let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s1, s2 being 0 -started State of SCM+FSA
for I being really-closed Program of SCM+FSA st I is_halting_on s1,P1 & I c= P1 & I c= P2 & ex k being Nat st Comput (P1,s1,k) = s2 holds
Result (P1,s1) = Result (P2,s2)
let s1, s2 be 0 -started State of SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA st I is_halting_on s1,P1 & I c= P1 & I c= P2 & ex k being Nat st Comput (P1,s1,k) = s2 holds
Result (P1,s1) = Result (P2,s2)
set D = Data-Locations ;
let I be really-closed Program of SCM+FSA; :: thesis: ( I is_halting_on s1,P1 & I c= P1 & I c= P2 & ex k being Nat st Comput (P1,s1,k) = s2 implies Result (P1,s1) = Result (P2,s2) )
assume A1: I is_halting_on s1,P1 ; :: thesis: ( not I c= P1 or not I c= P2 or for k being Nat holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
A2: Start-At (0,SCM+FSA) c= s1 by MEMSTR_0:29;
A3: s2 = Initialize s2 by MEMSTR_0:44;
assume I c= P1 ; :: thesis: ( not I c= P2 or for k being Nat holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
then A4: P1 = P1 +* I by FUNCT_4:98;
assume I c= P2 ; :: thesis: ( for k being Nat holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
then A5: P2 = P2 +* I by FUNCT_4:98;
s1 = Initialize s1 by A2, FUNCT_4:98;
then A6: P1 halts_on s1 by A1, A4;
then consider n being Nat such that
A7: CurInstr (P1,(Comput (P1,s1,n))) = halt SCM+FSA ;
given k being Nat such that A8: Comput (P1,s1,k) = s2 ; :: thesis: Result (P1,s1) = Result (P2,s2)
set s3 = Comput (P1,s1,k);
set P3 = P1;
A9: IC in dom (Comput (P1,s1,k)) by MEMSTR_0:2;
IC (Comput (P1,s1,k)) = IC s2 by A8
.= IC (Initialize s2) by A3
.= 0 by FUNCT_4:113 ;
then (IC ) .--> 0 c= Comput (P1,s1,k) by A9, FUNCOP_1:73;
then Start-At (0,SCM+FSA) c= Comput (P1,s1,k) ;
then A10: Comput (P1,s1,k) = Initialize (Comput (P1,s1,k)) by FUNCT_4:98;
A11: Comput (P1,s1,(n + k)) = Comput (P1,(Comput (P1,s1,k)),n) by EXTPRO_1:4;
A12: Comput (P1,s1,(n + k)) = Comput (P1,s1,n) by A7, EXTPRO_1:5, NAT_1:11;
P1 halts_on Comput (P1,s1,k) by A7, A12, A11, EXTPRO_1:29;
then A13: I is_halting_on Comput (P1,s1,k),P1 by A10, A4;
A14: DataPart (Comput (P1,s1,k)) = DataPart s2 by A8;
consider k being Nat such that
A15: CurInstr (P1,(Comput (P1,s1,k))) = halt SCM+FSA by A6;
A16: P1 . (IC (Comput (P1,s1,k))) = halt SCM+FSA by A15, PBOOLE:143;
Result (P1,(Comput (P1,s1,k))) = Result (P2,s2) by A3, A14, A10, A13, Th63, A4, A5;
hence Result (P1,s1) = Result (P2,s2) by A16, EXTPRO_1:8; :: thesis: verum