Volume 9, 1997

University of Bialystok

Copyright (c) 1997 Association of Mizar Users

### The abstract of the Mizar article:

### On Same Equivalents of Well-foundedness

**by****Piotr Rudnicki, and****Andrzej Trybulec**- Received February 25, 1997
- MML identifier: WELLFND1

- [ Mizar article, MML identifier index ]

environ vocabulary PARTFUN1, FRAENKEL, FUNCT_1, RELAT_1, BOOLE, TARSKI, CARD_1, CARD_5, ORDINAL2, ORDERS_1, WELLORD1, WAYBEL_0, SETFAM_1, CAT_1, FUNCT_4, CARD_2, FINSET_1, ORDINAL1, REALSET1, FUNCOP_1, NEWTON, NORMSP_1, WELLFND1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, NAT_1, RELAT_1, FUNCT_1, SETFAM_1, STRUCT_0, RELSET_1, PARTFUN1, FUNCT_2, FUNCT_4, FUNCOP_1, ORDINAL1, WELLORD1, ORDINAL2, FINSET_1, CARD_1, CARD_2, ORDERS_1, FRAENKEL, REALSET1, CQC_LANG, NORMSP_1, CARD_5, RFUNCT_3, WAYBEL_0; constructors NAT_1, WELLORD1, DOMAIN_1, CARD_2, REALSET1, FUNCT_4, CQC_LANG, NORMSP_1, CARD_5, RFUNCT_3, WAYBEL_0, FRAENKEL, MEMBERED; clusters SUBSET_1, STRUCT_0, RELSET_1, ORDERS_1, FRAENKEL, REALSET1, NORMSP_1, CARD_5, ALTCAT_1, WAYBEL_0, FUNCOP_1, CQC_LANG, PARTFUN1, CARD_1, MEMBERED, NUMBERS, ORDINAL2; requirements NUMERALS, SUBSET, BOOLE; begin :: Preliminaries :: General preliminaries ::::::::::::::::::::::::::::::::::::::::::::: definition let A, B be set; cluster PFuncs(A,B) -> functional; end; definition let R be 1-sorted, X be set, p be PartFunc of the carrier of R, X; redefine func dom p -> Subset of R; end; theorem :: WELLFND1:1 for X being set, f, g being Function st f c= g & X c= dom f holds f|X = g|X; theorem :: WELLFND1:2 for X being functional set st for f, g being Function st f in X & g in X holds f tolerates g holds union X is Function; scheme PFSeparation {X, Y() -> set, P[set]}: ex PFS being Subset of PFuncs(X(), Y()) st for pfs being PartFunc of X(), Y() holds pfs in PFS iff P[pfs]; :: Cardinals :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: definition let X be set; cluster nextcard X -> non empty; end; definition cluster regular Aleph; end; theorem :: WELLFND1:3 for M being regular Aleph, X being set st X c= M & Card X in M holds sup X in M; :: Relational structures :::::::::::::::::::::::::::::::::::::::::::::: theorem :: WELLFND1:4 for R being RelStr, x being set holds (the InternalRel of R)-Seg x c= the carrier of R; definition let R be RelStr, X be Subset of R; redefine attr X is lower means :: WELLFND1:def 1 for x, y being set st x in X & [y, x] in the InternalRel of R holds y in X; end; theorem :: WELLFND1:5 for R being RelStr, X being Subset of R, x being set st X is lower & x in X holds (the InternalRel of R)-Seg x c= X; theorem :: WELLFND1:6 for R being RelStr, X being lower Subset of R, Y being Subset of R, x being set st Y = X \/ {x} & (the InternalRel of R)-Seg x c= X holds Y is lower; begin :: Well-founded relational structures definition let R be RelStr; attr R is well_founded means :: WELLFND1:def 2 the InternalRel of R is_well_founded_in the carrier of R; end; definition cluster non empty well_founded RelStr; end; definition let R be RelStr, X be Subset of R; attr X is well_founded means :: WELLFND1:def 3 the InternalRel of R is_well_founded_in X; end; definition let R be RelStr; cluster well_founded Subset of R; end; definition let R be RelStr; func well_founded-Part R -> Subset of R means :: WELLFND1:def 4 it = union {S where S is Subset of R : S is well_founded lower}; end; definition let R be RelStr; cluster well_founded-Part R -> lower well_founded; end; theorem :: WELLFND1:7 for R being non empty RelStr, x be Element of R holds {x} is well_founded Subset of R; theorem :: WELLFND1:8 for R being RelStr, X, Y being well_founded Subset of R st X is lower holds X \/ Y is well_founded Subset of R; theorem :: WELLFND1:9 for R being RelStr holds R is well_founded iff well_founded-Part R = the carrier of R; theorem :: WELLFND1:10 for R being non empty RelStr, x being Element of R st (the InternalRel of R)-Seg x c= well_founded-Part R holds x in well_founded-Part R; :: Well-founded induction ::::::::::::::::::::::::::::::::::::::::::::: scheme WFMin {R() -> non empty RelStr, x() -> Element of R(), P[set]}: ex x being Element of R() st P[x] & not ex y being Element of R() st x <> y & P[y] & [y,x] in the InternalRel of R() provided P[x()] and R() is well_founded; theorem :: WELLFND1:11 :: WF iff WFInduction for R being non empty RelStr holds R is well_founded iff for S being set st for x being Element of R st (the InternalRel of R)-Seg x c= S holds x in S holds the carrier of R c= S; scheme WFInduction {R() -> non empty RelStr, P[set]}: for x being Element of R() holds P[x] provided for x being Element of R() st for y being Element of R() st y <> x & [y,x] in the InternalRel of R() holds P[y] holds P[x] and R() is well_founded; :: Well-foundedness and recursive definitions ::::::::::::::::::::::::: definition let R be non empty RelStr, V be non empty set, H be Function of [:the carrier of R, PFuncs(the carrier of R, V):], V, F be Function; pred F is_recursively_expressed_by H means :: WELLFND1:def 5 for x being Element of R holds F.x = H.[x, F|(the InternalRel of R)-Seg x]; end; theorem :: WELLFND1:12 :: Well foundedness and existence for R being non empty RelStr holds R is well_founded iff for V being non empty set, H being Function of [:the carrier of R, PFuncs(the carrier of R, V):], V ex F being Function of the carrier of R, V st F is_recursively_expressed_by H; theorem :: WELLFND1:13 :: Uniqueness implies well-foundedness for R being non empty RelStr, V being non trivial set st for H being Function of [:the carrier of R, PFuncs(the carrier of R, V):], V, F1, F2 being Function of the carrier of R, V st F1 is_recursively_expressed_by H & F2 is_recursively_expressed_by H holds F1 = F2 holds R is well_founded; theorem :: WELLFND1:14 :: Well-foundedness implies uniqueness for R being non empty well_founded RelStr, V being non empty set, H being Function of [:the carrier of R, PFuncs(the carrier of R, V):], V, F1, F2 being Function of the carrier of R, V st F1 is_recursively_expressed_by H & F2 is_recursively_expressed_by H holds F1 = F2; :: Well-foundedness and omega chains ::::::::::::::::::::::::::::::::: definition let R be RelStr, f be sequence of R; canceled; attr f is descending means :: WELLFND1:def 7 for n being Nat holds f.(n+1) <> f.n & [f.(n+1), f.n] in the InternalRel of R; end; theorem :: WELLFND1:15 :: omega chains for R being non empty RelStr holds R is well_founded iff not ex f being sequence of R st f is descending;

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