YES
(ignored inputs)COMMENT the following rules are removed from the original TRS not ( T ( ) ) -> F ( ) not ( F ( ) ) -> T ( ) imply ( x , y ) -> or ( not ( x ) , y )
Rewrite Rules:
   [ or(?x,?y) -> or(?y,?x),
     or(?x,T) -> T,
     or(?x,F) -> ?x,
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,T) -> ?x,
     and(?x,F) -> F,
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
Apply Direct Methods...
Inner CPs:
   [ or(or(?y,?x),?z_3) = or(?x,or(?y,?z_3)),
     or(T,?z_3) = or(?x_1,or(T,?z_3)),
     or(?x_2,?z_3) = or(?x_2,or(F,?z_3)),
     and(?x_4,?z_7) = and(?x_4,and(T,?z_7)),
     and(F,?z_7) = and(?x_5,and(F,?z_7)),
     and(and(?y_6,?x_6),?z_7) = and(?x_6,and(?y_6,?z_7)),
     or(or(?x,or(?y,?z)),?z_1) = or(or(?x,?y),or(?z,?z_1)),
     and(and(?x,and(?y,?z)),?z_1) = and(and(?x,?y),and(?z,?z_1)) ]
Outer CPs:
   [ or(T,?x) = T,
     or(F,?x) = ?x,
     or(?y,or(?x_3,?y_3)) = or(?x_3,or(?y_3,?y)),
     T = or(?x_3,or(?y_3,T)),
     or(?x_3,?y_3) = or(?x_3,or(?y_3,F)),
     ?x_4 = and(T,?x_4),
     and(?x_7,?y_7) = and(?x_7,and(?y_7,T)),
     F = and(F,?x_5),
     F = and(?x_7,and(?y_7,F)),
     and(?y_6,and(?x_7,?y_7)) = and(?x_7,and(?y_7,?y_6)) ]
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:
   [ 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),
     or(T,?x) = T,
     or(?x_3,or(?y_3,T)) = T,
     or(?x,or(T,?z_4)) = or(T,?z_4),
     or(F,?x) = ?x,
     or(?x_3,or(?y_3,F)) = or(?x_3,?y_3),
     or(?x,or(F,?z_4)) = or(?x,?z_4),
     or(?z,or(?x_1,or(?y_1,?y))) = or(or(?x_1,?y_1),or(?y,?z)),
     or(?z,or(?y,?x)) = or(?x,or(?y,?z)),
     or(?z,T) = or(?x,or(T,?z)),
     or(?z,?x) = or(?x,or(F,?z)),
     T = or(or(?x_1,?y_1),or(?y,T)),
     T = or(?x,or(?y,T)),
     T = or(?x,or(T,T)),
     T = or(?x,or(F,T)),
     or(?x_1,or(?y_1,?y)) = or(or(?x_1,?y_1),or(?y,F)),
     or(?y,?x) = or(?x,or(?y,F)),
     T = or(?x,or(T,F)),
     ?x = or(?x,or(F,F)),
     or(?z,or(?x,?y)) = or(?x,or(?y,?z)),
     or(?x,?y) = or(?x,or(?y,F)),
     or(or(?x_1,or(?y_1,?y)),?z) = or(or(?x_1,?y_1),or(?y,?z)),
     or(or(?y,?x),?z) = or(?x,or(?y,?z)),
     or(T,?z) = or(?x,or(T,?z)),
     or(?x,?z) = or(?x,or(F,?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(or(?y,?x),or(?z,?z_1)) = or(or(?x,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(?x,?y),or(?z,?z_1)) = or(or(?x,or(?y,?z)),?z_1),
     and(T,?x) = ?x,
     and(?x_7,and(?y_7,T)) = and(?x_7,?y_7),
     and(?x,and(T,?z_8)) = and(?x,?z_8),
     and(F,?x) = F,
     and(?x_7,and(?y_7,F)) = F,
     and(?x,and(F,?z_8)) = and(F,?z_8),
     ?x = and(T,?x),
     F = and(F,?x),
     and(?x_7,and(?y_7,?y)) = and(?y,and(?x_7,?y_7)),
     and(?x,and(?y,?z_8)) = and(and(?y,?x),?z_8),
     and(?x_1,and(?y_1,?y)) = and(and(?x_1,?y_1),and(?y,T)),
     ?x = and(?x,and(T,T)),
     F = and(?x,and(F,T)),
     and(?y,?x) = and(?x,and(?y,T)),
     F = and(and(?x_1,?y_1),and(?y,F)),
     F = and(?x,and(T,F)),
     F = and(?x,and(F,F)),
     F = and(?x,and(?y,F)),
     and(?z,and(?x_1,and(?y_1,?y))) = and(and(?x_1,?y_1),and(?y,?z)),
     and(?z,?x) = and(?x,and(T,?z)),
     and(?z,F) = and(?x,and(F,?z)),
     and(?z,and(?y,?x)) = and(?x,and(?y,?z)),
     and(?x,?y) = and(?x,and(?y,T)),
     and(?z,and(?x,?y)) = and(?x,and(?y,?z)),
     and(and(?x_1,and(?y_1,?y)),?z) = and(and(?x_1,?y_1),and(?y,?z)),
     and(?x,?z) = and(?x,and(T,?z)),
     and(F,?z) = and(?x,and(F,?z)),
     and(and(?y,?x),?z) = and(?x,and(?y,?z)),
     and(and(?x_2,and(?y_2,?y)),and(?z,?z_1)) = and(and(and(?x_2,?y_2),and(?y,?z)),?z_1),
     and(?x,and(?z,?z_1)) = and(and(?x,and(T,?z)),?z_1),
     and(F,and(?z,?z_1)) = and(and(?x,and(F,?z)),?z_1),
     and(and(?y,?x),and(?z,?z_1)) = and(and(?x,and(?y,?z)),?z_1),
     and(and(?x,?y),and(?z,?z_1)) = and(and(?x,and(?y,?z)),?z_1) ]
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(or(?y,?x),?z_3), or(?x,or(?y,?z_3))> by Rules <0, 3> preceded by [(or,1)]
 joinable by a reduction of rules <[([(or,1)],0),([],3)], []>
 joinable by a reduction of rules <[([],3),([(or,2)],0)], [([],0),([],3)]>
Critical Pair <or(T,?z_3), or(?x_1,or(T,?z_3))> by Rules <1, 3> preceded by [(or,1)]
 joinable by a reduction of rules <[([],0),([],1)], [([(or,2)],0),([(or,2)],1),([],1)]>
Critical Pair <or(?x_2,?z_3), or(?x_2,or(F,?z_3))> by Rules <2, 3> preceded by [(or,1)]
 joinable by a reduction of rules <[], [([(or,2)],0),([(or,2)],2)]>
Critical Pair <and(?x_4,?z_7), and(?x_4,and(T,?z_7))> by Rules <4, 7> preceded by [(and,1)]
 joinable by a reduction of rules <[], [([(and,2)],6),([(and,2)],4)]>
Critical Pair <and(F,?z_7), and(?x_5,and(F,?z_7))> by Rules <5, 7> preceded by [(and,1)]
 joinable by a reduction of rules <[([],6),([],5)], [([(and,2)],6),([(and,2)],5),([],5)]>
Critical Pair <and(and(?y_6,?x_6),?z_7), and(?x_6,and(?y_6,?z_7))> by Rules <6, 7> preceded by [(and,1)]
 joinable by a reduction of rules <[([(and,1)],6),([],7)], []>
 joinable by a reduction of rules <[([],7),([(and,2)],6)], [([],6),([],7)]>
Critical Pair <or(or(?x,or(?y,?z)),?z_1), or(or(?x,?y),or(?z,?z_1))> by Rules <3, 3> preceded by [(or,1)]
 joinable by a reduction of rules <[([],3),([(or,2)],3)], [([],3)]>
Critical Pair <and(and(?x,and(?y,?z)),?z_1), and(and(?x,?y),and(?z,?z_1))> by Rules <7, 7> preceded by [(and,1)]
 joinable by a reduction of rules <[([],7),([(and,2)],7)], [([],7)]>
Critical Pair <T, or(T,?x_1)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],1)]>
Critical Pair <?x_2, or(F,?x_2)> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],2)]>
Critical Pair <or(?x_3,or(?y_3,?z_3)), or(?z_3,or(?x_3,?y_3))> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],3)]>
Critical Pair <or(?x_3,or(?y_3,T)), T> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([(or,2)],1),([],1)], []>
Critical Pair <or(?x_3,or(?y_3,F)), or(?x_3,?y_3)> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[([(or,2)],2)], []>
Critical Pair <and(T,?x_6), ?x_6> by Rules <6, 4> preceded by []
 joinable by a reduction of rules <[([],6),([],4)], []>
Critical Pair <and(?x_7,and(?y_7,T)), and(?x_7,?y_7)> by Rules <7, 4> preceded by []
 joinable by a reduction of rules <[([(and,2)],4)], []>
Critical Pair <and(F,?x_6), F> by Rules <6, 5> preceded by []
 joinable by a reduction of rules <[([],6),([],5)], []>
Critical Pair <and(?x_7,and(?y_7,F)), F> by Rules <7, 5> preceded by []
 joinable by a reduction of rules <[([(and,2)],5),([],5)], []>
Critical Pair <and(?x_7,and(?y_7,?z_7)), and(?z_7,and(?x_7,?y_7))> by Rules <7, 6> preceded by []
 joinable by a reduction of rules <[], [([],6),([],7)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 10 rules by 3 steps unfolding
obtain 100 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
Check relative termination:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     and(?x,T) -> ?x,
     and(?x,F) -> F ]
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
Polynomial Interpretation:
  F:= (1)
  T:= 0
  or:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  and:= (1)*x1+(1)*x2
retract or(?x,T) -> T
retract or(?x,F) -> ?x
Polynomial Interpretation:
  F:= 0
  T:= 0
  or:= (1)*x1+(1)*x2
  and:= (1)+(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,
     and(?x,T) -> ?x,
     and(?x,F) -> F ]
P:
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
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
<T, or(T,?x)> --> <T, or(T,?x)> => no
<or(?x_1,T), or(or(?x_1,?x),T)> --> <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,or(F,?z_1))> => no
<?x, or(F,?x)> --> <?x, or(F,?x)> => no
<or(?x_1,?x), or(or(?x_1,?x),F)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<and(?x_1,?y_1), and(?x_1,and(?y_1,T))> --> <and(?x_1,?y_1), and(?x_1,?y_1)> => yes
<and(?x,?z_1), and(?x,and(T,?z_1))> --> <and(?x,?z_1), and(?x,and(T,?z_1))> => no
<?x, and(T,?x)> --> <?x, and(T,?x)> => no
<and(?x_1,?x), and(and(?x_1,?x),T)> --> <and(?x_1,?x), and(?x_1,?x)> => yes
<F, and(?x_1,and(?y_1,F))> --> <F, F> => yes
<and(F,?z_1), and(?x,and(F,?z_1))> --> <and(F,?z_1), and(?x,and(F,?z_1))> => no
<F, and(F,?x)> --> <F, and(F,?x)> => no
<and(?x_1,F), and(and(?x_1,?x),F)> --> <F, F> => yes
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
check modulo reachablity from and(?x,?z_1) to and(?x,and(T,?z_1)): maybe not reachable
check modulo reachablity from ?x to and(T,?x): maybe not reachable
check modulo reachablity from and(F,?z_1) to and(?x,and(F,?z_1)): maybe not reachable
check modulo reachablity from F to and(F,?x): maybe not reachable
failed
failure(Step 1)
   [ or(T,?x) -> T,
     or(F,?x) -> ?x,
     and(T,?x) -> ?x,
     and(F,?x) -> F ]
Added S-Rules:
   [ or(T,?x) -> T,
     or(F,?x) -> ?x,
     and(T,?x) -> ?x,
     and(F,?x) -> F ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     and(?x,T) -> ?x,
     and(?x,F) -> F ]
P:
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
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
<T, or(T,?x)> --> <T, or(T,?x)> => no
<or(?x_1,T), or(or(?x_1,?x),T)> --> <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,or(F,?z_1))> => no
<?x, or(F,?x)> --> <?x, or(F,?x)> => no
<or(?x_1,?x), or(or(?x_1,?x),F)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<and(?x_1,?y_1), and(?x_1,and(?y_1,T))> --> <and(?x_1,?y_1), and(?x_1,?y_1)> => yes
<and(?x,?z_1), and(?x,and(T,?z_1))> --> <and(?x,?z_1), and(?x,and(T,?z_1))> => no
<?x, and(T,?x)> --> <?x, and(T,?x)> => no
<and(?x_1,?x), and(and(?x_1,?x),T)> --> <and(?x_1,?x), and(?x_1,?x)> => yes
<F, and(?x_1,and(?y_1,F))> --> <F, F> => yes
<and(F,?z_1), and(?x,and(F,?z_1))> --> <and(F,?z_1), and(?x,and(F,?z_1))> => no
<F, and(F,?x)> --> <F, and(F,?x)> => no
<and(?x_1,F), and(and(?x_1,?x),F)> --> <F, F> => yes
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
check modulo reachablity from and(?x,?z_1) to and(?x,and(T,?z_1)): maybe not reachable
check modulo reachablity from ?x to and(T,?x): maybe not reachable
check modulo reachablity from and(F,?z_1) to and(?x,and(F,?z_1)): maybe not reachable
