:: MSUALG_6 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
let S be non empty ManySortedSign ;
let A be MSAlgebra of S;
let s be SortSymbol of S;
mode Element of A,s is Element of the Sorts of A . s;
end;

definition
let I be set ;
let A be ManySortedSet of I;
let h1, h2 be ManySortedFunction of A,A;
:: original: **
redefine func h2 ** h1 -> ManySortedFunction of A,A;
coherence
h2 ** h1 is ManySortedFunction of A,A
proof end;
end;

theorem Th1: :: MSUALG_6:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for o being OperSymbol of S
for a being set st a in Args o,A holds
a is Function
proof end;

theorem Th2: :: MSUALG_6:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for o being OperSymbol of S
for a being Function st a in Args o,A holds
( dom a = dom (the_arity_of o) & ( for i being set st i in dom (the_arity_of o) holds
a . i in the Sorts of A . ((the_arity_of o) /. i) ) )
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra of S;
attr A is feasible means :Def1: :: MSUALG_6:def 1
for o being OperSymbol of S st Args o,A <> {} holds
Result o,A <> {} ;
end;

:: deftheorem Def1 defines feasible MSUALG_6:def 1 :
for S being non empty non void ManySortedSign
for A being MSAlgebra of S holds
( A is feasible iff for o being OperSymbol of S st Args o,A <> {} holds
Result o,A <> {} );

theorem Th3: :: MSUALG_6:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra of S holds
( Args o,A <> {} iff for i being Nat st i in dom (the_arity_of o) holds
the Sorts of A . ((the_arity_of o) /. i) <> {} )
proof end;

registration
let S be non empty non void ManySortedSign ;
cluster non-empty -> feasible MSAlgebra of S;
coherence
for b1 being MSAlgebra of S st b1 is non-empty holds
b1 is feasible
proof end;
end;

registration
let S be non empty non void ManySortedSign ;
cluster non-empty feasible MSAlgebra of S;
existence
ex b1 being MSAlgebra of S st b1 is non-empty
proof end;
end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra of S;
mode Endomorphism of A -> ManySortedFunction of A,A means :Def2: :: MSUALG_6:def 2
it is_homomorphism A,A;
existence
ex b1 being ManySortedFunction of A,A st b1 is_homomorphism A,A
proof end;
end;

:: deftheorem Def2 defines Endomorphism MSUALG_6:def 2 :
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for b3 being ManySortedFunction of A,A holds
( b3 is Endomorphism of A iff b3 is_homomorphism A,A );

theorem Th4: :: MSUALG_6:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being MSAlgebra of S holds id the Sorts of A is Endomorphism of A
proof end;

theorem Th5: :: MSUALG_6:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for h1, h2 being ManySortedFunction of A,A
for o being OperSymbol of S
for a being Element of Args o,A st a in Args o,A holds
h2 # (h1 # a) = (h2 ** h1) # a
proof end;

theorem Th6: :: MSUALG_6:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for h1, h2 being Endomorphism of A holds h2 ** h1 is Endomorphism of A
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra of S;
let h1, h2 be Endomorphism of A;
:: original: **
redefine func h2 ** h1 -> Endomorphism of A;
coherence
h2 ** h1 is Endomorphism of A
by Th6;
end;

definition
let S be non empty non void ManySortedSign ;
func TranslationRel S -> Relation of the carrier of S means :Def3: :: MSUALG_6:def 3
for s1, s2 being SortSymbol of S holds
( [s1,s2] in it iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 ) ) );
existence
ex b1 being Relation of the carrier of S st
for s1, s2 being SortSymbol of S holds
( [s1,s2] in b1 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 ) ) )
proof end;
correctness
uniqueness
for b1, b2 being Relation of the carrier of S st ( for s1, s2 being SortSymbol of S holds
( [s1,s2] in b1 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 ) ) ) ) & ( for s1, s2 being SortSymbol of S holds
( [s1,s2] in b2 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 ) ) ) ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def3 defines TranslationRel MSUALG_6:def 3 :
for S being non empty non void ManySortedSign
for b2 being Relation of the carrier of S holds
( b2 = TranslationRel S iff for s1, s2 being SortSymbol of S holds
( [s1,s2] in b2 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 ) ) ) );

theorem Th7: :: MSUALG_6:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra of S
for a being Function st a in Args o,A holds
for i being Nat
for x being Element of A,((the_arity_of o) /. i) holds a +* i,x in Args o,A
proof end;

theorem Th8: :: MSUALG_6:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A1, A2 being MSAlgebra of S
for h being ManySortedFunction of A1,A2
for o being OperSymbol of S st Args o,A1 <> {} & Args o,A2 <> {} holds
for i being Nat st i in dom (the_arity_of o) holds
for x being Element of A1,((the_arity_of o) /. i) holds (h . ((the_arity_of o) /. i)) . x in the Sorts of A2 . ((the_arity_of o) /. i)
proof end;

theorem Th9: :: MSUALG_6:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for i being Nat st i in dom (the_arity_of o) holds
for A1, A2 being MSAlgebra of S
for h being ManySortedFunction of A1,A2
for a, b being Element of Args o,A1 st a in Args o,A1 & h # a in Args o,A2 holds
for f, g1, g2 being Function st f = a & g1 = h # a & g2 = h # b holds
for x being Element of A1,((the_arity_of o) /. i) st b = f +* i,x holds
( g2 . i = (h . ((the_arity_of o) /. i)) . x & h # b = g1 +* i,(g2 . i) )
proof end;

definition
let S be non empty non void ManySortedSign ;
let o be OperSymbol of S;
let i be Nat;
let A be MSAlgebra of S;
let a be Function;
func transl o,i,a,A -> Function means :Def4: :: MSUALG_6:def 4
( dom it = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
it . x = (Den o,A) . (a +* i,x) ) );
existence
ex b1 being Function st
( dom b1 = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
b1 . x = (Den o,A) . (a +* i,x) ) )
proof end;
uniqueness
for b1, b2 being Function st dom b1 = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
b1 . x = (Den o,A) . (a +* i,x) ) & dom b2 = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
b2 . x = (Den o,A) . (a +* i,x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines transl MSUALG_6:def 4 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for i being Nat
for A being MSAlgebra of S
for a, b6 being Function holds
( b6 = transl o,i,a,A iff ( dom b6 = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
b6 . x = (Den o,A) . (a +* i,x) ) ) );

theorem Th10: :: MSUALG_6:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for i being Nat st i in dom (the_arity_of o) holds
for A being feasible MSAlgebra of S
for a being Function st a in Args o,A holds
transl o,i,a,A is Function of the Sorts of A . ((the_arity_of o) /. i),the Sorts of A . (the_result_sort_of o)
proof end;

definition
let S be non empty non void ManySortedSign ;
let s1, s2 be SortSymbol of S;
let A be MSAlgebra of S;
let f be Function;
pred f is_e.translation_of A,s1,s2 means :Def5: :: MSUALG_6:def 5
ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 & ex a being Function st
( a in Args o,A & f = transl o,i,a,A ) ) );
end;

:: deftheorem Def5 defines is_e.translation_of MSUALG_6:def 5 :
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being MSAlgebra of S
for f being Function holds
( f is_e.translation_of A,s1,s2 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Nat st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 & ex a being Function st
( a in Args o,A & f = transl o,i,a,A ) ) ) );

theorem Th11: :: MSUALG_6:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being feasible MSAlgebra of S
for f being Function st f is_e.translation_of A,s1,s2 holds
( f is Function of the Sorts of A . s1,the Sorts of A . s2 & the Sorts of A . s1 <> {} & the Sorts of A . s2 <> {} )
proof end;

theorem Th12: :: MSUALG_6:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being MSAlgebra of S
for f being Function st f is_e.translation_of A,s1,s2 holds
[s1,s2] in TranslationRel S
proof end;

theorem :: MSUALG_6:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being non-empty MSAlgebra of S st [s1,s2] in TranslationRel S holds
ex f being Function st f is_e.translation_of A,s1,s2
proof end;

theorem Th14: :: MSUALG_6:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being feasible MSAlgebra of S
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for q being RedSequence of TranslationRel S
for p being Function-yielding FinSequence st len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Nat
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) holds
( compose p,(the Sorts of A . s1) is Function of the Sorts of A . s1,the Sorts of A . s2 & ( p <> {} implies ( the Sorts of A . s1 <> {} & the Sorts of A . s2 <> {} ) ) )
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let s1, s2 be SortSymbol of S;
assume A1: TranslationRel S reduces s1,s2 ;
mode Translation of A,s1,s2 -> Function of the Sorts of A . s1,the Sorts of A . s2 means :Def6: :: MSUALG_6:def 6
ex q being RedSequence of TranslationRel S ex p being Function-yielding FinSequence st
( it = compose p,(the Sorts of A . s1) & len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Nat
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) );
existence
ex b1 being Function of the Sorts of A . s1,the Sorts of A . s2 ex q being RedSequence of TranslationRel S ex p being Function-yielding FinSequence st
( b1 = compose p,(the Sorts of A . s1) & len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Nat
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) )
proof end;
end;

:: deftheorem Def6 defines Translation MSUALG_6:def 6 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for b5 being Function of the Sorts of A . s1,the Sorts of A . s2 holds
( b5 is Translation of A,s1,s2 iff ex q being RedSequence of TranslationRel S ex p being Function-yielding FinSequence st
( b5 = compose p,(the Sorts of A . s1) & len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Nat
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) ) );

theorem :: MSUALG_6:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for q being RedSequence of TranslationRel S
for p being Function-yielding FinSequence st len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Nat
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) holds
compose p,(the Sorts of A . s1) is Translation of A,s1,s2
proof end;

theorem Th16: :: MSUALG_6:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s being SortSymbol of S holds id (the Sorts of A . s) is Translation of A,s,s
proof end;

theorem Th17: :: MSUALG_6:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s1, s2 being SortSymbol of S
for f being Function st f is_e.translation_of A,s1,s2 holds
( TranslationRel S reduces s1,s2 & f is Translation of A,s1,s2 )
proof end;

theorem Th18: :: MSUALG_6:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s1,s2 & TranslationRel S reduces s2,s3 holds
for t1 being Translation of A,s1,s2
for t2 being Translation of A,s2,s3 holds t2 * t1 is Translation of A,s1,s3
proof end;

theorem Th19: :: MSUALG_6:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for t being Translation of A,s1,s2
for f being Function st f is_e.translation_of A,s2,s3 holds
f * t is Translation of A,s1,s3
proof end;

theorem :: MSUALG_6:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s2,s3 holds
for f being Function st f is_e.translation_of A,s1,s2 holds
for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3
proof end;

scheme :: MSUALG_6:sch 1
TranslationInd{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra of F1(), P1[ set , set , set ] } :
for s1, s2 being SortSymbol of F1() st TranslationRel F1() reduces s1,s2 holds
for t being Translation of F2(),s1,s2 holds P1[t,s1,s2]
provided
A1: for s being SortSymbol of F1() holds P1[ id (the Sorts of F2() . s),s,s] and
A2: for s1, s2, s3 being SortSymbol of F1() st TranslationRel F1() reduces s1,s2 holds
for t being Translation of F2(),s1,s2 st P1[t,s1,s2] holds
for f being Function st f is_e.translation_of F2(),s2,s3 holds
P1[f * t,s1,s3]
proof end;

theorem Th21: :: MSUALG_6:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A1, A2 being non-empty MSAlgebra of S
for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 holds
for o being OperSymbol of S
for i being Nat st i in dom (the_arity_of o) holds
for a being Element of Args o,A1 holds (h . (the_result_sort_of o)) * (transl o,i,a,A1) = (transl o,i,(h # a),A2) * (h . ((the_arity_of o) /. i))
proof end;

theorem :: MSUALG_6:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for h being Endomorphism of A
for o being OperSymbol of S
for i being Nat st i in dom (the_arity_of o) holds
for a being Element of Args o,A holds (h . (the_result_sort_of o)) * (transl o,i,a,A) = (transl o,i,(h # a),A) * (h . ((the_arity_of o) /. i))
proof end;

theorem Th23: :: MSUALG_6:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A1, A2 being non-empty MSAlgebra of S
for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 holds
for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A1,s1,s2 holds
ex T being Function of the Sorts of A2 . s1,the Sorts of A2 . s2 st
( T is_e.translation_of A2,s1,s2 & T * (h . s1) = (h . s2) * t )
proof end;

theorem :: MSUALG_6:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for h being Endomorphism of A
for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A,s1,s2 holds
ex T being Function of the Sorts of A . s1,the Sorts of A . s2 st
( T is_e.translation_of A,s1,s2 & T * (h . s1) = (h . s2) * t )
proof end;

theorem Th25: :: MSUALG_6:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A1, A2 being non-empty MSAlgebra of S
for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 holds
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for t being Translation of A1,s1,s2 ex T being Translation of A2,s1,s2 st T * (h . s1) = (h . s2) * t
proof end;

theorem Th26: :: MSUALG_6:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for h being Endomorphism of A
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for t being Translation of A,s1,s2 ex T being Translation of A,s1,s2 st T * (h . s1) = (h . s2) * t
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra of S;
let R be ManySortedRelation of A;
attr R is compatible means :Def7: :: MSUALG_6:def 7
for o being OperSymbol of S
for a, b being Function st a in Args o,A & b in Args o,A & ( for n being Nat st n in dom (the_arity_of o) holds
[(a . n),(b . n)] in R . ((the_arity_of o) /. n) ) holds
[((Den o,A) . a),((Den o,A) . b)] in R . (the_result_sort_of o);
attr R is invariant means :Def8: :: MSUALG_6:def 8
for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A,s1,s2 holds
for a, b being set st [a,b] in R . s1 holds
[(t . a),(t . b)] in R . s2;
attr R is stable means :Def9: :: MSUALG_6:def 9
for h being Endomorphism of A
for s being SortSymbol of S
for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s;
end;

:: deftheorem Def7 defines compatible MSUALG_6:def 7 :
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for R being ManySortedRelation of A holds
( R is compatible iff for o being OperSymbol of S
for a, b being Function st a in Args o,A & b in Args o,A & ( for n being Nat st n in dom (the_arity_of o) holds
[(a . n),(b . n)] in R . ((the_arity_of o) /. n) ) holds
[((Den o,A) . a),((Den o,A) . b)] in R . (the_result_sort_of o) );

:: deftheorem Def8 defines invariant MSUALG_6:def 8 :
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for R being ManySortedRelation of A holds
( R is invariant iff for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A,s1,s2 holds
for a, b being set st [a,b] in R . s1 holds
[(t . a),(t . b)] in R . s2 );

:: deftheorem Def9 defines stable MSUALG_6:def 9 :
for S being non empty non void ManySortedSign
for A being MSAlgebra of S
for R being ManySortedRelation of A holds
( R is stable iff for h being Endomorphism of A
for s being SortSymbol of S
for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s );

theorem :: MSUALG_6:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being MSEquivalence-like ManySortedRelation of A holds
( R is compatible iff R is MSCongruence of A )
proof end;

theorem Th28: :: MSUALG_6:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A holds
( R is invariant iff for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for f being Translation of A,s1,s2
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 )
proof end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
cluster MSEquivalence-like invariant -> MSEquivalence-like compatible ManySortedRelation of the Sorts of A,the Sorts of A;
coherence
for b1 being MSEquivalence-like ManySortedRelation of A st b1 is invariant holds
b1 is compatible
proof end;
cluster MSEquivalence-like compatible -> MSEquivalence-like invariant ManySortedRelation of the Sorts of A,the Sorts of A;
coherence
for b1 being MSEquivalence-like ManySortedRelation of A st b1 is compatible holds
b1 is invariant
proof end;
end;

registration
let X be non empty set ;
cluster id X -> non empty ;
coherence
not id X is empty
;
end;

scheme :: MSUALG_6:sch 2
MSRExistence{ F1() -> non empty set , F2() -> V3 ManySortedSet of F1(), P1[ set , set , set ] } :
ex R being ManySortedRelation of F2() st
for i being Element of F1()
for a, b being Element of F2() . i holds
( [a,b] in R . i iff P1[i,a,b] )
proof end;

