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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 Overlay + Local Confluence (⇔)
↳6 QTRS
↳7 DependencyPairsProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 AND
    ↳11 QDP
        ↳12 UsableRulesProof (⇔)
        ↳13 QDP
        ↳14 QReductionProof (⇔)
        ↳15 QDP
        ↳16 NonTerminationProof (⇔)
        ↳17 NO
    ↳18 QDP
        ↳19 UsableRulesProof (⇔)
        ↳20 QDP
        ↳21 QReductionProof (⇔)
        ↳22 QDP
        ↳23 QDPSizeChangeProof (⇔)
        ↳24 YES
    ↳25 QDP
        ↳26 UsableRulesProof (⇔)
        ↳27 QDP
        ↳28 QReductionProof (⇔)
        ↳29 QDP
        ↳30 QDPSizeChangeProof (⇔)
        ↳31 YES

(0) Obligation:

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

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(nil) → nil
adx(cons(X, L)) → incr(cons(X, adx(L)))
nats → adx(zeros)
zeros → cons(0, zeros)
head(cons(X, L)) → X
tail(cons(X, L)) → L

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(adx(x1)) = 2·x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(head(x1)) = 2 + x1   
POL(incr(x1)) = x1   
POL(nats) = 0   
POL(nil) = 2   
POL(s(x1)) = x1   
POL(tail(x1)) = 2 + x1   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

adx(nil) → nil
head(cons(X, L)) → X
tail(cons(X, L)) → L


(2) Obligation:

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

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
nats → adx(zeros)
zeros → cons(0, zeros)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(adx(x1)) = 2·x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(incr(x1)) = x1   
POL(nats) = 2   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

nats → adx(zeros)


(4) Obligation:

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

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
zeros → cons(0, zeros)

Q is empty.

(5) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(6) Obligation:

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

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
zeros → cons(0, zeros)

The set Q consists of the following terms:

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

(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:

INCR(cons(X, L)) → INCR(L)
ADX(cons(X, L)) → INCR(cons(X, adx(L)))
ADX(cons(X, L)) → ADX(L)
ZEROS → ZEROS

The TRS R consists of the following rules:

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
zeros → cons(0, zeros)

The set Q consists of the following terms:

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 1 less node.

(10) Complex Obligation (AND)

(11) Obligation:

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

ZEROS → ZEROS

The TRS R consists of the following rules:

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
zeros → cons(0, zeros)

The set Q consists of the following terms:

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(12) 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.

(13) Obligation:

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

ZEROS → ZEROS

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

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(14) 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].

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

(15) Obligation:

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

ZEROS → ZEROS

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

(16) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ZEROS evaluates to t =ZEROS

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ZEROS to ZEROS.



(17) NO

(18) Obligation:

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

INCR(cons(X, L)) → INCR(L)

The TRS R consists of the following rules:

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
zeros → cons(0, zeros)

The set Q consists of the following terms:

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(19) 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.

(20) Obligation:

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

INCR(cons(X, L)) → INCR(L)

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

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(21) 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].

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

(22) Obligation:

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

INCR(cons(X, L)) → INCR(L)

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

(23) 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:

  • INCR(cons(X, L)) → INCR(L)
    The graph contains the following edges 1 > 1

(24) YES

(25) Obligation:

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

ADX(cons(X, L)) → ADX(L)

The TRS R consists of the following rules:

incr(nil) → nil
incr(cons(X, L)) → cons(s(X), incr(L))
adx(cons(X, L)) → incr(cons(X, adx(L)))
zeros → cons(0, zeros)

The set Q consists of the following terms:

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(26) 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.

(27) Obligation:

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

ADX(cons(X, L)) → ADX(L)

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

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

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

(28) 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].

incr(nil)
incr(cons(x0, x1))
adx(cons(x0, x1))
zeros

(29) Obligation:

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

ADX(cons(X, L)) → ADX(L)

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

(30) 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:

  • ADX(cons(X, L)) → ADX(L)
    The graph contains the following edges 1 > 1

(31) YES