:: Algebra of Morphisms
:: by Grzegorz Bancerek
::
:: Received January 28, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users

definition

let I be set ;
let A, f be Function;

func f -MSF (I,A) -> ManySortedFunction of I means :Def1: :: CATALG_1:def 1
for i being object st i in I holds
it . i = f | (A . i);

existence
ex b₁ being ManySortedFunction of I st
for i being object st i in I holds
b₁ . i = f | (A . i)

proof end;

uniqueness
for b₁, b₂ being ManySortedFunction of I st ( for i being object st i in I holds
b₁ . i = f | (A . i) ) & ( for i being object st i in I holds
b₂ . i = f | (A . i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines -MSF CATALG_1:def 1 :
for I being set
for A, f being Function
for b₄ being ManySortedFunction of I holds
( b₄ = f -MSF (I,A) iff for i being object st i in I holds
b₄ . i = f | (A . i) );

theorem :: CATALG_1:1

for I being set
for A being ManySortedSet of I holds (id (Union A)) -MSF (I,A) = id A

proof end;

theorem :: CATALG_1:2

for I being set
for A, B being ManySortedSet of I
for f, g being Function st rngs (f -MSF (I,A)) c= B holds
(g * f) -MSF (I,A) = (g -MSF (I,B)) ** (f -MSF (I,A))

proof end;

theorem :: CATALG_1:3

for f being Function
for I being set
for A, B being ManySortedSet of I st ( for i being set st i in I holds
( A . i c= dom f & f .: (A . i) c= B . i ) ) holds
f -MSF (I,A) is ManySortedFunction of A,B

proof end;

Lm1: now :: thesis: for x, y being set st <*x*> = <*y*> holds
x = y

let x, y be set ; :: thesis: ( <*x*> = <*y*> implies x = y )
assume <*x*> = <*y*> ; :: thesis: x = y
then <*x*> . 1 = y by FINSEQ_1:40;
hence x = y by FINSEQ_1:40; :: thesis: verum

end;

definition

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

attr A is empty means :Def2: :: CATALG_1:def 2
the Sorts of A is V4();

end;

:: deftheorem Def2 defines empty CATALG_1:def 2 :
for S being non empty ManySortedSign
for A being MSAlgebra over S holds
( A is empty iff the Sorts of A is V4() );

registration

let S be non empty ManySortedSign ;

cluster non-empty -> non empty for MSAlgebra over S;

coherence
for b₁ being MSAlgebra over S st b₁ is non-empty holds
not b₁ is empty ;

end;

registration

let S be non empty non void ManySortedSign ;

cluster strict non-empty disjoint_valued for MSAlgebra over S;

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

proof end;

end;

registration

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

cluster the Sorts of A -> V4() ;

coherence
not the Sorts of A is empty-yielding by Def2;

end;

registration

cluster Relation-like non empty-yielding Function-like for set ;

existence
not for b₁ being Function holds b₁ is empty-yielding

proof end;

end;

definition

let A be set ;

func CatSign A -> strict ManySortedSign means :Def3: :: CATALG_1:def 3
( the carrier of it = [:{0},(2 -tuples_on A):] & the carrier' of it = [:{1},(1 -tuples_on A):] \/ [:{2},(3 -tuples_on A):] & ( for a being set st a in A holds
( the Arity of it . [1,<*a*>] = {} & the ResultSort of it . [1,<*a*>] = [0,<*a,a*>] ) ) & ( for a, b, c being set st a in A & b in A & c in A holds
( the Arity of it . [2,<*a,b,c*>] = <*[0,<*b,c*>],[0,<*a,b*>]*> & the ResultSort of it . [2,<*a,b,c*>] = [0,<*a,c*>] ) ) );

existence
ex b₁ being strict ManySortedSign st
( the carrier of b₁ = [:{0},(2 -tuples_on A):] & the carrier' of b₁ = [:{1},(1 -tuples_on A):] \/ [:{2},(3 -tuples_on A):] & ( for a being set st a in A holds
( the Arity of b₁ . [1,<*a*>] = {} & the ResultSort of b₁ . [1,<*a*>] = [0,<*a,a*>] ) ) & ( for a, b, c being set st a in A & b in A & c in A holds
( the Arity of b₁ . [2,<*a,b,c*>] = <*[0,<*b,c*>],[0,<*a,b*>]*> & the ResultSort of b₁ . [2,<*a,b,c*>] = [0,<*a,c*>] ) ) )

