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

(0) Obligation:

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

app(D, t) → 1
app(D, constant) → 0
app(D, app(app(+, x), y)) → app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) → app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) → app(app(-, app(D, x)), app(D, y))
app(D, app(minus, x)) → app(minus, app(D, x))
app(D, app(app(div, x), y)) → app(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
app(D, app(ln, x)) → app(app(div, app(D, x)), x)
app(D, app(app(pow, x), y)) → app(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
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(D, app(app(+, x), y)) → APP(app(+, app(D, x)), app(D, y))
APP(D, app(app(+, x), y)) → APP(+, app(D, x))
APP(D, app(app(+, x), y)) → APP(D, x)
APP(D, app(app(+, x), y)) → APP(D, y)
APP(D, app(app(*, x), y)) → APP(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
APP(D, app(app(*, x), y)) → APP(+, app(app(*, y), app(D, x)))
APP(D, app(app(*, x), y)) → APP(app(*, y), app(D, x))
APP(D, app(app(*, x), y)) → APP(*, y)
APP(D, app(app(*, x), y)) → APP(D, x)
APP(D, app(app(*, x), y)) → APP(app(*, x), app(D, y))
APP(D, app(app(*, x), y)) → APP(D, y)
APP(D, app(app(-, x), y)) → APP(app(-, app(D, x)), app(D, y))
APP(D, app(app(-, x), y)) → APP(-, app(D, x))
APP(D, app(app(-, x), y)) → APP(D, x)
APP(D, app(app(-, x), y)) → APP(D, y)
APP(D, app(minus, x)) → APP(minus, app(D, x))
APP(D, app(minus, x)) → APP(D, x)
APP(D, app(app(div, x), y)) → APP(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
APP(D, app(app(div, x), y)) → APP(-, app(app(div, app(D, x)), y))
APP(D, app(app(div, x), y)) → APP(app(div, app(D, x)), y)
APP(D, app(app(div, x), y)) → APP(div, app(D, x))
APP(D, app(app(div, x), y)) → APP(D, x)
APP(D, app(app(div, x), y)) → APP(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2))
APP(D, app(app(div, x), y)) → APP(div, app(app(*, x), app(D, y)))
APP(D, app(app(div, x), y)) → APP(app(*, x), app(D, y))
APP(D, app(app(div, x), y)) → APP(*, x)
APP(D, app(app(div, x), y)) → APP(D, y)
APP(D, app(app(div, x), y)) → APP(app(pow, y), 2)
APP(D, app(app(div, x), y)) → APP(pow, y)
APP(D, app(ln, x)) → APP(app(div, app(D, x)), x)
APP(D, app(ln, x)) → APP(div, app(D, x))
APP(D, app(ln, x)) → APP(D, x)
APP(D, app(app(pow, x), y)) → APP(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
APP(D, app(app(pow, x), y)) → APP(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x)))
APP(D, app(app(pow, x), y)) → APP(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))
APP(D, app(app(pow, x), y)) → APP(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1))))
APP(D, app(app(pow, x), y)) → APP(app(*, y), app(app(pow, x), app(app(-, y), 1)))
APP(D, app(app(pow, x), y)) → APP(*, y)
APP(D, app(app(pow, x), y)) → APP(app(pow, x), app(app(-, y), 1))
APP(D, app(app(pow, x), y)) → APP(app(-, y), 1)
APP(D, app(app(pow, x), y)) → APP(-, y)
APP(D, app(app(pow, x), y)) → APP(D, x)
APP(D, app(app(pow, x), y)) → APP(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y))
APP(D, app(app(pow, x), y)) → APP(*, app(app(*, app(app(pow, x), y)), app(ln, x)))
APP(D, app(app(pow, x), y)) → APP(app(*, app(app(pow, x), y)), app(ln, x))
APP(D, app(app(pow, x), y)) → APP(*, app(app(pow, x), y))
APP(D, app(app(pow, x), y)) → APP(ln, x)
APP(D, app(app(pow, x), y)) → APP(D, 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(D, t) → 1
app(D, constant) → 0
app(D, app(app(+, x), y)) → app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) → app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) → app(app(-, app(D, x)), app(D, y))
app(D, app(minus, x)) → app(minus, app(D, x))
app(D, app(app(div, x), y)) → app(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
app(D, app(ln, x)) → app(app(div, app(D, x)), x)
app(D, app(app(pow, x), y)) → app(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
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 2 SCCs with 46 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(D, app(app(+, x), y)) → APP(D, y)
APP(D, app(app(+, x), y)) → APP(D, x)
APP(D, app(app(*, x), y)) → APP(D, x)
APP(D, app(app(*, x), y)) → APP(D, y)
APP(D, app(app(-, x), y)) → APP(D, x)
APP(D, app(app(-, x), y)) → APP(D, y)
APP(D, app(minus, x)) → APP(D, x)
APP(D, app(app(div, x), y)) → APP(D, x)
APP(D, app(app(div, x), y)) → APP(D, y)
APP(D, app(ln, x)) → APP(D, x)
APP(D, app(app(pow, x), y)) → APP(D, x)
APP(D, app(app(pow, x), y)) → APP(D, y)

The TRS R consists of the following rules:

app(D, t) → 1
app(D, constant) → 0
app(D, app(app(+, x), y)) → app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) → app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) → app(app(-, app(D, x)), app(D, y))
app(D, app(minus, x)) → app(minus, app(D, x))
app(D, app(app(div, x), y)) → app(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
app(D, app(ln, x)) → app(app(div, app(D, x)), x)
app(D, app(app(pow, x), y)) → app(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
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) 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:
D  =  D
+  =  +
*  =  *
-  =  -
minus  =  minus
div  =  div
ln  =  ln
pow  =  pow
app(x1, x2)  =  app(x1, x2)

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

  • APP(D, app(app(+, x), y)) → APP(D, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(+, x), y)) → APP(D, x) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(*, x), y)) → APP(D, x) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(*, x), y)) → APP(D, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(-, x), y)) → APP(D, x) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(-, x), y)) → APP(D, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

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

  • APP(D, app(app(div, x), y)) → APP(D, x) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(div, x), y)) → APP(D, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

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

  • APP(D, app(app(pow, x), y)) → APP(D, x) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(D, app(app(pow, x), y)) → APP(D, y) (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(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(D, t) → 1
app(D, constant) → 0
app(D, app(app(+, x), y)) → app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) → app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) → app(app(-, app(D, x)), app(D, y))
app(D, app(minus, x)) → app(minus, app(D, x))
app(D, app(app(div, x), y)) → app(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
app(D, app(ln, x)) → app(app(div, app(D, x)), x)
app(D, app(app(pow, x), y)) → app(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
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.

(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:
map  =  map
cons  =  cons
filter  =  filter
filter2  =  filter2
true  =  true
false  =  false
app(x1, x2)  =  app(x1, x2)

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

  • APP(app(filter, 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(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

  • APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 2 >= 2

  • APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 2 >= 2

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

(10) TRUE