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

0 QTRS
  ↳1 QTRS Reverse (⇔)
    ↳2 QTRS
  ↳3 QTRS Reverse (⇔)
    ↳4 QTRS
  ↳5 AAECC Innermost (⇔)
    ↳6 QTRS
    ↳7 DependencyPairsProof (⇔)
    ↳8 QDP
    ↳9 DependencyGraphProof (⇔)
    ↳10 QDP
      ↳11 UsableRulesProof (⇔)
        ↳12 QDP
        ↳13 QReductionProof (⇔)
        ↳14 QDP
      ↳15 UsableRulesProof (⇔)
        ↳16 QDP
        ↳17 QReductionProof (⇔)
        ↳18 QDP
        ↳19 MNOCProof (⇔)
        ↳20 QDP
        ↳21 NonTerminationProof (⇔)
        ↳22 FALSE

(0) Obligation:

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

g(X) → h(X)
c → d
h(d) → g(c)

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

g(x) → h(x)
c'(x) → d'(x)
d'(h(x)) → c'(g(x))

Q is empty.

(3) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(4) Obligation:

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

g(x) → h(x)
c'(x) → d'(x)
d'(h(x)) → c'(g(x))

Q is empty.

(5) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

c → d

The TRS R 2 is

g(X) → h(X)
h(d) → g(c)

The signature Sigma is {g, h}

(6) Obligation:

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

g(X) → h(X)
c → d
h(d) → g(c)

The set Q consists of the following terms:

g(x0)
c
h(d)

(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:

G(X) → H(X)
H(d) → G(c)
H(d) → C

The TRS R consists of the following rules:

g(X) → h(X)
c → d
h(d) → g(c)

The set Q consists of the following terms:

g(x0)
c
h(d)

We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(d) → G(c)
G(X) → H(X)

The TRS R consists of the following rules:

g(X) → h(X)
c → d
h(d) → g(c)

The set Q consists of the following terms:

g(x0)
c
h(d)

We have to consider all minimal (P,Q,R)-chains.

(11) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(d) → G(c)
G(X) → H(X)

The TRS R consists of the following rules:

c → d

The set Q consists of the following terms:

g(x0)
c
h(d)

We have to consider all minimal (P,Q,R)-chains.

(13) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

g(x0)
h(d)

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(d) → G(c)
G(X) → H(X)

The TRS R consists of the following rules:

c → d

The set Q consists of the following terms:

c

We have to consider all minimal (P,Q,R)-chains.

(15) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(d) → G(c)
G(X) → H(X)

The TRS R consists of the following rules:

c → d

The set Q consists of the following terms:

g(x0)
c
h(d)

We have to consider all minimal (P,Q,R)-chains.

(17) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

g(x0)
h(d)

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(d) → G(c)
G(X) → H(X)

The TRS R consists of the following rules:

c → d

The set Q consists of the following terms:

c

We have to consider all minimal (P,Q,R)-chains.

(19) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(d) → G(c)
G(X) → H(X)

The TRS R consists of the following rules:

c → d

Q is empty.
We have to consider all (P,Q,R)-chains.

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = G(c) evaluates to t =G(c)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

G(c) → G(d)
with rule c → d at position [0] and matcher [ ]

G(d) → H(d)
with rule G(X) → H(X) at position [] and matcher [X / d]

H(d) → G(c)
with rule H(d) → G(c)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(22) FALSE