proof end;

correctness
uniqueness
for b₁, b₂ being strict ManySortedSign st the carrier of b₁ = [:{0},(2 -tuples_on A):] & the carrier' of b₁ = [:{1},(1 -tuples_on A):] \/ [:{2},(3 -tuples_on A):] & ( for a being set st a in A holds
( the Arity of b₁ . [1,<*a*>] = {} & the ResultSort of b₁ . [1,<*a*>] = [0,<*a,a*>] ) ) & ( for a, b, c being set st a in A & b in A & c in A holds
( the Arity of b₁ . [2,<*a,b,c*>] = <*[0,<*b,c*>],[0,<*a,b*>]*> & the ResultSort of b₁ . [2,<*a,b,c*>] = [0,<*a,c*>] ) ) & the carrier of b₂ = [:{0},(2 -tuples_on A):] & the carrier' of b₂ = [:{1},(1 -tuples_on A):] \/ [:{2},(3 -tuples_on A):] & ( for a being set st a in A holds
( the Arity of b₂ . [1,<*a*>] = {} & the ResultSort of b₂ . [1,<*a*>] = [0,<*a,a*>] ) ) & ( for a, b, c being set st a in A & b in A & c in A holds
( the Arity of b₂ . [2,<*a,b,c*>] = <*[0,<*b,c*>],[0,<*a,b*>]*> & the ResultSort of b₂ . [2,<*a,b,c*>] = [0,<*a,c*>] ) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def3 defines CatSign CATALG_1:def 3 :
for A being set
for b₂ being strict ManySortedSign holds
( b₂ = CatSign A iff ( the carrier of b₂ = [:{0},(2 -tuples_on A):] & the carrier' of b₂ = [:{1},(1 -tuples_on A):] \/ [:{2},(3 -tuples_on A):] & ( for a being set st a in A holds
( the Arity of b₂ . [1,<*a*>] = {} & the ResultSort of b₂ . [1,<*a*>] = [0,<*a,a*>] ) ) & ( for a, b, c being set st a in A & b in A & c in A holds
( the Arity of b₂ . [2,<*a,b,c*>] = <*[0,<*b,c*>],[0,<*a,b*>]*> & the ResultSort of b₂ . [2,<*a,b,c*>] = [0,<*a,c*>] ) ) ) );

registration

let A be set ;

cluster CatSign A -> strict feasible ;

coherence
CatSign A is feasible

proof end;

end;

registration

let A be non empty set ;

cluster CatSign A -> non empty non void strict ;

coherence
( not CatSign A is empty & not CatSign A is void )

proof end;

end;

definition

mode Signature is feasible ManySortedSign ;

end;

definition

let S be Signature;

attr S is Categorial means :Def4: :: CATALG_1:def 4
ex A being set st
( CatSign A is Subsignature of S & the carrier of S = [:{0},(2 -tuples_on A):] );

end;

:: deftheorem Def4 defines Categorial CATALG_1:def 4 :
for S being Signature holds
( S is Categorial iff ex A being set st
( CatSign A is Subsignature of S & the carrier of S = [:{0},(2 -tuples_on A):] ) );

registration

cluster non empty feasible Categorial -> non empty non void for ManySortedSign ;

coherence
for b₁ being non empty Signature st b₁ is Categorial holds
not b₁ is void

proof end;

end;

registration

cluster non empty strict feasible Categorial for ManySortedSign ;

existence
ex b₁ being Signature st
( b₁ is Categorial & not b₁ is empty & b₁ is strict )

proof end;

end;

definition

mode CatSignature is Categorial Signature;

end;

definition

let A be set ;

mode CatSignature of A -> Signature means :Def5: :: CATALG_1:def 5
( CatSign A is Subsignature of it & the carrier of it = [:{0},(2 -tuples_on A):] );

existence
ex b₁ being Signature st
( CatSign A is Subsignature of b₁ & the carrier of b₁ = [:{0},(2 -tuples_on A):] )

proof end;

end;

:: deftheorem Def5 defines CatSignature CATALG_1:def 5 :
for A being set
for b₂ being Signature holds
( b₂ is CatSignature of A iff ( CatSign A is Subsignature of b₂ & the carrier of b₂ = [:{0},(2 -tuples_on A):] ) );

theorem :: CATALG_1:4

for A1, A2 being set
for S being CatSignature of A1 st S is CatSignature of A2 holds
A1 = A2

proof end;

registration

let A be set ;

cluster -> Categorial for CatSignature of A;

coherence
for b₁ being CatSignature of A holds b₁ is Categorial

proof end;

end;

registration

let A be non empty set ;

cluster -> non empty for CatSignature of A;

coherence
for b₁ being CatSignature of A holds not b₁ is empty

proof end;

end;

registration

let A be set ;

cluster strict feasible Categorial for CatSignature of A;

existence
ex b₁ being CatSignature of A st b₁ is strict

proof end;

end;

definition

let A be set ;
:: original: CatSign
redefine func CatSign A -> strict CatSignature of A;
coherence
CatSign A is strict CatSignature of A

proof end;

end;

definition

let S be ManySortedSign ;
assume A1: for x being object st x in proj2 ( the carrier of S \/ the carrier' of S) holds
x is Function ;

func underlay S -> set means :Def6: :: CATALG_1:def 6
for x being object holds
( x in it iff ex a being set ex f being Function st
( [a,f] in the carrier of S \/ the carrier' of S & x in rng f ) );

existence
ex b₁ being set st
for x being object holds
( x in b₁ iff ex a being set ex f being Function st
( [a,f] in the carrier of S \/ the carrier' of S & x in rng f ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being object holds
( x in b₁ iff ex a being set ex f being Function st
( [a,f] in the carrier of S \/ the carrier' of S & x in rng f ) ) ) & ( for x being object holds
( x in b₂ iff ex a being set ex f being Function st
( [a,f] in the carrier of S \/ the carrier' of S & x in rng f ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines underlay CATALG_1:def 6 :
for S being ManySortedSign st ( for x being object st x in proj2 ( the carrier of S \/ the carrier' of S) holds
x is Function ) holds
for b₂ being set holds
( b₂ = underlay S iff for x being object holds
( x in b₂ iff ex a being set ex f being Function st
( [a,f] in the carrier of S \/ the carrier' of S & x in rng f ) ) );

definition

let S be ManySortedSign ;

attr S is delta-concrete means :Def7: :: CATALG_1:def 7
ex f being sequence of NAT st
( ( for s being object st s in the carrier of S holds
ex i being Element of NAT ex p being FinSequence st
( s = [i,p] & len p = f . i & [:{i},((f . i) -tuples_on (underlay S)):] c= the carrier of S ) ) & ( for o being object st o in the carrier' of S holds
ex i being Element of NAT ex p being FinSequence st
( o = [i,p] & len p = f . i & [:{i},((f . i) -tuples_on (underlay S)):] c= the carrier' of S ) ) );

end;

:: deftheorem Def7 defines delta-concrete CATALG_1:def 7 :
for S being ManySortedSign holds
( S is delta-concrete iff ex f being sequence of NAT st
( ( for s being object st s in the carrier of S holds
ex i being Element of NAT ex p being FinSequence st
( s = [i,p] & len p = f . i & [:{i},((f . i) -tuples_on (underlay S)):] c= the carrier of S ) ) & ( for o being object st o in the carrier' of S holds
ex i being Element of NAT ex p being FinSequence st
( o = [i,p] & len p = f . i & [:{i},((f . i) -tuples_on (underlay S)):] c= the carrier' of S ) ) ) );

