R
↳Dependency Pair Analysis
-'(s(x), s(y)) -> -'(x, y)
+'(s(x), y) -> +'(x, y)
*'(x, s(y)) -> +'(x, *(x, y))
*'(x, s(y)) -> *'(x, y)
F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))
F(s(x)) -> -'(*(s(s(0)), s(x)), s(s(x)))
F(s(x)) -> *'(s(s(0)), s(x))
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
-'(s(x), s(y)) -> -'(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
-'(s(x), s(y)) -> -'(x, y)
-'(x1, x2) -> -'(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 5
↳Dependency Graph
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
+'(s(x), y) -> +'(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
+'(s(x), y) -> +'(x, y)
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 6
↳Dependency Graph
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Argument Filtering and Ordering
→DP Problem 4
↳Rw
*'(x, s(y)) -> *'(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
*'(x, s(y)) -> *'(x, y)
*'(x1, x2) -> *'(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 7
↳Dependency Graph
→DP Problem 4
↳Rw
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rewriting Transformation
F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
one new Dependency Pair is created:
F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))
F(s(x)) -> F(-(+(s(s(0)), *(s(s(0)), x)), s(s(x))))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
→DP Problem 8
↳Rewriting Transformation
F(s(x)) -> F(-(+(s(s(0)), *(s(s(0)), x)), s(s(x))))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
one new Dependency Pair is created:
F(s(x)) -> F(-(+(s(s(0)), *(s(s(0)), x)), s(s(x))))
F(s(x)) -> F(-(s(+(s(0), *(s(s(0)), x))), s(s(x))))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
→DP Problem 8
↳Rw
...
→DP Problem 9
↳Rewriting Transformation
F(s(x)) -> F(-(s(+(s(0), *(s(s(0)), x))), s(s(x))))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
one new Dependency Pair is created:
F(s(x)) -> F(-(s(+(s(0), *(s(s(0)), x))), s(s(x))))
F(s(x)) -> F(-(+(s(0), *(s(s(0)), x)), s(x)))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
→DP Problem 8
↳Rw
...
→DP Problem 10
↳Rewriting Transformation
F(s(x)) -> F(-(+(s(0), *(s(s(0)), x)), s(x)))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
one new Dependency Pair is created:
F(s(x)) -> F(-(+(s(0), *(s(s(0)), x)), s(x)))
F(s(x)) -> F(-(s(+(0, *(s(s(0)), x))), s(x)))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
→DP Problem 8
↳Rw
...
→DP Problem 11
↳Rewriting Transformation
F(s(x)) -> F(-(s(+(0, *(s(s(0)), x))), s(x)))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
one new Dependency Pair is created:
F(s(x)) -> F(-(s(+(0, *(s(s(0)), x))), s(x)))
F(s(x)) -> F(-(+(0, *(s(s(0)), x)), x))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
→DP Problem 8
↳Rw
...
→DP Problem 12
↳Rewriting Transformation
F(s(x)) -> F(-(+(0, *(s(s(0)), x)), x))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost
one new Dependency Pair is created:
F(s(x)) -> F(-(+(0, *(s(s(0)), x)), x))
F(s(x)) -> F(-(*(s(s(0)), x), x))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳AFS
→DP Problem 4
↳Rw
→DP Problem 8
↳Rw
...
→DP Problem 13
↳Remaining Obligation(s)
F(s(x)) -> F(-(*(s(s(0)), x), x))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
innermost