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 ATransformationProof (⇔)
    ↳9 QDP
    ↳10 QDPSizeChangeProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 UsableRulesProof (⇔)
    ↳14 QDP
    ↳15 ATransformationProof (⇔)
    ↳16 QDP
    ↳17 QDPSizeChangeProof (⇔)
    ↳18 TRUE
  ↳19 QDP
    ↳20 QDPSizeChangeProof (⇔)
    ↳21 TRUE

(0) Obligation:

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

app(f, 0) → true
app(f, 1) → false
app(f, app(s, x)) → app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) → app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) → app(s, y)
app(app(g, x), app(c, y)) → app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) → app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), 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(f, app(s, x)) → APP(f, x)
APP(app(g, x), app(c, y)) → APP(c, app(app(g, x), y))
APP(app(g, x), app(c, y)) → APP(app(g, x), y)
APP(app(g, x), app(c, y)) → APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) → APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) → APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) → APP(if, app(f, x))
APP(app(g, x), app(c, y)) → APP(f, x)
APP(app(g, x), app(c, y)) → APP(c, app(app(g, app(s, x)), y))
APP(app(g, x), app(c, y)) → APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) → APP(g, app(s, x))
APP(app(g, x), app(c, y)) → APP(s, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(cons, x), app(app(filter, fun), xs))
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, true), fun), x), xs) → APP(filter, fun)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)

The TRS R consists of the following rules:

app(f, 0) → true
app(f, 1) → false
app(f, app(s, x)) → app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) → app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) → app(s, y)
app(app(g, x), app(c, y)) → app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) → app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), 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 18 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(f, app(s, x)) → APP(f, x)

The TRS R consists of the following rules:

app(f, 0) → true
app(f, 1) → false
app(f, app(s, x)) → app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) → app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) → app(s, y)
app(app(g, x), app(c, y)) → app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) → app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), 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(f, app(s, x)) → APP(f, x)

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

(8) ATransformationProof (EQUIVALENT transformation)

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

(9) Obligation:

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

f(s(x)) → f(x)

R is empty.
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:

  • f(s(x)) → f(x)
    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(g, x), app(c, y)) → APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) → APP(app(g, x), y)

The TRS R consists of the following rules:

app(f, 0) → true
app(f, 1) → false
app(f, app(s, x)) → app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) → app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) → app(s, y)
app(app(g, x), app(c, y)) → app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) → app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)

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

(13) 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.

(14) Obligation:

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

APP(app(g, x), app(c, y)) → APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) → APP(app(g, x), y)

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

(15) ATransformationProof (EQUIVALENT transformation)

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

(16) Obligation:

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

g(x, c(y)) → g(s(x), y)
g(x, c(y)) → g(x, y)

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

(17) 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:

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

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

(18) TRUE

(19) Obligation:

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

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

The TRS R consists of the following rules:

app(f, 0) → true
app(f, 1) → false
app(f, app(s, x)) → app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) → app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) → app(s, y)
app(app(g, x), app(c, y)) → app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) → app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)

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

(20) 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, fun), app(app(cons, x), xs)) → APP(fun, x)
    The graph contains the following edges 1 > 1, 2 > 2

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

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

  • APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
    The graph contains the following edges 2 > 2

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

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

(21) TRUE