(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
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:
B(y, z) → C(c(y, z, z), a, a)
B(y, z) → C(y, z, z)
C(f(z), f(c(a, x, a)), y) → C(f(b(x, z)), c(z, y, a), a)
C(f(z), f(c(a, x, a)), y) → B(x, z)
C(f(z), f(c(a, x, a)), y) → C(z, y, a)
The TRS R consists of the following rules:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
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 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, z) → C(y, z, z)
C(f(z), f(c(a, x, a)), y) → C(f(b(x, z)), c(z, y, a), a)
C(f(z), f(c(a, x, a)), y) → B(x, z)
C(f(z), f(c(a, x, a)), y) → C(z, y, a)
The TRS R consists of the following rules:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
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.
C(f(z), f(c(a, x, a)), y) → C(f(b(x, z)), c(z, y, a), a)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
B(
x0,
x1,
x2) =
B(
x0)
C(
x0,
x1,
x2,
x3) =
C(
x0,
x2,
x3)
Tags:
B has argument tags [1,0,0] and root tag 0
C has argument tags [1,0,0,1] and root tag 0
Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(B(x1, x2)) = 0
POL(C(x1, x2, x3)) = 0
POL(a) = 1
POL(b(x1, x2)) = x2
POL(c(x1, x2, x3)) = 0
POL(f(x1)) = 1
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, z) → C(y, z, z)
C(f(z), f(c(a, x, a)), y) → B(x, z)
C(f(z), f(c(a, x, a)), y) → C(z, y, a)
The TRS R consists of the following rules:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
Q is empty.
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.
B(y, z) → C(y, z, z)
C(f(z), f(c(a, x, a)), y) → B(x, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
B(
x0,
x1,
x2) =
B(
x0,
x1)
C(
x0,
x1,
x2,
x3) =
C(
x0,
x1,
x2)
Tags:
B has argument tags [0,6,2] and root tag 1
C has argument tags [5,4,5,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(B(x1, x2)) = 1 + x2
POL(C(x1, x2, x3)) = x3
POL(a) = 1
POL(c(x1, x2, x3)) = x2
POL(f(x1)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
none
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(f(z), f(c(a, x, a)), y) → C(z, y, a)
The TRS R consists of the following rules:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
Q is empty.
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.
C(f(z), f(c(a, x, a)), y) → C(z, y, a)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
C(
x0,
x1,
x2,
x3) =
C(
x0,
x2,
x3)
Tags:
C has argument tags [1,0,0,1] and root tag 0
Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(C(x1, x2, x3)) = x1 + x3
POL(a) = 0
POL(c(x1, x2, x3)) = 1
POL(f(x1)) = x1
The following usable rules [FROCOS05] were oriented:
none
(10) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
Q is empty.
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