Termination Proof using AProVE

Term Rewriting System R:
[x, y, n, m]
app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), 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'(app'(eq, app'(s, x)), app'(s, y)) -> APP'(app'(eq, x), y)
APP'(app'(eq, app'(s, x)), app'(s, y)) -> APP'(eq, x)
APP'(app'(le, app'(s, x)), app'(s, y)) -> APP'(app'(le, x), y)
APP'(app'(le, app'(s, x)), app'(s, y)) -> APP'(le, x)
APP'(app'(app, app'(app'(add, n), x)), y) -> APP'(app'(add, n), app'(app'(app, x), y))
APP'(app'(app, app'(app'(add, n), x)), y) -> APP'(app'(app, x), y)
APP'(app'(app, app'(app'(add, n), x)), y) -> APP'(app, x)
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) -> APP'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) -> APP'(if_min, app'(app'(le, n), m))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) -> APP'(app'(le, n), m)
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) -> APP'(le, n)
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> APP'(min, app'(app'(add, n), x))
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> APP'(app'(add, n), x)
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> APP'(min, app'(app'(add, m), x))
APP'(app'(rm, n), app'(app'(add, m), x)) -> APP'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
APP'(app'(rm, n), app'(app'(add, m), x)) -> APP'(app'(if_rm, app'(app'(eq, n), m)), n)
APP'(app'(rm, n), app'(app'(add, m), x)) -> APP'(if_rm, app'(app'(eq, n), m))
APP'(app'(rm, n), app'(app'(add, m), x)) -> APP'(app'(eq, n), m)
APP'(app'(rm, n), app'(app'(add, m), x)) -> APP'(eq, n)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> APP'(rm, n)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> APP'(app'(add, m), app'(app'(rm, n), x))
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> APP'(rm, n)
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x))
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x))))
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(app'(eq, n), app'(min, app'(app'(add, n), x)))
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(eq, n)
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(min, app'(app'(add, n), x))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(minsort, app'(app'(app, app'(app'(rm, n), x)), y))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(app'(app, app'(app'(rm, n), x)), y)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(app, app'(app'(rm, n), x))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(app'(rm, n), x)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(rm, n)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> APP'(app'(minsort, x), app'(app'(add, n), y))
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> APP'(minsort, x)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> APP'(app'(add, n), y)

Furthermore, R contains six SCCs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP'(app'(eq, app'(s, x)), app'(s, y)) -> APP'(app'(eq, x), y)

Rules:

app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 21 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 7

                 ↳A-Transformation

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP'(app'(eq, app'(s, x)), app'(s, y)) -> APP'(app'(eq, 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 8

                 ↳Size-Change Principle

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP'(app'(le, app'(s, x)), app'(s, y)) -> APP'(app'(le, x), y)

Rules:

app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 21 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

...

               →DP Problem 9

                 ↳A-Transformation

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP'(app'(le, app'(s, x)), app'(s, y)) -> APP'(app'(le, 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 10

                 ↳Size-Change Principle

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

             ↳Usable Rules (Innermost)

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP'(app'(app, app'(app'(add, n), x)), y) -> APP'(app'(app, x), y)

Rules:

app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 21 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 11

                 ↳A-Transformation

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP'(app'(app, app'(app'(add, n), x)), y) -> APP'(app'(app, 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 3

             ↳UsableRules

...

               →DP Problem 12

                 ↳Size-Change Principle

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pair:

APP(add(n, x), y) -> APP(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

APP(add(n, x), y) -> APP(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:

add(x₁, x₂) -> add(x₁, x₂)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳Usable Rules (Innermost)

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pairs:

APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> APP'(app'(rm, n), x)
APP'(app'(rm, n), app'(app'(add, m), x)) -> APP'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))

Rules:

app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 17 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 13

                 ↳A-Transformation

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pairs:

Rules:

app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(eq, 0), 0) -> true
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, 0), app'(s, x)) -> false

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 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 14

