let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic standard-ins homogeneous regular J/A-independent halting relocable AMI-Struct of N
for INS being Instruction of S
for s being State of S
for j, k being Element of NAT st IC s = j + k holds
Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k)
let S be non empty stored-program IC-Ins-separated definite realistic standard-ins homogeneous regular J/A-independent halting relocable AMI-Struct of N; :: thesis: for INS being Instruction of S
for s being State of S
for j, k being Element of NAT st IC s = j + k holds
Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k)
let INS be Instruction of S; :: thesis: for s being State of S
for j, k being Element of NAT st IC s = j + k holds
Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k)
let s be State of S; :: thesis: for j, k being Element of NAT st IC s = j + k holds
Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k)
let j, k be Element of NAT ; :: thesis: ( IC s = j + k implies Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k) )
assume A1: IC s = j + k ; :: thesis: Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k)
set s1 = s +* (Start-At (j,S));
A2: IncIC ((s +* (Start-At (j,S))),k) = s +* (Start-At ((IC s),S)) by A1, COMPOS_1:163
.= s by COMPOS_1:165 ;
A3: Exec ((IncAddr (INS,k)),(IncIC ((s +* (Start-At (j,S))),k))) = IncIC ((Exec (INS,(s +* (Start-At (j,S))))),k) by Th4;
thus Exec (INS,(DecIC (s,k))) = Exec (INS,(s +* (Start-At (j,S)))) by A1, NAT_D:34
.= (Exec (INS,(s +* (Start-At (j,S))))) +* (Start-At ((IC (Exec (INS,(s +* (Start-At (j,S)))))),S)) by COMPOS_1:165
.= (IncIC ((Exec (INS,(s +* (Start-At (j,S))))),k)) +* (Start-At ((IC (Exec (INS,(s +* (Start-At (j,S)))))),S)) by FUNCT_4:122
.= DecIC ((Exec ((IncAddr (INS,k)),s)),k) by A2, A3, COMPOS_1:166 ; :: thesis: verum