YES
(ignored inputs)COMMENT translated from Cops 11
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Constructors: {0,s}
Defined function symbols: {+}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
Rule part & Conj Part:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat}
Signature:
   [ + : Nat,Nat -> Nat,
     0 : Nat,
     s : Nat -> Nat ]
Rule Part:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjecture Part:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)) ]
Precedence (by weight): {(+,3),(0,2),(s,0)}
Rule part is confluent.
R0 is ground confluent.
Check conj part consists of inductive theorems of R0.
Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjectures:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)) ]
STEP 0
ES:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)) ]
HS:
   [ ]
ES0:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)) ]
HS0:
   [ ]
ES1:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)) ]
HS1:
   [ ]
Expand +(0,?y) = ?y
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2) ]
ES2:
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS2:
   [ +(0,?y) -> ?y ]
STEP 1
ES:
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS:
   [ +(0,?y) -> ?y ]
ES0:
   [ 0 = 0,
     s(?y_2) = s(?y_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS0:
   [ +(0,?y) -> ?y ]
ES1:
   [ +(s(?x),?y) = s(+(?x,?y)) ]
HS1:
   [ +(0,?y) -> ?y ]
Expand +(s(?x),?y) = s(+(?x,?y))
   [ s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))) ]
ES2:
   [ s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))) ]
HS2:
   [ +(s(?x),?y) -> s(+(?x,?y)),
     +(0,?y) -> ?y ]
STEP 2
ES:
   [ s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))) ]
HS:
   [ +(s(?x),?y) -> s(+(?x,?y)),
     +(0,?y) -> ?y ]
ES0:
   [ s(?x) = s(?x),
     s(s(+(?x,?y_2))) = s(s(+(?x,?y_2))) ]
HS0:
   [ +(s(?x),?y) -> s(+(?x,?y)),
     +(0,?y) -> ?y ]
ES1:
   [ ]
HS1:
   [ +(s(?x),?y) -> s(+(?x,?y)),
     +(0,?y) -> ?y ]
Conj part consisits of inductive theorems of R0.
examples/fromCops/cr/11.trs: Success(GCR)
(6 msec.)