/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) not(true()) -> false() not(false()) -> true() if(true(),x,y) -> x if(false(),x,y) -> y ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() log(x) -> -(log'(x),1(#())) log'(#()) -> #() log'(1(x)) -> +(log'(x),1(#())) log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#()) Proof: Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + x1, [if](x0, x1, x2) = x0 + x1 + x2, [0](x0) = 2x0, [1](x0) = 2x0, [true] = 0, [log](x0) = 4x0 + 4, [#] = 0, [false] = 0, [log'](x0) = 4x0 + 1, [ge](x0, x1) = x0 + x1, [not](x0) = 2x0, [+](x0, x1) = x0 + x1 orientation: 0(#()) = 0 >= 0 = #() +(#(),x) = x >= x = x +(x,#()) = x >= x = x +(0(x),0(y)) = 2x + 2y >= 2x + 2y = 0(+(x,y)) +(0(x),1(y)) = 2x + 2y >= 2x + 2y = 1(+(x,y)) +(1(x),0(y)) = 2x + 2y >= 2x + 2y = 1(+(x,y)) +(1(x),1(y)) = 2x + 2y >= 2x + 2y = 0(+(+(x,y),1(#()))) +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) -(#(),x) = x >= 0 = #() -(x,#()) = x >= x = x -(0(x),0(y)) = 2x + 2y >= 2x + 2y = 0(-(x,y)) -(0(x),1(y)) = 2x + 2y >= 2x + 2y = 1(-(-(x,y),1(#()))) -(1(x),0(y)) = 2x + 2y >= 2x + 2y = 1(-(x,y)) -(1(x),1(y)) = 2x + 2y >= 2x + 2y = 0(-(x,y)) not(true()) = 0 >= 0 = false() not(false()) = 0 >= 0 = true() if(true(),x,y) = x + y >= x = x if(false(),x,y) = x + y >= y = y ge(0(x),0(y)) = 2x + 2y >= x + y = ge(x,y) ge(0(x),1(y)) = 2x + 2y >= 2x + 2y = not(ge(y,x)) ge(1(x),0(y)) = 2x + 2y >= x + y = ge(x,y) ge(1(x),1(y)) = 2x + 2y >= x + y = ge(x,y) ge(x,#()) = x >= 0 = true() ge(#(),0(x)) = 2x >= x = ge(#(),x) ge(#(),1(x)) = 2x >= 0 = false() log(x) = 4x + 4 >= 4x + 1 = -(log'(x),1(#())) log'(#()) = 1 >= 0 = #() log'(1(x)) = 8x + 1 >= 4x + 1 = +(log'(x),1(#())) log'(0(x)) = 8x + 1 >= 5x + 1 = if(ge(x,1(#())),+(log'(x),1(#())),#()) problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) not(true()) -> false() not(false()) -> true() if(true(),x,y) -> x if(false(),x,y) -> y ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() log'(1(x)) -> +(log'(x),1(#())) log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#()) Matrix Interpretation Processor: dim=3 interpretation: [-](x0, x1) = x0 + x1 , [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + x1 + x2 [0 0 0] , [1 0 0] [0] [0](x0) = [1 0 0]x0 + [0] [0 1 1] [1], [1 0 0] [0] [1](x0) = [1 0 0]x0 + [0] [0 1 1] [1], [1] [true] = [0] [0], [0] [#] = [0] [0], [1] [false] = [0] [0], [1 1 1] [0] [log'](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1 0 0] [1] [ge](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [0] [1 0 1] [1 0 0] [0], [1 0 0] [not](x0) = [0 0 0]x0 [1 0 0] , [+](x0, x1) = x0 + x1 orientation: [0] [0] 0(#()) = [0] >= [0] = #() [1] [0] +(#(),x) = x >= x = x +(x,#()) = x >= x = x [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(0(x),0(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 0(+(x,y)) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(0(x),1(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 1(+(x,y)) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(1(x),0(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 1(+(x,y)) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(1(x),1(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 0(+(+(x,y),1(#()))) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [2] +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) [0] -(#(),x) = x >= [0] = #() [0] -(x,#()) = x >= x = x [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(0(x),0(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 0(-(x,y)) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(0(x),1(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 1(-(-(x,y),1(#()))) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [2] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(1(x),0(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 1(-(x,y)) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(1(x),1(y)) = [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]x + [1 0 0]y + [0] = 0(-(x,y)) [0 1 1] [0 1 1] [2] [0 1 1] [0 1 1] [1] [1] [1] not(true()) = [0] >= [0] = false() [1] [0] [1] [1] not(false()) = [0] >= [0] = true() [1] [0] [1] if(true(),x,y) = x + y + [0] >= x = x [0] [1] if(false(),x,y) = x + y + [0] >= y = y [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] ge(0(x),0(y)) = [0 0 0]x + [1 0 0]y + [0] >= [0 0 0]x + [1 0 0]y + [0] = ge(x,y) [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] ge(0(x),1(y)) = [0 0 0]x + [1 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = not(ge(y,x)) [1 1 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] ge(1(x),0(y)) = [0 0 0]x + [1 0 0]y + [0] >= [0 0 0]x + [1 0 0]y + [0] = ge(x,y) [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] ge(1(x),1(y)) = [0 0 0]x + [1 0 0]y + [0] >= [0 0 0]x + [1 0 0]y + [0] = ge(x,y) [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 0] [1] [1] ge(x,#()) = [0 0 0]x + [0] >= [0] = true() [1 0 1] [0] [0] [1 0 0] [1] [1 0 0] [1] ge(#(),0(x)) = [1 0 0]x + [0] >= [1 0 0]x + [0] = ge(#(),x) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1] [1] ge(#(),1(x)) = [1 0 0]x + [0] >= [0] = false() [1 0 0] [0] [0] [2 1 1] [1] [1 1 1] [0] log'(1(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = +(log'(x),1(#())) [0 1 1] [2] [0 0 1] [2] [2 1 1] [1] [2 1 1] [1] log'(0(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = if(ge(x,1(#())),+(log'(x),1(#())),#()) [0 1 1] [2] [0 0 1] [2] problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) not(true()) -> false() not(false()) -> true() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [-](x0, x1) = x0 + [0 1 0]x1 [0 1 0] , [1 0 0] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 1 0] [0] [0](x0) = [1 1 0]x0 + [0] [1 0 1] [1], [1 1 0] [0] [1](x0) = [1 1 0]x0 + [0] [1 0 1] [1], [0] [true] = [0] [0], [0] [#] = [0] [0], [0] [false] = [0] [0], [1 1 1] [0] [log'](x0) = [0 0 1]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [ge](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [not](x0) = [0 0 0]x0 [0 0 0] , [+](x0, x1) = x0 + x1 orientation: [0] [0] 0(#()) = [0] >= [0] = #() [1] [0] +(#(),x) = x >= x = x +(x,#()) = x >= x = x [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] +(0(x),0(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 0(+(x,y)) [1 0 1] [1 0 1] [2] [1 0 1] [1 0 1] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] +(0(x),1(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 1(+(x,y)) [1 0 1] [1 0 1] [2] [1 0 1] [1 0 1] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] +(1(x),0(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 1(+(x,y)) [1 0 1] [1 0 1] [2] [1 0 1] [1 0 1] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] +(1(x),1(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 0(+(+(x,y),1(#()))) [1 0 1] [1 0 1] [2] [1 0 1] [1 0 1] [2] +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) [1 0 0] [0] -(#(),x) = [0 1 0]x >= [0] = #() [0 1 0] [0] -(x,#()) = x >= x = x [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] -(0(x),0(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 0(-(x,y)) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] -(0(x),1(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 1(-(-(x,y),1(#()))) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] -(1(x),0(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 1(-(x,y)) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] -(1(x),1(y)) = [1 1 0]x + [1 1 0]y + [0] >= [1 1 0]x + [1 1 0]y + [0] = 0(-(x,y)) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [1] [0] [0] not(true()) = [0] >= [0] = false() [0] [0] [0] [0] not(false()) = [0] >= [0] = true() [0] [0] [1 1 0] [1 1 0] [1 0 0] [1 0 0] ge(0(x),0(y)) = [0 0 0]x + [1 1 0]y >= [0 0 0]x + [0 1 0]y = ge(x,y) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 0 0] ge(0(x),1(y)) = [0 0 0]x + [1 1 0]y >= [0 0 0]x + [0 0 0]y = not(ge(y,x)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 0 0] ge(1(x),0(y)) = [0 0 0]x + [1 1 0]y >= [0 0 0]x + [0 1 0]y = ge(x,y) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 0 0] ge(1(x),1(y)) = [0 0 0]x + [1 1 0]y >= [0 0 0]x + [0 1 0]y = ge(x,y) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] ge(x,#()) = [0 0 0]x >= [0] = true() [0 0 0] [0] [1 1 0] [1 0 0] ge(#(),0(x)) = [1 1 0]x >= [0 1 0]x = ge(#(),x) [0 0 0] [0 0 0] [1 1 0] [0] ge(#(),1(x)) = [1 1 0]x >= [0] = false() [0 0 0] [0] [3 