(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
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:
TIMES(x, plus(y, 1)) → PLUS(times(x, plus(y, times(1, 0))), x)
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
TIMES(x, plus(y, 1)) → PLUS(y, times(1, 0))
TIMES(x, plus(y, 1)) → TIMES(1, 0)
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
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 1 SCC with 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TIMES(
x1,
x2) =
TIMES(
x2)
Tags:
TIMES has tags [0,1]
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
plus(
x1,
x2) =
plus(
x1,
x2)
1 =
1
times(
x1,
x2) =
times
0 =
0
Lexicographic path order with status [LPO].
Quasi-Precedence:
1 > times > [plus2, 0]
Status:
plus2: [2,1]
1: []
times: []
0: []
The following usable rules [FROCOS05] were oriented:
times(x, 0) → 0
plus(x, 0) → x
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE