YES Problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Proof: DP Processor: DPs: a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) a__U11#(tt()) -> a__U12#(tt()) a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> a__U11#(mark(X)) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNePal(X)) -> mark#(X) mark#(isNePal(X)) -> a__isNePal#(mark(X)) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) EDG Processor: DPs: a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) a__U11#(tt()) -> a__U12#(tt()) a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> a__U11#(mark(X)) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNePal(X)) -> mark#(X) mark#(isNePal(X)) -> a__isNePal#(mark(X)) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) graph: a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) -> a__U11#(tt()) -> a__U12#(tt()) mark#(isNePal(X)) -> a__isNePal#(mark(X)) -> a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) mark#(isNePal(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) mark#(isNePal(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) mark#(isNePal(X)) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(isNePal(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(isNePal(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(isNePal(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(isNePal(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNePal(X)) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) mark#(isNePal(X)) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) mark#(U12(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U12(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(U11(X)) -> a__U11#(mark(X)) -> a__U11#(tt()) -> a__U12#(tt()) mark#(U11(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) mark#(U11(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U11(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X2) -> mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X2) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U11(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(isNePal(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(isNePal(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> mark#(Z) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> mark#(Y) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(X,nil()) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(nil(),X) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) a____#(nil(),X) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a____#(nil(),X) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(nil(),X) -> mark#(X) -> mark#(U11(X)) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(nil(),X) -> mark#(X) -> mark#(U12(X)) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(nil(),X) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(__(X1,X2)) -> mark#(X1) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U11(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(isNePal(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(__(X1,X2)) -> mark#(X1) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U11(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(isNePal(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a____#(__(X,Y),Z) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U11(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(X,nil()) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(nil(),X) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(X,nil()) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(nil(),X) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) a____#(X,nil()) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a____#(X,nil()) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(X,nil()) -> mark#(X) -> mark#(U11(X)) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(X,nil()) -> mark#(X) -> mark#(U12(X)) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(X,nil()) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) SCC Processor: #sccs: 1 #rules: 13 #arcs: 114/324 DPs: mark#(isNePal(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(nil(),X) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X2) a____#(X,nil()) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> mark#(Z) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [a____#](x0, x1) = x0 + x1 + 1, [isNePal](x0) = x0 + 1, [U12](x0) = x0, [U11](x0) = x0, [a__isNePal](x0) = x0 + 1, [a__U12](x0) = x0, [a__U11](x0) = x0, [tt] = 1, [nil] = 1, [mark](x0) = x0, [a____](x0, x1) = x0 + x1, [__](x0, x1) = x0 + x1 orientation: mark#(isNePal(X)) = X + 2 >= X + 1 = mark#(X) mark#(U12(X)) = X + 1 >= X + 1 = mark#(X) mark#(U11(X)) = X + 1 >= X + 1 = mark#(X) mark#(__(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a____#(mark(X1),mark(X2)) a____#(nil(),X) = X + 2 >= X + 1 = mark#(X) mark#(__(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(__(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) a____#(X,nil()) = X + 2 >= X + 1 = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 1 >= X + Y + Z + 1 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) = X + Y + Z + 1 >= X + 1 = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 1 >= Y + Z + 1 = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = X + Y + Z + 1 >= Y + 1 = mark#(Y) a____#(__(X,Y),Z) = X + Y + Z + 1 >= Z + 1 = mark#(Z) a____(__(X,Y),Z) = X + Y + Z >= X + Y + Z = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 1 >= X = mark(X) a____(nil(),X) = X + 1 >= X = mark(X) a__U11(tt()) = 1 >= 1 = a__U12(tt()) a__U12(tt()) = 1 >= 1 = tt() a__isNePal(__(I,__(P,I))) = 2I + P + 1 >= 1 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 >= X1 + X2 = a____(mark(X1),mark(X2)) mark(U11(X)) = X >= X = a__U11(mark(X)) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNePal(X)) = X + 1 >= X + 1 = a__isNePal(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(tt()) = 1 >= 1 = tt() a____(X1,X2) = X1 + X2 >= X1 + X2 = __(X1,X2) a__U11(X) = X >= X = U11(X) a__U12(X) = X >= X = U12(X) a__isNePal(X) = X + 1 >= X + 1 = isNePal(X) problem: DPs: mark#(U12(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> mark#(Z) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [a____#](x0, x1) = x0 + x1 + 1, [isNePal](x0) = 1, [U12](x0) = x0 + 1, [U11](x0) = x0 + 1, [a__isNePal](x0) = 1, [a__U12](x0) = x0 + 1, [a__U11](x0) = x0 + 1, [tt] = 0, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = x0 + x1, [__](x0, x1) = x0 + x1 orientation: mark#(U12(X)) = X + 2 >= X + 1 = mark#(X) mark#(U11(X)) = X + 2 >= X + 1 = mark#(X) mark#(__(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(__(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) a____#(__(X,Y),Z) = X + Y + Z + 1 >= X + Y + Z + 1 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) = X + Y + Z + 1 >= X + 1 = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 1 >= Y + Z + 1 = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = X + Y + Z + 1 >= Y + 1 = mark#(Y) a____#(__(X,Y),Z) = X + Y + Z + 1 >= Z + 1 = mark#(Z) a____(__(X,Y),Z) = X + Y + Z >= X + Y + Z = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X >= X = mark(X) a____(nil(),X) = X >= X = mark(X) a__U11(tt()) = 1 >= 1 = a__U12(tt()) a__U12(tt()) = 1 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 1 >= 1 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 >= X1 + X2 = a____(mark(X1),mark(X2)) mark(U11(X)) = X + 1 >= X + 1 = a__U11(mark(X)) mark(U12(X)) = X + 1 >= X + 1 = a__U12(mark(X)) mark(isNePal(X)) = 1 >= 1 = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = X1 + X2 >= X1 + X2 = __(X1,X2) a__U11(X) = X + 1 >= X + 1 = U11(X) a__U12(X) = X + 1 >= X + 1 = U12(X) a__isNePal(X) = 1 >= 1 = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> mark#(Z) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [a____#](x0, x1) = x0 + x1 + 1, [isNePal](x0) = x0, [U12](x0) = 0, [U11](x0) = 1, [a__isNePal](x0) = x0, [a__U12](x0) = 0, [a__U11](x0) = 1, [tt] = 0, [nil] = 1, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 1, [__](x0, x1) = x0 + x1 + 1 orientation: mark#(__(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(__(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) a____#(__(X,Y),Z) = X + Y + Z + 2 >= X + Y + Z + 2 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) = X + Y + Z + 2 >= X = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 2 >= Y + Z + 1 = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = X + Y + Z + 2 >= Y = mark#(Y) a____#(__(X,Y),Z) = X + Y + Z + 2 >= Z = mark#(Z) a____(__(X,Y),Z) = X + Y + Z + 2 >= X + Y + Z + 2 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 2 >= X = mark(X) a____(nil(),X) = X + 2 >= X = mark(X) a__U11(tt()) = 1 >= 0 = a__U12(tt()) a__U12(tt()) = 0 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 2I + P + 2 >= 1 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a____(mark(X1),mark(X2)) mark(U11(X)) = 1 >= 1 = a__U11(mark(X)) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNePal(X)) = X >= X = a__isNePal(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = __(X1,X2) a__U11(X) = 1 >= 1 = U11(X) a__U12(X) = 0 >= 0 = U12(X) a__isNePal(X) = X >= X = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [a____#](x0, x1) = 0, [isNePal](x0) = 0, [U12](x0) = 0, [U11](x0) = 0, [a__isNePal](x0) = 0, [a__U12](x0) = 0, [a__U11](x0) = 0, [tt] = 0, [nil] = 1, [mark](x0) = x0, [a____](x0, x1) = x0 + x1, [__](x0, x1) = x0 + x1 orientation: mark#(__(X1,X2)) = 1 >= 0 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = 0 >= 0 = a____#(mark(X),a____(mark(Y),mark(Z))) a____(__(X,Y),Z) = X + Y + Z >= X + Y + Z = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 1 >= X = mark(X) a____(nil(),X) = X + 1 >= X = mark(X) a__U11(tt()) = 0 >= 0 = a__U12(tt()) a__U12(tt()) = 0 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 0 >= 0 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 >= X1 + X2 = a____(mark(X1),mark(X2)) mark(U11(X)) = 0 >= 0 = a__U11(mark(X)) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNePal(X)) = 0 >= 0 = a__isNePal(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = X1 + X2 >= X1 + X2 = __(X1,X2) a__U11(X) = 0 >= 0 = U11(X) a__U12(X) = 0 >= 0 = U12(X) a__isNePal(X) = 0 >= 0 = isNePal(X) problem: DPs: a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a____#](x0, x1) = x0, [isNePal](x0) = 0, [U12](x0) = 0, [U11](x0) = 0, [a__isNePal](x0) = 0, [a__U12](x0) = 0, [a__U11](x0) = 0, [tt] = 0, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 1, [__](x0, x1) = x0 + x1 + 1 orientation: a____#(__(X,Y),Z) = X + Y + 1 >= X = a____#(mark(X),a____(mark(Y),mark(Z))) a____(__(X,Y),Z) = X + Y + Z + 2 >= X + Y + Z + 2 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 1 >= X = mark(X) a____(nil(),X) = X + 1 >= X = mark(X) a__U11(tt()) = 0 >= 0 = a__U12(tt()) a__U12(tt()) = 0 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 0 >= 0 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a____(mark(X1),mark(X2)) mark(U11(X)) = 0 >= 0 = a__U11(mark(X)) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNePal(X)) = 0 >= 0 = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = __(X1,X2) a__U11(X) = 0 >= 0 = U11(X) a__U12(X) = 0 >= 0 = U12(X) a__isNePal(X) = 0 >= 0 = isNePal(X) problem: DPs: TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Qed