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 QDPSizeChangeProof (⇔)
        ↳7 TRUE
    ↳8 QDP
        ↳9 QDPSizeChangeProof (⇔)
        ↳10 TRUE
    ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
    ↳14 QDP
        ↳15 QDPSizeChangeProof (⇔)
        ↳16 TRUE

(0) Obligation:

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

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(append, xs), nil) → xs
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) → yss
app(app(zip, xss), nil) → xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) → xs
app(app(combine, xs), app(app(cons, ys), yss)) → app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) → app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), 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(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(append, app(app(cons, x), xs)), ys) → APP(app(cons, x), app(app(append, xs), ys))
APP(app(append, app(app(cons, x), xs)), ys) → APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) → APP(append, xs)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(cons, app(app(append, xs), ys))
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(app(append, xs), ys)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(append, xs)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(app(zip, xss), yss)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(zip, xss)
APP(app(combine, xs), app(app(cons, ys), yss)) → APP(app(combine, app(app(zip, xs), ys)), yss)
APP(app(combine, xs), app(app(cons, ys), yss)) → APP(combine, app(app(zip, xs), ys))
APP(app(combine, xs), app(app(cons, ys), yss)) → APP(app(zip, xs), ys)
APP(app(combine, xs), app(app(cons, ys), yss)) → APP(zip, xs)
APP(levels, app(app(node, x), xs)) → APP(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))
APP(levels, app(app(node, x), xs)) → APP(cons, app(app(cons, x), nil))
APP(levels, app(app(node, x), xs)) → APP(app(cons, x), nil)
APP(levels, app(app(node, x), xs)) → APP(cons, x)
APP(levels, app(app(node, x), xs)) → APP(app(combine, nil), app(app(map, levels), xs))
APP(levels, app(app(node, x), xs)) → APP(combine, nil)
APP(levels, app(app(node, x), xs)) → APP(app(map, levels), xs)
APP(levels, app(app(node, x), xs)) → APP(map, levels)

The TRS R consists of the following rules:

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(append, xs), nil) → xs
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) → yss
app(app(zip, xss), nil) → xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) → xs
app(app(combine, xs), app(app(cons, ys), yss)) → app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) → app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), 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 4 SCCs with 19 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(append, app(app(cons, x), xs)), ys) → APP(app(append, xs), ys)

The TRS R consists of the following rules:

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(append, xs), nil) → xs
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) → yss
app(app(zip, xss), nil) → xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) → xs
app(app(combine, xs), app(app(cons, ys), yss)) → app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) → app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))

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

(6) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
append  =  append
cons  =  cons
app(x1, x2)  =  app(x2)

From the DPs we obtained the following set of size-change graphs:

  • APP(app(append, app(app(cons, x), xs)), ys) → APP(app(append, xs), ys) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 2 >= 2

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(7) TRUE

(8) Obligation:

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

APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(app(zip, xss), yss)

The TRS R consists of the following rules:

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(append, xs), nil) → xs
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) → yss
app(app(zip, xss), nil) → xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) → xs
app(app(combine, xs), app(app(cons, ys), yss)) → app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) → app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))

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

(9) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
zip  =  zip
cons  =  cons
app(x1, x2)  =  app(x2)

From the DPs we obtained the following set of size-change graphs:

  • APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → APP(app(zip, xss), yss) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 2 > 2

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(10) TRUE

(11) Obligation:

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

APP(app(combine, xs), app(app(cons, ys), yss)) → APP(app(combine, app(app(zip, xs), ys)), yss)

The TRS R consists of the following rules:

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(append, xs), nil) → xs
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) → yss
app(app(zip, xss), nil) → xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) → xs
app(app(combine, xs), app(app(cons, ys), yss)) → app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) → app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
combine  =  combine
cons  =  cons
app(x1, x2)  =  app(x2)

From the DPs we obtained the following set of size-change graphs:

  • APP(app(combine, xs), app(app(cons, ys), yss)) → APP(app(combine, app(app(zip, xs), ys)), yss) (allowed arguments on rhs = {2})
    The graph contains the following edges 2 > 2

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(13) TRUE

(14) 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(levels, app(app(node, x), xs)) → APP(app(map, levels), xs)

The TRS R consists of the following rules:

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(append, xs), nil) → xs
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) → yss
app(app(zip, xss), nil) → xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) → app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) → xs
app(app(combine, xs), app(app(cons, ys), yss)) → app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) → app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))

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

(15) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
map  =  map
cons  =  cons
levels  =  levels
node  =  node
app(x1, x2)  =  app(x1, x2)

From the DPs we obtained the following set of size-change graphs:

  • APP(levels, app(app(node, x), xs)) → APP(app(map, levels), xs) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 2 > 2

  • APP(app(map, f), app(app(cons, x), xs)) → APP(f, x) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 2 > 2

  • APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(16) TRUE