Volume 14, 2002

University of Bialystok

Copyright (c) 2002 Association of Mizar Users

### The abstract of the Mizar article:

### Some Remarks on Clockwise Oriented Sequences on Go-boards

**by****Adam Naumowicz, and****Robert Milewski**- Received March 1, 2002
- MML identifier: JORDAN1I

- [ Mizar article, MML identifier index ]

environ vocabulary JORDAN8, PSCOMP_1, BOOLE, JORDAN1A, JORDAN9, GOBOARD1, GOBOARD5, JORDAN6, SPRECT_2, JORDAN2C, ABSVALUE, REALSET1, JORDAN1E, FINSEQ_4, TOPS_1, TOPREAL2, CONNSP_1, COMPTS_1, TOPREAL1, SPPOL_1, FINSEQ_1, TREES_1, EUCLID, RELAT_1, MCART_1, MATRIX_1, GOBOARD9, PRE_TOPC, RELAT_2, SEQM_3, SUBSET_1, FUNCT_1, ARYTM_1, FINSEQ_6, GOBOARD2, ARYTM_3, ORDINAL2, FUNCT_5, JORDAN5D; notation TARSKI, XBOOLE_0, SUBSET_1, XREAL_0, REAL_1, NAT_1, ABSVALUE, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_4, MATRIX_1, FINSEQ_6, REALSET1, STRUCT_0, PRE_TOPC, TOPS_1, TOPREAL2, BINARITH, CONNSP_1, COMPTS_1, EUCLID, PSCOMP_1, SPRECT_2, GOBOARD1, TOPREAL1, GOBOARD2, GOBOARD5, GOBOARD9, GOBRD13, SPPOL_1, JORDAN8, JORDAN2C, JORDAN6, JORDAN9, JORDAN5D, JORDAN1A, JORDAN1E; constructors JORDAN8, REALSET1, GOBOARD9, JORDAN6, REAL_1, CARD_4, PSCOMP_1, BINARITH, JORDAN2C, CONNSP_1, JORDAN9, JORDAN1A, ABSVALUE, FINSEQ_4, GOBRD13, TOPREAL2, SPRECT_1, GOBOARD2, INT_1, TOPS_1, JORDAN1E, JORDAN5D; clusters SPRECT_1, TOPREAL6, JORDAN8, REVROT_1, INT_1, RELSET_1, SPRECT_3, GOBOARD2, JORDAN1A, MEMBERED; requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM; begin :: Preliminaries reserve i,j,k,n for Nat; theorem :: JORDAN1I:1 for A,B being Subset of TOP-REAL n holds A is Bounded or B is Bounded implies A /\ B is Bounded; theorem :: JORDAN1I:2 for A,B being Subset of TOP-REAL n holds A is not Bounded & B is Bounded implies A \ B is not Bounded; theorem :: JORDAN1I:3 for C being compact connected non vertical non horizontal Subset of TOP-REAL 2 holds (W-min L~Cage(C,n))..Cage(C,n) > 1; theorem :: JORDAN1I:4 for C being compact connected non vertical non horizontal Subset of TOP-REAL 2 holds (E-max L~Cage(C,n))..Cage(C,n) > 1; theorem :: JORDAN1I:5 for C being compact connected non vertical non horizontal Subset of TOP-REAL 2 holds (S-max L~Cage(C,n))..Cage(C,n) > 1; begin :: On bounding points of circular sequences theorem :: JORDAN1I:6 for f being non constant standard special_circular_sequence, p being Point of TOP-REAL 2 st p in rng f holds left_cell(f,p..f) = left_cell(Rotate(f,p),1); theorem :: JORDAN1I:7 for f being non constant standard special_circular_sequence, p being Point of TOP-REAL 2 st p in rng f holds right_cell(f,p..f) = right_cell(Rotate(f,p),1); theorem :: JORDAN1I:8 for C being compact connected non vertical non horizontal non empty Subset of TOP-REAL 2 holds W-min C in right_cell(Rotate(Cage(C,n),W-min L~Cage(C,n)),1); theorem :: JORDAN1I:9 for C being compact connected non vertical non horizontal non empty Subset of TOP-REAL 2 holds E-max C in right_cell(Rotate(Cage(C,n),E-max L~Cage(C,n)),1); theorem :: JORDAN1I:10 for C being compact connected non vertical non horizontal non empty Subset of TOP-REAL 2 holds S-max C in right_cell(Rotate(Cage(C,n),S-max L~Cage(C,n)),1); begin :: On clockwise oriented sequences theorem :: JORDAN1I:11 for f being clockwise_oriented non constant standard special_circular_sequence for p being Point of TOP-REAL 2 st p`1 < W-bound (L~f) holds p in LeftComp f; theorem :: JORDAN1I:12 for f being clockwise_oriented non constant standard special_circular_sequence for p being Point of TOP-REAL 2 st p`1 > E-bound (L~f) holds p in LeftComp f; theorem :: JORDAN1I:13 for f being clockwise_oriented non constant standard special_circular_sequence for p being Point of TOP-REAL 2 st p`2 < S-bound (L~f) holds p in LeftComp f; theorem :: JORDAN1I:14 for f being clockwise_oriented non constant standard special_circular_sequence for p being Point of TOP-REAL 2 st p`2 > N-bound (L~f) holds p in LeftComp f; theorem :: JORDAN1I:15 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i+1,j] in Indices G & f/.k = G*(i+1,j) & f/.