(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
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(plus(X, Y), Z) → PLUS(X, plus(Y, Z))
PLUS(plus(X, Y), Z) → PLUS(Y, Z)
TIMES(X, s(Y)) → PLUS(X, times(Y, X))
TIMES(X, s(Y)) → TIMES(Y, X)
The TRS R consists of the following rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(plus(X, Y), Z) → PLUS(Y, Z)
PLUS(plus(X, Y), Z) → PLUS(X, plus(Y, Z))
The TRS R consists of the following rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) 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:Homeomorphic Embedding Order
AFS:
plus(x1, x2) = plus(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- PLUS(plus(X, Y), Z) → PLUS(Y, Z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- PLUS(plus(X, Y), Z) → PLUS(X, plus(Y, Z)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(7) TRUE
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(X, s(Y)) → TIMES(Y, X)
The TRS R consists of the following rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) 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:Homeomorphic Embedding Order
AFS:
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- TIMES(X, s(Y)) → TIMES(Y, X) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 > 1, 1 >= 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(10) TRUE