YES Input TRS: 1: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) 2: thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) 3: half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) 4: half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) 5: half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) 6: sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) 7: sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) 8: p(p(s(x1))) -> p(x1) 9: p(s(x1)) -> x1 10: p(0(x1)) -> 0(s(s(s(s(x1))))) 11: 0(x1) -> x1 Number of strict rules: 11 Direct POLO(bPol) ... removes: 1 3 11 6 2 s w: x1 half w: x1 + 1 p w: x1 0 w: x1 + 5744 sixtimes w: x1 + 1 thrice w: x1 + 3 Number of strict rules: 6 Direct POLO(bPol) ... failed. Uncurrying p 4: half(s(x1)) -> p^1_s(p(p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1)))))))))))) 5: half(s(s(x1))) -> p^1_s(p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1))))))))))) 7: sixtimes(s(x1)) -> p(p^1_s(s(s(s(s(s(s(p(p^1_s(p^1_s(s(s(sixtimes(p^1_s(p(p(p^1_s(s(s(x1)))))))))))))))))))) 8: p(p^1_s(x1)) -> p(x1) 9: p^1_s(x1) -> x1 10: p^1_0(x1) -> 0(s(s(s(s(x1))))) 12: p(0(_1)) ->= p^1_0(_1) 13: p(s(_1)) ->= p^1_s(_1) Number of strict rules: 6 Direct POLO(bPol) ... failed. Dependency Pairs: #1: #p(s(_1)) ->? #p^1_s(_1) #2: #p(0(_1)) ->? #p^1_0(_1) #3: #sixtimes(s(x1)) -> #p(p^1_s(s(s(s(s(s(s(p(p^1_s(p^1_s(s(s(sixtimes(p^1_s(p(p(p^1_s(s(s(x1)))))))))))))))))))) #4: #sixtimes(s(x1)) -> #p^1_s(s(s(s(s(s(s(p(p^1_s(p^1_s(s(s(sixtimes(p^1_s(p(p(p^1_s(s(s(x1))))))))))))))))))) #5: #sixtimes(s(x1)) -> #p(p^1_s(p^1_s(s(s(sixtimes(p^1_s(p(p(p^1_s(s(s(x1)))))))))))) #6: #sixtimes(s(x1)) -> #p^1_s(p^1_s(s(s(sixtimes(p^1_s(p(p(p^1_s(s(s(x1))))))))))) #7: #sixtimes(s(x1)) -> #p^1_s(s(s(sixtimes(p^1_s(p(p(p^1_s(s(s(x1)))))))))) #8: #sixtimes(s(x1)) -> #sixtimes(p^1_s(p(p(p^1_s(s(s(x1))))))) #9: #sixtimes(s(x1)) -> #p^1_s(p(p(p^1_s(s(s(x1)))))) #10: #sixtimes(s(x1)) -> #p(p(p^1_s(s(s(x1))))) #11: #sixtimes(s(x1)) -> #p(p^1_s(s(s(x1)))) #12: #sixtimes(s(x1)) -> #p^1_s(s(s(x1))) #13: #half(s(s(x1))) -> #p^1_s(p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1))))))))))) #14: #half(s(s(x1))) -> #p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1)))))))))) #15: #half(s(s(x1))) -> #p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1)))))))) #16: #half(s(s(x1))) -> #p^1_s(s(half(p(p^1_s(s(p^1_s(x1))))))) #17: #half(s(s(x1))) -> #half(p(p^1_s(s(p^1_s(x1))))) #18: #half(s(s(x1))) -> #p(p^1_s(s(p^1_s(x1)))) #19: #half(s(s(x1))) -> #p^1_s(s(p^1_s(x1))) #20: #half(s(s(x1))) -> #p^1_s(x1) #21: #p(p^1_s(x1)) -> #p(x1) #22: #half(s(x1)) -> #p^1_s(p(p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1)))))))))))) #23: #half(s(x1)) -> #p(p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1))))))))))) #24: #half(s(x1)) -> #p^1_s(s(p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1)))))))))) #25: #half(s(x1)) -> #p(p^1_s(s(half(p(p^1_s(s(p^1_s(x1)))))))) #26: #half(s(x1)) -> #p^1_s(s(half(p(p^1_s(s(p^1_s(x1))))))) #27: #half(s(x1)) -> #half(p(p^1_s(s(p^1_s(x1))))) #28: #half(s(x1)) -> #p(p^1_s(s(p^1_s(x1)))) #29: #half(s(x1)) -> #p^1_s(s(p^1_s(x1))) #30: #half(s(x1)) -> #p^1_s(x1) Number of SCCs: 3, DPs: 4 SCC { #21 } POLO(Sum)... succeeded. s w: 0 #p^1_0 w: 0 #p^1_s w: 0 p^1_0 w: 0 #sixtimes w: 0 p^1_s w: x1 + 1 #half w: 0 #p w: x1 half w: 0 p w: 0 0 w: 0 sixtimes w: 0 thrice w: 0 USABLE RULES: { } Removed DPs: #21 Number of SCCs: 2, DPs: 3 SCC { #8 } POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded. s w: max(x1 + 129145, 0) #p^1_0 w: max(x1 - 1, 0) #p^1_s w: max(x1 - 1, 0) p^1_0 w: 0 #sixtimes w: max(x1 - 32287, 0) p^1_s w: max(x1 + 32286, 0) #half w: max(x1 - 1, 0) #p w: 0 half w: 0 p w: max(x1 - 96859, 0) 0 w: 0 sixtimes w: max(x1 - 1, 0) thrice w: 0 USABLE RULES: { 8..10 12 13 } Removed DPs: #8 Number of SCCs: 1, DPs: 2 SCC { #17 #27 } POLO(Sum)... succeeded. s w: x1 + 1 #p^1_0 w: 0 #p^1_s w: 0 p^1_0 w: 33954 #sixtimes w: 0 p^1_s w: x1 #half w: x1 #p w: 0 half w: 0 p w: x1 0 w: 33954 sixtimes w: 0 thrice w: 0 USABLE RULES: { 8..10 12 13 } Removed DPs: #17 Number of SCCs: 1, DPs: 1 SCC { #27 } POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded. s w: max(x1 + 4, 0) #p^1_0 w: max(x1 - 1, 0) #p^1_s w: max(x1 - 1, 0) p^1_0 w: 0 #sixtimes w: max(x1 - 32287, 0) p^1_s w: max(x1 + 1, 0) #half w: max(x1 + 4, 0) #p w: 0 half w: 0 p w: max(x1 - 3, 0) 0 w: 0 sixtimes w: max(x1 - 1, 0) thrice w: 0 USABLE RULES: { 8..10 12 13 } Removed DPs: #27 Number of SCCs: 0, DPs: 0