/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() and(x,true()) -> x and(x,false()) -> false() if(true(),x,y) -> x if(false(),x,y) -> y ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(0()) -> 0() Log'(I(x)) -> +(Log'(x),I(0())) Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Log(x) -> -(Log'(x),I(0())) Val(L(x)) -> x Val(N(x,l(),r())) -> x Min(L(x)) -> x Min(N(x,l(),r())) -> Min(l()) Max(L(x)) -> x Max(N(x,l(),r())) -> Max(r()) BS(L(x)) -> true() BS(N(x,l(),r())) -> and(and(ge(x,Max(l())),ge(Min(r()),x)),and(BS(l()),BS(r()))) Size(L(x)) -> I(0()) Size(N(x,l(),r())) -> +(+(Size(l()),Size(r())),I(1())) WB(L(x)) -> true() WB(N(x,l(),r())) -> and(if(ge(Size(l()),Size(r())),ge(I(0()),-(Size(l()),Size(r()))),ge(I(0()),-(Size(r()),Size(l())))), and(WB(l()),WB(r()))) Proof: Matrix Interpretation Processor: dim=1 interpretation: [N](x0, x1, x2) = 4x0 + x1 + 2x2 + 2, [-](x0, x1) = x0 + 2x1, [Val](x0) = x0, [false] = 0, [O](x0) = 4x0, [I](x0) = 4x0, [l] = 0, [BS](x0) = 4x0, [Min](x0) = 4x0, [r] = 0, [Size](x0) = x0, [1] = 0, [if](x0, x1, x2) = 2x0 + 2x1 + 4x2, [0] = 0, [not](x0) = x0, [Max](x0) = 4x0, [L](x0) = 2x0, [Log'](x0) = 2x0, [WB](x0) = x0 + 2, [ge](x0, x1) = 2x0 + 2x1, [and](x0, x1) = x0 + x1, [Log](x0) = 2x0 + 1, [true] = 0, [+](x0, x1) = x0 + 2x1 orientation: O(0()) = 0 >= 0 = 0() +(0(),x) = 2x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = 4x + 8y >= 4x + 8y = O(+(x,y)) +(O(x),I(y)) = 4x + 8y >= 4x + 8y = I(+(x,y)) +(I(x),O(y)) = 4x + 8y >= 4x + 8y = I(+(x,y)) +(I(x),I(y)) = 4x + 8y >= 4x + 8y = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + 2y + 4z >= x + 2y + 2z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = 2x >= 0 = 0() -(O(x),O(y)) = 4x + 8y >= 4x + 8y = O(-(x,y)) -(O(x),I(y)) = 4x + 8y >= 4x + 8y = I(-(-(x,y),I(1()))) -(I(x),O(y)) = 4x + 8y >= 4x + 8y = I(-(x,y)) -(I(x),I(y)) = 4x + 8y >= 4x + 8y = O(-(x,y)) not(true()) = 0 >= 0 = false() not(false()) = 0 >= 0 = true() and(x,true()) = x >= x = x and(x,false()) = x >= 0 = false() if(true(),x,y) = 2x + 4y >= x = x if(false(),x,y) = 2x + 4y >= y = y ge(O(x),O(y)) = 8x + 8y >= 2x + 2y = ge(x,y) ge(O(x),I(y)) = 8x + 8y >= 2x + 2y = not(ge(y,x)) ge(I(x),O(y)) = 8x + 8y >= 2x + 2y = ge(x,y) ge(I(x),I(y)) = 8x + 8y >= 2x + 2y = ge(x,y) ge(x,0()) = 2x >= 0 = true() ge(0(),O(x)) = 8x >= 2x = ge(0(),x) ge(0(),I(x)) = 8x >= 0 = false() Log'(0()) = 0 >= 0 = 0() Log'(I(x)) = 8x >= 2x = +(Log'(x),I(0())) Log'(O(x)) = 8x >= 8x = if(ge(x,I(0())),+(Log'(x),I(0())),0()) Log(x) = 2x + 1 >= 2x = -(Log'(x),I(0())) Val(L(x)) = 2x >= x = x Val(N(x,l(),r())) = 4x + 2 >= x = x Min(L(x)) = 8x >= x = x Min(N(x,l(),r())) = 16x + 8 >= 0 = Min(l()) Max(L(x)) = 8x >= x = x Max(N(x,l(),r())) = 16x + 8 >= 0 = Max(r()) BS(L(x)) = 8x >= 0 = true() BS(N(x,l(),r())) = 16x + 8 >= 4x = and(and(ge(x,Max(l())),ge(Min(r()),x)),and(BS(l()),BS(r()))) Size(L(x)) = 2x >= 0 = I(0()) Size(N(x,l(),r())) = 4x + 2 >= 0 = +(+(Size(l()),Size(r())),I(1())) WB(L(x)) = 2x + 2 >= 0 = true() WB(N(x,l(),r())) = 4x + 4 >= 4 = and(if(ge(Size(l()),Size(r())),ge(I(0()),-(Size(l()),Size(r()))),ge(I(0()), - ( Size(r()), Size (l())))), and(WB(l()),WB(r()))) problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() and(x,true()) -> x and(x,false()) -> false() if(true(),x,y) -> x if(false(),x,y) -> y ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(0()) -> 0() Log'(I(x)) -> +(Log'(x),I(0())) Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Val(L(x)) -> x Min(L(x)) -> x Max(L(x)) -> x BS(L(x)) -> true() Size(L(x)) -> I(0()) WB(N(x,l(),r())) -> and(if(ge(Size(l()),Size(r())),ge(I(0()),-(Size(l()),Size(r()))),ge ( I(0()),-(Size(r()),Size(l())))), and(WB(l()),WB(r()))) Matrix Interpretation Processor: dim=1 interpretation: [N](x0, x1, x2) = 4x0 + x1 + x2 + 3, [-](x0, x1) = x0 + x1, [Val](x0) = 4x0, [false] = 0, [O](x0) = 3x0, [I](x0) = 3x0, [l] = 0, [BS](x0) = x0 + 1, [Min](x0) = x0 + 4, [r] = 0, [Size](x0) = x0, [1] = 0, [if](x0, x1, x2) = x0 + 2x1 + x2, [0] = 0, [not](x0) = x0, [Max](x0) = x0, [L](x0) = 2x0 + 2, [Log'](x0) = 2x0, [WB](x0) = 6x0, [ge](x0, x1) = x0 + 2x1, [and](x0, x1) = x0 + 4x1, [true] = 0, [+](x0, x1) = x0 + x1 orientation: O(0()) = 0 >= 0 = 0() +(0(),x) = x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = 3x + 3y >= 3x + 3y = O(+(x,y)) +(O(x),I(y)) = 3x + 3y >= 3x + 3y = I(+(x,y)) +(I(x),O(y)) = 3x + 3y >= 3x + 3y = I(+(x,y)) +(I(x),I(y)) = 3x + 3y >= 3x + 3y = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = x >= 0 = 0() -(O(x),O(y)) = 3x + 3y >= 3x + 3y = O(-(x,y)) -(O(x),I(y)) = 3x + 3y >= 3x + 3y = I(-(-(x,y),I(1()))) -(I(x),O(y)) = 3x + 3y >= 3x + 3y = I(-(x,y)) -(I(x),I(y)) = 3x + 3y >= 3x + 3y = O(-(x,y)) not(true()) = 0 >= 0 = false() not(false()) = 0 >= 0 = true() and(x,true()) = x >= x = x and(x,false()) = x >= 0 = false() if(true(),x,y) = 2x + y >= x = x if(false(),x,y) = 2x + y >= y = y ge(O(x),O(y)) = 3x + 6y >= x + 2y = ge(x,y) ge(O(x),I(y)) = 3x + 6y >= 2x + y = not(ge(y,x)) ge(I(x),O(y)) = 3x + 6y >= x + 2y = ge(x,y) ge(I(x),I(y)) = 3x + 6y >= x + 2y = ge(x,y) ge(x,0()) = x >= 0 = true() ge(0(),O(x)) = 6x >= 2x = ge(0(),x) ge(0(),I(x)) = 6x >= 0 = false() Log'(0()) = 0 >= 0 = 0() Log'(I(x)) = 6x >= 2x = +(Log'(x),I(0())) Log'(O(x)) = 6x >= 5x = if(ge(x,I(0())),+(Log'(x),I(0())),0()) Val(L(x)) = 8x + 8 >= x = x Min(L(x)) = 2x + 6 >= x = x Max(L(x)) = 2x + 2 >= x = x BS(L(x)) = 2x + 3 >= 0 = true() Size(L(x)) = 2x + 2 >= 0 = I(0()) WB(N(x,l(),r())) = 24x + 18 >= 0 = and(if(ge(Size(l()),Size(r())),ge(I(0()),-(Size(l()),Size(r()))),ge(I(0()), - ( Size(r()), Size (l())))), and(WB(l()),WB(r()))) problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() and(x,true()) -> x and(x,false()) -> false() if(true(),x,y) -> x if(false(),x,y) -> y ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(0()) -> 0() Log'(I(x)) -> +(Log'(x),I(0())) Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + x1, [false] = 0, [O](x0) = 4x0, [I](x0) = 4x0, [1] = 0, [if](x0, x1, x2) = x0 + 2x1 + 4x2, [0] = 0, [not](x0) = x0, [Log'](x0) = x0, [ge](x0, x1) = 2x0 + 2x1, [and](x0, x1) = x0 + 2x1 + 1, [true] = 0, [+](x0, x1) = x0 + x1 orientation: O(0()) = 0 >= 0 = 0() +(0(),x) = x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = 4x + 4y >= 4x + 4y = O(+(x,y)) +(O(x),I(y)) = 4x + 4y >= 4x + 4y = I(+(x,y)) +(I(x),O(y)) = 4x + 4y >= 4x + 4y = I(+(x,y)) +(I(x),I(y)) = 4x + 4y >= 4x + 4y = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = x >= 0 = 0() -(O(x),O(y)) = 4x + 4y >= 4x + 4y = O(-(x,y)) -(O(x),I(y)) = 