MAYBE (ignored inputs)COMMENT the following rules are removed from the original TRS c ( a ( x ) ) -> c ( a ( x ) ) Rewrite Rules: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ b(c(b(?x_1))) = a(a(b(?x_1))), b(a(a(?x_3))) = a(a(c(?x_3))), b(a(a(?x_4))) = a(a(a(?x_4))), b(a(a(?x))) = c(b(b(?x))), b(a(a(?x_3))) = c(b(c(?x_3))), b(a(a(?x_4))) = c(b(a(?x_4))), a(a(a(?x))) = c(b(b(?x))), a(c(b(?x_1))) = c(b(b(?x_1))), a(a(a(?x_3))) = c(b(c(?x_3))), a(a(a(?x_4))) = c(b(a(?x_4))), b(c(b(?x_5))) = a(a(a(?x_5))), b(a(a(?x_7))) = a(a(c(?x_7))), b(c(b(?x_2))) = a(a(b(?x_2))), b(c(b(?x_6))) = a(a(a(?x_6))), c(c(b(?x_2))) = c(b(b(?x_2))), c(c(b(?x_6))) = c(b(a(?x_6))), a(c(b(?x_2))) = c(b(b(?x_2))), c(c(b(?x_5))) = a(a(a(?x_5))), b(a(a(?x))) = a(a(b(?x))), b(c(b(?x))) = c(b(b(?x))), a(c(b(?x))) = c(b(a(?x))), c(a(a(?x))) = a(a(c(?x))) ] Outer CPs: [ a(a(?x)) = c(b(?x)) ] not Overlay, check Termination... unknown/not Terminating unknown Knuth & Bendix Linear unknown Development Closed unknown Strongly Closed unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow inner CP cond (upside-parallel) innter CP Cond (outside) unknown Upside-Parallel-Closed/Outside-Closed (inner) Parallel CPs: (not computed) unknown Toyama (Parallel CPs) Simultaneous CPs: [ c(b(?x)) = a(a(?x)), b(a(a(?x_1))) = a(a(b(?x_1))), b(c(b(?x_2))) = a(a(b(?x_2))), b(a(a(?x_4))) = a(a(c(?x_4))), b(a(a(?x_5))) = a(a(a(?x_5))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(b(?x_2)))) = b(a(a(b(?x_2)))), a(a(a(a(?x_4)))) = b(a(a(c(?x_4)))), a(a(a(a(?x_5)))) = b(a(a(a(?x_5)))), c(b(a(a(?x_1)))) = b(a(a(b(?x_1)))), c(b(c(b(?x_2)))) = b(a(a(b(?x_2)))), c(b(a(a(?x_4)))) = b(a(a(c(?x_4)))), c(b(a(a(?x_5)))) = b(a(a(a(?x_5)))), c(b(a(a(?x_1)))) = a(a(a(b(?x_1)))), c(b(c(b(?x_2)))) = a(a(a(b(?x_2)))), c(b(a(a(?x_4)))) = a(a(a(c(?x_4)))), c(b(a(a(?x_5)))) = a(a(a(a(?x_5)))), a(a(b(?x))) = b(a(a(?x))), c(b(b(?x))) = b(a(a(?x))), c(b(b(?x))) = a(a(a(?x))), a(a(?x)) = c(b(?x)), b(c(b(?x_1))) = c(b(b(?x_1))), b(a(a(?x_2))) = c(b(b(?x_2))), b(a(a(?x_4))) = c(b(c(?x_4))), b(a(a(?x_5))) = c(b(a(?x_5))), c(b(c(b(?x_1)))) = b(c(b(b(?x_1)))), c(b(a(a(?x_2)))) = b(c(b(b(?x_2)))), c(b(a(a(?x_4)))) = b(c(b(c(?x_4)))), c(b(a(a(?x_5)))) = b(c(b(a(?x_5)))), a(a(c(b(?x_1)))) = b(c(b(b(?x_1)))), a(a(a(a(?x_2)))) = b(c(b(b(?x_2)))), a(a(a(a(?x_4)))) = b(c(b(c(?x_4)))), a(a(a(a(?x_5)))) = b(c(b(a(?x_5)))), c(b(c(b(?x_1)))) = a(c(b(b(?x_1)))), c(b(a(a(?x_2)))) = a(c(b(b(?x_2)))), c(b(a(a(?x_4)))) = a(c(b(c(?x_4)))), c(b(a(a(?x_5)))) = a(c(b(a(?x_5)))), c(b(b(?x))) = b(c(b(?x))), a(a(b(?x))) = b(c(b(?x))), c(b(b(?x))) = a(c(b(?x))), a(a(a(?x_2))) = c(b(b(?x_2))), a(c(b(?x_3))) = c(b(b(?x_3))), a(a(a(?x_4))) = c(b(c(?x_4))), a(a(a(?x_5))) = c(b(a(?x_5))), c(b(a(a(?x_2)))) = c(c(b(b(?x_2)))), c(b(c(b(?x_3)))) = c(c(b(b(?x_3)))), c(b(a(a(?x_4)))) = c(c(b(c(?x_4)))), c(b(a(a(?x_5)))) = c(c(b(a(?x_5)))), c(b(b(?x))) = c(c(b(?x))), b(c(b(?x_6))) = a(a(a(?x_6))), a(a(c(b(?x_6)))