(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(x, f(s(s(y)), f(z, w))) → f(s(x), f(y, f(s(z), w)))
L(f(s(s(y)), f(z, w))) → L(f(s(0), f(y, f(s(z), w))))
f(x, f(s(s(y)), nil)) → f(s(x), f(y, f(s(0), nil)))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(s(s(z1)), f(z2, z3))) → f(s(z0), f(z1, f(s(z2), z3)))
f(z0, f(s(s(z1)), nil)) → f(s(z0), f(z1, f(s(0), nil)))
L(f(s(s(z0)), f(z1, z2))) → L(f(s(0), f(z0, f(s(z1), z2))))
Tuples:
F(z0, f(s(s(z1)), f(z2, z3))) → c(F(s(z0), f(z1, f(s(z2), z3))), F(z1, f(s(z2), z3)), F(s(z2), z3))
F(z0, f(s(s(z1)), nil)) → c1(F(s(z0), f(z1, f(s(0), nil))), F(z1, f(s(0), nil)), F(s(0), nil))
L'(f(s(s(z0)), f(z1, z2))) → c2(L'(f(s(0), f(z0, f(s(z1), z2)))), F(s(0), f(z0, f(s(z1), z2))), F(z0, f(s(z1), z2)), F(s(z1), z2))
S tuples:
F(z0, f(s(s(z1)), f(z2, z3))) → c(F(s(z0), f(z1, f(s(z2), z3))), F(z1, f(s(z2), z3)), F(s(z2), z3))
F(z0, f(s(s(z1)), nil)) → c1(F(s(z0), f(z1, f(s(0), nil))), F(z1, f(s(0), nil)), F(s(0), nil))
L'(f(s(s(z0)), f(z1, z2))) → c2(L'(f(s(0), f(z0, f(s(z1), z2)))), F(s(0), f(z0, f(s(z1), z2))), F(z0, f(s(z1), z2)), F(s(z1), z2))
K tuples:none
Defined Rule Symbols:
f, L
Defined Pair Symbols:
F, L'
Compound Symbols:
c, c1, c2
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(z0, f(s(s(z1)), f(z2, z3))) → c(F(s(z0), f(z1, f(s(z2), z3))), F(z1, f(s(z2), z3)), F(s(z2), z3))
F(z0, f(s(s(z1)), nil)) → c1(F(s(z0), f(z1, f(s(0), nil))), F(z1, f(s(0), nil)), F(s(0), nil))
L'(f(s(s(z0)), f(z1, z2))) → c2(L'(f(s(0), f(z0, f(s(z1), z2)))), F(s(0), f(z0, f(s(z1), z2))), F(z0, f(s(z1), z2)), F(s(z1), z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(s(s(z1)), f(z2, z3))) → f(s(z0), f(z1, f(s(z2), z3)))
f(z0, f(s(s(z1)), nil)) → f(s(z0), f(z1, f(s(0), nil)))
L(f(s(s(z0)), f(z1, z2))) → L(f(s(0), f(z0, f(s(z1), z2))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, L
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))