:: UNIALG_1 semantic presentation begin definition attrc1 is strict ; struct UAStr -> 1-sorted ; aggrUAStr(# carrier, charact #) -> UAStr ; sel charact c1 -> PFuncFinSequence of the carrier of c1; end; registration cluster non empty strict for UAStr ; existence ex b1 being UAStr st ( not b1 is empty & b1 is strict ) proof set D = the non empty set ; set c = the PFuncFinSequence of the non empty set ; take UAStr(# the non empty set , the PFuncFinSequence of the non empty set #) ; ::_thesis: ( not UAStr(# the non empty set , the PFuncFinSequence of the non empty set #) is empty & UAStr(# the non empty set , the PFuncFinSequence of the non empty set #) is strict ) thus not the carrier of UAStr(# the non empty set , the PFuncFinSequence of the non empty set #) is empty ; :: according to STRUCT_0:def_1 ::_thesis: UAStr(# the non empty set , the PFuncFinSequence of the non empty set #) is strict thus UAStr(# the non empty set , the PFuncFinSequence of the non empty set #) is strict ; ::_thesis: verum end; end; registration let D be non empty set ; let c be PFuncFinSequence of D; cluster UAStr(# D,c #) -> non empty ; coherence not UAStr(# D,c #) is empty ; end; definition let IT be UAStr ; attrIT is partial means :Def1: :: UNIALG_1:def 1 the charact of IT is homogeneous ; attrIT is quasi_total means :Def2: :: UNIALG_1:def 2 the charact of IT is quasi_total ; attrIT is non-empty means :Def3: :: UNIALG_1:def 3 ( the charact of IT <> {} & the charact of IT is non-empty ); end; :: deftheorem Def1 defines partial UNIALG_1:def_1_:_ for IT being UAStr holds ( IT is partial iff the charact of IT is homogeneous ); :: deftheorem Def2 defines quasi_total UNIALG_1:def_2_:_ for IT being UAStr holds ( IT is quasi_total iff the charact of IT is quasi_total ); :: deftheorem Def3 defines non-empty UNIALG_1:def_3_:_ for IT being UAStr holds ( IT is non-empty iff ( the charact of IT <> {} & the charact of IT is non-empty ) ); registration cluster non empty strict partial quasi_total non-empty for UAStr ; existence ex b1 being UAStr st ( b1 is quasi_total & b1 is partial & b1 is non-empty & b1 is strict & not b1 is empty ) proof set A = the non empty set ; set a = the Element of the non empty set ; reconsider w = (<*> the non empty set ) .--> the Element of the non empty set as Element of PFuncs (( the non empty set *), the non empty set ) by MARGREL1:19; set U1 = UAStr(# the non empty set ,<*w*> #); take UAStr(# the non empty set ,<*w*> #) ; ::_thesis: ( UAStr(# the non empty set ,<*w*> #) is quasi_total & UAStr(# the non empty set ,<*w*> #) is partial & UAStr(# the non empty set ,<*w*> #) is non-empty & UAStr(# the non empty set ,<*w*> #) is strict & not UAStr(# the non empty set ,<*w*> #) is empty ) A1: ( the charact of UAStr(# the non empty set ,<*w*> #) is non-empty & the charact of UAStr(# the non empty set ,<*w*> #) <> {} ) by MARGREL1:20; ( the charact of UAStr(# the non empty set ,<*w*> #) is quasi_total & the charact of UAStr(# the non empty set ,<*w*> #) is homogeneous ) by MARGREL1:20; hence ( UAStr(# the non empty set ,<*w*> #) is quasi_total & UAStr(# the non empty set ,<*w*> #) is partial & UAStr(# the non empty set ,<*w*> #) is non-empty & UAStr(# the non empty set ,<*w*> #) is strict & not UAStr(# the non empty set ,<*w*> #) is empty ) by A1, Def1, Def2, Def3; ::_thesis: verum end; end; registration let U1 be partial UAStr ; cluster the charact of U1 -> homogeneous ; coherence the charact of U1 is homogeneous by Def1; end; registration let U1 be quasi_total UAStr ; cluster the charact of U1 -> quasi_total ; coherence the charact of U1 is quasi_total by Def2; end; registration let U1 be non-empty UAStr ; cluster the charact of U1 -> non-empty non empty ; coherence ( the charact of U1 is non-empty & not the charact of U1 is empty ) by Def3; end; definition mode Universal_Algebra is non empty partial quasi_total non-empty UAStr ; end; theorem Th1: :: UNIALG_1:1 for n being Nat for U1 being non empty partial non-empty UAStr st n in dom the charact of U1 holds the charact of U1 . n is PartFunc of ( the carrier of U1 *), the carrier of U1 proof let n be Nat; ::_thesis: for U1 being non empty partial non-empty UAStr st n in dom the charact of U1 holds the charact of U1 . n is PartFunc of ( the carrier of U1 *), the carrier of U1 let U1 be non empty partial non-empty UAStr ; ::_thesis: ( n in dom the charact of U1 implies the charact of U1 . n is PartFunc of ( the carrier of U1 *), the carrier of U1 ) set o = the charact of U1; assume n in dom the charact of U1 ; ::_thesis: the charact of U1 . n is PartFunc of ( the carrier of U1 *), the carrier of