NO Termination w.r.t. Q proof of EEG_IJCAR_12_enger-nonloop-swapXY.ari

(0) Obligation:

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

g(tt, x, y) → g(swap(x, y), s(x), s(y))
swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

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

swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

The TRS R 2 is

g(tt, x, y) → g(swap(x, y), s(x), s(y))

The signature Sigma is {g}

(2) Obligation:

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

g(tt, x, y) → g(swap(x, y), s(x), s(y))
swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

The set Q consists of the following terms:

g(tt, x0, x1)
swap(0, x0)
swap(s(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:

G(tt, x, y) → G(swap(x, y), s(x), s(y))
G(tt, x, y) → SWAP(x, y)
SWAP(s(x), y) → SWAP(x, s(y))

The TRS R consists of the following rules:

g(tt, x, y) → g(swap(x, y), s(x), s(y))
swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

The set Q consists of the following terms:

g(tt, x0, x1)
swap(0, x0)
swap(s(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 1 less node.

(6) Complex Obligation (AND)

(7) Obligation:

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

SWAP(s(x), y) → SWAP(x, s(y))

The TRS R consists of the following rules:

g(tt, x, y) → g(swap(x, y), s(x), s(y))
swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

The set Q consists of the following terms:

g(tt, x0, x1)
swap(0, x0)
swap(s(x0), x1)

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

(8) 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.

(9) Obligation:

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

SWAP(s(x), y) → SWAP(x, s(y))

R is empty.
The set Q consists of the following terms:

g(tt, x0, x1)
swap(0, x0)
swap(s(x0), x1)

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

(10) 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(tt, x0, x1)
swap(0, x0)
swap(s(x0), x1)

(11) Obligation:

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

SWAP(s(x), y) → SWAP(x, s(y))

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • SWAP(s(x), y) → SWAP(x, s(y))
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

G(tt, x, y) → G(swap(x, y), s(x), s(y))

The TRS R consists of the following rules:

g(tt, x, y) → g(swap(x, y), s(x), s(y))
swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

The set Q consists of the following terms:

g(tt, x0, x1)
swap(0, x0)
swap(s(x0), x1)

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

(15) MNOCProof (EQUIVALENT transformation)

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

(16) Obligation:

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

G(tt, x, y) → G(swap(x, y), s(x), s(y))

The TRS R consists of the following rules:

g(tt, x, y) → g(swap(x, y), s(x), s(y))
swap(0, y) → tt
swap(s(x), y) → swap(x, s(y))

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

(17) NonLoopProof (COMPLETE transformation)

By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 1,
σ' = [ ], and μ' = [x0 / s(x0)] on the rule
G(tt, s(zr0), s(x0))[zr0 / s(zr0)]n[zr0 / 0] → G(tt, s(s(zr0)), s(x0))[zr0 / s(zr0)]n[zr0 / 0, x0 / s(x0)]
This rule is correct for the QDP as the following derivation shows:

G(tt, s(zr0), s(x0))[zr0 / s(zr0)]n[zr0 / 0] → G(tt, s(s(zr0)), s(x0))[zr0 / s(zr0)]n[zr0 / 0, x0 / s(x0)]
    by Equivalency by Simplifying Mu with µ1: [x0 / s(x0)] µ2: [zr0 / 0]
    intermediate steps: Instantiate mu - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
    G(tt, s(zl1), x1)[zr2 / s(zr2), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / x1, zr3 / 0, zl1 / 0] → G(tt, s(s(zr3)), s(x1))[zr2 / s(zr2), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / x1, zr3 / 0, zl1 / 0]
        by Narrowing at position: [0]
        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (lhs)
        G(tt, s(zs1), x1)[zt1 / s(zt1), zs1 / s(zs1)]n[zt1 / x1, zs1 / y1] → G(swap(y1, s(zt1)), s(s(zs1)), s(x1))[zt1 / s(zt1), zs1 / s(zs1)]n[zt1 / x1, zs1 / y1]
            by Narrowing at position: [0]
            intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
            G(tt, x, y)[ ]n[ ] → G(swap(x, y), s(x), s(y))[ ]n[ ]
                by Rule from TRS P

            intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
            swap(s(x), y)[x / s(x)]n[ ] → swap(x, s(y))[y / s(y)]n[ ]
                by PatternCreation I with delta: [ ], theta: [y / s(y)], sigma: [x / s(x)]
                swap(s(x), y)[ ]n[ ] → swap(x, s(y))[ ]n[ ]
                    by Rule from TRS R

        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
        swap(0, y)[ ]n[ ] → tt[ ]n[ ]
            by Rule from TRS R

(18) NO