YES
(ignored inputs)COMMENT translated from Cops 170
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     s(s(?x)) -> s(?x),
     s(?x) -> s(s(?x)) ]
Constructors: {0,s}
Defined function symbols: {+}
Constructor subsystem:
   [ s(s(?x)) -> s(?x) ]
Rule part & Conj Part:
   [ s(s(?x)) -> s(?x),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     s(?x) -> s(s(?x)) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat}
Signature:
   [ + : Nat,Nat -> Nat,
     0 : Nat,
     s : Nat -> Nat ]
Rule Part:
   [ s(s(?x)) -> s(?x),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
Conjecture Part:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x),
     s(?x) = s(s(?x)) ]
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:
   [ s(s(?x)) -> s(?x),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
Conjectures:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x),
     s(?x) = s(s(?x)) ]
STEP 0
ES:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x),
     s(?x) = s(s(?x)) ]
HS:
   [ ]
ES0:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x),
     s(?x) = s(?x) ]
HS0:
   [ ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x) ]
HS1:
   [ ]
Expand +(?x,?y) = +(?y,?x)
   [ ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)) ]
ES2:
   [ ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS2:
   [ +(?x,?y) -> +(?y,?x) ]
STEP 1
ES:
   [ ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS:
   [ +(?x,?y) -> +(?y,?x) ]
ES0:
   [ ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS0:
   [ +(?x,?y) -> +(?y,?x) ]
ES1:
   [ ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS1:
   [ +(?x,?y) -> +(?y,?x) ]
Expand +(?y_2,0) = ?y_2
   [ 0 = 0,
     s(+(?x_3,0)) = s(?x_3) ]
ES2:
   [ 0 = 0,
     s(+(?x_3,0)) = s(?x_3),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)) ]
HS2:
   [ +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
STEP 2
ES:
   [ 0 = 0,
     s(+(?x_3,0)) = s(?x_3),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)) ]
HS:
   [ +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ 0 = 0,
     s(?x_3) = s(?x_3),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)) ]
HS0:
   [ +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)) ]
HS1:
   [ +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
Expand +(?y_3,s(?x_3)) = s(+(?x_3,?y_3))
   [ s(?x) = s(+(?x,0)),
     s(+(?x_3,s(?x))) = s(+(?x,s(?x_3))) ]
ES2:
   [ s(?x) = s(+(?x,0)),
     s(+(?x_3,s(?x))) = s(+(?x,s(?x_3))),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS2:
   [ +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
STEP 3
ES:
   [ s(?x) = s(+(?x,0)),
     s(+(?x_3,s(?x))) = s(+(?x,s(?x_3))),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS:
   [ +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ s(?x) = s(?x),
     s(+(?x,?x_3)) = s(+(?x_3,?x)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS0:
   [ +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS1:
   [ +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
Expand +(+(?x,?y),?z) = +(?x,+(?y,?z))
   [ +(?y_2,?z) = +(0,+(?y_2,?z)),
     +(s(+(?x_3,?y_3)),?z) = +(s(?x_3),+(?y_3,?z)) ]
ES2:
   [ +(?y_2,?z) = +(0,+(?y_2,?z)),
     s(+(+(?x_3,?y_3),?z)) = +(s(?x_3),+(?y_3,?z)) ]
HS2:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
STEP 4
ES:
   [ +(?y_2,?z) = +(0,+(?y_2,?z)),
     s(+(+(?x_3,?y_3),?z)) = +(s(?x_3),+(?y_3,?z)) ]
HS:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ +(?y_2,?z) = +(?y_2,?z),
     s(+(+(?x_3,?y_3),?z)) = s(+(?x_3,+(?y_3,?z))) ]
HS0:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ ]
HS1:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?y_3,s(?x_3)) -> s(+(?x_3,?y_3)),
     +(?y_2,0) -> ?y_2,
     +(?x,?y) -> +(?y,?x) ]
Conj part consisits of inductive theorems of R0.
examples/fromCops/cr/170.trs: Success(GCR)
(12 msec.)