YES Problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = x0, [proper](x0) = x0, [false] = 1, [mark](x0) = x0, [if](x0, x1, x2) = x0 + 2x1 + x2, [true] = 0, [c] = 0, [active](x0) = x0, [f](x0) = 4x0 + 2 orientation: active(f(X)) = 4X + 2 >= X + 2 = mark(if(X,c(),f(true()))) active(if(true(),X,Y)) = 2X + Y >= X = mark(X) active(if(false(),X,Y)) = 2X + Y + 1 >= Y = mark(Y) active(f(X)) = 4X + 2 >= 4X + 2 = f(active(X)) active(if(X1,X2,X3)) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = if(active(X1),X2,X3) active(if(X1,X2,X3)) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = if(X1,active(X2),X3) f(mark(X)) = 4X + 2 >= 4X + 2 = mark(f(X)) if(mark(X1),X2,X3) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = mark(if(X1,X2,X3)) if(X1,mark(X2),X3) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = mark(if(X1,X2,X3)) proper(f(X)) = 4X + 2 >= 4X + 2 = f(proper(X)) proper(if(X1,X2,X3)) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = if(proper(X1),proper(X2),proper(X3)) proper(c()) = 0 >= 0 = ok(c()) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) f(ok(X)) = 4X + 2 >= 4X + 2 = ok(f(X)) if(ok(X1),ok(X2),ok(X3)) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = ok(if(X1,X2,X3)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(if(true(),X,Y)) -> mark(X) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 1 0]x0 [0 0 0] , [proper](x0) = x0 , [0] [false] = [0] [0], [1 1 0] [mark](x0) = [0 0 0]x0 [1 0 0] , [1 1 0] [1 1 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [1 1 0] [1 1 0] [1 0 1] , [0] [true] = [1] [0], [0] [c] = [0] [0], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [f](x0) = [0 1 0]x0 [1 1 0] orientation: [1 1 0] [1 1 0] active(f(X)) = [0 0 0]X >= [0 0 0]X = mark(if(X,c(),f(true()))) [1 1 0] [1 1 0] [1 1 0] [1 0 0] [1] [1 1 0] active(if(true(),X,Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X = mark(X) [1 1 0] [1 0 1] [1] [1 0 0] [1 1 0] [1 1 0] active(f(X)) = [0 0 0]X >= [0 0 0]X = f(active(X)) [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] active(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(active(X1),X2,X3) [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] active(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,active(X2),X3) [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] f(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(f(X)) [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(if(X1,X2,X3)) [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(if(X1,X2,X3)) [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] proper(f(X)) = [0 1 0]X >= [0 1 0]X = f(proper(X)) [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] proper(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(proper(X1),proper(X2),proper(X3)) [1 1 0] [1 1 0] [1 0 1] [1 1 0] [1 1 0] [1 0 1] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [0] [0] [0] [0] proper(true()) = [1] >= [1] = ok(true()) [0] [0] [0] [0] proper(false()) = [0] >= [0] = ok(false()) [0] [0] [1 0 0] [1 0 0] f(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(f(X)) [1 1 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(if(X1,X2,X3)) [1 1 0] [1 1 0] [1 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] top(mark(X)) = [1 1 0]X >= [1 1 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] top(ok(X)) = [1 1 0]X >= [1 1 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) DP Processor: DPs: active#(f(X)) -> f#(true()) active#(f(X)) -> if#(X,c(),f(true())) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(if(X1,X2,X3)) -> active#(X2) active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) f#(mark(X)) -> f#(X) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) f#(ok(X)) -> f#(X) if#(ok(X1),ok(X2),ok(X3)) -> if#(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(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(f(X)) -> f#(true()) active#(f(X)) -> if#(X,c(),f(true())) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(if(X1,X2,X3)) -> active#(X2) active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) f#(mark(X)) -> f#(X) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) f#(ok(X)) -> f#(X) if#(ok(X1),ok(X2),ok(X3)) -> if#(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(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) 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#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> if#(X,c(),f(true())) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(true()) 