Term Rewriting System R:
[l, h, t, f, l1, l2, l3]
app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(append, app(app(cons, h), t)), l) -> APP(app(cons, h), app(app(append, t), l))
APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)
APP(app(append, app(app(cons, h), t)), l) -> APP(append, t)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(cons, app(f, h))
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l1), app(app(append, l2), l3))
APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(append, app(app(append, l1), l2)), l3) -> APP(append, l2)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(append, app(app(map, f), l1)), app(app(map, f), l2))
APP(app(map, f), app(app(append, l1), l2)) -> APP(append, app(app(map, f), l1))
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l1)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳A-Transformation
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳ATrans
             ...
               →DP Problem 4
                 ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

APPEND(append(l1, l2), l3) -> APPEND(l2, l3)
APPEND(cons(h, t), l) -> APPEND(t, l)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APPEND(append(l1, l2), l3) -> APPEND(l2, l3)
  2. APPEND(cons(h, t), l) -> APPEND(t, l)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
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)
append(x1, x2) -> append(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)


Dependency Pairs:

APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 5
             ↳Size-Change Principle


Dependency Pairs:

APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
  2. APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
  3. APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
and get the following Size-Change Graph(s):
{1, 2, 3} , {1, 2, 3}
1=1
2>2
{1, 2, 3} , {1, 2, 3}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3} , {1, 2, 3}
1>1
2>2
{1, 2, 3} , {1, 2, 3}
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:
app(x1, x2) -> app(x1, x2)

We obtain no new DP problems.

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