Term Rewriting System R:
[X, Y, X1, X2, Z]
and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(nadd(activate(X), activate(Y)))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nadd(X1, X2)) -> add(activate(X1), X2)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(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

add(s(X), Y) -> s(nadd(activate(X), activate(Y)))
first(s(X), cons(Y, Z)) -> cons(activate(Y), nfirst(activate(X), activate(Z)))


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AND(true, X) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(ns(X)) -> S(X)
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ADD(0, X) -> ACTIVATE(X)
FROM(X) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

FROM(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), X2)


Rules:


and(true, X) -> activate(X)
and(false, Y) -> false
activate(nadd(X1, X2)) -> add(activate(X1), X2)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(X)
activate(X) -> X
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)





We number the DPs as follows:
  1. FROM(X) -> ACTIVATE(X)
  2. ACTIVATE(nfrom(X)) -> FROM(X)
  3. ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
  4. ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
  5. ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
  6. ADD(0, X) -> ACTIVATE(X)
  7. ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), X2)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
{2} , {2}
1>1
{5, 4, 3} , {5, 4, 3}
1>1
{6} , {6}
2=1
{7} , {7}
1>2

which lead(s) to this/these maximal multigraph(s):
{5, 4, 3} , {5, 4, 3}
1>1
{7} , {6}
1>1
{1} , {2}
1>1
{2} , {1}
1>1
{6} , {7}
2>2
{5, 4, 3} , {6}
1>1
{5, 4, 3} , {1}
1>1
{2} , {5, 4, 3}
1>1
{7} , {5, 4, 3}
1>1
{7} , {1}
1>1
{2} , {6}
1>1

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
nfrom(x1) -> nfrom(x1)
nadd(x1, x2) -> nadd(x1, x2)
nfirst(x1, x2) -> nfirst(x1, x2)

We obtain no new DP problems.

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