Term Rewriting System R:
[X, Y, X1, X2]
fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

add(s(X), Y) -> s(add(X, Y))
prod(s(X), Y) -> add(Y, prod(X, Y))
zero(s(X)) -> false
p(s(X)) -> X


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FACT(X) -> IF(zero(X), ns(0), nprod(X, fact(p(X))))
FACT(X) -> ZERO(X)
FACT(X) -> FACT(p(X))
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(ns(X)) -> S(X)
ACTIVATE(nprod(X1, X2)) -> PROD(X1, X2)

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Modular Removal of Rules


Dependency Pair:

FACT(X) -> FACT(p(X))


Rules:


fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
add(0, X) -> X
prod(0, X) -> 0
prod(X1, X2) -> nprod(X1, X2)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X
s(X) -> ns(X)





We have the following set of usable rules: none
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(FACT(x1))=  x1  
  POL(p(x1))=  x1  

We have the following set D of usable symbols: {FACT, p}
No Dependency Pairs can be deleted.
11 non usable rules have been deleted.

The result of this processor delivers one new DP problem.



   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳MRR
             ...
               →DP Problem 2
                 ↳Non-Overlappingness Check


Dependency Pair:

FACT(X) -> FACT(p(X))


Rule:

none





R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle: 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳MRR
             ...
               →DP Problem 3
                 ↳Non Termination


Dependency Pair:

FACT(X) -> FACT(p(X))


Rule:

none


Strategy:

innermost




Found an infinite P-chain over R:
P =

FACT(X) -> FACT(p(X))

R = none

s = FACT(X)
evaluates to t =FACT(p(X))

Thus, s starts an infinite chain as s matches t.

Innermost Non-Termination of R could be shown.
Duration:
0:02 minutes