(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) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
PLUS(plus(X, Y), Z) → PLUS(Y, Z)
PLUS(plus(X, Y), Z) → PLUS(X, plus(Y, Z))
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
[times2, s1] > [PLUS2, plus2]
Status:
PLUS2: [1,2]
plus2: [1,2]
times2: multiset
s1: multiset
The following usable rules [FROCOS05] were oriented:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
(7) Obligation:
Q DP problem:
P is empty.
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.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) 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.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
TIMES(X, s(Y)) → TIMES(Y, X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TIMES(
x1,
x2) =
TIMES(
x1,
x2)
s(
x1) =
s(
x1)
plus(
x1,
x2) =
plus
times(
x1,
x2) =
times(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
times2 > plus
Status:
TIMES2: multiset
s1: [1]
plus: multiset
times2: multiset
The following usable rules [FROCOS05] were oriented:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
(12) Obligation:
Q DP problem:
P is empty.
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.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE