YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex16_Luc06_C.ari

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

0 QTRS
↳1 QTRSToCSRProof (⇔, 0 ms)
↳2 CSR
↳3 CSRRRRProof (⇔, 0 ms)
↳4 CSR
↳5 CSDependencyPairsProof (⇔, 0 ms)
↳6 QCSDP
↳7 QCSDependencyGraphProof (⇔, 0 ms)
↳8 TRUE

(0) Obligation:

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

active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
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(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
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}
a: empty set
b: empty set
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(X, X) → f(a, b)
b → a

The replacement map contains the following entries:

f: {1}
a: empty set
b: empty set

(3) CSRRRRProof (EQUIVALENT transformation)

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

f(X, X) → f(a, b)
b → a

The replacement map contains the following entries:

f: {1}
a: empty set
b: empty set
Used ordering:
Polynomial interpretation [POLO]:


POL(a) = 0   
POL(b) = 1   
POL(f(x1, x2)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

b → a


(4) Obligation:

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

f(X, X) → f(a, b)

The replacement map contains the following entries:

f: {1}
a: empty set
b: empty set

(5) CSDependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f, F} we have µ(f) = {1}.

The ordinary context-sensitive dependency pairs DPo are:

F(X, X) → F(a, b)

The TRS R consists of the following rules:

f(X, X) → f(a, b)

Q is empty.

(7) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(z0, z0) → F(a, b) and F(x0, x0) → F(a, b) form no chain, because ECapµ(F(a, b)) = F(a, b) does not unify with F(x0, x0).

(8) TRUE