Termination w.r.t. Q proof of EEG_IJCAR_12

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

0 QTRS

↳1 AAECC Innermost (⇔, 0 ms)

↳2 QTRS

↳3 DependencyPairsProof (⇔, 9 ms)

↳4 QDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 QDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 QDP

        ↳10 QReductionProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 QDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 QDP

        ↳17 QReductionProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 QDP

        ↳22 MNOCProof (⇔, 0 ms)

        ↳23 QDP

        ↳24 NonLoopProof (⇒, 43 ms)

        ↳25 NO

(0) Obligation:

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

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

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

eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

The TRS R 2 is

f(tt, x) → f(eq(toOne(x), s(0)), s(x))

The signature Sigma is {f}

(2) Obligation:

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

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

The set Q consists of the following terms:

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

(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:

F(tt, x) → F(eq(toOne(x), s(0)), s(x))
F(tt, x) → EQ(toOne(x), s(0))
F(tt, x) → TOONE(x)
EQ(s(x), s(y)) → EQ(x, y)
TOONE(s(s(x))) → TOONE(s(x))

The TRS R consists of the following rules:

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

The set Q consists of the following terms:

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

TOONE(s(s(x))) → TOONE(s(x))

The TRS R consists of the following rules:

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

The set Q consists of the following terms:

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

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:

TOONE(s(s(x))) → TOONE(s(x))

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

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

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].

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

(11) Obligation:

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

TOONE(s(s(x))) → TOONE(s(x))

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:

TOONE(s(s(x))) → TOONE(s(x))
The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

EQ(s(x), s(y)) → EQ(x, y)

The TRS R consists of the following rules:

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

The set Q consists of the following terms:

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

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:

EQ(s(x), s(y)) → EQ(x, y)

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

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

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].

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

(18) Obligation:

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

EQ(s(x), s(y)) → EQ(x, y)

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

(19) 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:

EQ(s(x), s(y)) → EQ(x, y)
The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

F(tt, x) → F(eq(toOne(x), s(0)), s(x))

The TRS R consists of the following rules:

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

The set Q consists of the following terms:

f(tt, x0)
eq(s(x0), s(x1))
eq(0, 0)
toOne(s(s(x0)))
toOne(s(0))

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

(22) MNOCProof (EQUIVALENT transformation)

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

(23) Obligation:

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

F(tt, x) → F(eq(toOne(x), s(0)), s(x))

The TRS R consists of the following rules:

f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)

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

(24) 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' = [ ] on the rule
F(tt, s(s(zr0)))[zr0 / s(zr0)]ⁿ[zr0 / 0] → F(tt, s(s(s(zr0))))[zr0 / s(zr0)]ⁿ[zr0 / 0]
This rule is correct for the QDP as the following derivation shows:

F(tt, s(s(zr0)))[zr0 / s(zr0)]ⁿ[zr0 / 0] → F(tt, s(s(s(zr0))))[zr0 / s(zr0)]ⁿ[zr0 / 0]
    by Equivalence by Domain Renaming of the lhs with [zl0 / zr0]
    intermediate steps: Equiv DR (rhs) - Equiv DR (lhs)
    F(tt, s(s(zl1)))[zl1 / s(zl1)]ⁿ[zl1 / 0] → F(tt, s(s(s(zr1))))[zr1 / s(zr1)]ⁿ[zr1 / 0]
        by Rewrite t with the rewrite sequence : [([0],eq(s(x), s(y)) -> eq(x, y)), ([0],eq(0, 0) -> tt)]
        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs)
        F(tt, s(s(zl1)))[zr1 / s(zr1), zl1 / s(zl1)]ⁿ[zr1 / 0, zl1 / 0] → F(eq(s(0), s(0)), s(s(s(zr1))))[zr1 / s(zr1), zl1 / s(zl1)]ⁿ[zr1 / 0, zl1 / 0]
            by Narrowing at position: [0,0]
            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)
            F(tt, s(s(zs1)))[zs1 / s(zs1)]ⁿ[zs1 / y0] → F(eq(toOne(s(y0)), s(0)), s(s(s(zs1))))[zs1 / s(zs1)]ⁿ[zs1 / y0]
                by Narrowing at position: [0,0]
                intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation
                F(tt, x)[ ]ⁿ[ ] → F(eq(toOne(x), s(0)), s(x))[ ]ⁿ[ ]
                    by Rule from TRS P

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

            intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
            toOne(s(0))[ ]ⁿ[ ] → s(0)[ ]ⁿ[ ]
                by Rule from TRS R

(0) Obligation:

(1) AAECC Innermost (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) QReductionProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) MNOCProof (EQUIVALENT transformation)

(23) Obligation:

(24) NonLoopProof (COMPLETE transformation)

(25) NO