(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) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) 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
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) 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
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(6) 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
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(7) 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: Combined order from the following AFS and order.
TIMES(
x1,
x2) =
TIMES(
x1,
x2)
plus(
x1,
x2) =
plus(
x1,
x2)
1 =
1
times(
x1,
x2) =
times(
x2)
0 =
0
Lexicographic path order with status [LPO].
Quasi-Precedence:
1 > [TIMES2, plus2] > 0
1 > times1 > 0
Status:
TIMES2: [2,1]
plus2: [2,1]
1: []
times1: [1]
0: []
The following usable rules [FROCOS05] were oriented:
times(x, 0) → 0
plus(x, 0) → x
(8) 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
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE