:: MSAFREE semantic presentation

theorem Th1: :: MSAFREE:1

for I being set
for J being non empty set
for f being Function of I,J *
for X being ManySortedSet of J
for p being Element of J *
for x being set st x in I & p = f . x holds
((X # ) * f) . x = product (X * p)

proof end;

definition

let I be set ;
let A, B be ManySortedSet of I;
let C be ManySortedSubset of A;
let F be ManySortedFunction of A,B;
func F || C -> ManySortedFunction of C,B means :Def1: :: MSAFREE:def 1
for i being set st i in I holds
for f being Function of A . i,B . i st f = F . i holds
it . i = f | (C . i);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines || MSAFREE:def 1 :

definition

let I be set ;
let X be ManySortedSet of I;
let i be set ;
assume A1: i in I ;
func coprod i,X -> set means :Def2: :: MSAFREE:def 2
for x being set holds
( x in it iff ex a being set st
( a in X . i & x = [a,i] ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines coprod MSAFREE:def 2 :

notation

let I be set ;
let X be ManySortedSet of I;
synonym coprod X for disjoin I;

end;

definition

let I be set ;
let X be ManySortedSet of I;
:: original: coprod
redefine func coprod X -> ManySortedSet of I means :Def3: :: MSAFREE:def 3
for i being set st i in I holds
it . i = coprod i,X;
coherence

proof end;

compatibility

proof end;

end;

:: deftheorem Def3 defines coprod MSAFREE:def 3 :

registration

let I be non empty set ;
let X be V3 ManySortedSet of I;
cluster coprod -> non-empty ;
coherence

proof end;

end;

registration

let I be non empty set ;
let X be V3 ManySortedSet of I;
cluster Union X -> non empty ;
coherence

proof end;

end;

theorem :: MSAFREE:2

for I being set
for X being ManySortedSet of I
for i being set st i in I holds
( X . i <> {} iff (coprod X) . i <> {} )

proof end;

definition

let S be non empty non void ManySortedSign ;
let U0 be MSAlgebra of S;
mode GeneratorSet of U0 -> MSSubset of U0 means :Def4: :: MSAFREE:def 4
the Sorts of (GenMSAlg it) = the Sorts of U0;
existence

proof end;

end;

:: deftheorem Def4 defines GeneratorSet MSAFREE:def 4 :

theorem :: MSAFREE:3

for S being non empty non void ManySortedSign
for U0 being strict non-empty MSAlgebra of S
for A being MSSubset of U0 holds
( A is GeneratorSet of U0 iff GenMSAlg A = U0 )

proof end;

definition

let S be non empty non void ManySortedSign ;
let U0 be MSAlgebra of S;
let IT be GeneratorSet of U0;
attr IT is free means :Def5: :: MSAFREE:def 5
for U1 being non-empty MSAlgebra of S
for f being ManySortedFunction of IT,the Sorts of U1 ex h being ManySortedFunction of U0,U1 st
( h is_homomorphism U0,U1 & h || IT = f );

end;

:: deftheorem Def5 defines free MSAFREE:def 5 :

definition

let S be non empty non void ManySortedSign ;
let IT be MSAlgebra of S;
attr IT is free means :Def6: :: MSAFREE:def 6
ex G being GeneratorSet of IT st G is free;

end;

:: deftheorem Def6 defines free MSAFREE:def 6 :

theorem Th4: :: MSAFREE:4

for S being non empty non void ManySortedSign
for X being ManySortedSet of the carrier of S holds Union (coprod X) misses [:the OperSymbols of S,{the carrier of S}:]

proof end;

registration

let S be non void ManySortedSign ;
cluster the OperSymbols of S -> non empty ;
coherence by MSUALG_1:def 5;

end;

definition

let S be non empty non void ManySortedSign ;
let X be ManySortedSet of the carrier of S;
canceled;
canceled;
func REL X -> Relation of [:the OperSymbols of S,{the carrier of S}:] \/ (Union (coprod X)),([:the OperSymbols of S,{the carrier of S}:] \/ (Union (coprod X))) * means :Def9: :: MSAFREE:def 9
for a being Element of [:the OperSymbols of S,{the carrier of S}:] \/ (Union (coprod X))
for b being Element of ([:the OperSymbols of S,{the carrier of S}:] \/ (Union (coprod X))) * holds
( [a,b] in it iff ( a in [:the OperSymbols of S,{the carrier of S}:] & ( for o being OperSymbol of S st [o,the carrier of S] = a holds
( len b = len (the_arity_of o) & ( for x being set st x in dom b holds
( ( b . x in [:the OperSymbols of S,{the carrier of S}:] implies for o1 being OperSymbol of S st [o1,the carrier of S] = b . x holds
the_result_sort_of o1 = (the_arity_of o) . x ) & ( b . x in Union (coprod X) implies b . x in coprod ((the_arity_of o) . x),X ) ) ) ) ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem MSAFREE:def 7 :

:: deftheorem MSAFREE:def 8 :

:: deftheorem Def9 defines REL MSAFREE:def 9 :

theorem Th5: :: MSAFREE:5

for S being non empty non void ManySortedSign
for X being ManySortedSet of the carrier of S
for o being OperSymbol of S
for b being Element of ([:the OperSymbols of S,{the carrier of S}:] \/ (Union (coprod X))) * holds
( [[o,the carrier of S],b] in REL X iff ( len b = len (the_arity_of o) & ( for x being set st x in dom b holds
( ( b . x in [:the OperSymbols of S,{the carrier of S}:] implies for o1 being OperSymbol of S st [o1,the carrier of S] = b . x holds
the_result_sort_of o1 = (the_arity_of o) . x ) & ( b . x in Union (coprod X) implies b . x in coprod ((the_arity_of o) . x),X ) ) ) ) )

proof end;

definition

let S be non empty non void ManySortedSign ;
let X be ManySortedSet of the carrier of S;
func DTConMSA X -> DTConstrStr equals :: MSAFREE:def 10
DTConstrStr(# ([:the OperSymbols of S,{the carrier of S}:] \/ (Union (coprod X))),(REL X) #);
correctness ;

end;

:: deftheorem defines DTConMSA MSAFREE:def 10 :

registration

let S be non empty non void ManySortedSign ;
let X be ManySortedSet of the carrier of S;
cluster DTConMSA X -> non empty strict ;
coherence ;

end;

theorem Th6: :: MSAFREE:6

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S holds
( NonTerminals (DTConMSA X) = [:the OperSymbols of S,{the carrier of S}:] & Terminals (DTConMSA X) = Union (coprod X) )

proof end;

registration

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
cluster DTConMSA X -> non empty strict with_terminals with_nonterminals with_useful_nonterminals ;
coherence

proof end;

end;

theorem Th7: :: MSAFREE:7

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S
for t being set holds
( t in Terminals (DTConMSA X) iff ex s being SortSymbol of S ex x being set st
( x in X . s & t = [x,s] ) )

proof end;

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let o be OperSymbol of S;
func Sym o,X -> Symbol of (DTConMSA X) equals :: MSAFREE:def 11
[o,the carrier of S];
coherence

proof end;

end;

:: deftheorem defines Sym MSAFREE:def 11 :

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let s be SortSymbol of S;
func FreeSort X,s -> Subset of (TS (DTConMSA X)) equals :: MSAFREE:def 12
{ a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) } ;
coherence

proof end;

end;

:: deftheorem defines FreeSort MSAFREE:def 12 :

registration

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let s be SortSymbol of S;
cluster FreeSort X,s -> non empty ;
coherence

proof end;

end;

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
func FreeSort X -> ManySortedSet of the carrier of S means :Def13: :: MSAFREE:def 13
for s being SortSymbol of S holds it . s = FreeSort X,s;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def13 defines FreeSort MSAFREE:def 13 :

registration

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
cluster FreeSort X -> V3 ;
coherence

proof end;

end;

theorem Th8: :: MSAFREE:8

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S
for o being OperSymbol of S
for x being set st x in (((FreeSort X) # ) * the Arity of S) . o holds
x is FinSequence of TS (DTConMSA X)

proof end;

theorem Th9: :: MSAFREE:9

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S
for o being OperSymbol of S
for p being FinSequence of TS (DTConMSA X) holds
( p in (((FreeSort X) # ) * the Arity of S) . o iff ( dom p = dom (the_arity_of o) & ( for n being Nat st n in dom p holds
p . n in FreeSort X,((the_arity_of o) /. n) ) ) )

proof end;

theorem Th10: :: MSAFREE:10

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S
for o being OperSymbol of S
for p being FinSequence of TS (DTConMSA X) holds
( Sym o,X ==> roots p iff p in (((FreeSort X) # ) * the Arity of S) . o )

proof end;

theorem :: MSAFREE:11

canceled;

theorem Th12: :: MSAFREE:12

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S holds union (rng (FreeSort X)) = TS (DTConMSA X)

proof end;

theorem Th13: :: MSAFREE:13

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S
for s1, s2 being SortSymbol of S st s1 <> s2 holds
(FreeSort X) . s1 misses (FreeSort X) . s2

proof end;

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let o be OperSymbol of S;
func DenOp o,X -> Function of (((FreeSort X) # ) * the Arity of S) . o,((FreeSort X) * the ResultSort of S) . o means :Def14: :: MSAFREE:def 14
for p being FinSequence of TS (DTConMSA X) st Sym o,X ==> roots p holds
it . p = (Sym o,X) -tree p;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def14 defines DenOp MSAFREE:def 14 :

