MAYBE (ignored inputs)COMMENT the following rules are removed from the original TRS a ( b ( x ) ) -> a ( b ( x ) ) c ( a ( x ) ) -> c ( a ( x ) ) Rewrite Rules: [ b(c(?x)) -> b(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(a(?x)), b(b(?x)) -> b(c(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> a(b(?x)), c(c(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ b(b(a(?x_2))) = b(a(b(?x_2))), b(b(b(?x_6))) = b(a(c(?x_6))), b(b(a(?x_2))) = c(b(b(?x_2))), b(b(b(?x_6))) = c(b(c(?x_6))), c(b(a(?x))) = b(a(c(?x))), c(c(b(?x_1))) = b(a(c(?x_1))), c(b(c(?x_3))) = b(a(b(?x_3))), b(b(a(?x))) = b(c(c(?x))), b(c(b(?x_1))) = b(c(c(?x_1))), a(a(b(?x_5))) = a(b(c(?x_5))), a(b(a(?x_2))) = a(b(b(?x_2))), a(b(b(?x_6))) = a(b(c(?x_6))), c(b(a(?x_2))) = b(b(b(?x_2))), b(b(c(?x))) = b(c(b(?x))), a(a(b(?x))) = a(b(a(?x))), c(b(b(?x))) = b(b(c(?x))) ] Outer CPs: [ b(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)) = b(a(?x)), b(b(a(?x_3))) = b(a(b(?x_3))), b(b(b(?x_7))) = b(a(c(?x_7))), b(a(b(a(?x_3)))) = c(b(a(b(?x_3)))), b(a(b(b(?x_7)))) = c(b(a(c(?x_7)))), b(c(b(a(?x_3)))) = b(b(a(b(?x_3)))), b(c(b(b(?x_7)))) = b(b(a(c(?x_7)))), b(a(c(?x))) = c(b(a(?x))), b(c(c(?x))) = b(b(a(?x))), b(a(?x)) = c(b(?x)), b(b(a(?x_3))) = c(b(b(?x_3))), b(b(b(?x_7))) = c(b(c(?x_7))), b(a(b(a(?x_3)))) = c(c(b(b(?x_3)))), b(a(b(b(?x_7)))) = c(c(b(c(?x_7)))), b(c(b(a(?x_3)))) = b(c(b(b(?x_3)))), b(c(b(b(?x_7)))) = b(c(b(c(?x_7)))), b(a(c(?x))) = c(c(b(?x))), b(c(c(?x))) = b(c(b(?x))), c(b(a(?x_2))) = b(a(c(?x_2))), c(c(b(?x_3))) = b(a(c(?x_3))), c(b(c(?x_4))) = b(a(b(?x_4))), b(a(b(a(?x_2)))) = b(b(a(c(?x_2)))), b(a(c(b(?x_3)))) = b(b(a(c(?x_3)))), b(a(b(c(?x_4)))) = b(b(a(b(?x_4)))), c(b(b(a(?x_2)))) = b(b(a(c(?x_2)))), c(b(c(b(?x_3)))) = b(b(a(c(?x_3)))), c(b(b(c(?x_4)))) = b(b(a(b(?x_4)))), a(b(b(a(?x_2)))) = a(b(a(c(?x_2)))), a(b(c(b(?x_3)))) = a(b(a(c(?x_3)))), a(b(b(c(?x_4)))) = a(b(a(b(?x_4)))), b(b(b(a(?x_2)))) = c(b(a(c(?x_2)))), b(b(c(b(?x_3)))) = c(b(a(c(?x_3)))), b(b(b(c(?x_4)))) = c(b(a(b(?x_4)))), b(a(b(?x))) = b(b(a(?x))), c(b(b(?x))) = b(b(a(?x))), a(b(b(?x))) = a(b(a(?x))), b(b(b(?x))) = c(b(a(?x))), b(b(c(?x_1))) = b(c(b(?x_1))), b(b(a(?x_2))) = b(c(c(?x_2))), b(c(b(?x_3))) = b(c(c(?x_3))), b(c(b(c(?x_1)))) = b(b(c(b(?x_1)))), b(c(b(a(?x_2)))) = b(b(c(c(?x_2)))), b(c(c(b(?x_3)))) = b(b(c(c(?x_3)))), b(a(b(c(?x_1)))) = c(b(c(b(?x_1)))), b(a(b(a(?x_2)))) = c(b(c(c(?x_2)))), b(a(c(b(?x_3)))) = c(b(c(c(?x_3)))), b(c(b(?x))) = b(b(c(?x))), b(a(b(?x))) = c(b(c(?x))), a(a(b(?x_1))) = a(b(a(?x_1))), a(a(b(?x_6))) = a(b(c(?x_6))), a(b(a(b(?x_1)))) = a(a(b(a(?x_1)))), a(b(a(b(?x_6)))) = a(a(b(c(?x_6)))), a(b(a(?x))) = a(a(b(?x))), a(b(a(?x_4))) = a(b(b(?x_4))), a(b(b(?x_7))) = a(b(c(?x_7))), a(b(b(a(?x_4)))) = a(a(b(b(?x_4)))), a(b(b(b(?x_7)))) = a(a(b(c(?x_7)))), a(b(c(?x))) = a(a(b(?x))), c(b(b(?x_1))) = b(b(c(?x_1))), c(b(a(?x_4))) = b(b(b(?x_4))), b(b(b(b(?x_1)))) = c(b(b(c(?x_1)))), b(b(b(a(?x_4)))) = c(b(b(b(?x_4)))), b(a(b(b(?x_1)))) = b(b(b(c(?x_1)))), b(a(b(a(?x_4)))) = b(b(b(b(?x_4)))), c(b(b(b(?x_1)))) = b(b(b(c(?x_1)))), c(b(b(a(?x_4)))) = b(b(b(b(?x_4)))), a(b(b(b(?x_1)))) = a(b(b(c(?x_1)))), a(b(b(a(?x_4)))) = a(b(b(b(?x_4)))), b(b(c(?x))) = c(b(b(?x))), b(a(c(?x))) = b(b(b(?x))), c(b(c(?x))) = b(b(b(?x))), a(b(c(?x))) = a(b(b(?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 <2, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([],0),([(b,1)],4)], []> Critical Pair by Rules <6, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([],0)], [([(b,1)],5)]> Critical Pair by Rules <2, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([],1)], [([(c,1)],3),([(c,1)],0)]> Critical Pair by Rules <6, 1> preceded by [(b,1)] joinable by a reduction of rules <[], [([(c,1)],1),([],6)]> joinable by a reduction of rules <[([],3),([],0)], [([],2),([(b,1)],5)]> Critical Pair by Rules <0, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(b,1)],4)], [([(b,1)],5)]> Critical Pair by Rules <1, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2),([],2),([(b,1)],4)], [([(b,1)],5)]> joinable by a reduction of rules <[([],6),([],3),([],0)], [([(b,1)],5)]> Critical Pair by Rules <3, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(b,1)],5)], []> Critical Pair by Rules <0, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([],1)], [([],1),([(c,1)],0)]> Critical Pair by Rules <1, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],1),([(c,1)],3)], [([],1)]> joinable by a reduction of rules <[], [([(b,1)],6),([],3)]> joinable by a reduction of rules <[([],0)], [([],0),([(b,1)],5)]> joinable by a reduction of rules <[([],1),([],2)], [([],0),([(b,1)],5)]> Critical Pair by Rules <5, 4> preceded by [(a,1)] joinable by a reduction of rules <[([],4),([(a,1)],3)], []> joinable by a reduction of rules <[([],4)], [([(a,1)],1),([],5)]> Critical Pair by Rules <2, 5> preceded by [(a,1)] joinable by a reduction of rules <[], [([(a,1)],3),([(a,1)],0)]> Critical Pair by Rules <6, 5> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3)], []> Critical Pair by Rules <2, 6> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(b,1)],4)], [([],3),([],0)]> Critical Pair by Rules <3, 3> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],1)], []> joinable by a reduction of rules <[([(b,1)],0)], [([(b,1)],2)]> Critical Pair by Rules <4, 4> preceded by [(a,1)] joinable by a reduction of rules <[([],4),([(a,1)],3),([(a,1)],0)], []> Critical Pair by Rules <6, 6> preceded by [(c,1)] joinable by a reduction of rules <[], [([(b,1)],1),([],1)]> joinable by a reduction of rules <[([(c,1)],3)], [([],3),([],1)]> joinable by a reduction of rules <[([],2)], [([(b,1)],1),([],0)]> joinable by a reduction of rules <[([(c,1)],3),([],2)], [([],3),([],0)]> Critical Pair by Rules <1, 0> preceded by [] joinable by a reduction of rules <[([],2)], []> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 100 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(c(?x)) -> b(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(a(?x)), b(b(?x)) -> b(c(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> a(b(?x)), c(c(?x)) -> b(b(?x)) ] E-part: [ ] ...failed to find a suitable LPO. unknown Confluence by Ordered Rewriting Direct Methods: Can't judge Try Persistent Decomposition for... [ b(c(?x)) -> b(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(a(?x)), b(b(?x)) -> b(c(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> a(b(?x)), c(c(?x)) -> b(b(?x)) ] Sort Assignment: a : 14=>14 b : 14=>14 c : 14=>14 maximal types: {14} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ b(c(?x)) -> b(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(a(?x)), b(b(?x)) -> b(c(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> a(b(?x)), c(c(?x)) -> b(b(?x)) ] Layer Preserving Decomposition failed: Can't judge Try Commutative Decomposition for... [ b(c(?x)) -> b(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(a(?x)), b(b(?x)) -> b(c(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> a(b(?x)), c(c(?x)) -> b(b(?x)) ] Outside Critical Pair: by Rules <1, 0> develop reducts from lhs term... <{2}, b(a(?x_1))> <{}, c(b(?x_1))> develop reducts from rhs term... <{}, b(a(?x_1))> Inside Critical Pair: by Rules <2, 0> develop reducts from lhs term... <{3}, b(c(a(?x_2)))> <{}, b(b(a(?x_2)))> develop reducts from rhs term... <{}, b(a(b(?x_2)))> Inside Critical Pair: by Rules <6, 0> develop reducts from lhs term... <{3}, b(c(b(?x_6)))> <{3}, b(b(c(?x_6)))> <{}, b(b(b(?x_6)))> develop reducts from rhs term... <{5}, b(a(b(?x_6)))> <{}, b(a(c(?x_6)))> Inside Critical Pair: by Rules <2, 1> develop reducts from lhs term... <{3}, b(c(a(?x_2)))> <{}, b(b(a(?x_2)))> develop reducts from rhs term... <{2}, b(a(b(?x_2)))> <{3}, c(b(c(?x_2)))> <{}, c(b(b(?x_2)))> Inside Critical Pair: by Rules <6, 1> develop reducts from lhs term... <{3}, b(c(b(?x_6)))> <{3}, b(b(c(?x_6)))> <{}, b(b(b(?x_6)))> develop reducts from rhs term... <{2}, b(a(c(?x_6)))> <{1}, c(c(b(?x_6)))> <{0}, c(b(a(?x_6)))> <{}, c(b(c(?x_6)))> Inside Critical Pair: by Rules <0, 2> develop reducts from lhs term... <{2}, b(a(a(?x)))> <{}, c(b(a(?x)))> develop reducts from rhs term... <{5}, b(a(b(?x)))> <{}, b(a(c(?x)))> Inside Critical Pair: by Rules <1, 2> develop reducts from lhs term... <{6}, b(b(b(?x_1)))> <{2}, c(b(a(?x_1)))> <{}, c(c(b(?x_1)))> develop reducts from rhs term... <{5}, b(a(b(?x_1)))> <{}, b(a(c(?x_1)))> Inside Critical Pair: by Rules <3, 2> develop reducts from lhs term... <{2}, b(a(c(?x_3)))> <{1}, c(c(b(?x_3)))> <{0}, c(b(a(?x_3)))> <{}, c(b(c(?x_3)))> develop reducts from rhs term... <{}, b(a(b(?x_3)))> Inside Critical Pair: by Rules <0, 3> develop reducts from lhs term... <{3}, b(c(a(?x)))> <{}, b(b(a(?x)))> develop reducts from rhs term... <{1}, c(b(c(?x)))> <{0}, b(a(c(?x)))> <{6}, b(b(b(?x)))> <{}, b(c(c(?x)))> Inside Critical Pair: by Rules <1, 3> develop reducts from lhs term... <{1}, c(b(b(?x_1)))> <{0}, b(a(b(?x_1)))> <{2}, b(b(a(?x_1)))> <{}, b(c(b(?x_1)))> develop reducts from rhs term... <{1}, c(b(c(?x_1)))> <{0}, b(a(c(?x_1)))> <{6}, b(b(b(?x_1)))> <{}, b(c(c(?x_1)))> Inside Critical Pair: by Rules <5, 4> develop reducts from lhs term... <{4}, a(b(b(?x_5)))> <{}, a(a(b(?x_5)))> develop reducts from rhs term... <{1}, a(c(b(?x_5)))> <{0}, a(b(a(?x_5)))> <{}, a(b(c(?x_5)))> Inside Critical Pair: by Rules <2, 5> develop reducts from lhs term... <{}, a(b(a(?x_2)))> develop reducts from rhs term... <{3}, a(b(c(?x_2)))> <{}, a(b(b(?x_2)))> Inside Critical Pair: by Rules <6, 5> develop reducts from lhs term... <{3}, a(b(c(?x_6)))> <{}, a(b(b(?x_6)))> develop reducts from rhs term... <{1}, a(c(b(?x_6)))> <{0}, a(b(a(?x_6)))> <{}, a(b(c(?x_6)))> Inside Critical Pair: by Rules <2, 6> develop reducts from lhs term... <{2}, b(a(a(?x_2)))> <{}, c(b(a(?x_2)))> develop reducts from rhs term... <{3}, b(c(b(?x_2)))> <{3}, b(b(c(?x_2)))> <{}, b(b(b(?x_2)))> Commutative Decomposition failed: Can't judge No further decomposition possible Combined result: Can't judge /tmp/fileohUP8X.trs: Failure(unknown CR) (1966 msec.)