Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Rotating and Reversing

**by****Andrzej Trybulec**- Received January 21, 1999
- MML identifier: REVROT_1

- [ Mizar article, MML identifier index ]

environ vocabulary FINSEQ_1, REALSET1, SEQM_3, RELAT_1, FINSEQ_4, FUNCT_1, FINSEQ_5, RFINSEQ, BOOLE, ARYTM_1, FINSEQ_6, EUCLID, JORDAN2C, FINSEQ_2, GOBOARD1, MCART_1, PRE_TOPC, TOPREAL1, GOBOARD2, CARD_1, GOBOARD5, TARSKI, MATRIX_1, ABSVALUE, GOBOARD9, CONNSP_1, SUBSET_1, TOPS_1, PSCOMP_1, SPRECT_2, COMPTS_1, JORDAN5D, TREES_1; notation TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XREAL_0, NAT_1, ABSVALUE, BINARITH, CARD_1, REALSET1, FUNCT_1, FINSEQ_1, FINSEQ_2, MATRIX_1, FINSEQ_4, RFINSEQ, FINSEQ_5, FINSEQ_6, STRUCT_0, PRE_TOPC, TOPS_1, CONNSP_1, COMPTS_1, EUCLID, TOPREAL1, GOBOARD1, GOBOARD2, GOBOARD5, GOBOARD9, PSCOMP_1, JORDAN5D, SPRECT_2, JORDAN2C; constructors TOPS_1, GOBOARD9, SPRECT_2, PSCOMP_1, RFINSEQ, BINARITH, REAL_1, CONNSP_1, GOBOARD2, FINSEQ_4, INT_1, ABSVALUE, JORDAN5D, JORDAN2C, REALSET1, COMPTS_1, FINSEQOP; clusters RELSET_1, SPRECT_1, SPRECT_2, EUCLID, FINSEQ_6, FINSEQ_5, GOBOARD9, GOBOARD2, PSCOMP_1, INT_1, FINSEQ_1, TEX_1, REALSET1, BINARITH, MEMBERED; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; begin ::Preliminaries reserve i,j,k,m,n for Nat, D for non empty set, p for Element of D, f for FinSequence of D; definition let S be non trivial 1-sorted; cluster the carrier of S -> non trivial; end; definition let D be non empty set; let f be FinSequence of D; redefine attr f is constant means :: REVROT_1:def 1 :: GOBOARD1:def 2 for n,m st n in dom f & m in dom f holds f/.n=f/.m; end; theorem :: REVROT_1:1 for D being non empty set, f being FinSequence of D st f just_once_values f/.len f holds f/.len f..f = len f; theorem :: REVROT_1:2 for D being non empty set, f being FinSequence of D holds f /^ len f = {}; theorem :: REVROT_1:3 for D being non empty set, f being non empty FinSequence of D holds f/.len f in rng f; definition let D be non empty set, f be FinSequence of D, y be set; redefine pred f just_once_values y means :: REVROT_1:def 2 ex x being set st x in dom f & y = f/.x & for z being set st z in dom f & z <> x holds f/.z <> y; end; theorem :: REVROT_1:4 for D being non empty set, f being FinSequence of D st f just_once_values f/.len f holds f-:f/.len f = f; theorem :: REVROT_1:5 for D being non empty set, f being FinSequence of D st f just_once_values f/.len f holds f:-f/.len f = <*f/.len f*>; theorem :: REVROT_1:6 1 <= len (f:-p); theorem :: REVROT_1:7 for D being non empty set, p being Element of D, f being FinSequence of D st p in rng f holds len(f:-p) <= len f; theorem :: REVROT_1:8 for D being non empty set, f being circular non empty FinSequence of D holds Rev f is circular; begin :: About Rotation reserve D for non empty set, p for Element of D, f for FinSequence of D; theorem :: REVROT_1:9 p in rng f & 1 <= i & i <= len(f:-p) implies (Rotate(f,p))/.i = f/.(i -' 1 + p..f); theorem :: REVROT_1:10 p in rng f & p..f <= i & i <= len f implies f/.i = (Rotate(f,p))/.(i+1 -' p..f); theorem :: REVROT_1:11 p in rng f implies (Rotate(f,p))/.len(f:-p) = f/.len f; theorem :: REVROT_1:12 p in rng f & len(f:-p) < i & i <= len f implies (Rotate(f,p))/.i = f/.(i + p..f -' len f); theorem :: REVROT_1:13 p in rng f & 1 < i & i <= p..f implies f/.i = (Rotate(f,p))/.(i + len f -' p..f); theorem :: REVROT_1:14 len Rotate(f,p) = len f; theorem :: REVROT_1:15 dom Rotate(f,p) = dom f; theorem :: REVROT_1:16 for D being non empty set, f being circular FinSequence of D, p be Element of D st for i st 1 < i & i < len f holds f/.i <> f/.1 holds Rotate(Rotate(f,p),f/.1) = f; begin :: Rotating circular reserve f for circular FinSequence of D; theorem :: REVROT_1:17 p in rng f & len(f:-p) <= i & i <= len f implies (Rotate(f,p))/.i = f/.