let R be Ring; :: thesis: for il being Nat holds SUCC (il,(SCM R)) = {il,(il + 1)}
let il be Nat; :: thesis: SUCC (il,(SCM R)) = {il,(il + 1)}
set X = { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ;
set N = {il,(il + 1)};
now :: thesis: for x being object holds
( ( x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } implies x in {il,(il + 1)} ) & ( x in {il,(il + 1)} implies x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ) )
let x be object ; :: thesis: ( ( x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } implies x in {il,(il + 1)} ) & ( x in {il,(il + 1)} implies b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } ) )
hereby :: thesis: ( x in {il,(il + 1)} implies b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } )
assume x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ; :: thesis: x in {il,(il + 1)}
then consider Y being set such that
A1: x in Y and
A2: Y in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } by TARSKI:def 4;
consider i being Element of the InstructionsF of (SCM R) such that
A3: Y = (NIC (i,il)) \ (JUMP i) by A2;
per cases ( i = [0,{},{}] or ex a, b being Data-Location of R st i = a := b or ex a, b being Data-Location of R st i = AddTo (a,b) or ex a, b being Data-Location of R st i = SubFrom (a,b) or ex a, b being Data-Location of R st i = MultBy (a,b) or ex i1 being Nat st i = goto (i1,R) or ex a being Data-Location of R ex i1 being Nat st i = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st i = a := r ) by SCMRING2:7;
suppose i = [0,{},{}] ; :: thesis: x in {il,(il + 1)}
then i = halt (SCM R) ;
then x in {il} \ (JUMP (halt (SCM R))) by ;
then x = il by TARSKI:def 1;
hence x in {il,(il + 1)} by TARSKI:def 2; :: thesis: verum
end;
suppose ex a, b being Data-Location of R st i = a := b ; :: thesis: x in {il,(il + 1)}
then consider a, b being Data-Location of R such that
A4: i = a := b ;
x in {(il + 1)} \ (JUMP (a := b)) by ;
then x = il + 1 by TARSKI:def 1;
hence x in {il,(il + 1)} by TARSKI:def 2; :: thesis: verum
end;
suppose ex a, b being Data-Location of R st i = AddTo (a,b) ; :: thesis: x in {il,(il + 1)}
then consider a, b being Data-Location of R such that
A5: i = AddTo (a,b) ;
x in {(il + 1)} \ (JUMP (AddTo (a,b))) by ;
then x = il + 1 by TARSKI:def 1;
hence x in {il,(il + 1)} by TARSKI:def 2; :: thesis: verum
end;
suppose ex a, b being Data-Location of R st i = SubFrom (a,b) ; :: thesis: x in {il,(il + 1)}
then consider a, b being Data-Location of R such that
A6: i = SubFrom (a,b) ;
x in {(il + 1)} \ (JUMP (SubFrom (a,b))) by ;
then x = il + 1 by TARSKI:def 1;
hence x in {il,(il + 1)} by TARSKI:def 2; :: thesis: verum
end;
suppose ex a, b being Data-Location of R st i = MultBy (a,b) ; :: thesis: x in {il,(il + 1)}
then consider a, b being Data-Location of R such that
A7: i = MultBy (a,b) ;
x in {(il + 1)} \ (JUMP (MultBy (a,b))) by ;
then x = il + 1 by TARSKI:def 1;
hence x in {il,(il + 1)} by TARSKI:def 2; :: thesis: verum
end;
suppose ex i1 being Nat st i = goto (i1,R) ; :: thesis: x in {il,(il + 1)}
then consider i1 being Nat such that
A8: i = goto (i1,R) ;
x in {i1} \ (JUMP i) by A1, A3, A8, Th29;
then x in {i1} \ {i1} by ;
hence x in {il,(il + 1)} by XBOOLE_1:37; :: thesis: verum
end;
suppose ex a being Data-Location of R ex i1 being Nat st i = a =0_goto i1 ; :: thesis: x in {il,(il + 1)}
then consider a being Data-Location of R, i1 being Nat such that
A9: i = a =0_goto i1 ;
A10: NIC (i,il) c= {i1,(il + 1)} by ;
x in NIC (i,il) by ;
then A11: ( x = i1 or x = il + 1 ) by ;
x in (NIC (i,il)) \ {i1} by A1, A3, A9, Th33;
then not x in {i1} by XBOOLE_0:def 5;
hence x in {il,(il + 1)} by ; :: thesis: verum
end;
suppose ex a being Data-Location of R ex r being Element of R st i = a := r ; :: thesis: x in {il,(il + 1)}
then consider a being Data-Location of R, r being Element of R such that
A12: i = a := r ;
x in {(il + 1)} \ (JUMP (a := r)) by ;
then x = il + 1 by TARSKI:def 1;
hence x in {il,(il + 1)} by TARSKI:def 2; :: thesis: verum
end;
end;
end;
assume A13: x in {il,(il + 1)} ; :: thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum }
per cases ( x = il or x = il + 1 ) by ;
suppose A14: x = il ; :: thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum }
set i = halt (SCM R);
(NIC ((halt (SCM R)),il)) \ (JUMP (halt (SCM R))) = {il} by AMISTD_1:2;
then A15: {il} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ;
x in {il} by ;
hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } by ; :: thesis: verum
end;
suppose A16: x = il + 1 ; :: thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum }
set a = the Data-Location of R;
set i = AddTo ( the Data-Location of R, the Data-Location of R);
(NIC ((AddTo ( the Data-Location of R, the Data-Location of R)),il)) \ (JUMP (AddTo ( the Data-Location of R, the Data-Location of R))) = {(il + 1)} by AMISTD_1:12;
then A17: {(il + 1)} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ;
x in {(il + 1)} by ;
hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } by ; :: thesis: verum
end;
end;
end;
hence SUCC (il,(SCM R)) = {il,(il + 1)} by TARSKI:2; :: thesis: verum