Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   PLUS(s(x), y) -> PLUS(x, s(y))
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(s(plus(x, y)), z) -> PLUS(plus(x, y), z)
   PLUS(s(x), y) -> PLUS(minus(x, y), double(y))
   PLUS(s(x), y) -> MINUS(x, y)
   PLUS(s(x), y) -> DOUBLE(y)
   MINUS(s(x), s(y)) -> MINUS(x, y)
   DOUBLE(s(x)) -> DOUBLE(x)

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:

DOUBLE(s(x)) -> DOUBLE(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(s(x_1)) = 1 + x_1
POL(DOUBLE(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   DOUBLE(s(x)) -> DOUBLE(x)



 This transformation is resulting in no new subcycles.


SCC3:

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

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

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


   PLUS(s(plus(s(x''), y'')), z) -> PLUS(plus(x'', s(y'')), z)
   PLUS(s(plus(s(x''), y'')), z) -> PLUS(s(plus(x'', y'')), z)
   PLUS(s(plus(s(plus(x'', y'')), y0)), z) -> PLUS(s(plus(plus(x'', y''), y0)), z)
   PLUS(s(plus(0, y'')), z) -> PLUS(y'', z)
   PLUS(s(plus(s(x''), y'')), z) -> PLUS(s(plus(minus(x'', y''), double(y''))), z)

The transformation is resulting in one subcycle:


SCC3.Nar1:

PLUS(s(plus(s(x''), y'')), z) -> PLUS(s(plus(minus(x'', y''), double(y''))), z)
PLUS(s(plus(0, y'')), z) -> PLUS(y'', z)
PLUS(s(plus(s(plus(x'', y'')), y0)), z) -> PLUS(s(plus(plus(x'', y''), y0)), z)
PLUS(s(plus(s(x''), y'')), z) -> PLUS(s(plus(x'', y'')), z)
PLUS(s(plus(s(x''), y'')), z) -> PLUS(plus(x'', s(y'')), z)
PLUS(s(x), y) -> PLUS(x, y)
PLUS(s(x), y) -> PLUS(x, s(y))
PLUS(s(x), y) -> PLUS(minus(x, y), double(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: 

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


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

 resulting in one subcycle.


SCC3.Nar1.Polo1:

PLUS(s(plus(s(plus(x'', y'')), y0)), z) -> PLUS(s(plus(plus(x'', y''), y0)), z)
PLUS(s(plus(s(x''), y'')), z) -> PLUS(s(plus(x'', y'')), z)
PLUS(s(plus(s(x''), y'')), z) -> PLUS(plus(x'', s(y'')), z)
PLUS(s(x), y) -> PLUS(minus(x, y), double(y))
PLUS(s(x), y) -> PLUS(x, y)
PLUS(s(x), y) -> PLUS(x, s(y))
PLUS(s(plus(s(x''), y'')), z) -> PLUS(s(plus(minus(x'', y''), double(y''))), z)

At least one Dependency Pair of this SCC can be strictly oriented.


The following rules can be oriented: 

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
PLUS > double > s
plus > double > s

 resulting in no new SCCs.

Used Argument Filtering System: ([PLUS(x_1, x_2) -> PLUS(x_1, x_2), s(x_1) -> s(x_1), plus(x_1, x_2) -> plus(x_1, x_2), minus(x_1, x_2) -> x_1, double(x_1) -> double(x_1)])

Termination of R successfully shown.


Duration: 23.226 seconds.