Lm2: for A being set
for x being object st x in proj2 ( the carrier of (CatSign A) \/ the carrier' of (CatSign A)) holds
x is Function

proof end;

theorem Th5: :: CATALG_1:5

for A being set holds underlay (CatSign A) = A

proof end;

registration

let A be set ;

cluster CatSign A -> strict delta-concrete ;

coherence
CatSign A is delta-concrete

proof end;

end;

registration

cluster non empty strict feasible Categorial delta-concrete for ManySortedSign ;

existence
ex b₁ being CatSignature st
( b₁ is delta-concrete & not b₁ is empty & b₁ is strict )

proof end;

let A be set ;

cluster strict feasible Categorial delta-concrete for CatSignature of A;

existence
ex b₁ being CatSignature of A st
( b₁ is delta-concrete & b₁ is strict )

proof end;

end;

theorem :: CATALG_1:6

for A being set holds underlay (CatSign A) = A

proof end;

registration

let A be set ;

cluster CatSign A -> strict delta-concrete ;

coherence
CatSign A is delta-concrete ;

end;

registration

cluster non empty strict feasible Categorial delta-concrete for ManySortedSign ;

existence
ex b₁ being CatSignature st
( b₁ is delta-concrete & not b₁ is empty & b₁ is strict )

proof end;

let A be set ;

cluster strict feasible Categorial delta-concrete for CatSignature of A;

existence
ex b₁ being CatSignature of A st
( b₁ is delta-concrete & b₁ is strict )

proof end;

end;

Lm3: for S being delta-concrete ManySortedSign
for x being object st x in proj2 ( the carrier of S \/ the carrier' of S) holds
x is Function

proof end;

theorem Th7: :: CATALG_1:7

for S being delta-concrete ManySortedSign
for x being set st ( x in the carrier of S or x in the carrier' of S ) holds
ex i being Element of NAT ex p being FinSequence st
( x = [i,p] & rng p c= underlay S )

proof end;

theorem :: CATALG_1:8

for S being delta-concrete ManySortedSign
for i being set
for p1, p2 being FinSequence st ( ( [i,p1] in the carrier of S & [i,p2] in the carrier of S ) or ( [i,p1] in the carrier' of S & [i,p2] in the carrier' of S ) ) holds
len p1 = len p2

proof end;

theorem :: CATALG_1:9

for S being delta-concrete ManySortedSign
for i being set
for p1, p2 being FinSequence st len p2 = len p1 & rng p2 c= underlay S holds
( ( [i,p1] in the carrier of S implies [i,p2] in the carrier of S ) & ( [i,p1] in the carrier' of S implies [i,p2] in the carrier' of S ) )

proof end;

theorem :: CATALG_1:10

for S being non empty Categorial delta-concrete Signature holds S is CatSignature of underlay S

proof end;

registration

let S be non empty CatSignature;
let s be SortSymbol of S;

cluster s `2 -> Relation-like Function-like for set ;

coherence
for b₁ being set st b₁ = s `2 holds
( b₁ is Relation-like & b₁ is Function-like )

proof end;

end;

registration

let S be non empty delta-concrete ManySortedSign ;
let s be SortSymbol of S;

cluster s `2 -> Relation-like Function-like for set ;

coherence
for b₁ being set st b₁ = s `2 holds
( b₁ is Relation-like & b₁ is Function-like )

proof end;

end;

registration

let S be non void delta-concrete ManySortedSign ;
let o be Element of the carrier' of S;

cluster o `2 -> Relation-like Function-like for set ;

coherence
for b₁ being set st b₁ = o `2 holds
( b₁ is Relation-like & b₁ is Function-like )

proof end;

end;

registration

let S be non empty CatSignature;
let s be SortSymbol of S;

cluster s `2 -> FinSequence-like for Function;

coherence
for b₁ being Function st b₁ = s `2 holds
b₁ is FinSequence-like

proof end;

end;

registration

let S be non empty delta-concrete ManySortedSign ;
let s be SortSymbol of S;

cluster s `2 -> FinSequence-like for Function;

coherence
for b₁ being Function st b₁ = s `2 holds
b₁ is FinSequence-like

proof end;

end;

registration

let S be non void delta-concrete ManySortedSign ;
let o be Element of the carrier' of S;

cluster o `2 -> FinSequence-like for Function;

coherence
for b₁ being Function st b₁ = o `2 holds
b₁ is FinSequence-like

proof end;

end;

definition

let a be set ;

func idsym a -> set equals :: CATALG_1:def 8
[1,<*a*>];

correctness
coherence
[1,<*a*>] is set ;
;
let b be set ;

func homsym (a,b) -> set equals :: CATALG_1:def 9
[0,<*a,b*>];

correctness
coherence
[0,<*a,b*>] is set ;
;
let c be set ;

func compsym (a,b,c) -> set equals :: CATALG_1:def 10
[2,<*a,b,c*>];

correctness
coherence
[2,<*a,b,c*>] is set ;
;

end;