scheme :: MSUALG_6:sch 3
MSRLambdaU{ F1() -> set , F2() -> ManySortedSet of F1(), F3( set ) -> set } :
( ex R being ManySortedRelation of F2() st
for i being set st i in F1() holds
R . i = F3(i) & ( for R1, R2 being ManySortedRelation of F2() st ( for i being set st i in F1() holds
R1 . i = F3(i) ) & ( for i being set st i in F1() holds
R2 . i = F3(i) ) holds
R1 = R2 ) )
provided
A1: for i being set st i in F1() holds
F3(i) is Relation of F2() . i,F2() . i
proof end;

definition
let I be set ;
let A be ManySortedSet of I;
func id I,A -> ManySortedRelation of A means :Def10: :: MSUALG_6:def 10
for i being set st i in I holds
it . i = id (A . i);
correctness
existence
ex b1 being ManySortedRelation of A st
for i being set st i in I holds
b1 . i = id (A . i)
;
uniqueness
for b1, b2 being ManySortedRelation of A st ( for i being set st i in I holds
b1 . i = id (A . i) ) & ( for i being set st i in I holds
b2 . i = id (A . i) ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def10 defines id MSUALG_6:def 10 :
for I being set
for A being ManySortedSet of I
for b3 being ManySortedRelation of A holds
( b3 = id I,A iff for i being set st i in I holds
b3 . i = id (A . i) );

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
cluster MSEquivalence-like -> V3 ManySortedRelation of the Sorts of A,the Sorts of A;
coherence
for b1 being ManySortedRelation of A st b1 is MSEquivalence-like holds
b1 is non-empty
proof end;
end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
cluster V3 MSEquivalence-like compatible invariant stable ManySortedRelation of the Sorts of A,the Sorts of A;
existence
ex b1 being ManySortedRelation of A st
( b1 is invariant & b1 is stable & b1 is MSEquivalence-like )
proof end;
end;

scheme :: MSUALG_6:sch 4
MSRelCl{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra of F1(), P1[ set , set , set ], P2[ set ], F3() -> ManySortedRelation of F2(), F4() -> ManySortedRelation of F2() } :
( P2[F4()] & F3() c= F4() & ( for P being ManySortedRelation of F2() st P2[P] & F3() c= P holds
F4() c= P ) )
provided
A1: for R being ManySortedRelation of F2() holds
( P2[R] iff for s1, s2 being SortSymbol of F1()
for f being Function of the Sorts of F2() . s1,the Sorts of F2() . s2 st P1[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) and
A2: for s1, s2, s3 being SortSymbol of F1()
for f1 being Function of the Sorts of F2() . s1,the Sorts of F2() . s2
for f2 being Function of the Sorts of F2() . s2,the Sorts of F2() . s3 st P1[f1,s1,s2] & P1[f2,s2,s3] holds
P1[f2 * f1,s1,s3] and
A3: for s being SortSymbol of F1() holds P1[ id (the Sorts of F2() . s),s,s] and
A4: for s being SortSymbol of F1()
for a, b being Element of F2(),s holds
( [a,b] in F4() . s iff ex s' being SortSymbol of F1() ex f being Function of the Sorts of F2() . s',the Sorts of F2() . s ex x, y being Element of F2(),s' st
( P1[f,s',s] & [x,y] in F3() . s' & a = f . x & b = f . y ) )
proof end;

Lm1: now
let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra of S
for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let A be non-empty MSAlgebra of S; :: thesis: for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let R, Q be ManySortedRelation of A; :: thesis: ( ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ) implies ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) )

defpred S1[ ManySortedRelation of A] means $1 is invariant;
defpred S2[ Function, SortSymbol of S, SortSymbol of S] means ( TranslationRel S reduces $2,$3 & $1 is Translation of A,$2,$3 );
assume A1: for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ; :: thesis: ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

A2: now
let R be ManySortedRelation of A; :: thesis: ( ( S1[R] implies for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) & ( ( for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] ) )

thus ( S1[R] implies for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) by Th28; :: thesis: ( ( for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] )