2 1] [1] [2 1 1] log'(0(x)) = [1 0 1]x + [2] >= [0 0 0]x = if(ge(x,1(#())),+(log'(x),1(#())),#()) [0 0 0] [0] [0 0 0] problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) not(true()) -> false() not(false()) -> true() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() Matrix Interpretation Processor: dim=3 interpretation: [-](x0, x1) = x0 + x1 , [1 0 0] [0](x0) = [0 0 1]x0 [1 0 0] , [1 0 0] [1](x0) = [0 0 1]x0 [1 0 0] , [0] [true] = [0] [0], [0] [#] = [0] [0], [0] [false] = [0] [0], [1 0 0] [1 0 0] [1] [ge](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [not](x0) = [0 0 0]x0 [0 0 0] , [+](x0, x1) = x0 + x1 orientation: [0] [0] 0(#()) = [0] >= [0] = #() [0] [0] +(#(),x) = x >= x = x +(x,#()) = x >= x = x [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(0(x),0(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 0(+(x,y)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(0(x),1(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 1(+(x,y)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(1(x),0(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 1(+(x,y)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(1(x),1(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 0(+(+(x,y),1(#()))) [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) [0] -(#(),x) = x >= [0] = #() [0] -(x,#()) = x >= x = x [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(0(x),0(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 0(-(x,y)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(0(x),1(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 1(-(-(x,y),1(#()))) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(1(x),0(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 1(-(x,y)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(1(x),1(y)) = [0 0 1]x + [0 0 1]y >= [0 0 1]x + [0 0 1]y = 0(-(x,y)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [0] [0] not(true()) = [0] >= [0] = false() [0] [0] [0] [0] not(false()) = [0] >= [0] = true() [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] ge(0(x),0(y)) = [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]x + [0 0 0]y + [1] = ge(x,y) [0 0 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] [1] ge(0(x),1(y)) = [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]x + [0 0 0]y + [0] = not(ge(y,x)) [0 0 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] [1] ge(1(x),0(y)) = [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]x + [0 0 0]y + [1] = ge(x,y) [0 0 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] [1] ge(1(x),1(y)) = [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]x + [0 0 0]y + [1] = ge(x,y) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1] [0] ge(x,#()) = [0 0 0]x + [1] >= [0] = true() [0 0 0] [0] [0] [1 0 0] [1] [1 0 0] [1] ge(#(),0(x)) = [0 0 0]x + [1] >= [0 0 0]x + [1] = ge(#(),x) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [0] ge(#(),1(x)) = [0 0 0]x + [1] >= [0] = false() [0 0 0] [0] [0] problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) not(true()) -> false() not(false()) -> true() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(#(),0(x)) -> ge(#(),x) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + x1, [0](x0) = 4x0, [1](x0) = 4x0, [true] = 1, [#] = 0, [false] = 4, [ge](x0, x1) = 4x0 + 4x1, [not](x0) = 4x0, [+](x0, x1) = x0 + x1 orientation: 0(#()) = 0 >= 0 = #() +(#(),x) = x >= x = x +(x,#()) = x >= x = x +(0(x),0(y)) = 4x + 4y >= 4x + 4y = 0(+(x,y)) +(0(x),1(y)) = 4x + 4y >= 4x + 4y = 1(+(x,y)) +(1(x),0(y)) = 4x + 4y >= 4x + 4y = 1(+(x,y)) +(1(x),1(y)) = 4x + 4y >= 4x + 4y = 0(+(+(x,y),1(#()))) +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) -(#(),x) = x >= 0 = #() -(x,#()) = x >= x = x -(0(x),0(y)) = 4x + 4y >= 4x + 4y = 0(-(x,y)) -(0(x),1(y)) = 4x + 4y >= 4x + 4y = 1(-(-(x,y),1(#()))) -(1(x),0(y)) = 4x + 4y >= 4x + 4y = 1(-(x,y)) -(1(x),1(y)) = 4x + 4y >= 4x + 4y = 0(-(x,y)) not(true()) = 4 >= 4 = false() not(false()) = 16 >= 1 = true() ge(0(x),0(y)) = 16x + 16y >= 4x + 4y = ge(x,y) ge(0(x),1(y)) = 16x + 16y >= 16x + 16y = not(ge(y,x)) ge(1(x),0(y)) = 16x + 16y >= 4x + 4y = ge(x,y) ge(1(x),1(y)) = 16x + 16y >= 4x + 4y = ge(x,y) ge(#(),0(x)) = 16x >= 4x = ge(#(),x) problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) not(true()) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(#(),0(x)) -> ge(#(),x) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [-](x0, x1) = x0 + [1 0 1]x1 [0 0 0] , [1 0 0] [0](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1](x0) = [0 0 0]x0 [0 0 1] , [1] [true] = [0] [0], [0] [#] = [0] [0], [0] [false] = [0] [0], [1 0 1] [1 0 1] [ge](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 0] [1 0 0] , [1 0 0] [not](x0) = [0 0 0]x0 [0 0 1] , [+](x0, x1) = x0 + x1 orientation: [0] [0] 0(#()) = [0] >= [0] = #() [0] [0] +(#(),x) = x >= x = x +(x,#()) = x >= x = x [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(0(x),0(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = 0(+(x,y)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(0(x),1(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = 1(+(x,y)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(1(x),0(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = 1(+(x,y)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] +(1(x),1(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = 0(+(+(x,y),1(#()))) [0 0 1] [0 0 1] [0 0 1] [0 0 1] +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) [1 0 0] [0] -(#(),x) = [1 0 1]x >= [0] = #() [0 0 0] [0] -(x,#()) = x >= x = x [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(0(x),0(y)) = [0 0 0]x + [1 0 1]y >= [0 0 0]x + [0 0 0]y = 0(-(x,y)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(0(x),1(y)) = [0 0 0]x + [1 0 1]y >= [0 0 0]x + [0 0 0]y = 1(-(-(x,y),1(#()))) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(1(x),0(y)) = [0 0 0]x + [1 0 1]y >= [0 0 0]x + [0 0 0]y = 1(-(x,y)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] -(1(x),1(y)) = [0 0 0]x + [1 0 1]y >= [0 0 0]x + [0 0 0]y = 0(-(x,y)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1] [0] not(true()) = [0] >= [0] = false() [0] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] ge(0(x),0(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = ge(x,y) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] ge(0(x),1(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = not(ge(y,x)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] ge(1(x),0(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = ge(x,y) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] ge(1(x),1(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = ge(x,y) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 1] ge(#(),0(x)) = [0 0 0]x >= [0 0 0]x = ge(#(),x) [1 0 0] [1 0 0] problem: 0(#()) -> #() +(#(),x) -> x +(x,#()) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) ge(#(),0(x)) -> ge(#(),x) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + 6x1, [0](x0) = x0 + 2, [1](x0) = x0 + 2, [#] = 0, [ge](x0, x1) = 4x0 + 4x1, [not](x0) = x0 + 7, [+](x0, x1) = x0 + x1 orientation: 0(#()) = 2 >= 0 = #() +(#(),x) = x >= x = x +(x,#()) = x >= x = x +(0(x),0(y)) = x + y + 4 >= x + y + 2 = 0(+(x,y)) +(0(x),1(y)) = x + y + 4 >= x + y + 2 = 1(+(x,y)) +(1(x),0(y)) = x + y + 4 >= x + y + 2 = 1(+(x,y)) +(1(x),1(y)) = x + y + 4 >= x + y + 4 = 0(+(+(x,y),1(#()))) +(+(x,y),z) = x + y + z >= x + y + z = +(x,+(y,z)) -(#(),x) = 6x >= 0 = #() -(x,#()) = x >= x = x -(0(x),0(y)) = x + 6y + 14 >= x + 6y + 2 = 0(-(x,y)) -(0(x),1(y)) = x + 6y + 14 >= x + 6y + 14 = 1(-(-(x,y),1(#()))) -(1(x),0(y)) = x + 6y + 14 >= x + 6y + 2 = 1(-(x,y)) -(1(x),1(y)) = x + 6y + 14 >= x + 6y + 2 = 0(-(x,y)) ge(0(x),0(y)) = 4x + 4y + 16 >= 4x + 4y = ge(x,y) ge(0(x),1(y)) = 4x + 4y + 16 >= 4x + 4y + 7 = not(ge(y,x)) ge(1(x),0(y)) = 4x + 4y + 16 >= 4x + 4y = ge(x,y) ge(1(x),1(y)) = 4x + 4y + 16 >= 4x + 4y = ge(x,y) ge(#(),0(x)) = 4x + 8 >= 4x = ge(#(),x) problem: +(#(),x) -> x +(x,#()) -> x +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) +(+(x,y),z) -> +(x,+(y,z)) -(#(),x) -> #() -(x,#()) -> x -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) WPO Processor: algebra: Sum weight function: w0 = 0 w(1) = w(0) = 1 w(-) = w(+) = w(#) = 0 status function: st(-) = [1, 0] st(1) = [0] st(+) = [0, 1] st(0) = [0] st(#) = [] precedence: + > - > 1 ~ 0 ~ # problem: Qed