YES Problem: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(X,nil())) -> mark(X) active(__(nil(),X)) -> mark(X) active(and(tt(),X)) -> mark(X) active(isNePal(__(I,__(P,I)))) -> mark(tt()) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(mark(X1),X2) -> mark(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [isNePal](x0) = 4x0, [and](x0, x1) = x0 + 4x1 + 2, [tt] = 0, [nil] = 2, [mark](x0) = x0, [active](x0) = x0, [__](x0, x1) = 2x0 + x1 orientation: active(__(__(X,Y),Z)) = 4X + 2Y + Z >= 2X + 2Y + Z = mark(__(X,__(Y,Z))) active(__(X,nil())) = 2X + 2 >= X = mark(X) active(__(nil(),X)) = X + 4 >= X = mark(X) active(and(tt(),X)) = 4X + 2 >= X = mark(X) active(isNePal(__(I,__(P,I)))) = 12I + 8P >= 0 = mark(tt()) active(__(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = __(active(X1),X2) active(__(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = __(X1,active(X2)) active(and(X1,X2)) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = and(active(X1),X2) active(isNePal(X)) = 4X >= 4X = isNePal(active(X)) __(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(__(X1,X2)) __(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = mark(__(X1,X2)) and(mark(X1),X2) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = mark(and(X1,X2)) isNePal(mark(X)) = 4X >= 4X = mark(isNePal(X)) proper(__(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = __(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(and(X1,X2)) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNePal(X)) = 4X >= 4X = isNePal(proper(X)) __(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(__(X1,X2)) and(ok(X1),ok(X2)) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = ok(and(X1,X2)) isNePal(ok(X)) = 4X >= 4X = ok(isNePal(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(isNePal(__(I,__(P,I)))) -> mark(tt()) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(mark(X1),X2) -> mark(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [isNePal](x0) = x0, [and](x0, x1) = 2x0 + x1, [tt] = 0, [nil] = 0, [mark](x0) = x0, [active](x0) = x0, [__](x0, x1) = x0 + x1 + 3 orientation: active(__(__(X,Y),Z)) = X + Y + Z + 6 >= X + Y + Z + 6 = mark(__(X,__(Y,Z))) active(isNePal(__(I,__(P,I)))) = 2I + P + 6 >= 0 = mark(tt()) active(__(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = __(active(X1),X2) active(__(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = __(X1,active(X2)) active(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(active(X1),X2) active(isNePal(X)) = X >= X = isNePal(active(X)) __(mark(X1),X2) = X1 + X2 + 3 >= X1 + X2 + 3 = mark(__(X1,X2)) __(X1,mark(X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = mark(__(X1,X2)) and(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(and(X1,X2)) isNePal(mark(X)) = X >= X = mark(isNePal(X)) proper(__(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = __(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNePal(X)) = X >= X = isNePal(proper(X)) __(ok(X1),ok(X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = ok(__(X1,X2)) and(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(and(X1,X2)) isNePal(ok(X)) = X >= X = ok(isNePal(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(mark(X1),X2) -> mark(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [isNePal](x0) = x0, [and](x0, x1) = 5x0 + 2x1, [tt] = 0, [nil] = 0, [mark](x0) = x0 + 4, [active](x0) = x0, [__](x0, x1) = 2x0 + x1 + 4 orientation: active(__(__(X,Y),Z)) = 4X + 2Y + Z + 12 >= 2X + 2Y + Z + 12 = mark(__(X,__(Y,Z))) active(__(X1,X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = __(active(X1),X2) active(__(X1,X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = __(X1,active(X2)) active(and(X1,X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = and(active(X1),X2) active(isNePal(X)) = X >= X = isNePal(active(X)) __(mark(X1),X2) = 2X1 + X2 + 12 >= 2X1 + X2 + 8 = mark(__(X1,X2)) __(X1,mark(X2)) = 2X1 + X2 + 8 >= 2X1 + X2 + 8 = mark(__(X1,X2)) and(mark(X1),X2) = 5X1 + 2X2 + 20 >= 5X1 + 2X2 + 4 = mark(and(X1,X2)) isNePal(mark(X)) = X + 4 >= X + 4 = mark(isNePal(X)) proper(__(X1,X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = __(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(and(X1,X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNePal(X)) = X >= X = isNePal(proper(X)) __(ok(X1),ok(X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = ok(__(X1,X2)) and(ok(X1),ok(X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = ok(and(X1,X2)) isNePal(ok(X)) = X >= X = ok(isNePal(X)) top(mark(X)) = X + 4 >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(X1,mark(X2)) -> mark(__(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [0 1 1]x0 [0 0 0] , [1 0 0] [0] [ok](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [1 0 0] [0] [proper](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [1] [isNePal](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0], [1] [nil] = [0] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [active](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [__](x0, x1) = [0 1 