4x + 4y >= 4x + 4y = I(-(-(x,y),I(1()))) -(I(x),O(y)) = 4x + 4y >= 4x + 4y = I(-(x,y)) -(I(x),I(y)) = 4x + 4y >= 4x + 4y = O(-(x,y)) not(true()) = 0 >= 0 = false() not(false()) = 0 >= 0 = true() and(x,true()) = x + 1 >= x = x and(x,false()) = x + 1 >= 0 = false() if(true(),x,y) = 2x + 4y >= x = x if(false(),x,y) = 2x + 4y >= y = y ge(O(x),O(y)) = 8x + 8y >= 2x + 2y = ge(x,y) ge(O(x),I(y)) = 8x + 8y >= 2x + 2y = not(ge(y,x)) ge(I(x),O(y)) = 8x + 8y >= 2x + 2y = ge(x,y) ge(I(x),I(y)) = 8x + 8y >= 2x + 2y = ge(x,y) ge(x,0()) = 2x >= 0 = true() ge(0(),O(x)) = 8x >= 2x = ge(0(),x) ge(0(),I(x)) = 8x >= 0 = false() Log'(0()) = 0 >= 0 = 0() Log'(I(x)) = 4x >= x = +(Log'(x),I(0())) Log'(O(x)) = 4x >= 4x = if(ge(x,I(0())),+(Log'(x),I(0())),0()) problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() if(true(),x,y) -> x if(false(),x,y) -> y ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(0()) -> 0() Log'(I(x)) -> +(Log'(x),I(0())) Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + 4x1, [false] = 0, [O](x0) = 4x0, [I](x0) = 4x0, [1] = 0, [if](x0, x1, x2) = x0 + x1 + x2, [0] = 0, [not](x0) = x0, [Log'](x0) = x0 + 1, [ge](x0, x1) = 3x0 + x1, [true] = 0, [+](x0, x1) = x0 + x1 orientation: O(0()) = 0 >= 0 = 0() +(0(),x) = x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = 4x + 4y >= 4x + 4y = O(+(x,y)) +(O(x),I(y)) = 4x + 4y >= 4x + 4y = I(+(x,y)) +(I(x),O(y)) = 4x + 4y >= 4x + 4y = I(+(x,y)) +(I(x),I(y)) = 4x + 4y >= 4x + 4y = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = 4x >= 0 = 0() -(O(x),O(y)) = 4x + 16y >= 4x + 16y = O(-(x,y)) -(O(x),I(y)) = 4x + 16y >= 4x + 16y = I(-(-(x,y),I(1()))) -(I(x),O(y)) = 4x + 16y >= 4x + 16y = I(-(x,y)) -(I(x),I(y)) = 4x + 16y >= 4x + 16y = O(-(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(O(x),O(y)) = 12x + 4y >= 3x + y = ge(x,y) ge(O(x),I(y)) = 12x + 4y >= x + 3y = not(ge(y,x)) ge(I(x),O(y)) = 12x + 4y >= 3x + y = ge(x,y) ge(I(x),I(y)) = 12x + 4y >= 3x + y = ge(x,y) ge(x,0()) = 3x >= 0 = true() ge(0(),O(x)) = 4x >= x = ge(0(),x) ge(0(),I(x)) = 4x >= 0 = false() Log'(0()) = 1 >= 0 = 0() Log'(I(x)) = 4x + 1 >= x + 1 = +(Log'(x),I(0())) Log'(O(x)) = 4x + 1 >= 4x + 1 = if(ge(x,I(0())),+(Log'(x),I(0())),0()) problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() if(true(),x,y) -> x if(false(),x,y) -> y ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(I(x)) -> +(Log'(x),I(0())) Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [-](x0, x1) = x0 + [0 0 0]x1 [0 0 1] , [0] [false] = [0] [1], [1 0 0] [0] [O](x0) = [0 0 0]x0 + [0] [1 1 1] [1], [1 0 0] [0] [I](x0) = [0 0 0]x0 + [0] [1 1 1] [1], [0] [1] = [0] [0], [1 0 1] [if](x0, x1, x2) = [0 0 0]x0 + x1 + x2 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [0] [not](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [1] [Log'](x0) = [0 0 0]x0 + [0] [1 1 1] [0], [1 0 0] [1 0 0] [0] [ge](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [true] = [0] [0], [+](x0, x1) = x0 + x1 orientation: [0] [0] O(0()) = [0] >= [0] = 0() [1] [0] +(0(),x) = x >= x = x +(x,0()) = x >= x = x [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(O(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = O(+(x,y)) [1 1 1] [1 1 1] [2] [1 1 1] [1 