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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

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


Rules:


app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, 0), x) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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

none


It is proper and hence, it can be A-transformed which results in the DP problem

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

none


We number the DPs as follows:
  1. PLUS(s(x), y) -> PLUS(x, y)
  2. PLUS(x, s(y)) -> PLUS(x, y)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2
{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
{1, 2} , {1, 2}
1>1
2=2
{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:
s(x1) -> s(x1)

We obtain no new DP problems. 


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


Dependency Pair:

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


Rules:


app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, 0), x) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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

none


It is proper and hence, it can be A-transformed which results in the DP problem

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

none


We number the DPs as follows:
  1. TIMES(x, s(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.

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