Term Rewriting System R:
[X, Y, X1, X2]
p(0) -> 0
p(s(X)) -> X
p(X) -> np(X)
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
diff(X, Y) -> if(leq(X, Y), n0, ns(ndiff(np(X), Y)))
diff(X1, X2) -> ndiff(X1, X2)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(activate(X))
activate(ndiff(X1, X2)) -> diff(activate(X1), activate(X2))
activate(np(X)) -> p(activate(X))
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

p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n0) -> 0'
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(ndiff(X1, X2)) -> DIFF(activate(X1), activate(X2))
ACTIVATE(ndiff(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ndiff(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(np(X)) -> P(activate(X))
ACTIVATE(np(X)) -> ACTIVATE(X)
DIFF(X, Y) -> IF(leq(X, Y), n0, ns(ndiff(np(X), Y)))

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


Dependency Pairs:

ACTIVATE(np(X)) -> ACTIVATE(X)
ACTIVATE(ndiff(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(ndiff(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ns(X)) -> ACTIVATE(X)


Rules:


p(X) -> np(X)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
activate(n0) -> 0
activate(ns(X)) -> s(activate(X))
activate(ndiff(X1, X2)) -> diff(activate(X1), activate(X2))
activate(np(X)) -> p(activate(X))
activate(X) -> X
diff(X, Y) -> if(leq(X, Y), n0, ns(ndiff(np(X), Y)))
diff(X1, X2) -> ndiff(X1, X2)
0 -> n0
s(X) -> ns(X)





We number the DPs as follows:
  1. ACTIVATE(np(X)) -> ACTIVATE(X)
  2. ACTIVATE(ndiff(X1, X2)) -> ACTIVATE(X2)
  3. ACTIVATE(ndiff(X1, X2)) -> ACTIVATE(X1)
  4. ACTIVATE(ns(X)) -> ACTIVATE(X)
and get the following Size-Change Graph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
ndiff(x1, x2) -> ndiff(x1, x2)
ns(x1) -> ns(x1)
np(x1) -> np(x1)

We obtain no new DP problems.

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