:: OSALG_1 semantic presentation
Lemma1:
for b1 being set
for b2 being ManySortedSet of b1
for b3 being FinSequence of b1 holds
( dom (b2 * b3) = dom b3 & len (b2 * b3) = len b3 )
theorem Th1: :: OSALG_1:1
for b
1, b
2 being non
empty set for b
3 being
Order of b
1for b
4 being
Equivalence_Relation of b
2for b
5 being
Function of b
2,b
1 * for b
6 being
Function of b
2,b
1 holds
( not
OverloadedRSSign(# b
1,b
3,b
2,b
4,b
5,b
6 #) is
empty & not
OverloadedRSSign(# b
1,b
3,b
2,b
4,b
5,b
6 #) is
void &
OverloadedRSSign(# b
1,b
3,b
2,b
4,b
5,b
6 #) is
reflexive &
OverloadedRSSign(# b
1,b
3,b
2,b
4,b
5,b
6 #) is
transitive &
OverloadedRSSign(# b
1,b
3,b
2,b
4,b
5,b
6 #) is
antisymmetric )
registration
let c
1 be non
empty set ;
let c
2 be
Order of c
1;
let c
3 be non
empty set ;
let c
4 be
Equivalence_Relation of c
3;
let c
5 be
Function of c
3,c
1 * ;
let c
6 be
Function of c
3,c
1;
cluster OverloadedRSSign(# a
1,a
2,a
3,a
4,a
5,a
6 #)
-> non
empty reflexive transitive antisymmetric ;
coherence
( OverloadedRSSign(# c1,c2,c3,c4,c5,c6 #) is strict & not OverloadedRSSign(# c1,c2,c3,c4,c5,c6 #) is empty & OverloadedRSSign(# c1,c2,c3,c4,c5,c6 #) is reflexive & OverloadedRSSign(# c1,c2,c3,c4,c5,c6 #) is transitive & OverloadedRSSign(# c1,c2,c3,c4,c5,c6 #) is antisymmetric )
by Th1;
end;
:: deftheorem Def1 OSALG_1:def 1 :
canceled;
:: deftheorem Def2 defines order-sorted OSALG_1:def 2 :
:: deftheorem Def3 defines ~= OSALG_1:def 3 :
theorem Th2: :: OSALG_1:2
:: deftheorem Def4 defines discernable OSALG_1:def 4 :
:: deftheorem Def5 defines op-discrete OSALG_1:def 5 :
theorem Th3: :: OSALG_1:3
theorem Th4: :: OSALG_1:4
:: deftheorem Def6 defines OSSign OSALG_1:def 6 :
theorem Th5: :: OSALG_1:5
:: deftheorem Def7 defines <= OSALG_1:def 7 :
theorem Th6: :: OSALG_1:6
theorem Th7: :: OSALG_1:7
theorem Th8: :: OSALG_1:8
:: deftheorem Def8 defines monotone OSALG_1:def 8 :
:: deftheorem Def9 defines monotone OSALG_1:def 9 :
theorem Th9: :: OSALG_1:9
theorem Th10: :: OSALG_1:10
:: deftheorem Def10 defines has_least_args_for OSALG_1:def 10 :
:: deftheorem Def11 defines has_least_sort_for OSALG_1:def 11 :
:: deftheorem Def12 defines has_least_rank_for OSALG_1:def 12 :
:: deftheorem Def13 defines regular OSALG_1:def 13 :
:: deftheorem Def14 defines regular OSALG_1:def 14 :
theorem Th11: :: OSALG_1:11
theorem Th12: :: OSALG_1:12
theorem Th13: :: OSALG_1:13
:: deftheorem Def15 defines LBound OSALG_1:def 15 :
theorem Th14: :: OSALG_1:14
:: deftheorem Def16 defines ConstOSSet OSALG_1:def 16 :
theorem Th15: :: OSALG_1:15
:: deftheorem Def17 OSALG_1:def 17 :
