(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,
x2)
C(
x0,
x1,
x2,
x3) =
C(
x0,
x2,
x3)
Tags:
B has argument tags [4,0,3] and root tag 0
C has argument tags [4,1,3,4] and root tag 0
Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
B(
x1,
x2) =
B
C(
x1,
x2,
x3) =
C
f(
x1) =
f
c(
x1,
x2,
x3) =
c
a =
a
b(
x1,
x2) =
b
Recursive path order with status [RPO].
Quasi-Precedence:
b > [f, a] > [B, C, c]
Status:
B: []
C: []
f: multiset
c: []
a: multiset
b: multiset
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(
x1,
x2)
C(
x0,
x1,
x2,
x3) =
C(
x0,
x2,
x3)
Tags:
B has argument tags [0,2,5] and root tag 0
C has argument tags [0,5,0,0] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
B(
x1,
x2) =
x2
C(
x1,
x2,
x3) =
x1
f(
x1) =
f(
x1)
c(
x1,
x2,
x3) =
c(
x1,
x2)
a =
a
Recursive path order with status [RPO].
Quasi-Precedence:
c2 > [f1, a]
Status:
f1: [1]
c2: multiset
a: multiset
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(
x2,
x3)
Tags:
C has argument tags [1,0,0,0] and root tag 0
Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
C(
x1,
x2,
x3) =
C(
x1,
x3)
f(
x1) =
x1
c(
x1,
x2,
x3) =
c(
x1,
x2)
a =
a
Recursive path order with status [RPO].
Quasi-Precedence:
[C2, c2, a]
Status:
C2: [1,2]
c2: [1,2]
a: multiset
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