let N be non empty with_non-empty_elements set ; :: thesis: for S being standard-ins homogeneous regular J/A-independent non empty IC-Ins-separated 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 standard-ins homogeneous regular J/A-independent non empty IC-Ins-separated 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));
B3: IncIC ((s +* (Start-At (j,S))),k) = s +* (Start-At ((IC s),S)) by A1, MEMSTR_0:58
.= s by MEMSTR_0:31 ;
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 MEMSTR_0:31
.= (IncIC ((Exec (INS,(s +* (Start-At (j,S))))),k)) +* (Start-At ((IC (Exec (INS,(s +* (Start-At (j,S)))))),S)) by FUNCT_4:114
.= DecIC ((IncIC ((Exec (INS,(s +* (Start-At (j,S))))),k)),k) by MEMSTR_0:59
.= DecIC ((IncIC ((Exec (INS,(s +* (Start-At (j,S))))),k)),k)
.= DecIC ((Exec ((IncAddr (INS,k)),s)),k) by B3, Th4
.= DecIC ((Exec ((IncAddr (INS,k)),s)),k) ; :: thesis: verum