YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ or(T,?z_2) = or(?x,or(T,?z_2)),
     or(?x_1,?z_2) = or(?x_1,or(F,?z_2)),
     or(or(?y_3,?x_3),?z_2) = or(?x_3,or(?y_3,?z_2)),
     or(or(?x,or(?y,?z)),?z_1) = or(or(?x,?y),or(?z,?z_1)) ]
Outer CPs:
   [ T = or(?x_2,or(?y_2,T)),
     T = or(T,?x),
     or(?x_2,?y_2) = or(?x_2,or(?y_2,F)),
     ?x_1 = or(F,?x_1),
     or(?x_2,or(?y_2,?z_2)) = or(?z_2,or(?x_2,?y_2)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ or(?x_2,or(?y_2,T)) = T,
     or(T,?x) = T,
     or(?x,or(T,?z_3)) = or(T,?z_3),
     or(?x_2,or(?y_2,F)) = or(?x_2,?y_2),
     or(F,?x) = ?x,
     or(?x,or(F,?z_3)) = or(?x,?z_3),
     T = or(or(?x_1,?y_1),or(?y,T)),
     T = or(?x,or(T,T)),
     T = or(?x,or(F,T)),
     T = or(?x,or(?y,T)),
     or(?x_1,or(?y_1,?y)) = or(or(?x_1,?y_1),or(?y,F)),
     T = or(?x,or(T,F)),
     ?x = or(?x,or(F,F)),
     or(?y,?x) = or(?x,or(?y,F)),
     or(?z,or(?x_1,or(?y_1,?y))) = or(or(?x_1,?y_1),or(?y,?z)),
     or(?z,T) = or(?x,or(T,?z)),
     or(?z,?x) = or(?x,or(F,?z)),
     or(?z,or(?y,?x)) = or(?x,or(?y,?z)),
     or(?x,?y) = or(?x,or(?y,F)),
     or(?z,or(?x,?y)) = or(?x,or(?y,?z)),
     or(or(?x_1,or(?y_1,?y)),?z) = or(or(?x_1,?y_1),or(?y,?z)),
     or(T,?z) = or(?x,or(T,?z)),
     or(?x,?z) = or(?x,or(F,?z)),
     or(or(?y,?x),?z) = or(?x,or(?y,?z)),
     or(or(?x_2,or(?y_2,?y)),or(?z,?z_1)) = or(or(or(?x_2,?y_2),or(?y,?z)),?z_1),
     or(T,or(?z,?z_1)) = or(or(?x,or(T,?z)),?z_1),
     or(?x,or(?z,?z_1)) = or(or(?x,or(F,?z)),?z_1),
     or(or(?y,?x),or(?z,?z_1)) = or(or(?x,or(?y,?z)),?z_1),
     or(or(?x,?y),or(?z,?z_1)) = or(or(?x,or(?y,?z)),?z_1),
     T = or(T,?x),
     ?x = or(F,?x),
     or(?x_3,or(?y_3,?y)) = or(?y,or(?x_3,?y_3)),
     or(?x,or(?y,?z_4)) = or(or(?y,?x),?z_4) ]
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 <or(T,?z_2), or(?x,or(T,?z_2))> by Rules <0, 2> preceded by [(or,1)]
 joinable by a reduction of rules <[([],3),([],0)], [([(or,2)],3),([(or,2)],0),([],0)]>
Critical Pair <or(?x_1,?z_2), or(?x_1,or(F,?z_2))> by Rules <1, 2> preceded by [(or,1)]
 joinable by a reduction of rules <[], [([(or,2)],3),([(or,2)],1)]>
Critical Pair <or(or(?y_3,?x_3),?z_2), or(?x_3,or(?y_3,?z_2))> by Rules <3, 2> preceded by [(or,1)]
 joinable by a reduction of rules <[([(or,1)],3),([],2)], []>
 joinable by a reduction of rules <[([],2),([(or,2)],3)], [([],3),([],2)]>
Critical Pair <or(or(?x,or(?y,?z)),?z_1), or(or(?x,?y),or(?z,?z_1))> by Rules <2, 2> preceded by [(or,1)]
 joinable by a reduction of rules <[([],2),([(or,2)],2)], [([],2)]>
Critical Pair <or(?x_2,or(?y_2,T)), T> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([(or,2)],0),([],0)], []>
Critical Pair <or(T,?x_3), T> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([],3),([],0)], []>
Critical Pair <or(?x_2,or(?y_2,F)), or(?x_2,?y_2)> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([(or,2)],1)], []>
Critical Pair <or(F,?x_3), ?x_3> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([],3),([],1)], []>
Critical Pair <or(?y_3,or(?x_2,?y_2)), or(?x_2,or(?y_2,?y_3))> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[([],3),([],2)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 14 rules by 3 steps unfolding
strenghten or(?x_8,F) and ?x_8
strenghten or(?x_10,T) and T
strenghten or(F,?x_3) and ?x_3
strenghten or(T,?x_3) and T
strenghten or(?x_13,?y_13) and or(?y_13,?x_13)
strenghten or(?x_2,or(?y_2,T)) and T
strenghten or(?x,or(T,?z_2)) and or(T,?z_2)
strenghten or(?x_1,or(F,?z_2)) and or(?x_1,?z_2)
strenghten or(?x_2,or(?y_2,F)) and or(?x_2,?y_2)
strenghten or(F,or(?x_8,?z_2)) and or(?x_8,?z_2)
strenghten or(T,or(?x_10,?z_2)) and or(T,?z_2)
strenghten or(?x_2,or(?y_2,?y_3)) and or(?y_3,or(?x_2,?y_2))
strenghten or(?x_2,or(?y_2,?y_12)) and or(or(?x_2,?y_2),?y_12)
strenghten or(?x_3,or(?y_3,?z_2)) and or(or(?y_3,?x_3),?z_2)
strenghten or(or(?x,?y),or(?z,?z_1)) and or(or(?x,or(?y,?z)),?z_1)
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-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
Check relative termination:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x ]
