Term Rewriting System R:

   [x, y]
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   div(x, y) -> ify(ge(y, s(0)), x, y)
   ify(false, x, y) -> divByZeroError
   ify(true, x, y) -> if(ge(x, y), x, y)
   if(false, x, y) -> 0
   if(true, x, y) -> s(div(minus(x, y), y))

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: 


   IF(true, x, y) -> DIV(minus(x, y), y)
   IF(true, x, y) -> MINUS(x, y)
   DIV(x, y) -> IFY(ge(y, s(0)), x, y)
   DIV(x, y) -> GE(y, s(0))
   MINUS(s(x), s(y)) -> MINUS(x, y)
   GE(s(x), s(y)) -> GE(x, y)
   IFY(true, x, y) -> IF(ge(x, y), x, y)
   IFY(true, x, y) -> GE(x, y)

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:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

IFY(true, x, y) -> IF(ge(x, y), x, y)
DIV(x, y) -> IFY(ge(y, s(0)), x, y)
IF(true, x, y) -> DIV(minus(x, y), y)

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

   DIV(x, y) -> IFY(ge(y, s(0)), x, y)
two new Dependency Pairs
are created:


   DIV(x, s(x'')) -> IFY(ge(x'', 0), x, s(x''))
   DIV(x, 0) -> IFY(false, x, 0)

The transformation is resulting in one subcycle:


SCC3.Nar1:

DIV(x, s(x'')) -> IFY(ge(x'', 0), x, s(x''))
IF(true, x, y) -> DIV(minus(x, y), y)
IFY(true, x, y) -> IF(ge(x, y), x, y)

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

   DIV(x, s(x'')) -> IFY(ge(x'', 0), x, s(x''))
one new Dependency Pair
is created:


   DIV(x, s(x'')) -> IFY(true, x, s(x''))

The transformation is resulting in one subcycle:


SCC3.Nar1.Rewr1:

IFY(true, x, y) -> IF(ge(x, y), x, y)
DIV(x, s(x'')) -> IFY(true, x, s(x''))
IF(true, x, y) -> DIV(minus(x, y), y)

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

   IFY(true, x, y) -> IF(ge(x, y), x, y)
three new Dependency Pairs
are created:


   IFY(true, x'', 0) -> IF(true, x'', 0)
   IFY(true, s(x''), s(y'')) -> IF(ge(x'', y''), s(x''), s(y''))
   IFY(true, 0, s(x'')) -> IF(false, 0, s(x''))

The transformation is resulting in one subcycle:


SCC3.Nar1.Rewr1.Nar1:

IF(true, x, y) -> DIV(minus(x, y), y)
IFY(true, s(x''), s(y'')) -> IF(ge(x'', y''), s(x''), s(y''))
DIV(x, s(x'')) -> IFY(true, x, s(x''))

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

   IF(true, x, y) -> DIV(minus(x, y), y)
one new Dependency Pair
is created:


   IF(true, s(x'''''), s(y'''')) -> DIV(minus(s(x'''''), s(y'''')), s(y''''))

The transformation is resulting in one subcycle:


SCC3.Nar1.Rewr1.Nar1.Inst1:

DIV(x, s(x'')) -> IFY(true, x, s(x''))
IF(true, s(x'''''), s(y'''')) -> DIV(minus(s(x'''''), s(y'''')), s(y''''))
IFY(true, s(x''), s(y'')) -> IF(ge(x'', y''), s(x''), s(y''))

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

   IF(true, s(x'''''), s(y'''')) -> DIV(minus(s(x'''''), s(y'''')), s(y''''))
one new Dependency Pair
is created:


   IF(true, s(x'''''), s(y'''')) -> DIV(minus(x''''', y''''), s(y''''))

The transformation is resulting in one subcycle:


SCC3.Nar1.Rewr1.Nar1.Inst1.Rewr1:

IF(true, s(x'''''), s(y'''')) -> DIV(minus(x''''', y''''), s(y''''))
IFY(true, s(x''), s(y'')) -> IF(ge(x'', y''), s(x''), s(y''))
DIV(x, s(x'')) -> IFY(true, x, s(x''))

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(DIV(x_1, x_2)) = x_1
POL(minus(x_1, x_2)) = x_1
POL(ge(x_1, x_2)) = 0
POL(true) = 0
POL(IFY(x_1, x_2, x_3)) = x_2
POL(0) = 0
POL(false) = 0
POL(IF(x_1, x_2, x_3)) = x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 11.273 seconds.