) = b(a(a(a(?x_6)))), c(b(c(b(?x_6)))) = b(a(a(a(?x_6)))), c(b(c(b(?x_6)))) = a(a(a(a(?x_6)))), a(a(c(?x))) = b(a(a(?x))), c(b(c(?x))) = b(a(a(?x))), c(b(c(?x))) = a(a(a(?x))), a(a(a(?x))) = b(a(a(?x))), c(b(a(?x))) = b(a(a(?x))), c(b(a(?x))) = a(a(a(?x))), c(c(b(?x_4))) = c(b(b(?x_4))), c(c(b(?x_7))) = c(b(a(?x_7))), a(a(c(b(?x_7)))) = b(c(b(a(?x_7)))), a(a(c(b(?x_4)))) = c(c(b(b(?x_4)))), a(a(c(b(?x_7)))) = c(c(b(a(?x_7)))), a(a(a(?x))) = b(c(b(?x))), a(a(a(?x))) = c(c(b(?x))), a(c(b(?x_1))) = c(b(a(?x_1))), c(b(c(b(?x_1)))) = a(c(b(a(?x_1)))), c(b(c(b(?x_1)))) = c(c(b(a(?x_1)))), c(b(a(?x))) = a(c(b(?x))), c(b(a(?x))) = c(c(b(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(c(b(?x_7))) = a(a(a(?x_7))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(c(b(?x_7)))) = c(a(a(a(?x_7)))), a(a(c(?x))) = c(a(a(?x))) ] unknown Okui (Simultaneous CPs) unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping check Locally Decreasing Diagrams by Rule Labelling... Critical Pair by Rules <1, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <3, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6),([(c,1)],4)], [([],6),([(c,1)],3)]> joinable by a reduction of rules <[([],4),([],6)], [([],6),([(c,1)],3),([],5)]> Critical Pair by Rules <4, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],4)], []> Critical Pair by Rules <0, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],3)], [([(c,1)],1),([],7)]> joinable by a reduction of rules <[([],4),([],6)], [([(c,1)],0),([],5)]> Critical Pair by Rules <3, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6)], [([(c,1)],3),([],5)]> Critical Pair by Rules <4, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6)], []> Critical Pair by Rules <0, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],6),([(c,1)],4)], [([(c,1)],0)]> joinable by a reduction of rules <[([],6)], [([(c,1)],0),([],5)]> Critical Pair by Rules <1, 2> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],1),([],7),([(a,1)],2)]> Critical Pair by Rules <3, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],6),([(c,1)],4)], [([(c,1)],3)]> joinable by a reduction of rules <[([],6)], [([(c,1)],3),([],5)]> Critical Pair by Rules <4, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],6)], []> Critical Pair by Rules <5, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([(a,1)],2)], [([(a,1)],6)]> Critical Pair by Rules <7, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6),([(c,1)],4)], [([],6),([(c,1)],3)]> joinable by a reduction of rules <[([],4),([],6)], [([],6),([(c,1)],3),([],5)]> Critical Pair by Rules <2, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <6, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([(a,1)],2)], [([(a,1)],6)]> Critical Pair by Rules <2, 5> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],1)]> Critical Pair by Rules <6, 5> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],4),([(c,1)],6)]> Critical Pair by Rules <2, 6> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],1),([],7),([(a,1)],2)]> Critical Pair by Rules <5, 7> preceded by [(c,1)] joinable by a reduction of rules <[([],7),([(a,1)],2)], [([(a,1)],6)]> Critical Pair by Rules <0, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],3)], []> joinable by a reduction of rules <[([],4),([(a,1)],6)], [([(a,1)],2)]> Critical Pair by Rules <1, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([],6)], []> joinable by a reduction of rules <[([],3)], [([(c,1)],1),([],7)]> Critical Pair by Rules <6, 6> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],4),([(c,1)],6),([],7),([(a,1)],2)]> Critical Pair by Rules <7, 7> preceded by [(c,1)] joinable by a reduction of rules <[], [([],6),([(c,1)],3)]> Critical Pair by Rules <1, 0> preceded by [] joinable by a reduction of rules <[], [([],6)]> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 75 candidates for checking non-joinability check by TCAP-Approximation (failure) check by Ordering(rpo), check by Tree-Automata Approximation (failure) check by Interpretation(mod2) (failure) check by Descendants-Approximation, check by Ordering(poly) (failure) unknown Non-Confluence unknown Huet (modulo AC) check by Reduction-Preserving Completion... failure(empty P) unknown Reduction-Preserving Completion check by Ordered Rewriting... remove redundants rules and split R-part: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] E-part: [ ] ...failed to find a suitable LPO. unknown Confluence by Ordered Rewriting Direct Methods: Can't judge Try Persistent Decomposition for... [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] Sort Assignment: a : 15=>15 b : 15=>15 c : 15=>15 maximal types: {15} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] Layer Preserving Decomposition failed: Can't judge Try Commutative Decomposition for... [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] Outside Critical Pair: by Rules <1, 0> develop reducts from lhs term... <{}, c(b(?x_1))> develop reducts from rhs term... <{6}, c(b(?x_1))> <{}, a(a(?x_1))> Inside Critical Pair: by Rules <1, 0> develop reducts from lhs term... <{3}, a(a(b(?x_1)))> <{}, b(c(b(?x_1)))> develop reducts from rhs term... <{6}, c(b(b(?x_1)))> <{2}, a(c(b(?x_1)))> <{}, a(a(b(?x_1)))> Inside Critical Pair: by Rules <3, 0> develop reducts from lhs term... <{4}, a(a(a(?x_3)))> <{6}, b(c(b(?x_3)))> <{}, b(a(a(?x_3)))> develop reducts from rhs term... <{6}, c(b(c(?x_3)))> <{}, a(a(c(?x_3)))> Inside Critical Pair: by Rules <4, 0> develop reducts from lhs term... <{4}, a(a(a(?x_4)))> <{6}, b(c(b(?x_4)))> <{}, b(a(a(?x_4)))> develop reducts from rhs term... <{6}, c(b(a(?x_4)))> <{6}, a(c(b(?x_4)))> <{}, a(a(a(?x_4)))> Inside Critical Pair: by Rules <0, 1> develop reducts from lhs term... <{4}, a(a(a(?x)))> <{6}, b(c(b(?x)))> <{}, b(a(a(?x)))> develop reducts from rhs term... <{1}, c(c(b(?x)))> <{0}, c(a(a(?x)))> <{}, c(b(b(?x)))> Inside Critical Pair: by Rules <3, 1> develop