MAYBE Problem: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(f(X,X)) -> f#(a(),b()) active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> f#(active(X1),X2) f#(mark(X1),X2) -> f#(X1,X2) proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) f#(ok(X1),ok(X2)) -> f#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(f(X,X)) -> f#(a(),b()) active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> f#(active(X1),X2) f#(mark(X1),X2) -> f#(X1,X2) proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) f#(ok(X1),ok(X2)) -> f#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(f(X,X)) -> f#(a(),b()) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(mark(X1),X2) -> f#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(mark(X1),X2) -> f#(X1,X2) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> f#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> active#(X1) -> active#(f(X,X)) -> f#(a(),b()) active#(f(X,X)) -> f#(a(),b()) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(f(X,X)) -> f#(a(),b()) -> f#(mark(X1),X2) -> f#(X1,X2) EDG Processor: DPs: active#(f(X,X)) -> f#(a(),b()) active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> f#(active(X1),X2) f#(mark(X1),X2) -> f#(X1,X2) proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) f#(ok(X1),ok(X2)) -> f#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(f(X,X)) -> f#(a(),b()) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(mark(X1),X2) -> f#(X1,X2) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(mark(X1),X2) -> f#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(f(X1,X2)) -> active#(X1) -> active#(f(X,X)) -> f#(a(),b()) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> f#(active(X1),X2) SCC Processor: #sccs: 4 #rules: 7 #arcs: 31/144 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [mark](x0) = 0, [b] = 0, [a] = 1, [active](x0) = x0 + 1, [f](x0, x1) = x0 + x1 + 1 orientation: proper#(f(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(f(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(f(X,X)) = 2X + 2 >= 0 = mark(f(a(),b())) active(b()) = 1 >= 0 = mark(a()) active(f(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = f(active(X1),X2) f(mark(X1),X2) = X2 + 1 >= 0 = mark(f(X1,X2)) proper(f(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = f(proper(X1),proper(X2)) proper(a()) = 1 >= 1 = ok(a()) proper(b()) = 0 >= 0 = ok(b()) f(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(f(X1,X2)) -> active#(X1) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0 + 1, [mark](x0) = x0 + 1, [b] = 0, [a] = 0, [active](x0) = x0 + 1, [f](x0, x1) = x0 + 1 orientation: active#(f(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(f(X,X)) = X + 2 >= 2 = mark(f(a(),b())) active(b()) = 1 >= 1 = mark(a()) active(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(active(X1),X2) f(mark(X1),X2) = X1 + 2 >= X1 + 2 = mark(f(X1,X2)) proper(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(proper(X1),proper(X2)) proper(a()) = 1 >= 0 = ok(a()) proper(b()) = 1 >= 0 = ok(b()) f(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1) = x0 + x1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [mark](x0) = x0, [b] = 0, [a] = 0, [active](x0) = 0, [f](x0, x1) = x0 orientation: f#(ok(X1),ok(X2)) = X1 + X2 + 2 >= X1 + X2 = f#(X1,X2) f#(mark(X1),X2) = X1 + X2 >= X1 + X2 = f#(X1,X2) active(f(X,X)) = 0 >= 0 = mark(f(a(),b())) active(b()) = 0 >= 0 = mark(a()) active(f(X1,X2)) = 0 >= 0 = f(active(X1),X2) f(mark(X1),X2) = X1 >= X1 = mark(f(X1,X2)) proper(f(X1,X2)) = 1 >= 1 = f(proper(X1),proper(X2)) proper(a()) = 1 >= 1 = ok(a()) proper(b()) = 1 >= 1 = ok(b()) f(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(f(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: f#(mark(X1),X2) -> f#(X1,X2) TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [mark](x0) = x0 + 1, [b] = 0, [a] = 0, [active](x0) = 1, [f](x0, x1) = x0 orientation: f#(mark(X1),X2) = X1 + 1 >= X1 = f#(X1,X2) active(f(X,X)) = 1 >= 1 = mark(f(a(),b())) active(b()) = 1 >= 1 = mark(a()) active(f(X1,X2)) = 1 >= 1 = f(active(X1),X2) f(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(f(X1,X2)) proper(f(X1,X2)) = 0 >= 0 = f(proper(X1),proper(X2)) proper(a()) = 0 >= 0 = ok(a()) proper(b()) = 0 >= 0 = ok(b()) f(ok(X1),ok(X2)) = 0 >= 0 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(X,X)) -> mark(f(a(),b())) active(b()) -> mark(a()) active(f(X1,X2)) -> f(active(X1),X2) f(mark(X1),X2) -> mark(f(X1,X2)) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed