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
         ↳Narrowing Transformation


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




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

DIFF(X, Y) -> IF(leq(X, Y), n0, ns(ndiff(np(X), Y)))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pairs:

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




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
np(x1) -> np(x1)
ns(x1) -> ns(x1)
ndiff(x1, x2) -> ndiff(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


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




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes