YES Input TRS: 1: tower(0(x1)) -> s(0(p(s(p(s(x1)))))) 2: tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 3: twoto(0(x1)) -> s(0(x1)) 4: twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 5: twice(0(x1)) -> 0(x1) 6: twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 7: p(p(s(x1))) -> p(x1) 8: p(s(x1)) -> x1 9: p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) Number of strict rules: 9 Direct POLO(bPol) ... removes: 1 tower w: x1 + 24773 s w: x1 twice w: x1 p w: x1 0 w: x1 + 5599 twoto w: x1 Number of strict rules: 8 Direct POLO(bPol) ... failed. Uncurrying p 2: tower(s(x1)) -> p^1_s(p^1_s(twoto(p^1_s(p^1_s(tower(p^1_s(p^1_s(x1)))))))) 3: twoto(0(x1)) -> s(0(x1)) 4: twoto(s(x1)) -> p(p^1_s(p(p(p^1_s(s(p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))))))))) 5: twice(0(x1)) -> 0(x1) 6: twice(s(x1)) -> p(p(p^1_s(s(s(s(s(twice(p(p(p^1_s(s(s(x1))))))))))))) 7: p(p^1_s(x1)) -> p(x1) 8: p^1_s(x1) -> x1 9: p^1_0(x1) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 10: p(0(_1)) ->= p^1_0(_1) 11: p(s(_1)) ->= p^1_s(_1) Number of strict rules: 8 Direct POLO(bPol) ... failed. Dependency Pairs: #1: #tower(s(x1)) -> #p^1_s(p^1_s(twoto(p^1_s(p^1_s(tower(p^1_s(p^1_s(x1)))))))) #2: #tower(s(x1)) -> #p^1_s(twoto(p^1_s(p^1_s(tower(p^1_s(p^1_s(x1))))))) #3: #tower(s(x1)) -> #twoto(p^1_s(p^1_s(tower(p^1_s(p^1_s(x1)))))) #4: #tower(s(x1)) -> #p^1_s(p^1_s(tower(p^1_s(p^1_s(x1))))) #5: #tower(s(x1)) -> #p^1_s(tower(p^1_s(p^1_s(x1)))) #6: #tower(s(x1)) -> #tower(p^1_s(p^1_s(x1))) #7: #tower(s(x1)) -> #p^1_s(p^1_s(x1)) #8: #tower(s(x1)) -> #p^1_s(x1) #9: #twice(s(x1)) -> #p(p(p^1_s(s(s(s(s(twice(p(p(p^1_s(s(s(x1))))))))))))) #10: #twice(s(x1)) -> #p(p^1_s(s(s(s(s(twice(p(p(p^1_s(s(s(x1)))))))))))) #11: #twice(s(x1)) -> #p^1_s(s(s(s(s(twice(p(p(p^1_s(s(s(x1))))))))))) #12: #twice(s(x1)) -> #twice(p(p(p^1_s(s(s(x1)))))) #13: #twice(s(x1)) -> #p(p(p^1_s(s(s(x1))))) #14: #twice(s(x1)) -> #p(p^1_s(s(s(x1)))) #15: #twice(s(x1)) -> #p^1_s(s(s(x1))) #16: #p(s(_1)) ->? #p^1_s(_1) #17: #p(p^1_s(x1)) -> #p(x1) #18: #p(0(_1)) ->? #p^1_0(_1) #19: #twoto(s(x1)) -> #p(p^1_s(p(p(p^1_s(s(p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))))))))) #20: #twoto(s(x1)) -> #p^1_s(p(p(p^1_s(s(p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))))))))))))) #21: #twoto(s(x1)) -> #p(p(p^1_s(s(p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))))))) #22: #twoto(s(x1)) -> #p(p^1_s(s(p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))))))))))) #23: #twoto(s(x1)) -> #p^1_s(s(p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))))) #24: #twoto(s(x1)) -> #p^1_s(s(p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))))) #25: #twoto(s(x1)) -> #p^1_s(s(p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))))) #26: #twoto(s(x1)) -> #p^1_s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))))) #27: #twoto(s(x1)) -> #twice(p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))))) #28: #twoto(s(x1)) -> #p^1_s(p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))))) #29: #twoto(s(x1)) -> #p^1_s(p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))))) #30: #twoto(s(x1)) -> #p(p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))))) #31: #twoto(s(x1)) -> #p(p^1_s(s(s(twoto(p^1_s(p^1_s(x1))))))) #32: #twoto(s(x1)) -> #p^1_s(s(s(twoto(p^1_s(p^1_s(x1)))))) #33: #twoto(s(x1)) -> #twoto(p^1_s(p^1_s(x1))) #34: #twoto(s(x1)) -> #p^1_s(p^1_s(x1)) #35: #twoto(s(x1)) -> #p^1_s(x1) Number of SCCs: 4, DPs: 4 SCC { #17 } POLO(Sum)... succeeded. tower w: 0 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 #tower w: 0 twoto w: 0 #twice w: 0 #twoto w: 0 USABLE RULES: { } Removed DPs: #17 Number of SCCs: 3, DPs: 3 SCC { #6 } POLO(Sum)... succeeded. tower w: 0 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 #tower w: x1 twoto w: 0 #twice w: 0 #twoto w: 0 USABLE RULES: { 8 } Removed DPs: #6 Number of SCCs: 2, DPs: 2 SCC { #33 } POLO(Sum)... succeeded. tower w: 0 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 #tower w: x1 twoto w: 0 #twice w: 0 #twoto w: x1 USABLE RULES: { 8 } Removed DPs: #33 Number of SCCs: 1, DPs: 1 SCC { #12 } POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded. tower w: max(x1 - 1, 0) s w: max(x1 + 5, 0) #p^1_0 w: max(x1 - 1, 0) twice w: max(x1 - 1, 0) #p^1_s w: 0 p^1_0 w: 0 p^1_s w: max(x1 + 1, 0) #p w: 0 p w: max(x1 - 4, 0) 0 w: 0 #tower w: max(x1 - 1, 0) twoto w: max(x1 - 1, 0) #twice w: max(x1 - 4, 0) #twoto w: 0 USABLE RULES: { 7..11 } Removed DPs: #12 Number of SCCs: 0, DPs: 0