YES
(ignored inputs)COMMENT translated from Cops 105
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Constructors: {0,s}
Defined function symbols: {+}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
*** 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:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjectures:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x) ]
STEP 0
ES:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x) ]
HS:
   [ ]
ES0:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x) ]
HS0:
   [ ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?y,?x) ]
HS1:
   [ ]
Expand +(?x,?y) = +(?y,?x)
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2) ]
ES2:
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS2:
   [ +(?x,?y) -> +(?y,?x) ]
STEP 1
ES:
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS:
   [ +(?x,?y) -> +(?y,?x) ]
ES0:
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS0:
   [ +(?x,?y) -> +(?y,?x) ]
ES1:
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS1:
   [ +(?x,?y) -> +(?y,?x) ]
Expand +(0,?x_1) = ?x_1
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2) ]
ES2:
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2) ]
HS2:
   [ +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
STEP 2
ES:
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2) ]
HS:
   [ +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ 0 = 0,
     s(?y_2) = s(?y_2),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2) ]
HS0:
   [ +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2) ]
HS1:
   [ +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
Expand +(s(?y_2),?x_2) = s(+(?x_2,?y_2))
   [ s(?y) = s(+(0,?y)),
     s(+(s(?y),?y_2)) = s(+(s(?y_2),?y)) ]
ES2:
   [ s(?y) = s(+(0,?y)),
     s(+(s(?y),?y_2)) = s(+(s(?y_2),?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS2:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
STEP 3
ES:
   [ s(?y) = s(+(0,?y)),
     s(+(s(?y),?y_2)) = s(+(s(?y_2),?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ s(?y) = s(?y),
     s(s(+(?y_2,?y))) = s(s(+(?y,?y_2))),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS0:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS1:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
Expand +(+(?x,?y),?z) = +(?x,+(?y,?z))
   [ +(?x_1,?z) = +(?x_1,+(0,?z)),
     +(s(+(?x_2,?y_2)),?z) = +(?x_2,+(s(?y_2),?z)) ]
ES2:
   [ +(?x_1,?z) = +(?x_1,+(0,?z)),
     s(+(?z,+(?x_2,?y_2))) = +(?x_2,+(s(?y_2),?z)) ]
HS2:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
STEP 4
ES:
   [ +(?x_1,?z) = +(?x_1,+(0,?z)),
     s(+(?z,+(?x_2,?y_2))) = +(?x_2,+(s(?y_2),?z)) ]
HS:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ +(?x_1,?z) = +(?x_1,?z),
     s(+(?z,+(?x_2,?y_2))) = s(+(?x_2,+(?z,?y_2))) ]
HS0:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ s(+(?z,+(?x_2,?y_2))) = s(+(?x_2,+(?z,?y_2))) ]
HS1:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
Expand s(+(?z,+(?x_2,?y_2))) = s(+(?x_2,+(?z,?y_2)))
   [ s(+(?z,?x_3)) = s(+(?x_3,+(?z,0))),
     s(+(?z,s(+(?x_4,?y_4)))) = s(+(?x_4,+(?z,s(?y_4)))) ]
ES2:
   [ s(+(?z,?x_3)) = s(+(?x_3,+(?z,0))),
     s(s(+(?z,+(?x_4,?y_4)))) = s(+(?x_4,+(?z,s(?y_4)))) ]
HS2:
   [ s(+(?z,+(?x_2,?y_2))) -> s(+(?x_2,+(?z,?y_2))),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
STEP 5
ES:
   [ s(+(?z,?x_3)) = s(+(?x_3,+(?z,0))),
     s(s(+(?z,+(?x_4,?y_4)))) = s(+(?x_4,+(?z,s(?y_4)))) ]
HS:
   [ s(+(?z,+(?x_2,?y_2))) -> s(+(?x_2,+(?z,?y_2))),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ s(+(?z,?x_3)) = s(+(?x_3,?z)),
     s(s(+(?z,+(?x_4,?y_4)))) = s(s(+(?x_4,+(?z,?y_4)))) ]
HS0:
   [ s(+(?z,+(?x_2,?y_2))) -> s(+(?x_2,+(?z,?y_2))),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ ]
HS1:
   [ s(+(?z,+(?x_2,?y_2))) -> s(+(?x_2,+(?z,?y_2))),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(0,?x_1) -> ?x_1,
     +(?x,?y) -> +(?y,?x) ]
Conj part consisits of inductive theorems of R0.
examples/fromCops/cr/105.trs: Success(GCR)
(8 msec.)