(k+1) = G*(i,j) holds j < width G; theorem :: JORDAN1I:16 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i,j+1] in Indices G & f/.k = G*(i,j) & f/.(k+1) = G*(i,j+1) holds i < len G; theorem :: JORDAN1I:17 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i+1,j] in Indices G & f/.k = G*(i,j) & f/.(k+1) = G*(i+1,j) holds j > 1; theorem :: JORDAN1I:18 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i,j+1] in Indices G & f/.k = G*(i,j+1) & f/.(k+1) = G*(i,j) holds i > 1; theorem :: JORDAN1I:19 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i+1,j] in Indices G & f/.k = G*(i+1,j) & f/.(k+1) = G*(i,j) holds (f/.k)`2 <> N-bound L~f; theorem :: JORDAN1I:20 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i,j+1] in Indices G & f/.k = G*(i,j) & f/.(k+1) = G*(i,j+1) holds (f/.k)`1 <> E-bound L~f; theorem :: JORDAN1I:21 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i+1,j] in Indices G & f/.k = G*(i,j) & f/.(k+1) = G*(i+1,j) holds (f/.k)`2 <> S-bound L~f; theorem :: JORDAN1I:22 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i,j+1] in Indices G & f/.k = G*(i,j+1) & f/.(k+1) = G*(i,j) holds (f/.k)`1 <> W-bound L~f; theorem :: JORDAN1I:23 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board for k being Nat st f is_sequence_on G & 1 <= k & k+1 <= len f & f/.k = W-min L~f ex i,j be Nat st [i,j] in Indices G & [i,j+1] in Indices G & f/.k = G*(i,j) & f/.(k+1) = G*(i,j+1); theorem :: JORDAN1I:24 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board for k being Nat st f is_sequence_on G & 1 <= k & k+1 <= len f & f/.k = N-min L~f ex i,j be Nat st [i,j] in Indices G & [i+1,j] in Indices G & f/.k = G*(i,j) & f/.(k+1) = G*(i+1,j); theorem :: JORDAN1I:25 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board for k being Nat st f is_sequence_on G & 1 <= k & k+1 <= len f & f/.k = E-max L~f ex i,j be Nat st [i,j+1] in Indices G & [i,j] in Indices G & f/.k = G*(i,j+1) & f/.(k+1) = G*(i,j); theorem :: JORDAN1I:26 for f being clockwise_oriented non constant standard special_circular_sequence for G being Go-board for k being Nat st f is_sequence_on G & 1 <= k & k+1 <= len f & f/.k = S-max L~f ex i,j be Nat st [i+1,j] in Indices G & [i,j] in Indices G & f/.k = G*(i+1,j) & f/.(k+1) = G*(i,j); theorem :: JORDAN1I:27 for f being non constant standard special_circular_sequence holds f is clockwise_oriented iff Rotate(f,W-min L~f)/.2 in W-most L~f; theorem :: JORDAN1I:28 for f being non constant standard special_circular_sequence holds f is clockwise_oriented iff Rotate(f,E-max L~f)/.2 in E-most L~f; theorem :: JORDAN1I:29 for f being non constant standard special_circular_sequence holds f is clockwise_oriented iff Rotate(f,S-max L~f)/.2 in S-most L~f; theorem :: JORDAN1I:30 for C being compact non vertical non horizontal non empty being_simple_closed_curve Subset of TOP-REAL 2 for p being Point of TOP-REAL 2 holds p`1 = (W-bound C + E-bound C)/2 & i > 0 & 1 <= k & k <= width Gauge(C,i) & Gauge(C,i)*(Center Gauge(C,i),k) in Upper_Arc L~Cage(C,i) & p`2 = sup(proj2.:(LSeg(Gauge(C,1)*(Center Gauge(C,1),1), Gauge(C,i)*(Center Gauge(C,i),k)) /\ Lower_Arc L~Cage(C,i))) implies ex j st 1 <= j & j <= width Gauge(C,i) & p = Gauge(C,i)*(Center Gauge(C,i),j) ;

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