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

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 QDPSizeChangeProof (⇔)
    ↳17 TRUE
  ↳18 QDP
    ↳19 UsableRulesProof (⇔)
    ↳20 QDP
    ↳21 QReductionProof (⇔)
    ↳22 QDP
    ↳23 QDPSizeChangeProof (⇔)
    ↳24 TRUE

(0) Obligation:

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

int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + x2   
POL(0) = 0   
POL(int(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(int_list(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

int(0, 0) → .(0, nil)
int(s(x), 0) → nil


(2) Obligation:

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

int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

int_list(nil) → nil


(4) Obligation:

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

int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), s(y)) → int_list(int(x, y))
int_list(.(x, y)) → .(s(x), int_list(y))

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:

int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), s(y)) → int_list(int(x, y))
int_list(.(x, y)) → .(s(x), int_list(y))

The set Q consists of the following terms:

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

(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:

INT(0, s(y)) → INT(s(0), s(y))
INT(s(x), s(y)) → INT_LIST(int(x, y))
INT(s(x), s(y)) → INT(x, y)
INT_LIST(.(x, y)) → INT_LIST(y)

The TRS R consists of the following rules:

int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), s(y)) → int_list(int(x, y))
int_list(.(x, y)) → .(s(x), int_list(y))

The set Q consists of the following terms:

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(10) Complex Obligation (AND)

(11) Obligation:

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

INT_LIST(.(x, y)) → INT_LIST(y)

The TRS R consists of the following rules:

int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), s(y)) → int_list(int(x, y))
int_list(.(x, y)) → .(s(x), int_list(y))

The set Q consists of the following terms:

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

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:

INT_LIST(.(x, y)) → INT_LIST(y)

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

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

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].

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

(15) Obligation:

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

INT_LIST(.(x, y)) → INT_LIST(y)

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

(16) 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:

  • INT_LIST(.(x, y)) → INT_LIST(y)
    The graph contains the following edges 1 > 1

(17) TRUE

(18) Obligation:

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

INT(s(x), s(y)) → INT(x, y)
INT(0, s(y)) → INT(s(0), s(y))

The TRS R consists of the following rules:

int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), s(y)) → int_list(int(x, y))
int_list(.(x, y)) → .(s(x), int_list(y))

The set Q consists of the following terms:

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

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:

INT(s(x), s(y)) → INT(x, y)
INT(0, s(y)) → INT(s(0), s(y))

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

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

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].

int(0, s(x0))
int(s(x0), s(x1))
int_list(.(x0, x1))

(22) Obligation:

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

INT(s(x), s(y)) → INT(x, y)
INT(0, s(y)) → INT(s(0), s(y))

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:

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

  • INT(0, s(y)) → INT(s(0), s(y))
    The graph contains the following edges 2 >= 2

(24) TRUE