assume for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ; :: thesis: S1[R]
then for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for f being Translation of A,s1,s2
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ;
hence S1[R] by Th28; :: thesis: verum
end;
A3: for s1, s2, s3 being SortSymbol of S
for f1 being Function of the Sorts of A . s1,the Sorts of A . s2
for f2 being Function of the Sorts of A . s2,the Sorts of A . s3 st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3] by Th18, REWRITE1:17;
A4: for s being SortSymbol of S holds S2[ id (the Sorts of A . s),s,s] by Th16, REWRITE1:13;
thus ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) from MSUALG_6:sch 4( S A R s , A2, A3, A4, A1); :: thesis: verum
end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be ManySortedRelation of the Sorts of A;
func InvCl R -> invariant ManySortedRelation of A means :Def11: :: MSUALG_6:def 11
( R c= it & ( for Q being invariant ManySortedRelation of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being invariant ManySortedRelation of A st R c= b1 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being invariant ManySortedRelation of A st
( R c= b1 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def11 defines InvCl MSUALG_6:def 11 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for b4 being invariant ManySortedRelation of A holds
( b4 = InvCl R iff ( R c= b4 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b4 c= Q ) ) );

theorem Th29: :: MSUALG_6:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (InvCl R) . s iff ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in R . s' & a = t . x & b = t . y ) )
proof end;

theorem Th30: :: MSUALG_6:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being stable ManySortedRelation of A holds InvCl R is stable
proof end;

Lm2: now
let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra of S
for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let A be non-empty MSAlgebra of S; :: thesis: for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let R, Q be ManySortedRelation of A; :: thesis: ( ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ) implies ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) )

defpred S1[ ManySortedRelation of A] means $1 is stable;
defpred S2[ Function, SortSymbol of S, SortSymbol of S] means ( $2 = $3 & ex h being Endomorphism of A st $1 = h . $2 );
assume A1: for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s' being SortSymbol of S ex f being Function of the Sorts of A . s',the Sorts of A . s ex x, y being Element of A,s' st
( S2[f,s',s] & [x,y] in R . s' & a = f . x & b = f . y ) ) ; :: thesis: ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

A2: for R being ManySortedRelation of A holds
( S1[R] iff for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 )
proof
let R be ManySortedRelation of A; :: thesis: ( S1[R] iff for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 )

thus ( R is stable implies for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st s1 = s2 & ex h being Endomorphism of A st f = h . s1 holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) by Def9; :: thesis: ( ( for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] )

assume A3: for s1, s2 being SortSymbol of S
for f being Function of the Sorts of A . s1,the Sorts of A . s2 st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ; :: thesis: S1[R]
thus R is stable :: thesis: verum
proof
let h be Endomorphism of A; :: according to MSUALG_6:def 9 :: thesis: for s being SortSymbol of S
for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s

let s be SortSymbol of S; :: thesis: for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s

thus for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s by A3; :: thesis: verum
end;
end;
A4: for s1, s2, s3 being SortSymbol of S
for f1 being Function of the Sorts of A . s1,the Sorts of A . s2
for f2 being Function of the Sorts of A . s2,the Sorts of A . s3 st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3]
proof
let s1, s2, s3 be SortSymbol of S; :: thesis: for f1 being Function of the Sorts of A . s1,the Sorts of A . s2
for f2 being Function of the Sorts of A . s2,the Sorts of A . s3 st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3]

let f1 be Function of the Sorts of A . s1,the Sorts of A . s2; :: thesis: for f2 being Function of the Sorts of A . s2,the Sorts of A . s3 st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3]

