R
↳Dependency Pair Analysis
-'(s(x), s(y)) -> -'(x, y)
F(s(x)) -> -'(s(x), g(f(x)))
F(s(x)) -> G(f(x))
F(s(x)) -> F(x)
G(s(x)) -> -'(s(x), f(g(x)))
G(s(x)) -> F(g(x))
G(s(x)) -> G(x)
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↳DPs
→DP Problem 1
↳Polynomial Ordering
→DP Problem 2
↳Nar
-'(s(x), s(y)) -> -'(x, y)
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
-'(s(x), s(y)) -> -'(x, y)
POL(-'(x1, x2)) = x1 POL(s(x1)) = 1 + x1
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↳DPs
→DP Problem 1
↳Polo
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Nar
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
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↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Narrowing Transformation
G(s(x)) -> G(x)
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
F(s(x)) -> G(f(x))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
two new Dependency Pairs are created:
F(s(x)) -> G(f(x))
F(s(0)) -> G(0)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Nar
→DP Problem 4
↳Narrowing Transformation
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
G(s(x)) -> G(x)
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
two new Dependency Pairs are created:
G(s(x)) -> F(g(x))
G(s(0)) -> F(s(0))
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
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↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 5
↳Polynomial Ordering
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
F(s(x)) -> F(x)
G(s(x)) -> G(x)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
POL(0) = 0 POL(g(x1)) = 0 POL(G(x1)) = x1 POL(s(x1)) = 1 + x1 POL(-(x1, x2)) = x1 POL(f(x1)) = 0 POL(F(x1)) = x1
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↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 6
↳Dependency Graph
G(s(0)) -> F(s(0))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))