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
func FreeOper X -> ManySortedFunction of ((FreeSort X) # ) * the Arity of S,(FreeSort X) * the ResultSort of S means :Def15: :: MSAFREE:def 15
for o being OperSymbol of S holds it . o = DenOp o,X;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def15 defines FreeOper MSAFREE:def 15 :

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
func FreeMSA X -> MSAlgebra of S equals :: MSAFREE:def 16
MSAlgebra(# (FreeSort X),(FreeOper X) #);
coherence ;

end;

:: deftheorem defines FreeMSA MSAFREE:def 16 :

registration

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
cluster FreeMSA X -> strict non-empty ;
coherence by MSUALG_1:def 8;

end;

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let s be SortSymbol of S;
func FreeGen s,X -> Subset of ((FreeSort X) . s) means :Def17: :: MSAFREE:def 17
for x being set holds
( x in it iff ex a being set st
( a in X . s & x = root-tree [a,s] ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def17 defines FreeGen MSAFREE:def 17 :

registration

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let s be SortSymbol of S;
cluster FreeGen s,X -> non empty ;
coherence

proof end;

end;

theorem Th14: :: MSAFREE:14

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S
for s being SortSymbol of S holds FreeGen s,X = { (root-tree t) where t is Symbol of (DTConMSA X) : ( t in Terminals (DTConMSA X) & t `2 = s ) }

proof end;

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
func FreeGen X -> GeneratorSet of FreeMSA X means :Def18: :: MSAFREE:def 18
for s being SortSymbol of S holds it . s = FreeGen s,X;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def18 defines FreeGen MSAFREE:def 18 :

theorem :: MSAFREE:15

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S holds FreeGen X is non-empty

proof end;

theorem :: MSAFREE:16

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S holds union (rng (FreeGen X)) = { (root-tree t) where t is Symbol of (DTConMSA X) : t in Terminals (DTConMSA X) }

proof end;

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let s be SortSymbol of S;
func Reverse s,X -> Function of FreeGen s,X,X . s means :Def19: :: MSAFREE:def 19
for t being Symbol of (DTConMSA X) st root-tree t in FreeGen s,X holds
it . (root-tree t) = t `1 ;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def19 defines Reverse MSAFREE:def 19 :

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
func Reverse X -> ManySortedFunction of FreeGen X,X means :Def20: :: MSAFREE:def 20
for s being SortSymbol of S holds it . s = Reverse s,X;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def20 defines Reverse MSAFREE:def 20 :

definition

let S be non empty non void ManySortedSign ;
let X, A be V3 ManySortedSet of the carrier of S;
let F be ManySortedFunction of FreeGen X,A;
let t be Symbol of (DTConMSA X);
assume A1: t in Terminals (DTConMSA X) ;
func pi F,A,t -> Element of Union A means :Def21: :: MSAFREE:def 21
for f being Function st f = F . (t `2 ) holds
it = f . (root-tree t);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def21 defines pi MSAFREE:def 21 :

definition

let S be non empty non void ManySortedSign ;
let X be V3 ManySortedSet of the carrier of S;
let t be Symbol of (DTConMSA X);
assume A1: ex p being FinSequence st t ==> p ;
func @ X,t -> OperSymbol of S means :Def22: :: MSAFREE:def 22
[it,the carrier of S] = t;
existence

proof end;

uniqueness by ZFMISC_1:33;

end;

:: deftheorem Def22 defines @ MSAFREE:def 22 :

definition

let S be non empty non void ManySortedSign ;
let U0 be non-empty MSAlgebra of S;
let o be OperSymbol of S;
let p be FinSequence;
assume A1: p in Args o,U0 ;
func pi o,U0,p -> Element of Union the Sorts of U0 equals :Def23: :: MSAFREE:def 23
(Den o,U0) . p;
coherence

proof end;

end;

:: deftheorem Def23 defines pi MSAFREE:def 23 :

theorem Th17: :: MSAFREE:17

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S holds FreeGen X is free

proof end;

theorem Th18: :: MSAFREE:18

for S being non empty non void ManySortedSign
for X being V3 ManySortedSet of the carrier of S holds FreeMSA X is free

proof end;

registration

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

proof end;

end;

registration

let S be non empty non void ManySortedSign ;
let U0 be free MSAlgebra of S;
cluster free GeneratorSet of U0;
existence by Def6;

end;

theorem Th19: :: MSAFREE:19

for S being non empty non void ManySortedSign
for U1 being non-empty MSAlgebra of S ex U0 being strict non-empty free MSAlgebra of S ex F being ManySortedFunction of U0,U1 st F is_epimorphism U0,U1

proof end;

theorem :: MSAFREE:20

for S being non empty non void ManySortedSign
for U1 being strict non-empty MSAlgebra of S ex U0 being strict non-empty free MSAlgebra of S ex F being ManySortedFunction of U0,U1 st
( F is_epimorphism U0,U1 & Image F = U1 )

proof end;