(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
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:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sum1(0)
sum1(s(x0))
(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:
SUM(s(x)) → SUM(x)
SUM1(s(x)) → SUM1(x)
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sum1(0)
sum1(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SUM1(s(x)) → SUM1(x)
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sum1(0)
sum1(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(8) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
SUM1(s(x)) → SUM1(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SUM1(
x1) =
x1
s(
x1) =
s(
x1)
sum(
x1) =
x1
0 =
0
+(
x1,
x2) =
+(
x1)
sum1(
x1) =
sum1(
x1)
Lexicographic Path Order [LPO].
Precedence:
[0, sum11] > [s1, +1]
The following usable rules [FROCOS05] were oriented:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
(9) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sum1(0)
sum1(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(10) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(11) TRUE
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SUM(s(x)) → SUM(x)
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sum1(0)
sum1(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
SUM(s(x)) → SUM(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SUM(
x1) =
x1
s(
x1) =
s(
x1)
sum(
x1) =
x1
0 =
0
+(
x1,
x2) =
+(
x1)
sum1(
x1) =
sum1(
x1)
Lexicographic Path Order [LPO].
Precedence:
[0, sum11] > [s1, +1]
The following usable rules [FROCOS05] were oriented:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
(14) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sum1(0)
sum1(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(15) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(16) TRUE