Termination Proof using AProVE

Term Rewriting System R:
[x, y]
fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FAC(0) -> 1'
FAC(s(x)) -> *'(s(x), fac(x))
FAC(s(x)) -> FAC(x)
FLOOP(s(x), y) -> FLOOP(x, *(s(x), y))
FLOOP(s(x), y) -> *'(s(x), y)
*'(x, s(y)) -> +'(*(x, y), x)
*'(x, s(y)) -> *'(x, y)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
+'(x, s(y)) -> +'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
*'(x, s(y)) -> *'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FAC(s(x)) -> FAC(x)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FLOOP(s(x), y) -> FLOOP(x, *(s(x), y))

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
+'(x, s(y)) -> +'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
*'(x, s(y)) -> *'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FAC(s(x)) -> FAC(x)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FLOOP(s(x), y) -> FLOOP(x, *(s(x), y))

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
+'(x, s(y)) -> +'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
*'(x, s(y)) -> *'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FAC(s(x)) -> FAC(x)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FLOOP(s(x), y) -> FLOOP(x, *(s(x), y))

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
+'(x, s(y)) -> +'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
*'(x, s(y)) -> *'(x, y)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FAC(s(x)) -> FAC(x)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)
Dependency Pair:
FLOOP(s(x), y) -> FLOOP(x, *(s(x), y))

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Termination of R could not be shown.
Duration:
0:02 minutes