Termination Proof using AProVE

Term Rewriting System R:
[x, y]
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)))

Termination of R to be shown.

     ↳Overlay and local confluence Check

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

     ↳OC

       →TRS2

         ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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)

Furthermore, R contains one SCC.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

Dependency Pairs:

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

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)))

Strategy:

innermost

As we are in the innermost case, we can delete all 9 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 2

                 ↳A-Transformation

Dependency Pairs:

Rule:

none

Strategy:

innermost

We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 3

                 ↳Size-Change Principle

Dependency Pairs:

D'(pow(x, y)) -> D'(y)
D'(pow(x, y)) -> D'(x)
D'(ln(x)) -> D'(x)
D'(div(x, y)) -> D'(y)
D'(div(x, y)) -> D'(x)
D'(minus(x)) -> D'(x)
D'(-(x, y)) -> D'(y)
D'(-(x, y)) -> D'(x)
D'(*(x, y)) -> D'(y)
D'(*(x, y)) -> D'(x)
D'(+(x, y)) -> D'(y)
D'(+(x, y)) -> D'(x)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

D'(pow(x, y)) -> D'(y)
D'(pow(x, y)) -> D'(x)
D'(ln(x)) -> D'(x)
D'(div(x, y)) -> D'(y)
D'(div(x, y)) -> D'(x)
D'(minus(x)) -> D'(x)
D'(-(x, y)) -> D'(y)
D'(-(x, y)) -> D'(x)
D'(*(x, y)) -> D'(y)
D'(*(x, y)) -> D'(x)
D'(+(x, y)) -> D'(y)
D'(+(x, y)) -> D'(x)

and get the following Size-Change Graph(s):

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}	,	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}	,	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
1	>	1

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

pow(x₁, x₂) -> pow(x₁, x₂)
*(x₁, x₂) -> *(x₁, x₂)
minus(x₁) -> minus(x₁)
-(x₁, x₂) -> -(x₁, x₂)
+(x₁, x₂) -> +(x₁, x₂)
div(x₁, x₂) -> div(x₁, x₂)
ln(x₁) -> ln(x₁)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:03 minutes