YES Input TRS: 1: twoto(0(x1)) -> p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1)))))))))))))))) 2: twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 3: twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 4: twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 5: p(p(s(x1))) -> p(x1) 6: p(s(x1)) -> x1 7: p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 8: 0(x1) -> x1 Number of strict rules: 8 Direct POLO(bPol) ... removes: 8 1 s w: x1 twice w: x1 p w: x1 0 w: x1 + 5599 twoto w: x1 + 38870 Number of strict rules: 6 Direct POLO(bPol) ... failed. Uncurrying p 2: twoto(s(x1)) -> p(p^1_s(s(p(p(p^1_s(s(s(twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))))))) 3: twice(0(x1)) -> p^1_s(p^1_s(0(s(p^1_s(s(s(s(p^1_s(x1))))))))) 4: twice(s(x1)) -> s(p(p(p(p^1_s(s(s(s(s(twice(p^1_s(p^1_s(p^1_s(p^1_s(x1)))))))))))))) 5: p(p^1_s(x1)) -> p(x1) 6: p^1_s(x1) -> x1 7: p^1_0(x1) -> 0(s(s(s(s(p^1_s(x1)))))) 9: p(0(_1)) ->= p^1_0(_1) 10: p(s(_1)) ->= p^1_s(_1) Number of strict rules: 6 Direct POLO(bPol) ... failed. Dependency Pairs: #1: #twoto(s(x1)) -> #p(p^1_s(s(p(p(p^1_s(s(s(twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))))))) #2: #twoto(s(x1)) -> #p^1_s(s(p(p(p^1_s(s(s(twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))))))))))) #3: #twoto(s(x1)) -> #p(p(p^1_s(s(s(twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))))))))) #4: #twoto(s(x1)) -> #p(p^1_s(s(s(twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))) #5: #twoto(s(x1)) -> #p^1_s(s(s(twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))))))) #6: #twoto(s(x1)) -> #twice(p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))) #7: #twoto(s(x1)) -> #p(p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))) #8: #twoto(s(x1)) -> #p^1_s(s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))) #9: #twoto(s(x1)) -> #p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))) #10: #twoto(s(x1)) -> #p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))) #11: #twoto(s(x1)) -> #p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))) #12: #twoto(s(x1)) -> #twoto(p^1_s(p^1_s(x1))) #13: #twoto(s(x1)) -> #p^1_s(p^1_s(x1)) #14: #twoto(s(x1)) -> #p^1_s(x1) #15: #p(0(_1)) ->? #p^1_0(_1) #16: #p^1_0(x1) -> #p^1_s(x1) #17: #p(s(_1)) ->? #p^1_s(_1) #18: #p(p^1_s(x1)) -> #p(x1) #19: #twice(0(x1)) -> #p^1_s(p^1_s(0(s(p^1_s(s(s(s(p^1_s(x1))))))))) #20: #twice(0(x1)) -> #p^1_s(0(s(p^1_s(s(s(s(p^1_s(x1)))))))) #21: #twice(0(x1)) -> #p^1_s(s(s(s(p^1_s(x1))))) #22: #twice(0(x1)) -> #p^1_s(x1) #23: #twice(s(x1)) -> #p(p(p(p^1_s(s(s(s(s(twice(p^1_s(p^1_s(p^1_s(p^1_s(x1))))))))))))) #24: #twice(s(x1)) -> #p(p(p^1_s(s(s(s(s(twice(p^1_s(p^1_s(p^1_s(p^1_s(x1)))))))))))) #25: #twice(s(x1)) -> #p(p^1_s(s(s(s(s(twice(p^1_s(p^1_s(p^1_s(p^1_s(x1))))))))))) #26: #twice(s(x1)) -> #p^1_s(s(s(s(s(twice(p^1_s(p^1_s(p^1_s(p^1_s(x1)))))))))) #27: #twice(s(x1)) -> #twice(p^1_s(p^1_s(p^1_s(p^1_s(x1))))) #28: #twice(s(x1)) -> #p^1_s(p^1_s(p^1_s(p^1_s(x1)))) #29: #twice(s(x1)) -> #p^1_s(p^1_s(p^1_s(x1))) #30: #twice(s(x1)) -> #p^1_s(p^1_s(x1)) #31: #twice(s(x1)) -> #p^1_s(x1) Number of SCCs: 3, DPs: 3 SCC { #18 } POLO(Sum)... succeeded. s w: 0 #p^1_0 w: 0 twice w: 0 #p^1_s w: 0 p^1_0 w: 0 p^1_s w: x1 + 1 #p w: x1 p w: 0 0 w: 0 twoto w: 0 #twice w: 0 #twoto w: 0 USABLE RULES: { } Removed DPs: #18 Number of SCCs: 2, DPs: 2 SCC { #12 } POLO(Sum)... succeeded. s w: x1 + 3 #p^1_0 w: 0 twice w: 0 #p^1_s w: 0 p^1_0 w: 0 p^1_s w: x1 + 1 #p w: 0 p w: 0 0 w: 0 twoto w: 0 #twice w: 0 #twoto w: x1 USABLE RULES: { 6 } Removed DPs: #12 Number of SCCs: 1, DPs: 1 SCC { #27 } POLO(Sum)... succeeded. s w: x1 + 5 #p^1_0 w: 0 twice w: 0 #p^1_s w: 0 p^1_0 w: 0 p^1_s w: x1 + 1 #p w: 0 p w: 0 0 w: 0 twoto w: 0 #twice w: x1 #twoto w: x1 USABLE RULES: { 6 } Removed DPs: #27 Number of SCCs: 0, DPs: 0