:: The Correspondence Between Monotonic Many Sorted Signatures and Well-Founded Graphs. {P}art {II}
:: by Czes{\l}aw Byli\'nski and Piotr Rudnicki
::
:: Received April 10, 1996
:: Copyright (c) 1996-2011 Association of Mizar Users

begin

definition

let S be ManySortedSign ;
defpred S₁[ set ] means ex op, v being set st
( $1 = [op,v] & op in the carrier' of S & v in the carrier of S & ex n being Nat ex args being Element of the carrier of S * st
( the Arity of S . op = args & n in dom args & args . n = v ) );
func InducedEdges S -> set means :Def1: :: MSSCYC_2:def 1
for x being set holds
( x in it iff ex op, v being set st
( x = [op,v] & op in the carrier' of S & v in the carrier of S & ex n being Nat ex args being Element of the carrier of S * st
( the Arity of S . op = args & n in dom args & args . n = v ) ) );
existence
ex b₁ being set st
for x being set holds
( x in b₁ iff ex op, v being set st
( x = [op,v] & op in the carrier' of S & v in the carrier of S & ex n being Nat ex args being Element of the carrier of S * st
( the Arity of S . op = args & n in dom args & args . n = v ) ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff ex op, v being set st
( x = [op,v] & op in the carrier' of S & v in the carrier of S & ex n being Nat ex args being Element of the carrier of S * st
( the Arity of S . op = args & n in dom args & args . n = v ) ) ) ) & ( for x being set holds
( x in b₂ iff ex op, v being set st
( x = [op,v] & op in the carrier' of S & v in the carrier of S & ex n being Nat ex args being Element of the carrier of S * st
( the Arity of S . op = args & n in dom args & args . n = v ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines InducedEdges MSSCYC_2:def 1 :
for S being ManySortedSign
for b₂ being set holds
( b₂ = InducedEdges S iff for x being set holds
( x in b₂ iff ex op, v being set st
( x = [op,v] & op in the carrier' of S & v in the carrier of S & ex n being Nat ex args being Element of the carrier of S * st
( the Arity of S . op = args & n in dom args & args . n = v ) ) ) );

theorem Th1: :: MSSCYC_2:1

for S being ManySortedSign holds InducedEdges S c= [: the carrier' of S, the carrier of S:]

proof end;

definition

let S be ManySortedSign ;
set IE = InducedEdges S;
set IV = the carrier of S;
func InducedSource S -> Function of (InducedEdges S), the carrier of S means :Def2: :: MSSCYC_2:def 2
for e being set st e in InducedEdges S holds
it . e = e `2 ;
existence
ex b<sub>1</sub> being Function of (InducedEdges S), the carrier of S st
for e being set st e in InducedEdges S holds
b<sub>1</sub> . e = e `2

proof end;

uniqueness
for b₁, b₂ being Function of (InducedEdges S), the carrier of S st ( for e being set st e in InducedEdges S holds
b₁ . e = e `2 ) & ( for e being set st e in InducedEdges S holds
b<sub>2</sub> . e = e `2 ) holds
b₁ = b₂

proof end;

set OS = the carrier' of S;
set RS = the ResultSort of S;
func InducedTarget S -> Function of (InducedEdges S), the carrier of S means :Def3: :: MSSCYC_2:def 3
for e being set st e in InducedEdges S holds
it . e = the ResultSort of S . (e `1);
existence
ex b<sub>1</sub> being Function of (InducedEdges S), the carrier of S st
for e being set st e in InducedEdges S holds
b<sub>1</sub> . e = the ResultSort of S . (e `1)

proof end;

uniqueness
for b₁, b₂ being Function of (InducedEdges S), the carrier of S st ( for e being set st e in InducedEdges S holds
b₁ . e = the ResultSort of S . (e `1) ) & ( for e being set st e in InducedEdges S holds
b<sub>2</sub> . e = the ResultSort of S . (e `1) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines InducedSource MSSCYC_2:def 2 :
for S being ManySortedSign
for b₂ being Function of (InducedEdges S), the carrier of S holds
( b₂ = InducedSource S iff for e being set st e in InducedEdges S holds
b₂ . e = e `2 );

:: deftheorem Def3 defines InducedTarget MSSCYC_2:def 3 :
for S being ManySortedSign
for b₂ being Function of (InducedEdges S), the carrier of S holds
( b₂ = InducedTarget S iff for e being set st e in InducedEdges S holds
b₂ . e = the ResultSort of S . (e `1) );

definition

let S be non empty ManySortedSign ;
func InducedGraph S -> Graph equals :: MSSCYC_2:def 4
MultiGraphStruct(# the carrier of S,(InducedEdges S),(InducedSource S),(InducedTarget S) #);
coherence
MultiGraphStruct(# the carrier of S,(InducedEdges S),(InducedSource S),(InducedTarget S) #) is Graph ;

end;

:: deftheorem defines InducedGraph MSSCYC_2:def 4 :
for S being non empty ManySortedSign holds InducedGraph S = MultiGraphStruct(# the carrier of S,(InducedEdges S),(InducedSource S),(InducedTarget S) #);

theorem Th2: :: MSSCYC_2:2

for S being non empty non void ManySortedSign
for X being V2() ManySortedSet of the carrier of S
for v being SortSymbol of S
for n being Nat st 1 <= n holds
( ex t being Element of the Sorts of (FreeMSA X) . v st depth t = n iff ex c being directed Chain of InducedGraph S st
( len c = n & (vertex-seq c) . ((len c) + 1) = v ) )

proof end;

theorem :: MSSCYC_2:3

for S being non empty void ManySortedSign holds
( S is monotonic iff InducedGraph S is well-founded )

proof end;

theorem :: MSSCYC_2:4

for S being non empty non void ManySortedSign st S is monotonic holds
InducedGraph S is well-founded

proof end;

theorem Th5: :: MSSCYC_2:5

for S being non empty non void ManySortedSign
for X being V2() V31() ManySortedSet of the carrier of S st S is finitely_operated holds
for n being Nat
for v being SortSymbol of S holds { t where t is Element of the Sorts of (FreeMSA X) . v : depth t <= n } is finite

proof end;

theorem :: MSSCYC_2:6

for S being non empty non void ManySortedSign st S is finitely_operated & InducedGraph S is well-founded holds
S is monotonic

proof end;