0.00/0.77 YES 0.00/0.77 0.00/0.77 Problem: 0.00/0.77 filter(n, r, nil()) -> pair(mo(), nil()) 0.00/0.77 filter(n, r, cons(x, xs)) -> pair(x, xs) <= eq(div(x, n), pair(q, r)) = true() 0.00/0.77 filter(n, r, cons(x, xs)) -> pair(y, cons(x, ys())) <= filter(n, r, xs) = pair(y, ys()), eq(div(x, n), pair(q, r)) = false() 0.00/0.77 0.00/0.77 Proof: 0.00/0.77 This system is confluent. 0.00/0.77 By \cite{SMI95}, Corollary 4.7 or 5.3. 0.00/0.77 This system is oriented. 0.00/0.77 This system is of type 3 or smaller. 0.00/0.77 This system is right-stable. 0.00/0.78 This system is properly oriented. 0.00/0.78 This is an overlay system. 0.00/0.78 This system is left-linear. 0.00/0.78 All 2 critical pairs are trivial or infeasible. 0.00/0.78 CP: pair(x, $3) = pair($7, cons(x, ys())) <= filter($5, $8, $3) = pair($7, ys()), eq(div(x, $5), pair(y, $8)) = false(), eq(div(x, $5), pair($4, $8)) = true(): 0.00/0.78 This critical pair is infeasible. 0.00/0.78 This critical pair is conditional. 0.00/0.78 This critical pair has some non-trivial conditions. 0.00/0.78 'tcap(conds(filter($5, $8, $3), eq(div(x, $5), pair(y, $8)), eq(div(x, $5), pair($4, $8))))' and 'conds(pair($7, ys()), false(), true())' are not unifiable. 0.00/0.78 CP: pair($3, cons($1, ys())) = pair($1, $8) <= eq(div($1, x), pair($4, $5)) = true(), filter(x, $5, $8) = pair($3, ys()), eq(div($1, x), pair($7, $5)) = false(): 0.00/0.78 This critical pair is infeasible. 0.00/0.78 This critical pair is conditional. 0.00/0.78 This critical pair has some non-trivial conditions. 0.00/0.78 usable rules 0.00/0.78 '\Sigma(eq(div($1, x), pair($7, $5))) \cap (->^*_R)[\Sigma(REN(false()))]' is empty. 0.00/0.78 0.00/0.78 EOF