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] active(__(__(X,Y),Z)) = [0 1 0]X + [1 0 0]Y + [1 0 0]Z + [0] >= [0 0 0]X + [0 0 0]Y + [0 0 0]Z = mark(__(X,__(Y,Z))) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] active(__(X1,X2)) = [0 1 0]X1 + [1 0 0]X2 + [0] >= [0 1 0]X1 + [1 0 0]X2 = __(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] active(__(X1,X2)) = [0 1 0]X1 + [1 0 0]X2 + [0] >= [0 1 0]X1 + [1 0 0]X2 = __(X1,active(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [0] >= [0 1 0]X1 + [0 0 0]X2 = and(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] active(isNePal(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = isNePal(active(X)) [0 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] __(X1,mark(X2)) = [0 1 0]X1 + [1 0 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(__(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1] [1 0 0] [1] isNePal(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(isNePal(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(__(X1,X2)) = [0 1 0]X1 + [1 0 0]X2 + [1] >= [0 1 0]X1 + [1 0 0]X2 + [1] = __(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] proper(nil()) = [1] >= [1] = ok(nil()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(and(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] proper(tt()) = [1] >= [1] = ok(tt()) [0] [0] [1 0 0] [1] [1 0 0] [1] proper(isNePal(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = isNePal(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] __(ok(X1),ok(X2)) = [0 1 1]X1 + [1 0 0]X2 + [1] >= [0 1 0]X1 + [1 0 0]X2 + [1] = ok(__(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(ok(X1),ok(X2)) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = ok(and(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] isNePal(ok(X)) = [0 1 1]X + [1] >= [0 1 0]X + [1] = ok(isNePal(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 0] [0] top(ok(X)) = [0 1 1]X + [1] >= [0 1 0]X + [1] = top(active(X)) [0 0 0] [0] [0 0 0] [0] problem: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(X1,mark(X2)) -> mark(__(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 1 1] , [1 0 1] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [isNePal](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 1] , [0] [tt] = [0] [1], [0] [nil] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1] [active](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 1 1] [0 0 1] orientation: [1 1 1] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 0] [1 0 0] active(__(__(X,Y),Z)) = [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [0] >= [0 0 0]X + [0 0 0]Y + [0 0 0]Z = mark(__(X,__(Y,Z))) [0 1 1] [0 0 1] [0 0 1] [0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 1] [1] [1 0 1] [1 0 0] [1] active(__(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = __(active(X1),X2) [0 1 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [1 1 1] [1 0 1] [1] [1 0 0] [1 0 1] [1] active(__(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = __(X1,active(X2)) [0 1 1] [0 0 1] [0] [0 1 1] [0 0 1] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 0] [1] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(active(X1),X2) [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [1 0 1] [1] [1 0 1] [1] active(isNePal(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNePal(active(X)) [0 0 1] [0] [0 0 1] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] __(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(__(X1,X2)) [0 1 1] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNePal(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(isNePal(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 1] [1 0 1] [1 0 1] proper(__(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = __(proper(X1),proper(X2)) [0 1 1] [0 0 1] [0 0 1] [0 0 1] [1] [0] proper(nil()) = [0] >= [0] = ok(nil()) [1] [1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 1] [1 0 1] proper(isNePal(X)) = [0 0 0]X >= [0 0 0]X = isNePal(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] __(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(__(X1,X2)) [0 1 1] [0 1 1] [0 1 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 1 1] [0 1 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] isNePal(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNePal(X)) [0 1 1] [0 0 1] problem: active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(X1,mark(X2)) -> mark(__(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = x0, [proper](x0) = 4x0, [isNePal](x0) = 4x0, [and](x0, x1) = x0 + x1, [mark](x0) = x0 + 2, [active](x0) = 4x0, [__](x0, x1) = x0 + x1 + 2 orientation: active(__(X1,X2)) = 4X1 + 4X2 + 8 >= 4X1 + X2 + 2 = __(active(X1),X2) active(__(X1,X2)) = 4X1 + 4X2 + 8 >= X1 + 4X2 + 2 = __(X1,active(X2)) active(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = and(active(X1),X2) active(isNePal(X)) = 16X >= 16X = isNePal(active(X)) __(X1,mark(X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = mark(__(X1,X2)) isNePal(mark(X)) = 4X + 8 >= 4X + 2 = mark(isNePal(X)) proper(__(X1,X2)) = 4X1 + 4X2 + 