Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Scott-Continuous Functions. Part II

**by****Adam Grabowski**- Received June 22, 1999
- MML identifier: WAYBEL24

- [ Mizar article, MML identifier index ]

environ vocabulary WAYBEL_0, WAYBEL11, WAYBEL_9, WAYBEL17, LATTICES, ORDINAL2, PRE_TOPC, YELLOW_1, ORDERS_1, RELAT_2, SEQM_3, FUNCT_1, RELAT_1, FUNCOP_1, LATTICE3, BOOLE, GROUP_1, BHSP_3, YELLOW_0, FUNCT_2, BORSUK_1, FUNCT_5, MCART_1, QUANTAL1, CAT_1, MONOID_0, WAYBEL_3, PBOOLE, CARD_3, FUNCT_4, WAYBEL24; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, RELAT_1, FUNCT_1, FUNCT_2, FUNCT_5, SEQM_3, MONOID_0, CARD_3, FUNCOP_1, BORSUK_1, PRE_TOPC, ORDERS_1, LATTICE3, PBOOLE, FUNCT_7, YELLOW_0, ORDERS_3, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_3, WAYBEL_9, WAYBEL11, WAYBEL17, YELLOW_3; constructors FUNCT_7, SEQM_3, ORDERS_3, WAYBEL_3, WAYBEL_5, WAYBEL_1, WAYBEL17, MONOID_0, BORSUK_1; clusters STRUCT_0, WAYBEL_0, WAYBEL_3, RELSET_1, YELLOW_1, LATTICE3, BORSUK_2, WAYBEL10, WAYBEL11, WAYBEL17, YELLOW_3, YELLOW_0, MONOID_0, SUBSET_1, FUNCT_2, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Preliminaries theorem :: WAYBEL24:1 for S, T being up-complete Scott TopLattice for M being Subset of SCMaps (S,T) holds "\/" (M, SCMaps (S,T)) is continuous map of S, T; definition let S be non empty RelStr, T be non empty reflexive RelStr; cluster constant -> monotone map of S, T; end; definition let S be non empty RelStr, T be reflexive non empty RelStr, a be Element of T; cluster S --> a -> monotone; end; theorem :: WAYBEL24:2 for S being non empty RelStr, T being lower-bounded antisymmetric reflexive non empty RelStr holds Bottom MonMaps(S, T) = S --> Bottom T; theorem :: WAYBEL24:3 for S being non empty RelStr, T being upper-bounded antisymmetric reflexive non empty RelStr holds Top MonMaps(S, T) = S --> Top T; theorem :: WAYBEL24:4 for S, T being complete LATTICE, f being monotone map of S, T holds for x being Element of S holds f.x = sup (f.:downarrow x); theorem :: WAYBEL24:5 for S, T being complete lower-bounded LATTICE, f being monotone map of S, T holds ( for x being Element of S holds f.x = "\/"({ f.w where w is Element of S : w <= x },T) ); theorem :: WAYBEL24:6 for S being RelStr, T being non empty RelStr, F being Subset of (T |^ the carrier of S) holds sup F is map of S, T; begin :: On the Scott continuity of maps definition let X1, X2, Y be non empty RelStr; let f be map of [:X1, X2:], Y; let x be Element of X1; func Proj (f, x) -> map of X2, Y equals :: WAYBEL24:def 1 (curry f).x; end; reserve X1, X2, Y for non empty RelStr, f for map of [:X1, X2:], Y, x for Element of X1, y for Element of X2; theorem :: WAYBEL24:7 for y being Element of X2 holds (Proj (f, x)). y = f. [x, y]; definition let X1, X2, Y be non empty RelStr; let f be map of [:X1, X2:], Y; let y be Element of X2; func Proj (f, y) -> map of X1, Y equals :: WAYBEL24:def 2 (curry' f).y; end; theorem :: WAYBEL24:8 for x being Element of X1 holds (Proj (f, y)). x = f. [x, y]; theorem :: WAYBEL24:9 for R, S, T being non empty RelStr, f being map of [:R,S:], T, a being Element of R, b being Element of S holds Proj (f, a). b = Proj (f, b). a; definition let S be non empty RelStr, T be non empty reflexive RelStr; cluster antitone map of S, T; end; theorem :: WAYBEL24:10 for R, S, T being non empty reflexive RelStr, f being map of [:R,S:], T, a being Element of R, b being Element of S st f is monotone holds Proj (f, a) is monotone & Proj (f, b) is monotone; theorem :: WAYBEL24:11 for R, S, T being non empty reflexive RelStr, f being map of [:R,S:], T, a being Element of R, b being Element of S st f is antitone holds Proj (f, a) is antitone & Proj (f, b) is antitone; definition let R, S, T be non empty reflexive RelStr; let f be monotone map of [:R, S:], T; let a be Element of R; cluster Proj (f, a) -> monotone; end; definition let R, S, T be non empty reflexive RelStr; let f be monotone map of [:R, S:], T; let b be Element of S; cluster Proj (f, b) -> monotone; end; definition let R, S, T be non empty reflexive RelStr; let f be antitone map of [:R, S:], T; let a be Element of R; cluster Proj (f, a) -> antitone; end; definition let R, S, T be non empty reflexive RelStr; let f be antitone map of [:R, S:], T; let b be Element of S; cluster Proj (f, b) -> antitone; end; theorem :: WAYBEL24:12 for R, S, T being LATTICE, f being map of [:R,S:], T st ( for a being Element of R, b being Element of S holds Proj (f, a) is monotone & Proj (f, b) is monotone) holds f is monotone; theorem :: WAYBEL24:13 for R, S, T being LATTICE, f being map of [:R,S:], T st ( for a being Element of R, b being Element of S holds Proj (f, a) is antitone & Proj (f, b) is antitone) holds f is antitone; theorem :: WAYBEL24:14 for R, S, T being LATTICE, f being map of [:R,S:], T, b being Element of S, X being Subset of R holds Proj (f, b).:X = f.:[:X, {b}:]; theorem :: WAYBEL24:15 for R, S, T being LATTICE, f being map of [:R,S:], T, b being Element of R, X being Subset of S holds Proj (f, b).:X = f.:[:{b}, X:]; theorem :: WAYBEL24:16 :: Lemma 2.9 p. 116 (1) => (2) for R, S, T being LATTICE, f being map of [:R,S:], T, a being Element of R, b being Element of S st f is directed-sups-preserving holds Proj (f, a) is directed-sups-preserving & Proj (f, b) is directed-sups-preserving; theorem :: WAYBEL24:17 for R, S, T being LATTICE, f being monotone map of [:R, S:], T, a being Element of R, b being Element of S, X being directed Subset of [:R, S:] st ex_sup_of f.:X, T & a in proj1 X & b in proj2 X holds f. [a, b] <= sup (f.:X); theorem :: WAYBEL24:18 :: Lemma 2.9 p. 116 (2) => (1) for R, S, T being complete LATTICE, f being map of [:R, S:], T st ( for a being Element of R, b being Element of S holds Proj (f, a) is directed-sups-preserving & Proj (f, b) is directed-sups-preserving ) holds f is directed-sups-preserving; theorem :: WAYBEL24:19 for S being non empty 1-sorted, T being non empty RelStr, f be set holds f is Element of (T |^ the carrier of S) iff f is map of S, T; begin :: The poset of continuous maps definition let S be TopStruct, T be non empty TopRelStr; func ContMaps (S, T) -> strict RelStr means :: WAYBEL24:def 3 it is full SubRelStr of T |^ the carrier of S & for x being set holds x in the carrier of it iff ex f being map of S, T st x = f & f is continuous; end; definition let S be non empty TopSpace, T be non empty TopSpace-like TopRelStr; cluster ContMaps (S, T) -> non empty; end; definition let S be non empty TopSpace, T be non empty TopSpace-like TopRelStr; cluster ContMaps (S, T) -> constituted-Functions; end; theorem :: WAYBEL24:20 for S being non empty TopSpace, T being non empty reflexive TopSpace-like TopRelStr, x, y being Element of ContMaps (S, T) holds x <= y iff for i being Element of S holds [x.i, y.i] in the InternalRel of T; theorem :: WAYBEL24:21 for S being non empty TopSpace, T being non empty reflexive TopSpace-like TopRelStr, x being set holds x is continuous map of S, T iff x is Element of ContMaps (S, T); definition let S be non empty TopSpace, T be non empty reflexive TopSpace-like TopRelStr; cluster ContMaps (S, T) -> reflexive; end; definition let S be non empty TopSpace, T be non empty transitive TopSpace-like TopRelStr; cluster ContMaps (S, T) -> transitive; end; definition let S be non empty TopSpace, T be non empty antisymmetric TopSpace-like TopRelStr; cluster ContMaps (S, T) -> antisymmetric; end; definition let S be non empty 1-sorted, T be non empty TopSpace-like TopRelStr; cluster T |^ the carrier of S -> constituted-Functions; end; theorem :: WAYBEL24:22 for S being non empty 1-sorted, T being complete LATTICE for f, g, h being map of S, T, i being Element of S st h = "\/" ({f, g}, T |^ the carrier of S) holds h.i = sup {f.i, g.i}; theorem :: WAYBEL24:23 for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is complete LATTICE for X being Subset of product J, i being Element of I holds (inf X).i = inf pi(X,i); theorem :: WAYBEL24:24 for S being non empty 1-sorted, T being complete LATTICE for f, g, h being map of S, T, i being Element of S st h = "/\" ({f, g}, T |^ the carrier of S) holds h.i = inf {f.i, g.i}; definition let S be non empty 1-sorted, T be LATTICE; cluster -> Function-like Relation-like Element of (T |^ the carrier of S); end; definition let S, T be TopLattice; cluster -> Function-like Relation-like Element of ContMaps (S, T); end; theorem :: WAYBEL24:25 for S being non empty RelStr, T being complete LATTICE for F being non empty Subset of (T |^ the carrier of S), i being Element of S holds (sup F).i = "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ); theorem :: WAYBEL24:26 for S, T being complete TopLattice for F being non empty Subset of ContMaps (S, T), i being Element of S holds "\/" (F, (T |^ the carrier of S)).i = "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ); reserve S for non empty RelStr, T for complete LATTICE; theorem :: WAYBEL24:27 for F being non empty Subset of (T |^ the carrier of S), D being non empty Subset of S holds (sup F).:D = { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S : i in D }; theorem :: WAYBEL24:28 for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T), D being non empty Subset of S holds ("\/" (F, (T |^ the carrier of S))).:D = { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S : i in D }; scheme FraenkelF'RSS{ B() -> non empty RelStr, F(set) -> set, G(set) -> set, P[set] } : { F(v1) where v1 is Element of B() : P[v1] } = { G(v2) where v2 is Element of B() : P[v2] } provided for v being Element of B() st P[v] holds F(v) = G(v); theorem :: WAYBEL24:29 for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T) holds "\/" (F, (T |^ the carrier of S)) is monotone map of S, T; theorem :: WAYBEL24:30 for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T), D being directed non empty Subset of S holds "\/"({ "\/"({g.i where i is Element of S : i in D }, T ) where g is Element of (T |^ the carrier of S) : g in F }, T ) = "\/"({ "\/" ({g'.i' where g' is Element of (T |^ the carrier of S) : g' in F }, T ) where i' is Element of S : i' in D }, T); theorem :: WAYBEL24:31 for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T), D being directed non empty Subset of S holds "\/" (("\/"(F, (T |^ the carrier of S))).:D, T) = "\/" (F, (T |^ the carrier of S)).sup D; theorem :: WAYBEL24:32 for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T) holds "\/"(F, (T |^ the carrier of S)) in the carrier of ContMaps (S, T); theorem :: WAYBEL24:33 for S being non empty RelStr, T being lower-bounded antisymmetric non empty RelStr holds Bottom (T |^ the carrier of S) = S --> Bottom T; theorem :: WAYBEL24:34 for S being non empty RelStr, T being upper-bounded antisymmetric non empty RelStr holds Top (T |^ the carrier of S) = S --> Top T; definition let S be non empty reflexive RelStr, T be complete LATTICE, a be Element of T; cluster S --> a -> directed-sups-preserving; end; theorem :: WAYBEL24:35 :: Lemma 2.4, p. 115 for S, T being complete Scott TopLattice holds ContMaps (S, T) is sups-inheriting SubRelStr of (T |^ the carrier of S); definition let S, T be complete Scott TopLattice; cluster ContMaps (S, T) -> complete; end; theorem :: WAYBEL24:36 for S, T being non empty Scott complete TopLattice holds Bottom ContMaps (S, T) = S --> Bottom T; theorem :: WAYBEL24:37 for S, T being non empty Scott complete TopLattice holds Top ContMaps (S, T) = S --> Top T; theorem :: WAYBEL24:38 for S, T being Scott complete TopLattice holds SCMaps (S, T) = ContMaps (S, T);

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