let I be Instruction of S; :: thesis: ( I is halting implies I is IC-good )
assume A1: I is halting ; :: thesis: I is IC-good
let k be natural number ; :: according to AMISTD_2:def 18 :: thesis: for s1 being State of S holds (IC (Exec (I,s1))) + k = IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k))))
let s1 be State of S; :: thesis: (IC (Exec (I,s1))) + k = IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k))))
set s2 = IncrIC (s1,k);
dom ((IC S) .--> ((IC s1) + k)) = {(IC S)} by FUNCOP_1:19;
then A3: IC S in dom ((IC S) .--> ((IC s1) + k)) by TARSKI:def 1;
thus (IC (Exec (I,s1))) + k = (IC s1) + k by A1, EXTPRO_1:def 3
.= ((IC S) .--> ((IC s1) + k)) . (IC S) by FUNCOP_1:87
.= IC (IncrIC (s1,k)) by A3, FUNCT_4:14
.= IC (Exec (I,(IncrIC (s1,k)))) by A1, EXTPRO_1:def 3
.= IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k)))) by A1, COMPOS_1:92 ; :: thesis: verum