(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
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:
C(z, x, a) → F(b(b(f(z), z), x))
C(z, x, a) → B(b(f(z), z), x)
C(z, x, a) → B(f(z), z)
C(z, x, a) → F(z)
B(y, b(z, a)) → F(b(c(f(a), y, z), z))
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
B(y, b(z, a)) → F(a)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
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 4 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(z, x, a) → B(b(f(z), z), x)
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
C(z, x, a) → B(f(z), z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
C(
z,
x,
a) →
B(
b(
f(
z),
z),
x) we obtained the following new rules [LPAR04]:
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
C(z, x, a) → B(f(z), z)
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
C(
z,
x,
a) →
B(
f(
z),
z) we obtained the following new rules [LPAR04]:
C(f(a), y_0, a) → B(f(f(a)), f(a))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
C(f(a), y_0, a) → B(f(f(a)), f(a))
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → C(f(a), y, z)
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
B(y, b(z, a)) → B(c(f(a), y, z), z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
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.
B(y, b(z, a)) → C(f(a), y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(B(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(b(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(C(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(c(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
The following usable rules [FROCOS05] were oriented:
f(c(c(z, a, a), x, a)) → z
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
c(z, x, a) → f(b(b(f(z), z), x))
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
B(y, b(z, a)) → B(c(f(a), y, z), z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → B(c(f(a), y, z), z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- B(y, b(z, a)) → B(c(f(a), y, z), z)
The graph contains the following edges 2 > 2
(16) TRUE