                 ↳Size-Change Principle

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

Dependency Pairs:

IF_RM(false, n, add(m, x)) -> RM(n, x)
IF_RM(true, n, add(m, x)) -> RM(n, x)
RM(n, add(m, x)) -> IF_RM(eq(n, m), n, add(m, x))

Rules:

eq(s(x), s(y)) -> eq(x, y)
eq(0, 0) -> true
eq(s(x), 0) -> false
eq(0, s(x)) -> false

Strategy:

innermost

We number the DPs as follows:

IF_RM(false, n, add(m, x)) -> RM(n, x)
IF_RM(true, n, add(m, x)) -> RM(n, x)
RM(n, add(m, x)) -> IF_RM(eq(n, m), n, add(m, x))

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

{1, 2}	,	{1, 2}
2	=	1
3	>	2

{3}	,	{3}
1	=	2
2	=	3

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

{1, 2}	,	{3}
2	=	2
3	>	3

{3}	,	{1, 2}
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:

add(x₁, x₂) -> add(x₁, x₂)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳Usable Rules (Innermost)

           →DP Problem 6

             ↳UsableRules

Dependency Pairs:

APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> APP'(min, app'(app'(add, m), x))
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> APP'(min, app'(app'(add, n), x))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) -> APP'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))

Rules:

app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 18 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

...

               →DP Problem 15

                 ↳A-Transformation

           →DP Problem 6

             ↳UsableRules

Dependency Pairs:

Rules:

app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, 0), y) -> true

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 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

...

               →DP Problem 16

                 ↳Size-Change Principle

           →DP Problem 6

             ↳UsableRules

Dependency Pairs:

IF_MIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IF_MIN(true, add(n, add(m, x))) -> MIN(add(n, x))
MIN(add(n, add(m, x))) -> IF_MIN(le(n, m), add(n, add(m, x)))

Rules:

le(s(x), s(y)) -> le(x, y)
le(s(x), 0) -> false
le(0, y) -> true

Strategy:

innermost

We number the DPs as follows:

IF_MIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IF_MIN(true, add(n, add(m, x))) -> MIN(add(n, x))
MIN(add(n, add(m, x))) -> IF_MIN(le(n, m), add(n, add(m, x)))

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

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

{3}	,	{3}
1	=	2

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

{1, 2}	,	{3}
2	>	2

{3}	,	{1, 2}
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:

add(x₁, x₂) -> add(x₁, x₂)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳Usable Rules (Innermost)

Dependency Pairs:

APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> APP'(app'(minsort, x), app'(app'(add, n), y))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> APP'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil)
APP'(app'(minsort, app'(app'(add, n), x)), y) -> APP'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)

Rules:

app'(app'(eq, 0), 0) -> true
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(le, 0), y) -> true
app'(app'(le, app'(s, x)), 0) -> false
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(app, nil), y) -> y
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) -> n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) -> nil
app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) -> nil
app'(app'(minsort, app'(app'(add, n), x)), y) -> app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) -> app'(app'(minsort, x), app'(app'(add, n), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 4 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

...

               →DP Problem 17

                 ↳A-Transformation

Dependency Pairs:

Rules:

app'(app'(rm, n), app'(app'(add, m), x)) -> app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app, nil), y) -> y
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) -> app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(eq, app'(s, x)), app'(s, y)) -> app'(app'(eq, x), y)
app'(app'(rm, n), nil) -> nil
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) -> app'(app'(rm, n), x)
app'(app'(eq, 0), 0) -> true
app'(app'(eq, app'(s, x)), 0) -> false
app'(app'(eq, 0), app'(s, x)) -> false
app'(app'(app, app'(app'(add, n), x)), y) -> app'(app'(add, n), app'(app'(app, x), y))
app'(app'(le, app'(s, x)), app'(s, y)) -> app'(app'(le, x), y)
app'(app'(le, app'(s, x)), 0) -> false
app'(min, app'(app'(add, n), app'(app'(add, m), x))) -> app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(le, 0), y) -> true
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, m), x))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) -> app'(min, app'(app'(add, n), x))
app'(min, app'(app'(add, n), nil)) -> n

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 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

...

