Term Rewriting System R:

   [x, y, z]
   p(s(x)) -> x
   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   div(div(x, y), z) -> div(x, times(y, z))
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   eq(0, 0) -> true
   eq(s(x), 0) -> false
   eq(0, s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0)) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   PR(x, s(s(y))) -> IF(divides(s(s(y)), x), x, s(y))
   PR(x, s(s(y))) -> DIVIDES(s(s(y)), x)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(x, s(y)) -> PLUS(x, p(s(y)))
   PLUS(x, s(y)) -> P(s(y))
   PLUS(s(x), y) -> PLUS(p(s(x)), y)
   PLUS(s(x), y) -> P(s(x))
   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
   QUOT(x, 0, s(z)) -> DIV(x, s(z))
   EQ(s(x), s(y)) -> EQ(x, y)
   TIMES(s(x), y) -> PLUS(y, times(x, y))
   TIMES(s(x), y) -> TIMES(x, y)
   DIV(x, y) -> QUOT(x, y, y)
   DIV(div(x, y), z) -> DIV(x, times(y, z))
   DIV(div(x, y), z) -> TIMES(y, z)
   PRIME(s(s(x))) -> PR(s(s(x)), s(x))
   IF(false, x, y) -> PR(x, y)
   DIVIDES(y, x) -> EQ(x, times(div(x, y), y))
   DIVIDES(y, x) -> TIMES(div(x, y), y)
   DIVIDES(y, x) -> DIV(x, y)

Furthermore, R contains five SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

PLUS(s(x), y) -> PLUS(p(s(x)), y)
PLUS(x, s(y)) -> PLUS(x, p(s(y)))
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(p(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   pr(x, s(0)) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   plus(s(x), y) -> s(plus(x, y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, 0) -> x
   plus(0, y) -> y
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   eq(0, 0) -> true
   eq(s(x), s(y)) -> eq(x, y)
   eq(s(x), 0) -> false
   eq(0, s(y)) -> false
   times(s(x), y) -> plus(y, times(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   div(div(x, y), z) -> div(x, times(y, z))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

PLUS(x, s(y)) -> PLUS(x, p(s(y)))
PLUS(s(x), y) -> PLUS(x, y)
PLUS(s(x), y) -> PLUS(p(s(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)) = 1 + x_1
POL(p(x_1)) = x_1

The following Dependency Pairs can be deleted:

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

The following rules of R can be deleted: 

   p(s(x)) -> x


 This transformation is resulting in one new subcycle:


SCC2.MRR1.MRR1:

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

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC2.MRR1.MRR1.NOC1:

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

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

   PLUS(s(x), y) -> PLUS(p(s(x)), y)
one new Dependency Pair
is created:


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

The transformation is resulting in no subcycles.


SCC3:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC4:

DIV(div(x, y), z) -> DIV(x, times(y, z))
QUOT(x, 0, s(z)) -> DIV(x, s(z))
QUOT(s(x), s(y), z) -> QUOT(x, y, z)
DIV(x, y) -> QUOT(x, y, y)

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = x_1
POL(plus(x_1, x_2)) = 0
POL(DIV(x_1, x_2)) = x_1
POL(div(x_1, x_2)) = 1 + x_1
POL(QUOT(x_1, x_2, x_3)) = x_1
POL(times(x_1, x_2)) = 0
POL(0) = 0
POL(p(x_1)) = 0

 resulting in one subcycle.


SCC4.Polo1:

QUOT(s(x), s(y), z) -> QUOT(x, y, z)
DIV(x, y) -> QUOT(x, y, y)
QUOT(x, 0, s(z)) -> DIV(x, s(z))

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(DIV(x_1, x_2)) = x_1
POL(QUOT(x_1, x_2, x_3)) = x_1
POL(0) = 0

 resulting in one subcycle.


SCC4.Polo1.Polo1:

QUOT(x, 0, s(z)) -> DIV(x, s(z))
DIV(x, y) -> QUOT(x, y, y)

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

   DIV(x, y) -> QUOT(x, y, y)
one new Dependency Pair
is created:


   DIV(x'', s(z'')) -> QUOT(x'', s(z''), s(z''))

The transformation is resulting in no subcycles.


SCC5:

IF(false, x, y) -> PR(x, y)
PR(x, s(s(y))) -> IF(divides(s(s(y)), x), x, s(y))

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(plus(x_1, x_2)) = 0
POL(eq(x_1, x_2)) = 0
POL(0) = 0
POL(p(x_1)) = 0
POL(IF(x_1, x_2, x_3)) = 1 + x_3
POL(divides(x_1, x_2)) = 0
POL(quot(x_1, x_2, x_3)) = 0
POL(div(x_1, x_2)) = 0
POL(true) = 0
POL(times(x_1, x_2)) = 0
POL(PR(x_1, x_2)) = x_2
POL(false) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.717 seconds.