Termination w.r.t. Q proof of Applicative_first_order_05

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

0 QTRS

↳1 DependencyPairsProof (⇔, 5 ms)

↳2 QDP

↳3 DependencyGraphProof (⇔, 0 ms)

↳4 AND

    ↳5 QDP

        ↳6 UsableRulesProof (⇔, 0 ms)

        ↳7 QDP

        ↳8 UsableRulesReductionPairsProof (⇔, 0 ms)

        ↳9 QDP

        ↳10 PisEmptyProof (⇔, 0 ms)

        ↳11 YES

    ↳12 QDP

        ↳13 QDPApplicativeOrderProof (⇔, 0 ms)

        ↳14 QDP

        ↳15 QDPOrderProof (⇔, 11 ms)

        ↳16 QDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 ATransformationProof (⇔, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 QDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

(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.

The following dependency pairs can be deleted:

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

The following rules are removed from R:

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

Used ordering: POLO with Polynomial interpretation [POLO]:

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

(9) Obligation:

Q DP problem:
P is empty.
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) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) YES

(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 [LPAR04,JAR06] 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))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(x, +(y, z)) → +(*(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] with respect to the argument filtering of the ordering [JAR06] were oriented:

app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, x), app(app(+, y), z)) → 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,JAR06]. 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: Combined order from the following AFS and order.
*1(x1, x2) = x1
*(x1, x2) = *(x1, x2)
+(x1, x2) = +

Knuth-Bendix order [KBO] with precedence:

trivial

and weight map:

+=1
*_2=1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] 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)

(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) YES

(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)

(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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) UsableRulesProof (EQUIVALENT transformation)

(7) Obligation:

(8) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(9) Obligation:

(10) PisEmptyProof (EQUIVALENT transformation)

(11) YES

(12) Obligation:

(13) QDPApplicativeOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) ATransformationProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES