MAYBE Problem: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(f(a(),b(),X)) -> f#(X,X,X) active#(f(X1,X2,X3)) -> active#(X1) active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) active#(f(X1,X2,X3)) -> active#(X3) active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) 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(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) EDG Processor: DPs: active#(f(a(),b(),X)) -> f#(X,X,X) active#(f(X1,X2,X3)) -> active#(X1) active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) active#(f(X1,X2,X3)) -> active#(X3) active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) 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(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) 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(a(),b(),X)) -> f#(X,X,X) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> active#(X3) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) 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,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(a(),b(),X)) -> f#(X,X,X) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> active#(X3) -> active#(f(a(),b(),X)) -> f#(X,X,X) active#(f(X1,X2,X3)) -> active#(X3) -> active#(f(X1,X2,X3)) -> active#(X1) active#(f(X1,X2,X3)) -> active#(X3) -> active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) active#(f(X1,X2,X3)) -> active#(X3) -> active#(f(X1,X2,X3)) -> active#(X3) active#(f(X1,X2,X3)) -> active#(X3) -> active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) active#(f(X1,X2,X3)) -> active#(X1) -> active#(f(a(),b(),X)) -> f#(X,X,X) active#(f(X1,X2,X3)) -> active#(X1) -> active#(f(X1,X2,X3)) -> active#(X1) active#(f(X1,X2,X3)) -> active#(X1) -> active#(f(X1,X2,X3)) -> f#(active(X1),X2,X3) active#(f(X1,X2,X3)) -> active#(X1) -> active#(f(X1,X2,X3)) -> active#(X3) active#(f(X1,X2,X3)) -> active#(X1) -> active#(f(X1,X2,X3)) -> f#(X1,X2,active(X3)) SCC Processor: #sccs: 4 #rules: 10 #arcs: 56/256 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X3) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) 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) = 0, [proper](x0) = x0, [c] = 1, [mark](x0) = 0, [active](x0) = x0, [f](x0, x1, x2) = x0 + x1 + x2 + 1, [b] = 0, [a] = 0 orientation: proper#(f(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(f(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(f(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) active(f(a(),b(),X)) = X + 1 >= 0 = mark(f(X,X,X)) active(c()) = 1 >= 0 = mark(a()) active(c()) = 1 >= 0 = mark(b()) active(f(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = f(active(X1),X2,X3) active(f(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = f(X1,X2,active(X3)) f(mark(X1),X2,X3) = X2 + X3 + 1 >= 0 = mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) = X1 + X2 + 1 >= 0 = mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = f(proper(X1),proper(X2),proper(X3)) proper(a()) = 0 >= 0 = ok(a()) proper(b()) = 0 >= 0 = ok(b()) proper(c()) = 1 >= 0 = ok(c()) f(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(f(X1,X2,X3)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(f(X1,X2,X3)) -> active#(X3) active#(f(X1,X2,X3)) -> active#(X1) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) 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, [c] = 0, [mark](x0) = 0, [active](x0) = x0, [f](x0, x1, x2) = x0 + x2 + 1, [b] = 0, [a] = 0 orientation: active#(f(X1,X2,X3)) = X1 + X3 + 2 >= X3 + 1 = active#(X3) active#(f(X1,X2,X3)) = X1 + X3 + 2 >= X1 + 1 = active#(X1) active(f(a(),b(),X)) = X + 1 >= 0 = mark(f(X,X,X)) active(c()) = 0 >= 0 = mark(a()) active(c()) = 0 >= 0 = mark(b()) active(f(X1,X2,X3)) = X1 + X3 + 1 >= X1 + X3 + 1 = f(active(X1),X2,X3) active(f(X1,X2,X3)) = X1 + X3 + 1 >= X1 + X3 + 1 = f(X1,X2,active(X3)) f(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) = X1 + 1 >= 0 = mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) = X1 + X3 + 1 >= X1 + X3 + 1 = f(proper(X1),proper(X2),proper(X3)) proper(a()) = 0 >= 0 = ok(a()) proper(b()) = 0 >= 0 = ok(b()) proper(c()) = 0 >= 0 = ok(c()) f(ok(X1),ok(X2),ok(X3)) = X1 + X3 + 1 >= X1 + X3 + 1 = ok(f(X1,X2,X3)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1, x2) = x1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [c] = 0, [mark](x0) = 0, [active](x0) = x0, [f](x0, x1, x2) = x2 + 1, [b] = 1, [a] = 1 orientation: f#(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 = f#(X1,X2,X3) f#(X1,X2,mark(X3)) = X2 >= X2 = f#(X1,X2,X3) f#(mark(X1),X2,X3) = X2 >= X2 = f#(X1,X2,X3) active(f(a(),b(),X)) = X + 1 >= 0 = mark(f(X,X,X)) active(c()) = 0 >= 0 = mark(a()) active(c()) = 0 >= 0 = mark(b()) active(f(X1,X2,X3)) = X3 + 1 >= X3 + 1 = f(active(X1),X2,X3) active(f(X1,X2,X3)) = X3 + 1 >= X3 + 1 = f(X1,X2,active(X3)) f(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) = 1 >= 0 = mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) = X3 + 2 >= X3 + 2 = f(proper(X1),proper(X2),proper(X3)) proper(a()) = 2 >= 2 = ok(a()) proper(b()) = 2 >= 2 = ok(b()) proper(c()) = 1 >= 1 = ok(c()) f(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(f(X1,X2,X3)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(active(X1),X2,X3) active(f(X1,X2,X3)) -> f(X1,X2,active(X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open