reducts from lhs term... <{4}, a(a(a(?x_3)))> <{6}, b(c(b(?x_3)))> <{}, b(a(a(?x_3)))> develop reducts from rhs term... <{3}, c(a(a(?x_3)))> <{}, c(b(c(?x_3)))> Inside Critical Pair: by Rules <4, 1> develop reducts from lhs term... <{4}, a(a(a(?x_4)))> <{6}, b(c(b(?x_4)))> <{}, b(a(a(?x_4)))> develop reducts from rhs term... <{4}, c(a(a(?x_4)))> <{}, c(b(a(?x_4)))> Inside Critical Pair: by Rules <0, 2> develop reducts from lhs term... <{6}, c(b(a(?x)))> <{6}, a(c(b(?x)))> <{}, a(a(a(?x)))> develop reducts from rhs term... <{1}, c(c(b(?x)))> <{0}, c(a(a(?x)))> <{}, c(b(b(?x)))> Inside Critical Pair: by Rules <1, 2> develop reducts from lhs term... <{}, a(c(b(?x_1)))> develop reducts from rhs term... <{1}, c(c(b(?x_1)))> <{0}, c(a(a(?x_1)))> <{}, c(b(b(?x_1)))> Inside Critical Pair: by Rules <3, 2> develop reducts from lhs term... <{6}, c(b(a(?x_3)))> <{6}, a(c(b(?x_3)))> <{}, a(a(a(?x_3)))> develop reducts from rhs term... <{3}, c(a(a(?x_3)))> <{}, c(b(c(?x_3)))> Inside Critical Pair: by Rules <4, 2> develop reducts from lhs term... <{6}, c(b(a(?x_4)))> <{6}, a(c(b(?x_4)))> <{}, a(a(a(?x_4)))> develop reducts from rhs term... <{4}, c(a(a(?x_4)))> <{}, c(b(a(?x_4)))> Inside Critical Pair: by Rules <5, 3> develop reducts from lhs term... <{3}, a(a(b(?x_5)))> <{}, b(c(b(?x_5)))> develop reducts from rhs term... <{6}, c(b(a(?x_5)))> <{6}, a(c(b(?x_5)))> <{}, a(a(a(?x_5)))> Inside Critical Pair: by Rules <7, 3> develop reducts from lhs term... <{4}, a(a(a(?x_7)))> <{6}, b(c(b(?x_7)))> <{}, b(a(a(?x_7)))> develop reducts from rhs term... <{6}, c(b(c(?x_7)))> <{}, a(a(c(?x_7)))> Inside Critical Pair: by Rules <2, 4> develop reducts from lhs term... <{3}, a(a(b(?x_2)))> <{}, b(c(b(?x_2)))> develop reducts from rhs term... <{6}, c(b(b(?x_2)))> <{2}, a(c(b(?x_2)))> <{}, a(a(b(?x_2)))> Inside Critical Pair: by Rules <6, 4> develop reducts from lhs term... <{3}, a(a(b(?x_6)))> <{}, b(c(b(?x_6)))> develop reducts from rhs term... <{6}, c(b(a(?x_6)))> <{6}, a(c(b(?x_6)))> <{}, a(a(a(?x_6)))> Inside Critical Pair: by Rules <2, 5> develop reducts from lhs term... <{7}, a(a(b(?x_2)))> <{}, c(c(b(?x_2)))> develop reducts from rhs term... <{1}, c(c(b(?x_2)))> <{0}, c(a(a(?x_2)))> <{}, c(b(b(?x_2)))> Inside Critical Pair: by Rules <6, 5> develop reducts from lhs term... <{7}, a(a(b(?x_6)))> <{}, c(c(b(?x_6)))> develop reducts from rhs term... <{4}, c(a(a(?x_6)))> <{}, c(b(a(?x_6)))> Inside Critical Pair: by Rules <2, 6> develop reducts from lhs term... <{}, a(c(b(?x_2)))> develop reducts from rhs term... <{1}, c(c(b(?x_2)))> <{0}, c(a(a(?x_2)))> <{}, c(b(b(?x_2)))> Inside Critical Pair: by Rules <5, 7> develop reducts from lhs term... <{7}, a(a(b(?x_5)))> <{}, c(c(b(?x_5)))> develop reducts from rhs term... <{6}, c(b(a(?x_5)))> <{6}, a(c(b(?x_5)))> <{}, a(a(a(?x_5)))> Commutative Decomposition failed: Can't judge No further decomposition possible Combined result: Can't judge /tmp/fileMzqW2o.trs: Failure(unknown CR) (2843 msec.)