let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let s be State of SCM+FSA; :: thesis: for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let I, J be really-closed parahalting MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let a be read-write Int-Location; :: thesis: ( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A1: I is_halting_on Initialized s,P by SCMFSA7B:19;
for s being 0 -started State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st if>0 (a,I,J) c= P holds
P halts_on s

proof

let s be 0 -started State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA st if>0 (a,I,J) c= P holds
P halts_on s
Start-At (0,SCM+FSA) c= s by MEMSTR_0:29;
then A2: s = Initialize s by FUNCT_4:98;
let Q be Instruction-Sequence of SCM+FSA; :: thesis: ( if>0 (a,I,J) c= Q implies Q halts_on s )
assume A3: if>0 (a,I,J) c= Q ; :: thesis: Q halts_on s
A4: if>0 (a,I,J) c= Q by A3;
A5: I is_halting_on s,Q by SCMFSA7B:19;
A6: J is_halting_on s,Q by SCMFSA7B:19;
A7: Q +* (if>0 (a,I,J)) = Q by A4, FUNCT_4:98;

per cases ( s . a > 0 or s . a <= 0 ) ;

suppose s . a > 0 ; :: thesis: Q halts_on s

then if>0 (a,I,J) is_halting_on s,Q by A5, Th13;
hence Q halts_on s by A2, A7, SCMFSA7B:def 7; :: thesis: verum

end;

suppose s . a <= 0 ; :: thesis: Q halts_on s

then if>0 (a,I,J) is_halting_on s,Q by A6, Th15;
hence Q halts_on s by A2, A7, SCMFSA7B:def 7; :: thesis: verum

end;

end;

end;

hence if>0 (a,I,J) is parahalting by AMISTD_1:def 11; :: thesis: ( ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
thus ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by A1, Th14; :: thesis: ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
J is_halting_on Initialized s,P by SCMFSA7B:19;
hence ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by Th16; :: thesis: verum