Term Rewriting System R:

   [x, y, 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)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0)))
   plus(plus(x, s(0)), plus(y, s(s(z)))) -> plus(plus(y, s(s(z))), plus(x, s(0)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)
   MINUS(s(x), s(y)) -> MINUS(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(plus(x, s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), plus(x, s(0)))
   PLUS(minus(x, s(0)), minus(y, s(s(z)))) -> PLUS(minus(y, s(s(z))), minus(x, s(0)))

Furthermore, R contains three SCCs.


SCC1:

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.


SCC2:

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


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.


SCC3:

PLUS(minus(x, s(0)), minus(y, s(s(z)))) -> PLUS(minus(y, s(s(z))), minus(x, s(0)))
PLUS(plus(x, s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), plus(x, s(0)))
PLUS(s(x), y) -> PLUS(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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(plus(x_1, x_2)) = x_1 + x_2
POL(minus(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

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


 This transformation is resulting in one new subcycle:


SCC3.MRR1:

PLUS(plus(x, s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), plus(x, s(0)))
PLUS(s(x), y) -> PLUS(x, y)
PLUS(minus(x, s(0)), minus(y, s(s(z)))) -> PLUS(minus(y, s(s(z))), minus(x, s(0)))

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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(plus(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(minus(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   plus(0, y) -> y


 This transformation is resulting in one new subcycle:


SCC3.MRR1.MRR1:

PLUS(minus(x, s(0)), minus(y, s(s(z)))) -> PLUS(minus(y, s(s(z))), minus(x, s(0)))
PLUS(s(x), y) -> PLUS(x, y)
PLUS(plus(x, s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), plus(x, s(0)))

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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1
POL(plus(x_1, x_2)) = x_1 + x_2
POL(minus(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0

The following Dependency Pairs can be deleted:

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

The following rules of R can be deleted: 

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


 This transformation is resulting in one new subcycle:


SCC3.MRR1.MRR1.MRR1:

PLUS(plus(x, s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), plus(x, s(0)))

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

   PLUS(plus(x, s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), plus(x, s(0)))
two new Dependency Pairs
are created:


   PLUS(plus(x, s(0)), plus(s(x''), s(s(z')))) -> PLUS(s(plus(x'', s(s(z')))), plus(x, s(0)))
   PLUS(plus(s(x''), s(0)), plus(y, s(s(z)))) -> PLUS(plus(y, s(s(z))), s(plus(x'', s(0))))

The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.20 seconds.