:: deftheorem defines idsym CATALG_1:def 8 :
for a being set holds idsym a = [1,<*a*>];

:: deftheorem defines homsym CATALG_1:def 9 :
for a, b being set holds homsym (a,b) = [0,<*a,b*>];

:: deftheorem defines compsym CATALG_1:def 10 :
for a, b, c being set holds compsym (a,b,c) = [2,<*a,b,c*>];

theorem Th11: :: CATALG_1:11

for A being non empty set
for S being CatSignature of A
for a being Element of A holds
( idsym a in the carrier' of S & ( for b being Element of A holds
( homsym (a,b) in the carrier of S & ( for c being Element of A holds compsym (a,b,c) in the carrier' of S ) ) ) )

proof end;

definition

let A be non empty set ;
let a be Element of A;
:: original: idsym
redefine func idsym a -> OperSymbol of (CatSign A);
correctness
coherence
idsym a is OperSymbol of (CatSign A);
by Th11;
let b be Element of A;
:: original: homsym
redefine func homsym (a,b) -> SortSymbol of (CatSign A);
correctness
coherence
homsym (a,b) is SortSymbol of (CatSign A);
by Th11;
let c be Element of A;
:: original: compsym
redefine func compsym (a,b,c) -> OperSymbol of (CatSign A);
correctness
coherence
compsym (a,b,c) is OperSymbol of (CatSign A);
by Th11;

end;

theorem Th12: :: CATALG_1:12

for a, b being set st idsym a = idsym b holds
a = b

proof end;

theorem Th13: :: CATALG_1:13

for a1, b1, a2, b2 being set st homsym (a1,a2) = homsym (b1,b2) holds
( a1 = b1 & a2 = b2 )

proof end;

theorem Th14: :: CATALG_1:14

for a1, b1, a2, b2, a3, b3 being set st compsym (a1,a2,a3) = compsym (b1,b2,b3) holds
( a1 = b1 & a2 = b2 & a3 = b3 )

proof end;

theorem Th15: :: CATALG_1:15

for A being non empty set
for S being CatSignature of A
for s being SortSymbol of S ex a, b being Element of A st s = homsym (a,b)

proof end;

theorem Th16: :: CATALG_1:16

for A being non empty set
for o being OperSymbol of (CatSign A) holds
( ( o `1 = 1 & len (o `2) = 1 ) or ( o `1 = 2 & len (o `2) = 3 ) )

proof end;

theorem Th17: :: CATALG_1:17

for A being non empty set
for o being OperSymbol of (CatSign A) st ( o `1 = 1 or len (o `2) = 1 ) holds
ex a being Element of A st o = idsym a

proof end;

theorem Th18: :: CATALG_1:18

for A being non empty set
for o being OperSymbol of (CatSign A) st ( o `1 = 2 or len (o `2) = 3 ) holds
ex a, b, c being Element of A st o = compsym (a,b,c)

proof end;

theorem :: CATALG_1:19

for A being non empty set
for a being Element of A holds
( the_arity_of (idsym a) = {} & the_result_sort_of (idsym a) = homsym (a,a) ) by Def3;

theorem :: CATALG_1:20

for A being non empty set
for a, b, c being Element of A holds
( the_arity_of (compsym (a,b,c)) = <*(homsym (b,c)),(homsym (a,b))*> & the_result_sort_of (compsym (a,b,c)) = homsym (a,c) ) by Def3;

definition

let C1, C2 be Category;
let F be Functor of C1,C2;

func Upsilon F -> Function of the carrier of (CatSign the carrier of C1), the carrier of (CatSign the carrier of C2) means :Def11: :: CATALG_1:def 11
for s being SortSymbol of (CatSign the carrier of C1) holds it . s = [0,((Obj F) * (s `2))];

uniqueness
for b₁, b₂ being Function of the carrier of (CatSign the carrier of C1), the carrier of (CatSign the carrier of C2) st ( for s being SortSymbol of (CatSign the carrier of C1) holds b₁ . s = [0,((Obj F) * (s `2))] ) & ( for s being SortSymbol of (CatSign the carrier of C1) holds b<sub>2</sub> . s = [0,((Obj F) * (s `2))] ) holds
b₁ = b₂

