(ignored inputs)COMMENT doi:10.23638/LMCS-13 ( 2:4 ) 2017 [131] Example 5.11 submitted by: Julian Nagele
Rewrite Rules:
   [ a -> b,
     f(a,b) -> f(a,a),
     f(b,a) -> f(a,a),
     f(a,a) -> c,
     g(?x) -> f(?x,?x) ]
Apply Direct Methods...
Inner CPs:
   [ f(b,b) = f(a,a),
     f(b,b) = f(a,a),
     f(b,a) = c,
     f(a,b) = 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(a,a) = f(b,b),
     c = f(b,a),
     c = f(a,b),
     f(b,b) = f(a,a),
     f(b,b) = c,
     f(b,a) = c,
     f(a,b) = c ]
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(b,b), f(a,a)> by Rules <0, 1> preceded by [(f,1)]
 joinable by a reduction of rules <[], [([(f,2)],0),([(f,1)],0)]>
 joinable by a reduction of rules <[], [([(f,1)],0),([(f,2)],0)]>
Critical Pair <f(b,b), f(a,a)> by Rules <0, 2> preceded by [(f,2)]
 joinable by a reduction of rules <[], [([(f,2)],0),([(f,1)],0)]>
 joinable by a reduction of rules <[], [([(f,1)],0),([(f,2)],0)]>
Critical Pair <f(b,a), c> by Rules <0, 3> preceded by [(f,1)]
 joinable by a reduction of rules <[([],2),([],3)], []>
Critical Pair <f(a,b), c> by Rules <0, 3> preceded by [(f,2)]
 joinable by a reduction of rules <[([],1),([],3)], []>
unknown Diagram Decreasing
   [ a -> b,
     f(a,b) -> f(a,a),
     f(b,a) -> f(a,a),
     f(a,a) -> c,
     g(?x) -> f(?x,?x) ]
Sort Assignment:
 a : =>6
 b : =>6
 c : =>10
 f : 6*6=>10
 g : 6=>10
non-linear variables: {?x}
non-linear types: {6}
types leq non-linear types: {6}
rules applicable to terms of non-linear types:
   [ a -> b ]
Rnl:
0: {}
1: {}
2: {}
3: {}
4: {0}
terms of non-linear types are innermost terminating
Critical Pair <f(b,b), f(a,a)> by Rules <0, 1>
 convertible by a reduction of rules [(0)<-,->(2)]
 convertible by a reduction of rules [(0)<-,(0)<-]
 convertible by a reduction of rules [(4)<-,(0)<-,->(4)]
Critical Pair <f(b,b), f(a,a)> by Rules <0, 2>
 convertible by a reduction of rules [(0)<-,->(1)]
 convertible by a reduction of rules [(0)<-,(0)<-]
 convertible by a reduction of rules [(4)<-,(0)<-,->(4)]
Critical Pair <f(b,a), c> by Rules <0, 3>
 convertible by a reduction of rules [->(2),->(3)]
 convertible by a reduction of rules [->(0),(0)<-,->(1),->(3)]
 convertible by a reduction of rules [->(0),(4)<-,(0)<-,->(4),->(3)]
Critical Pair <f(a,b), c> by Rules <0, 3>
 convertible by a reduction of rules [->(1),->(3)]
 convertible by a reduction of rules [->(0),(0)<-,->(2),->(3)]
 convertible by a reduction of rules [->(0),(4)<-,(0)<-,->(4),->(3)]
Not Satisfiable
unknown Quasi-Linear & Linearized-Decreasing
   [ a -> b,
     f(a,b) -> f(a,a),
     f(b,a) -> f(a,a),
     f(a,a) -> c,
     g(?x) -> f(?x,?x) ]
Sort Assignment:
 a : =>6
 b : =>6
 c : =>10
 f : 6*6=>10
 g : 6=>10
non-linear variables: {?x}
non-linear types: {6}
types leq non-linear types: {6}
rules applicable to terms of non-linear types:
   [ a -> b ]
terms of non-linear types are terminating
Check Joinablility of CP from NLR:
done.
Critical Pair <f(b,b), f(a,a)> by Rules <0, 1>
 convertible by a reduction of rules [(0)<-,->(2)]
 convertible by a reduction of rules [(0)<-,(0)<-]
 convertible by a reduction of rules [(4)<-,(0)<-,->(4)]
Critical Pair <f(b,b), f(a,a)> by Rules <0, 2>
 convertible by a reduction of rules [(0)<-,->(1)]
 convertible by a reduction of rules [(0)<-,(0)<-]
 convertible by a reduction of rules [(4)<-,(0)<-,->(4)]
Critical Pair <f(b,a), c> by Rules <0, 3>
 convertible by a reduction of rules [->(2),->(3)]
 convertible by a reduction of rules [->(0),(0)<-,->(1),->(3)]
 convertible by a reduction of rules [->(0),(4)<-,(0)<-,->(4),->(3)]
Critical Pair <f(a,b), c> by Rules <0, 3>
 convertible by a reduction of rules [->(1),->(3)]
 convertible by a reduction of rules [->(0),(0)<-,->(2),->(3)]
 convertible by a reduction of rules [->(0),(4)<-,(0)<-,->(4),->(3)]
Not Satisfiable
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 12 rules by 3 steps unfolding
obtain 21 candidates for checking non-joinability
check by TCAP-Approximation (success)
Witness for Non-Confluence: <f(b,b) <- g(a) -> c>
Direct Methods: not CR

Combined result: not CR
762.trs: Success(not CR)
NO
(29 msec.)