MAYBE (ignored inputs)COMMENT the following rules are removed from the original TRS a ( a ( x ) ) -> a ( a ( x ) ) Rewrite Rules: [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] Apply Direct Methods... Inner CPs: [ a(c(b(?x_3))) = c(a(a(?x_3))), a(a(b(?x_6))) = c(a(b(?x_6))), a(a(c(?x_4))) = c(a(b(?x_4))), a(a(c(?x_5))) = c(a(a(?x_5))), a(c(a(?x))) = a(c(c(?x))), a(c(a(?x_1))) = a(c(b(?x_1))), a(c(a(?x_7))) = a(c(a(?x_7))), c(c(a(?x))) = c(b(c(?x))), c(c(a(?x_1))) = c(b(b(?x_1))), c(a(c(?x_2))) = c(b(a(?x_2))), c(c(a(?x_7))) = c(b(a(?x_7))), b(a(c(?x_5))) = a(c(a(?x_5))), b(c(a(?x))) = a(c(c(?x))), b(c(a(?x_1))) = a(c(b(?x_1))), b(a(c(?x_2))) = a(c(a(?x_2))), b(c(a(?x_7))) = a(c(a(?x_7))), c(a(c(?x_4))) = a(b(b(?x_4))), c(a(c(?x_5))) = a(b(a(?x_5))), a(c(a(?x))) = c(a(c(?x))), a(c(a(?x_1))) = c(a(b(?x_1))), a(a(c(?x_2))) = c(a(a(?x_2))), a(a(c(?x))) = a(c(a(?x))), b(a(c(?x))) = a(c(b(?x))), a(c(a(?x))) = c(a(a(?x))) ] Outer CPs: [ a(c(?x_2)) = c(a(?x_2)) ] 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: [ a(c(b(?x_4))) = c(a(a(?x_4))), a(a(b(?x_7))) = c(a(b(?x_7))), a(c(c(b(?x_4)))) = a(c(a(a(?x_4)))), a(c(a(b(?x_7)))) = a(c(a(b(?x_7)))), c(b(c(b(?x_4)))) = c(c(a(a(?x_4)))), c(b(a(b(?x_7)))) = c(c(a(b(?x_7)))), a(c(c(b(?x_4)))) = b(c(a(a(?x_4)))), a(c(a(b(?x_7)))) = b(c(a(b(?x_7)))), c(a(c(b(?x_4)))) = a(c(a(a(?x_4)))), c(a(a(b(?x_7)))) = a(c(a(b(?x_7)))), a(c(c(?x))) = a(c(a(?x))), c(b(c(?x))) = c(c(a(?x))), a(c(c(?x))) = b(c(a(?x))), c(a(c(?x))) = a(c(a(?x))), a(a(c(?x_5))) = c(a(b(?x_5))), a(a(c(?x_6))) = c(a(a(?x_6))), a(c(a(c(?x_5)))) = a(c(a(b(?x_5)))), a(c(a(c(?x_6)))) = a(c(a(a(?x_6)))), c(b(a(c(?x_5)))) = c(c(a(b(?x_5)))), c(b(a(c(?x_6)))) = c(c(a(a(?x_6)))), a(c(a(c(?x_5)))) = b(c(a(b(?x_5)))), a(c(a(c(?x_6)))) = b(c(a(a(?x_6)))), c(a(a(c(?x_5)))) = a(c(a(b(?x_5)))), c(a(a(c(?x_6)))) = a(c(a(a(?x_6)))), a(c(b(?x))) = a(c(a(?x))), c(b(b(?x))) = c(c(a(?x))), a(c(b(?x))) = b(c(a(?x))), c(a(b(?x))) = a(c(a(?x))), c(a(?x)) = a(c(?x)), a(a(c(?x_1))) = a(c(a(?x_1))), a(c(a(?x_2))) = a(c(c(?x_2))), a(c(a(?x_3))) = a(c(b(?x_3))), a(c(a(?x_8))) = a(c(a(?x_8))), a(c(a(c(?x_1)))) = a(a(c(a(?x_1)))), a(c(c(a(?x_2)))) = a(a(c(c(?x_2)))), a(c(c(a(?x_3)))) = a(a(c(b(?x_3)))), a(c(c(a(?x_8)))) = a(a(c(a(?x_8)))), c(b(a(c(?x_1)))) = c(a(c(a(?x_1)))), c(b(c(a(?x_2)))) = c(a(c(c(?x_2)))), c(b(c(a(?x_3)))) = c(a(c(b(?x_3)))), c(b(c(a(?x_8)))) = c(a(c(a(?x_8)))), a(c(a(c(?x_1)))) = b(a(c(a(?x_1)))), a(c(c(a(?x_2)))) = b(a(c(c(?x_2)))), a(c(c(a(?x_3)))) = b(a(c(b(?x_3)))), a(c(c(a(?x_8)))) = b(a(c(a(?x_8)))), c(a(a(c(?x_1)))) = a(a(c(a(?x_1)))), c(a(c(a(?x_2)))) = a(a(c(c(?x_2)))), c(a(c(a(?x_3)))) = a(a(c(b(?x_3)))), c(a(c(a(?x_8)))) = a(a(c(a(?x_8)))), a(c(a(?x))) = a(a(c(?x))), c(