(ignored inputs)COMMENT doi:10.1007/3-540-45744-5_49 [153] Example 20 submitted by: Raul Gutierrez and Salvador Lucas
Rewrite Rules:
   [ f(a,b,?x) -> f(?x,?x,?x),
     a -> c ]
Apply Direct Methods...
Inner CPs:
   [ f(c,b,?x) = f(?x,?x,?x) ]
Outer CPs:
   [  ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Left-Linear, not Right-Linear
unknown Development 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:
   [ f(c,b,?x) = f(?x,?x,?x),
     f(?x,?x,?x) = f(c,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 <f(c,b,?x), f(?x,?x,?x)> by Rules <1, 0> preceded by [(f,1)]
unknown Diagram Decreasing
   [ f(a,b,?x) -> f(?x,?x,?x),
     a -> c ]
Sort Assignment:
 a : =>9
 b : =>9
 c : =>9
 f : 9*9*9=>10
non-linear variables: {?x}
non-linear types: {9}
types leq non-linear types: {9}
rules applicable to terms of non-linear types:
   [ a -> c ]
Rnl:
0: {1}
1: {}
terms of non-linear types are innermost terminating
Critical Pair <f(c,b,?x), f(?x,?x,?x)> by Rules <1, 0>
no joinable sequence for some critical pairs
unknown Quasi-Linear & Linearized-Decreasing
   [ f(a,b,?x) -> f(?x,?x,?x),
     a -> c ]
Sort Assignment:
 a : =>9
 b : =>9
 c : =>9
 f : 9*9*9=>10
non-linear variables: {?x}
non-linear types: {9}
types leq non-linear types: {9}
rules applicable to terms of non-linear types:
   [ a -> c ]
terms of non-linear types are terminating
Check Joinablility of CP from NLR:
done.
Critical Pair <f(c,b,?x), f(?x,?x,?x)> by Rules <1, 0>
no joinable sequence for some critical pairs
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 10 rules by 3 steps unfolding
obtain 71 candidates for checking non-joinability
check by TCAP-Approximation (success)
Witness for Non-Confluence: <f(a,c,a) <- f(a,b,a) -> f(c,b,a)>
Direct Methods: not CR

Combined result: not CR
802.trs: Success(not CR)
NO
(26 msec.)