Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(0))) → mark(0)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(0))) → mark(0)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

f: {1}
0: empty set
cons: {1}
s: {1}
p: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

f(0) → cons(0, f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0

The replacement map contains the following entries:

f: {1}
0: empty set
cons: {1}
s: {1}
p: {1}

(3) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

f(0) → cons(0, f(s(0)))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(cons(x₁, x₂)) = 2·x₁   
POL(f(x₁)) = 1 + 2·x₁   
POL(p(x₁)) = 2·x₁   
POL(s(x₁)) = 2·x₁   

(4) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

f(s(0)) → f(p(s(0)))
p(s(0)) → 0

The replacement map contains the following entries:

f: {1}
0: empty set
s: {1}
p: {1}

(5) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

p(s(0)) → 0

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(f(x₁)) = x₁   
POL(p(x₁)) = x₁   
POL(s(x₁)) = 1 + 2·x₁   

(6) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

f(s(0)) → f(p(s(0)))

The replacement map contains the following entries:

f: {1}
0: empty set
s: {1}
p: {1}

(7) CSRInnermostProof (EQUIVALENT transformation)

The CSR is orthogonal. By [CS_Inn] we can switch to innermost.

(8) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

f(s(0)) → f(p(s(0)))

The replacement map contains the following entries:

f: {1}
0: empty set
s: {1}
p: {1}

Innermost Strategy.

(9) CSDependencyPairsProof (EQUIVALENT transformation)

Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.

(10) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {f, s, p, F} are replacing on all positions.

The ordinary context-sensitive dependency pairs DP_o are:

F(s(0)) → F(p(s(0)))

The TRS R consists of the following rules:

f(s(0)) → f(p(s(0)))

The set Q consists of the following terms:

f(s(0))

(11) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(s(0)) → F(p(s(0))) and F(s(0)) → F(p(s(0))) form no chain, because ECap^µ(F(p(s(0)))) = F(p(s(0))) does not unify with F(s(0)).

(12) TRUE