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 UsableRulesReductionPairsProof (⇔)
    ↳9 QDP
    ↳10 QDPSizeChangeProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPApplicativeOrderProof (⇔)
    ↳14 QDP
    ↳15 QDPOrderProof (⇔)
    ↳16 QDP
    ↳17 UsableRulesProof (⇔)
    ↳18 QDP
    ↳19 ATransformationProof (⇔)
    ↳20 QDP
    ↳21 QDPSizeChangeProof (⇔)
    ↳22 TRUE
  ↳23 QDP
    ↳24 UsableRulesProof (⇔)
    ↳25 QDP
    ↳26 QDPSizeChangeProof (⇔)
    ↳27 TRUE

(0) Obligation:

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

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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:

APP(app(*, x), app(app(+, y), z)) → APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) → APP(+, app(app(*, x), y))
APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) → APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, app(app(+, y), z)), x) → APP(+, app(app(*, x), y))
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), y)
APP(app(*, app(app(+, y), z)), x) → APP(*, x)
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), z)
APP(app(*, app(app(*, x), y)), z) → APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(*, x), y)), z) → APP(app(*, y), z)
APP(app(*, app(app(*, x), y)), z) → APP(*, y)
APP(app(+, app(app(+, x), y)), z) → APP(app(+, x), app(app(+, y), z))
APP(app(+, app(app(+, x), y)), z) → APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) → APP(+, y)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)

The TRS R consists of the following rules:

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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 3 SCCs with 17 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(+, app(app(+, x), y)), z) → APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) → APP(app(+, x), app(app(+, y), z))

The TRS R consists of the following rules:

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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:

APP(app(+, app(app(+, x), y)), z) → APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) → APP(app(+, x), app(app(+, y), z))

The TRS R consists of the following rules:

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

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

(8) UsableRulesReductionPairsProof (EQUIVALENT transformation)

First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem.
The pairs P are:

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

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

+(+(x, y), z) → +(x, +(y, z))

Q is empty.

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(+1(x1, x2)) = 2·x1 + 2·x2   

(9) 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.

(10) 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

(11) TRUE

(12) Obligation:

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

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

The TRS R consists of the following rules:

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(13) QDPApplicativeOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

*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 a-transformed usable rules are

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


The following pairs can be oriented strictly and are deleted.


APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), y)
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), z)
The remaining pairs can at least be oriented weakly.

APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), y)
APP(app(*, app(app(*, x), y)), z) → APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(*, x), y)), z) → APP(app(*, y), z)
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:

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

(14) Obligation:

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

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

The TRS R consists of the following rules:

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

*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 a-transformed usable rules are

*(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))


The following pairs can be oriented strictly and are deleted.


APP(app(*, app(app(*, x), y)), z) → APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(*, x), y)), z) → APP(app(*, y), z)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(*1(x1, x2)) =
/0\
\1/
 +
/01\
\01/
·x1 +
/00\
\00/
·x2

POL(+(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/00\
\10/
·x2

POL(*(x1, x2)) =
/0\
\1/
 +
/00\
\01/
·x1 +
/10\
\01/
·x2

The following usable rules [FROCOS05] were oriented:

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

(16) Obligation:

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

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

The TRS R consists of the following rules:

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(17) 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.

(18) Obligation:

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

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

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

(19) ATransformationProof (EQUIVALENT transformation)

We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem.

(20) Obligation:

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

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

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

(21) 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:

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

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

(22) TRUE

(23) Obligation:

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

APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

The TRS R consists of the following rules:

app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(24) 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.

(25) Obligation:

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

APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

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

(26) 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:

  • APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
    The graph contains the following edges 1 > 1, 2 > 2

  • APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
    The graph contains the following edges 1 > 1, 2 > 2

  • APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
    The graph contains the following edges 2 >= 2

  • APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
    The graph contains the following edges 2 >= 2

(27) TRUE