Termination Proof using AProVE

Term Rewriting System R:
[y, x]
leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

LEQ(s(x), s(y)) -> LEQ(x, y)
-'(s(x), s(y)) -> -'(x, y)
MOD(s(x), s(y)) -> IF(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
MOD(s(x), s(y)) -> LEQ(y, x)
MOD(s(x), s(y)) -> MOD(-(s(x), s(y)), s(y))
MOD(s(x), s(y)) -> -'(s(x), s(y))

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
LEQ(s(x), s(y)) -> LEQ(x, y)

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
Dependency Pair:
-'(s(x), s(y)) -> -'(x, y)

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
Dependency Pair:
MOD(s(x), s(y)) -> MOD(-(s(x), s(y)), s(y))

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
LEQ(s(x), s(y)) -> LEQ(x, y)

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
Dependency Pair:
-'(s(x), s(y)) -> -'(x, y)

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
Dependency Pair:
MOD(s(x), s(y)) -> MOD(-(s(x), s(y)), s(y))

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
LEQ(s(x), s(y)) -> LEQ(x, y)

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
Dependency Pair:
-'(s(x), s(y)) -> -'(x, y)

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
Dependency Pair:
MOD(s(x), s(y)) -> MOD(-(s(x), s(y)), s(y))

Rules:

leq(0, y) -> true
leq(s(x), 0) -> false
leq(s(x), s(y)) -> leq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
mod(0, y) -> 0
mod(s(x), 0) -> 0
mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Termination of R could not be shown.
Duration:
0:13 minutes