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: RT 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 interpretation: [1] = 0, [+](x0, x1) = x0 + x1 + 9, [minus](x0) = x0 + 23, [0] = 0 orientation: minus(0()) = 23 >= 0 = 0() +(x,0()) = x + 9 >= x = x +(0(),y) = y + 9 >= y = y +(minus(1()),1()) = 32 >= 0 = 0() minus(minus(x)) = x + 46 >= x = x +(x,minus(y)) = x + y + 32 >= x + y + 55 = minus(+(minus(x),y)) +(x,+(y,z)) = x + y + z + 18 >= x + y + z + 18 = +(+(x,y),z) +(minus(+(x,1())),1()) = x + 41 >= x + 23 = minus(x) problem: strict: +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) weak: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(minus(+(x,1())),1()) -> minus(x) Matrix Interpretation Processor: dimension: 2 interpretation: [0] [1] = [7], [1 2] [0] [+](x0, x1) = x0 + [0 1]x1 + [1], [minus](x0) = x0, [0 ] [0] = [15] orientation: [1 2] [0] [1 2] [0] +(x,minus(y)) = x + [0 1]y + [1] >= x + [0 1]y + [1] = minus(+(minus(x),y)) [1 2] [1 4] [2] [1 2] [1 2] [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()) = [15] >= [15] = 0() [30] +(x,0()) = x + [16] >= x = x [1 2] [0 ] +(0(),y) = [0 1]y + [16] >= y = y [14] [0 ] +(minus(1()),1()) = [15] >= [15] = 0() minus(minus(x)) = x >= x = x [28] +(minus(+(x,1())),1()) = x + [16] >= x = minus(x) problem: strict: +(x,minus(y)) -> minus(+(minus(x),y)) weak: +(x,+(y,z)) -> +(+(x,y),z) minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(minus(+(x,1())),1()) -> minus(x) Open