:: MSAFREE2 semantic presentation
:: deftheorem Def1 defines SortsWithConstants MSAFREE2:def 1 :
:: deftheorem Def2 defines InputVertices MSAFREE2:def 2 :
:: deftheorem Def3 defines InnerVertices MSAFREE2:def 3 :
theorem Th1: :: MSAFREE2:1
theorem Th2: :: MSAFREE2:2
theorem Th3: :: MSAFREE2:3
theorem Th4: :: MSAFREE2:4
theorem Th5: :: MSAFREE2:5
theorem Th6: :: MSAFREE2:6
:: deftheorem Def4 defines with_input_V MSAFREE2:def 4 :
:: deftheorem Def5 defines InputValues MSAFREE2:def 5 :
:: deftheorem Def6 defines Circuit-like MSAFREE2:def 6 :
:: deftheorem Def7 defines action_at MSAFREE2:def 7 :
theorem Th7: :: MSAFREE2:7
:: deftheorem Def8 defines FreeEnv MSAFREE2:def 8 :
theorem Th8: :: MSAFREE2:8
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
non-empty MSAlgebra of c
1;
func Eval c
2 -> ManySortedFunction of
(FreeEnv a2),a
2 means :: MSAFREE2:def 9
( a
3 is_homomorphism FreeEnv a
2,a
2 & ( for b
1 being
SortSymbol of a
1for b
2, b
3 being
set holds
( b
3 in FreeSort the
Sorts of a
2,b
1 & b
3 = root-tree [b2,b1] & b
2 in the
Sorts of a
2 . b
1 implies
(a3 . b1) . b
3 = b
2 ) ) );
existence
ex b1 being ManySortedFunction of (FreeEnv c2),c2 st
( b1 is_homomorphism FreeEnv c2,c2 & ( for b2 being SortSymbol of c1
for b3, b4 being set holds
( b4 in FreeSort the Sorts of c2,b2 & b4 = root-tree [b3,b2] & b3 in the Sorts of c2 . b2 implies (b1 . b2) . b4 = b3 ) ) )
uniqueness
for b1, b2 being ManySortedFunction of (FreeEnv c2),c2 holds
( b1 is_homomorphism FreeEnv c2,c2 & ( for b3 being SortSymbol of c1
for b4, b5 being set holds
( b5 in FreeSort the Sorts of c2,b3 & b5 = root-tree [b4,b3] & b4 in the Sorts of c2 . b3 implies (b1 . b3) . b5 = b4 ) ) & b2 is_homomorphism FreeEnv c2,c2 & ( for b3 being SortSymbol of c1
for b4, b5 being set holds
( b5 in FreeSort the Sorts of c2,b3 & b5 = root-tree [b4,b3] & b4 in the Sorts of c2 . b3 implies (b2 . b3) . b5 = b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines Eval MSAFREE2:def 9 :
theorem Th9: :: MSAFREE2:9
:: deftheorem Def10 defines finitely-generated MSAFREE2:def 10 :
:: deftheorem Def11 defines locally-finite MSAFREE2:def 11 :
:: deftheorem Def12 defines Trivial_Algebra MSAFREE2:def 12 :
:: deftheorem Def13 defines monotonic MSAFREE2:def 13 :
theorem Th10: :: MSAFREE2:10
theorem Th11: :: MSAFREE2:11
theorem Th12: :: MSAFREE2:12
theorem Th13: :: MSAFREE2:13
theorem Th14: :: MSAFREE2:14
theorem Th15: :: MSAFREE2:15
:: deftheorem Def14 defines depth MSAFREE2:def 14 :