(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 1
POL(cons(x1, x2)) = x1 + 2·x2
POL(f(x1)) = x1
POL(g(x1)) = x1
POL(h(x1)) = 2·x1
POL(s(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
g(cons(0, Y)) → g(Y)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(X)) → f(X)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(cons(x1, x2)) = 2·x1 + x2
POL(f(x1)) = x1
POL(g(x1)) = x1
POL(h(x1)) = x1
POL(s(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(s(X)) → f(X)
g(cons(s(X), Y)) → s(X)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
h(cons(X, Y)) → h(g(cons(X, Y)))
Q is empty.
(5) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
h(cons(X, Y)) → h(g(cons(X, Y)))
The signature Sigma is {
h}
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
h(cons(x0, x1))
(7) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(cons(X, Y)) → H(g(cons(X, Y)))
The TRS R consists of the following rules:
h(cons(X, Y)) → h(g(cons(X, Y)))
The set Q consists of the following terms:
h(cons(x0, x1))
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(10) TRUE
(11) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(cons(x1, x2)) = 1 + x1 + x2
POL(f(x1)) = x1
POL(g(x1)) = x1
POL(h(x1)) = x1
POL(s(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
(12) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
h(cons(X, Y)) → h(g(cons(X, Y)))
Q is empty.