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

(0) Obligation:

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

app(app(mapbt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) → app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

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(mapbt, f), app(leaf, x)) → APP(leaf, app(f, x))
APP(app(mapbt, f), app(leaf, x)) → APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(branch, app(f, x)), app(app(mapbt, f), l))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(branch, app(f, x))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), r)

The TRS R consists of the following rules:

app(app(mapbt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) → app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

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 1 SCC with 4 less nodes.

(4) Obligation:

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

APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(f, x)
APP(app(mapbt, f), app(leaf, x)) → APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), r)

The TRS R consists of the following rules:

app(app(mapbt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) → app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(f, x)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2)  =  APP(x0, x1)

Tags:
APP has argument tags [2,0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(APP(x₁, x₂)) = x₂   
POL(app(x₁, x₂)) = x₁ + x₂   
POL(branch) = 1   
POL(leaf) = 0   
POL(mapbt) = 1   

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

APP(app(mapbt, f), app(leaf, x)) → APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), r)

The TRS R consists of the following rules:

app(app(mapbt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) → app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(app(mapbt, f), app(leaf, x)) → APP(f, x)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2)  =  APP(x0, x2)

Tags:
APP has argument tags [0,3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(APP(x₁, x₂)) = x₁ + x₂   
POL(app(x₁, x₂)) = x₁ + x₂   
POL(branch) = 0   
POL(leaf) = 0   
POL(mapbt) = 1   

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), r)

The TRS R consists of the following rules:

app(app(mapbt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) → app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) → APP(app(mapbt, f), r)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2)  =  APP(x2)

Tags:
APP has argument tags [3,1,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(APP(x₁, x₂)) = 0   
POL(app(x₁, x₂)) = x₁ + x₂   
POL(branch) = 1   
POL(mapbt) = 0   

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app(app(mapbt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) → app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

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

(11) PisEmptyProof (EQUIVALENT transformation)

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

(12) TRUE