let S be locally_directed OrderSortedSign; :: thesis:
let X be non-empty ManySortedSet of S; :: thesis:
let t be Element of TS (DTConOSA X); :: thesis:
set PTA = ParsedTermsOSA X;
set SPTA = the Sorts of (ParsedTermsOSA X);
set D = DTConOSA X;
set C = bool [:(TS (DTConOSA X)), the carrier of S:];
set F = PTClasses X;
defpred S₁[ set ] means for s being Element of S holds
( $1 in the Sorts of (ParsedTermsOSA X) . s iff [$1,s] in (PTClasses X) . $1 );
defpred S₂[ set ] means for s being Element of S
for y being Element of TS (DTConOSA X) st [y,s] in (PTClasses X) . $1 holds
[$1,s] in (PTClasses X) . y;
defpred S₃[ DecoratedTree of the carrier of (DTConOSA X)] means ( S₁[$1] & S₂[$1] );
A1: for nt being Symbol of (DTConOSA X)
for ts being FinSequence of TS (DTConOSA X) st nt ==> roots ts & ( for t being DecoratedTree of the carrier of (DTConOSA X) st t in rng ts holds
S₃[t] ) holds
S₃[nt -tree ts]

proof

let nt be Symbol of (DTConOSA X); :: thesis:
let ts be FinSequence of TS (DTConOSA X); :: thesis:
assume that
A2: nt ==> roots ts and
A3: for t being DecoratedTree of the carrier of (DTConOSA X) st t in rng ts holds
( S₁[t] & S₂[t] ) ; :: thesis:
consider o being OperSymbol of S such that
A4: nt = [o, the carrier of S] and
A5: ts in Args (o,(ParsedTermsOSA X)) and
A6: nt -tree ts = (Den (o,(ParsedTermsOSA X))) . ts and
A7: for s1 being Element of S holds
( nt -tree ts in the Sorts of (ParsedTermsOSA X) . s1 iff the_result_sort_of o <= s1 ) by A2, Th12;
reconsider x = (PTClasses X) * ts as FinSequence of bool [:(TS (DTConOSA X)), the carrier of S:] ;
A8: (PTClasses X) . (nt -tree ts) = @ (nt,x) by A2, Def21
.= { [((Den (o2,(ParsedTermsOSA X))) . x2),s3] where o2 is OperSymbol of S, x2 is Element of Args (o2,(ParsedTermsOSA X)), s3 is Element of S : ( ex o1 being OperSymbol of S st
( nt = [o1, the carrier of S] & o1 ~= o2 & len (the_arity_of o1) = len (the_arity_of o2) & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 ) & ex w3 being Element of the carrier of S * st
( dom w3 = dom x & ( for y being Nat st y in dom x holds
[(x2 . y),(w3 /. y)] in x . y ) ) ) } ;
reconsider ts1 = ts as Element of Args (o,(ParsedTermsOSA X)) by A5;
set w = the_arity_of o;
A9: rng ts c= TS (DTConOSA X) by FINSEQ_1:def 4;
dom (PTClasses X) = TS (DTConOSA X) by FUNCT_2:def 1;
then len x = len ts by A9, FINSEQ_2:29;
then A10: dom x = dom ts by FINSEQ_3:29;
A11: dom (the_arity_of o) = dom ts by A5, MSUALG_3:6;
A12: for y being Nat st y in dom x holds
[(ts1 . y),((the_arity_of o) /. y)] in x . y

proof

let y be Nat; :: thesis:
assume A13: y in dom x ; :: thesis:
A14: ts1 . y in rng ts1 by A10, A13, FUNCT_1:3;
then reconsider t1 = ts1 . y as Element of TS (DTConOSA X) by A9;
ts1 . y in the Sorts of (ParsedTermsOSA X) . ((the_arity_of o) /. y) by A10, A11, A13, MSUALG_6:2;
then [t1,((the_arity_of o) /. y)] in (PTClasses X) . t1 by A3, A14;
hence [(ts1 . y),((the_arity_of o) /. y)] in x . y by A10, A13, FUNCT_1:13; :: thesis:

end;

thus S₁[nt -tree ts] :: thesis:

proof

let s1 be Element of S; :: thesis:

hereby :: thesis:

assume nt -tree ts in the Sorts of (ParsedTermsOSA X) . s1 ; :: thesis:
then A15: the_result_sort_of o <= s1 by A7;
len (the_arity_of o) = len (the_arity_of o) ;
hence [(nt -tree ts),s1] in (PTClasses X) . (nt -tree ts) by A4, A6, A10, A11, A12, A8, A15; :: thesis:

end;