canceled;
:: deftheorem Def18 defines order-sorted OSALG_1:def 18 :
theorem Th16: :: OSALG_1:16
:: deftheorem Def19 defines order-sorted OSALG_1:def 19 :
theorem Th17: :: OSALG_1:17
definition
let c
1 be
OrderSortedSign;
let c
2 be non
empty set ;
let c
3 be
ManySortedFunction of
((ConstOSSet c1,c2) # ) * the
Arity of c
1,
(ConstOSSet c1,c2) * the
ResultSort of c
1;
func ConstOSA c
1,c
2,c
3 -> strict non-empty MSAlgebra of a
1 means :
Def20:
:: OSALG_1:def 20
( the
Sorts of a
4 = ConstOSSet a
1,a
2 & the
Charact of a
4 = a
3 );
existence
ex b1 being strict non-empty MSAlgebra of c1 st
( the Sorts of b1 = ConstOSSet c1,c2 & the Charact of b1 = c3 )
uniqueness
for b1, b2 being strict non-empty MSAlgebra of c1 holds
( the Sorts of b1 = ConstOSSet c1,c2 & the Charact of b1 = c3 & the Sorts of b2 = ConstOSSet c1,c2 & the Charact of b2 = c3 implies b1 = b2 )
;
end;
:: deftheorem Def20 defines ConstOSA OSALG_1:def 20 :
theorem Th18: :: OSALG_1:18
theorem Th19: :: OSALG_1:19
theorem Th20: :: OSALG_1:20
:: deftheorem Def21 defines OSAlg OSALG_1:def 21 :
theorem Th21: :: OSALG_1:21
:: deftheorem Def22 defines <= OSALG_1:def 22 :
theorem Th22: :: OSALG_1:22
theorem Th23: :: OSALG_1:23
theorem Th24: :: OSALG_1:24
theorem Th25: :: OSALG_1:25
theorem Th26: :: OSALG_1:26
:: deftheorem Def23 defines monotone OSALG_1:def 23 :
theorem Th27: :: OSALG_1:27
theorem Th28: :: OSALG_1:28
definition
let c
1 be
OrderSortedSign;
let c
2 be non
empty set ;
let c
3 be
Element of c
2;
func TrivialOSA c
1,c
2,c
3 -> strict OSAlgebra of a
1 means :
Def24:
:: OSALG_1:def 24
( the
Sorts of a
4 = ConstOSSet a
1,a
2 & ( for b
1 being
OperSymbol of a
1 holds
Den b
1,a
4 = (Args b1,a4) --> a
3 ) );
existence
ex b1 being strict OSAlgebra of c1 st
( the Sorts of b1 = ConstOSSet c1,c2 & ( for b2 being OperSymbol of c1 holds Den b2,b1 = (Args b2,b1) --> c3 ) )
uniqueness
for b1, b2 being strict OSAlgebra of c1 holds
( the Sorts of b1 = ConstOSSet c1,c2 & ( for b3 being OperSymbol of c1 holds Den b3,b1 = (Args b3,b1) --> c3 ) & the Sorts of b2 = ConstOSSet c1,c2 & ( for b3 being OperSymbol of c1 holds Den b3,b2 = (Args b3,b2) --> c3 ) implies b1 = b2 )
end;
:: deftheorem Def24 defines TrivialOSA OSALG_1:def 24 :
theorem Th29: :: OSALG_1:29
:: deftheorem Def25 defines OperNames OSALG_1:def 25 :
:: deftheorem Def26 defines Name OSALG_1:def 26 :
theorem Th30: :: OSALG_1:30
theorem Th31: :: OSALG_1:31
theorem Th32: :: OSALG_1:32
theorem Th33: :: OSALG_1:33
theorem Th34: :: OSALG_1:34
:: deftheorem Def27 defines LBound OSALG_1:def 27 :
theorem Th35: :: OSALG_1:35