(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
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(c(c(y))) → C(c(a(a(c(b(0, y)), 0), 0)))
C(c(c(y))) → C(a(a(c(b(0, y)), 0), 0))
C(c(c(y))) → A(a(c(b(0, y)), 0), 0)
C(c(c(y))) → A(c(b(0, y)), 0)
C(c(c(y))) → C(b(0, y))
C(c(c(y))) → B(0, y)
A(y, 0) → B(y, 0)
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
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 5 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(y))) → C(a(a(c(b(0, y)), 0), 0))
C(c(c(y))) → C(c(a(a(c(b(0, y)), 0), 0)))
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
C(c(c(y))) → C(a(a(c(b(0, y)), 0), 0))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(C(x1)) = 2·x1
POL(a(x1, x2)) = x1 + 2·x2
POL(b(x1, x2)) = x1 + x2
POL(c(x1)) = 2 + 2·x1
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(y))) → C(c(a(a(c(b(0, y)), 0), 0)))
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) TransformationProof (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
C(
c(
c(
y))) →
C(
c(
a(
a(
c(
b(
0,
y)),
0),
0))) at position [0] we obtained the following new rules [LPAR04]:
C(c(c(y0))) → C(c(b(a(c(b(0, y0)), 0), 0))) → C(c(c(y0))) → C(c(b(a(c(b(0, y0)), 0), 0)))
C(c(c(y0))) → C(c(a(b(c(b(0, y0)), 0), 0))) → C(c(c(y0))) → C(c(a(b(c(b(0, y0)), 0), 0)))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(y0))) → C(c(b(a(c(b(0, y0)), 0), 0)))
C(c(c(y0))) → C(c(a(b(c(b(0, y0)), 0), 0)))
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) TransformationProof (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
C(
c(
c(
y0))) →
C(
c(
b(
a(
c(
b(
0,
y0)),
0),
0))) at position [0] we obtained the following new rules [LPAR04]:
C(c(c(y0))) → C(c(b(b(c(b(0, y0)), 0), 0))) → C(c(c(y0))) → C(c(b(b(c(b(0, y0)), 0), 0)))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(y0))) → C(c(a(b(c(b(0, y0)), 0), 0)))
C(c(c(y0))) → C(c(b(b(c(b(0, y0)), 0), 0)))
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(y0))) → C(c(a(b(c(b(0, y0)), 0), 0)))
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) TransformationProof (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
C(
c(
c(
y0))) →
C(
c(
a(
b(
c(
b(
0,
y0)),
0),
0))) at position [0] we obtained the following new rules [LPAR04]:
C(c(c(y0))) → C(c(b(b(c(b(0, y0)), 0), 0))) → C(c(c(y0))) → C(c(b(b(c(b(0, y0)), 0), 0)))
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(y0))) → C(c(b(b(c(b(0, y0)), 0), 0)))
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(16) TRUE