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 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 PisEmptyProof (⇔)
        ↳9 TRUE
    ↳10 QDP
        ↳11 QDPOrderProof (⇔)
        ↳12 QDP
        ↳13 DependencyGraphProof (⇔)
        ↳14 TRUE

(0) Obligation:

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

app(app(neq, 0), 0) → false
app(app(neq, 0), app(s, y)) → true
app(app(neq, app(s, x)), 0) → true
app(app(neq, app(s, x)), app(s, y)) → app(app(neq, x), y)
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, y), ys)) → app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) → app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) → app(app(filter, f), ys)
nonzero → app(filter, app(neq, 0))

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(neq, app(s, x)), app(s, y)) → APP(app(neq, x), y)
APP(app(neq, app(s, x)), app(s, y)) → APP(neq, x)
APP(app(filter, f), app(app(cons, y), ys)) → APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
APP(app(filter, f), app(app(cons, y), ys)) → APP(app(filtersub, app(f, y)), f)
APP(app(filter, f), app(app(cons, y), ys)) → APP(filtersub, app(f, y))
APP(app(filter, f), app(app(cons, y), ys)) → APP(f, y)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) → APP(app(cons, y), app(app(filter, f), ys))
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) → APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) → APP(filter, f)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) → APP(app(filter, f), ys)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) → APP(filter, f)
NONZERO → APP(filter, app(neq, 0))
NONZERO → APP(neq, 0)

The TRS R consists of the following rules:

app(app(neq, 0), 0) → false
app(app(neq, 0), app(s, y)) → true
app(app(neq, app(s, x)), 0) → true
app(app(neq, app(s, x)), app(s, y)) → app(app(neq, x), y)
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, y), ys)) → app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) → app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) → app(app(filter, f), ys)
nonzero → app(filter, app(neq, 0))

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 2 SCCs with 8 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(neq, app(s, x)), app(s, y)) → APP(app(neq, x), y)

The TRS R consists of the following rules:

app(app(neq, 0), 0) → false
app(app(neq, 0), app(s, y)) → true
app(app(neq, app(s, x)), 0) → true
app(app(neq, app(s, x)), app(s, y)) → app(app(neq, x), y)
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, y), ys)) → app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) → app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) → app(app(filter, f), ys)
nonzero → app(filter, app(neq, 0))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(neq, app(s, x)), app(s, y)) → APP(app(neq, x), y)
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 [0,0,3] and root tag 0

Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2)  =  APP
app(x1, x2)  =  app(x1, x2)
neq  =  neq
s  =  s

Lexicographic path order with status [LPO].
Quasi-Precedence:
APP > [app2, neq]

Status:
APP: []
app2: [2,1]
neq: []
s: []


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

app(app(neq, 0), 0) → false
app(app(neq, 0), app(s, y)) → true
app(app(neq, app(s, x)), 0) → true
app(app(neq, app(s, x)), app(s, y)) → app(app(neq, x), y)
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, y), ys)) → app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) → app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) → app(app(filter, f), ys)
nonzero → app(filter, app(neq, 0))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

APP(app(filter, f), app(app(cons, y), ys)) → APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) → APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) → APP(f, y)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) → APP(app(filter, f), ys)

The TRS R consists of the following rules:

app(app(neq, 0), 0) → false
app(app(neq, 0), app(s, y)) → true
app(app(neq, app(s, x)), 0) → true
app(app(neq, app(s, x)), app(s, y)) → app(app(neq, x), y)
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, y), ys)) → app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) → app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) → app(app(filter, f), ys)
nonzero → app(filter, app(neq, 0))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(app(filtersub, true), f), app(app(cons, y), ys)) → APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) → APP(f, y)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) → APP(app(filter, f), ys)
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 [3,0,3] and root tag 0

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2)  =  APP
app(x1, x2)  =  app(x1, x2)
filter  =  filter
cons  =  cons
filtersub  =  filtersub
true  =  true
false  =  false
neq  =  neq
0  =  0
s  =  s
nil  =  nil

Lexicographic path order with status [LPO].
Quasi-Precedence:
APP > [filter, filtersub] > cons > app2
neq > app2
0 > true > app2
0 > false > [filter, filtersub] > cons > app2
s > app2
nil > app2

Status:
APP: []
app2: [1,2]
filter: []
cons: []
filtersub: []
true: []
false: []
neq: []
0: []
s: []
nil: []


The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

APP(app(filter, f), app(app(cons, y), ys)) → APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))

The TRS R consists of the following rules:

app(app(neq, 0), 0) → false
app(app(neq, 0), app(s, y)) → true
app(app(neq, app(s, x)), 0) → true
app(app(neq, app(s, x)), app(s, y)) → app(app(neq, x), y)
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, y), ys)) → app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) → app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) → app(app(filter, f), ys)
nonzero → app(filter, app(neq, 0))

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE