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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [isNePal](x0) = 4x0 + 4, [and](x0, x1) = x0 + x1 + 7, [a__isNePal](x0) = 4x0 + 4, [a__and](x0, x1) = x0 + x1 + 7, [tt] = 1, [nil] = 6, [mark](x0) = x0, [a____](x0, x1) = 2x0 + x1, [__](x0, x1) = 2x0 + x1 orientation: a____(__(X,Y),Z) = 4X + 2Y + Z >= 2X + 2Y + Z = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = 2X + 6 >= X = mark(X) a____(nil(),X) = X + 12 >= X = mark(X) a__and(tt(),X) = X + 8 >= X = mark(X) a__isNePal(__(I,__(P,I))) = 12I + 8P + 4 >= 1 = tt() mark(__(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a____(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 7 >= X1 + X2 + 7 = a__and(mark(X1),X2) mark(isNePal(X)) = 4X + 4 >= 4X + 4 = a__isNePal(mark(X)) mark(nil()) = 6 >= 6 = nil() mark(tt()) = 1 >= 1 = tt() a____(X1,X2) = 2X1 + X2 >= 2X1 + X2 = __(X1,X2) a__and(X1,X2) = X1 + X2 + 7 >= X1 + X2 + 7 = and(X1,X2) a__isNePal(X) = 4X + 4 >= 4X + 4 = isNePal(X) problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dim=1 interpretation: [isNePal](x0) = x0, [and](x0, x1) = 2x0 + x1, [a__isNePal](x0) = x0, [a__and](x0, x1) = 2x0 + x1, [tt] = 4, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = 2x0 + x1 + 1, [__](x0, x1) = 2x0 + x1 + 1 orientation: a____(__(X,Y),Z) = 4X + 2Y + Z + 3 >= 2X + 2Y + Z + 2 = a____(mark(X),a____(mark(Y),mark(Z))) mark(__(X1,X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = a____(mark(X1),mark(X2)) mark(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__and(mark(X1),X2) mark(isNePal(X)) = X >= X = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 4 >= 4 = tt() a____(X1,X2) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = __(X1,X2) a__and(X1,X2) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) a__isNePal(X) = X >= X = isNePal(X) problem: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [isNePal](x0) = x0 + [1] [1], [1 0 0] [0] [and](x0, x1) = x0 + [0 1 0]x1 + [1] [0 1 0] [1], [0] [a__isNePal](x0) = x0 + [1] [1], [1 1 0] [0] [a__and](x0, x1) = x0 + [0 1 0]x1 + [1] [0 1 0] [1], [0] [tt] = [0] [0], [0] [nil] = [1] [1], [1 1 0] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 1 0] [1 1 0] [0] [a____](x0, x1) = [0 0 1]x0 + [0 0 1]x1 + [1] [0 1 0] [0 1 0] [1], [1 1 0] [1 1 0] [0] [__](x0, x1) = [0 0 1]x0 + [0 0 1]x1 + [1] [0 1 0] [0 1 0] [1] orientation: [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [0] mark(__(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = a____(mark(X1),mark(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] mark(and(X1,X2)) = [0 0 1]X1 + [0 1 0]X2 + [1] >= [0 0 1]X1 + [0 1 0]X2 + [1] = a__and(mark(X1),X2) [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [1] [1 1 0] [1] [1 1 0] [0] mark(isNePal(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__isNePal(mark(X)) [0 1 0] [1] [0 1 0] [1] [1] [0] mark(nil()) = [1] >= [1] = nil() [1] [1] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] a____(X1,X2) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [1] = __(X1,X2) [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [1] [1 1 0] [0] [1 0 0] [0] a__and(X1,X2) = X1 + [0 1 0]X2 + [1] >= X1 + [0 1 0]X2 + [1] = and(X1,X2) [0 1 0] [1] [0 1 0] [1] [0] [0] a__isNePal(X) = X + [1] >= X + [1] = isNePal(X) [1] [1] problem: mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [a__isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [a__and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [1] [0], [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [a____](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1] [0] mark(tt()) = [1] >= [1] = tt() [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a____(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = __(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__and(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__isNePal(X) = [0 0 0]X >= [0 0 0]X = isNePal(X) [0 0 0] [0 0 0] problem: a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1] [a__isNePal](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [a__and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [a____](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1 0 0] [1 0 0] a____(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = __(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__and(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] a__isNePal(X) = [0 0 0]X + [0] >= [0 0 0]X = isNePal(X) [0 0 0] [0] [0 0 0] problem: a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [a__and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [a____](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1 0 0] [1 0 0] a____(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = __(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] a__and(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] problem: a____(X1,X2) -> __(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1] [a____](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [0], [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] a____(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = __(X1,X2) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 0] problem: Qed