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#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(f(X)) -> f#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(f(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(f(X)) -> f#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(f(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(f(X)) -> f#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(f(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X) proper#(f(X)) -> f#(proper(X)) -> f#(mark(X)) -> f#(X) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> active#(X2) -> active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) active#(if(X1,X2,X3)) -> active#(X2) -> active#(if(X1,X2,X3)) -> active#(X2) active#(if(X1,X2,X3)) -> active#(X2) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(if(X1,X2,X3)) -> active#(X2) -> active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X2) -> active#(f(X)) -> f#(active(X)) active#(if(X1,X2,X3)) -> active#(X2) -> active#(f(X)) -> active#(X) active#(if(X1,X2,X3)) -> active#(X2) -> active#(f(X)) -> if#(X,c(),f(true())) active#(if(X1,X2,X3)) -> active#(X2) -> active#(f(X)) -> f#(true()) active#(if(X1,X2,X3)) -> active#(X1) -> active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) active#(if(X1,X2,X3)) -> active#(X1) -> active#(if(X1,X2,X3)) -> active#(X2) active#(if(X1,X2,X3)) -> active#(X1) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(if(X1,X2,X3)) -> active#(X1) -> active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X1) -> active#(f(X)) -> f#(active(X)) active#(if(X1,X2,X3)) -> active#(X1) -> active#(f(X)) -> active#(X) active#(if(X1,X2,X3)) -> active#(X1) -> active#(f(X)) -> if#(X,c(),f(true())) active#(if(X1,X2,X3)) -> active#(X1) -> active#(f(X)) -> f#(true()) active#(f(X)) -> if#(X,c(),f(true())) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) active#(f(X)) -> if#(X,c(),f(true())) -> if#(X1,mark(X2),X3) -> if#(X1,X2,X3) active#(f(X)) -> if#(X,c(),f(true())) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) active#(f(X)) -> f#(true()) -> f#(ok(X)) -> f#(X) active#(f(X)) -> f#(true()) -> f#(mark(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) active#(f(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) active#(f(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X2) active#(f(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(f(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X1) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> if#(X,c(),f(true())) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(true()) SCC Processor: #sccs: 5 #rules: 14 #arcs: 101/529 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) Semantic Labeling Processor: dimension: 1 usable rules: interpretation: [ok](x0) = x0, [proper](x0) = x0, [false] = 0, [mark](x0) = x0 + 1, [if](x0, x1, x2) = x0 + x1, [true] = 0, [c] = 0, [active](x0) = x0, [f](x0) = x0 + 1 labeled: top# ok proper usable (for model): top# ok active mark proper f if c true false argument filtering: pi(f) = 0 pi(active) = 0 pi(c) = [] pi(true) = [] pi(if) = 2 pi(mark) = [] pi(false) = [] pi(proper) = 0 pi(ok) = 0 pi(top#) = [] precedence: top# ~ ok ~ proper ~ false ~ mark ~ if ~ true ~ c ~ active ~ f problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) Restore Modifier: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {1} transitions: f660() -> 2* top{#,0}(3) -> 1* active0(2) -> 3* if0(3,2,2) -> 3* if0(7,2,2) -> 7* if0(2,6,5) -> 7* if0(2,3,2) -> 3* if0(2,7,2) -> 7* f0(7) -> 7* f0(4) -> 5* f0(3) -> 3* mark0(7) -> 3* c0() -> 6* true0() -> 4* 3 -> 7* problem: DPs: TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(f(X)) -> active#(X) active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X2) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(f(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(if#) = 2 problem: DPs: if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(X1,mark(X2),X3) -> if#(X1,X2,X3) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(if#) = 1 problem: DPs: if#(mark(X1),X2,X3) -> if#(X1,X2,X3) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(if#) = 0 problem: DPs: TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: f#(mark(X)) -> f#(X) f#(ok(X)) -> f#(X) TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(f#) = 0 problem: DPs: TRS: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed