MAYBE Problem: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Proof: DP Processor: DPs: active#(f(a(),b(),X)) -> f#(X,X,X) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) active#(c()) -> mark#(a()) active#(c()) -> mark#(b()) mark#(f(X1,X2,X3)) -> mark#(X3) mark#(f(X1,X2,X3)) -> mark#(X1) mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) mark#(a()) -> active#(a()) mark#(b()) -> active#(b()) mark#(c()) -> active#(c()) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) EDG Processor: DPs: active#(f(a(),b(),X)) -> f#(X,X,X) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) active#(c()) -> mark#(a()) active#(c()) -> mark#(b()) mark#(f(X1,X2,X3)) -> mark#(X3) mark#(f(X1,X2,X3)) -> mark#(X1) mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) mark#(a()) -> active#(a()) mark#(b()) -> active#(b()) mark#(c()) -> active#(c()) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) graph: mark#(c()) -> active#(c()) -> active#(c()) -> mark#(a()) mark#(c()) -> active#(c()) -> active#(c()) -> mark#(b()) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(f(X1,X2,X3)) -> mark#(X3) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(f(X1,X2,X3)) -> mark#(X1) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(a()) -> active#(a()) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(b()) -> active#(b()) mark#(f(X1,X2,X3)) -> mark#(X3) -> mark#(c()) -> active#(c()) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(f(X1,X2,X3)) -> mark#(X3) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(f(X1,X2,X3)) -> mark#(X1) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(a()) -> active#(a()) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(b()) -> active#(b()) mark#(f(X1,X2,X3)) -> mark#(X1) -> mark#(c()) -> active#(c()) mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) -> f#(active(X1),X2,X3) -> f#(X1,X2,X3) mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) -> f#(X1,X2,active(X3)) -> f#(X1,X2,X3) mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) -> active#(f(a(),b(),X)) -> f#(X,X,X) mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) -> active#(f(a(),b(),X)) -> mark#(f(X,X,X)) 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,mark(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#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) -> f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) -> f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) -> f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) -> f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) -> f#(X1,X2,active(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,mark(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#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) -> f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) -> f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) -> f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) -> f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) -> f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) -> f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) -> f#(X1,X2,active(X3)) -> f#(X1,X2,X3) active#(c()) -> mark#(b()) -> mark#(b()) -> active#(b()) active#(c()) -> mark#(a()) -> mark#(a()) -> active#(a()) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) -> mark#(f(X1,X2,X3)) -> mark#(X3) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) -> mark#(f(X1,X2,X3)) -> mark#(X1) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) -> mark#(f(X1,X2,X3)) -> f#(mark(X1),X2,mark(X3)) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) -> mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) SCC Processor: #sccs: 2 #rules: 10 #arcs: 64/289 DPs: mark#(f(X1,X2,X3)) -> mark#(X3) mark#(f(X1,X2,X3)) -> active#(f(mark(X1),X2,mark(X3))) active#(f(a(),b(),X)) -> mark#(f(X,X,X)) mark#(f(X1,X2,X3)) -> mark#(X1) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Open DPs: f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(X1,mark(X2),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()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1, x2) = x0, [c] = 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [f](x0, x1, x2) = 0, [b] = 0, [a] = 0 orientation: f#(X1,X2,active(X3)) = X1 >= X1 = f#(X1,X2,X3) f#(X1,active(X2),X3) = X1 >= X1 = f#(X1,X2,X3) f#(active(X1),X2,X3) = X1 + 1 >= X1 = f#(X1,X2,X3) f#(X1,X2,mark(X3)) = X1 >= X1 = f#(X1,X2,X3) f#(X1,mark(X2),X3) = X1 >= X1 = f#(X1,X2,X3) f#(mark(X1),X2,X3) = X1 + 1 >= X1 = f#(X1,X2,X3) active(f(a(),b(),X)) = 1 >= 1 = mark(f(X,X,X)) active(c()) = 2 >= 1 = mark(a()) active(c()) = 2 >= 1 = mark(b()) mark(f(X1,X2,X3)) = 1 >= 1 = active(f(mark(X1),X2,mark(X3))) mark(a()) = 1 >= 1 = active(a()) mark(b()) = 1 >= 1 = active(b()) mark(c()) = 2 >= 2 = active(c()) f(mark(X1),X2,X3) = 0 >= 0 = f(X1,X2,X3) f(X1,mark(X2),X3) = 0 >= 0 = f(X1,X2,X3) f(X1,X2,mark(X3)) = 0 >= 0 = f(X1,X2,X3) f(active(X1),X2,X3) = 0 >= 0 = f(X1,X2,X3) f(X1,active(X2),X3) = 0 >= 0 = f(X1,X2,X3) f(X1,X2,active(X3)) = 0 >= 0 = f(X1,X2,X3) problem: DPs: f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(X1,mark(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()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1, x2) = x1, [c] = 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [f](x0, x1, x2) = 1, [b] = 0, [a] = 1 orientation: f#(X1,X2,active(X3)) = X2 >= X2 = f#(X1,X2,X3) f#(X1,active(X2),X3) = X2 + 1 >= X2 = f#(X1,X2,X3) f#(X1,X2,mark(X3)) = X2 >= X2 = f#(X1,X2,X3) f#(X1,mark(X2),X3) = X2 + 1 >= X2 = f#(X1,X2,X3) active(f(a(),b(),X)) = 2 >= 2 = mark(f(X,X,X)) active(c()) = 2 >= 2 = mark(a()) active(c()) = 2 >= 1 = mark(b()) mark(f(X1,X2,X3)) = 2 >= 2 = active(f(mark(X1),X2,mark(X3))) mark(a()) = 2 >= 2 = active(a()) mark(b()) = 1 >= 1 = active(b()) mark(c()) = 2 >= 2 = active(c()) f(mark(X1),X2,X3) = 1 >= 1 = f(X1,X2,X3) f(X1,mark(X2),X3) = 1 >= 1 = f(X1,X2,X3) f(X1,X2,mark(X3)) = 1 >= 1 = f(X1,X2,X3) f(active(X1),X2,X3) = 1 >= 1 = f(X1,X2,X3) f(X1,active(X2),X3) = 1 >= 1 = f(X1,X2,X3) f(X1,X2,active(X3)) = 1 >= 1 = f(X1,X2,X3) problem: DPs: f#(X1,X2,active(X3)) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1, x2) = x2, [c] = 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [f](x0, x1, x2) = 0, [b] = 1, [a] = 1 orientation: f#(X1,X2,active(X3)) = X3 + 1 >= X3 = f#(X1,X2,X3) f#(X1,X2,mark(X3)) = X3 + 1 >= X3 = f#(X1,X2,X3) active(f(a(),b(),X)) = 1 >= 1 = mark(f(X,X,X)) active(c()) = 2 >= 2 = mark(a()) active(c()) = 2 >= 2 = mark(b()) mark(f(X1,X2,X3)) = 1 >= 1 = active(f(mark(X1),X2,mark(X3))) mark(a()) = 2 >= 2 = active(a()) mark(b()) = 2 >= 2 = active(b()) mark(c()) = 2 >= 2 = active(c()) f(mark(X1),X2,X3) = 0 >= 0 = f(X1,X2,X3) f(X1,mark(X2),X3) = 0 >= 0 = f(X1,X2,X3) f(X1,X2,mark(X3)) = 0 >= 0 = f(X1,X2,X3) f(active(X1),X2,X3) = 0 >= 0 = f(X1,X2,X3) f(X1,active(X2),X3) = 0 >= 0 = f(X1,X2,X3) f(X1,X2,active(X3)) = 0 >= 0 = f(X1,X2,X3) problem: DPs: TRS: active(f(a(),b(),X)) -> mark(f(X,X,X)) active(c()) -> mark(a()) active(c()) -> mark(b()) mark(f(X1,X2,X3)) -> active(f(mark(X1),X2,mark(X3))) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(c()) -> active(c()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Qed