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
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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(X, Y) -> LEQ(X, 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)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes