(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
g(cons(s(X), Y)) → s(X)
g(cons(0, Y)) → g(Y)
The TRS R 2 is
f(s(X)) → f(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The signature Sigma is {
f,
h}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
(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(s(X)) → F(X)
G(cons(0, Y)) → G(Y)
H(cons(X, Y)) → H(g(cons(X, Y)))
H(cons(X, Y)) → G(cons(X, Y))
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(cons(0, Y)) → G(Y)
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
We have to consider all minimal (P,Q,R)-chains.
(8) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
G(cons(0, Y)) → G(Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
cons2 > [G1, 0]
Status:
cons2: multiset
G1: multiset
0: multiset
The following usable rules [FROCOS05] were oriented:
none
(9) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
We have to consider all minimal (P,Q,R)-chains.
(10) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(11) TRUE
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(X)) → F(X)
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(s(X)) → F(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
s1 > F1
Status:
s1: multiset
F1: multiset
The following usable rules [FROCOS05] were oriented:
none
(14) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
We have to consider all minimal (P,Q,R)-chains.
(15) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(16) TRUE