(ignored inputs)COMMENT from Example 4 of \cite{OO04} Rewrite Rules: [ h(f,a,a) -> h(g,a,a), h(g,a,a) -> h(f,a,a), a -> a', h(?x,a',?y) -> h(?x,?y,?y), h(?x,?y,a') -> h(?x,?y,?y) ] Apply Direct Methods... Inner CPs: [ h(f,a',a) = h(g,a,a), h(f,a,a') = h(g,a,a), h(g,a',a) = h(f,a,a), h(g,a,a') = h(f,a,a) ] Outer CPs: [ h(?x,a',a') = h(?x,a',a') ] not Overlay, check Termination... unknown/not Terminating unknown Knuth&Bendix Left-Linear, not Right-Linear unknown Development Closed inner CP cond (upside-parallel) innter CP Cond (outside) unknown Upside-Parallel-Closed/Outside-Closed (inner) Parallel CPs: [ h(f,a,a') = h(g,a,a) {}, h(f,a',a) = h(g,a,a) {}, h(f,a',a') = h(g,a,a) {}, h(g,a,a') = h(f,a,a) {}, h(g,a',a) = h(f,a,a) {}, h(g,a',a') = h(f,a,a) {} ] unknown Toyama (Parallel CPs) Simultaneous CPs: [ h(?x,a',a') = h(?x,a',a'), h(f,a,a) = h(g,a,a'), h(f,a,a) = h(g,a',a), h(g,a,a) = h(f,a,a'), h(g,a,a) = h(f,a',a), h(g,a,a') = h(f,a,a), h(g,a',a) = h(f,a,a), h(g,a',a') = h(f,a,a), h(f,a,a') = h(g,a,a), h(f,a',a) = h(g,a,a), h(f,a',a') = h(g,a,a) ] unknown Okui (Simultaneous CPs) unknown Strongly Depth-Preserving & Non-E-Overlapping check Locally Decreasing Diagrams by Rule Labelling... Critical Pair by Rules <2, 0> preceded by [(h,2)] joinable by a reduction of rules <[([],3)], [([],1)]> Critical Pair by Rules <2, 0> preceded by [(h,3)] joinable by a reduction of rules <[([],4)], [([],1)]> Critical Pair by Rules <2, 1> preceded by [(h,2)] joinable by a reduction of rules <[([],3)], [([],0)]> Critical Pair by Rules <2, 1> preceded by [(h,3)] joinable by a reduction of rules <[([],4)], [([],0)]> Critical Pair by Rules <4, 3> preceded by [] joinable by a reduction of rules <[], []> unknown Diagram Decreasing check by Reduction-Preserving Completion... STEP: 1 (parallel) S: [ a -> a', h(?x,a',?y) -> h(?x,?y,?y), h(?x,?y,a') -> h(?x,?y,?y) ] P: [ h(f,a,a) -> h(g,a,a), h(g,a,a) -> h(f,a,a) ] S: unknown termination failure(Step 1) STEP: 2 (linear) S: [ a -> a', h(?x,a',?y) -> h(?x,?y,?y), h(?x,?y,a') -> h(?x,?y,?y) ] P: [ h(f,a,a) -> h(g,a,a), h(g,a,a) -> h(f,a,a) ] S: not linear failure(Step 2) STEP: 3 (relative) S: [ a -> a', h(?x,a',?y) -> h(?x,?y,?y), h(?x,?y,a') -> h(?x,?y,?y) ] P: [ h(f,a,a) -> h(g,a,a), h(g,a,a) -> h(f,a,a) ] Check relative termination: [ a -> a', h(?x,a',?y) -> h(?x,?y,?y), h(?x,?y,a') -> h(?x,?y,?y) ] [ h(f,a,a) -> h(g,a,a), h(g,a,a) -> h(f,a,a) ] unknown relative termination S/P: unknown relative termination failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion check Non-Confluence...Find Non-Joinable CP reducts: from h(g,a,a) Not Confluent Direct Methods: not CR Final result: not CR /local-scratch/hzankl/2012/cops2012/64.trs: Success(not CR) (105 msec.)