Term Rewriting System R:
[x, y]
prime(0) -> false
prime(s(0)) -> false
prime(s(s(x))) -> prime1(s(s(x)), s(x))
prime1(x, 0) -> false
prime1(x, s(0)) -> true
prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
divp(x, y) -> =(rem(x, y), 0)

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

PRIME(s(s(x))) -> PRIME1(s(s(x)), s(x))
PRIME1(x, s(s(y))) -> DIVP(s(s(y)), x)
PRIME1(x, s(s(y))) -> PRIME1(x, s(y))

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pair:

PRIME1(x, s(s(y))) -> PRIME1(x, s(y))


Rules:


prime(0) -> false
prime(s(0)) -> false
prime(s(s(x))) -> prime1(s(s(x)), s(x))
prime1(x, 0) -> false
prime1(x, s(0)) -> true
prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
divp(x, y) -> =(rem(x, y), 0)


Strategy:

innermost




As we are in the innermost case, we can delete all 7 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pair:

PRIME1(x, s(s(y))) -> PRIME1(x, s(y))


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. PRIME1(x, s(s(y))) -> PRIME1(x, s(y))
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes