let I, J be Program of SCM+FSA ; :: thesis: for a being Int-Location holds (if>0 a,I,J) . (((card I) + (card J)) + 3) = halt SCM+FSA
let a be Int-Location ; :: thesis: (if>0 a,I,J) . (((card I) + (card J)) + 3) = halt SCM+FSA
A1: if>0 a,I,J = ((((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) ';' (Stop SCM+FSA ) by SCMFSA8B:def 2;
set II = (((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I;
A2: card ((((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) = (card (((Macro (a >0_goto ((card J) + 3))) ';' J) ';' (Goto ((card I) + 1)))) + (card I) by SCMFSA6A:61
.= ((card ((Macro (a >0_goto ((card J) + 3))) ';' J)) + (card (Goto ((card I) + 1)))) + (card I) by SCMFSA6A:61
.= ((card ((Macro (a >0_goto ((card J) + 3))) ';' J)) + 1) + (card I) by SCMFSA8A:29
.= (((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1) + (card I) by SCMFSA6A:61
.= ((2 + (card J)) + 1) + (card I) by SCMFSA7B:6
.= ((card I) + (card J)) + 3 ;
then A3: (((card I) + (card J)) + 3) -' (card ((((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I)) = 0 by XREAL_1:234;
y: (Stop SCM+FSA ) . 0 = halt SCM+FSA by AFINSQ_1:38;
card (Stop SCM+FSA ) = 1 by SCMNORM:3;
then ((card I) + (card J)) + 3 < (card ((((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I)) + (card (Stop SCM+FSA )) by A2, NAT_1:13;
hence (if>0 a,I,J) . (((card I) + (card J)) + 3) = IncAddr (halt SCM+FSA ),(card ((((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I)) by A1, A2, A3, Th13, y
.= halt SCM+FSA by SCMFSA_4:8 ;
:: thesis: verum