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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 UsableRulesProof (⇔)
    ↳7 QDP
    ↳8 QDPSizeChangeProof (⇔)
    ↳9 TRUE
  ↳10 QDP
    ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 QDPSizeChangeProof (⇔)
    ↳14 TRUE

(0) Obligation:

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

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

*1(x, +(y, z)) → +1(*(x, y), *(x, z))
*1(x, +(y, z)) → *1(x, y)
*1(x, +(y, z)) → *1(x, z)
*1(+(y, z), x) → +1(*(x, y), *(x, z))
*1(+(y, z), x) → *1(x, y)
*1(+(y, z), x) → *1(x, z)
*1(*(x, y), z) → *1(x, *(y, z))
*1(*(x, y), z) → *1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))
+1(+(x, y), z) → +1(y, z)

The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Complex Obligation (AND)

(5) Obligation:

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

+1(+(x, y), z) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))

The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))

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

(6) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(7) Obligation:

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

+1(+(x, y), z) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))

The TRS R consists of the following rules:

+(+(x, y), z) → +(x, +(y, z))

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

(8) 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:

  • +1(+(x, y), z) → +1(y, z)
    The graph contains the following edges 1 > 1, 2 >= 2

  • +1(+(x, y), z) → +1(x, +(y, z))
    The graph contains the following edges 1 > 1

(9) TRUE

(10) Obligation:

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

*1(x, +(y, z)) → *1(x, z)
*1(x, +(y, z)) → *1(x, y)
*1(+(y, z), x) → *1(x, y)
*1(+(y, z), x) → *1(x, z)
*1(*(x, y), z) → *1(x, *(y, z))
*1(*(x, y), z) → *1(y, z)

The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


*1(+(y, z), x) → *1(x, y)
*1(+(y, z), x) → *1(x, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(*(x1, x2)) = x1 + x1·x2 + x2   
POL(*1(x1, x2)) = x1 + x1·x2   
POL(+(x1, x2)) = 1 + x1 + x2   

The following usable rules [FROCOS05] were oriented:

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))

(12) Obligation:

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

*1(x, +(y, z)) → *1(x, z)
*1(x, +(y, z)) → *1(x, y)
*1(*(x, y), z) → *1(x, *(y, z))
*1(*(x, y), z) → *1(y, z)

The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))

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

(13) 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:

  • *1(x, +(y, z)) → *1(x, z)
    The graph contains the following edges 1 >= 1, 2 > 2

  • *1(x, +(y, z)) → *1(x, y)
    The graph contains the following edges 1 >= 1, 2 > 2

  • *1(*(x, y), z) → *1(x, *(y, z))
    The graph contains the following edges 1 > 1

  • *1(*(x, y), z) → *1(y, z)
    The graph contains the following edges 1 > 1, 2 >= 2

(14) TRUE