/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem: +(0(),0()) -> 0() +(0(),1()) -> 1() +(0(),2()) -> 2() +(0(),3()) -> 3() +(0(),4()) -> 4() +(0(),5()) -> 5() +(0(),6()) -> 6() +(0(),7()) -> 7() +(0(),8()) -> 8() +(0(),9()) -> 9() +(1(),0()) -> 1() +(1(),1()) -> 2() +(1(),2()) -> 3() +(1(),3()) -> 4() +(1(),4()) -> 5() +(1(),5()) -> 6() +(1(),6()) -> 7() +(1(),7()) -> 8() +(1(),8()) -> 9() +(1(),9()) -> c(1(),0()) +(2(),0()) -> 2() +(2(),1()) -> 3() +(2(),2()) -> 4() +(2(),3()) -> 5() +(2(),4()) -> 6() +(2(),5()) -> 7() +(2(),6()) -> 8() +(2(),7()) -> 9() +(2(),8()) -> c(1(),0()) +(2(),9()) -> c(1(),1()) +(3(),0()) -> 3() +(3(),1()) -> 4() +(3(),2()) -> 5() +(3(),3()) -> 6() +(3(),4()) -> 7() +(3(),5()) -> 8() +(3(),6()) -> 9() +(3(),7()) -> c(1(),0()) +(3(),8()) -> c(1(),1()) +(3(),9()) -> c(1(),2()) +(4(),0()) -> 4() +(4(),1()) -> 5() +(4(),2()) -> 6() +(4(),3()) -> 7() +(4(),4()) -> 8() +(4(),5()) -> 9() +(4(),6()) -> c(1(),0()) +(4(),7()) -> c(1(),1()) +(4(),8()) -> c(1(),2()) +(4(),9()) -> c(1(),3()) +(5(),0()) -> 5() +(5(),1()) -> 6() +(5(),2()) -> 7() +(5(),3()) -> 8() +(5(),4()) -> 9() +(5(),5()) -> c(1(),0()) +(5(),6()) -> c(1(),1()) +(5(),7()) -> c(1(),2()) +(5(),8()) -> c(1(),3()) +(5(),9()) -> c(1(),4()) +(6(),0()) -> 6() +(6(),1()) -> 7() +(6(),2()) -> 8() +(6(),3()) -> 9() +(6(),4()) -> c(1(),0()) +(6(),5()) -> c(1(),1()) +(6(),6()) -> c(1(),2()) +(6(),7()) -> c(1(),3()) +(6(),8()) -> c(1(),4()) +(6(),9()) -> c(1(),5()) +(7(),0()) -> 7() +(7(),1()) -> 8() +(7(),2()) -> 9() +(7(),3()) -> c(1(),0()) +(7(),4()) -> c(1(),1()) +(7(),5()) -> c(1(),2()) +(7(),6()) -> c(1(),3()) +(7(),7()) -> c(1(),4()) +(7(),8()) -> c(1(),5()) +(7(),9()) -> c(1(),6()) +(8(),0()) -> 8() +(8(),1()) -> 9() +(8(),2()) -> c(1(),0()) +(8(),3()) -> c(1(),1()) +(8(),4()) -> c(1(),2()) +(8(),5()) -> c(1(),3()) +(8(),6()) -> c(1(),4()) +(8(),7()) -> c(1(),5()) +(8(),8()) -> c(1(),6()) +(8(),9()) -> c(1(),7()) +(9(),0()) -> 9() +(9(),1()) -> c(1(),0()) +(9(),2()) -> c(1(),1()) +(9(),3()) -> c(1(),2()) +(9(),4()) -> c(1(),3()) +(9(),5()) -> c(1(),4()) +(9(),6()) -> c(1(),5()) +(9(),7()) -> c(1(),6()) +(9(),8()) -> c(1(),7()) +(9(),9()) -> c(1(),8()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(0(),x) -> x c(x,c(y,z)) -> c(+(x,y),z) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [c](x0, x1) = [0 0 0]x0 + x1 [0 0 0] , [0] [9] = [1] [0], [0] [8] = [1] [0], [0] [7] = [1] [0], [0] [6] = [1] [0], [0] [5] = [1] [0], [0] [4] = [1] [0], [0] [3] = [1] [0], [0] [2] = [1] [0], [0] [1] = [1] [0], [1 1 0] [1 0 0] [+](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 1 0] , [0] [0] = [1] [0] orientation: [1] [0] +(0(),0()) = [1] >= [1] = 0() [1] [0] [1] [0] +(0(),1()) = [1] >= [1] = 1() [1] [0] [1] [0] +(0(),2()) = [1] >= [1] = 2() [1] [0] [1] [0] +(0(),3()) = [1] >= [1] = 3() [1] [0] [1] [0] +(0(),4()) = [1] >= [1] = 4() [1] [0] [1] [0] +(0(),5()) = [1] >= [1] = 5() [1] [0] [1] [0] +(0(),6()) = [1] >= [1] = 6() [1] [0] [1] [0] +(0(),7()) = [1] >= [1] = 7() [1] [0] [1] [0] +(0(),8()) = [1] >= [1] = 8() [1] [0] [1] [0] +(0(),9()) = [1] >= [1] = 9() [1] [0] [1] [0] +(1(),0()) = [1] >= [1] = 1() [1] [0] [1] [0] +(1(),1()) = [1] >= [1] = 2() [1] [0] [1] [0] +(1(),2()) = [1] >= [1] = 3() [1] [0] [1] [0] +(1(),3()) = [1] >= [1] = 4() [1] [0] [1] [0] +(1(),4()) = [1] >= [1] = 5() [1] [0] [1] [0] +(1(),5()) = [1] >= [1] = 6() [1] [0] [1] [0] +(1(),6()) = [1] >= [1] = 7() [1] [0] [1] [0] +(1(),7()) = [1] >= [1] = 8() [1] [0] [1] [0] +(1(),8()) = [1] >= [1] = 9() [1] [0] [1] [1] +(1(),9()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [0] +(2(),0()) = [1] >= [1] = 2() [1] [0] [1] [0] +(2(),1()) = [1] >= [1] = 3() [1] [0] [1] [0] +(2(),2()) = [1] >= [1] = 4() [1] [0] [1] [0] +(2(),3()) = [1] >= [1] = 5() [1] [0] [1] [0] +(2(),4()) = [1] >= [1] = 6() [1] [0] [1] [0] +(2(),5()) = [1] >= [1] = 7() [1] [0] [1] [0] +(2(),6()) = [1] >= [1] = 8() [1] [0] [1] [0] +(2(),7()) = [1] >= [1] = 9() [1] [0] [1] [1] +(2(),8()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(2(),9()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [0] +(3(),0()) = [1] >= [1] = 3() [1] [0] [1] [0] +(3(),1()) = [1] >= [1] = 4() [1] [0] [1] [0] +(3(),2()) = [1] >= [1] = 5() [1] [0] [1] [0] +(3(),3()) = [1] >= [1] = 6() [1] [0] [1] [0] +(3(),4()) = [1] >= [1] = 7() [1] [0] [1] [0] +(3(),5()) = [1] >= [1] = 8() [1] [0] [1] [0] +(3(),6()) = [1] >= [1] = 9() [1] [0] [1] [1] +(3(),7()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(3(),8()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(3(),9()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [0] +(4(),0()) = [1] >= [1] = 4() [1] [0] [1] [0] +(4(),1()) = [1] >= [1] = 5() [1] [0] [1] [0] +(4(),2()) = [1] >= [1] = 6() [1] [0] [1] [0] +(4(),3()) = [1] >= [1] = 7() [1] [0] [1] [0] +(4(),4()) = [1] >= [1] = 8() [1] [0] [1] [0] +(4(),5()) = [1] >= [1] = 9() [1] [0] [1] [1] +(4(),6()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(4(),7()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(4(),8()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [1] +(4(),9()) = [1] >= [1] = c(1(),3()) [1] [0] [1] [0] +(5(),0()) = [1] >= [1] = 5() [1] [0] [1] [0] +(5(),1()) = [1] >= [1] = 6() [1] [0] [1] [0] +(5(),2()) = [1] >= [1] = 7() [1] [0] [1] [0] +(5(),3()) = [1] >= [1] = 8() [1] [0] [1] [0] +(5(),4()) = [1] >= [1] = 9() [1] [0] [1] [1] +(5(),5()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(5(),6()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(5(),7()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [1] +(5(),8()) = [1] >= [1] = c(1(),3()) [1] [0] [1] [1] +(5(),9()) = [1] >= [1] = c(1(),4()) [1] [0] [1] [0] +(6(),0()) = [1] >= [1] = 6() [1] [0] [1] [0] +(6(),1()) = [1] >= [1] = 7() [1] [0] [1] [0] +(6(),2()) = [1] >= [1] = 8() [1] [0] [1] [0] +(6(),3()) = [1] >= [1] = 9() [1] [0] [1] [1] +(6(),4()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(6(),5()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(6(),6()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [1] +(6(),7()) = [1] >= [1] = c(1(),3()) [1] [0] [1] [1] +(6(),8()) = [1] >= [1] = c(1(),4()) [1] [0] [1] [1] +(6(),9()) = [1] >= [1] = c(1(),5()) [1] [0] [1] [0] +(7(),0()) = [1] >= [1] = 7() [1] [0] [1] [0] +(7(),1()) = [1] >= [1] = 8() [1] [0] [1] [0] +(7(),2()) = [1] >= [1] = 9() [1] [0] [1] [1] +(7(),3()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(7(),4()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(7(),5()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [1] +(7(),6()) = [1] >= [1] = c(1(),3()) [1] [0] [1] [1] +(7(),7()) = [1] >= [1] = c(1(),4()) [1] [0] [1] [1] +(7(),8()) = [1] >= [1] = c(1(),5()) [1] [0] [1] [1] +(7(),9()) = [1] >= [1] = c(1(),6()) [1] [0] [1] [0] +(8(),0()) = [1] >= [1] = 8() [1] [0] [1] [0] +(8(),1()) = [1] >= [1] = 9() [1] [0] [1] [1] +(8(),2()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(8(),3()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(8(),4()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [1] +(8(),5()) = [1] >= [1] = c(1(),3()) [1] [0] [1] [1] +(8(),6()) = [1] >= [1] = c(1(),4()) [1] [0] [1] [1] +(8(),7()) = [1] >= [1] = c(1(),5()) [1] [0] [1] [1] +(8(),8()) = [1] >= [1] = c(1(),6()) [1] [0] [1] [1] +(8(),9()) = [1] >= [1] = c(1(),7()) [1] [0] [1] [0] +(9(),0()) = [1] >= [1] = 9() [1] [0] [1] [1] +(9(),1()) = [1] >= [1] = c(1(),0()) [1] [0] [1] [1] +(9(),2()) = [1] >= [1] = c(1(),1()) [1] [0] [1] [1] +(9(),3()) = [1] >= [1] = c(1(),2()) [1] [0] [1] [1] +(9(),4()) = [1] >= [1] = c(1(),3()) [1] [0] [1] [1] +(9(),5()) = [1] >= [1] = c(1(),4()) [1] [0] [1] [1] +(9(),6()) = [1] >= [1] = c(1(),5()) [1] [0] [1] [1] +(9(),7()) = [1] >= [1] = c(1(),6()) [1] [0] [1] [1] +(9(),8()) = [1] >= [1] = c(1(),7()) [1] [0] [1] [1] +(9(),9()) = [1] >= [1] = c(1(),8()) [1] [0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] +(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 1 0]z >= [0 0 0]x + [0 0 0]y + [0 1 0]z = c(y,+(x,z)) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] +(c(x,y),z) = [0 0 0]x + [0 0 0]y + [0 1 0]z >= [0 0 0]x + [0 0 0]y + [0 1 0]z = c(x,+(y,z)) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1] c(0(),x) = x + [0] >= x = x [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] c(x,c(y,z)) = [0 0 0]x + [0 0 0]y + z >= [0 0 0]x + [0 0 0]y + z = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: +(1(),9()) -> c(1(),0()) +(2(),8()) -> c(1(),0()) +(2(),9()) -> c(1(),1()) +(3(),7()) -> c(1(),0()) +(3(),8()) -> c(1(),1()) +(3(),9()) -> c(1(),2()) +(4(),6()) -> c(1(),0()) +(4(),7()) -> c(1(),1()) +(4(),8()) -> c(1(),2()) +(4(),9()) -> c(1(),3()) +(5(),5()) -> c(1(),0()) +(5(),6()) -> c(1(),1()) +(5(),7()) -> c(1(),2()) +(5(),8()) -> c(1(),3()) +(5(),9()) -> c(1(),4()) +(6(),4()) -> c(1(),0()) +(6(),5()) -> c(1(),1()) +(6(),6()) -> c(1(),2()) +(6(),7()) -> c(1(),3()) +(6(),8()) -> c(1(),4()) +(6(),9()) -> c(1(),5()) +(7(),3()) -> c(1(),0()) +(7(),4()) -> c(1(),1()) +(7(),5()) -> c(1(),2()) +(7(),6()) -> c(1(),3()) +(7(),7()) -> c(1(),4()) +(7(),8()) -> c(1(),5()) +(7(),9()) -> c(1(),6()) +(8(),2()) -> c(1(),0()) +(8(),3()) -> c(1(),1()) +(8(),4()) -> c(1(),2()) +(8(),5()) -> c(1(),3()) +(8(),6()) -> c(1(),4()) +(8(),7()) -> c(1(),5()) +(8(),8()) -> c(1(),6()) +(8(),9()) -> c(1(),7()) +(9(),1()) -> c(1(),0()) +(9(),2()) -> c(1(),1()) +(9(),3()) -> c(1(),2()) +(9(),4()) -> c(1(),3()) +(9(),5()) -> c(1(),4()) +(9(),6()) -> c(1(),5()) +(9(),7()) -> c(1(),6()) +(9(),8()) -> c(1(),7()) +(9(),9()) -> c(1(),8()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [c](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [0] [9] = [0] [1], [0] [8] = [0] [0], [0] [7] = [0] [0], [0] [6] = [0] [0], [0] [5] = [0] [0], [0] [4] = [0] [0], [0] [3] = [0] [0], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 1] [1 0 0] [0] [+](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0] orientation: [0] [0] +(1(),9()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(2(),8()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(2(),9()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(3(),7()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(3(),8()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(3(),9()) = [1] >= [0] = c(1(),2()) [0] [0] [0] [0] +(4(),6()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(4(),7()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(4(),8()) = [1] >= [0] = c(1(),2()) [0] [0] [0] [0] +(4(),9()) = [1] >= [0] = c(1(),3()) [0] [0] [0] [0] +(5(),5()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(5(),6()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(5(),7()) = [1] >= [0] = c(1(),2()) [0] [0] [0] [0] +(5(),8()) = [1] >= [0] = c(1(),3()) [0] [0] [0] [0] +(5(),9()) = [1] >= [0] = c(1(),4()) [0] [0] [0] [0] +(6(),4()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(6(),5()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(6(),6()) = [1] >= [0] = c(1(),2()) [0] [0] [0] [0] +(6(),7()) = [1] >= [0] = c(1(),3()) [0] [0] [0] [0] +(6(),8()) = [1] >= [0] = c(1(),4()) [0] [0] [0] [0] +(6(),9()) = [1] >= [0] = c(1(),5()) [0] [0] [0] [0] +(7(),3()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(7(),4()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(7(),5()) = [1] >= [0] = c(1(),2()) [0] [0] [0] [0] +(7(),6()) = [1] >= [0] = c(1(),3()) [0] [0] [0] [0] +(7(),7()) = [1] >= [0] = c(1(),4()) [0] [0] [0] [0] +(7(),8()) = [1] >= [0] = c(1(),5()) [0] [0] [0] [0] +(7(),9()) = [1] >= [0] = c(1(),6()) [0] [0] [0] [0] +(8(),2()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(8(),3()) = [1] >= [0] = c(1(),1()) [0] [0] [0] [0] +(8(),4()) = [1] >= [0] = c(1(),2()) [0] [0] [0] [0] +(8(),5()) = [1] >= [0] = c(1(),3()) [0] [0] [0] [0] +(8(),6()) = [1] >= [0] = c(1(),4()) [0] [0] [0] [0] +(8(),7()) = [1] >= [0] = c(1(),5()) [0] [0] [0] [0] +(8(),8()) = [1] >= [0] = c(1(),6()) [0] [0] [0] [0] +(8(),9()) = [1] >= [0] = c(1(),7()) [0] [0] [1] [0] +(9(),1()) = [1] >= [0] = c(1(),0()) [0] [0] [1] [0] +(9(),2()) = [1] >= [0] = c(1(),1()) [0] [0] [1] [0] +(9(),3()) = [1] >= [0] = c(1(),2()) [0] [0] [1] [0] +(9(),4()) = [1] >= [0] = c(1(),3()) [0] [0] [1] [0] +(9(),5()) = [1] >= [0] = c(1(),4()) [0] [0] [1] [0] +(9(),6()) = [1] >= [0] = c(1(),5()) [0] [0] [1] [0] +(9(),7()) = [1] >= [0] = c(1(),6()) [0] [0] [1] [0] +(9(),8()) = [1] >= [0] = c(1(),7()) [0] [0] [1] [0] +(9(),9()) = [1] >= [0] = c(1(),8()) [0] [0] [1 0 1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 1] [1 0 0] +(x,c(y,z)) = [1 0 0]x + [0 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(y,+(x,z)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 1] [1 0 0] +(c(x,y),z) = [1 0 1]x + [1 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(x,+(y,z)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] c(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] problem: +(1(),9()) -> c(1(),0()) +(2(),8()) -> c(1(),0()) +(2(),9()) -> c(1(),1()) +(3(),7()) -> c(1(),0()) +(3(),8()) -> c(1(),1()) +(3(),9()) -> c(1(),2()) +(4(),6()) -> c(1(),0()) +(4(),7()) -> c(1(),1()) +(4(),8()) -> c(1(),2()) +(4(),9()) -> c(1(),3()) +(5(),5()) -> c(1(),0()) +(5(),6()) -> c(1(),1()) +(5(),7()) -> c(1(),2()) +(5(),8()) -> c(1(),3()) +(5(),9()) -> c(1(),4()) +(6(),4()) -> c(1(),0()) +(6(),5()) -> c(1(),1()) +(6(),6()) -> c(1(),2()) +(6(),7()) -> c(1(),3()) +(6(),8()) -> c(1(),4()) +(6(),9()) -> c(1(),5()) +(7(),3()) -> c(1(),0()) +(7(),4()) -> c(1(),1()) +(7(),5()) -> c(1(),2()) +(7(),6()) -> c(1(),3()) +(7(),7()) -> c(1(),4()) +(7(),8()) -> c(1(),5()) +(7(),9()) -> c(1(),6()) +(8(),2()) -> c(1(),0()) +(8(),3()) -> c(1(),1()) +(8(),4()) -> c(1(),2()) +(8(),5()) -> c(1(),3()) +(8(),6()) -> c(1(),4()) +(8(),7()) -> c(1(),5()) +(8(),8()) -> c(1(),6()) +(8(),9()) -> c(1(),7()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [0] [c](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 1 1] [0], [1] [9] = [0] [0], [0] [8] = [0] [0], [0] [7] = [0] [0], [0] [6] = [0] [0], [0] [5] = [0] [0], [0] [4] = [0] [0], [0] [3] = [0] [0], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 0] [1 0 0] [0] [+](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 1 1] [0 1 1] [0], [0] [0] = [0] [0] orientation: [1] [0] +(1(),9()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(2(),8()) = [1] >= [1] = c(1(),0()) [0] [0] [1] [0] +(2(),9()) = [1] >= [1] = c(1(),1()) [0] [0] [0] [0] +(3(),7()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(3(),8()) = [1] >= [1] = c(1(),1()) [0] [0] [1] [0] +(3(),9()) = [1] >= [1] = c(1(),2()) [0] [0] [0] [0] +(4(),6()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(4(),7()) = [1] >= [1] = c(1(),1()) [0] [0] [0] [0] +(4(),8()) = [1] >= [1] = c(1(),2()) [0] [0] [1] [0] +(4(),9()) = [1] >= [1] = c(1(),3()) [0] [0] [0] [0] +(5(),5()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(5(),6()) = [1] >= [1] = c(1(),1()) [0] [0] [0] [0] +(5(),7()) = [1] >= [1] = c(1(),2()) [0] [0] [0] [0] +(5(),8()) = [1] >= [1] = c(1(),3()) [0] [0] [1] [0] +(5(),9()) = [1] >= [1] = c(1(),4()) [0] [0] [0] [0] +(6(),4()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(6(),5()) = [1] >= [1] = c(1(),1()) [0] [0] [0] [0] +(6(),6()) = [1] >= [1] = c(1(),2()) [0] [0] [0] [0] +(6(),7()) = [1] >= [1] = c(1(),3()) [0] [0] [0] [0] +(6(),8()) = [1] >= [1] = c(1(),4()) [0] [0] [1] [0] +(6(),9()) = [1] >= [1] = c(1(),5()) [0] [0] [0] [0] +(7(),3()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(7(),4()) = [1] >= [1] = c(1(),1()) [0] [0] [0] [0] +(7(),5()) = [1] >= [1] = c(1(),2()) [0] [0] [0] [0] +(7(),6()) = [1] >= [1] = c(1(),3()) [0] [0] [0] [0] +(7(),7()) = [1] >= [1] = c(1(),4()) [0] [0] [0] [0] +(7(),8()) = [1] >= [1] = c(1(),5()) [0] [0] [1] [0] +(7(),9()) = [1] >= [1] = c(1(),6()) [0] [0] [0] [0] +(8(),2()) = [1] >= [1] = c(1(),0()) [0] [0] [0] [0] +(8(),3()) = [1] >= [1] = c(1(),1()) [0] [0] [0] [0] +(8(),4()) = [1] >= [1] = c(1(),2()) [0] [0] [0] [0] +(8(),5()) = [1] >= [1] = c(1(),3()) [0] [0] [0] [0] +(8(),6()) = [1] >= [1] = c(1(),4()) [0] [0] [0] [0] +(8(),7()) = [1] >= [1] = c(1(),5()) [0] [0] [0] [0] +(8(),8()) = [1] >= [1] = c(1(),6()) [0] [0] [1] [0] +(8(),9()) = [1] >= [1] = c(1(),7()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] +(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] = c(y,+(x,z)) [0 1 1] [0 0 0] [0 1 1] [1] [0 1 1] [0 0 0] [0 1 1] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] +(c(x,y),z) = [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] = c(x,+(y,z)) [0 0 0] [0 1 1] [0 1 1] [1] [0 