The Runtime Complexity (full) of the given

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 1677 ms)

↳2 BOUNDS(n^1, INF)

↳3 RenamingProof (⇔, 0 ms)

↳4 CpxRelTRS

↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 typed CpxTrs

↳7 OrderProof (LOWER BOUND(ID), 0 ms)

↳8 typed CpxTrs

↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 553 ms)

↳10 typed CpxTrs

The TRS R consists of the following rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)

f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)

f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5)

f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5)

f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5)

f(0, 0, 0, 0, 0) → 0

Rewrite Strategy: FULL

The rewrite sequence

f(s(x1), x2, x3, x4, x5) →

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x1 / s(x1)].

The result substitution is [ ].

The TRS R consists of the following rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)

f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)

f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)

f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)

f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)

f(0', 0', 0', 0', 0') → 0'

S is empty.

Rewrite Strategy: FULL

Rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)

f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)

f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)

f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)

f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)

f(0', 0', 0', 0', 0') → 0'

Types:

f :: s:0' → s:0' → s:0' → s:0' → s:0' → s:0'

s :: s:0' → s:0'

0' :: s:0'

hole_s:0'1_0 :: s:0'

gen_s:0'2_0 :: Nat → s:0'

f

Rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)

f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)

f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)

f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)

f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)

f(0', 0', 0', 0', 0') → 0'

Types:

f :: s:0' → s:0' → s:0' → s:0' → s:0' → s:0'

s :: s:0' → s:0'

0' :: s:0'

hole_s:0'1_0 :: s:0'

gen_s:0'2_0 :: Nat → s:0'

Generator Equations:

gen_s:0'2_0(0) ⇔ 0'

gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:

f

Rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)

f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)

f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)

f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)

f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)

f(0', 0', 0', 0', 0') → 0'

Types:

f :: s:0' → s:0' → s:0' → s:0' → s:0' → s:0'

s :: s:0' → s:0'

0' :: s:0'

hole_s:0'1_0 :: s:0'

gen_s:0'2_0 :: Nat → s:0'

Generator Equations:

gen_s:0'2_0(0) ⇔ 0'

gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.