let R be Ring; :: thesis: for i1 being Nat holds not goto (i1,R) is halting
let i1 be Nat; :: thesis: not goto (i1,R) is halting
reconsider i5 = i1 as Element of NAT by ORDINAL1:def 12;
set s = the SCM-State of R;
set t = the SCM-State of R +* (NAT .--> (i1 + 1));
set f = the_Values_of (SCM R);
A1: {NAT} c= SCM-Memory by AMI_2:22, ZFMISC_1:31;
A2: dom ( the SCM-State of R +* (NAT .--> (i1 + 1))) = (dom the SCM-State of R) \/ (dom (NAT .--> (i1 + 1))) by FUNCT_4:def 1
.= SCM-Memory \/ (dom (NAT .--> (i1 + 1))) by SCMRING1:19
.= SCM-Memory \/ {NAT}
.= SCM-Memory by A1, XBOOLE_1:12 ;
A3: the_Values_of (SCM R) = (SCM-VAL R) * SCM-OK by Lm1;
NAT in dom (NAT .--> (i1 + 1)) by TARSKI:def 1;
then A5: ( the SCM-State of R +* (NAT .--> (i1 + 1))) . NAT = (NAT .--> (i1 + 1)) . NAT by FUNCT_4:13
.= i5 + 1 by FUNCOP_1:72 ;
A6: dom ( the SCM-State of R +* (NAT .--> (i1 + 1))) = the carrier of (SCM R) by A2, Def1
.= dom (the_Values_of (SCM R)) by PARTFUN1:def 2 ;
A7: for x being object st x in dom ( the SCM-State of R +* (NAT .--> (i1 + 1))) holds
( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x

proof

let x be object ; :: thesis: ( x in dom ( the SCM-State of R +* (NAT .--> (i1 + 1))) implies ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x )
assume A8: x in dom ( the SCM-State of R +* (NAT .--> (i1 + 1))) ; :: thesis: ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x

per cases ( x = NAT or x <> NAT ) ;

suppose A9: x = NAT ; :: thesis: ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x

then (the_Values_of (SCM R)) . x = NAT by A3, SCMRING1:2;
hence ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x by A5, A9, ORDINAL1:def 12; :: thesis: verum

end;

suppose x <> NAT ; :: thesis: ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x

then not x in dom (NAT .--> (i1 + 1)) by TARSKI:def 1;
then ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x = the SCM-State of R . x by FUNCT_4:11;
hence ( the SCM-State of R +* (NAT .--> (i1 + 1))) . x in (the_Values_of (SCM R)) . x by A3, A8, A6, CARD_3:9; :: thesis: verum

end;

end;

end;

A10: the_Values_of (SCM R) = (SCM-VAL R) * SCM-OK by Lm1;
dom ( the SCM-State of R +* (NAT .--> (i1 + 1))) = the carrier of (SCM R) by A2, Def1;
then reconsider t = the SCM-State of R +* (NAT .--> (i1 + 1)) as PartState of (SCM R) by A7, FUNCT_1:def 14, RELAT_1:def 18;
dom t = the carrier of (SCM R) by A2, Def1;
then reconsider t = t as State of (SCM R) by PARTFUN1:def 2;
reconsider w = t as SCM-State of R by A10, CARD_3:107;
A11: i1 in NAT by ORDINAL1:def 12;
NAT in dom (NAT .--> i1) by TARSKI:def 1;
then A12: (w +* (NAT .--> i1)) . NAT = (NAT .--> i1) . NAT by FUNCT_4:13
.= i1 by FUNCOP_1:72 ;
reconsider V = goto (i1,R) as Element of SCM-Instr R by Def1;
assume A13: goto (i1,R) is halting ; :: thesis: contradiction
A14: 6 is Element of Segm 8 by NAT_1:44;
w +* (NAT .--> i1) = SCM-Chg (w,i5)
.= SCM-Chg (w,(V jump_address)) by A14, SCMRINGI:2
.= SCM-Exec-Res (V,w) by SCMRING1:def 14, A11
.= Exec ((goto (i1,R)),t) by Th10
.= t by A13 ;
hence contradiction by A5, A12; :: thesis: verum