MAYBE (ignored inputs)COMMENT the following rules are removed from the original TRS a ( b ( x ) ) -> a ( b ( x ) ) Rewrite Rules: [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] Apply Direct Methods... Inner CPs: [ c(c(c(?x))) = a(a(a(?x))), c(a(a(?x_1))) = c(a(a(?x_1))), c(a(c(?x_2))) = c(a(b(?x_2))), a(c(c(?x))) = c(c(a(?x))), c(c(a(?x))) = c(a(c(?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: [ a(c(c(?x_1))) = c(c(a(?x_1))), c(c(c(c(?x_1)))) = a(c(c(a(?x_1)))), a(a(c(c(?x_1)))) = c(c(c(a(?x_1)))), c(c(a(?x))) = a(c(c(?x))), a(a(a(?x))) = c(c(c(?x))), c(c(c(?x_2))) = a(a(a(?x_2))), c(a(c(c(?x_2)))) = c(a(a(a(?x_2)))), c(a(a(?x))) = c(a(a(?x))), c(a(b(?x))) = c(a(c(?x))), c(c(a(?x_1))) = c(a(c(?x_1))), c(a(c(?x_4))) = c(a(b(?x_4))), c(a(c(a(?x_1)))) = c(c(a(c(?x_1)))), c(a(a(a(?x_3)))) = c(c(a(a(?x_3)))), c(a(a(c(?x_4)))) = c(c(a(b(?x_4)))), c(a(c(?x))) = c(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 <0, 1> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],3)], [([],0)]> Critical Pair by Rules <1, 3> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <2, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([],1),([],0),([(c,1)],2)]> Critical Pair by Rules <0, 0> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3),([(a,1)],1)], [([(c,1)],1),([],1)]> joinable by a reduction of rules <[([(a,1)],3),([(a,1)],1)], [([],3),([],1)]> Critical Pair by Rules <3, 3> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],1),([(c,1)],0)], [([],1),([],0)]> joinable by a reduction of rules <[([],3),([(c,1)],0)], [([],1),([],0)]> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 74 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(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?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(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] Sort Assignment: a : 11=>11 b : 11=>11 c : 11=>11 maximal types: {11} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] Layer Preserving Decomposition failed: Can't judge Try Commutative Decomposition for... [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] Inside Critical Pair: by Rules <0, 1> develop reducts from lhs term... <{3}, c(a(c(?x)))> <{3}, c(c(a(?x)))> <{}, c(c(c(?x)))> develop reducts from rhs term... <{0}, c(c(a(?x)))> <{0}, a(c(c(?x)))> <{}, a(a(a(?x)))> Inside Critical Pair: by Rules <1, 3> develop reducts from lhs term... <{1}, a(a(a(?x_1)))> <{0}, c(c(c(?x_1)))> <{}, c(a(a(?x_1)))> develop reducts from rhs term... <{1}, a(a(a(?x_1)))> <{0}, c(c(c(?x_1)))> <{}, c(a(a(?x_1)))> Inside Critical Pair: by Rules <2, 3> develop reducts from lhs term... <{1}, a(a(c(?x_2)))> <{}, c(a(c(?x_2)))> develop reducts from rhs term... <{1}, a(a(b(?x_2)))> <{}, c(a(b(?x_2)))> Try A Minimal Decomposition {2,1,3}{0} {2,1,3} (cm)Rewrite Rules: [ c(b(?x)) -> a(c(?x)), c(a(?x)) -> a(a(?x)), c(c(?x)) -> c(a(?x)) ] Apply Direct Methods... Inner CPs: [ c(a(c(?x))) = c(a(b(?x))), c(a(a(?x_1))) = c(a(a(?x_1))), c(c(a(?x))) = c(a(c(?x))) ] Outer CPs: [ ] not Overlay, check Termination... Terminating, not WCR Knuth & Bendix Direct Methods: not CR {0} (cm)Rewrite Rules: [ a(a(?x)) -> c(c(?x)) ] Apply Direct Methods... Inner CPs: [ a(c(c(?x))) = c(c(a(?x))) ] Outer CPs: [ ] not Overlay, check Termination... Terminating, not WCR Knuth & Bendix Direct Methods: not CR Try A Minimal Decomposition {0,1}{2,3} {0,1} (cm)Rewrite Rules: [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ c(c(c(?x))) = a(a(a(?x))), a(c(c(?x))) = c(c(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: [ a(c(c(?x_1))) = c(c(a(?x_1))), c(c(c(c(?x_1)))) = a(c(c(a(?x_1)))), a(a(c(c(?x_1)))) = c(c(c(a(?x_1)))), c(c(a(?x))) = a(c(c(?x))), a(a(a(?x))) = c(c(c(?x))), c(c(c(?x_2))) = a(a(a(?x_2))) ] 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 <0, 1> preceded by [(c,1)] joinable by a reduction of rules <[], [([],0),([(c,1)],1),([(c,1)],0)]> Critical Pair by Rules <0, 0> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],1),([],1),([(a,1)],0)]> unknown Diagram Decreasing check Non-Confluence... obtain 8 rules by 3 steps unfolding obtain 100 candidates for checking non-joinability check by TCAP-Approximation (success) (success) (success) (failure) check by Ordering(rpo), check by Tree-Automata Approximation (success) Witness for Non-Confluence: c(c(c(c_1)))> Direct Methods: not CR {2,3} (cm)Rewrite Rules: [ c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] Apply Direct Methods... Inner CPs: [ c(a(c(?x))) = c(a(b(?x))), c(c(a(?x))) = c(a(c(?x))) ] Outer CPs: [ ] not Overlay, check Termination... Terminating, not WCR Knuth & Bendix Direct Methods: not CR Commutative Decomposition failed: Can't judge No further decomposition possible Combined result: Can't judge /tmp/file9sy6dE.trs: Failure(unknown CR) (1858 msec.)