YES
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?y,?x)),
     +(?x,?y) -> +(?y,?x),
     +(+(?x,?x),?x) -> +(?x,+(?x,?x)) ]
Constructors: {0,s}
Defined function symbols: {+}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?y,?x)) ]
   [ +(?x,?y) -> +(?y,?x),
     +(+(?x,?x),?x) -> +(?x,+(?x,?x)) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat}
Signature:
   [ + : Nat,Nat -> Nat,
     s : Nat -> Nat,
     0 : Nat ]
Rule Part:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?y,?x)) ]
Conjecture Part:
   [ +(?x,?y) = +(?y,?x),
     +(+(?x,?x),?x) = +(?x,+(?x,?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:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?y,?x)) ]
Conjectures:
   [ +(?x,?y) = +(?y,?x),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
STEP 0
ES:
   [ +(?x,?y) = +(?y,?x),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
HS:
   [ ]
ES0:
   [ +(?x,?y) = +(?y,?x),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
HS0:
   [ ]
ES1:
   [ +(?x,?y) = +(?y,?x),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
HS1:
   [ ]
Expand +(?x,?y) = +(?y,?x)
   [ ?y_1 = +(?y_1,0),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)) ]
ES2:
   [ ?y_1 = +(?y_1,0),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
HS2:
   [ +(?x,?y) -> +(?y,?x) ]
STEP 1
ES:
   [ ?y_1 = +(?y_1,0),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
HS:
   [ +(?x,?y) -> +(?y,?x) ]
ES0:
   [ ?y_1 = +(?y_1,0),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)),
     +(+(?x,?x),?x) = +(?x,+(?x,?x)) ]
HS0:
   [ +(?x,?y) -> +(?y,?x) ]
ES1:
   [ ?y_1 = +(?y_1,0),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)) ]
HS1:
   [ +(?x,?y) -> +(?y,?x) ]
Expand +(?y_1,0) = ?y_1
   [ 0 = 0,
     s(+(0,?x_2)) = s(?x_2) ]
ES2:
   [ 0 = 0,
     s(?x_2) = s(?x_2),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)) ]
HS2:
   [ +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
STEP 2
ES:
   [ 0 = 0,
     s(?x_2) = s(?x_2),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)) ]
HS:
   [ +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ 0 = 0,
     s(?x_2) = s(?x_2),
     s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)) ]
HS0:
   [ +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ s(+(?y_2,?x_2)) = +(?y_2,s(?x_2)) ]
HS1:
   [ +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
Expand +(?y_2,s(?x_2)) = s(+(?y_2,?x_2))
   [ s(?x) = s(+(0,?x)),
     s(+(s(?x),?x_2)) = s(+(s(?x_2),?x)) ]
ES2:
   [ s(?x) = s(+(0,?x)),
     s(s(+(?x_2,?x))) = s(+(s(?x_2),?x)) ]
HS2:
   [ +(?y_2,s(?x_2)) -> s(+(?y_2,?x_2)),
     +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
STEP 3
ES:
   [ s(?x) = s(+(0,?x)),
     s(s(+(?x_2,?x))) = s(+(s(?x_2),?x)) ]
HS:
   [ +(?y_2,s(?x_2)) -> s(+(?y_2,?x_2)),
     +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
ES0:
   [ s(?x) = s(?x),
     s(s(+(?x_2,?x))) = s(s(+(?x,?x_2))) ]
HS0:
   [ +(?y_2,s(?x_2)) -> s(+(?y_2,?x_2)),
     +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
ES1:
   [ ]
HS1:
   [ +(?y_2,s(?x_2)) -> s(+(?y_2,?x_2)),
     +(?y_1,0) -> ?y_1,
     +(?x,?y) -> +(?y,?x) ]
Conj part consisits of inductive theorems of R0.
examples/additions/plus13.trs: Success(GCR)
(8 msec.)