Termination Proof using AProVE

Term Rewriting System R:
[x, y, f, g, h, t, l]
app(app(max, 0), x) -> x
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), x) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) -> app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, h), t)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) -> app(app(app(sort, min), max), l)
app(descending_sort, l) -> app(app(app(sort, max), min), l)

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(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(max, x)
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(min, app(s, x)), app(s, y)) -> APP(min, x)
APP(app(app(app(insert, f), g), nil), x) -> APP(app(cons, x), nil)
APP(app(app(app(insert, f), g), nil), x) -> APP(cons, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(cons, app(app(f, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(app(app(insert, f), g), t), app(app(g, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(app(insert, f), g), t)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(g, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(insert, f), g), app(app(app(sort, f), g), t))
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(insert, f), g)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(insert, f)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(sort, f), g), t)
APP(ascending_sort, l) -> APP(app(app(sort, min), max), l)
APP(ascending_sort, l) -> APP(app(sort, min), max)
APP(ascending_sort, l) -> APP(sort, min)
APP(descending_sort, l) -> APP(app(app(sort, max), min), l)
APP(descending_sort, l) -> APP(app(sort, max), min)
APP(descending_sort, l) -> APP(sort, max)

Furthermore, R contains three SCCs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Nar

Dependency Pair:

APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)

Rules:

app(app(max, 0), x) -> x
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), x) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) -> app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, h), t)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) -> app(app(app(sort, min), max), l)
app(descending_sort, l) -> app(app(app(sort, max), min), l)

Strategy:

innermost

As we are in the innermost case, we can delete all 12 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 4

                 ↳A-Transformation

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Nar

Dependency Pair:

APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)

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 5

                 ↳Size-Change Principle

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Nar

Dependency Pair:

MAX(s(x), s(y)) -> MAX(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

MAX(s(x), s(y)) -> MAX(x, y)

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

{1}	,	{1}
1	>	1
2	>	2

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

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳Usable Rules (Innermost)

           →DP Problem 3

             ↳Nar

Dependency Pair:

APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)

Rules:

app(app(max, 0), x) -> x
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), x) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) -> app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, h), t)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) -> app(app(app(sort, min), max), l)
app(descending_sort, l) -> app(app(app(sort, max), min), l)

Strategy:

innermost

As we are in the innermost case, we can delete all 12 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

...

               →DP Problem 6

                 ↳A-Transformation

           →DP Problem 3

             ↳Nar

Dependency Pair:

APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)

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 2

             ↳UsableRules

...

               →DP Problem 7

                 ↳Size-Change Principle

           →DP Problem 3

             ↳Nar

Dependency Pair:

MIN(s(x), s(y)) -> MIN(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

MIN(s(x), s(y)) -> MIN(x, y)

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

{1}	,	{1}
1	>	1
2	>	2

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

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Narrowing Transformation

Dependency Pairs:

APP(descending_sort, l) -> APP(app(app(sort, max), min), l)
APP(ascending_sort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(sort, f), g), t)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(g, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(app(app(insert, f), g), t), app(app(g, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(f, x), h)

Rules:

app(app(max, 0), x) -> x
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), x) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) -> app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, h), t)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) -> app(app(app(sort, min), max), l)
app(descending_sort, l) -> app(app(app(sort, max), min), l)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)

two new Dependency Pairs are created:

APP(app(app(sort, f''), g''), app(app(cons, h), nil)) -> APP(app(app(app(insert, f''), g''), nil), h)
APP(app(app(sort, f''), g''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(app(app(insert, f''), g''), app(app(app(app(insert, f''), g''), app(app(app(sort, f''), g''), t'')), h'')), h)

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Nar

...

               →DP Problem 8

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(ascending_sort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(g, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(app(app(insert, f), g), t), app(app(g, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(app(sort, f''), g''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(app(app(insert, f''), g''), app(app(app(app(insert, f''), g''), app(app(app(sort, f''), g''), t'')), h'')), h)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(sort, f), g), t)
APP(descending_sort, l) -> APP(app(app(sort, max), min), l)

Rules:

app(app(max, 0), x) -> x
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), x) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) -> app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, h), t)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) -> app(app(app(sort, min), max), l)
app(descending_sort, l) -> app(app(app(sort, max), min), l)

Strategy:

innermost

The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:01 minutes