:: Many Sorted Algebras
:: by Andrzej Trybulec
::
:: Received April 21, 1994
:: Copyright (c) 1994-2021 Association of Mizar Users

definition

attr c₁ is strict ;
struct ManySortedSign -> 2-sorted ;
aggr ManySortedSign(# carrier, carrier', Arity, ResultSort #) -> ManySortedSign ;
sel Arity c₁ -> Function of the carrier' of c₁,( the carrier of c₁ *);
sel ResultSort c₁ -> Function of the carrier' of c₁, the carrier of c₁;

end;

registration

cluster non empty void strict for ManySortedSign ;

existence
ex b₁ being ManySortedSign st
( b₁ is void & b₁ is strict & not b₁ is empty )

proof end;

cluster non empty non void strict for ManySortedSign ;

existence
ex b₁ being ManySortedSign st
( not b₁ is void & b₁ is strict & not b₁ is empty )

proof end;

end;

definition

let S be non empty ManySortedSign ;
mode SortSymbol of S is Element of S;
mode OperSymbol of S is Element of the carrier' of S;

end;

definition

let S be non empty non void ManySortedSign ;
let o be OperSymbol of S;

func the_arity_of o -> Element of the carrier of S * equals :: MSUALG_1:def 1
the Arity of S . o;

coherence
the Arity of S . o is Element of the carrier of S * ;

func the_result_sort_of o -> Element of S equals :: MSUALG_1:def 2
the ResultSort of S . o;

coherence
the ResultSort of S . o is Element of S ;

end;

:: deftheorem defines the_arity_of MSUALG_1:def 1 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S holds the_arity_of o = the Arity of S . o;

:: deftheorem defines the_result_sort_of MSUALG_1:def 2 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S holds the_result_sort_of o = the ResultSort of S . o;

definition

let S be 1-sorted ;
attr c₂ is strict ;
struct many-sorted over S -> ;
aggr many-sorted(# Sorts #) -> many-sorted over S;
sel Sorts c₂ -> ManySortedSet of the carrier of S;

end;

definition

let S be non empty ManySortedSign ;
attr c₂ is strict ;
struct MSAlgebra over S -> many-sorted over S;
aggr MSAlgebra(# Sorts, Charact #) -> MSAlgebra over S;
sel Charact c₂ -> ManySortedFunction of ( the Sorts of c₂ #) * the Arity of S, the Sorts of c₂ * the ResultSort of S;

end;

definition

let S be 1-sorted ;
let A be many-sorted over S;

attr A is non-empty means :Def3: :: MSUALG_1:def 3
the Sorts of A is V5();

end;

:: deftheorem Def3 defines non-empty MSUALG_1:def 3 :
for S being 1-sorted
for A being many-sorted over S holds
( A is non-empty iff the Sorts of A is V5() );

registration

let S be non empty ManySortedSign ;

cluster strict non-empty for MSAlgebra over S;

existence
ex b₁ being MSAlgebra over S st
( b₁ is strict & b₁ is non-empty )

proof end;

end;

registration

let S be 1-sorted ;

cluster strict non-empty for many-sorted over S;

existence
ex b₁ being many-sorted over S st
( b₁ is strict & b₁ is non-empty )

proof end;

end;

registration

let S be 1-sorted ;
let A be non-empty many-sorted over S;

cluster the Sorts of A -> V5() ;

coherence
the Sorts of A is non-empty by Def3;

end;

registration

let S be non empty ManySortedSign ;
let A be non-empty MSAlgebra over S;

cluster -> non empty for Element of rng the Sorts of A;

coherence
for b₁ being Component of the Sorts of A holds not b₁ is empty

proof end;

cluster -> non empty for Element of rng ( the Sorts of A #);

coherence
for b₁ being Component of ( the Sorts of A #) holds not b₁ is empty

proof end;

end;

definition

let S be non empty non void ManySortedSign ;
let o be OperSymbol of S;
let A be MSAlgebra over S;

func Args (o,A) -> Component of ( the Sorts of A #) equals :: MSUALG_1:def 4
(( the Sorts of A #) * the Arity of S) . o;

coherence
(( the Sorts of A #) * the Arity of S) . o is Component of ( the Sorts of A #)

proof end;

correctness
;

func Result (o,A) -> Component of the Sorts of A equals :: MSUALG_1:def 5
( the Sorts of A * the ResultSort of S) . o;

coherence
( the Sorts of A * the ResultSort of S) . o is Component of the Sorts of A

proof end;

correctness
;

end;

:: deftheorem defines Args MSUALG_1:def 4 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra over S holds Args (o,A) = (( the Sorts of A #) * the Arity of S) . o;

