Input TRS: 1: o(lambda(x),y) -> lambda(o(x,d(1(),o(y,p())))) 2: o(d(x,y),z) -> d(o(x,z),o(y,z)) 3: o(o(x,y),z) -> o(x,o(y,z)) 4: lambda(x) -> x 5: o(x,y) -> x 6: o(x,y) -> y 7: d(x,y) -> x 8: d(x,y) -> y e1: o(x,y) ->= d(x,y) [relative] e2: o(x,y) ->= d(y,x) [relative] Number of Rules: 8 Direct POLO(Sum) ...Direct Mat2b ... failed. Dependency Pairs: #1: # o(d(x,y),z) -> # d(o(x,z),o(y,z)) #2: # o(d(x,y),z) -> # o(x,z) #3: # o(d(x,y),z) -> # o(y,z) #4: # o(x,y) -> # d(y,x) [relative] #5: # o(x,y) -> # d(x,y) [relative] #6: # o(o(x,y),z) -> # o(x,o(y,z)) #7: # o(o(x,y),z) -> # o(y,z) #8: # o(lambda(x),y) -> # lambda(o(x,d(1(),o(y,p())))) #9: # o(lambda(x),y) -> # o(x,d(1(),o(y,p()))) #10: # o(lambda(x),y) -> # d(1(),o(y,p())) #11: # o(lambda(x),y) -> # o(y,p()) Number of SCCs: 1 SCC { #2 #3 #6 #7 #9 #11 } POLO(Sum)... QLPOS... QWPOpS(mSum)... removes: #9 #11 I(1) = 0 I(d) = max(x1, x2) sigma(d) = [2,1] I(lambda) = x1 + 4 sigma(lambda) = [1] I(# o) = x1 + x2 + 5 sigma(# o) = [] I(o) = x1 + x2 sigma(o) = [1,2] I(p) = 0 PREC: 1 > lambda > d > o > p > # o USABLE RULES: { 4 8 1 3 5 10 7 9 6 2 } Number of SCCs: 1 SCC { #2 #3 #6 #7 } POLO(Sum)... removes: #2 #3 #6 #7 I(1) = 1 I(d) = max(x1 + x2 + 2, 0) I(lambda) = max(x1 + 3, 0) I(# o) = max(x1 - 1, 0) I(o) = max(x1 + x2 + 2, 0) I(p) = 1 USABLE RULES: { } Number of SCCs: 0