Runtime Complexity (full) proof of /tmp/tmppU6FDk/Ex15_Luc06

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 93 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), 8 ms)

    ↳10 typed CpxTrs

    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

    ↳12 typed CpxTrs

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(activate(X))
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
activate(n__g(X)) →⁺ g(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__g(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
f(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__a:n__f:n__g → n__a:n__f:n__g
n__f :: n__a:n__f:n__g → n__a:n__f:n__g
n__a :: n__a:n__f:n__g
n__g :: n__a:n__f:n__g → n__a:n__f:n__g
a :: n__a:n__f:n__g
g :: n__a:n__f:n__g → n__a:n__f:n__g
activate :: n__a:n__f:n__g → n__a:n__f:n__g
hole_n__a:n__f:n__g1_0 :: n__a:n__f:n__g
gen_n__a:n__f:n__g2_0 :: Nat → n__a:n__f:n__g

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f, activate

They will be analysed ascendingly in the following order:
f < activate

(8) Obligation:

Generator Equations:
gen_n__a:n__f:n__g2_0(0) ⇔ n__a
gen_n__a:n__f:n__g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:n__g2_0(x))

The following defined symbols remain to be analysed:
f, activate

They will be analysed ascendingly in the following order:
f < activate

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(10) Obligation:

Generator Equations:
gen_n__a:n__f:n__g2_0(0) ⇔ n__a
gen_n__a:n__f:n__g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:n__g2_0(x))

The following defined symbols remain to be analysed:
activate

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol activate.

(12) Obligation:

Generator Equations:
gen_n__a:n__f:n__g2_0(0) ⇔ n__a
gen_n__a:n__f:n__g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:n__g2_0(x))

No more defined symbols left to analyse.