Term Rewriting System R:
[x, y]
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(f, 0) -> app(s, 0)
app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x)))
app(g, 0) -> 0
app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x)))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(minus, app(s, x)), app(s, y)) -> APP(minus, x)
APP(f, 0) -> APP(s, 0)
APP(f, app(s, x)) -> APP(app(minus, app(s, x)), app(g, app(f, x)))
APP(f, app(s, x)) -> APP(minus, app(s, x))
APP(f, app(s, x)) -> APP(g, app(f, x))
APP(f, app(s, x)) -> APP(f, x)
APP(g, app(s, x)) -> APP(app(minus, app(s, x)), app(f, app(g, x)))
APP(g, app(s, x)) -> APP(minus, app(s, x))
APP(g, app(s, x)) -> APP(f, app(g, x))
APP(g, app(s, x)) -> APP(g, x)

Furthermore, R contains two SCCs.


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


Dependency Pair:

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


Rules:


app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(f, 0) -> app(s, 0)
app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x)))
app(g, 0) -> 0
app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x)))


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
ATrans


Dependency Pair:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, 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 3
ATrans
             ...
               →DP Problem 4
Size-Change Principle
       →DP Problem 2
ATrans


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. MINUS(s(x), s(y)) -> MINUS(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
A-Transformation


Dependency Pairs:

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


Rules:


app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(f, 0) -> app(s, 0)
app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x)))
app(g, 0) -> 0
app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x)))


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
ATrans
           →DP Problem 5
Negative Polynomial Order


Dependency Pairs:

G(s(x)) -> G(x)
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
F(s(x)) -> G(f(x))


Rules:


minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
f(0) -> s(0)
f(s(x)) -> minus(s(x), g(f(x)))
g(0) -> 0
g(s(x)) -> minus(s(x), f(g(x)))


Strategy:

innermost




The following Dependency Pairs can be strictly oriented using the given order.

G(s(x)) -> G(x)
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

f(0) -> s(0)
minus(s(x), s(y)) -> minus(x, y)
g(0) -> 0
f(s(x)) -> minus(s(x), g(f(x)))
minus(x, 0) -> x
g(s(x)) -> minus(s(x), f(g(x)))


Used ordering:
Polynomial Order with Interpretation:

POL( G(x1) ) = x1

POL( s(x1) ) = x1 + 1

POL( F(x1) ) = x1

POL( f(x1) ) = x1 + 1

POL( g(x1) ) = x1

POL( 0 ) = 0

POL( minus(x1, x2) ) = x1


This results in one new DP problem.


   R
DPs
       →DP Problem 1
UsableRules
       →DP Problem 2
ATrans
           →DP Problem 5
Neg POLO
             ...
               →DP Problem 6
Dependency Graph


Dependency Pair:

F(s(x)) -> G(f(x))


Rules:


minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
f(0) -> s(0)
f(s(x)) -> minus(s(x), g(f(x)))
g(0) -> 0
g(s(x)) -> minus(s(x), f(g(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes