YES Input TRS: 1: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) 2: active(U12(tt(),M,N)) -> mark(s(plus(N,M))) 3: active(plus(N,0())) -> mark(N) 4: active(plus(N,s(M))) -> mark(U11(tt(),M,N)) 5: active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) 6: active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) 7: active(s(X)) -> s(active(X)) 8: active(plus(X1,X2)) -> plus(active(X1),X2) 9: active(plus(X1,X2)) -> plus(X1,active(X2)) 10: U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) 11: U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) 12: s(mark(X)) -> mark(s(X)) 13: plus(mark(X1),X2) -> mark(plus(X1,X2)) 14: plus(X1,mark(X2)) -> mark(plus(X1,X2)) 15: proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) 16: proper(tt()) -> ok(tt()) 17: proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) 18: proper(s(X)) -> s(proper(X)) 19: proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) 20: proper(0()) -> ok(0()) 21: U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) 22: U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) 23: s(ok(X)) -> ok(s(X)) 24: plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) 25: top(mark(X)) -> top(proper(X)) 26: top(ok(X)) -> top(active(X)) Number of strict rules: 26 Direct POLO(bPol) ... removes: 3 U11 w: x1 + x2 + x3 s w: x1 + 3578 top w: x1 + 1 U12 w: x1 + x2 + x3 proper w: x1 ok w: x1 0 w: 0 mark w: x1 plus w: x1 + x2 + 1 active w: x1 tt w: 3579 Number of strict rules: 25 Direct POLO(bPol) ... removes: 4 1 25 14 13 2 U11 w: x1 + 2 * x2 + 2 * x3 + 23349 s w: x1 + 17342 top w: x1 + 2560 U12 w: x1 + 2 * x2 + 2 * x3 + 7810 proper w: x1 ok w: x1 0 w: 26666 mark w: x1 + 1 plus w: 2 * x1 + 2 * x2 + 1 active w: x1 tt w: 11334 Number of strict rules: 19 Direct POLO(bPol) ... removes: 10 11 U11 w: 2 * x1 + 2 * x2 + x3 + 73504 s w: x1 + 76518 top w: x1 + 33312 U12 w: 2 * x1 + 2 * x2 + x3 + 23715 proper w: x1 ok w: x1 0 w: 17573 mark w: x1 + 1 plus w: x1 + 2 * x2 + 1 active w: x1 tt w: 39766 Number of strict rules: 17 Direct POLO(bPol) ... failed. Uncurrying ... failed. Dependency Pairs: #1: #active(U12(X1,X2,X3)) -> #U12(active(X1),X2,X3) #2: #active(U12(X1,X2,X3)) -> #active(X1) #3: #active(plus(X1,X2)) -> #plus(X1,active(X2)) #4: #active(plus(X1,X2)) -> #active(X2) #5: #plus(ok(X1),ok(X2)) -> #plus(X1,X2) #6: #s(ok(X)) -> #s(X) #7: #s(mark(X)) -> #s(X) #8: #active(s(X)) -> #s(active(X)) #9: #active(s(X)) -> #active(X) #10: #active(U11(X1,X2,X3)) -> #U11(active(X1),X2,X3) #11: #active(U11(X1,X2,X3)) -> #active(X1) #12: #U12(ok(X1),ok(X2),ok(X3)) -> #U12(X1,X2,X3) #13: #proper(U12(X1,X2,X3)) -> #U12(proper(X1),proper(X2),proper(X3)) #14: #proper(U12(X1,X2,X3)) -> #proper(X1) #15: #proper(U12(X1,X2,X3)) -> #proper(X2) #16: #proper(U12(X1,X2,X3)) -> #proper(X3) #17: #proper(plus(X1,X2)) -> #plus(proper(X1),proper(X2)) #18: #proper(plus(X1,X2)) -> #proper(X1) #19: #proper(plus(X1,X2)) -> #proper(X2) #20: #top(ok(X)) -> #top(active(X)) #21: #top(ok(X)) -> #active(X) #22: #U11(ok(X1),ok(X2),ok(X3)) -> #U11(X1,X2,X3) #23: #active(plus(X1,X2)) -> #plus(active(X1),X2) #24: #active(plus(X1,X2)) -> #active(X1) #25: #proper(U11(X1,X2,X3)) -> #U11(proper(X1),proper(X2),proper(X3)) #26: #proper(U11(X1,X2,X3)) -> #proper(X1) #27: #proper(U11(X1,X2,X3)) -> #proper(X2) #28: #proper(U11(X1,X2,X3)) -> #proper(X3) #29: #proper(s(X)) -> #s(proper(X)) #30: #proper(s(X)) -> #proper(X) Number of SCCs: 7, DPs: 20 SCC { #20 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 1 top w: 0 #plus w: 0 #top w: x1 U12 w: x1 + x2 + x3 + 33428 #U12 w: 0 proper w: 0 ok w: x1 + 2 0 w: 0 #s w: 0 mark w: x1 + 16304 #proper w: 0 plus w: x1 + x2 + 58713 #U11 w: 0 active w: x1 + 1 #active w: 0 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #20 Number of SCCs: 6, DPs: 19 SCC { #5 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 23626 top w: 0 #plus w: x1 + x2 #top w: x1 U12 w: x1 + x2 + x3 + 1 #U12 w: 0 proper w: 0 ok w: x1 + 1 0 w: 0 #s w: 0 mark w: x1 + 16304 #proper w: 0 plus w: x1 + x2 + 29283 #U11 w: 0 active w: x1 + 1 #active w: 0 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #5 Number of SCCs: 5, DPs: 18 SCC { #6 #7 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 33215 top w: 0 #plus w: 0 #top w: x1 U12 w: x1 + x2 + x3 + 40113 #U12 w: 0 proper w: 0 ok w: x1 + 1 0 w: 0 #s w: x1 mark w: x1 + 16304 #proper w: 0 plus w: x1 + x2 + 54788 #U11 w: 0 active w: x1 + 1 #active w: 0 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #6 #7 Number of SCCs: 4, DPs: 16 SCC { #12 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 33215 top w: 0 #plus w: 0 #top w: x1 U12 w: x1 + x2 + x3 + 22078 #U12 w: x1 + x2 + x3 proper w: 0 ok w: x1 + 1 0 w: 0 #s w: 0 mark w: x1 + 35495 #proper w: 0 plus w: x1 + x2 + 54788 #U11 w: 0 active w: x1 + 1 #active w: 0 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #12 Number of SCCs: 3, DPs: 15 SCC { #22 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 1 top w: 0 #plus w: 0 #top w: x1 U12 w: x1 + x2 + x3 + 1 #U12 w: 0 proper w: 0 ok w: x1 + 1 0 w: 0 #s w: 0 mark w: x1 + 35495 #proper w: 0 plus w: x1 + x2 + 49485 #U11 w: x1 + x3 active w: x1 + 1 #active w: 0 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #22 Number of SCCs: 2, DPs: 14 SCC { #2 #4 #9 #11 #24 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 1 top w: 0 #plus w: 0 #top w: x1 U12 w: x1 + x2 + x3 + 1 #U12 w: 0 proper w: 0 ok w: x1 + 1 0 w: 0 #s w: 0 mark w: x1 + 35495 #proper w: 0 plus w: x1 + x2 + 49485 #U11 w: 0 active w: x1 + 1 #active w: x1 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #2 #4 #9 #11 #24 Number of SCCs: 1, DPs: 9 SCC { #14..16 #18 #19 #26..28 #30 } POLO(Sum)... succeeded. U11 w: x1 + x2 + x3 + 1 s w: x1 + 1 top w: 0 #plus w: 0 #top w: x1 U12 w: x1 + x2 + x3 + 1 #U12 w: 0 proper w: 0 ok w: x1 + 1 0 w: 0 #s w: 0 mark w: x1 + 37034 #proper w: x1 plus w: x1 + x2 + 49485 #U11 w: 0 active w: x1 + 1 #active w: 0 tt w: 0 USABLE RULES: { 5..9 12 21..24 } Removed DPs: #14..16 #18 #19 #26..28 #30 Number of SCCs: 0, DPs: 0