YES Input TRS: 1: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) 2: sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) 3: twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) 4: twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(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(s(s(s(s(s(s(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 + 120 sq w: x1 + 18748 Number of strict rules: 6 Direct POLO(bPol) ... failed. Uncurrying p 2: sq(s(x1)) -> s(p^1_s(p^1_s(p(p^1_s(s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))))))))))))))))) 3: twice(0(x1)) -> p(p(p(p^1_s(s(p^1_s(s(s(0(p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1)))))))))))))))))))) 4: twice(s(x1)) -> p(p^1_s(s(s(p(p^1_s(s(s(twice(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(s(s(s(s(s(s(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: #sq(s(x1)) -> #p^1_s(p^1_s(p(p^1_s(s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1))))))))))))))))))))))))) #2: #sq(s(x1)) -> #p^1_s(p(p^1_s(s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))))))))))))))) #3: #sq(s(x1)) -> #p(p^1_s(s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1))))))))))))))))))))))) #4: #sq(s(x1)) -> #p^1_s(s(twice(p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))))))))))))) #5: #sq(s(x1)) -> #twice(p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))))))))))) #6: #sq(s(x1)) -> #p^1_s(p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1))))))))))))))))))) #7: #sq(s(x1)) -> #p^1_s(p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))))))))) #8: #sq(s(x1)) -> #p(p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1))))))))))))))))) #9: #sq(s(x1)) -> #p(p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))))))) #10: #sq(s(x1)) -> #p^1_s(s(s(sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1))))))))))))))) #11: #sq(s(x1)) -> #sq(p(p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))))) #12: #sq(s(x1)) -> #p(p(p(p(p(p^1_s(s(s(s(s(s(x1))))))))))) #13: #sq(s(x1)) -> #p(p(p(p(p^1_s(s(s(s(s(s(x1)))))))))) #14: #sq(s(x1)) -> #p(p(p(p^1_s(s(s(s(s(s(x1))))))))) #15: #sq(s(x1)) -> #p(p(p^1_s(s(s(s(s(s(x1)))))))) #16: #sq(s(x1)) -> #p(p^1_s(s(s(s(s(s(x1))))))) #17: #sq(s(x1)) -> #p^1_s(s(s(s(s(s(x1)))))) #18: #p(0(_1)) ->? #p^1_0(_1) #19: #p(s(_1)) ->? #p^1_s(_1) #20: #p(p^1_s(x1)) -> #p(x1) #21: #twice(0(x1)) -> #p(p(p(p^1_s(s(p^1_s(s(s(0(p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1)))))))))))))))))))) #22: #twice(0(x1)) -> #p(p(p^1_s(s(p^1_s(s(s(0(p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1))))))))))))))))))) #23: #twice(0(x1)) -> #p(p^1_s(s(p^1_s(s(s(0(p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1)))))))))))))))))) #24: #twice(0(x1)) -> #p^1_s(s(p^1_s(s(s(0(p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1))))))))))))))))) #25: #twice(0(x1)) -> #p^1_s(s(s(0(p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1))))))))))))))) #26: #twice(0(x1)) -> #p(p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1))))))))))) #27: #twice(0(x1)) -> #p(p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1)))))))))) #28: #twice(0(x1)) -> #p^1_s(s(s(p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1))))))))) #29: #twice(0(x1)) -> #p(p^1_s(s(p^1_s(p^1_s(p^1_s(x1)))))) #30: #twice(0(x1)) -> #p^1_s(s(p^1_s(p^1_s(p^1_s(x1))))) #31: #twice(0(x1)) -> #p^1_s(p^1_s(p^1_s(x1))) #32: #twice(0(x1)) -> #p^1_s(p^1_s(x1)) #33: #twice(0(x1)) -> #p^1_s(x1) #34: #twice(s(x1)) -> #p(p^1_s(s(s(p(p^1_s(s(s(twice(p^1_s(p^1_s(x1))))))))))) #35: #twice(s(x1)) -> #p^1_s(s(s(p(p^1_s(s(s(twice(p^1_s(p^1_s(x1)))))))))) #36: #twice(s(x1)) -> #p(p^1_s(s(s(twice(p^1_s(p^1_s(x1))))))) #37: #twice(s(x1)) -> #p^1_s(s(s(twice(p^1_s(p^1_s(x1)))))) #38: #twice(s(x1)) -> #twice(p^1_s(p^1_s(x1))) #39: #twice(s(x1)) -> #p^1_s(p^1_s(x1)) #40: #twice(s(x1)) -> #p^1_s(x1) Number of SCCs: 3, DPs: 3 SCC { #20 } POLO(Sum)... succeeded. s w: 0 #p^1_0 w: 0 twice w: 0 #p^1_s w: 0 #sq w: 0 p^1_0 w: 0 p^1_s w: x1 + 1 #p w: x1 p w: 0 0 w: 0 #twice w: 0 sq w: 0 USABLE RULES: { } Removed DPs: #20 Number of SCCs: 2, DPs: 2 SCC { #38 } POLO(Sum)... succeeded. s w: x1 + 3 #p^1_0 w: 0 twice w: 0 #p^1_s w: 0 #sq w: 0 p^1_0 w: 0 p^1_s w: x1 + 1 #p w: 0 p w: 0 0 w: 0 #twice w: x1 sq w: 0 USABLE RULES: { 6 } Removed DPs: #38 Number of SCCs: 1, DPs: 1 SCC { #11 } POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded. s w: max(x1 + 8, 0) #p^1_0 w: max(x1 - 1, 0) twice w: max(x1 - 1, 0) #p^1_s w: max(x1 - 1, 0) #sq w: max(x1 - 7, 0) p^1_0 w: 0 p^1_s w: max(x1 + 1, 0) #p w: 0 p w: max(x1 - 7, 0) 0 w: 0 #twice w: max(x1 - 1, 0) sq w: 0 USABLE RULES: { 5..7 9 10 } Removed DPs: #11 Number of SCCs: 0, DPs: 0