Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### The Class of Series-Parallel Graphs. Part II

**by****Krzysztof Retel**- Received May 29, 2003
- MML identifier: NECKLA_2

- [ Mizar article, MML identifier index ]

environ vocabulary NECKLA_2, NECKLACE, CLASSES1, CLASSES2, ORDERS_1, RELAT_1, REALSET1, FUNCT_1, PRE_TOPC, BOOLE, SETFAM_1, CANTOR_1, CARD_1, PROB_1, SQUARE_1, ARYTM, FINSET_1, ORDINAL2, ORDINAL1; notation TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, ZFMISC_1, RELSET_1, FINSET_1, CARD_1, ORDINAL1, ORDINAL2, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, STRUCT_0, GROUP_1, REALSET1, SQUARE_1, ORDERS_1, RELAT_1, NECKLACE, PRE_TOPC, FUNCT_1, FUNCT_2, CLASSES2, SETFAM_1, CANTOR_1, CLASSES1, PROB_1, CARD_LAR; constructors ENUMSET1, NECKLACE, NAT_1, CLASSES2, CANTOR_1, GROUP_1, PROB_1, SQUARE_1, COHSP_1, CARD_LAR, REALSET1, MEMBERED; clusters RELSET_1, ORDERS_1, STRUCT_0, FINSET_1, NAT_1, NECKLACE, ORDINAL1, CLASSES2, YELLOW13, XREAL_0, COHSP_1, XBOOLE_0, CARD_1, MEMBERED, NUMBERS, ORDINAL2; requirements BOOLE, SUBSET, REAL, NUMERALS, ARITHM; begin reserve U for Universe; theorem :: NECKLA_2:1 for X,Y being set st X in U & Y in U for R being Relation of X,Y holds R in U; theorem :: NECKLA_2:2 the InternalRel of Necklace 4 = {[0,1],[1,0],[1,2],[2,1],[2,3],[3,2]}; definition let n be natural number; cluster -> finite Element of Rank n; end; theorem :: NECKLA_2:3 for x be set st x in FinSETS holds x is finite; definition cluster -> finite Element of FinSETS; end; definition cluster finite ordinal -> natural number; end; definition let G be non empty RelStr; attr G is N-free means :: NECKLA_2:def 1 not G embeds Necklace 4; end; definition cluster strict finite N-free (non empty RelStr); end; definition let R,S be RelStr; func union_of(R,S) -> strict RelStr means :: NECKLA_2:def 2 the carrier of it = (the carrier of R) \/ (the carrier of S) & the InternalRel of it = (the InternalRel of R) \/ (the InternalRel of S); end; definition let R, S be RelStr; func sum_of(R,S) -> strict RelStr means :: NECKLA_2:def 3 the carrier of it = (the carrier of R) \/ (the carrier of S) & the InternalRel of it = (the InternalRel of R) \/ (the InternalRel of S) \/ [:the carrier of R, the carrier of S:] \/ [:the carrier of S, the carrier of R:]; end; definition func fin_RelStr means :: NECKLA_2:def 4 for X being set holds X in it iff ex R being strict RelStr st X = R & the carrier of R in FinSETS; end; definition cluster fin_RelStr -> non empty; end; definition func fin_RelStr_sp -> Subset of fin_RelStr means :: NECKLA_2:def 5 (for R be strict RelStr st the carrier of R is non empty trivial & the carrier of R in FinSETS holds R in it) & (for H1,H2 be strict RelStr st (the carrier of H1) misses (the carrier of H2) & H1 in it & H2 in it holds union_of(H1,H2) in it & sum_of(H1,H2) in it) & for M be Subset of fin_RelStr st ( (for R be strict RelStr st the carrier of R is non empty trivial & the carrier of R in FinSETS holds R in M) & for H1,H2 be strict RelStr st (the carrier of H1) misses (the carrier of H2) & H1 in M & H2 in M holds union_of(H1,H2) in M & sum_of(H1,H2) in M ) holds it c= M; end; definition cluster fin_RelStr_sp -> non empty; end; theorem :: NECKLA_2:4 for X being set st X in fin_RelStr_sp holds X is finite strict non empty RelStr; theorem :: NECKLA_2:5 for R being RelStr st R in fin_RelStr_sp holds (the carrier of R) in FinSETS; theorem :: NECKLA_2:6 for X being set st X in fin_RelStr_sp holds X is strict non empty trivial RelStr or ex H1,H2 being strict RelStr st (the carrier of H1) misses (the carrier of H2) & H1 in fin_RelStr_sp & H2 in fin_RelStr_sp & (X = union_of(H1,H2) or X = sum_of(H1,H2) ); theorem :: NECKLA_2:7 for R being strict non empty RelStr st R in fin_RelStr_sp holds R is N-free;

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