0 0] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] c(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] = c(+(x,y),z) [0 0 0] [0 0 0] [0 1 1] [1] [0 0 0] [0 0 0] [0 1 1] [0] problem: +(2(),8()) -> c(1(),0()) +(3(),7()) -> c(1(),0()) +(3(),8()) -> c(1(),1()) +(4(),6()) -> c(1(),0()) +(4(),7()) -> c(1(),1()) +(4(),8()) -> c(1(),2()) +(5(),5()) -> c(1(),0()) +(5(),6()) -> c(1(),1()) +(5(),7()) -> c(1(),2()) +(5(),8()) -> c(1(),3()) +(6(),4()) -> c(1(),0()) +(6(),5()) -> c(1(),1()) +(6(),6()) -> c(1(),2()) +(6(),7()) -> c(1(),3()) +(6(),8()) -> c(1(),4()) +(7(),3()) -> c(1(),0()) +(7(),4()) -> c(1(),1()) +(7(),5()) -> c(1(),2()) +(7(),6()) -> c(1(),3()) +(7(),7()) -> c(1(),4()) +(7(),8()) -> c(1(),5()) +(8(),2()) -> c(1(),0()) +(8(),3()) -> c(1(),1()) +(8(),4()) -> c(1(),2()) +(8(),5()) -> c(1(),3()) +(8(),6()) -> c(1(),4()) +(8(),7()) -> c(1(),5()) +(8(),8()) -> c(1(),6()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=1 interpretation: [c](x0, x1) = x0 + x1, [8] = 0, [7] = 4, [6] = 0, [5] = 4, [4] = 0, [3] = 2, [2] = 0, [1] = 0, [+](x0, x1) = x0 + x1, [0] = 0 orientation: +(2(),8()) = 0 >= 0 = c(1(),0()) +(3(),7()) = 6 >= 0 = c(1(),0()) +(3(),8()) = 2 >= 0 = c(1(),1()) +(4(),6()) = 0 >= 0 = c(1(),0()) +(4(),7()) = 4 >= 0 = c(1(),1()) +(4(),8()) = 0 >= 0 = c(1(),2()) +(5(),5()) = 8 >= 0 = c(1(),0()) +(5(),6()) = 4 >= 0 = c(1(),1()) +(5(),7()) = 8 >= 0 = c(1(),2()) +(5(),8()) = 4 >= 2 = c(1(),3()) +(6(),4()) = 0 >= 0 = c(1(),0()) +(6(),5()) = 4 >= 0 = c(1(),1()) +(6(),6()) = 0 >= 0 = c(1(),2()) +(6(),7()) = 4 >= 2 = c(1(),3()) +(6(),8()) = 0 >= 0 = c(1(),4()) +(7(),3()) = 6 >= 0 = c(1(),0()) +(7(),4()) = 4 >= 0 = c(1(),1()) +(7(),5()) = 8 >= 0 = c(1(),2()) +(7(),6()) = 4 >= 2 = c(1(),3()) +(7(),7()) = 8 >= 0 = c(1(),4()) +(7(),8()) = 4 >= 4 = c(1(),5()) +(8(),2()) = 0 >= 0 = c(1(),0()) +(8(),3()) = 2 >= 0 = c(1(),1()) +(8(),4()) = 0 >= 0 = c(1(),2()) +(8(),5()) = 4 >= 2 = c(1(),3()) +(8(),6()) = 0 >= 0 = c(1(),4()) +(8(),7()) = 4 >= 4 = c(1(),5()) +(8(),8()) = 0 >= 0 = c(1(),6()) +(x,c(y,z)) = x + y + z >= x + y + z = c(y,+(x,z)) +(c(x,y),z) = x + y + z >= x + y + z = c(x,+(y,z)) c(x,c(y,z)) = x + y + z >= x + y + z = c(+(x,y),z) problem: +(2(),8()) -> c(1(),0()) +(4(),6()) -> c(1(),0()) +(4(),8()) -> c(1(),2()) +(6(),4()) -> c(1(),0()) +(6(),6()) -> c(1(),2()) +(6(),8()) -> c(1(),4()) +(7(),8()) -> c(1(),5()) +(8(),2()) -> c(1(),0()) +(8(),4()) -> c(1(),2()) +(8(),6()) -> c(1(),4()) +(8(),7()) -> c(1(),5()) +(8(),8()) -> c(1(),6()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 1] [c](x0, x1) = [0 0 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [0] [8] = [0] [0], [1] [7] = [0] [0], [0] [6] = [0] [0], [0] [5] = [0] [0], [0] [4] = [0] [0], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 0] [1 0 0] [+](x0, x1) = [1 0 1]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0] orientation: [0] [0] +(2(),8()) = [0] >= [0] = c(1(),0()) [0] [0] [0] [0] +(4(),6()) = [0] >= [0] = c(1(),0()) [0] [0] [0] [0] +(4(),8()) = [0] >= [0] = c(1(),2()) [0] [0] [0] [0] +(6(),4()) = [0] >= [0] = c(1(),0()) [0] [0] [0] [0] +(6(),6()) = [0] >= [0] = c(1(),2()) [0] [0] [0] [0] +(6(),8()) = [0] >= [0] = c(1(),4()) [0] [0] [1] [0] +(7(),8()) = [1] >= [0] = c(1(),5()) [0] [0] [0] [0] +(8(),2()) = [0] >= [0] = c(1(),0()) [0] [0] [0] [0] +(8(),4()) = [0] >= [0] = c(1(),2()) [0] [0] [0] [0] +(8(),6()) = [0] >= [0] = c(1(),4()) [0] [0] [1] [0] +(8(),7()) = [1] >= [0] = c(1(),5()) [0] [0] [0] [0] +(8(),8()) = [0] >= [0] = c(1(),6()) [0] [0] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] +(x,c(y,z)) = [1 0 1]x + [1 0 1]y + [1 0 1]z >= [1 0 0]x + [0 0 0]y + [1 0 0]z = c(y,+(x,z)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] +(c(x,y),z) = [1 0 1]x + [1 0 1]y + [1 0 0]z >= [0 0 0]x + [1 0 0]y + [1 0 0]z = c(x,+(y,z)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 1] c(x,c(y,z)) = [0 0 0]x + [1 0 1]y + [1 0 1]z >= [0 0 0]x + [0 0 0]y + [1 0 0]z = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: +(2(),8()) -> c(1(),0()) +(4(),6()) -> c(1(),0()) +(4(),8()) -> c(1(),2()) +(6(),4()) -> c(1(),0()) +(6(),6()) -> c(1(),2()) +(6(),8()) -> c(1(),4()) +(8(),2()) -> c(1(),0()) +(8(),4()) -> c(1(),2()) +(8(),6()) -> c(1(),4()) +(8(),8()) -> c(1(),6()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1 0 1] [0] [c](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [8] = [1] [0], [0] [6] = [0] [0], [0] [4] = [0] [0], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 1] [1 1 1] [0] [+](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [0] = [0] [0] orientation: [1] [0] +(2(),8()) = [0] >= [0] = c(1(),0()) [1] [1] [0] [0] +(4(),6()) = [0] >= [0] = c(1(),0()) [1] [1] [1] [0] +(4(),8()) = [0] >= [0] = c(1(),2()) [1] [1] [0] [0] +(6(),4()) = [0] >= [0] = c(1(),0()) [1] [1] [0] [0] +(6(),6()) = [0] >= [0] = c(1(),2()) [1] [1] [1] [0] +(6(),8()) = [0] >= [0] = c(1(),4()) [1] [1] [0] [0] +(8(),2()) = [0] >= [0] = c(1(),0()) [1] [1] [0] [0] +(8(),4()) = [0] >= [0] = c(1(),2()) [1] [1] [0] [0] +(8(),6()) = [0] >= [0] = c(1(),4()) [1] [1] [1] [0] +(8(),8()) = [0] >= [0] = c(1(),6()) [1] [1] [1 0 1] [1 1 1] [1 1 1] [1] [1 0 1] [1 1 1] [1 1 1] [1] +(x,c(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 + [0] = c(y,+(x,z)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 1 1] [1 0 1] [1 1 1] [1] [1 1 1] [1 0 1] [1 1 1] [1] +(c(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 + [0] = c(x,+(y,z)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 1 1] [1 1 1] [1 0 1] [1] [1 0 1] [1 1 1] [1 0 1] [1] c(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 1 0]z + [0] >= [0 0 0]x + [0 0 0]y + [0 1 0]z + [0] = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] problem: +(4(),6()) -> c(1(),0()) +(6(),4()) -> c(1(),0()) +(6(),6()) -> c(1(),2()) +(8(),2()) -> c(1(),0()) +(8(),4()) -> c(1(),2()) +(8(),6()) -> c(1(),4()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [c](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1] [8] = [0] [0], [0] [6] = [0] [0], [0] [4] = [0] [0], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 0] [1 0 0] [0] [+](x0, x1) = [0 0 1]x0 + [1 0 0]x1 + [1] [0 0 1] [0 0 0] [0], [0] [0] = [0] [0] orientation: [0] [0] +(4(),6()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(6(),4()) = [1] >= [0] = c(1(),0()) [0] [0] [0] [0] +(6(),6()) = [1] >= [0] = c(1(),2()) [0] [0] [1] [0] +(8(),2()) = [1] >= [0] = c(1(),0()) [0] [0] [1] [0] +(8(),4()) = [1] >= [0] = c(1(),2()) [0] [0] [1] [0] +(8(),6()) = [1] >= [0] = c(1(),4()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] +(x,c(y,z)) = [0 0 1]x + [1 0 1]y + [1 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(y,+(x,z)) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [0] [1 0 1] [1 0 0] [1 0 0] +(c(x,y),z) = [0 0 0]x + [0 0 0]y + [1 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(x,+(y,z)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] c(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: +(4(),6()) -> c(1(),0()) +(6(),4()) -> c(1(),0()) +(6(),6()) -> c(1(),2()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 1] [0] [c](x0, x1) = [0 0 0]x0 + [1 0 1]x1 + [0] [0 0 0] [0 0 0] [1], [0] [6] = [0] [0], [0] [4] = [0] [1], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 1] [1 0 1] [0] [+](x0, x1) = [1 0 1]x0 + [1 0 1]x1 + [0] [0 0 0] [0 0 0] [1], [0] [0] = [0] [0] orientation: [1] [0] +(4(),6()) = [1] >= [0] = c(1(),0()) [1] [1] [1] [0] +(6(),4()) = [1] >= [0] = c(1(),0()) [1] [1] [0] [0] +(6(),6()) = [0] >= [0] = c(1(),2()) [1] [1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] +(x,c(y,z)) = [1 0 1]x + [1 0 1]y + [1 0 1]z + [1] >= [1 0 1]x + [0 0 0]y + [1 0 1]z + [1] = c(y,+(x,z)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] +(c(x,y),z) = [1 0 1]x + [1 0 1]y + [1 0 1]z + [1] >= [0 0 0]x + [1 0 1]y + [1 0 1]z + [1] = c(x,+(y,z)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] c(x,c(y,z)) = [0 0 0]x + [1 0 1]y + [1 0 1]z + [1] >= [0 0 0]x + [0 0 0]y + [1 0 1]z + [0] = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] problem: +(6(),6()) -> c(1(),2()) +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [c](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [0] [6] = [0] [1], [0] [2] = [0] [0], [0] [1] = [0] [0], [1 0 1] [1 0 0] [+](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1] [0] +(6(),6()) = [0] >= [0] = c(1(),2()) [0] [0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] +(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(y,+(x,z)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] +(c(x,y),z) = [0 0 0]x + [0 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(x,+(y,z)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] c(x,c(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = c(+(x,y),z) [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] problem: +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) c(x,c(y,z)) -> c(+(x,y),z) Matrix Interpretation Processor: dim=1 interpretation: [c](x0, x1) = x0 + x1 + 2, [+](x0, x1) = x0 + x1 orientation: +(x,c(y,z)) = x + y + z + 2 >= x + y + z + 2 = c(y,+(x,z)) +(c(x,y),z) = x + y + z + 2 >= x + y + z + 2 = c(x,+(y,z)) c(x,c(y,z)) = x + y + z + 4 >= x + y + z + 2 = c(+(x,y),z) problem: +(x,c(y,z)) -> c(y,+(x,z)) +(c(x,y),z) -> c(x,+(y,z)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [c](x0, x1) = [0 0 1]x0 + x1 + [0] [0 0 0] [1], [1 0 1] [1 0 1] [+](x0, x1) = [1 0 0]x0 + [0 1 0]x1 [0 1 1] [0 0 1] orientation: [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] +(x,c(y,z)) = [1 0 0]x + [0 0 1]y + [0 1 0]z + [0] >= [1 0 0]x + [0 0 1]y + [0 1 0]z + [0] = c(y,+(x,z)) [0 1 1] [0 0 0] [0 0 1] [1] [0 1 1] [0 0 0] [0 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] +(c(x,y),z) = [1 0 1]x + [1 0 0]y + [0 1 0]z + [0] >= [0 0 1]x + [1 0 0]y + [0 1 0]z + [0] = c(x,+(y,z)) [0 0 1] [0 1 1] [0 0 1] [1] [0 0 0] [0 1 1] [0 0 1] [1] problem: Qed