Term Rewriting System R:
[x, y]
p(s(x)) -> x
fact(0) -> s(0)
fact(s(x)) -> *(s(x), fact(p(s(x))))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FACT(s(x)) -> *'(s(x), fact(p(s(x))))
FACT(s(x)) -> FACT(p(s(x)))
FACT(s(x)) -> P(s(x))
*'(s(x), y) -> +'(*(x, y), y)
*'(s(x), y) -> *'(x, y)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Remaining


Dependency Pair:

+'(x, s(y)) -> +'(x, y)


Rules:


p(s(x)) -> x
fact(0) -> s(0)
fact(s(x)) -> *(s(x), fact(p(s(x))))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

+'(x, s(y)) -> +'(x, y)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Remaining


Dependency Pair:


Rules:


p(s(x)) -> x
fact(0) -> s(0)
fact(s(x)) -> *(s(x), fact(p(s(x))))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳Remaining


Dependency Pair:

*'(s(x), y) -> *'(x, y)


Rules:


p(s(x)) -> x
fact(0) -> s(0)
fact(s(x)) -> *(s(x), fact(p(s(x))))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

*'(s(x), y) -> *'(x, y)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
*'(x1, x2) -> *'(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 3
         ↳Remaining


Dependency Pair:


Rules:


p(s(x)) -> x
fact(0) -> s(0)
fact(s(x)) -> *(s(x), fact(p(s(x))))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

FACT(s(x)) -> FACT(p(s(x)))


Rules:


p(s(x)) -> x
fact(0) -> s(0)
fact(s(x)) -> *(s(x), fact(p(s(x))))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes