Term Rewriting System R:
[y, x, f, g]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(times, app(s, x)), y) -> APP(plus, app(app(times, x), y))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(times, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(twice, f) -> APP(app(comp, f), f)
APP(twice, f) -> APP(comp, f)

Furthermore, R contains three SCCs.


   R
DPs
       →DP Problem 1
Usable Rules (Innermost)
       →DP Problem 2
UsableRules
       →DP Problem 3
SCP


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


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 4
A-Transformation
       →DP Problem 2
UsableRules
       →DP Problem 3
SCP


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


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 4
ATrans
             ...
               →DP Problem 5
Size-Change Principle
       →DP Problem 2
UsableRules
       →DP Problem 3
SCP


Dependency Pair:

PLUS(s(x), y) -> PLUS(x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. PLUS(s(x), y) -> PLUS(x, y)
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:
s(x1) -> s(x1)

We obtain no new DP problems.


   R
DPs
       →DP Problem 1
UsableRules
       →DP Problem 2
Usable Rules (Innermost)
       →DP Problem 3
SCP


Dependency Pair:

APP(app(times, app(s, x)), y) -> APP(app(times, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


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 6
A-Transformation
       →DP Problem 3
SCP


Dependency Pair:

APP(app(times, app(s, x)), y) -> APP(app(times, x), y)


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 2
UsableRules
           →DP Problem 6
ATrans
             ...
               →DP Problem 7
Size-Change Principle
       →DP Problem 3
SCP


Dependency Pair:

TIMES(s(x), y) -> TIMES(x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. TIMES(s(x), y) -> TIMES(x, y)
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:
s(x1) -> s(x1)

We obtain no new DP problems.


   R
DPs
       →DP Problem 1
UsableRules
       →DP Problem 2
UsableRules
       →DP Problem 3
Size-Change Principle


Dependency Pairs:

APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(app(comp, f), g), x) -> APP(g, x)
  2. APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2=2
{2, 1} , {2, 1}
1>1

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

We obtain no new DP problems.

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