(ignored inputs)COMMENT from p.246 of \cite{Klop80}
Rewrite Rules:
   [ Ap(Ap(Ap(S,?x),?y),?z) -> Ap(Ap(?x,?z),Ap(?y,?z)),
     Ap(Ap(K,?x),?y) -> ?x,
     Ap(I,?x) -> ?x,
     B(true,?x,?y) -> ?x,
     B(false,?x,?y) -> ?y,
     B(?z,?x,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?y_3 = ?y_3,
     ?y_4 = ?y_4 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
not Left-Linear, not Right-Linear
unknown Strongly Depth-Preserving & Non-E-Overlapping
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ Ap(Ap(Ap(S,?x),?y),?z) -> Ap(Ap(?x,?z),Ap(?y,?z)),
     Ap(Ap(K,?x),?y) -> ?x,
     Ap(I,?x) -> ?x,
     B(true,?x,?y) -> ?x,
     B(false,?x,?y) -> ?y,
     B(?z,?x,?x) -> ?x ]
Sort Assignment:
 B : 13*25*25=>25
 I : =>28
 K : =>28
 S : =>28
 Ap : 28*28=>28
 true : =>13
 false : =>13
maximal types: {13,25}{28}
{13,25}
(ps)Rewrite Rules:
   [ B(true,?x,?y) -> ?x,
     B(false,?x,?y) -> ?y,
     B(?z,?x,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?y = ?y,
     ?y_1 = ?y_1 ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{28}
(ps)Rewrite Rules:
   [ Ap(Ap(Ap(S,?x),?y),?z) -> Ap(Ap(?x,?z),Ap(?y,?z)),
     Ap(Ap(K,?x),?y) -> ?x,
     Ap(I,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
Development Closed
Direct Methods: CR
Result by Persistent Decomposition: CR

Final result: CR
/local-scratch/hzankl/2012/cops2012/53.trs: Success(CR)
(78 msec.)