8 >= 4X1 + 4X2 + 2 = __(proper(X1),proper(X2)) proper(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = and(proper(X1),proper(X2)) proper(isNePal(X)) = 16X >= 16X = isNePal(proper(X)) __(ok(X1),ok(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = ok(__(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNePal(ok(X)) = 4X >= 4X = ok(isNePal(X)) problem: active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) __(X1,mark(X2)) -> mark(__(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [ok](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1] [isNePal](x0) = x0 + [0] [0], [1 0 0] [1 1 0] [0] [and](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [0] [0 0 1] [0 0 1] [1], [1 0 1] [0] [mark](x0) = [0 1 0]x0 + [1] [0 1 0] [1], [active](x0) = x0 , [1 0 0] [1 0 1] [__](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 1 0] orientation: [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] active(and(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 + [0] >= [0 0 0]X1 + [0 1 0]X2 + [0] = and(active(X1),X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1] [1] active(isNePal(X)) = X + [0] >= X + [0] = isNePal(active(X)) [0] [0] [1 0 0] [1 1 1] [1] [1 0 0] [1 1 1] [0] __(X1,mark(X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = mark(__(X1,X2)) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 1] [1 1 1] [1] [1 0 1] [1 0 1] [1] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(proper(X1),proper(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [1] proper(isNePal(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNePal(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 0 1] [1 1 1] [0] [1 0 0] [1 1 1] [0] __(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = ok(__(X1,X2)) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = ok(and(X1,X2)) [0 1 0] [0 1 0] [1] [0 0 0] [0 1 0] [0] [1 0 1] [1] [1 0 1] [1] isNePal(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(isNePal(X)) [0 1 0] [0] [0 1 0] [0] problem: active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(X)) -> isNePal(active(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 1 0]x0 [0 1 0] , [proper](x0) = x0 , [1 0 0] [0] [isNePal](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1 1 0] [0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 1] [0 0 0] [1], [1 0 1] [0] [active](x0) = [1 0 0]x0 + [1] [0 0 1] [0], [1 1 0] [__](x0, x1) = x0 + [1 0 0]x1 [1 0 0] orientation: [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [0] active(and(X1,X2)) = [1 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(active(X1),X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] active(isNePal(X)) = [1 0 0]X + [1] >= [0 0 0]X + [0] = isNePal(active(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(proper(X1),proper(X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(isNePal(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNePal(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] __(ok(X1),ok(X2)) = [0 1 0]X1 + [1 0 0]X2 >= [0 1 0]X1 + [1 0 0]X2 = ok(__(X1,X2)) [0 1 0] [1 0 0] [0 1 0] [1 0 0] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(and(X1,X2)) [0 1 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] isNePal(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X = ok(isNePal(X)) [0 1 0] [1] [0 0 0] problem: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [1 0 1] [proper](x0) = [0 1 0]x0 [0 0 1] , [0] [isNePal](x0) = x0 + [0] [1], [1 0 0] [0] [and](x0, x1) = [0 0 1]x0 + x1 + [0] [0 0 1] [1], [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 1] [1 0 0] orientation: [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [0] proper(and(X1,X2)) = [0 0 1]X1 + [0 1 0]X2 + [0] >= [0 0 1]X1 + [0 1 0]X2 + [0] = and(proper(X1),proper(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [0] proper(isNePal(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = isNePal(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] __(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(__(X1,X2)) [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(ok(X1),ok(X2)) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(and(X1,X2)) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [0] [1 0 0] [0] isNePal(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(isNePal(X)) [1 0 0] [1] [1 0 0] [0] problem: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(ok(X)) -> ok(isNePal(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [ok](x0) = x0 + [0] [1], [1 0 1] [0] [isNePal](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 0 1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [1 0 1] [1 0 0] [0] [__](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [1] [0 0 1] [0 0 0] [0] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] __(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = ok(__(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(and(X1,X2)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [1 0 1] [1] [1 0 1] [0] isNePal(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(isNePal(X)) [0 0 1] [1] [0 0 1] [1] problem: Qed