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(O(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = I(+(x,y)) [1 1 1] [1 1 1] [2] [1 1 1] [1 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(I(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = I(+(x,y)) [1 1 1] [1 1 1] [2] [1 1 1] [1 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(I(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = O(+(+(x,y),I(0()))) [1 1 1] [1 1 1] [2] [1 1 1] [1 1 1] [2] +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x [1 0 0] [0] -(0(),x) = [0 0 0]x >= [0] = 0() [0 0 1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(O(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = O(-(x,y)) [1 1 1] [1 1 1] [2] [1 1 1] [1 0 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(O(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = I(-(-(x,y),I(1()))) [1 1 1] [1 1 1] [2] [1 1 1] [1 0 1] [2] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(I(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = I(-(x,y)) [1 1 1] [1 1 1] [2] [1 1 1] [1 0 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(I(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = O(-(x,y)) [1 1 1] [1 1 1] [2] [1 1 1] [1 0 1] [1] [0] [0] not(true()) = [0] >= [0] = false() [1] [1] [0] [0] not(false()) = [0] >= [0] = true() [1] [0] if(true(),x,y) = x + y >= x = x [1] if(false(),x,y) = x + y + [0] >= y = y [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] ge(O(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = ge(x,y) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] ge(O(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = not(ge(y,x)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] ge(I(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = ge(x,y) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] ge(I(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = ge(x,y) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [0] ge(x,0()) = [0 0 0]x + [0] >= [0] = true() [0 0 0] [1] [0] [1 0 0] [0] [1 0 0] [0] ge(0(),O(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = ge(0(),x) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [0] ge(0(),I(x)) = [0 0 0]x + [0] >= [0] = false() [0 0 0] [1] [1] [2 1 1] [2] [1 0 1] [1] Log'(I(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = +(Log'(x),I(0())) [2 1 1] [1] [1 1 1] [1] [2 1 1] [2] [2 0 1] [2] Log'(O(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = if(ge(x,I(0())),+(Log'(x),I(0())),0()) [2 1 1] [1] [1 1 1] [1] problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() if(true(),x,y) -> x ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [0] [-](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [0] [1 0 1] [1 0 0] [1], [0] [false] = [0] [0], [1 0 0] [0] [O](x0) = [1 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [I](x0) = [1 1 0]x0 + [1] [0 0 0] [0], [0] [1] = [0] [0], [1 0 0] [1 0 0] [1] [if](x0, x1, x2) = [0 0 0]x0 + x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [not](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1] [Log'](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [ge](x0, x1) = [1 1 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [0] [true] = [0] [0], [+](x0, x1) = x0 + x1 orientation: [0] [0] O(0()) = [1] >= [0] = 0() [0] [0] +(0(),x) = x >= x = x +(x,0()) = x >= x = x [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(O(x),O(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = O(+(x,y)) [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] +(O(x),I(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = I(+(x,y)) [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] +(I(x),O(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = I(+(x,y)) [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] +(I(x),I(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [2] = O(+(+(x,y),I(0()))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) [1 0 0] [0] -(x,0()) = [0 1 0]x + [0] >= x = x [1 0 1] [1] [1 0 0] [0] [0] -(0(),x) = [0 1 0]x + [0] >= [0] = 0() [1 0 0] [1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(O(x),O(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = O(-(x,y)) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(O(x),I(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [2] = I(-(-(x,y),I(1()))) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(I(x),O(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = I(-(x,y)) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(I(x),I(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = O(-(x,y)) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [0] [0] not(true()) = [0] >= [0] = false() [0] [0] [0] [0] not(false()) = [0] >= [0] = true() [0] [0] [1 0 0] [1] if(true(),x,y) = x + [0 0 0]y + [0] >= x = x [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] ge(O(x),O(y)) = [2 1 0]x + [1 0 0]y + [1] >= [1 1 0]x + [1 0 0]y = ge(x,y) [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] ge(O(x),I(y)) = [2 1 0]x + [1 0 0]y + [1] >= [0 0 0]x + [0 0 0]y = not(ge(y,x)) [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] ge(I(x),O(y)) = [2 1 0]x + [1 0 0]y + [1] >= [1 1 0]x + [1 0 0]y = ge(x,y) [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] ge(I(x),I(y)) = [2 1 0]x + [1 0 0]y + [1] >= [1 1 0]x + [1 0 0]y = ge(x,y) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [0] ge(x,0()) = [1 1 0]x >= [0] = true() [0 0 0] [0] [1 0 0] [1 0 0] ge(0(),O(x)) = [1 0 0]x >= [1 0 0]x = ge(0(),x) [0 0 0] [0 0 0] [1 0 0] [0] ge(0(),I(x)) = [1 0 0]x >= [0] = false() [0 0 0] [0] [2 1 0] [2] [2 1 0] [2] Log'(O(x)) = [1 1 0]x + [1] >= [0 1 0]x + [1] = if(ge(x,I(0())),+(Log'(x),I(0())),0()) [0 0 0] [1] [0 0 0] [1] problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Log'(O(x)) -> if(ge(x,I(0())),+(Log'(x),I(0())),0()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [-](x0, x1) = x0 + [0 0 1]x1 [0 1 0] , [0] [false] = [0] [0], [1 0 0] [0] [O](x0) = [1 1 0]x0 + [1] [1 0 1] [0], [1 0 0] [0] [I](x0) = [1 1 0]x0 + [1] [1 0 1] [0], [0] [1] = [0] [0], [1 0 0] [1 0 0] [1 0 0] [0] [if](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 0 1] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [not](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [Log'](x0) = [1 1 0]x0 [1 0 0] , [1 0 0] [1 0 0] [ge](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [true] = [0] [0], [+](x0, x1) = x0 + x1 orientation: [0] [0] O(0()) = [1] >= [0] = 0() [0] [0] +(0(),x) = x >= x = x +(x,0()) = x >= x = x [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(O(x),O(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = O(+(x,y)) [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(O(x),I(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = I(+(x,y)) [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(I(x),O(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [1] = I(+(x,y)) [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] +(I(x),I(y)) = [1 1 0]x + [1 1 0]y + [2] >= [1 1 0]x + [1 1 0]y + [2] = O(+(+(x,y),I(0()))) [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x [1 0 0] [0] -(0(),x) = [0 0 1]x >= [0] = 0() [0 1 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(O(x),O(y)) = [1 1 0]x + [1 0 1]y + [1] >= [1 1 0]x + [1 0 1]y + [1] = O(-(x,y)) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(O(x),I(y)) = [1 1 0]x + [1 0 1]y + [1] >= [1 1 0]x + [1 0 1]y + [1] = I(-(-(x,y),I(1()))) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(I(x),O(y)) = [1 1 0]x + [1 0 1]y + [1] >= [1 1 0]x + [1 0 1]y + [1] = I(-(x,y)) [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] -(I(x),I(y)) = [1 1 0]x + [1 0 1]y + [1] >= [1 1 0]x + [1 0 1]y + [1] = O(-(x,y)) [1 0 1] [1 1 0] [1] [1 0 1] [1 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 0 0] [1 0 0] ge(O(x),O(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = ge(x,y) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] ge(O(x),I(y)) = [0 0 0]x + [0 0 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 0 0] [1 0 0] [1 0 0] [1 0 0] ge(I(x),O(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = ge(x,y) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] ge(I(x),I(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 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 0]x >= [0] = true() [0 0 0] [0] [1 0 0] [1 0 0] ge(0(),O(x)) = [0 0 0]x >= [0 0 0]x = ge(0(),x) [0 0 0] [0 0 0] [1 0 0] [0] ge(0(),I(x)) = [0 0 0]x >= [0] = false() [0 0 0] [0] [2 1 0] [1] [2 1 0] [0] Log'(O(x)) = [2 1 0]x + [1] >= [1 1 0]x + [1] = if(ge(x,I(0())),+(Log'(x),I(0())),0()) [1 0 0] [0] [1 0 0] [0] problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(x,0()) -> true() ge(0(),O(x)) -> ge(0(),x) ge(0(),I(x)) -> false() Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [-](x0, x1) = x0 + [0 1 0]x1 [0 0 0] , [0] [false] = [0] [0], [1 1 0] [O](x0) = [1 0 0]x0 [0 0 0] , [1 1 0] [I](x0) = [1 0 0]x0 [0 0 0] , [0] [1] = [0] [0], [0] [0] = [0] [0], [1 0 0] [not](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1] [ge](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [1 0 0] [0 0 0] [0], [0] [true] = [0] [0], [+](x0, x1) = x0 + x1 orientation: [0] [0] O(0()) = [0] >= [0] = 0() [0] [0] +(0(),x) = x >= x = x +(x,0()) = x >= x = x [1 1 0] [1 1 0] [1 1 0] [1 1 0] +(O(x),O(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = O(+(x,y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] +(O(x),I(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = I(+(x,y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] +(I(x),O(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = I(+(x,y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] +(I(x),I(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = O(+(+(x,y),I(0()))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x [1 0 0] [0] -(0(),x) = [0 