R
↳Dependency Pair Analysis
F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)
F(f(a, x), a) -> F(f(a, a), f(x, a))
F(f(a, x), a) -> F(a, a)
F(f(a, x), a) -> F(x, a)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
F(f(a, x), a) -> F(x, a)
F(f(a, x), a) -> F(f(a, a), f(x, a))
F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)
F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Narrowing Transformation
F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)
F(f(a, x), a) -> F(f(a, a), f(x, a))
F(f(a, x), a) -> F(x, a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, x), a) -> F(f(a, a), f(x, a))
F(f(a, f(a, x'')), a) -> F(f(a, a), f(f(f(a, a), f(x'', a)), a))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 3
↳Narrowing Transformation
F(f(a, f(a, x'')), a) -> F(f(a, a), f(f(f(a, a), f(x'', a)), a))
F(f(a, x), a) -> F(x, a)
F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)
F(f(a, f(a, f(a, x'))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 4
↳Narrowing Transformation
F(f(a, f(a, f(a, x'))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)
F(f(a, x), a) -> F(x, a)
F(f(a, f(a, x'')), a) -> F(f(a, a), f(f(f(a, a), f(x'', a)), a))
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, f(a, x'')), a) -> F(f(a, a), f(f(f(a, a), f(x'', a)), a))
F(f(a, f(a, f(a, x'))), a) -> F(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 5
↳Forward Instantiation Transformation
F(f(a, f(a, f(a, x'))), a) -> F(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a))
F(f(a, x), a) -> F(x, a)
F(f(a, f(a, f(a, x'))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
two new Dependency Pairs are created:
F(f(a, x), a) -> F(x, a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
F(f(a, f(a, f(a, f(a, x''')))), a) -> F(f(a, f(a, f(a, x'''))), a)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 6
↳Forward Instantiation Transformation
F(f(a, f(a, f(a, f(a, x''')))), a) -> F(f(a, f(a, f(a, x'''))), a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
F(f(a, f(a, f(a, x'))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)
F(f(a, f(a, f(a, x'))), a) -> F(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a))
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
no new Dependency Pairs are created.
F(f(a, f(a, f(a, x'))), a) -> F(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 7
↳Narrowing Transformation
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
F(f(a, f(a, f(a, x'))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)
F(f(a, f(a, f(a, f(a, x''')))), a) -> F(f(a, f(a, f(a, x'''))), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, f(a, f(a, x'))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)
F(f(a, f(a, f(a, f(a, x'')))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)), a)), a)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 8
↳Narrowing Transformation
F(f(a, f(a, f(a, f(a, x'')))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)), a)), a)
F(f(a, f(a, f(a, f(a, x''')))), a) -> F(f(a, f(a, f(a, x'''))), a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, f(a, f(a, f(a, x'')))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)), a)), a)
F(f(a, f(a, f(a, f(a, f(a, x'))))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)), a)), a)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 9
↳Narrowing Transformation
F(f(a, f(a, f(a, f(a, f(a, x'))))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)), a)), a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
F(f(a, f(a, f(a, f(a, x''')))), a) -> F(f(a, f(a, f(a, x'''))), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost
one new Dependency Pair is created:
F(f(a, f(a, f(a, f(a, f(a, x'))))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x', a)), a)), a)), a)), a)), a)
F(f(a, f(a, f(a, f(a, f(a, f(a, x'')))))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)), a)), a)), a)), a)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 10
↳Remaining Obligation(s)
F(f(a, f(a, f(a, f(a, f(a, f(a, x'')))))), a) -> F(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)), a)), a)), a)), a)
F(f(a, f(a, f(a, f(a, x''')))), a) -> F(f(a, f(a, f(a, x'''))), a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)
innermost