Term Rewriting System R:
[p, x, xs]
app(app(and, true), true) -> true
app(app(and, true), false) -> false
app(app(and, false), true) -> false
app(app(and, false), false) -> false
app(app(or, true), true) -> true
app(app(or, true), false) -> true
app(app(or, false), true) -> true
app(app(or, false), false) -> false
app(app(forall, p), nil) -> true
app(app(forall, p), app(app(cons, x), xs)) -> app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) -> false
app(app(forsome, p), app(app(cons, x), xs)) -> app(app(or, app(p, x)), app(app(forsome, p), xs))

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:

APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(and, app(p, x)), app(app(forall, p), xs))
APP(app(forall, p), app(app(cons, x), xs)) -> APP(and, app(p, x))
APP(app(forall, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(forall, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(or, app(p, x)), app(app(forsome, p), xs))
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(or, app(p, x))
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(forsome, p), xs)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(forsome, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(forall, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(p, x)


Rules:


app(app(and, true), true) -> true
app(app(and, true), false) -> false
app(app(and, false), true) -> false
app(app(and, false), false) -> false
app(app(or, true), true) -> true
app(app(or, true), false) -> true
app(app(or, false), true) -> true
app(app(or, false), false) -> false
app(app(forall, p), nil) -> true
app(app(forall, p), app(app(cons, x), xs)) -> app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) -> false
app(app(forsome, p), app(app(cons, x), xs)) -> app(app(or, app(p, x)), app(app(forsome, p), xs))


Strategy:

innermost




As we are in the innermost case, we can delete all 12 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pairs:

APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(forsome, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(forall, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(p, x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(forsome, p), xs)
  2. APP(app(forsome, p), app(app(cons, x), xs)) -> APP(p, x)
  3. APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(forall, p), xs)
  4. APP(app(forall, p), app(app(cons, x), xs)) -> APP(p, x)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2
{2} , {2}
1>1
2>2
{3} , {3}
1=1
2>2
{4} , {4}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{2} , {2}
1>1
2>2
{3} , {3}
1=1
2>2
{4} , {4}
1>1
2>2
{1} , {1}
1=1
2>2
{3} , {4}
1>1
2>2
{4} , {3}
1>1
2>2
{2} , {4}
1>1
2>2
{4} , {2}
1>1
2>2
{1} , {2}
1>1
2>2
{2} , {1}
1>1
2>2
{1} , {4}
1>1
2>2
{3} , {3}
1>1
2>2
{3} , {2}
1>1
2>2
{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:
app(x1, x2) -> app(x1, x2)

We obtain no new DP problems.

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