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

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

0 QTRS
↳1 AAECC Innermost (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 0 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 (⇒, 419 ms)
        ↳25 NO

(0) Obligation:

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

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

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

isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

The TRS R 2 is

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))

The signature Sigma is {add}

(2) Obligation:

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

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

(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:

ADD(true, x, xs) → ADD(and(isNat(x), isList(xs)), x, Cons(tt, xs))
ADD(true, x, xs) → AND(isNat(x), isList(xs))
ADD(true, x, xs) → ISNAT(x)
ADD(true, x, xs) → ISLIST(xs)
ISLIST(Cons(x, xs)) → ISLIST(xs)
ISNAT(s(x)) → ISNAT(x)

The TRS R consists of the following rules:

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

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 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ISNAT(s(x)) → ISNAT(x)

The TRS R consists of the following rules:

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

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:

ISNAT(s(x)) → ISNAT(x)

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

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

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].

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

(11) Obligation:

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

ISNAT(s(x)) → ISNAT(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:

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

(13) YES

(14) Obligation:

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

ISLIST(Cons(x, xs)) → ISLIST(xs)

The TRS R consists of the following rules:

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

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:

ISLIST(Cons(x, xs)) → ISLIST(xs)

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

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

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].

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

(18) Obligation:

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

ISLIST(Cons(x, xs)) → ISLIST(xs)

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:

  • ISLIST(Cons(x, xs)) → ISLIST(xs)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

ADD(true, x, xs) → ADD(and(isNat(x), isList(xs)), x, Cons(tt, xs))

The TRS R consists of the following rules:

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

add(true, x0, x1)
isList(Cons(x0, x1))
isList(nil)
isNat(s(x0))
isNat(0)
and(true, true)
and(false, x0)
and(x0, false)

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:

ADD(true, x, xs) → ADD(and(isNat(x), isList(xs)), x, Cons(tt, xs))

The TRS R consists of the following rules:

add(true, x, xs) → add(and(isNat(x), isList(xs)), x, Cons(tt, xs))
isList(Cons(x, xs)) → isList(xs)
isList(nil) → true
isNat(s(x)) → isNat(x)
isNat(0) → true
and(true, true) → true
and(false, x) → false
and(x, false) → false

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
ADD(true, 0, Cons(tt, zr1))[zr1 / Cons(tt, zr1)]n[zr1 / nil] → ADD(true, 0, Cons(tt, Cons(tt, zr1)))[zr1 / Cons(tt, zr1)]n[zr1 / nil]
This rule is correct for the QDP as the following derivation shows:

ADD(true, 0, Cons(tt, zr1))[zr1 / Cons(tt, zr1)]n[zr1 / nil] → ADD(true, 0, Cons(tt, Cons(tt, zr1)))[zr1 / Cons(tt, zr1)]n[zr1 / nil]
    by Equivalence by Domain Renaming of the lhs with [zl0 / zr1]
    intermediate steps: Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
    ADD(true, 0, Cons(x0, zl1))[zr1 / Cons(x0, zr1), zl1 / Cons(x0, zl1)]n[zr1 / nil, zl1 / nil] → ADD(true, 0, Cons(tt, Cons(x0, zr1)))[zr1 / Cons(x0, zr1), zl1 / Cons(x0, zl1)]n[zr1 / nil, zl1 / nil]
        by Narrowing at position: [0]
        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
        ADD(true, 0, Cons(x1, zl1))[zr1 / Cons(x1, zr1), zl1 / Cons(x1, zl1)]n[zr1 / nil, zl1 / nil] → ADD(and(true, true), 0, Cons(tt, Cons(x1, zr1)))[zr1 / Cons(x1, zr1), zl1 / Cons(x1, zl1)]n[zr1 / nil, zl1 / nil]
            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) - Equiv IPS (rhs) - Equiv IPS (lhs)
            ADD(true, x2, Cons(x0, zl1))[zr1 / Cons(x0, zr1), zl1 / Cons(x0, zl1)]n[zr1 / nil, zl1 / nil] → ADD(and(isNat(x2), true), x2, Cons(tt, Cons(x0, zr1)))[zr1 / Cons(x0, zr1), zl1 / Cons(x0, zl1)]n[zr1 / nil, zl1 / nil]
                by Narrowing at position: [0,1]
                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)
                ADD(true, x1, Cons(y0, zs1))[zs1 / Cons(y0, zs1)]n[zs1 / y1] → ADD(and(isNat(x1), isList(y1)), x1, Cons(tt, Cons(y0, zs1)))[zs1 / Cons(y0, zs1)]n[zs1 / y1]
                    by Narrowing at position: [0,1]
                    intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation
                    ADD(true, x, xs)[ ]n[ ] → ADD(and(isNat(x), isList(xs)), x, Cons(tt, xs))[ ]n[ ]
                        by Rule from TRS P

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

                intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
                isList(nil)[ ]n[ ] → true[ ]n[ ]
                    by Rule from TRS R

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

        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
        and(true, true)[ ]n[ ] → true[ ]n[ ]
            by Rule from TRS R

(25) NO