(ignored inputs)COMMENT [103] Example 2.5
Rewrite Rules:
   [ f(h(?x,d),?y) -> f(h(?y,d),?x),
     h(c,?x) -> h(?x,?x) ]
Apply Direct Methods...
Inner CPs:
   [ f(h(d,d),?y) = f(h(?y,d),c) ]
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(h(d,d),?y) = f(h(?y,d),c),
     f(h(?y_2,d),c) = f(h(d,d),?y_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 <f(h(d,d),?y), f(h(?y,d),c)> by Rules <1, 0> preceded by [(f,1)]
 joinable by a reduction of rules <[], [([],0),([(f,1)],1)]>
unknown Diagram Decreasing
   [ f(h(?x,d),?y) -> f(h(?y,d),?x),
     h(c,?x_1) -> h(?x_1,?x_1) ]
Sort Assignment:
 c : =>9
 d : =>9
 f : 12*9=>13
 h : 9*9=>12
non-linear variables: {?x_1}
non-linear types: {9}
types leq non-linear types: {9}
rules applicable to terms of non-linear types:
   [ ]
Rnl:
0: {}
1: {}
terms of non-linear types are innermost terminating
Critical Pair <f(h(d,d),?y), f(h(?y,d),c)> by Rules <1, 0>
 convertible by a reduction of rules [(1)<-,(0)<-]
Not Satisfiable
unknown Quasi-Linear & Linearized-Decreasing
   [ f(h(?x,d),?y) -> f(h(?y,d),?x),
     h(c,?x_1) -> h(?x_1,?x_1) ]
Sort Assignment:
 c : =>9
 d : =>9
 f : 12*9=>13
 h : 9*9=>12
non-linear variables: {?x_1}
non-linear types: {9}
types leq non-linear types: {9}
rules applicable to terms of non-linear types:
   [ ]
terms of non-linear types are terminating
Check Joinablility of CP from NLR:
done.
Critical Pair <f(h(d,d),?y), f(h(?y,d),c)> by Rules <1, 0>
 convertible by a reduction of rules [(1)<-,(0)<-]
Not Satisfiable
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 6 rules by 3 steps unfolding
obtain 18 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...
STEP: 1 (parallel)
S:
   [ h(c,?x) -> h(?x,?x) ]
P:
   [ f(h(?x,d),?y) -> f(h(?y,d),?x) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ h(c,?x) -> h(?x,?x) ]
P:
   [ f(h(?x,d),?y) -> f(h(?y,d),?x) ]
S: not linear
failure(Step 2)
STEP: 3 (relative)
S:
   [ h(c,?x) -> h(?x,?x) ]
P:
   [ f(h(?x,d),?y) -> f(h(?y,d),?x) ]
Check relative termination:
   [ h(c,?x) -> h(?x,?x) ]
   [ f(h(?x,d),?y) -> f(h(?y,d),?x) ]
unknown relative termination
S/P: unknown relative termination
failure(Step 3)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ f(h(?x,d),?y) -> f(h(?y,d),?x),
     h(c,?x) -> h(?x,?x) ]
Sort Assignment:
 c : =>9
 d : =>9
 f : 12*9=>13
 h : 9*9=>12
maximal types: {9,12,13}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ f(h(?x,d),?y) -> f(h(?y,d),?x),
     h(c,?x) -> h(?x,?x) ]
Layer Preserving Decomposition failed: Can't judge

Try Commutative Decomposition for...
   [ f(h(?x,d),?y) -> f(h(?y,d),?x),
     h(c,?x) -> h(?x,?x) ]
Inside Critical Pair: <f(h(d,d),?y), f(h(?y,d),c)> by Rules <1, 0>
develop reducts from lhs term...
<{0}, f(h(?y,d),d)>
<{}, f(h(d,d),?y)>
develop reducts from rhs term...
<{0}, f(h(c,d),?y)>
<{}, f(h(?y,d),c)>
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
498.trs: Failure(unknown CR)
MAYBE
(893 msec.)