:: 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 S be non
empty non
void ManySortedSign ;
let MSA be
non-empty MSAlgebra of
S;
func Eval c2 -> ManySortedFunction of
(FreeEnv a2),
a2 means :: MSAFREE2:def 9
(
it is_homomorphism FreeEnv MSA,
MSA & ( for
s being
SortSymbol of
S for
x,
y being
set st
y in FreeSort the
Sorts of
MSA,
s &
y = root-tree [x,s] &
x in the
Sorts of
MSA . s holds
(it . s) . y = x ) );
existence
ex b1 being ManySortedFunction of (FreeEnv MSA),MSA st
( b1 is_homomorphism FreeEnv MSA,MSA & ( for s being SortSymbol of S
for x, y being set st y in FreeSort the Sorts of MSA,s & y = root-tree [x,s] & x in the Sorts of MSA . s holds
(b1 . s) . y = x ) )
uniqueness
for b1, b2 being ManySortedFunction of (FreeEnv MSA),MSA st b1 is_homomorphism FreeEnv MSA,MSA & ( for s being SortSymbol of S
for x, y being set st y in FreeSort the Sorts of MSA,s & y = root-tree [x,s] & x in the Sorts of MSA . s holds
(b1 . s) . y = x ) & b2 is_homomorphism FreeEnv MSA,MSA & ( for s being SortSymbol of S
for x, y being set st y in FreeSort the Sorts of MSA,s & y = root-tree [x,s] & x in the Sorts of MSA . s holds
(b2 . s) . y = x ) holds
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 :