:: CATALG_1 semantic presentation

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 set st i in I holds
it . i = f | (A . i);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines -MSF CATALG_1:def 1 :

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;

theorem Th4: :: CATALG_1:4

for A being set
for i being Nat
for p being FinSequence holds
( p in i -tuples_on A iff ( len p = i & rng p c= A ) )

proof end;

theorem Th5: :: CATALG_1:5

for A being set
for i being Nat
for p being FinSequence of A holds
( p in i -tuples_on A iff len p = i )

proof end;

theorem :: CATALG_1:6

for A being set
for i being Nat holds i -tuples_on A c= A *

proof end;

Lm1: now

let x, y be set ; :: thesis:
assume <*x*> = <*y*> ; :: thesis:
then <*x*> . 1 = y by FINSEQ_1:57;
hence x = y by FINSEQ_1:57; :: thesis:

end;

theorem Th7: :: CATALG_1:7

for A being set
for i being Nat holds
( ( i <> 0 & A = {} ) iff i -tuples_on A = {} )

proof end;

theorem Th8: :: CATALG_1:8

for A, x being set holds
( x in 1 -tuples_on A iff ex a being set st
( a in A & x = <*a*> ) )

proof end;

theorem :: CATALG_1:9

for A, a being set st <*a*> in 1 -tuples_on A holds
a in A

proof end;

theorem Th10: :: CATALG_1:10

for A, x being set holds
( x in 2 -tuples_on A iff ex a, b being set st
( a in A & b in A & x = <*a,b*> ) )

proof end;

theorem Th11: :: CATALG_1:11

for A, a, b being set st <*a,b*> in 2 -tuples_on A holds
( a in A & b in A )

proof end;

theorem Th12: :: CATALG_1:12

for A, x being set holds
( x in 3 -tuples_on A iff ex a, b, c being set st
( a in A & b in A & c in A & x = <*a,b,c*> ) )

proof end;

theorem :: CATALG_1:13

for A, a, b, c being set st <*a,b,c*> in 3 -tuples_on A holds
( a in A & b in A & c in A )

proof end;

definition

let S be non empty ManySortedSign ;
let A be MSAlgebra of S;
canceled;
attr A is empty means :Def3: :: CATALG_1:def 3
the Sorts of A is empty-yielding;

end;

:: deftheorem CATALG_1:def 2 :

:: deftheorem Def3 defines empty CATALG_1:def 3 :

registration

let S be non empty ManySortedSign ;
cluster non-empty -> non empty MSAlgebra of S;
coherence

proof end;

end;

registration

let S be non empty non void ManySortedSign ;
let X be V2 ManySortedSet of the carrier of S;
cluster FreeMSA X -> disjoint_valued non empty ;
coherence

proof end;

end;

registration

let S be non empty non void ManySortedSign ;
cluster strict non-empty disjoint_valued non empty MSAlgebra of S;
existence

proof end;

end;

registration

let S be non empty non void ManySortedSign ;
let A be non empty MSAlgebra of S;
cluster the Sorts of A -> V3 ;
coherence by Def3;

end;

registration

cluster non empty-yielding set ;
existence

proof end;

end;

definition

let A be set ;
canceled;
func CatSign A -> strict ManySortedSign means :Def5: :: CATALG_1:def 5
( the carrier of it = [:{0},(2 -tuples_on A):] & the OperSymbols 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

proof end;

correctness

proof end;

end;

:: deftheorem CATALG_1:def 4 :

:: deftheorem Def5 defines CatSign CATALG_1:def 5 :

registration

let A be set ;
cluster CatSign A -> strict feasible ;
coherence

proof end;

end;

registration

let A be non empty set ;
cluster CatSign A -> non empty strict non void feasible ;
coherence

proof end;

end;

definition

mode Signature is feasible ManySortedSign ;

end;

definition

let S be Signature;
attr S is Categorial means :Def6: :: CATALG_1:def 6
ex A being set st
( CatSign A is Subsignature of S & the carrier of S = [:{0},(2 -tuples_on A):] );

end;

