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.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   APP(add(n, x), y) -> APP(x, y)
   MINUS(s(x), s(y)) -> MINUS(x, 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)
   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)
   CONCAT(cons(u, v), y) -> CONCAT(v, 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)

Furthermore, R contains seven SCCs.


SCC1:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(APP(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(MINUS(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(CONCAT(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC4:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(REVERSE(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC5:

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

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

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


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(minus(x_1, x_2)) = x_1
POL(QUOT(x_1, x_2)) = x_1 + x_2
POL(0) = 1

 resulting in no subcycles.


SCC6:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(concat(x_1, x_2)) = x_1 + x_2
POL(LESS_LEAVES(x_1, x_2)) = 1 + x_1 + x_2
POL(leaf) = 1
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   concat(leaf, y) -> y


 This transformation is resulting in one new subcycle:


SCC6.MRR1:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(concat(x_1, x_2)) = x_1 + x_2
POL(LESS_LEAVES(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC7:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 0
POL(reverse(x_1)) = x_1
POL(SHUFFLE(x_1)) = 1 + x_1
POL(app(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.60 seconds.