check modulo reachablity from F to and(F,?x): maybe not reachable
failed
failure(Step 2)
   [ or(T,?x) -> T,
     or(F,?x) -> ?x,
     and(T,?x) -> ?x,
     and(F,?x) -> F ]
Added S-Rules:
   [ or(T,?x) -> T,
     or(F,?x) -> ?x,
     and(T,?x) -> ?x,
     and(F,?x) -> F ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     and(?x,T) -> ?x,
     and(?x,F) -> F ]
P:
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
Check relative termination:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     and(?x,T) -> ?x,
     and(?x,F) -> F ]
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
Polynomial Interpretation:
  F:= (1)
  T:= 0
  or:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  and:= (1)*x1+(1)*x2
retract or(?x,T) -> T
retract or(?x,F) -> ?x
Polynomial Interpretation:
  F:= 0
  T:= 0
  or:= (1)*x1+(1)*x2
  and:= (1)+(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,
     and(?x,T) -> ?x,
     and(?x,F) -> F,
     or(T,?x) -> T,
     or(F,?x) -> ?x,
     and(T,?x) -> ?x,
     and(F,?x) -> F ]
P:
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
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
<T, T> --> <T, T> => yes
<F, F> --> <F, F> => yes
<F, F> --> <F, F> => yes
<F, F> --> <F, F> => yes
<T, T> --> <T, T> => yes
<T, T> --> <T, T> => yes
<T, T> --> <T, T> => yes
<F, F> --> <F, F> => yes
<T, T> --> <T, T> => yes
<F, F> --> <F, F> => yes
<F, F> --> <F, F> => 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
<T, or(T,?x)> --> <T, T> => yes
<or(?x_1,T), or(or(?x_1,?x),T)> --> <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
<?x, or(F,?x)> --> <?x, ?x> => yes
<or(?x_1,?x), or(or(?x_1,?x),F)> --> <or(?x_1,?x), or(?x_1,?x)> => yes
<and(?x_1,?y_1), and(?x_1,and(?y_1,T))> --> <and(?x_1,?y_1), and(?x_1,?y_1)> => yes
<and(?x,?z_1), and(?x,and(T,?z_1))> --> <and(?x,?z_1), and(?x,?z_1)> => yes
<?x, and(T,?x)> --> <?x, ?x> => yes
<and(?x_1,?x), and(and(?x_1,?x),T)> --> <and(?x_1,?x), and(?x_1,?x)> => yes
<F, and(?x_1,and(?y_1,F))> --> <F, F> => yes
<and(F,?z_1), and(?x,and(F,?z_1))> --> <F, F> => yes
<F, and(F,?x)> --> <F, F> => yes
<and(?x_1,F), and(and(?x_1,?x),F)> --> <F, F> => yes
<or(T,?z_1), or(T,or(?x,?z_1))> --> <T, T> => yes
<T, or(?x,T)> --> <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
<or(?x,?z_1), or(F,or(?x,?z_1))> --> <or(?x,?z_1), or(?x,?z_1)> => yes
<?x, or(?x,F)> --> <?x, ?x> => 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
<and(?x,?z_1), and(T,and(?x,?z_1))> --> <and(?x,?z_1), and(?x,?z_1)> => yes
<?x, and(?x,T)> --> <?x, ?x> => yes
<and(?y_1,?z_1), and(and(T,?y_1),?z_1)> --> <and(?y_1,?z_1), and(?y_1,?z_1)> => yes
<and(?x_1,?x), and(and(?x_1,T),?x)> --> <and(?x_1,?x), and(?x_1,?x)> => yes
<and(F,?z_1), and(F,and(?x,?z_1))> --> <F, F> => yes
<F, and(?x,F)> --> <F, F> => yes
<F, and(and(F,?y_1),?z_1)> --> <F, F> => yes
<and(?x_1,F), and(and(?x_1,F),?x)> --> <F, F> => yes
S:
   [ or(?x,T) -> T,
     or(?x,F) -> ?x,
     and(?x,T) -> ?x,
     and(?x,F) -> F,
     or(T,?x) -> T,
     or(F,?x) -> ?x,
     and(T,?x) -> ?x,
     and(F,?x) -> F ]
P:
   [ or(?x,?y) -> or(?y,?x),
     or(or(?x,?y),?z) -> or(?x,or(?y,?z)),
     and(?x,?y) -> and(?y,?x),
     and(and(?x,?y),?z) -> and(?x,and(?y,?z)) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Combined result: CR
/tmp/file5y2uJs.trs: Success(CR)
(2601 msec.)