:: deftheorem defines Result MSUALG_1:def 5 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra over S holds Result (o,A) = ( the Sorts of A * the ResultSort of S) . o;

definition

let S be non empty non void ManySortedSign ;
let o be OperSymbol of S;
let A be MSAlgebra over S;

func Den (o,A) -> Function of (Args (o,A)),(Result (o,A)) equals :: MSUALG_1:def 6
the Charact of A . o;

coherence
the Charact of A . o is Function of (Args (o,A)),(Result (o,A)) by PBOOLE:def 15;

end;

:: deftheorem defines Den MSUALG_1:def 6 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra over S holds Den (o,A) = the Charact of A . o;

theorem :: MSUALG_1:1

for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being non-empty MSAlgebra over S holds not Den (o,A) is empty ;

Lm1: for D being non empty set
for h being non empty homogeneous quasi_total PartFunc of (D *),D holds dom h = (arity h) -tuples_on D

proof end;

theorem Th2: :: MSUALG_1:2

for C being set
for A, B being non empty set
for F being PartFunc of C,A
for G being Function of A,B holds G * F is Function of (dom F),B

proof end;

theorem Th3: :: MSUALG_1:3

for D being non empty set
for h being non empty homogeneous quasi_total PartFunc of (D *),D holds dom h = Funcs ((Seg (arity h)),D)

proof end;

theorem Th4: :: MSUALG_1:4

for A being Universal_Algebra holds not signature A is empty

proof end;

definition

let IT be ManySortedSign ;

attr IT is segmental means :Def7: :: MSUALG_1:def 7
ex n being Nat st the carrier' of IT = Seg n;

end;

:: deftheorem Def7 defines segmental MSUALG_1:def 7 :
for IT being ManySortedSign holds
( IT is segmental iff ex n being Nat st the carrier' of IT = Seg n );

theorem Th5: :: MSUALG_1:5

for S being non empty ManySortedSign st S is trivial holds
for A being MSAlgebra over S
for c1, c2 being Component of the Sorts of A holds c1 = c2

proof end;

reconsider z = 0 as Element of {0} by TARSKI:def 1;

Lm2: for A being Universal_Algebra
for f being Function of (dom (signature A)),({0} *) holds
( ManySortedSign(# {0},(dom (signature A)),f,((dom (signature A)) --> z) #) is segmental & ManySortedSign(# {0},(dom (signature A)),f,((dom (signature A)) --> z) #) is 1 -element & not ManySortedSign(# {0},(dom (signature A)),f,((dom (signature A)) --> z) #) is void & ManySortedSign(# {0},(dom (signature A)),f,((dom (signature A)) --> z) #) is strict )

proof end;

registration

cluster non void 1 -element strict segmental for ManySortedSign ;

existence
ex b₁ being ManySortedSign st
( b₁ is segmental & not b₁ is void & b₁ is strict & b₁ is 1 -element )

proof end;

end;

definition

let A be Universal_Algebra;

func MSSign A -> trivial non void strict segmental ManySortedSign means :Def8: :: MSUALG_1:def 8
( the carrier of it = {0} & the carrier' of it = dom (signature A) & the Arity of it = (*--> 0) * (signature A) & the ResultSort of it = (dom (signature A)) --> 0 );

correctness
existence
ex b₁ being trivial non void strict segmental ManySortedSign st
( the carrier of b₁ = {0} & the carrier' of b₁ = dom (signature A) & the Arity of b₁ = (*--> 0) * (signature A) & the ResultSort of b₁ = (dom (signature A)) --> 0 );
uniqueness
for b₁, b₂ being trivial non void strict segmental ManySortedSign st the carrier of b₁ = {0} & the carrier' of b₁ = dom (signature A) & the Arity of b₁ = (*--> 0) * (signature A) & the ResultSort of b₁ = (dom (signature A)) --> 0 & the carrier of b₂ = {0} & the carrier' of b₂ = dom (signature A) & the Arity of b₂ = (*--> 0) * (signature A) & the ResultSort of b₂ = (dom (signature A)) --> 0 holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def8 defines MSSign MSUALG_1:def 8 :
for A being Universal_Algebra
for b₂ being trivial non void strict segmental ManySortedSign holds
( b₂ = MSSign A iff ( the carrier of b₂ = {0} & the carrier' of b₂ = dom (signature A) & the Arity of b₂ = (*--> 0) * (signature A) & the ResultSort of b₂ = (dom (signature A)) --> 0 ) );

registration

let A be Universal_Algebra;

cluster MSSign A -> trivial non void 1 -element strict segmental ;

coherence
MSSign A is 1 -element by Def8;

end;

definition

let A be Universal_Algebra;

func MSSorts A -> V5() ManySortedSet of the carrier of (MSSign A) equals :: MSUALG_1:def 9
0 .--> the carrier of A;

coherence
0 .--> the carrier of A is V5() ManySortedSet of the carrier of (MSSign A)

proof end;

correctness
;

end;

:: deftheorem defines MSSorts MSUALG_1:def 9 :
for A being Universal_Algebra holds MSSorts A = 0 .--> the carrier of A;

definition

let A be Universal_Algebra;

func MSCharact A -> ManySortedFunction of ((MSSorts A) #) * the Arity of (MSSign A),(MSSorts A) * the ResultSort of (MSSign A) equals :: MSUALG_1:def 10
the charact of A;

coherence
the charact of A is ManySortedFunction of ((MSSorts A) #) * the Arity of (MSSign A),(MSSorts A) * the ResultSort of (MSSign A)

proof end;

correctness
;

end;

:: deftheorem defines MSCharact MSUALG_1:def 10 :
for A being Universal_Algebra holds MSCharact A = the charact of A;

definition

let A be Universal_Algebra;

func MSAlg A -> strict MSAlgebra over MSSign A equals :: MSUALG_1:def 11
MSAlgebra(# (MSSorts A),(MSCharact A) #);

correctness
coherence
MSAlgebra(# (MSSorts A),(MSCharact A) #) is strict MSAlgebra over MSSign A;
;

end;

:: deftheorem defines MSAlg MSUALG_1:def 11 :
for A being Universal_Algebra holds MSAlg A = MSAlgebra(# (MSSorts A),(MSCharact A) #);

registration

let A be Universal_Algebra;

cluster MSAlg A -> strict non-empty ;

coherence
MSAlg A is non-empty ;

end;

:: Manysorted Algebras with 1 Sort Only

definition

let MS be 1 -element ManySortedSign ;
let A be MSAlgebra over MS;

func the_sort_of A -> set means :Def12: :: MSUALG_1:def 12
it is Component of the Sorts of A;

existence
ex b₁ being set st b₁ is Component of the Sorts of A

proof end;

uniqueness
for b₁, b₂ being set st b₁ is Component of the Sorts of A & b₂ is Component of the Sorts of A holds
b₁ = b₂ by Th5;

end;

:: deftheorem Def12 defines the_sort_of MSUALG_1:def 12 :
for MS being 1 -element ManySortedSign
for A being MSAlgebra over MS
for b₃ being set holds
( b₃ = the_sort_of A iff b₃ is Component of the Sorts of A );

registration

let MS be 1 -element ManySortedSign ;
let A be non-empty MSAlgebra over MS;

cluster the_sort_of A -> non empty ;

coherence
not the_sort_of A is empty

proof end;

end;

theorem Th6: :: MSUALG_1:6

for MS being non void 1 -element segmental ManySortedSign
for i being OperSymbol of MS
for A being non-empty MSAlgebra over MS holds Args (i,A) = (len (the_arity_of i)) -tuples_on (the_sort_of A)

proof end;

theorem Th7: :: MSUALG_1:7

for MS being non void 1 -element segmental ManySortedSign
for i being OperSymbol of MS
for A being non-empty MSAlgebra over MS holds Args (i,A) c= (the_sort_of A) *

proof end;

theorem Th8: :: MSUALG_1:8

for MS being non void 1 -element segmental ManySortedSign
for A being non-empty MSAlgebra over MS holds the Charact of A is FinSequence of PFuncs (((the_sort_of A) *),(the_sort_of A))

proof end;

definition

let MS be non void 1 -element segmental ManySortedSign ;
let A be non-empty MSAlgebra over MS;

func the_charact_of A -> PFuncFinSequence of (the_sort_of A) equals :: MSUALG_1:def 13
the Charact of A;

coherence
the Charact of A is PFuncFinSequence of (the_sort_of A) by Th8;

end;

:: deftheorem defines the_charact_of MSUALG_1:def 13 :
for MS being non void 1 -element segmental ManySortedSign
for A being non-empty MSAlgebra over MS holds the_charact_of A = the Charact of A;

definition

let MS be non void 1 -element segmental ManySortedSign ;
let A be non-empty MSAlgebra over MS;

func 1-Alg A -> strict Universal_Algebra equals :: MSUALG_1:def 14
UAStr(# (the_sort_of A),(the_charact_of A) #);

coherence
UAStr(# (the_sort_of A),(the_charact_of A) #) is strict Universal_Algebra

proof end;

correctness
;

end;

:: deftheorem defines 1-Alg MSUALG_1:def 14 :
for MS being non void 1 -element segmental ManySortedSign
for A being non-empty MSAlgebra over MS holds 1-Alg A = UAStr(# (the_sort_of A),(the_charact_of A) #);

theorem :: MSUALG_1:9

for A being strict Universal_Algebra holds A = 1-Alg (MSAlg A)

proof end;

theorem :: MSUALG_1:10

for A being Universal_Algebra
for f being Function of (dom (signature A)),({0} *)
for z being Element of {0} st f = (*--> 0) * (signature A) holds
MSSign A = ManySortedSign(# {0},(dom (signature A)),f,((dom (signature A)) --> z) #)

proof end;