b(a(?x))) = c(a(c(?x))), a(c(a(?x))) = b(a(c(?x))), c(a(a(?x))) = a(a(c(?x))), c(c(a(?x_2))) = c(b(c(?x_2))), c(c(a(?x_3))) = c(b(b(?x_3))), c(a(c(?x_4))) = c(b(a(?x_4))), c(c(a(?x_8))) = c(b(a(?x_8))), c(a(c(a(?x_2)))) = a(c(b(c(?x_2)))), c(a(c(a(?x_3)))) = a(c(b(b(?x_3)))), c(a(a(c(?x_4)))) = a(c(b(a(?x_4)))), c(a(c(a(?x_8)))) = a(c(b(a(?x_8)))), c(a(a(?x))) = a(c(b(?x))), b(a(c(?x_1))) = a(c(b(?x_1))), b(a(c(?x_6))) = a(c(a(?x_6))), a(c(a(c(?x_1)))) = b(a(c(b(?x_1)))), c(a(a(c(?x_1)))) = a(a(c(b(?x_1)))), a(b(a(c(?x_1)))) = c(a(c(b(?x_1)))), a(b(a(c(?x_6)))) = c(a(c(a(?x_6)))), a(c(b(?x))) = b(a(c(?x))), c(a(b(?x))) = a(a(c(?x))), a(b(b(?x))) = c(a(c(?x))), b(c(a(?x_2))) = a(c(c(?x_2))), b(c(a(?x_3))) = a(c(b(?x_3))), b(c(a(?x_8))) = a(c(a(?x_8))), a(b(c(a(?x_2)))) = c(a(c(c(?x_2)))), a(b(c(a(?x_3)))) = c(a(c(b(?x_3)))), a(b(c(a(?x_8)))) = c(a(c(a(?x_8)))), a(b(a(?x))) = c(a(c(?x))), c(a(c(?x_6))) = a(b(b(?x_6))), c(a(c(?x_7))) = a(b(a(?x_7))), c(a(a(c(?x_6)))) = a(a(b(b(?x_6)))), c(a(a(c(?x_7)))) = a(a(b(a(?x_7)))), c(a(b(?x))) = a(a(b(?x))), a(c(?x)) = c(a(?x)), a(c(a(?x_1))) = c(a(a(?x_1))), a(c(a(?x_2))) = c(a(c(?x_2))), a(c(a(?x_3))) = c(a(b(?x_3))), c(a(c(a(?x_1)))) = a(c(a(a(?x_1)))), c(a(c(a(?x_2)))) = a(c(a(c(?x_2)))), c(a(c(a(?x_3)))) = a(c(a(b(?x_3)))), a(c(c(a(?x_1)))) = a(c(a(a(?x_1)))), a(c(c(a(?x_2)))) = a(c(a(c(?x_2)))), a(c(c(a(?x_3)))) = a(c(a(b(?x_3)))), c(b(c(a(?x_1)))) = c(c(a(a(?x_1)))), c(b(c(a(?x_2)))) = c(c(a(c(?x_2)))), c(b(c(a(?x_3)))) = c(c(a(b(?x_3)))), a(c(c(a(?x_1)))) = b(c(a(a(?x_1)))), a(c(c(a(?x_2)))) = b(c(a(c(?x_2)))), a(c(c(a(?x_3)))) = b(c(a(b(?x_3)))), c(a(a(?x))) = a(c(a(?x))), c(b(a(?x))) = c(c(a(?x))), a(c(a(?x))) = b(c(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 <3, 0> preceded by [(a,1)] joinable by a reduction of rules <[([],0),([(c,1)],1)], [([(c,1)],7)]> joinable by a reduction of rules <[([],0),([(c,1)],1)], [([(c,1)],2),([(c,1)],0)]> Critical Pair by Rules <6, 0> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], []> Critical Pair by Rules <4, 1> preceded by [(a,1)] joinable by a reduction of rules <[([],7),([(c,1)],0)], [([(c,1)],1)]> joinable by a reduction of rules <[([],7)], [([],3),([(c,1)],4)]> joinable by a reduction of rules <[([],2),([],0)], [([],3),([(c,1)],4)]> Critical Pair by Rules <5, 1> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], [([(c,1)],2)]> Critical Pair by Rules <0, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],0),([(c,1)],2)], [([],0)]> joinable by a reduction of rules <[([],0),([(c,1)],7)], [([],0),([(c,1)],0)]> Critical Pair by Rules <1, 2> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3)], []> Critical Pair by Rules <7, 2> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <0, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([],6),([],1),([(c,1)],0)]> Critical Pair by Rules <1, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],4),([(c,1)],0)]> joinable by a reduction of rules <[([(c,1)],3),([(c,1)],6)], [([],6),([],1)]> Critical Pair by Rules <2, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],5)]> Critical Pair by Rules <7, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],5),([(c,1)],0)]> Critical Pair by Rules <5, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],5),([],0)], [([],0),([(c,1)],2)]> Critical Pair by Rules <0, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],3),([(b,1)],6),([],5),([],0)], [([],0),([(c,1)],0),([(c,1)],3),([(c,1)],6)]> Critical Pair by Rules <1, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],3),([(b,1)],6),([],5)], []> Critical Pair by Rules <2, 5> preceded by [(b,1)] joinable by a reduction of rules <[([],5),([],0)], [([],0),([(c,1)],2)]> Critical Pair by Rules <7, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],3),([(b,1)],6),([],5)], [([(a,1)],3)]> Critical Pair by Rules <4, 6> preceded by [(c,1)] joinable by a reduction of rules <[], [([(a,1)],4),([],7)]> joinable by a reduction of rules <[([(c,1)],0)], [([],1),([(c,1)],1)]> Critical Pair by Rules <5, 6> preceded by [(c,1)] joinable by a reduction of rules <[], [([(a,1)],5),([],7)]> joinable by a reduction of rules <[], [([],1),([(c,1)],2)]> joinable by a reduction of rules <[([(c,1)],0)], [([],1),([(c,1)],7)]> Critical Pair by Rules <0, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],0),([(c,1)],7)], [([(c,1)],0)]> joinable by a reduction of rules <[([],0),([(c,1)],2)], []> Critical Pair by Rules <1, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3),([],0)], []> joinable by a reduction of rules <[([],0),([(c,1)],7)], [([(c,1)],1)]> joinable by a reduction of rules <[([],0),([(c,1)],2)], [([],3),([(c,1)],4)]> Critical Pair by Rules <2, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], [([(c,1)],2)]> Critical Pair by Rules <2, 2> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], []> Critical Pair by Rules <4, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],5),([],0),([(c,1)],0)], [([],0),([(c,1)],1)]> joinable by a reduction of rules <[([],5),([],0)], [([],0),([],3),([(c,1)],4)]> joinable by a reduction of rules <[([],5),([],0),([(c,1)],0)], [([(a,1)],6),([],7),([(c,1)],1)]> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <7, 2> preceded by [] joinable by a reduction of rules <[], [([],0)]> unknown Diagram Decreasing check Non-Confluence... obtain 14 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: [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] E-part: [ ] ...failed to find a suitable