MAYBE (ignored inputs)COMMENT the following rules are removed from the original TRS b ( a ( x ) ) -> b ( a ( x ) ) Rewrite Rules: [ c(a(?x)) -> c(b(?x)), a(b(?x)) -> c(c(?x)), c(c(?x)) -> b(a(?x)), c(b(?x)) -> a(b(?x)), a(c(?x)) -> c(c(?x)), a(a(?x)) -> b(a(?x)) ] Apply Direct Methods... Inner CPs: [ c(c(c(?x_1))) = c(b(b(?x_1))), c(c(c(?x_4))) = c(b(c(?x_4))), c(b(a(?x_5))) = c(b(a(?x_5))), c(c(b(?x))) = b(a(a(?x))), c(a(b(?x_3))) = b(a(b(?x_3))), a(c(b(?x))) = c(c(a(?x))), a(b(a(?x_2))) = c(c(c(?x_2))), a(a(b(?x_3))) = c(c(b(?x_3))), a(c(c(?x_1))) = b(a(b(?x_1))), a(c(c(?x_4))) = b(a(c(?x_4))), c(b(a(?x))) = b(a(c(?x))), a(b(a(?x))) = b(a(a(?x))) ] Outer CPs: [ ] 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(c(c(?x_2))) = c(b(b(?x_2))), c(c(c(?x_5))) = c(b(c(?x_5))), c(b(a(?x_6))) = c(b(a(?x_6))), b(a(c(c(?x_2)))) = c(c(b(b(?x_2)))), b(a(c(c(?x_5)))) = c(c(b(c(?x_5)))), b(a(b(a(?x_6)))) = c(c(b(a(?x_6)))), c(c(c(c(?x_2)))) = a(c(b(b(?x_2)))), c(c(c(c(?x_5)))) = a(c(b(c(?x_5)))), c(c(b(a(?x_6)))) = a(c(b(a(?x_6)))), b(a(a(?x))) = c(c(b(?x))), c(c(a(?x))) = a(c(b(?x))), c(b(b(?x))) = c(c(c(?x))), b(a(b(?x))) = a(c(c(?x))), c(b(a(?x_1))) = b(a(c(?x_1))), c(c(b(?x_2))) = b(a(a(?x_2))), c(a(b(?x_4))) = b(a(b(?x_4))), b(a(b(a(?x_1)))) = c(b(a(c(?x_1)))), b(a(c(b(?x_2)))) = c(b(a(a(?x_2)))), b(a(a(b(?x_4)))) = c(b(a(b(?x_4)))), c(c(b(a(?x_1)))) = a(b(a(c(?x_1)))), c(c(c(b(?x_2)))) = a(b(a(a(?x_2)))), c(c(a(b(?x_4)))) = a(b(a(b(?x_4)))), b(a(c(?x))) = c(b(a(?x))), c(c(c(?x))) = a(b(a(?x))), b(a(b(?x))) = c(a(b(?x))), c(c(b(?x))) = a(a(b(?x))), a(c(b(?x_2))) = c(c(a(?x_2))), a(b(a(?x_4))) = c(c(c(?x_4))), a(a(b(?x_5))) = c(c(b(?x_5))), c(b(c(b(?x_2)))) = c(c(c(a(?x_2)))), c(b(b(a(?x_4)))) = c(c(c(c(?x_4)))), c(b(a(b(?x_5)))) = c(c(c(b(?x_5)))), b(a(c(b(?x_2)))) = a(c(c(a(?x_2)))), b(a(b(a(?x_4)))) = a(c(c(c(?x_4)))), b(a(a(b(?x_5)))) = a(c(c(b(?x_5)))), c(b(c(?x))) = c(c(c(?x))), b(a(c(?x))) = a(c(c(?x))), a(b(a(?x_1))) = b(a(a(?x_1))), a(c(c(?x_3))) = b(a(b(?x_3))), a(c(c(?x_6))) = b(a(c(?x_6))), b(a(b(a(?x_1)))) = a(b(a(a(?x_1)))), b(a(c(c(?x_3)))) = a(b(a(b(?x_3)))), b(a(c(c(?x_6)))) = a(b(a(c(?x_6)))), c(b(b(a(?x_1)))) = c(b(a(a(?x_1)))), c(b(c(c(?x_3)))) = c(b(a(b(?x_3)))), c(b(c(c(?x_6)))) = c(b(a(c(?x_6)))), b(a(a(?x))) = a(b(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 [(c,1)] joinable by a reduction of rules <[([(c,1)],2),([],3),([],1),([(c,1)],0)], [([],3),([],1)]> joinable by a reduction of rules <[], [([],3),([],1),([(c,1)],3),([(c,1)],1)]> joinable by a reduction of rules <[([],2),([(b,1)],4)], [([],3),([],1),([],2),([(b,1)],1)]> Critical Pair by Rules <4, 0> preceded by [(c,1)] joinable by a reduction of rules <[], [([],3),([],1)]> Critical Pair by Rules <5, 0> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <0, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(b,1)],1),([(b,1)],2)], [([(b,1)],5)]> Critical Pair by Rules <3, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],1),([],2),([(b,1)],4)], [([(b,1)],1)]> Critical Pair by Rules <0, 4> preceded by [(a,1)] joinable by a reduction of rules <[([],4)], [([(c,1)],0)]> Critical Pair by Rules <2, 4> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],2),([],3)]> Critical Pair by Rules <3, 4> preceded by [(a,1)] joinable by a reduction of rules <[([],5)], [([],2)]> Critical Pair by Rules <1, 5> preceded by [(a,1)] joinable by a reduction of rules <[([],4),([],2),([(b,1)],4)], [([(b,1)],1)]> Critical Pair by Rules <4, 5> preceded by [(a,1)] joinable by a reduction of