U1 then A1: the charact of U1 . n in rng the charact of U1 by FUNCT_1:def_3; rng the charact of U1 c= PFuncs (( the carrier of U1 *), the carrier of U1) by FINSEQ_1:def_4; hence the charact of U1 . n is PartFunc of ( the carrier of U1 *), the carrier of U1 by A1, PARTFUN1:47; ::_thesis: verum end; definition let U1 be non empty partial non-empty UAStr ; func signature U1 -> FinSequence of NAT means :: UNIALG_1:def 4 ( len it = len the charact of U1 & ( for n being Nat st n in dom it holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds it . n = arity h ) ); existence ex b1 being FinSequence of NAT st ( len b1 = len the charact of U1 & ( for n being Nat st n in dom b1 holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds b1 . n = arity h ) ) proof defpred S1[ Nat, set ] means for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . \$1 holds \$2 = arity h; A1: now__::_thesis:_for_m_being_Nat_st_m_in_Seg_(len_the_charact_of_U1)_holds_ ex_n_being_Element_of_NAT_st_S1[m,n] let m be Nat; ::_thesis: ( m in Seg (len the charact of U1) implies ex n being Element of NAT st S1[m,n] ) assume m in Seg (len the charact of U1) ; ::_thesis: ex n being Element of NAT st S1[m,n] then m in dom the charact of U1 by FINSEQ_1:def_3; then reconsider H = the charact of U1 . m as non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 by Th1; reconsider n = arity H as Element of NAT ; take n = n; ::_thesis: S1[m,n] thus S1[m,n] ; ::_thesis: verum end; consider p being FinSequence of NAT such that A2: dom p = Seg (len the charact of U1) and A3: for m being Nat st m in Seg (len the charact of U1) holds S1[m,p . m] from FINSEQ_1:sch_5(A1); take p ; ::_thesis: ( len p = len the charact of U1 & ( for n being Nat st n in dom p holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds p . n = arity h ) ) thus len p = len the charact of U1 by A2, FINSEQ_1:def_3; ::_thesis: for n being Nat st n in dom p holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds p . n = arity h let n be Nat; ::_thesis: ( n in dom p implies for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds p . n = arity h ) assume A4: n in dom p ; ::_thesis: for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds p . n = arity h let h be non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1; ::_thesis: ( h = the charact of U1 . n implies p . n = arity h ) assume h = the charact of U1 . n ; ::_thesis: p . n = arity h hence p . n = arity h by A2, A3, A4; ::_thesis: verum end; uniqueness for b1, b2 being FinSequence of NAT st len b1 = len the charact of U1 & ( for n being Nat st n in dom b1 holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds b1 . n = arity h ) & len b2 = len the charact of U1 & ( for n being Nat st n in dom b2 holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds b2 . n = arity h ) holds b1 = b2 proof let x, y be FinSequence of NAT ; ::_thesis: ( len x = len the charact of U1 & ( for n being Nat st n in dom x holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds x . n = arity h ) & len y = len the charact of U1 & ( for n being Nat st n in dom y holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds y . n = arity h ) implies x = y ) assume that A5: len x = len the charact of U1 and A6: for n being Nat st n in dom x holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds x . n = arity h and A7: len y = len the charact of U1 and A8: for n being Nat st n in dom y holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds y . n = arity h ; ::_thesis: x = y now__::_thesis:_for_m_being_Nat_st_1_<=_m_&_m_<=_len_x_holds_ x_._m_=_y_._m let m be Nat; ::_thesis: ( 1 <= m & m <= len x implies x . m = y . m ) assume ( 1 <= m & m <= len x ) ; ::_thesis: x . m = y . m then A9: m in Seg (len x) by FINSEQ_1:1; then m in dom the charact of U1 by A5, FINSEQ_1:def_3; then reconsider h = the charact of U1 . m as non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 by Th1; m in dom x by A9, FINSEQ_1:def_3; then A10: x . m = arity h by A6; m in dom y by A5, A7, A9, FINSEQ_1:def_3; hence x . m = y . m by A8, A10; ::_thesis: verum end; hence x = y by A5, A7, FINSEQ_1:14; ::_thesis: verum end; end; :: deftheorem defines signature UNIALG_1:def_4_:_ for U1 being non empty partial non-empty UAStr for b2 being FinSequence of NAT holds ( b2 = signature U1 iff ( len b2 = len the charact of U1 & ( for n being Nat st n in dom b2 holds for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds b2 . n = arity h ) ) ); begin registration let U0 be Universal_Algebra; cluster the charact of U0 -> Function-yielding ; coherence the charact of U0 is Function-yielding ; end;