Termination Proof using AProVE

Term Rewriting System R:
[x, y, n, u, v, w, z]
minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

MINUS(s(x), s(y)) -> MINUS(x, y)
QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
QUOT(s(x), s(y)) -> MINUS(x, y)
APP(add(n, x), y) -> APP(x, y)
REVERSE(add(n, x)) -> APP(reverse(x), add(n, nil))
REVERSE(add(n, x)) -> REVERSE(x)
SHUFFLE(add(n, x)) -> SHUFFLE(reverse(x))
SHUFFLE(add(n, x)) -> REVERSE(x)
CONCAT(cons(u, v), y) -> CONCAT(v, y)
LESS_LEAVES(cons(u, v), cons(w, z)) -> LESS_LEAVES(concat(u, v), concat(w, z))
LESS_LEAVES(cons(u, v), cons(w, z)) -> CONCAT(u, v)
LESS_LEAVES(cons(u, v), cons(w, z)) -> CONCAT(w, z)

Furthermore, R contains seven SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

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

one new Dependency Pair is created:

MINUS(s(s(x'')), s(s(y''))) -> MINUS(s(x''), s(y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 8

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

MINUS(s(s(x'')), s(s(y''))) -> MINUS(s(x''), s(y''))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

MINUS(s(s(x'')), s(s(y''))) -> MINUS(s(x''), s(y''))

one new Dependency Pair is created:

MINUS(s(s(s(x''''))), s(s(s(y'''')))) -> MINUS(s(s(x'''')), s(s(y'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 8

             ↳FwdInst

...

               →DP Problem 9

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

MINUS(s(s(s(x''''))), s(s(s(y'''')))) -> MINUS(s(s(x'''')), s(s(y'''')))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MINUS(s(s(s(x''''))), s(s(s(y'''')))) -> MINUS(s(s(x'''')), s(s(y'''')))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MINUS(x₁, x₂)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 8

             ↳FwdInst

...

               →DP Problem 10

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 11

             ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 12

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(APP(x₁, x₂)) = 1 + x₁

POL(add(x₁, x₂)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 13

                 ↳Dependency Graph

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

CONCAT(cons(u, v), y) -> CONCAT(v, y)

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

CONCAT(cons(u, v), y) -> CONCAT(v, y)

one new Dependency Pair is created:

CONCAT(cons(u, cons(u'', v'')), y'') -> CONCAT(cons(u'', v''), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 14

             ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

CONCAT(cons(u, cons(u'', v'')), y'') -> CONCAT(cons(u'', v''), y'')

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

CONCAT(cons(u, cons(u'', v'')), y'') -> CONCAT(cons(u'', v''), y'')

one new Dependency Pair is created:

CONCAT(cons(u, cons(u''0, cons(u'''', v''''))), y'''') -> CONCAT(cons(u''0, cons(u'''', v'''')), y'''')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 14

             ↳FwdInst

...

               →DP Problem 15

                 ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

CONCAT(cons(u, cons(u''0, cons(u'''', v''''))), y'''') -> CONCAT(cons(u''0, cons(u'''', v'''')), y'''')

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

CONCAT(cons(u, cons(u''0, cons(u'''', v''''))), y'''') -> CONCAT(cons(u''0, cons(u'''', v'''')), y'''')

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons(x₁, x₂)) = 1 + x₂

POL(CONCAT(x₁, x₂)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 14

             ↳FwdInst

...

               →DP Problem 16

                 ↳Dependency Graph

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Narrowing Transformation

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

two new Dependency Pairs are created:

QUOT(s(x''), s(0)) -> QUOT(x'', s(0))
QUOT(s(s(x'')), s(s(y''))) -> QUOT(minus(x'', y''), s(s(y'')))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 17

             ↳Forward Instantiation Transformation

           →DP Problem 18

             ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

QUOT(s(x''), s(0)) -> QUOT(x'', s(0))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

QUOT(s(x''), s(0)) -> QUOT(x'', s(0))

one new Dependency Pair is created:

QUOT(s(s(x'''')), s(0)) -> QUOT(s(x''''), s(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 17

             ↳FwdInst

...

               →DP Problem 19

                 ↳Polynomial Ordering

           →DP Problem 18

             ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

QUOT(s(s(x'''')), s(0)) -> QUOT(s(x''''), s(0))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

QUOT(s(s(x'''')), s(0)) -> QUOT(s(x''''), s(0))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUOT(x₁, x₂)) = 1 + x₁

POL(0) = 0

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 17

             ↳FwdInst

...

               →DP Problem 22

                 ↳Dependency Graph

           →DP Problem 18

             ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 17

             ↳FwdInst

           →DP Problem 18

             ↳Narrowing Transformation

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

QUOT(s(s(x'')), s(s(y''))) -> QUOT(minus(x'', y''), s(s(y'')))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

QUOT(s(s(x'')), s(s(y''))) -> QUOT(minus(x'', y''), s(s(y'')))

two new Dependency Pairs are created:

QUOT(s(s(x''')), s(s(0))) -> QUOT(x''', s(s(0)))
QUOT(s(s(s(x'))), s(s(s(y')))) -> QUOT(minus(x', y'), s(s(s(y'))))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 17

             ↳FwdInst

           →DP Problem 18

             ↳Nar

...

               →DP Problem 20

                 ↳Polynomial Ordering

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

QUOT(s(s(x''')), s(s(0))) -> QUOT(x''', s(s(0)))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

QUOT(s(s(x''')), s(s(0))) -> QUOT(x''', s(s(0)))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUOT(x₁, x₂)) = 1 + x₁

POL(0) = 0

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 17

             ↳FwdInst

           →DP Problem 18

             ↳Nar

...

               →DP Problem 21

                 ↳Polynomial Ordering

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

QUOT(s(s(s(x'))), s(s(s(y')))) -> QUOT(minus(x', y'), s(s(s(y'))))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

QUOT(s(s(s(x'))), s(s(s(y')))) -> QUOT(minus(x', y'), s(s(s(y'))))

Additionally, the following usable rules for innermost can be oriented:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUOT(x₁, x₂)) = 1 + x₁

POL(0) = 1

POL(minus(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳Forward Instantiation Transformation

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

REVERSE(add(n, x)) -> REVERSE(x)

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

REVERSE(add(n, x)) -> REVERSE(x)

one new Dependency Pair is created:

REVERSE(add(n, add(n'', x''))) -> REVERSE(add(n'', x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

           →DP Problem 25

             ↳Forward Instantiation Transformation

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

REVERSE(add(n, add(n'', x''))) -> REVERSE(add(n'', x''))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

REVERSE(add(n, add(n'', x''))) -> REVERSE(add(n'', x''))

one new Dependency Pair is created:

REVERSE(add(n, add(n'''', add(n''''', x'''')))) -> REVERSE(add(n'''', add(n''''', x'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

           →DP Problem 25

             ↳FwdInst

...

               →DP Problem 26

                 ↳Polynomial Ordering

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

REVERSE(add(n, add(n'''', add(n''''', x'''')))) -> REVERSE(add(n'''', add(n''''', x'''')))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

REVERSE(add(n, add(n'''', add(n''''', x'''')))) -> REVERSE(add(n'''', add(n''''', x'''')))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(REVERSE(x₁)) = 1 + x₁

POL(add(x₁, x₂)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

           →DP Problem 25

             ↳FwdInst

...

               →DP Problem 27

                 ↳Dependency Graph

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Narrowing Transformation

       →DP Problem 7

         ↳Nar

Dependency Pair:

LESS_LEAVES(cons(u, v), cons(w, z)) -> LESS_LEAVES(concat(u, v), concat(w, z))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

LESS_LEAVES(cons(u, v), cons(w, z)) -> LESS_LEAVES(concat(u, v), concat(w, z))

four new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(w, z)) -> LESS_LEAVES(v', concat(w, z))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESS_LEAVES(cons(u'', concat(v'', v0)), concat(w, z))
LESS_LEAVES(cons(u, v), cons(leaf, z')) -> LESS_LEAVES(concat(u, v), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

           →DP Problem 28

             ↳Narrowing Transformation

       →DP Problem 7

         ↳Nar

Dependency Pairs:

LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(u, v), cons(leaf, z')) -> LESS_LEAVES(concat(u, v), z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESS_LEAVES(cons(u'', concat(v'', v0)), concat(w, z))
LESS_LEAVES(cons(leaf, v'), cons(w, z)) -> LESS_LEAVES(v', concat(w, z))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

LESS_LEAVES(cons(leaf, v'), cons(w, z)) -> LESS_LEAVES(v', concat(w, z))

two new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

           →DP Problem 28

             ↳Nar

...