rules <[([],4),([],2)], []> Critical Pair by Rules <2, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],3),([],1),([],2),([(b,1)],5)], [([(b,1)],4),([(b,1)],2)]> Critical Pair by Rules <5, 5> preceded by [(a,1)] joinable by a reduction of rules <[([],1),([],2)], []> unknown Diagram Decreasing check Non-Confluence... obtain 11 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: [ c(a(?x)) -> c(b(?x)), a(b(?x)) -> c(c(?x)), c(c(?x)) -> b(a(?x)), c(b(?x)) -> a(b(?x)), a(c(?x)) -> c(c(?x)), a(a(?x)) -> b(a(?x)) ] E-part: [ ] ...failed to find a suitable LPO. unknown Confluence by Ordered Rewriting Direct Methods: Can't judge Try Persistent Decomposition for... [ c(a(?x)) -> c(b(?x)), a(b(?x)) -> c(c(?x)), c(c(?x)) -> b(a(?x)), c(b(?x)) -> a(b(?x)), a(c(?x)) -> c(c(?x)), a(a(?x)) -> b(a(?x)) ] Sort Assignment: a : 13=>13 b : 13=>13 c : 13=>13 maximal types: {13} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ c(a(?x)) -> c(b(?x)), a(b(?x)) -> c(c(?x)), c(c(?x)) -> b(a(?x)), c(b(?x)) -> a(b(?x)), a(c(?x)) -> c(c(?x)), a(a(?x)) -> b(a(?x)) ] Layer Preserving Decomposition failed: Can't judge Try Commutative Decomposition for... [ c(a(?x)) -> c(b(?x)), a(b(?x)) -> c(c(?x)), c(c(?x)) -> b(a(?x)), c(b(?x)) -> a(b(?x)), a(c(?x)) -> c(c(?x)), a(a(?x)) -> b(a(?x)) ] Inside Critical Pair: by Rules <1, 0> develop reducts from lhs term... <{2}, b(a(c(?x_1)))> <{2}, c(b(a(?x_1)))> <{}, c(c(c(?x_1)))> develop reducts from rhs term... <{3}, a(b(b(?x_1)))> <{}, c(b(b(?x_1)))> Inside Critical Pair: by Rules <4, 0> develop reducts from lhs term... <{2}, b(a(c(?x_4)))> <{2}, c(b(a(?x_4)))> <{}, c(c(c(?x_4)))> develop reducts from rhs term... <{3}, a(b(c(?x_4)))> <{}, c(b(c(?x_4)))> Inside Critical Pair: by Rules <5, 0> develop reducts from lhs term... <{3}, a(b(a(?x_5)))> <{}, c(b(a(?x_5)))> develop reducts from rhs term... <{3}, a(b(a(?x_5)))> <{}, c(b(a(?x_5)))> Inside Critical Pair: by Rules <0, 2> develop reducts from lhs term... <{2}, b(a(b(?x)))> <{3}, c(a(b(?x)))> <{}, c(c(b(?x)))> develop reducts from rhs term... <{5}, b(b(a(?x)))> <{}, b(a(a(?x)))> Inside Critical Pair: by Rules <3, 2> develop reducts from lhs term... <{0}, c(b(b(?x_3)))> <{1}, c(c(c(?x_3)))> <{}, c(a(b(?x_3)))> develop reducts from rhs term... <{1}, b(c(c(?x_3)))> <{}, b(a(b(?x_3)))> Inside Critical Pair: by Rules <0, 4> develop reducts from lhs term... <{4}, c(c(b(?x)))> <{3}, a(a(b(?x)))> <{}, a(c(b(?x)))> develop reducts from rhs term... <{2}, b(a(a(?x)))> <{0}, c(c(b(?x)))> <{}, c(c(a(?x)))> Inside Critical Pair: by Rules <2, 4> develop reducts from lhs term... <{1}, c(c(a(?x_2)))> <{}, a(b(a(?x_2)))> develop reducts from rhs term... <{2}, b(a(c(?x_2)))> <{2}, c(b(a(?x_2)))> <{}, c(c(c(?x_2)))> Inside Critical Pair: by Rules <3, 4> develop reducts from lhs term... <{5}, b(a(b(?x_3)))> <{1}, a(c(c(?x_3)))> <{}, a(a(b(?x_3)))> develop reducts from rhs term... <{2}, b(a(b(?x_3)))> <{3}, c(a(b(?x_3)))> <{}, c(c(b(?x_3)))> Inside Critical Pair: by Rules <1, 5> develop reducts from lhs term... <{4}, c(c(c(?x_1)))> <{2}, a(b(a(?x_1)))> <{}, a(c(c(?x_1)))> develop reducts from rhs term... <{1}, b(c(c(?x_1)))> <{}, b(a(b(?x_1)))> Inside Critical Pair: by Rules <4, 5> develop reducts from lhs term... <{4}, c(c(c(?x_4)))> <{2}, a(b(a(?x_4)))> <{}, a(c(c(?x_4)))> develop reducts from rhs term... <{4}, b(c(c(?x_4)))> <{}, b(a(c(?x_4)))> Commutative Decomposition failed: Can't judge No further decomposition possible Combined result: Can't judge /tmp/filem2LiAN.trs: Failure(unknown CR) (2317 msec.)