Term Rewriting System R:
[x, l, l1, l2]
isempty(nil) -> true
isempty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, isempty(l1))
ifappend(l1, l2, true) -> l2
ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APPEND(l1, l2) -> IFAPPEND(l1, l2, isempty(l1))
APPEND(l1, l2) -> ISEMPTY(l1)
IFAPPEND(l1, l2, false) -> HD(l1)
IFAPPEND(l1, l2, false) -> APPEND(tl(l1), l2)
IFAPPEND(l1, l2, false) -> TL(l1)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pairs:

IFAPPEND(l1, l2, false) -> APPEND(tl(l1), l2)
APPEND(l1, l2) -> IFAPPEND(l1, l2, isempty(l1))


Rules:


isempty(nil) -> true
isempty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, isempty(l1))
ifappend(l1, l2, true) -> l2
ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Narrowing Transformation


Dependency Pairs:

IFAPPEND(l1, l2, false) -> APPEND(tl(l1), l2)
APPEND(l1, l2) -> IFAPPEND(l1, l2, isempty(l1))


Rules:


tl(cons(x, l)) -> l
isempty(cons(x, l)) -> false
isempty(nil) -> true


Strategy:

innermost




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

APPEND(l1, l2) -> IFAPPEND(l1, l2, isempty(l1))
two new Dependency Pairs are created:

APPEND(cons(x', l'), l2) -> IFAPPEND(cons(x', l'), l2, false)
APPEND(nil, l2) -> IFAPPEND(nil, l2, true)

The transformation is resulting in one new DP problem: 



   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

APPEND(cons(x', l'), l2) -> IFAPPEND(cons(x', l'), l2, false)
IFAPPEND(l1, l2, false) -> APPEND(tl(l1), l2)


Rules:


tl(cons(x, l)) -> l
isempty(cons(x, l)) -> false
isempty(nil) -> true


Strategy:

innermost




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

IFAPPEND(l1, l2, false) -> APPEND(tl(l1), l2)
one new Dependency Pair is created:

IFAPPEND(cons(x', l'), l2, false) -> APPEND(l', l2)

The transformation is resulting in one new DP problem: 



   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 4
                 ↳Usable Rules (Innermost)


Dependency Pairs:

IFAPPEND(cons(x', l'), l2, false) -> APPEND(l', l2)
APPEND(cons(x', l'), l2) -> IFAPPEND(cons(x', l'), l2, false)


Rules:


tl(cons(x, l)) -> l
isempty(cons(x, l)) -> false
isempty(nil) -> true


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 5
                 ↳Size-Change Principle


Dependency Pairs:

IFAPPEND(cons(x', l'), l2, false) -> APPEND(l', l2)
APPEND(cons(x', l'), l2) -> IFAPPEND(cons(x', l'), l2, false)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. IFAPPEND(cons(x', l'), l2, false) -> APPEND(l', l2)
  2. APPEND(cons(x', l'), l2) -> IFAPPEND(cons(x', l'), l2, false)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2
{2} , {2}
1=1
2=2

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

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:06 minutes