theorem Th14:
(for R being Rule of S st R in D holds R is Correct) implies D is Correct
proof
set Q=S-sequents, O=OneStep D; {} null S is S-correct; then
reconsider e={} null Q as S-correct Subset of Q;
reconsider RO=rng O as Subset of bool Q by RELAT_1:def 19;
assume
A1: for R being Rule of S st R in D holds R is Correct;
defpred P[Nat] means for X being S-correct Subset of Q holds
($1,D)-derivables.X is S-correct;
A2: P[0]
proof
set f=(0,D)-derivables;
A3: f = id field O by FUNCT_7:68 .= id (bool Q\/RO) by FUNCT_2:def 1 .=
id(bool Q); let X be S-correct Subset of Q; thus thesis by A3;
end;
A4: for n st P[n] holds P[n+1]
proof
let n; assume
A5: P[n]; let X be S-correct Subset of Q;
set DM=(n+1,D)-derivables, Dm=(n,D)-derivables;
A6: dom Dm=bool Q by FUNCT_2:def 1;
reconsider oldSeqs=Dm.X as S-correct Subset of Q by A5;
A7: DM=O*Dm by FUNCT_7:71;
now
let U; set II=U-InterpretersOf S; let I be Element of II;
let H be I-satisfied set; let phi;
assume
A8: [H,phi] in DM.X;
set Fam={R.:{oldSeqs} where R is Subset of [:bool Q, bool Q:]: R in D};
DM.X=O.oldSeqs by A6, A7, FUNCT_1:13 .=
union union Fam by Lm5; then consider x such that
A9: [H,phi] in x & x in union Fam by A8, TARSKI:def 4; consider y such that
A10: x in y & y in Fam by A9, TARSKI:def 4; consider R being Subset of
[:bool Q, bool Q:] such that
A11: y=R.:{oldSeqs} & R in D by A10; reconsider RR=R as Correct Rule of S
by A1, A11; reconsider newSeqs=RR.oldSeqs as S-correct Subset of Q
by Def68;
dom RR=bool Q by FUNCT_2:def 1; then
y=Im(R,oldSeqs) &
Im(RR,oldSeqs)= {RR.oldSeqs} by FUNCT_1:59, A11; then
[H,phi] in newSeqs by A9, TARSKI:def 1, A10;
hence I-TruthEval phi = 1 by FOMODEL2:def 44;
end; hence thesis;
end;
A12: for n holds P[n] from NAT_1:sch 2(A2, A4);
now
let phi; let X; assume phi is (X,D)-provable; then
consider H being set, m such that
A13: H c= X & [H,phi] is (m,{},D)-derivable;
reconsider HH=H as Subset of X by A13;
reconsider seqt=[H,phi] as Element of Q by Def2, A13;
reconsider okSeqs=(m,D)-derivables.e as S-correct Subset of Q by A12;
hereby
let U; set II=U-InterpretersOf S; let I be Element of II; assume
X is I-satisfied; then reconsider XX=X as I-satisfied set;
reconsider HHH=HH as I-satisfied Subset of XX;
[HHH,phi] in okSeqs by A13;
hence I-TruthEval phi=1 by FOMODEL2:def 44;
end;
end; hence D is Correct;
end;