assume [(nt -tree ts),s1] in (PTClasses X) . (nt -tree ts) ; :: thesis:
then consider o2 being OperSymbol of S, x2 being Element of Args (o2,(ParsedTermsOSA X)), s3 being Element of S such that
A16: [(nt -tree ts),s1] = [((Den (o2,(ParsedTermsOSA X))) . x2),s3] and
A17: ex o1 being OperSymbol of S st
( nt = [o1, the carrier of S] & o1 ~= o2 & len (the_arity_of o1) = len (the_arity_of o2) & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 ) and
ex w3 being Element of the carrier of S * st
( dom w3 = dom x & ( for y being Nat st y in dom x holds
[(x2 . y),(w3 /. y)] in x . y ) ) by A8;
s1 = s3 by A16, XTUPLE_0:1;
then A18: the Sorts of (ParsedTermsOSA X) . (the_result_sort_of o2) c= the Sorts of (ParsedTermsOSA X) . s1 by A17, OSALG_1:def 16;
A19: (Den (o2,(ParsedTermsOSA X))) . x2 in the Sorts of (ParsedTermsOSA X) . (the_result_sort_of o2) by MSUALG_9:18;
nt -tree ts = (Den (o2,(ParsedTermsOSA X))) . x2 by A16, XTUPLE_0:1;
hence nt -tree ts in the Sorts of (ParsedTermsOSA X) . s1 by A19, A18; :: thesis:

end;

thus S₂[nt -tree ts] :: thesis:

proof

let s1 be Element of S; :: thesis:
let y be Element of TS (DTConOSA X); :: thesis:
assume [y,s1] in (PTClasses X) . (nt -tree ts) ; :: thesis:
then consider o2 being OperSymbol of S, x2 being Element of Args (o2,(ParsedTermsOSA X)), s3 being Element of S such that
A20: [y,s1] = [((Den (o2,(ParsedTermsOSA X))) . x2),s3] and
A21: ex o1 being OperSymbol of S st
( nt = [o1, the carrier of S] & o1 ~= o2 & len (the_arity_of o1) = len (the_arity_of o2) & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 ) and
A22: ex w3 being Element of the carrier of S * st
( dom w3 = dom x & ( for y being Nat st y in dom x holds
[(x2 . y),(w3 /. y)] in x . y ) ) by A8;
consider w3 being Element of the carrier of S * such that
A23: dom w3 = dom x and
A24: for y being Nat st y in dom x holds
[(x2 . y),(w3 /. y)] in x . y by A22;
consider o1 being OperSymbol of S such that
A25: nt = [o1, the carrier of S] and
A26: o1 ~= o2 and
A27: len (the_arity_of o1) = len (the_arity_of o2) and
A28: the_result_sort_of o1 <= s3 and
A29: the_result_sort_of o2 <= s3 by A21;
A30: y = (Den (o2,(ParsedTermsOSA X))) . x2 by A20, XTUPLE_0:1;
reconsider x3 = x2 as FinSequence of TS (DTConOSA X) by Th13;
reconsider xy = (PTClasses X) * x3 as FinSequence of bool [:(TS (DTConOSA X)), the carrier of S:] ;
A31: OSSym (o2,X) ==> roots x2 by Th13;
then consider o3 being OperSymbol of S such that
A32: OSSym (o2,X) = [o3, the carrier of S] and
x3 in Args (o3,(ParsedTermsOSA X)) and
A33: (OSSym (o2,X)) -tree x3 = (Den (o3,(ParsedTermsOSA X))) . x3 and
for s2 being Element of S holds
( (OSSym (o2,X)) -tree x3 in the Sorts of (ParsedTermsOSA X) . s2 iff the_result_sort_of o3 <= s2 ) by Th12;
o2 = o3 by A32, XTUPLE_0:1;
then A34: (PTClasses X) . y = @ ((OSSym (o2,X)),xy) by A30, A31, A33, Def21
.= { [((Den (o4,(ParsedTermsOSA X))) . x4),s4] where o4 is OperSymbol of S, x4 is Element of Args (o4,(ParsedTermsOSA X)), s4 is Element of S : ( ex o1 being OperSymbol of S st
( OSSym (o2,X) = [o1, the carrier of S] & o1 ~= o4 & len (the_arity_of o1) = len (the_arity_of o4) & the_result_sort_of o1 <= s4 & the_result_sort_of o4 <= s4 ) & ex w4 being Element of the carrier of S * st
( dom w4 = dom xy & ( for y being Nat st y in dom xy holds
[(x4 . y),(w4 /. y)] in xy . y ) ) ) } ;
A35: rng x3 c= TS (DTConOSA X) by FINSEQ_1:def 4;
then rng x3 c= dom (PTClasses X) by FUNCT_2:def 1;
then len xy = len x3 by FINSEQ_2:29;
then A36: dom x3 = dom xy by FINSEQ_3:29;
A37: o1 = o by A4, A25, XTUPLE_0:1;
then A38: dom (the_arity_of o2) = dom (the_arity_of o) by A27, FINSEQ_3:29;
then A39: dom w3 = dom xy by A10, A11, A23, A36, MSUALG_3:6;
A40: dom x2 = dom x by A10, A11, A38, MSUALG_3:6;
A41: for y being Nat st y in dom xy holds
[(ts1 . y),(w3 /. y)] in xy . y

