MAYBE Problem: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) Proof: Complexity Transformation Processor: strict: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) weak: Matrix Interpretation Processor: dimension: 1 max_matrix: 1 interpretation: [1] = 0, [+](x0, x1) = x0 + x1 + 1, [minus](x0) = x0, [0] = 0 orientation: minus(0()) = 0 >= 0 = 0() +(x,0()) = x + 1 >= x = x +(0(),y) = y + 1 >= y = y +(minus(1()),1()) = 1 >= 0 = 0() minus(minus(x)) = x >= x = x +(x,minus(y)) = x + y + 1 >= x + y + 1 = minus(+(minus(x),y)) +(x,+(y,z)) = x + y + z + 2 >= x + y + z + 2 = +(+(x,y),z) +(minus(+(x,1())),1()) = x + 2 >= x = minus(x) problem: strict: minus(0()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) weak: +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() +(minus(+(x,1())),1()) -> minus(x) Matrix Interpretation Processor: dimension: 1 max_matrix: 1 interpretation: [1] = 1, [+](x0, x1) = x0 + x1, [minus](x0) = x0 + 1, [0] = 0 orientation: minus(0()) = 1 >= 0 = 0() minus(minus(x)) = x + 2 >= x = x +(x,minus(y)) = x + y + 1 >= x + y + 2 = minus(+(minus(x),y)) +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(x,0()) = x >= x = x +(0(),y) = y >= y = y +(minus(1()),1()) = 3 >= 0 = 0() +(minus(+(x,1())),1()) = x + 3 >= x + 1 = minus(x) problem: strict: +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) weak: minus(0()) -> 0() minus(minus(x)) -> x +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() +(minus(+(x,1())),1()) -> minus(x) Matrix Interpretation Processor: dimension: 2 max_matrix: [1 1] [0 1] interpretation: [0] [1] = [0], [1 1] [0] [+](x0, x1) = x0 + [0 1]x1 + [1], [minus](x0) = x0, [0] [0] = [0] orientation: [1 1] [0] [1 1] [0] +(x,minus(y)) = x + [0 1]y + [1] >= x + [0 1]y + [1] = minus(+(minus(x),y)) [1 1] [1 2] [1] [1 1] [1 1] [0] +(x,+(y,z)) = x + [0 1]y + [0 1]z + [2] >= x + [0 1]y + [0 1]z + [2] = +(+(x,y),z) [0] [0] minus(0()) = [0] >= [0] = 0() minus(minus(x)) = x >= x = x [0] +(x,0()) = x + [1] >= x = x [1 1] [0] +(0(),y) = [0 1]y + [1] >= y = y [0] [0] +(minus(1()),1()) = [1] >= [0] = 0() [0] +(minus(+(x,1())),1()) = x + [2] >= x = minus(x) problem: strict: +(x,minus(y)) -> minus(+(minus(x),y)) weak: +(x,+(y,z)) -> +(+(x,y),z) minus(0()) -> 0() minus(minus(x)) -> x +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() +(minus(+(x,1())),1()) -> minus(x) Open