YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     or(?x,?y) -> or(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ T = or(T,?x),
     ?x_1 = or(F,?x_1) ]
Overlay, check Innermost Termination...
unknown Innermost 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(T,?x) = T,
     or(F,?x) = ?x,
     T = or(T,?x),
     ?x = or(F,?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 <or(T,?x_2), T> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <or(F,?x_2), ?x_2> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],1)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 rules by 3 steps unfolding
strenghten or(?x_7,F) and ?x_7
strenghten or(?x_9,T) and T
strenghten or(F,?x_2) and ?x_2
strenghten or(T,?x_2) and T
strenghten or(?x_12,?y_12) and or(?y_12,?x_12)
obtain 5 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(?x,?y) -> or(?y,?x) ]
Polynomial Interpretation:
  F:= (2)
  T:= 0
  or:= (10)+(1)*x1*x1+(15)*x1*x2+(1)*x2*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(?x,?y) -> or(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
<T, or(T,?x)> --> <T, or(T,?x)> => no
<?x, or(F,?x)> --> <?x, or(F,?x)> => no
check joinability condition:
check modulo reachablity from T to or(T,?x): 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(?x,?y) -> or(?y,?x) ]
S: terminating
CP(S,S):
CP_in(symP,S):
CP(S,symP):
<T, or(T,?x)> --> <T, or(T,?x)> => no
<?x, or(F,?x)> --> <?x, or(F,?x)> => no
check joinability condition:
check modulo reachablity from T to or(T,?x): 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(?x,?y) -> or(?y,?x) ]
Check relative termination:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x ]
   [ or(?x,?y) -> or(?y,?x) ]
Polynomial Interpretation:
  F:= (2)
  T:= 0
  or:= (10)+(1)*x1*x1+(15)*x1*x2+(1)*x2*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(?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(T,?x)> --> <T, T> => yes
<?x, or(F,?x)> --> <?x, ?x> => yes
<T, or(?x,T)> --> <T, T> => 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(?x,?y) -> or(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
190.trs: Success(CR)
(78 msec.)