let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s1, s2 being State of SCMPDS
for I being Program of SCMPDS st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
let s1, s2 be State of SCMPDS; :: thesis: for I being Program of SCMPDS st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
let I be Program of SCMPDS; :: thesis: ( DataPart s1 = DataPart s2 & I is_closed_on s1,P1 implies I is_closed_on s2,P2 )
set pI = stop I;
set S1 = Initialize s1;
set S2 = Initialize s2;
set E1 = P1 +* (stop I);
set E2 = P2 +* (stop I);
assume A3: DataPart s1 = DataPart s2 ; :: thesis: ( not I is_closed_on s1,P1 or I is_closed_on s2,P2 )
A4: Comput ((P2 +* (stop I)),(Initialize s2),0) = Initialize s2 by EXTPRO_1:2;
A5: Comput ((P1 +* (stop I)),(Initialize s1),0) = Initialize s1 by EXTPRO_1:2;
then A6: DataPart (Comput ((P1 +* (stop I)),(Initialize s1),0)) = DataPart s1 by MEMSTR_0:45
.= DataPart (Comput ((P2 +* (stop I)),(Initialize s2),0)) by A3, A4, MEMSTR_0:45 ;
defpred S₁[ Element of NAT ] means ( IC (Comput ((P1 +* (stop I)),(Initialize s1),$1)) = IC (Comput ((P2 +* (stop I)),(Initialize s2),$1)) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),$1))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),$1))) & DataPart (Comput ((P1 +* (stop I)),(Initialize s1),$1)) = DataPart (Comput ((P2 +* (stop I)),(Initialize s2),$1)) );
A7: 0 in dom (stop I) by COMPOS_1:36;
assume A10: I is_closed_on s1,P1 ; :: thesis: I is_closed_on s2,P2

A11: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
A13: Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) = Following ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) by EXTPRO_1:3;
assume A14: S₁[k] ; :: thesis: S₁[k + 1]
then A15: for a being Int_position holds (Comput ((P1 +* (stop I)),(Initialize s1),k)) . a = (Comput ((P2 +* (stop I)),(Initialize s2),k)) . a by SCMPDS_4:8;
stop I c= P2 +* (stop I) by FUNCT_4:25;
then A16: stop I c= P2 +* (stop I) ;
A17: IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1))) in dom (stop I) by A10, Def2;
A19: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Following ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) by EXTPRO_1:3;
Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) by A14, A15, SCMPDS_4:2;
then XX: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) by A14, A13, A19;
then A20: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) ;
A21: IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1))) by XX;
A22: (P1 +* (stop I)) /. (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) = (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) by PBOOLE:143;
A23: (P2 +* (stop I)) /. (IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) by PBOOLE:143;
stop I c= P1 +* (stop I) by FUNCT_4:25;
then stop I c= P1 +* (stop I) ;
then CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) = (stop I) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) by A17, A22, GRFUNC_1:2
.= CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) by A16, A21, A17, A23, GRFUNC_1:2 ;
hence S₁[k + 1] by A20; :: thesis: verum

end;

A25: IC (Comput ((P2 +* (stop I)),(Initialize s2),0)) = IC (Initialize s2) by A4
.= 0 by MEMSTR_0:def 8 ;
A26: (P1 +* (stop I)) /. (IC (Comput ((P1 +* (stop I)),(Initialize s1),0))) = (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),0))) by PBOOLE:143;
A27: (P2 +* (stop I)) /. (IC (Comput ((P2 +* (stop I)),(Initialize s2),0))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),0))) by PBOOLE:143;
A28: IC (Comput ((P1 +* (stop I)),(Initialize s1),0)) = IC (Initialize s1) by A5
.= 0 by MEMSTR_0:def 8 ;
then CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),0))) = (stop I) . 0 by A7, A26, FUNCT_4:13
.= CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),0))) by A25, A7, A27, FUNCT_4:13 ;
then A29: S₁[ 0 ] by A28, A25, A6;

now

let k be Element of NAT ; :: thesis: IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I)
A30: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A29, A11);
IC (Comput ((P1 +* (stop I)),(Initialize s1),k)) in dom (stop I) by A10, Def2;
hence IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I) by A30; :: thesis: verum

end;

hence I is_closed_on s2,P2 by Def2; :: thesis: verum