(i + p..f -' len f); theorem :: REVROT_1:18 p in rng f & 1 <= i & i <= p..f implies f/.i = (Rotate(f,p))/.(i + len f -' p..f); definition let D be non trivial set; cluster non constant circular FinSequence of D; end; definition let D be non trivial set, p be Element of D; let f be non constant circular FinSequence of D; cluster Rotate(f,p) -> non constant; end; begin :: Finite sequences on the plane theorem :: REVROT_1:19 for n being non empty Nat holds 0.REAL n <> 1.REAL n; definition let n be non empty Nat; cluster TOP-REAL n -> non trivial; end; reserve f,g for FinSequence of TOP-REAL 2; theorem :: REVROT_1:20 rng f c= rng g implies rng X_axis f c= rng X_axis g; theorem :: REVROT_1:21 rng f = rng g implies rng X_axis f = rng X_axis g; theorem :: REVROT_1:22 rng f c= rng g implies rng Y_axis f c= rng Y_axis g; theorem :: REVROT_1:23 rng f = rng g implies rng Y_axis f = rng Y_axis g; begin :: Rotating finite sequences on the plane reserve p for Point of TOP-REAL 2, f for FinSequence of TOP-REAL 2; definition let p be Point of TOP-REAL 2; let f be special circular FinSequence of TOP-REAL 2; cluster Rotate(f,p) -> special; end; theorem :: REVROT_1:24 p in rng f & 1 <= i & i < len(f:-p) implies LSeg(Rotate(f,p),i) = LSeg(f,i -' 1 + p..f); theorem :: REVROT_1:25 p in rng f & p..f <= i & i < len f implies LSeg(f,i) = LSeg(Rotate(f,p),i -' p..f+1); theorem :: REVROT_1:26 for f being circular FinSequence of TOP-REAL 2 holds Incr X_axis f = Incr X_axis Rotate(f,p); theorem :: REVROT_1:27 for f being circular FinSequence of TOP-REAL 2 holds Incr Y_axis f = Incr Y_axis Rotate(f,p); theorem :: REVROT_1:28 for f being non empty circular FinSequence of TOP-REAL 2 holds GoB Rotate(f,p) = GoB f; theorem :: REVROT_1:29 for f being non constant standard special_circular_sequence holds Rev Rotate(f,p) = Rotate(Rev f,p); begin :: Circular finite sequences of points of the plane reserve f for circular FinSequence of TOP-REAL 2; theorem :: REVROT_1:30 for f being circular s.c.c. FinSequence of TOP-REAL 2 st len f > 4 holds LSeg(f,len f -' 1) /\ LSeg(f,1) = {f/.1}; theorem :: REVROT_1:31 p in rng f & len(f:-p) <= i & i < len f implies LSeg(Rotate(f,p),i) = LSeg(f,i + p..f -' len f); definition let p be Point of TOP-REAL 2; let f be circular s.c.c. FinSequence of TOP-REAL 2; cluster Rotate(f,p) -> s.c.c.; end; definition let p be Point of TOP-REAL 2; let f be non constant standard special_circular_sequence; cluster Rotate(f,p) -> unfolded; end; theorem :: REVROT_1:32 p in rng f & 1 <= i & i < p..f implies LSeg(f,i) = LSeg(Rotate(f,p),i + len f -' p..f); theorem :: REVROT_1:33 L~Rotate(f,p) = L~f; theorem :: REVROT_1:34 for G being Go-board holds f is_sequence_on G iff Rotate(f,p) is_sequence_on G; definition let p be Point of TOP-REAL 2; let f be standard (non empty circular FinSequence of TOP-REAL 2); cluster Rotate(f,p) -> standard; end; theorem :: REVROT_1:35 for f being non constant standard special_circular_sequence, p,k st p in rng f & 1 <= k & k < p..f holds left_cell(f,k) = left_cell(Rotate(f,p),k + len f -' p..f); theorem :: REVROT_1:36 for f being non constant standard special_circular_sequence holds LeftComp Rotate(f,p) = LeftComp f; theorem :: REVROT_1:37 for f being non constant standard special_circular_sequence holds RightComp Rotate(f,p) = RightComp f; begin definition let p be Point of TOP-REAL 2; let f be clockwise_oriented (non constant standard special_circular_sequence); cluster Rotate(f,p) -> clockwise_oriented; end; theorem :: REVROT_1:38 for f being non constant standard special_circular_sequence holds f is clockwise_oriented or Rev f is clockwise_oriented;

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