YES TRS: { minus(x, 0()) -> x, minus(s(x), s(y)) -> minus(x, y), f(0()) -> s(0()), f(s(x)) -> minus(s(x), g(f(x))), g(0()) -> 0(), g(s(x)) -> minus(s(x), f(g(x)))} DP: Strict: {minus#(s(x), s(y)) -> minus#(x, y), f#(s(x)) -> minus#(s(x), g(f(x))), f#(s(x)) -> f#(x), f#(s(x)) -> g#(f(x)), g#(s(x)) -> minus#(s(x), f(g(x))), g#(s(x)) -> f#(g(x)), g#(s(x)) -> g#(x)} Weak: { minus(x, 0()) -> x, minus(s(x), s(y)) -> minus(x, y), f(0()) -> s(0()), f(s(x)) -> minus(s(x), g(f(x))), g(0()) -> 0(), g(s(x)) -> minus(s(x), f(g(x)))} EDG: {(g#(s(x)) -> minus#(s(x), f(g(x))), minus#(s(x), s(y)) -> minus#(x, y)) (f#(s(x)) -> f#(x), f#(s(x)) -> g#(f(x))) (f#(s(x)) -> f#(x), f#(s(x)) -> f#(x)) (f#(s(x)) -> f#(x), f#(s(x)) -> minus#(s(x), g(f(x)))) (f#(s(x)) -> g#(f(x)), g#(s(x)) -> g#(x)) (f#(s(x)) -> g#(f(x)), g#(s(x)) -> f#(g(x))) (f#(s(x)) -> g#(f(x)), g#(s(x)) -> minus#(s(x), f(g(x)))) (g#(s(x)) -> f#(g(x)), f#(s(x)) -> minus#(s(x), g(f(x)))) (g#(s(x)) -> f#(g(x)), f#(s(x)) -> f#(x)) (g#(s(x)) -> f#(g(x)), f#(s(x)) -> g#(f(x))) (g#(s(x)) -> g#(x), g#(s(x)) -> minus#(s(x), f(g(x)))) (g#(s(x)) -> g#(x), g#(s(x)) -> f#(g(x))) (g#(s(x)) -> g#(x), g#(s(x)) -> g#(x)) (minus#(s(x), s(y)) -> minus#(x, y), minus#(s(x), s(y)) -> minus#(x, y)) (f#(s(x)) -> minus#(s(x), g(f(x))), minus#(s(x), s(y)) -> minus#(x, y))} SCCS: Scc: {f#(s(x)) -> f#(x), f#(s(x)) -> g#(f(x)), g#(s(x)) -> f#(g(x)), g#(s(x)) -> g#(x)} Scc: {minus#(s(x), s(y)) -> minus#(x, y)} SCC: Strict: {f#(s(x)) -> f#(x), f#(s(x)) -> g#(f(x)), g#(s(x)) -> f#(g(x)), g#(s(x)) -> g#(x)} Weak: { minus(x, 0()) -> x, minus(s(x), s(y)) -> minus(x, y), f(0()) -> s(0()), f(s(x)) -> minus(s(x), g(f(x))), g(0()) -> 0(), g(s(x)) -> minus(s(x), f(g(x)))} POLY: Argument Filtering: pi(g#) = 0, pi(g) = 0, pi(f#) = 0, pi(f) = [0], pi(s) = [0], pi(0) = [], pi(minus) = 0 Usable Rules: {} Interpretation: [f](x0) = x0 + 1, [s](x0) = x0 + 1 Strict: {f#(s(x)) -> g#(f(x))} Weak: { minus(x, 0()) -> x, minus(s(x), s(y)) -> minus(x, y), f(0()) -> s(0()), f(s(x)) -> minus(s(x), g(f(x))), g(0()) -> 0(), g(s(x)) -> minus(s(x), f(g(x)))} EDG: {} SCCS: Qed SCC: Strict: {minus#(s(x), s(y)) -> minus#(x, y)} Weak: { minus(x, 0()) -> x, minus(s(x), s(y)) -> minus(x, y), f(0()) -> s(0()), f(s(x)) -> minus(s(x), g(f(x))), g(0()) -> 0(), g(s(x)) -> minus(s(x), f(g(x)))} SPSC: Simple Projection: pi(minus#) = 0 Strict: {} Qed