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

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 typed CpxTrs
↳5 OrderProof (LOWER BOUND(ID), 3 ms)
↳6 typed CpxTrs
↳7 NoRewriteLemmaProof (LOWER BOUND(ID), 92 ms)
↳8 typed CpxTrs
↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
↳10 typed CpxTrs
↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 506 ms)
↳12 typed CpxTrs
↳13 RewriteLemmaProof (LOWER BOUND(ID), 330 ms)
↳14 BEST
    ↳15 typed CpxTrs
        ↳16 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
        ↳17 typed CpxTrs
        ↳18 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
        ↳19 typed CpxTrs
        ↳20 NoRewriteLemmaProof (LOWER BOUND(ID), 512 ms)
        ↳21 typed CpxTrs
        ↳22 LowerBoundsProof (⇔, 0 ms)
        ↳23 BOUNDS(n^1, INF)
    ↳24 typed CpxTrs
        ↳25 LowerBoundsProof (⇔, 0 ms)
        ↳26 BOUNDS(n^1, INF)

(0) Obligation:

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

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Rewrite Strategy: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__h, a__g, mark, a__f

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(6) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
a__g, a__h, mark, a__f

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
a__f, a__h, mark

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

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

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

(10) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
a__h, mark

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

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

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

(12) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
mark

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

Induction Base:
mark(gen_a:b:h:g:f2_0(+(1, 0)))

Induction Step:
mark(gen_a:b:h:g:f2_0(+(1, +(n14323_0, 1)))) →RΩ(1)
a__h(mark(gen_a:b:h:g:f2_0(+(1, n14323_0)))) →IH
a__h(*3_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(14) Complex Obligation (BEST)

(15) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Lemmas:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
a__g, a__h, a__f

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Lemmas:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
a__f, a__h

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Lemmas:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
a__h

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Lemmas:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Lemmas:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_a:b:h:g:f2_0(+(1, n14323_0))) → *3_0, rt ∈ Ω(n143230)

(26) BOUNDS(n^1, INF)