proof

let y be Nat; :: thesis:
assume A42: y in dom xy ; :: thesis:
A43: ts1 . y in rng ts1 by A10, A23, A39, A42, FUNCT_1:3;
x2 . y in rng x3 by A36, A42, FUNCT_1:3;
then reconsider t1 = ts1 . y, t2 = x2 . y as Element of TS (DTConOSA X) by A9, A35, A43;
[(x2 . y),(w3 /. y)] in x . y by A24, A36, A40, A42;
then [(x2 . y),(w3 /. y)] in (PTClasses X) . (ts1 . y) by A10, A23, A39, A42, FUNCT_1:13;
then [t1,(w3 /. y)] in (PTClasses X) . t2 by A3, A43;
hence [(ts1 . y),(w3 /. y)] in xy . y by A42, FUNCT_1:12; :: thesis:

end;

A44: the_result_sort_of o2 <= s1 by A20, A29, XTUPLE_0:1;
the_result_sort_of o <= s1 by A20, A28, A37, XTUPLE_0:1;
hence [(nt -tree ts),s1] in (PTClasses X) . y by A6, A26, A27, A37, A39, A41, A34, A44; :: thesis:

end;

A45: for s being Symbol of (DTConOSA X) st s in Terminals (DTConOSA X) holds
S₃[ root-tree s]

proof

let sy be Symbol of (DTConOSA X); :: thesis:
assume A46: sy in Terminals (DTConOSA X) ; :: thesis:
reconsider sy1 = sy as Terminal of (DTConOSA X) by A46;
consider s being Element of S, x being set such that
A47: x in X . s and
A48: sy = [x,s] by A46, Th4;
A49: (PTClasses X) . (root-tree sy) = @ sy by A46, Def21
.= { [(root-tree sy),s1] where s1 is Element of S : ex s2 being Element of S ex x being set st
( x in X . s2 & sy = [x,s2] & s2 <= s1 ) } ;
root-tree sy1 in { a where a is Element of TS (DTConOSA X) : ( ex s1 being Element of S ex x being object st
( s1 <= s & x in X . s1 & a = root-tree [x,s1] ) or ex o being OperSymbol of S st
( [o, the carrier of S] = a . {} & the_result_sort_of o <= s ) ) } by A47, A48;
then A50: root-tree sy1 in the Sorts of (ParsedTermsOSA X) . s by Th9;
thus S₁[ root-tree sy] :: thesis:

proof

let s1 be Element of S; :: thesis:

hereby :: thesis:

assume root-tree sy in the Sorts of (ParsedTermsOSA X) . s1 ; :: thesis:
then s <= s1 by A47, A48, Th10;
hence [(root-tree sy),s1] in (PTClasses X) . (root-tree sy) by A47, A48, A49; :: thesis:

end;

assume [(root-tree sy),s1] in (PTClasses X) . (root-tree sy) ; :: thesis:
then consider s3 being Element of S such that
A51: [(root-tree sy),s1] = [(root-tree sy),s3] and
A52: ex s2 being Element of S ex x being set st
( x in X . s2 & sy = [x,s2] & s2 <= s3 ) by A49;
A53: s1 = s3 by A51, XTUPLE_0:1;
consider s2 being Element of S, x2 being set such that
x2 in X . s2 and
A54: sy = [x2,s2] and
A55: s2 <= s3 by A52;
s2 = s by A48, A54, XTUPLE_0:1;
then the Sorts of (ParsedTermsOSA X) . s c= the Sorts of (ParsedTermsOSA X) . s1 by A53, A55, OSALG_1:def 16;
hence root-tree sy in the Sorts of (ParsedTermsOSA X) . s1 by A50; :: thesis:

end;

thus S₂[ root-tree sy] :: thesis:

proof

let s1 be Element of S; :: thesis:
let y be Element of TS (DTConOSA X); :: thesis:
assume A56: [y,s1] in (PTClasses X) . (root-tree sy) ; :: thesis:
then ex s2 being Element of S st
( [y,s1] = [(root-tree sy),s2] & ex s3 being Element of S ex x being set st
( x in X . s3 & sy = [x,s3] & s3 <= s2 ) ) by A49;
then y = root-tree sy by XTUPLE_0:1;
hence [(root-tree sy),s1] in (PTClasses X) . y by A56; :: thesis:

end;

for t being DecoratedTree of the carrier of (DTConOSA X) st t in TS (DTConOSA X) holds
S₃[t] from DTCONSTR:sch 7(A45, A1);
hence ( ( for s being Element of S holds
( t in the Sorts of (ParsedTermsOSA X) . s iff [t,s] in (PTClasses X) . t ) ) & ( for s being Element of S
for y being Element of TS (DTConOSA X) st [y,s] in (PTClasses X) . t holds
[t,s] in (PTClasses X) . y ) ) ; :: thesis: