thus f :=<0,...,0> a is parahalting :: thesis: f :=<0,...,0> a is keeping_0

proof

set Ma = Macro (f :=<0,...,0> a);
let s be 0 -started State of SCM+FSA; :: according to AMISTD_1:def 11,SCMFSA6C:def 1 :: thesis: for b₁ being set holds
( not Macro (f :=<0,...,0> a) c= b₁ or b₁ halts_on s )
A11: Start-At (0,SCM+FSA) c= s by MEMSTR_0:29;
let P be Instruction-Sequence of SCM+FSA; :: thesis: ( not Macro (f :=<0,...,0> a) c= P or P halts_on s )
assume A12: Macro (f :=<0,...,0> a) c= P ; :: thesis: P halts_on s
take 1 ; :: according to EXTPRO_1:def 8 :: thesis: ( IC (Comput (P,s,1)) in dom P & CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA )
A13: dom P = NAT by PARTFUN1:def 2;
thus IC (Comput (P,s,1)) in dom P by A13; :: thesis: CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA
dom (Start-At (0,SCM+FSA)) = {(IC )} by FUNCOP_1:13;
then A14: IC in dom (Start-At (0,SCM+FSA)) by TARSKI:def 1;
A15: IC s = (Start-At (0,SCM+FSA)) . (IC ) by A14, A11, GRFUNC_1:2
.= 0 by FUNCOP_1:72 ;
then A16: IC (Exec ((f :=<0,...,0> a),s)) = succ 0 by SCMFSA_2:75
.= 0 + 1 ;
A17: 1 in dom (Macro (f :=<0,...,0> a)) by COMPOS_1:60;
A18: 0 in dom (Macro (f :=<0,...,0> a)) by COMPOS_1:60;
A19: P . 0 = (Macro (f :=<0,...,0> a)) . 0 by A12, A18, GRFUNC_1:2
.= f :=<0,...,0> a by COMPOS_1:58 ;
A20: P . 1 = (Macro (f :=<0,...,0> a)) . 1 by A12, A17, GRFUNC_1:2
.= halt SCM+FSA by COMPOS_1:59 ;
Comput (P,s,(0 + 1)) = Following (P,(Comput (P,s,0))) by EXTPRO_1:3
.= Following (P,s)
.= Exec ((f :=<0,...,0> a),s) by A15, A13, A19, PARTFUN1:def 6 ;
hence CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA by A16, A13, A20, PARTFUN1:def 6; :: thesis: verum

end;

thus f :=<0,...,0> a is keeping_0 :: thesis: verum

proof

set Ma = Macro (f :=<0,...,0> a);
let s be 0 -started State of SCM+FSA; :: according to SCMFSA6B:def 4,SCMFSA6C:def 2 :: thesis: for b₁ being set holds
( not Macro (f :=<0,...,0> a) c= b₁ or for b₂ being Element of NAT holds (Comput (b₁,s,b₂)) . (intloc 0) = s . (intloc 0) )
A21: Start-At (0,SCM+FSA) c= s by MEMSTR_0:29;
let P be Instruction-Sequence of SCM+FSA; :: thesis: ( not Macro (f :=<0,...,0> a) c= P or for b₁ being Element of NAT holds (Comput (P,s,b₁)) . (intloc 0) = s . (intloc 0) )
assume A22: Macro (f :=<0,...,0> a) c= P ; :: thesis: for b₁ being Element of NAT holds (Comput (P,s,b₁)) . (intloc 0) = s . (intloc 0)
let k be Element of NAT ; :: thesis: (Comput (P,s,k)) . (intloc 0) = s . (intloc 0)
dom (Start-At (0,SCM+FSA)) = {(IC )} by FUNCOP_1:13;
then A23: IC in dom (Start-At (0,SCM+FSA)) by TARSKI:def 1;
A24: IC s = (Start-At (0,SCM+FSA)) . (IC ) by A23, A21, GRFUNC_1:2
.= 0 by FUNCOP_1:72 ;
0 in dom (Macro (f :=<0,...,0> a)) by COMPOS_1:60;
then A25: (Macro (f :=<0,...,0> a)) . 0 = P . 0 by A22, GRFUNC_1:2;
A26: P /. (IC s) = P . (IC s) by PBOOLE:143;
A27: Comput (P,s,(0 + 1)) = Following (P,(Comput (P,s,0))) by EXTPRO_1:3
.= Following (P,s)
.= Exec ((f :=<0,...,0> a),s) by A24, A25, A26, COMPOS_1:58 ;
1 in dom (Macro (f :=<0,...,0> a)) by COMPOS_1:60;
then (Macro (f :=<0,...,0> a)) . 1 = P . 1 by A22, GRFUNC_1:2;
then A28: P . 1 = halt SCM+FSA by COMPOS_1:59;
IC (Exec ((f :=<0,...,0> a),s)) = succ 0 by A24, SCMFSA_2:75
.= 0 + 1 ;
then A29: CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA by A28, A27, PBOOLE:143;

per cases ( k = 0 or 1 <= k ) by NAT_1:14;

suppose k = 0 ; :: thesis: (Comput (P,s,k)) . (intloc 0) = s . (intloc 0)

hence (Comput (P,s,k)) . (intloc 0) = s . (intloc 0) by EXTPRO_1:2; :: thesis: verum

end;

suppose A30: 1 <= k ; :: thesis: (Comput (P,s,k)) . (intloc 0) = s . (intloc 0)

(Comput (P,s,1)) . (intloc 0) = s . (intloc 0) by A27, SCMFSA_2:75;
hence (Comput (P,s,k)) . (intloc 0) = s . (intloc 0) by A29, A30, EXTPRO_1:5; :: thesis: verum

end;

end;

end;