LPO. unknown Confluence by Ordered Rewriting Direct Methods: Can't judge Try Persistent Decomposition for... [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(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... [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] Layer Preserving Decomposition failed: Can't judge Try Commutative Decomposition for... [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] Outside Critical Pair: by Rules <7, 2> develop reducts from lhs term... <{3}, c(b(?x_7))> <{}, c(a(?x_7))> develop reducts from rhs term... <{0}, c(a(?x_7))> <{}, a(c(?x_7))> Inside Critical Pair: by Rules <3, 0> develop reducts from lhs term... <{0}, c(a(b(?x_3)))> <{6}, a(a(b(?x_3)))> <{}, a(c(b(?x_3)))> develop reducts from rhs term... <{3}, c(b(a(?x_3)))> <{7}, c(c(a(?x_3)))> <{2}, c(a(c(?x_3)))> <{}, c(a(a(?x_3)))> Inside Critical Pair: by Rules <6, 0> develop reducts from lhs term... <{7}, c(a(b(?x_6)))> <{2}, a(c(b(?x_6)))> <{1}, a(c(a(?x_6)))> <{}, a(a(b(?x_6)))> develop reducts from rhs term... <{3}, c(b(b(?x_6)))> <{1}, c(c(a(?x_6)))> <{}, c(a(b(?x_6)))> Inside Critical Pair: by Rules <4, 1> develop reducts from lhs term... <{7}, c(a(c(?x_4)))> <{2}, a(c(c(?x_4)))> <{0}, a(c(a(?x_4)))> <{}, a(a(c(?x_4)))> develop reducts from rhs term... <{3}, c(b(b(?x_4)))> <{1}, c(c(a(?x_4)))> <{}, c(a(b(?x_4)))> Inside Critical Pair: by Rules <5, 1> develop reducts from lhs term... <{7}, c(a(c(?x_5)))> <{2}, a(c(c(?x_5)))> <{0}, a(c(a(?x_5)))> <{}, a(a(c(?x_5)))> develop reducts from rhs term... <{3}, c(b(a(?x_5)))> <{7}, c(c(a(?x_5)))> <{2}, c(a(c(?x_5)))> <{}, c(a(a(?x_5)))> Inside Critical Pair: by Rules <0, 2> develop reducts from lhs term... <{0}, c(a(a(?x)))> <{3}, a(c(b(?x)))> <{}, a(c(a(?x)))> develop reducts from rhs term... <{0}, c(a(c(?x)))> <{}, a(c(c(?x)))> Inside Critical Pair: by Rules <1, 2> develop reducts from lhs term... <{0}, c(a(a(?x_1)))> <{3}, a(c(b(?x_1)))> <{}, a(c(a(?x_1)))> develop reducts from rhs term... <{0}, c(a(b(?x_1)))> <{6}, a(a(b(?x_1)))> <{}, a(c(b(?x_1)))> Inside Critical Pair: by Rules <7, 2> develop reducts from lhs term... <{0}, c(a(a(?x_7)))> <{3}, a(c(b(?x_7)))> <{}, a(c(a(?x_7)))> develop reducts from rhs term... <{0}, c(a(a(?x_7)))> <{3}, a(c(b(?x_7)))> <{}, a(c(a(?x_7)))> Inside Critical Pair: by Rules <0, 3> develop reducts from lhs term... <{3}, c(c(b(?x)))> <{}, c(c(a(?x)))> develop reducts from rhs term... <{6}, a(b(c(?x)))> <{}, c(b(c(?x)))> Inside Critical Pair: by Rules <1, 3> develop reducts from lhs term... <{3}, c(c(b(?x_1)))> <{}, c(c(a(?x_1)))> develop reducts from rhs term... <{6}, a(b(b(?x_1)))> <{4}, c(a(c(?x_1)))> <{}, c(b(b(?x_1)))> Inside Critical Pair: by Rules <2, 3> develop reducts from lhs term... <{3}, c(b(c(?x_2)))> <{0}, c(c(a(?x_2)))> <{}, c(a(c(?x_2)))> develop reducts from rhs term... <{6}, a(b(a(?x_2)))> <{5}, c(a(c(?x_2)))> <{}, c(b(a(?x_2)))> Inside Critical Pair: by Rules <7, 3> develop reducts from lhs term... <{3}, c(c(b(?x_7)))> <{}, c(c(a(?x_7)))> develop reducts from rhs term... <{6}, a(b(a(?x_7)))> <{5}, c(a(c(?x_7)))> <{}, c(b(a(?x_7)))> Inside Critical Pair: by Rules <5, 4> develop reducts from lhs term... <{5}, a(c(c(?x_5)))> <{0}, b(c(a(?x_5)))> <{}, b(a(c(?x_5)))> develop reducts from rhs term... <{0}, c(a(a(?x_5)))> <{3}, a(c(b(?x_5)))> <{}, a(c(a(?x_5)))> Inside Critical Pair: by Rules <0, 5> develop reducts from lhs term... <{3}, b(c(b(?x)))> <{}, b(c(a(?x)))> develop reducts from rhs term... <{0}, c(a(c(?x)))> <{}, a(c(c(?x)))> Inside Critical Pair: by Rules <1, 5> develop reducts from lhs term... <{3}, b(c(b(?x_1)))> <{}, b(c(a(?x_1)))> develop reducts from rhs term... <{0}, c(a(b(?x_1)))> <{6}, a(a(b(?x_1)))> <{}, a(c(b(?x_1)))> Inside Critical Pair: by Rules <2, 5> develop reducts from lhs term... <{5}, a(c(c(?x_2)))> <{0}, b(c(a(?x_2)))> <{}, b(a(c(?x_2)))> develop reducts from rhs term... <{0}, c(a(a(?x_2)))> <{3}, a(c(b(?x_2)))> <{}, a(c(a(?x_2)))> Inside Critical Pair: by Rules <7, 5> develop reducts from lhs term... <{3}, b(c(b(?x_7)))> <{}, b(c(a(?x_7)))> develop reducts from rhs term... <{0}, c(a(a(?x_7)))> <{3}, a(c(b(?x_7)))> <{}, a(c(a(?x_7)))> Inside Critical Pair: by Rules <4, 6> develop reducts from lhs term... <{3}, c(b(c(?x_4)))> <{0}, c(c(a(?x_4)))> <{}, c(a(c(?x_4)))> develop reducts from rhs term... <{1}, c(a(b(?x_4)))> <{4}, a(a(c(?x_4)))> <{}, a(b(b(?x_4)))> Inside Critical Pair: by Rules <5, 6> develop reducts from lhs term... <{3}, c(b(c(?x_5)))> <{0}, c(c(a(?x_5)))> <{}, c(a(c(?x_5)))> develop reducts from rhs term... <{1}, c(a(a(?x_5)))> <{5}, a(a(c(?x_5)))> <{}, a(b(a(?x_5)))> Inside Critical Pair: by Rules <0, 7> develop reducts from lhs term... <{0}, c(a(a(?x)))> <{3}, a(c(b(?x)))> <{}, a(c(a(?x)))> develop reducts from rhs term... <{3}, c(b(c(?x)))> <{0}, c(c(a(?x)))> <{}, c(a(c(?x)))> Inside Critical Pair: by Rules <1, 7> develop reducts from lhs term... <{0}, c(a(a(?x_1)))> <{3}, a(c(b(?x_1)))> <{}, a(c(a(?x_1)))> develop reducts from rhs term... <{3}, c(b(b(?x_1)))> <{1}, c(c(a(?x_1)))> <{}, c(a(b(?x_1)))> Inside Critical Pair: by Rules <2, 7> develop reducts from lhs term... <{7}, c(a(c(?x_2)))> <{2}, a(c(c(?x_2)))> <{0}, a(c(a(?x_2)))> <{}, a(a(c(?x_2)))> develop reducts from rhs term... <{3}, c(b(a(?x_2)))> <{7}, c(c(a(?x_2)))> <{2}, c(a(c(?x_2)))> <{}, c(a(a(?x_2)))> Commutative Decomposition failed: Can't judge No further decomposition possible Combined result: Can't judge /tmp/fileYggbB4.trs: Failure(unknown CR) (2903 msec.)