:: deftheorem Def6 defines Categorial CATALG_1:def 6 :

registration

cluster non empty Categorial -> non empty non void ManySortedSign ;
coherence

proof end;

end;

registration

cluster non empty strict non void Categorial ManySortedSign ;
existence

proof end;

end;

definition

mode CatSignature is Categorial Signature;

end;

definition

let A be set ;
mode CatSignature of A -> Signature means :Def7: :: CATALG_1:def 7
( CatSign A is Subsignature of it & the carrier of it = [:{0},(2 -tuples_on A):] );
existence

proof end;

end;

:: deftheorem Def7 defines CatSignature CATALG_1:def 7 :

theorem :: CATALG_1:14

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 CatSignature of A;
coherence

proof end;

end;

registration

let A be non empty set ;
cluster -> non empty non void Categorial CatSignature of A;
coherence

proof end;

end;

registration

let A be set ;
cluster strict Categorial CatSignature of A;
existence

proof end;

end;

definition

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

proof end;

end;

definition

let S be ManySortedSign ;
func underlay S -> set means :Def8: :: CATALG_1:def 8
for x being set holds
( x in it iff ex a being set ex f being Function st
( [a,f] in the carrier of S \/ the OperSymbols of S & x in rng f ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines underlay CATALG_1:def 8 :

theorem Th15: :: CATALG_1:15

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

proof end;

definition

let S be ManySortedSign ;
attr S is delta-concrete means :Def9: :: CATALG_1:def 9
ex f being Function of NAT , NAT st
( ( for s being set st s in the carrier of S holds
ex i being 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 set st o in the OperSymbols of S holds
ex i being Nat ex p being FinSequence st
( o = [i,p] & len p = f . i & [:{i},((f . i) -tuples_on (underlay S)):] c= the OperSymbols of S ) ) );

end;

:: deftheorem Def9 defines delta-concrete CATALG_1:def 9 :

registration

let A be set ;
cluster CatSign A -> strict feasible delta-concrete ;
coherence

proof end;

end;

registration

cluster non empty strict non void delta-concrete ManySortedSign ;
existence

proof end;

let A be set ;
cluster strict Categorial delta-concrete CatSignature of A;
existence

proof end;

end;

theorem Th16: :: CATALG_1:16

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

proof end;

theorem :: CATALG_1:17

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 OperSymbols of S & [i,p2] in the OperSymbols of S ) ) holds
len p1 = len p2

proof end;

theorem :: CATALG_1:18

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 OperSymbols of S implies [i,p2] in the OperSymbols of S ) )

proof end;

theorem :: CATALG_1:19

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 ;
coherence

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 ;
coherence

proof end;

end;

registration

let S be non void delta-concrete ManySortedSign ;
let o be Element of the OperSymbols of S;
cluster o `2 -> Relation-like Function-like ;
coherence

proof end;

end;

registration

let S be non empty CatSignature;
let s be SortSymbol of S;
cluster s `2 -> Relation-like Function-like FinSequence-like ;
coherence

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 FinSequence-like ;
coherence

proof end;

end;

registration

let S be non void delta-concrete ManySortedSign ;
let o be Element of the OperSymbols of S;
cluster o `2 -> Relation-like Function-like FinSequence-like ;
coherence

proof end;

end;

definition

let a be set ;
func idsym a -> set equals :: CATALG_1:def 10
[1,<*a*>];
correctness ;
let b be set ;
func homsym a,b -> set equals :: CATALG_1:def 11
[0,<*a,b*>];
correctness ;
let c be set ;
func compsym a,b,c -> set equals :: CATALG_1:def 12
[2,<*a,b,c*>];
correctness ;

end;

:: deftheorem defines idsym CATALG_1:def 10 :

:: deftheorem defines homsym CATALG_1:def 11 :

:: deftheorem defines compsym CATALG_1:def 12 :

theorem Th20: :: CATALG_1:20

for A being non empty set
for S being CatSignature of A
for a being Element of A holds
( idsym a in the OperSymbols 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 OperSymbols 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 by Th20;
let b be Element of A;
:: original: homsym
redefine func homsym a,b -> SortSymbol of (CatSign A);
correctness by Th20;
let c be Element of A;
:: original: compsym
redefine func compsym a,b,c -> OperSymbol of (CatSign A);
correctness by Th20;

end;

theorem Th21: :: CATALG_1:21

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

proof end;

theorem Th22: :: CATALG_1:22

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

proof end;

theorem Th23: :: CATALG_1:23

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 Th24: :: CATALG_1:24

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 Th25: :: CATALG_1:25

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 Th26: :: CATALG_1:26

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 Th27: :: CATALG_1:27

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 Th28: :: CATALG_1:28

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 Def5;

theorem Th29: :: CATALG_1:29

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 Def5;

definition

let C1, C2 be Category;
let F be Functor of C1,C2;
func Upsilon F -> Function of the carrier of (CatSign the Objects of C1),the carrier of (CatSign the Objects of C2) means :Def13: :: CATALG_1:def 13
for s being SortSymbol of (CatSign the Objects of C1) holds it . s = [0,((Obj F) * (s `2 ))];
uniqueness

proof end;

existence

proof end;

func Psi F -> Function of the OperSymbols of (CatSign the Objects of C1),the OperSymbols of (CatSign the Objects of C2) means :Def14: :: CATALG_1:def 14
for o being OperSymbol of (CatSign the Objects of C1) holds it . o = [(o `1 ),((Obj F) * (o `2 ))];
uniqueness

proof end;

existence

proof end;

end;

:: deftheorem Def13 defines Upsilon CATALG_1:def 13 :

:: deftheorem Def14 defines Psi CATALG_1:def 14 :

Lm2: now

let x be set ; :: thesis:
let f be Function; :: thesis:
assume x in dom f ; :: thesis:
then ( rng <*x*> = {x} & {x} c= dom f ) by FINSEQ_1:56, ZFMISC_1:37;
then A1: dom (f * <*x*>) = dom <*x*> by RELAT_1:46
.= Seg 1 by FINSEQ_1:55 ;
then reconsider p = f * <*x*> as FinSequence by FINSEQ_1:def 2;
( 1 in {1} & <*x*> . 1 = x ) by FINSEQ_1:57, TARSKI:def 1;
then ( len p = 1 & p . 1 = f . x ) by A1, FINSEQ_1:4, FINSEQ_1:def 3, FUNCT_1:22;
hence f * <*x*> = <*(f . x)*> by FINSEQ_1:57; :: thesis:

end;

theorem Th30: :: CATALG_1:30

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 Th31: :: CATALG_1:31

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 Th32: :: CATALG_1:32

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 Th33: :: CATALG_1:33

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

proof end;

theorem Th34: :: CATALG_1:34

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

proof end;

Lm3: for C being Category
for A being MSAlgebra of CatSign the Objects 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 , set , set ) -> set , F₅( set ) -> set } :

ex A being strict MSAlgebra of 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 of CatSign the Objects of C means :Def15: :: CATALG_1:def 15
( ( 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

proof end;

correctness

proof end;

end;

:: deftheorem Def15 defines MSAlg CATALG_1:def 15 :

theorem :: CATALG_1:35

canceled;

theorem Th36: :: CATALG_1:36

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

proof end;

theorem Th37: :: CATALG_1:37

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 feasible disjoint_valued ;
coherence

proof end;

end;

theorem Th38: :: CATALG_1:38

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

proof end;

theorem Th39: :: CATALG_1:39

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 Th40: :: CATALG_1:40

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 Objects of C1),(Upsilon F),(Psi F)) st H = F -MSF the carrier of (CatSign the Objects of C1),the Sorts of (MSAlg C1) holds
H # x = <*(F . g),(F . f)*>

proof end;

theorem :: CATALG_1:41

canceled;

theorem Th42: :: CATALG_1:42

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_1:43

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_1:44

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_1:45

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

proof end;