let s be State of SCMPDS; :: thesis: for a being Int_position holds
( (Exec ((return a),s)) . (IC ) = (abs (s . (DataLoc ((s . a),RetIC)))) + 2 & (Exec ((return a),s)) . a = s . (DataLoc ((s . a),RetSP)) & ( for b being Int_position st a <> b holds
(Exec ((return a),s)) . b = s . b ) )
let a be Int_position ; :: thesis: ( (Exec ((return a),s)) . (IC ) = (abs (s . (DataLoc ((s . a),RetIC)))) + 2 & (Exec ((return a),s)) . a = s . (DataLoc ((s . a),RetSP)) & ( for b being Int_position st a <> b holds
(Exec ((return a),s)) . b = s . b ) )
reconsider S = s as SCMPDS-State by CARD_3:107;
reconsider da = a as Element of SCM-Data-Loc by Def2;
reconsider I = return a as Element of SCMPDS-Instr ;
set A1 = Address_Add (S,(I address_1),RetSP);
set S1 = SCM-Chg (S,(I address_1),(S . (Address_Add (S,(I address_1),RetSP))));
set A2 = Address_Add (S,(I address_1),RetIC);
set lc = PopInstrLoc (S,(Address_Add (S,(I address_1),RetIC)));
reconsider i = 1 as Element of Segm 14 by NAT_1:44;
A1: I = [i,{},<*da*>] ;
then A2: I address_1 = da by SCMPDS_1:18;
A3: Exec ((return a),s) = SCM-Exec-Res (I,S) by SCMPDS_1:def 23
.= SCM-Chg ((SCM-Chg (S,(I address_1),(S . (Address_Add (S,(I address_1),RetSP))))),(PopInstrLoc (S,(Address_Add (S,(I address_1),RetIC))))) by A1, SCMPDS_1:def 22 ;
hence (Exec ((return a),s)) . (IC ) = (abs (s . (DataLoc ((s . a),RetIC)))) + 2 by A2, Th6, SCMPDS_1:13; :: thesis: ( (Exec ((return a),s)) . a = s . (DataLoc ((s . a),RetSP)) & ( for b being Int_position st a <> b holds
(Exec ((return a),s)) . b = s . b ) )
thus (Exec ((return a),s)) . a = (SCM-Chg (S,(I address_1),(S . (Address_Add (S,(I address_1),RetSP))))) . da by A3, SCMPDS_1:14
.= s . (DataLoc ((s . a),RetSP)) by A2, SCMPDS_1:16 ; :: thesis: for b being Int_position st a <> b holds
(Exec ((return a),s)) . b = s . b
let b be Int_position ; :: thesis: ( a <> b implies (Exec ((return a),s)) . b = s . b )
reconsider mn = b as Element of SCM-Data-Loc by Def2;
assume A4: b <> a ; :: thesis: (Exec ((return a),s)) . b = s . b
thus (Exec ((return a),s)) . b = (SCM-Chg (S,(I address_1),(S . (Address_Add (S,(I address_1),RetSP))))) . mn by A3, SCMPDS_1:14
.= s . b by A2, A4, SCMPDS_1:17 ; :: thesis: verum