let f2 be Function of the Sorts of A . s2,the Sorts of A . s3; :: thesis: ( S2[f1,s1,s2] & S2[f2,s2,s3] implies S2[f2 * f1,s1,s3] )
assume A5: s1 = s2 ; :: thesis: ( for h being Endomorphism of A holds not f1 = h . s1 or not S2[f2,s2,s3] or S2[f2 * f1,s1,s3] )
given h1 being Endomorphism of A such that A6: f1 = h1 . s1 ; :: thesis: ( not S2[f2,s2,s3] or S2[f2 * f1,s1,s3] )
assume A7: s2 = s3 ; :: thesis: ( for h being Endomorphism of A holds not f2 = h . s2 or S2[f2 * f1,s1,s3] )
given h2 being Endomorphism of A such that A8: f2 = h2 . s2 ; :: thesis: S2[f2 * f1,s1,s3]
thus s1 = s3 by A5, A7; :: thesis: ex h being Endomorphism of A st f2 * f1 = h . s1
reconsider h = h2 ** h1 as Endomorphism of A ;
take h ; :: thesis: f2 * f1 = h . s1
thus f2 * f1 = h . s1 by A5, A6, A8, MSUALG_3:2; :: thesis: verum
end;
A9: for s being SortSymbol of S holds S2[ id (the Sorts of A . s),s,s]
proof
let s be SortSymbol of S; :: thesis: S2[ id (the Sorts of A . s),s,s]
thus s = s ; :: thesis: ex h being Endomorphism of A st id (the Sorts of A . s) = h . s
reconsider h = id the Sorts of A as Endomorphism of A by Th4;
take h ; :: thesis: id (the Sorts of A . s) = h . s
thus id (the Sorts of A . s) = h . s by MSUALG_3:def 1; :: thesis: verum
end;
thus ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) from MSUALG_6:sch 4( S A R s , A2, A4, A9, A1); :: thesis: verum
end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be ManySortedRelation of the Sorts of A;
func StabCl R -> stable ManySortedRelation of A means :Def12: :: MSUALG_6:def 12
( R c= it & ( for Q being stable ManySortedRelation of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being stable ManySortedRelation of A st R c= b1 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being stable ManySortedRelation of A st
( R c= b1 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def12 defines StabCl MSUALG_6:def 12 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for b4 being stable ManySortedRelation of A holds
( b4 = StabCl R iff ( R c= b4 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b4 c= Q ) ) );

theorem Th31: :: MSUALG_6:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (StabCl R) . s iff ex x, y being Element of A,s ex h being Endomorphism of A st
( [x,y] in R . s & a = (h . s) . x & b = (h . s) . y ) )
proof end;

theorem :: MSUALG_6:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A holds InvCl (StabCl R) is stable by Th30;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be ManySortedRelation of the Sorts of A;
func TRS R -> invariant stable ManySortedRelation of A means :Def13: :: MSUALG_6:def 13
( R c= it & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being invariant stable ManySortedRelation of A st R c= b1 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being invariant stable ManySortedRelation of A st
( R c= b1 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def13 defines TRS MSUALG_6:def 13 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for b4 being invariant stable ManySortedRelation of A holds
( b4 = TRS R iff ( R c= b4 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b4 c= Q ) ) );

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be V3 ManySortedRelation of A;
cluster InvCl R -> V3 invariant ;
coherence
InvCl R is non-empty
proof end;
cluster StabCl R -> V3 stable ;
coherence
StabCl R is non-empty
proof end;
cluster TRS R -> V3 invariant stable ;
coherence
TRS R is non-empty
proof end;
end;

theorem :: MSUALG_6:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being invariant ManySortedRelation of A holds InvCl R = R
proof end;

theorem :: MSUALG_6:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being stable ManySortedRelation of A holds StabCl R = R
proof end;

theorem :: MSUALG_6:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being invariant stable ManySortedRelation of A holds TRS R = R
proof end;

theorem :: MSUALG_6:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A holds
( StabCl R c= TRS R & InvCl R c= TRS R & StabCl (InvCl R) c= TRS R )
proof end;

theorem Th37: :: MSUALG_6:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A holds InvCl (StabCl R) = TRS R
proof end;

theorem :: MSUALG_6:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (TRS R) . s iff ex s' being SortSymbol of S st
( TranslationRel S reduces s',s & ex l, r being Element of A,s' ex h being Endomorphism of A ex t being Translation of A,s',s st
( [l,r] in R . s' & a = t . ((h . s') . l) & b = t . ((h . s') . r) ) ) )
proof end;

theorem Th39: :: MSUALG_6:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for R, E being Relation of A st ( for a, b being set st a in A & b in A holds
( [a,b] in E iff a,b are_convertible_wrt R ) ) holds
( E is total & E is symmetric & E is transitive )
proof end;

theorem Th40: :: MSUALG_6:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for R being Relation of A
for E being Equivalence_Relation of A st R c= E holds
for a, b being set st a in A & b in A & a,b are_convertible_wrt R holds
[a,b] in E
proof end;

theorem Th41: :: MSUALG_6:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being non empty set
for R being Relation of A
for a, b being Element of A holds
( [a,b] in EqCl R iff a,b are_convertible_wrt R )
proof end;

theorem Th42: :: MSUALG_6:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty set
for A being V3 ManySortedSet of S
for R being ManySortedRelation of A
for s being Element of S
for a, b being Element of A . s holds
( [a,b] in (EqCl R) . s iff a,b are_convertible_wrt R . s )
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
mode EquationalTheory of A is MSEquivalence-like invariant stable ManySortedRelation of A;
let R be ManySortedRelation of A;
func EqCl R,A -> MSEquivalence-like ManySortedRelation of A equals :: MSUALG_6:def 14
EqCl R;
coherence
EqCl R is MSEquivalence-like ManySortedRelation of A
by MSUALG_4:def 5;
end;

:: deftheorem defines EqCl MSUALG_6:def 14 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A holds EqCl R,A = EqCl R;

theorem Th43: :: MSUALG_6:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A holds R c= EqCl R,A
proof end;

theorem Th44: :: MSUALG_6:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A
for E being MSEquivalence-like ManySortedRelation of A st R c= E holds
EqCl R,A c= E
proof end;

theorem Th45: :: MSUALG_6:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being stable ManySortedRelation of A
for s being SortSymbol of S
for a, b being Element of A,s st a,b are_convertible_wrt R . s holds
for h being Endomorphism of A holds (h . s) . a,(h . s) . b are_convertible_wrt R . s
proof end;

theorem Th46: :: MSUALG_6:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being stable ManySortedRelation of A holds EqCl R,A is stable
proof end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be stable ManySortedRelation of A;
cluster EqCl R,A -> V3 MSEquivalence-like stable ;
coherence
EqCl R,A is stable
by Th46;
end;

theorem Th47: :: MSUALG_6:47  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being invariant ManySortedRelation of A
for s1, s2 being SortSymbol of S
for a, b being Element of A,s1 st a,b are_convertible_wrt R . s1 holds
for t being Function st t is_e.translation_of A,s1,s2 holds
t . a,t . b are_convertible_wrt R . s2
proof end;

theorem Th48: :: MSUALG_6:48  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being invariant ManySortedRelation of A holds EqCl R,A is invariant
proof end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be invariant ManySortedRelation of A;
cluster EqCl R,A -> V3 MSEquivalence-like compatible invariant ;
coherence
EqCl R,A is invariant
by Th48;
end;

theorem Th49: :: MSUALG_6:49  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty set
for A being V3 ManySortedSet of S
for R, E being ManySortedRelation of A st ( for s being Element of S
for a, b being Element of A . s holds
( [a,b] in E . s iff a,b are_convertible_wrt R . s ) ) holds
E is MSEquivalence_Relation-like
proof end;

theorem Th50: :: MSUALG_6:50  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R, E being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in E . s iff a,b are_convertible_wrt (TRS R) . s ) ) holds
E is EquationalTheory of A
proof end;

theorem Th51: :: MSUALG_6:51  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty set
for A being V3 ManySortedSet of S
for R being ManySortedRelation of A
for E being MSEquivalence_Relation-like ManySortedRelation of A st R c= E holds
for s being Element of S
for a, b being Element of A . s st a,b are_convertible_wrt R . s holds
[a,b] in E . s
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra of S;
let R be ManySortedRelation of the Sorts of A;
func EqTh R -> EquationalTheory of A means :Def15: :: MSUALG_6:def 15
( R c= it & ( for Q being EquationalTheory of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being EquationalTheory of A st R c= b1 & ( for Q being EquationalTheory of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being EquationalTheory of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being EquationalTheory of A st
( R c= b1 & ( for Q being EquationalTheory of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def15 defines EqTh MSUALG_6:def 15 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of the Sorts of A
for b4 being EquationalTheory of A holds
( b4 = EqTh R iff ( R c= b4 & ( for Q being EquationalTheory of A st R c= Q holds
b4 c= Q ) ) );

theorem :: MSUALG_6:52  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A holds
( EqCl R,A c= EqTh R & InvCl R c= EqTh R & StabCl R c= EqTh R & TRS R c= EqTh R )
proof end;

theorem :: MSUALG_6:53  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (EqTh R) . s iff a,b are_convertible_wrt (TRS R) . s )
proof end;

theorem :: MSUALG_6:54  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra of S
for R being ManySortedRelation of A holds EqTh R = EqCl (TRS R),A
proof end;