1,r2,r1',r2'.] -> Subset of TOP-REAL 2 equals :: SPPOL_2:def 3 { p: p`1 = r1 & p`2 <= r2' & p`2 >= r1' or p`1 <= r2 & p`1 >= r1 & p`2 = r2' or p`1 <= r2 & p`1 >= r1 & p`2 = r1' or p`1 = r2 & p`2 <= r2' & p`2 >= r1'}; end; theorem :: SPPOL_2:58 r1<r2 & r1'<r2' implies [.r1,r2,r1',r2'.] = ( LSeg(|[r1,r1']|,|[r1,r2']|) \/ LSeg(|[r1,r2']|,|[r2,r2']|) ) \/ ( LSeg(|[r2,r2']|,|[r2,r1']|) \/ LSeg(|[r2,r1']|,|[r1,r1']|) ); theorem :: SPPOL_2:59 r1<r2 & r1'<r2' implies [.r1,r2,r1',r2'.] is special_polygonal; theorem :: SPPOL_2:60 R^2-unit_square = [.0,1,0,1.]; definition cluster special_polygonal Subset of TOP-REAL 2; end; theorem :: SPPOL_2:61 R^2-unit_square is special_polygonal; definition cluster special_polygonal Subset of TOP-REAL 2; cluster special_polygonal -> non empty Subset of TOP-REAL 2; cluster special_polygonal -> non trivial Subset of TOP-REAL 2; end; definition mode Special_polygon_in_R2 is special_polygonal Subset of TOP-REAL 2; end; theorem :: SPPOL_2:62 P is being_S-P_arc implies P is compact; theorem :: SPPOL_2:63 for G being Special_polygon_in_R2 holds G is compact; theorem :: SPPOL_2:64 P is special_polygonal implies for p1,p2 st p1 <> p2 & p1 in P & p2 in P holds p1,p2 split P; theorem :: SPPOL_2:65 P is special_polygonal implies for p1,p2 st p1 <> p2 & p1 in P & p2 in P ex P1,P2 being Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p1,p2 & P2 is_S-P_arc_joining p1,p2 & P1 /\ P2 = {p1,p2} & P = P1 \/ P2;

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