Term Rewriting System R:
[y, z, x]
f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

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

f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

COPY(0, y, z) -> F(z)
COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))
COPY(s(x), y, z) -> F(y)

Furthermore, R contains one SCC. 


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


Dependency Pair:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))


Rules:


f(cons(nil, y)) -> y
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))





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
             ↳NOC
             ...
               →DP Problem 2
                 ↳Usable Rules (Innermost)


Dependency Pair:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))


Rules:


f(cons(nil, y)) -> y
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳NOC
             ...
               →DP Problem 3
                 ↳Size-Change Principle


Dependency Pair:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))


Rule:


f(cons(nil, y)) -> y


Strategy:

innermost




We number the DPs as follows:
  1. COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)

We obtain no new DP problems.

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