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
Polynomial Interpretation:
  F:= 0
  T:= (4)
  or:= (2)+(1)*x1+(1)*x2
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x ]
P:
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
<T, or(?x_1,or(?y_1,T))> --> <T, T> => yes
<or(T,?z_1), or(?x,or(T,?z_1))> --> <or(T,?z_1), or(?x,or(T,?z_1))> => no
<or(?x_1,T), or(or(?x_1,?x),T)> --> <T, T> => yes
<T, or(T,?x)> --> <T, or(T,?x)> => no
<or(?x_1,?y_1), or(?x_1,or(?y_1,F))> --> <or(?x_1,?y_1), or(?x_1,?y_1)> => yes
<or(?x,?z_1), or(?x,or(F,?z_1))> --> <or(?x,?z_1), or(?x,or(F,?z_1))> => no
<or(?x_1,?x), or(or(?x_1,?x),F)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<?x, or(F,?x)> --> <?x, or(F,?x)> => no
check joinability condition:
check modulo reachablity from or(T,?z_1) to or(?x,or(T,?z_1)): maybe not reachable
check modulo reachablity from T to or(T,?x): maybe not reachable
check modulo reachablity from or(?x,?z_1) to or(?x,or(F,?z_1)): maybe not reachable
check modulo reachablity from ?x to or(F,?x): maybe not reachable
failed
failure(Step 1)
   [ or(T,?x) -> T,
     or(F,?x) -> ?x ]
Added S-Rules:
   [ or(T,?x) -> T,
     or(F,?x) -> ?x ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x ]
P:
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
S: terminating
CP(S,S):
CP_in(symP,S):
CP(S,symP):
<T, or(?x_1,or(?y_1,T))> --> <T, T> => yes
<or(T,?z_1), or(?x,or(T,?z_1))> --> <or(T,?z_1), or(?x,or(T,?z_1))> => no
<or(?x_1,T), or(or(?x_1,?x),T)> --> <T, T> => yes
<T, or(T,?x)> --> <T, or(T,?x)> => no
<or(?x_1,?y_1), or(?x_1,or(?y_1,F))> --> <or(?x_1,?y_1), or(?x_1,?y_1)> => yes
<or(?x,?z_1), or(?x,or(F,?z_1))> --> <or(?x,?z_1), or(?x,or(F,?z_1))> => no
<or(?x_1,?x), or(or(?x_1,?x),F)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<?x, or(F,?x)> --> <?x, or(F,?x)> => no
check joinability condition:
check modulo reachablity from or(T,?z_1) to or(?x,or(T,?z_1)): maybe not reachable
check modulo reachablity from T to or(T,?x): maybe not reachable
check modulo reachablity from or(?x,?z_1) to or(?x,or(F,?z_1)): maybe not reachable
check modulo reachablity from ?x to or(F,?x): maybe not reachable
failed
failure(Step 2)
   [ or(T,?x) -> T,
     or(F,?x) -> ?x ]
Added S-Rules:
   [ or(T,?x) -> T,
     or(F,?x) -> ?x ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x ]
P:
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
Check relative termination:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x ]
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
Polynomial Interpretation:
  F:= 0
  T:= (4)
  or:= (2)+(1)*x1+(1)*x2
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     or(T,?x) -> T,
     or(F,?x) -> ?x ]
P:
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
S: terminating
CP(S,S):
<T, T> --> <T, T> => yes
<T, T> --> <T, T> => yes
<T, T> --> <T, T> => yes
<F, F> --> <F, F> => yes
PCP_in(symP,S):
CP(S,symP):
<T, or(?x_1,or(?y_1,T))> --> <T, T> => yes
<or(T,?z_1), or(?x,or(T,?z_1))> --> <T, T> => yes
<or(?x_1,T), or(or(?x_1,?x),T)> --> <T, T> => yes
<T, or(T,?x)> --> <T, T> => yes
<or(?x_1,?y_1), or(?x_1,or(?y_1,F))> --> <or(?x_1,?y_1), or(?x_1,?y_1)> => yes
<or(?x,?z_1), or(?x,or(F,?z_1))> --> <or(?x,?z_1), or(?x,?z_1)> => yes
<or(?x_1,?x), or(or(?x_1,?x),F)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<?x, or(F,?x)> --> <?x, ?x> => yes
<or(T,?z_1), or(T,or(?x,?z_1))> --> <T, T> => yes
<T, or(or(T,?y_1),?z_1)> --> <T, T> => yes
<or(?x_1,T), or(or(?x_1,T),?x)> --> <T, T> => yes
<T, or(?x,T)> --> <T, T> => yes
<or(?x,?z_1), or(F,or(?x,?z_1))> --> <or(?x,?z_1), or(?x,?z_1)> => yes
<or(?y_1,?z_1), or(or(F,?y_1),?z_1)> --> <or(?y_1,?z_1), or(?y_1,?z_1)> => yes
<or(?x_1,?x), or(or(?x_1,F),?x)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<?x, or(?x,F)> --> <?x, ?x> => yes
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     or(T,?x) -> T,
     or(F,?x) -> ?x ]
P:
   [ or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     or(?x,?y) -> or(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
153.trs: Success(CR)
(2288 msec.)