(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
(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:
F(a, a) → F(a, b)
F(a, b) → F(s(a), c)
F(s(X), c) → F(X, c)
F(c, c) → F(a, a)
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
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.
F(c, c) → F(a, a)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1)
a =
a
b =
b
s(
x1) =
x1
c =
c
f(
x1,
x2) =
f
Lexicographic path order with status [LPO].
Precedence:
F1 > b > a
F1 > c > a
f > a
Status:
c: []
a: []
f: []
b: []
F1: [1]
The following usable rules [FROCOS05] were oriented:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, a) → F(a, b)
F(a, b) → F(s(a), c)
F(s(X), c) → F(X, c)
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(X), c) → F(X, c)
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
We have to consider all minimal (P,Q,R)-chains.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(s(X), c) → F(X, c)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1,
x2)
s(
x1) =
s(
x1)
c =
c
f(
x1,
x2) =
f
a =
a
b =
b
Lexicographic path order with status [LPO].
Precedence:
F2 > c
f > s1
a > s1
b > s1
Status:
c: []
a: []
f: []
s1: [1]
b: []
F2: [2,1]
The following usable rules [FROCOS05] were oriented:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
(10) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
We have to consider all minimal (P,Q,R)-chains.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) TRUE