let s1, s2 be State of SCMPDS ; :: thesis: for J being parahalting shiftable Program of SCMPDS st Initialize (stop J) c= s1 holds
for n being Element of NAT st Shift (stop J),n c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
let I be parahalting shiftable Program of SCMPDS ; :: thesis: ( Initialize (stop I) c= s1 implies for n being Element of NAT st Shift (stop I),n c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) )
set SI = stop I;
set II = Initialize (stop I);
assume A1: Initialize (stop I) c= s1 ; :: thesis: for n being Element of NAT st Shift (stop I),n c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
let n be Element of NAT ; :: thesis: ( Shift (stop I),n c= s2 & IC s2 = n & DataPart s1 = DataPart s2 implies for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) )
assume that
A2: Shift (stop I),n c= s2 and
A3: IC s2 = n and
A4: DataPart s1 = DataPart s2 ; :: thesis: for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
A5: 0 in dom (stop I) by Th75;
then A6: 0 + n in dom (Shift (stop I),n) by VALUED_1:25;
defpred S₁[ Nat] means ( (IC (Comput (ProgramPart s1),s1,$1)) + n = IC (Comput (ProgramPart s2),s2,$1) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,$1)),(Comput (ProgramPart s1),s1,$1) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,$1)),(Comput (ProgramPart s2),s2,$1) & DataPart (Comput (ProgramPart s1),s1,$1) = DataPart (Comput (ProgramPart s2),s2,$1) );
dom (stop I) misses dom (Start-At 0 ,SCMPDS ) by Th54;
then A7: stop I c= Initialize (stop I) by FUNCT_4:33;
then A8: dom (stop I) c= dom (Initialize (stop I)) by GRFUNC_1:8;
A9: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A10: S₁[k] ; :: thesis: S₁[k + 1]
reconsider m = IC (Comput (ProgramPart s1),s1,k) as Element of NAT ;
set i = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k);
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
A11: Comput (ProgramPart s1),s1,(k + 1) = Following (ProgramPart s1),(Comput (ProgramPart s1),s1,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k) by T ;
reconsider l = IC (Comput (ProgramPart s1),s1,(k + 1)) as Element of NAT ;
A13: IC (Comput (ProgramPart s1),s1,(k + 1)) in dom (stop I) by A1, Def9;
then A14: l + n in dom (Shift (stop I),n) by VALUED_1:25;
T: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,k) by AMI_1:123;
A15: Comput (ProgramPart s2),s2,(k + 1) = Following (ProgramPart s2),(Comput (ProgramPart s2),s2,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s2),s2,k)),(Comput (ProgramPart s2),s2,k)),(Comput (ProgramPart s2),s2,k) by T ;
A16: IC (Comput (ProgramPart s1),s1,k) in dom (stop I) by A1, Def9;
Y: (ProgramPart (Comput (ProgramPart s1),s1,k)) /. (IC (Comput (ProgramPart s1),s1,k)) = (Comput (ProgramPart s1),s1,k) . (IC (Comput (ProgramPart s1),s1,k)) by COMPOS_1:38;
A17: CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k) = s1 . (IC (Comput (ProgramPart s1),s1,k)) by Y, AMI_1:54
.= (Initialize (stop I)) . (IC (Comput (ProgramPart s1),s1,k)) by A1, A8, A16, GRFUNC_1:8
.= (stop I) . (IC (Comput (ProgramPart s1),s1,k)) by A7, A16, GRFUNC_1:8 ;
then A18: ( InsCode (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k)) <> 1 & InsCode (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k)) <> 3 ) by A16, Def12;
A19: CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k) valid_at m by A16, A17, Def12;
hence A20: (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Comput (ProgramPart s2),s2,(k + 1)) by A10, A11, A15, A18, Th83; :: thesis: ( CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1)) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))),(Comput (ProgramPart s2),s2,(k + 1)) & DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1)) )
Y: (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))) /. (IC (Comput (ProgramPart s1),s1,(k + 1))) = (Comput (ProgramPart s1),s1,(k + 1)) . (IC (Comput (ProgramPart s1),s1,(k + 1))) by COMPOS_1:38;
Z: (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))) /. (IC (Comput (ProgramPart s2),s2,(k + 1))) = (Comput (ProgramPart s2),s2,(k + 1)) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by COMPOS_1:38;
CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1)) = s1 . l by Y, AMI_1:54
.= (Initialize (stop I)) . l by A1, A8, A13, GRFUNC_1:8
.= (stop I) . l by A7, A13, GRFUNC_1:8 ;
hence CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1)) = (Shift (stop I),n) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A20, A13, VALUED_1:def 12
.= s2 . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A2, A20, A14, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))),(Comput (ProgramPart s2),s2,(k + 1)) by Z, AMI_1:54 ;
:: thesis: DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1))
thus DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1)) by A10, A11, A15, A18, A19, Th83; :: thesis: verum

end;

A21: IC SCMPDS in dom (Initialize (stop I)) by COMPOS_1:def 16;
then A22: s1 . (IC s1) = s1 . (IC (Initialize (stop I))) by A1, GRFUNC_1:8
.= s1 . 0 by COMPOS_1:def 16
.= (Initialize (stop I)) . 0 by A1, A8, A5, GRFUNC_1:8
.= (stop I) . 0 by A7, A5, GRFUNC_1:8 ;
let i be Element of NAT ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
A23: DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart s2 by A4, AMI_1:13
.= DataPart (Comput (ProgramPart s2),s2,0 ) by AMI_1:13 ;
A24: IC (Comput (ProgramPart s1),s1,0 ) = s1 . (IC SCMPDS ) by AMI_1:13
.= IC (Initialize (stop I)) by A1, A21, GRFUNC_1:8
.= 0 by COMPOS_1:def 16 ;
u: Comput (ProgramPart s1),s1,0 = s1 by AMI_1:13;
v: Comput (ProgramPart s2),s2,0 = s2 by AMI_1:13;
Y: (ProgramPart s2) /. (IC s2) = s2 . (IC s2) by COMPOS_1:38;
Z: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
CurInstr (ProgramPart (Comput (ProgramPart s1),s1,0 )),(Comput (ProgramPart s1),s1,0 ) = CurInstr (ProgramPart s1),s1 by u
.= (Shift (stop I),n) . (0 + n) by A5, A22, Z, VALUED_1:def 12
.= CurInstr (ProgramPart s2),s2 by A2, A3, A6, Y, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart s2),s2,0 )),(Comput (ProgramPart s2),s2,0 ) by v ;
then A25: S₁[ 0 ] by A3, A24, A23, AMI_1:13;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A25, A9);
hence ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ; :: thesis: verum