proof end;

existence
ex b₁ being Function of the carrier of (CatSign the carrier of C1), the carrier of (CatSign the carrier of C2) st
for s being SortSymbol of (CatSign the carrier of C1) holds b₁ . s = [0,((Obj F) * (s `2))]

proof end;

func Psi F -> Function of the carrier' of (CatSign the carrier of C1), the carrier' of (CatSign the carrier of C2) means :Def12: :: CATALG_1:def 12
for o being OperSymbol of (CatSign the carrier of C1) holds it . o = [(o `1),((Obj F) * (o `2))];

uniqueness
for b₁, b₂ being Function of the carrier' of (CatSign the carrier of C1), the carrier' of (CatSign the carrier of C2) st ( for o being OperSymbol of (CatSign the carrier of C1) holds b₁ . o = [(o `1),((Obj F) * (o `2))] ) & ( for o being OperSymbol of (CatSign the carrier of C1) holds b₂ . o = [(o `1),((Obj F) * (o `2))] ) holds
b₁ = b₂

proof end;

existence
ex b₁ being Function of the carrier' of (CatSign the carrier of C1), the carrier' of (CatSign the carrier of C2) st
for o being OperSymbol of (CatSign the carrier of C1) holds b₁ . o = [(o `1),((Obj F) * (o `2))]

proof end;

end;

:: deftheorem Def11 defines Upsilon CATALG_1:def 11 :
for C1, C2 being Category
for F being Functor of C1,C2
for b₄ being Function of the carrier of (CatSign the carrier of C1), the carrier of (CatSign the carrier of C2) holds
( b₄ = Upsilon F iff for s being SortSymbol of (CatSign the carrier of C1) holds b₄ . s = [0,((Obj F) * (s `2))] );

:: deftheorem Def12 defines Psi CATALG_1:def 12 :
for C1, C2 being Category
for F being Functor of C1,C2
for b₄ being Function of the carrier' of (CatSign the carrier of C1), the carrier' of (CatSign the carrier of C2) holds
( b₄ = Psi F iff for o being OperSymbol of (CatSign the carrier of C1) holds b₄ . o = [(o `1),((Obj F) * (o `2))] );

Lm4: now :: thesis: for x being set
for f being Function st x in dom f holds
f * <*x*> = <*(f . x)*>

let x be set ; :: thesis: for f being Function st x in dom f holds
f * <*x*> = <*(f . x)*>
let f be Function; :: thesis: ( x in dom f implies f * <*x*> = <*(f . x)*> )
assume x in dom f ; :: thesis: f * <*x*> = <*(f . x)*>
then ( rng <*x*> = {x} & {x} c= dom f ) by FINSEQ_1:39, ZFMISC_1:31;
then A1: dom (f * <*x*>) = dom <*x*> by RELAT_1:27
.= Seg 1 by FINSEQ_1:38 ;
then reconsider p = f * <*x*> as FinSequence by FINSEQ_1:def 2;
A2: len p = 1 by A1, FINSEQ_1:def 3;
( 1 in {1} & <*x*> . 1 = x ) by FINSEQ_1:40, TARSKI:def 1;
then p . 1 = f . x by A1, FINSEQ_1:2, FUNCT_1:12;
hence f * <*x*> = <*(f . x)*> by A2, FINSEQ_1:40; :: thesis: verum

end;

theorem Th21: :: CATALG_1:21

for C1, C2 being Category
for F being Functor of C1,C2
for a, b being Object of C1 holds (Upsilon F) . (homsym (a,b)) = homsym ((F . a),(F . b))

proof end;

theorem Th22: :: CATALG_1:22

for C1, C2 being Category
for F being Functor of C1,C2
for a being Object of C1 holds (Psi F) . (idsym a) = idsym (F . a)

proof end;

theorem Th23: :: CATALG_1:23

for C1, C2 being Category
for F being Functor of C1,C2
for a, b, c being Object of C1 holds (Psi F) . (compsym (a,b,c)) = compsym ((F . a),(F . b),(F . c))

proof end;

theorem Th24: :: CATALG_1:24

for C1, C2 being Category
for F being Functor of C1,C2 holds Upsilon F, Psi F form_morphism_between CatSign the carrier of C1, CatSign the carrier of C2

proof end;

theorem Th25: :: CATALG_1:25

for C being non empty set
for A being MSAlgebra over CatSign C
for a being Element of C holds Args ((idsym a),A) = {{}}

proof end;

