MAYBE Problem: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) Proof: DP Processor: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> mark#(M) a__x#(N,s(M)) -> mark#(N) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) TDG Processor: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> mark#(M) a__x#(N,s(M)) -> mark#(N) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) graph: a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> mark#(N) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> mark#(M) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,s(M)) -> mark#(N) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,s(M)) -> mark#(M) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,0()) -> mark#(N) a__x#(N,s(M)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__x#(N,s(M)) -> mark#(N) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__x#(N,s(M)) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X1) a__x#(N,s(M)) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X2) a__x#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__x#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__x#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__x#(N,s(M)) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__x#(N,s(M)) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__x#(N,s(M)) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__x#(N,s(M)) -> mark#(M) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__x#(N,s(M)) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X1) a__x#(N,s(M)) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X2) a__x#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__x#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__x#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__x#(N,s(M)) -> mark#(M) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__x#(N,s(M)) -> mark#(M) -> mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__plus#(N,s(M)) -> mark#(N) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X2) a__plus#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,s(M)) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__plus#(N,s(M)) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__plus#(N,s(M)) -> mark#(M) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X2) a__plus#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,s(M)) -> mark#(M) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__plus#(N,s(M)) -> mark#(M) -> mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__plus#(N,0()) -> mark#(N) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__plus#(N,0()) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X2) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,0()) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__plus#(N,0()) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> mark#(N) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> mark#(M) mark#(x(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(x(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(x(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> mark#(N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> mark#(M) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X2) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) Arctic Interpretation Processor: dimension: 1 interpretation: [a__x#](x0, x1) = 7x0 + x1 + 1, [a__plus#](x0, x1) = x0 + x1, [mark#](x0) = x0, [a__and#](x0, x1) = x0 + x1 + 4, [x](x0, x1) = 7x0 + x1 + 1, [plus](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1 + 4, [a__x](x0, x1) = 7x0 + x1 + 1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1 + 1, [0] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1 + 4, [tt] = 0 orientation: a__and#(tt(),X) = X + 4 >= X = mark#(X) a__plus#(N,0()) = N + 0 >= N = mark#(N) a__plus#(N,s(M)) = M + N >= M = mark#(M) a__plus#(N,s(M)) = M + N >= N = mark#(N) a__plus#(N,s(M)) = M + N >= M + N = a__plus#(mark(N),mark(M)) a__x#(N,s(M)) = M + 7N + 1 >= M = mark#(M) a__x#(N,s(M)) = M + 7N + 1 >= N = mark#(N) a__x#(N,s(M)) = M + 7N + 1 >= M + 7N + 1 = a__x#(mark(N),mark(M)) a__x#(N,s(M)) = M + 7N + 1 >= M + 7N + 1 = a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) = X1 + X2 + 4 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = a__and#(mark(X1),X2) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 = a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) = 7X1 + X2 + 1 >= X2 = mark#(X2) mark#(x(X1,X2)) = 7X1 + X2 + 1 >= X1 = mark#(X1) mark#(x(X1,X2)) = 7X1 + X2 + 1 >= 7X1 + X2 + 1 = a__x#(mark(X1),mark(X2)) mark#(s(X)) = X >= X = mark#(X) a__and(tt(),X) = X + 4 >= X = mark(X) a__plus(N,0()) = N + 1 >= N = mark(N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__x(N,0()) = 7N + 1 >= 0 = 0() a__x(N,s(M)) = M + 7N + 1 >= M + 7N + 1 = a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) = 7X1 + X2 + 1 >= 7X1 + X2 + 1 = a__x(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) a__and(X1,X2) = X1 + X2 + 4 >= X1 + X2 + 4 = and(X1,X2) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) a__x(X1,X2) = 7X1 + X2 + 1 >= 7X1 + X2 + 1 = x(X1,X2) problem: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> mark#(M) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__x#](x0, x1) = 5x0 + x1, [a__plus#](x0, x1) = x0 + 5x1 + 0, [mark#](x0) = x0, [a__and#](x0, x1) = 2x0 + x1, [x](x0, x1) = 5x0 + x1, [plus](x0, x1) = x0 + 5x1 + 0, [and](x0, x1) = 2x0 + x1, [a__x](x0, x1) = 5x0 + x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + 5x1 + 0, [0] = 2, [mark](x0) = x0, [a__and](x0, x1) = 2x0 + x1, [tt] = 2 orientation: a__and#(tt(),X) = X + 4 >= X = mark#(X) a__plus#(N,0()) = N + 7 >= N = mark#(N) a__plus#(N,s(M)) = 5M + N + 6 >= M = mark#(M) a__plus#(N,s(M)) = 5M + N + 6 >= N = mark#(N) a__plus#(N,s(M)) = 5M + N + 6 >= 5M + N + 0 = a__plus#(mark(N),mark(M)) a__x#(N,s(M)) = M + 5N + 1 >= M = mark#(M) a__x#(N,s(M)) = M + 5N + 1 >= M + 5N = a__x#(mark(N),mark(M)) a__x#(N,s(M)) = M + 5N + 1 >= M + 5N + 0 = a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__and#(mark(X1),X2) mark#(plus(X1,X2)) = X1 + 5X2 + 0 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + 5X2 + 0 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + 5X2 + 0 >= X1 + 5X2 + 0 = a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) = 5X1 + X2 >= X2 = mark#(X2) mark#(x(X1,X2)) = 5X1 + X2 >= 5X1 + X2 = a__x#(mark(X1),mark(X2)) mark#(s(X)) = X + 1 >= X = mark#(X) a__and(tt(),X) = X + 4 >= X = mark(X) a__plus(N,0()) = N + 7 >= N = mark(N) a__plus(N,s(M)) = 5M + N + 6 >= 5M + N + 1 = s(a__plus(mark(N),mark(M))) a__x(N,0()) = 5N + 2 >= 2 = 0() a__x(N,s(M)) = M + 5N + 1 >= M + 5N + 0 = a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + 5X2 + 0 >= X1 + 5X2 + 0 = a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) = 5X1 + X2 >= 5X1 + X2 = a__x(mark(X1),mark(X2)) mark(tt()) = 2 >= 2 = tt() mark(0()) = 2 >= 2 = 0() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) a__and(X1,X2) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) a__plus(X1,X2) = X1 + 5X2 + 0 >= X1 + 5X2 + 0 = plus(X1,X2) a__x(X1,X2) = 5X1 + X2 >= 5X1 + X2 = x(X1,X2) problem: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> mark#(M) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__x#](x0, x1) = 6x0 + 2x1 + 6, [a__plus#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0, [a__and#](x0, x1) = x0 + x1 + 0, [x](x0, x1) = 6x0 + 2x1 + 6, [plus](x0, x1) = x0 + x1 + 0, [and](x0, x1) = x0 + x1 + 0, [a__x](x0, x1) = 6x0 + 2x1 + 6, [s](x0) = x0 + 0, [a__plus](x0, x1) = x0 + x1 + 0, [0] = 0, [mark](x0) = x0 + 0, [a__and](x0, x1) = x0 + x1 + 0, [tt] = 0 orientation: a__and#(tt(),X) = X + 0 >= X = mark#(X) a__plus#(N,0()) = N + 0 >= N = mark#(N) a__plus#(N,s(M)) = M + N + 0 >= N = mark#(N) a__plus#(N,s(M)) = M + N + 0 >= M + N + 0 = a__plus#(mark(N),mark(M)) a__x#(N,s(M)) = 2M + 6N + 6 >= M = mark#(M) a__x#(N,s(M)) = 2M + 6N + 6 >= 2M + 6N + 6 = a__x#(mark(N),mark(M)) a__x#(N,s(M)) = 2M + 6N + 6 >= 2M + 6N + 6 = a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),X2) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) = 6X1 + 2X2 + 6 >= X2 = mark#(X2) mark#(x(X1,X2)) = 6X1 + 2X2 + 6 >= 6X1 + 2X2 + 6 = a__x#(mark(X1),mark(X2)) mark#(s(X)) = X + 0 >= X = mark#(X) a__and(tt(),X) = X + 0 >= X + 0 = mark(X) a__plus(N,0()) = N + 0 >= N + 0 = mark(N) a__plus(N,s(M)) = M + N + 0 >= M + N + 0 = s(a__plus(mark(N),mark(M))) a__x(N,0()) = 6N + 6 >= 0 = 0() a__x(N,s(M)) = 2M + 6N + 6 >= 2M + 6N + 6 = a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) = 6X1 + 2X2 + 6 >= 6X1 + 2X2 + 6 = a__x(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__plus(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = plus(X1,X2) a__x(X1,X2) = 6X1 + 2X2 + 6 >= 6X1 + 2X2 + 6 = x(X1,X2) problem: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__x#](x0, x1) = 2x0 + x1 + 0, [a__plus#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0 + 0, [a__and#](x0, x1) = 7x0 + 6x1, [x](x0, x1) = 2x0 + x1, [plus](x0, x1) = x0 + x1 + 0, [and](x0, x1) = 7x0 + 6x1 + 0, [a__x](x0, x1) = 2x0 + x1, [s](x0) = x0 + 0, [a__plus](x0, x1) = x0 + x1 + 0, [0] = 0, [mark](x0) = x0, [a__and](x0, x1) = 7x0 + 6x1 + 0, [tt] = 0 orientation: a__and#(tt(),X) = 6X + 7 >= X + 0 = mark#(X) a__plus#(N,0()) = N + 0 >= N + 0 = mark#(N) a__plus#(N,s(M)) = M + N + 0 >= N + 0 = mark#(N) a__plus#(N,s(M)) = M + N + 0 >= M + N + 0 = a__plus#(mark(N),mark(M)) a__x#(N,s(M)) = M + 2N + 0 >= M + 2N + 0 = a__x#(mark(N),mark(M)) a__x#(N,s(M)) = M + 2N + 0 >= M + 2N + 0 = a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) = 7X1 + 6X2 + 0 >= 7X1 + 6X2 = a__and#(mark(X1),X2) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = a__x#(mark(X1),mark(X2)) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__and(tt(),X) = 6X + 7 >= X = mark(X) a__plus(N,0()) = N + 0 >= N = mark(N) a__plus(N,s(M)) = M + N + 0 >= M + N + 0 = s(a__plus(mark(N),mark(M))) a__x(N,0()) = 2N + 0 >= 0 = 0() a__x(N,s(M)) = M + 2N + 0 >= M + 2N + 0 = a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) = 7X1 + 6X2 + 0 >= 7X1 + 6X2 + 0 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__x(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__and(X1,X2) = 7X1 + 6X2 + 0 >= 7X1 + 6X2 + 0 = and(X1,X2) a__plus(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = plus(X1,X2) a__x(X1,X2) = 2X1 + X2 >= 2X1 + X2 = x(X1,X2) problem: DPs: a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) SCC Processor: #sccs: 1 #rules: 9 #arcs: 129/100 DPs: a__x#(N,s(M)) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) a__plus#(N,0()) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(N) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) a__x(X1,X2) -> x(X1,X2) Open