               →DP Problem 18

                 ↳Negative Polynomial Order

Dependency Pairs:

IF_MINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
IF_MINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
MINSORT(add(n, x), y) -> IF_MINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
rm(n, nil) -> nil
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
eq(s(x), s(y)) -> eq(x, y)
eq(0, 0) -> true
eq(s(x), 0) -> false
eq(0, s(x)) -> false
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
le(s(x), s(y)) -> le(x, y)
le(s(x), 0) -> false
le(0, y) -> true
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
min(add(n, nil)) -> n
if_min(false, add(n, add(m, x))) -> min(add(m, x))
if_min(true, add(n, add(m, x))) -> min(add(n, x))

Strategy:

innermost

The following Dependency Pair can be strictly oriented using the given order.

IF_MINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

eq(0, 0) -> true
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
rm(n, nil) -> nil
eq(s(x), s(y)) -> eq(x, y)
eq(s(x), 0) -> false
eq(0, s(x)) -> false
if_rm(true, n, add(m, x)) -> rm(n, x)
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
le(s(x), 0) -> false
if_min(false, add(n, add(m, x))) -> min(add(m, x))

Used ordering:
Polynomial Order with Interpretation:

POL( IF_MINSORT(x₁, ..., x₃) ) = x₂ + x₃

POL( add(x₁, x₂) ) = x₁ + x₂ + 1

POL( MINSORT(x₁, x₂) ) = x₁ + x₂

POL( app(x₁, x₂) ) = x₁ + x₂

POL( rm(x₁, x₂) ) = x₂

POL( nil ) = 0

POL( eq(x₁, x₂) ) = 0

POL( true ) = 0

POL( if_rm(x₁, ..., x₃) ) = x₃

POL( false ) = 0

POL( le(x₁, x₂) ) = 0

POL( min(x₁) ) = x₁

POL( if_min(x₁, x₂) ) = x₁ + x₂

This results in one new DP problem.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

...

               →DP Problem 19

                 ↳Usable Rules (Innermost)

Dependency Pairs:

IF_MINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
MINSORT(add(n, x), y) -> IF_MINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
rm(n, nil) -> nil
if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
if_rm(true, n, add(m, x)) -> rm(n, x)
eq(s(x), s(y)) -> eq(x, y)
eq(0, 0) -> true
eq(s(x), 0) -> false
eq(0, s(x)) -> false
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
le(s(x), s(y)) -> le(x, y)
le(s(x), 0) -> false
le(0, y) -> true
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
min(add(n, nil)) -> n
if_min(false, add(n, add(m, x))) -> min(add(m, x))
if_min(true, add(n, add(m, x))) -> min(add(n, x))

Strategy:

innermost

As we are in the innermost case, we can delete all 6 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

           →DP Problem 6

             ↳UsableRules

...

               →DP Problem 20

                 ↳Size-Change Principle

Dependency Pairs:

IF_MINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
MINSORT(add(n, x), y) -> IF_MINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

eq(0, 0) -> true
eq(s(x), s(y)) -> eq(x, y)
eq(s(x), 0) -> false
eq(0, s(x)) -> false
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
if_min(true, add(n, add(m, x))) -> min(add(n, x))
if_min(false, add(n, add(m, x))) -> min(add(m, x))

Strategy:

innermost

We number the DPs as follows:

IF_MINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
MINSORT(add(n, x), y) -> IF_MINSORT(eq(n, min(add(n, x))), add(n, x), y)

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

{1}	,	{1}
2	>	1

{2}	,	{2}
1	=	2
2	=	3

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

{1}	,	{2}
2	>	2

{2}	,	{1}
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:

add(x₁, x₂) -> add(x₁, x₂)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:05 minutes