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))

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

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

MINUS(x₁, x₂) -> MINUS(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 8

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

APP(x₁, x₂) -> APP(x₁, x₂)
add(x₁, x₂) -> add(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 9

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

CONCAT(x₁, x₂) -> CONCAT(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 10

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →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))

The following dependency pair can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

QUOT(x₁, x₂) -> QUOT(x₁, x₂)
s(x₁) -> s(x₁)
minus(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 11

             ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →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))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →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))

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

REVERSE(x₁) -> REVERSE(x₁)
add(x₁, x₂) -> add(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 12

             ↳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))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 13

             ↳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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 13

             ↳Nar

...

               →DP Problem 14

                 ↳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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 13

             ↳Nar

...

               →DP Problem 15

                 ↳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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 13

             ↳Nar

...

               →DP Problem 16

                 ↳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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 13

             ↳Nar

...

               →DP Problem 17

                 ↳Argument Filtering and 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))

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')))

The following usable rules using the Ce-refinement can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{concat, cons}

resulting in one new DP problem.
Used Argument Filtering System:

LESS_LEAVES(x₁, x₂) -> LESS_LEAVES(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
concat(x₁, x₂) -> concat(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 13

             ↳Nar

...

               →DP Problem 18

                 ↳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))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳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))

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing 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))

On this DP problem, a Narrowing 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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

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))

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

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

two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', add(n''', nil)))) -> SHUFFLE(app(app(nil, add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 22

                 ↳Narrowing Transformation

Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', nil)))) -> SHUFFLE(app(app(nil, 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))

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 23

                 ↳Narrowing Transformation

Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n'''', nil)))) -> SHUFFLE(app(add(n'''', nil), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), 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))

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

SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 24

                 ↳Narrowing Transformation

Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n'''', nil)))) -> SHUFFLE(app(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))

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 25

                 ↳Narrowing Transformation

Dependency Pairs:

SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', app(nil, add(n''', nil))))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), 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))

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

SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))

one new Dependency Pair is created:

SHUFFLE(add(n, add(n'', add(n''', add(n'0', nil))))) -> SHUFFLE(app(app(add(n'0', nil), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 26

                 ↳Narrowing Transformation

Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0', nil))))) -> SHUFFLE(app(app(add(n'0', nil), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', 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))

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

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, nil)))))) -> SHUFFLE(app(app(app(app(nil, add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, add(n'2, x''))))))) -> SHUFFLE(app(app(app(app(app(reverse(x''), add(n'2, nil)), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

       →DP Problem 7

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 27

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, add(n'2, x''))))))) -> SHUFFLE(app(app(app(app(app(reverse(x''), add(n'2, nil)), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, nil)))))) -> SHUFFLE(app(app(app(app(nil, add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', app(nil, add(n''', nil))))
SHUFFLE(add(n, add(n'', add(n''', add(n'0', nil))))) -> SHUFFLE(app(app(add(n'0', nil), 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))

Termination of R could not be shown.
Duration:
0:35 minutes