(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(s(X), plus(Y, Z)) → plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) → plus(X1, plus(X3, plus(X2, X4)))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(X), plus(Y, Z)) → PLUS(X, plus(s(s(Y)), Z))
PLUS(s(X), plus(Y, Z)) → PLUS(s(s(Y)), Z)
PLUS(s(X1), plus(X2, plus(X3, X4))) → PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X1), plus(X2, plus(X3, X4))) → PLUS(X3, plus(X2, X4))
PLUS(s(X1), plus(X2, plus(X3, X4))) → PLUS(X2, X4)
The TRS R consists of the following rules:
plus(s(X), plus(Y, Z)) → plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) → plus(X1, plus(X3, plus(X2, X4)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Combined order from the following AFS and order.
plus(x1, x2) = plus(x2)
s(x1) = s(x1)
Homeomorphic Embedding Order
AFS:
plus(x1, x2) = plus(x2)
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- PLUS(s(X), plus(Y, Z)) → PLUS(X, plus(s(s(Y)), Z)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- PLUS(s(X), plus(Y, Z)) → PLUS(s(s(Y)), Z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 > 2
- PLUS(s(X1), plus(X2, plus(X3, X4))) → PLUS(X1, plus(X3, plus(X2, X4))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- PLUS(s(X1), plus(X2, plus(X3, X4))) → PLUS(X3, plus(X2, X4)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 > 2
- PLUS(s(X1), plus(X2, plus(X3, X4))) → PLUS(X2, X4) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 > 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
plus(s(X1), plus(X2, plus(X3, X4))) → plus(X1, plus(X3, plus(X2, X4)))
plus(s(X), plus(Y, Z)) → plus(X, plus(s(s(Y)), Z))
(4) TRUE