Lm5: for C being Category
for A being MSAlgebra over CatSign the carrier of C st ( for a, b being Object of C holds the Sorts of A . (homsym (a,b)) = Hom (a,b) ) holds
for a, b, c being Object of C holds
( Args ((compsym (a,b,c)),A) = product <*(Hom (b,c)),(Hom (a,b))*> & Result ((compsym (a,b,c)),A) = Hom (a,c) )

proof end;

scheme :: CATALG_1:sch 1
CatAlgEx{ F₁() -> non empty set , F₂() -> non empty set , F₃( set , set ) -> set , F₄( set , set , set , object , object ) -> set , F₅( set ) -> set } :

ex A being strict MSAlgebra over CatSign F₁() st
( ( for a, b being Element of F₁() holds the Sorts of A . (homsym (a,b)) = F₃(a,b) ) & ( for a being Element of F₁() holds (Den ((idsym a),A)) . {} = F₅(a) ) & ( for a, b, c being Element of F₁()
for f, g being Element of F₂() st f in F₃(a,b) & g in F₃(b,c) holds
(Den ((compsym (a,b,c)),A)) . <*g,f*> = F₄(a,b,c,g,f) ) )

provided

A1: for a, b being Element of F₁() holds F₃(a,b) c= F₂() and
A2: for a being Element of F₁() holds F₅(a) in F₃(a,a) and
A3: for a, b, c being Element of F₁()
for f, g being Element of F₂() st f in F₃(a,b) & g in F₃(b,c) holds
F₄(a,b,c,g,f) in F₃(a,c)

proof end;

definition

let C be Category;

func MSAlg C -> strict MSAlgebra over CatSign the carrier of C means :Def13: :: CATALG_1:def 13
( ( for a, b being Object of C holds the Sorts of it . (homsym (a,b)) = Hom (a,b) ) & ( for a being Object of C holds (Den ((idsym a),it)) . {} = id a ) & ( for a, b, c being Object of C
for f, g being Morphism of C st dom f = a & cod f = b & dom g = b & cod g = c holds
(Den ((compsym (a,b,c)),it)) . <*g,f*> = g (*) f ) );

uniqueness
for b₁, b₂ being strict MSAlgebra over CatSign the carrier of C st ( for a, b being Object of C holds the Sorts of b₁ . (homsym (a,b)) = Hom (a,b) ) & ( for a being Object of C holds (Den ((idsym a),b₁)) . {} = id a ) & ( for a, b, c being Object of C
for f, g being Morphism of C st dom f = a & cod f = b & dom g = b & cod g = c holds
(Den ((compsym (a,b,c)),b₁)) . <*g,f*> = g (*) f ) & ( for a, b being Object of C holds the Sorts of b₂ . (homsym (a,b)) = Hom (a,b) ) & ( for a being Object of C holds (Den ((idsym a),b₂)) . {} = id a ) & ( for a, b, c being Object of C
for f, g being Morphism of C st dom f = a & cod f = b & dom g = b & cod g = c holds
(Den ((compsym (a,b,c)),b₂)) . <*g,f*> = g (*) f ) holds
b₁ = b₂

proof end;

correctness
existence
ex b₁ being strict MSAlgebra over CatSign the carrier of C st
( ( for a, b being Object of C holds the Sorts of b₁ . (homsym (a,b)) = Hom (a,b) ) & ( for a being Object of C holds (Den ((idsym a),b₁)) . {} = id a ) & ( for a, b, c being Object of C
for f, g being Morphism of C st dom f = a & cod f = b & dom g = b & cod g = c holds
(Den ((compsym (a,b,c)),b₁)) . <*g,f*> = g (*) f ) );

proof end;

end;

