Term Rewriting System R:

   [x, y]
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
   if_quot(true, x, y) -> s(quot(minus(x, y), y))
   if_quot(false, x, y) -> 0

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: 


   LE(s(x), s(y)) -> LE(x, y)
   QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))
   QUOT(x, s(y)) -> LE(s(y), x)
   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
   IF_QUOT(true, x, y) -> MINUS(x, y)
   MINUS(s(x), s(y)) -> MINUS(x, y)

Furthermore, R contains three SCCs.


SCC1:

LE(s(x), s(y)) -> LE(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(LE(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   LE(s(x), s(y)) -> LE(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:

IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))

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

   QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))
two new Dependency Pairs
are created:


   QUOT(s(y''), s(y0)) -> IF_QUOT(le(y0, y''), s(y''), s(y0))
   QUOT(0, s(y')) -> IF_QUOT(false, 0, s(y'))

The transformation is resulting in one subcycle:


SCC3.Nar1:

QUOT(s(y''), s(y0)) -> IF_QUOT(le(y0, y''), s(y''), s(y0))
IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)

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

   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
two new Dependency Pairs
are created:


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

The transformation is resulting in one subcycle:


SCC3.Nar1.Nar1:

IF_QUOT(true, s(x''), s(y'')) -> QUOT(minus(x'', y''), s(y''))
QUOT(s(y''), s(y0)) -> IF_QUOT(le(y0, y''), s(y''), s(y0))

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(IF_QUOT(x_1, x_2, x_3)) = x_2
POL(s(x_1)) = 1 + x_1
POL(le(x_1, x_2)) = 0
POL(minus(x_1, x_2)) = x_1
POL(true) = 0
POL(QUOT(x_1, x_2)) = x_1
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 10.851 seconds.