YES
(ignored inputs)COMMENT translated from Cops 189
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(0) -> 0,
     dbl(s(?x)) -> s(s(dbl(?x))),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
Constructors: {0,s}
Defined function symbols: {+,dbl}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(0) -> 0,
     dbl(s(?x)) -> s(s(dbl(?x))) ]
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)),
     +(?x,?y) -> +(?y,?x) ]
Rule part & Conj Part:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     dbl(0) -> 0,
     dbl(s(?x)) -> s(s(dbl(?x))) ]
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)),
     +(?x,?y) -> +(?y,?x) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat}
Signature:
   [ + : Nat,Nat -> Nat,
     0 : Nat,
     s : Nat -> Nat,
     dbl : Nat -> Nat ]
Rule Part:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(0) -> 0,
     dbl(s(?x)) -> s(s(dbl(?x))) ]
Conjecture Part:
   [ +(?x,0) = ?x,
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(?x,?y) = +(?y,?x) ]
Precedence (by weight): {(+,3),(0,1),(s,0),(dbl,2)}
Rule part is confluent.
R0 is ground confluent.
Check conj part consists of inductive theorems of R0.
Rules:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(0) -> 0,
     dbl(s(?x)) -> s(s(dbl(?x))) ]
Conjectures:
   [ +(?x,0) = ?x,
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(?x,?y) = +(?y,?x) ]
STEP 0
ES:
   [ +(?x,0) = ?x,
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(?x,?y) = +(?y,?x) ]
HS:
   [ ]
ES0:
   [ +(?x,0) = ?x,
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(?x,?y) = +(?y,?x) ]
HS0:
   [ ]
ES1:
   [ +(?x,0) = ?x,
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(?x,?y) = +(?y,?x) ]
HS1:
   [ ]
Expand +(?x,0) = ?x
   [ 0 = 0,
     s(+(?x_2,0)) = s(?x_2) ]
ES2:
   [ 0 = 0,
     s(+(?x_2,0)) = s(?x_2),
     +(?x,?y) = +(?y,?x),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,s(?y)) = s(+(?x,?y)) ]
HS2:
   [ +(?x,0) -> ?x ]
STEP 1
ES:
   [ 0 = 0,
     s(+(?x_2,0)) = s(?x_2),
     +(?x,?y) = +(?y,?x),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,s(?y)) = s(+(?x,?y)) ]
HS:
   [ +(?x,0) -> ?x ]
ES0:
   [ 0 = 0,
     s(?x_2) = s(?x_2),
     +(?x,?y) = +(?y,?x),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,s(?y)) = s(+(?x,?y)) ]
HS0:
   [ +(?x,0) -> ?x ]
ES1:
   [ +(?x,?y) = +(?y,?x),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,s(?y)) = s(+(?x,?y)) ]
HS1:
   [ +(?x,0) -> ?x ]
Expand +(?x,?y) = +(?y,?x)
   [ ?y_1 = +(?y_1,0),
     s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)) ]
ES2:
   [ ?y_1 = +(?y_1,0),
     s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)),
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)) ]
HS2:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
STEP 2
ES:
   [ ?y_1 = +(?y_1,0),
     s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)),
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)) ]
HS:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES0:
   [ ?y_1 = ?y_1,
     s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)),
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)) ]
HS0:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES1:
   [ s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)),
     +(?x,s(?y)) = s(+(?x,?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)) ]
HS1:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
Expand +(?y_2,s(?x_2)) = s(+(?x_2,?y_2))
   [ s(?x) = s(+(?x,0)),
     s(+(?x_2,s(?x))) = s(+(?x,s(?x_2))) ]
ES2:
   [ s(?x) = s(+(?x,0)),
     s(+(?x_2,s(?x))) = s(+(?x,s(?x_2))),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,s(?y)) = s(+(?x,?y)) ]
HS2:
   [ +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
STEP 3
ES:
   [ s(?x) = s(+(?x,0)),
     s(+(?x_2,s(?x))) = s(+(?x,s(?x_2))),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     +(?x,s(?y)) = s(+(?x,?y)) ]
HS:
   [ +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES0:
   [ s(?x) = s(?x),
     s(s(+(?x,?x_2))) = s(s(+(?x_2,?x))),
     dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)),
     s(+(?y,?x)) = s(+(?x,?y)) ]
HS0:
   [ +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES1:
   [ dbl(+(?x,?y)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS1:
   [ +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
Expand +(dbl(?x),dbl(?y)) = dbl(+(?x,?y))
   [ +(0,dbl(?y)) = dbl(+(0,?y)),
     +(s(s(dbl(?x_3))),dbl(?y)) = dbl(+(s(?x_3),?y)) ]
ES2:
   [ dbl(?y) = dbl(+(0,?y)),
     s(s(+(dbl(?x_3),dbl(?y)))) = dbl(+(s(?x_3),?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS2:
   [ +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
STEP 4
ES:
   [ dbl(?y) = dbl(+(0,?y)),
     s(s(+(dbl(?x_3),dbl(?y)))) = dbl(+(s(?x_3),?y)),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS:
   [ +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES0:
   [ dbl(?y) = dbl(?y),
     s(s(dbl(+(?x_3,?y)))) = s(s(dbl(+(?x_3,?y)))),
     +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS0:
   [ +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES1:
   [ +(+(?x,?y),?z) = +(?x,+(?y,?z)) ]
HS1:
   [ +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
Expand +(+(?x,?y),?z) = +(?x,+(?y,?z))
   [ +(?y_1,?z) = +(0,+(?y_1,?z)),
     +(s(+(?x_2,?y_2)),?z) = +(s(?x_2),+(?y_2,?z)) ]
ES2:
   [ +(?y_1,?z) = +(0,+(?y_1,?z)),
     s(+(+(?x_2,?y_2),?z)) = +(s(?x_2),+(?y_2,?z)) ]
HS2:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
STEP 5
ES:
   [ +(?y_1,?z) = +(0,+(?y_1,?z)),
     s(+(+(?x_2,?y_2),?z)) = +(s(?x_2),+(?y_2,?z)) ]
HS:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES0:
   [ +(?y_1,?z) = +(?y_1,?z),
     s(+(+(?x_2,?y_2),?z)) = s(+(?x_2,+(?y_2,?z))) ]
HS0:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
ES1:
   [ ]
HS1:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(dbl(?x),dbl(?y)) -> dbl(+(?x,?y)),
     +(?y_2,s(?x_2)) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x ]
Conj part consisits of inductive theorems of R0.
examples/fromCops/cr/189.trs: Success(GCR)
(13 msec.)