1 0]x >= [0] = 0() [0 0 0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] -(O(x),O(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = O(-(x,y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] -(O(x),I(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = I(-(-(x,y),I(1()))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] -(I(x),O(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = I(-(x,y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] -(I(x),I(y)) = [1 0 0]x + [1 0 0]y >= [1 0 0]x + [1 0 0]y = O(-(x,y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [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] [1 0 0] [1 0 0] [1] ge(O(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = ge(x,y) [1 1 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] ge(O(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = not(ge(y,x)) [1 1 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] ge(I(x),O(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = ge(x,y) [1 1 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] ge(I(x),I(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = ge(x,y) [1 1 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 0 0] [1] [0] ge(x,0()) = [0 0 0]x + [0] >= [0] = true() [1 0 0] [0] [0] [1 1 0] [1] [1 0 0] [1] ge(0(),O(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = ge(0(),x) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [0] ge(0(),I(x)) = [0 0 0]x + [0] >= [0] = false() [0 0 0] [0] [0] problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(true()) -> false() not(false()) -> true() ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(0(),O(x)) -> ge(0(),x) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + 4x1, [false] = 2, [O](x0) = 4x0, [I](x0) = 4x0, [1] = 0, [0] = 0, [not](x0) = 2x0, [ge](x0, x1) = x0 + 2x1, [true] = 4, [+](x0, x1) = x0 + 4x1 orientation: O(0()) = 0 >= 0 = 0() +(0(),x) = 4x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = 4x + 16y >= 4x + 16y = O(+(x,y)) +(O(x),I(y)) = 4x + 16y >= 4x + 16y = I(+(x,y)) +(I(x),O(y)) = 4x + 16y >= 4x + 16y = I(+(x,y)) +(I(x),I(y)) = 4x + 16y >= 4x + 16y = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + 4y + 16z >= x + 4y + 4z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = 4x >= 0 = 0() -(O(x),O(y)) = 4x + 16y >= 4x + 16y = O(-(x,y)) -(O(x),I(y)) = 4x + 16y >= 4x + 16y = I(-(-(x,y),I(1()))) -(I(x),O(y)) = 4x + 16y >= 4x + 16y = I(-(x,y)) -(I(x),I(y)) = 4x + 16y >= 4x + 16y = O(-(x,y)) not(true()) = 8 >= 2 = false() not(false()) = 4 >= 4 = true() ge(O(x),O(y)) = 4x + 8y >= x + 2y = ge(x,y) ge(O(x),I(y)) = 4x + 8y >= 4x + 2y = not(ge(y,x)) ge(I(x),O(y)) = 4x + 8y >= x + 2y = ge(x,y) ge(I(x),I(y)) = 4x + 8y >= x + 2y = ge(x,y) ge(0(),O(x)) = 8x >= 2x = ge(0(),x) problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) not(false()) -> true() ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(0(),O(x)) -> ge(0(),x) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + 2x1, [false] = 2, [O](x0) = 4x0, [I](x0) = 4x0, [1] = 0, [0] = 0, [not](x0) = 2x0, [ge](x0, x1) = 2x0 + x1, [true] = 0, [+](x0, x1) = x0 + 2x1 orientation: O(0()) = 0 >= 0 = 0() +(0(),x) = 2x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = 4x + 8y >= 4x + 8y = O(+(x,y)) +(O(x),I(y)) = 4x + 8y >= 4x + 8y = I(+(x,y)) +(I(x),O(y)) = 4x + 8y >= 4x + 8y = I(+(x,y)) +(I(x),I(y)) = 4x + 8y >= 4x + 8y = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + 2y + 4z >= x + 2y + 2z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = 2x >= 0 = 0() -(O(x),O(y)) = 4x + 8y >= 4x + 8y = O(-(x,y)) -(O(x),I(y)) = 4x + 8y >= 4x + 8y = I(-(-(x,y),I(1()))) -(I(x),O(y)) = 4x + 8y >= 4x + 8y = I(-(x,y)) -(I(x),I(y)) = 4x + 8y >= 4x + 8y = O(-(x,y)) not(false()) = 4 >= 0 = true() ge(O(x),O(y)) = 8x + 4y >= 2x + y = ge(x,y) ge(O(x),I(y)) = 8x + 4y >= 2x + 4y = not(ge(y,x)) ge(I(x),O(y)) = 8x + 4y >= 2x + y = ge(x,y) ge(I(x),I(y)) = 8x + 4y >= 2x + y = ge(x,y) ge(0(),O(x)) = 4x >= x = ge(0(),x) problem: O(0()) -> 0() +(0(),x) -> x +(x,0()) -> x +(O(x),O(y)) -> O(+(x,y)) +(O(x),I(y)) -> I(+(x,y)) +(I(x),O(y)) -> I(+(x,y)) +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),O(y)) -> O(-(x,y)) -(O(x),I(y)) -> I(-(-(x,y),I(1()))) -(I(x),O(y)) -> I(-(x,y)) -(I(x),I(y)) -> O(-(x,y)) ge(O(x),O(y)) -> ge(x,y) ge(O(x),I(y)) -> not(ge(y,x)) ge(I(x),O(y)) -> ge(x,y) ge(I(x),I(y)) -> ge(x,y) ge(0(),O(x)) -> ge(0(),x) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + 2x1, [O](x0) = x0 + 1, [I](x0) = x0 + 1, [1] = 0, [0] = 0, [not](x0) = x0 + 2, [ge](x0, x1) = x0 + x1 + 7, [+](x0, x1) = x0 + x1 orientation: O(0()) = 1 >= 0 = 0() +(0(),x) = x >= x = x +(x,0()) = x >= x = x +(O(x),O(y)) = x + y + 2 >= x + y + 1 = O(+(x,y)) +(O(x),I(y)) = x + y + 2 >= x + y + 1 = I(+(x,y)) +(I(x),O(y)) = x + y + 2 >= x + y + 1 = I(+(x,y)) +(I(x),I(y)) = x + y + 2 >= x + y + 2 = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = 2x >= 0 = 0() -(O(x),O(y)) = x + 2y + 3 >= x + 2y + 1 = O(-(x,y)) -(O(x),I(y)) = x + 2y + 3 >= x + 2y + 3 = I(-(-(x,y),I(1()))) -(I(x),O(y)) = x + 2y + 3 >= x + 2y + 1 = I(-(x,y)) -(I(x),I(y)) = x + 2y + 3 >= x + 2y + 1 = O(-(x,y)) ge(O(x),O(y)) = x + y + 9 >= x + y + 7 = ge(x,y) ge(O(x),I(y)) = x + y + 9 >= x + y + 9 = not(ge(y,x)) ge(I(x),O(y)) = x + y + 9 >= x + y + 7 = ge(x,y) ge(I(x),I(y)) = x + y + 9 >= x + y + 7 = ge(x,y) ge(0(),O(x)) = x + 8 >= x + 7 = ge(0(),x) problem: +(0(),x) -> x +(x,0()) -> x +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),I(y)) -> I(-(-(x,y),I(1()))) ge(O(x),I(y)) -> not(ge(y,x)) Matrix Interpretation Processor: dim=1 interpretation: [-](x0, x1) = x0 + 4x1, [O](x0) = x0 + 2, [I](x0) = x0 + 2, [1] = 0, [0] = 0, [not](x0) = x0, [ge](x0, x1) = 4x0 + 4x1, [+](x0, x1) = x0 + 4x1 orientation: +(0(),x) = 4x >= x = x +(x,0()) = x >= x = x +(I(x),I(y)) = x + 4y + 10 >= x + 4y + 10 = O(+(+(x,y),I(0()))) +(x,+(y,z)) = x + 4y + 16z >= x + 4y + 4z = +(+(x,y),z) -(x,0()) = x >= x = x -(0(),x) = 4x >= 0 = 0() -(O(x),I(y)) = x + 4y + 10 >= x + 4y + 10 = I(-(-(x,y),I(1()))) ge(O(x),I(y)) = 4x + 4y + 16 >= 4x + 4y = not(ge(y,x)) problem: +(0(),x) -> x +(x,0()) -> x +(I(x),I(y)) -> O(+(+(x,y),I(0()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,0()) -> x -(0(),x) -> 0() -(O(x),I(y)) -> I(-(-(x,y),I(1()))) WPO Processor: algebra: Sum weight function: w0 = 0 w(I) = w(O) = 1 w(1) = w(-) = w(+) = w(0) = 0 status function: st(+) = [1, 0] st(1) = [] st(-) = [0, 1] st(I) = st(O) = [0] st(0) = [] precedence: + > - > 1 ~ I ~ O ~ 0 problem: Qed