:: deftheorem Def13 defines MSAlg CATALG_1:def 13 :
for C being Category
for b₂ being strict MSAlgebra over CatSign the carrier of C holds
( b₂ = MSAlg C iff ( ( for a, b being Object of C holds the Sorts of b₂ . (homsym (a,b)) = Hom (a,b) ) & ( for a being Object of C holds (Den ((idsym a),b₂)) . {} = id a ) & ( for a, b, c being Object of C
for f, g being Morphism of C st dom f = a & cod f = b & dom g = b & cod g = c holds
(Den ((compsym (a,b,c)),b₂)) . <*g,f*> = g (*) f ) ) );

theorem Th26: :: CATALG_1:26

for A being Category
for a being Object of A holds Result ((idsym a),(MSAlg A)) = Hom (a,a)

proof end;

theorem Th27: :: CATALG_1:27

for A being Category
for a, b, c being Object of A holds
( Args ((compsym (a,b,c)),(MSAlg A)) = product <*(Hom (b,c)),(Hom (a,b))*> & Result ((compsym (a,b,c)),(MSAlg A)) = Hom (a,c) )

proof end;

registration

let C be Category;

cluster MSAlg C -> strict disjoint_valued feasible ;

coherence
( MSAlg C is disjoint_valued & MSAlg C is feasible )

proof end;

end;

theorem Th28: :: CATALG_1:28

for C1, C2 being Category
for F being Functor of C1,C2 holds F -MSF ( the carrier of (CatSign the carrier of C1), the Sorts of (MSAlg C1)) is ManySortedFunction of (MSAlg C1),((MSAlg C2) | ((CatSign the carrier of C1),(Upsilon F),(Psi F)))

proof end;

theorem Th29: :: CATALG_1:29

for C being Category
for a, b, c being Object of C
for x being set holds
( x in Args ((compsym (a,b,c)),(MSAlg C)) iff ex g, f being Morphism of C st
( x = <*g,f*> & dom f = a & cod f = b & dom g = b & cod g = c ) )

proof end;

theorem Th30: :: CATALG_1:30

for C1, C2 being Category
for F being Functor of C1,C2
for a, b, c being Object of C1
for f, g being Morphism of C1 st f in Hom (a,b) & g in Hom (b,c) holds
for x being Element of Args ((compsym (a,b,c)),(MSAlg C1)) st x = <*g,f*> holds
for H being ManySortedFunction of (MSAlg C1),((MSAlg C2) | ((CatSign the carrier of C1),(Upsilon F),(Psi F))) st H = F -MSF ( the carrier of (CatSign the carrier of C1), the Sorts of (MSAlg C1)) holds
H # x = <*(F . g),(F . f)*>

proof end;

theorem Th31: :: CATALG_1:31

for C being Category
for a, b, c being Object of C
for f, g being Morphism of C st f in Hom (a,b) & g in Hom (b,c) holds
(Den ((compsym (a,b,c)),(MSAlg C))) . <*g,f*> = g (*) f

proof end;

theorem :: CATALG_1:32

for C being Category
for a, b, c, d being Object of C
for f, g, h being Morphism of C st f in Hom (a,b) & g in Hom (b,c) & h in Hom (c,d) holds
(Den ((compsym (a,c,d)),(MSAlg C))) . <*h,((Den ((compsym (a,b,c)),(MSAlg C))) . <*g,f*>)*> = (Den ((compsym (a,b,d)),(MSAlg C))) . <*((Den ((compsym (b,c,d)),(MSAlg C))) . <*h,g*>),f*>

proof end;

theorem :: CATALG_1:33

for C being Category
for a, b being Object of C
for f being Morphism of C st f in Hom (a,b) holds
( (Den ((compsym (a,b,b)),(MSAlg C))) . <*(id b),f*> = f & (Den ((compsym (a,a,b)),(MSAlg C))) . <*f,(id a)*> = f )

proof end;

theorem :: CATALG_1:34

for C1, C2 being Category
for F being Functor of C1,C2 ex H being ManySortedFunction of (MSAlg C1),((MSAlg C2) | ((CatSign the carrier of C1),(Upsilon F),(Psi F))) st
( H = F -MSF ( the carrier of (CatSign the carrier of C1), the Sorts of (MSAlg C1)) & H is_homomorphism MSAlg C1,(MSAlg C2) | ((CatSign the carrier of C1),(Upsilon F),(Psi F)) )

proof end;