(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w))))))
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w)))
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
t(t(x)) → t(c(t(x), x))
t(x) → x
t(x) → c(0, c(0, c(0, c(0, c(0, x)))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(c(z0, z1), c(s(z2), z2), t(z3)) → h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
h(z0, c(z1, z2), t(z3)) → h(c(s(z1), z0), z2, t(c(t(z3), z3)))
h(c(s(z0), c(s(0), z1)), z2, t(z0)) → h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
t(t(z0)) → t(c(t(z0), z0))
t(z0) → z0
t(z0) → c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
H(c(z0, z1), c(s(z2), z2), t(z3)) → c1(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3))
H(z0, c(z1, z2), t(z3)) → c2(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3))
H(c(s(z0), c(s(0), z1)), z2, t(z0)) → c3(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0))))
T(t(z0)) → c4(T(c(t(z0), z0)), T(z0))
S tuples:
H(c(z0, z1), c(s(z2), z2), t(z3)) → c1(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3))
H(z0, c(z1, z2), t(z3)) → c2(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3))
H(c(s(z0), c(s(0), z1)), z2, t(z0)) → c3(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0))))
T(t(z0)) → c4(T(c(t(z0), z0)), T(z0))
K tuples:none
Defined Rule Symbols:
h, t
Defined Pair Symbols:
H, T
Compound Symbols:
c1, c2, c3, c4
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
H(c(z0, z1), c(s(z2), z2), t(z3)) → c1(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3))
H(z0, c(z1, z2), t(z3)) → c2(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3))
H(c(s(z0), c(s(0), z1)), z2, t(z0)) → c3(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0))))
T(t(z0)) → c4(T(c(t(z0), z0)), T(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(c(z0, z1), c(s(z2), z2), t(z3)) → h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
h(z0, c(z1, z2), t(z3)) → h(c(s(z1), z0), z2, t(c(t(z3), z3)))
h(c(s(z0), c(s(0), z1)), z2, t(z0)) → h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
t(t(z0)) → t(c(t(z0), z0))
t(z0) → z0
t(z0) → c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
h, t
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))