               →DP Problem 29

                 ↳Narrowing Transformation

       →DP Problem 7

         ↳Nar

Dependency Pairs:

LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(u, v), cons(leaf, z')) -> LESS_LEAVES(concat(u, v), z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESS_LEAVES(cons(u'', concat(v'', v0)), concat(w, z))
LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

LESS_LEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESS_LEAVES(cons(u'', concat(v'', v0)), concat(w, z))

four new Dependency Pairs are created:

LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', v0'), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

           →DP Problem 28

             ↳Nar

...

               →DP Problem 30

                 ↳Narrowing Transformation

       →DP Problem 7

         ↳Nar

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', v0'), concat(w, z))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(u, v), cons(leaf, z')) -> LESS_LEAVES(concat(u, v), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

LESS_LEAVES(cons(u, v), cons(leaf, z')) -> LESS_LEAVES(concat(u, v), z')

two new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

           →DP Problem 28

             ↳Nar

...

               →DP Problem 31

                 ↳Narrowing Transformation

       →DP Problem 7

         ↳Nar

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', v0'), concat(w, z))
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))

four new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', z''))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

           →DP Problem 28

             ↳Nar

...

               →DP Problem 32

                 ↳Polynomial Ordering

       →DP Problem 7

         ↳Nar

Dependency Pairs:

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', z''))
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', v0'), concat(w, z))
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', z''))
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', v0'), concat(w, z))
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))

Additionally, the following usable rules for innermost can be oriented:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons(x₁, x₂)) = 1 + x₁ + x₂

POL(LESS_LEAVES(x₁, x₂)) = x₁

POL(leaf) = 0

POL(concat(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

           →DP Problem 28

             ↳Nar

...

               →DP Problem 33

                 ↳Dependency Graph

       →DP Problem 7

         ↳Nar

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Narrowing Transformation

Dependency Pair:

SHUFFLE(add(n, x)) -> SHUFFLE(reverse(x))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

SHUFFLE(add(n, x)) -> SHUFFLE(reverse(x))

two new Dependency Pairs are created:

SHUFFLE(add(n, nil)) -> SHUFFLE(nil)
SHUFFLE(add(n, add(n'', x''))) -> SHUFFLE(app(reverse(x''), add(n'', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 34

             ↳Narrowing Transformation

Dependency Pair:

SHUFFLE(add(n, add(n'', x''))) -> SHUFFLE(app(reverse(x''), add(n'', nil)))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

SHUFFLE(add(n, add(n'', x''))) -> SHUFFLE(app(reverse(x''), add(n'', nil)))

two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(app(nil, add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 34

             ↳Nar

...

               →DP Problem 35

                 ↳Rewriting Transformation

Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(app(nil, add(n'', nil)))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

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

SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(app(nil, add(n'', nil)))

one new Dependency Pair is created:

SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(add(n'', nil))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 34

             ↳Nar

...

               →DP Problem 36

                 ↳Polynomial Ordering

Dependency Pair:

SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))

Additionally, the following usable rules for innermost can be oriented:

app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(reverse(x₁)) = x₁

POL(SHUFFLE(x₁)) = 1 + x₁

POL(nil) = 0

POL(app(x₁, x₂)) = x₁ + x₂

POL(add(x₁, x₂)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳FwdInst

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 34

             ↳Nar

...

               →DP Problem 37

                 ↳Dependency Graph

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:09 minutes

POL(MINUS(x₁, x₂))	= 1 + x₁
POL(s(x₁))	= 1 + x₁

POL(APP(x₁, x₂))	= 1 + x₁
POL(add(x₁, x₂))	= 1 + x₂

POL(cons(x₁, x₂))	= 1 + x₂
POL(CONCAT(x₁, x₂))	= 1 + x₁

POL(QUOT(x₁, x₂))	= 1 + x₁
POL(0)	= 0
POL(s(x₁))	= 1 + x₁

POL(REVERSE(x₁))	= 1 + x₁
POL(add(x₁, x₂))	= 1 + x₂

POL(cons(x₁, x₂))	= 1 + x₁ + x₂
POL(LESS_LEAVES(x₁, x₂))	= x₁
POL(leaf)	= 0
POL(concat(x₁, x₂))	= x₁ + x₂

POL(reverse(x₁))	= x₁
POL(SHUFFLE(x₁))	= 1 + x₁
POL(nil)	= 0
POL(app(x₁, x₂))	= x₁ + x₂
POL(add(x₁, x₂))	= 1 + x₂