### (0) Obligation:

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

f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Types:
f :: s:0':cons → f
s :: s:0':cons → s:0':cons
g :: s:0':cons → s:0':cons
cons :: s:0':cons → s:0':cons → s:0':cons
0' :: s:0':cons
h :: s:0':cons → h
hole_f1_0 :: f
hole_s:0':cons2_0 :: s:0':cons
hole_h3_0 :: h
gen_s:0':cons4_0 :: Nat → s:0':cons

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

Heuristically decided to analyse the following defined symbols:
f, g, h

They will be analysed ascendingly in the following order:
g < h

### (8) Obligation:

TRS:
Rules:
f(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Types:
f :: s:0':cons → f
s :: s:0':cons → s:0':cons
g :: s:0':cons → s:0':cons
cons :: s:0':cons → s:0':cons → s:0':cons
0' :: s:0':cons
h :: s:0':cons → h
hole_f1_0 :: f
hole_s:0':cons2_0 :: s:0':cons
hole_h3_0 :: h
gen_s:0':cons4_0 :: Nat → s:0':cons

Generator Equations:
gen_s:0':cons4_0(0) ⇔ 0'
gen_s:0':cons4_0(+(x, 1)) ⇔ s(gen_s:0':cons4_0(x))

The following defined symbols remain to be analysed:
f, g, h

They will be analysed ascendingly in the following order:
g < h

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Induction Base:
f(gen_s:0':cons4_0(+(1, 0)))

Induction Step:
f(gen_s:0':cons4_0(+(1, +(n6_0, 1)))) →RΩ(1)
f(gen_s:0':cons4_0(+(1, n6_0))) →IH
*5_0

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

### (11) Obligation:

TRS:
Rules:
f(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Types:
f :: s:0':cons → f
s :: s:0':cons → s:0':cons
g :: s:0':cons → s:0':cons
cons :: s:0':cons → s:0':cons → s:0':cons
0' :: s:0':cons
h :: s:0':cons → h
hole_f1_0 :: f
hole_s:0':cons2_0 :: s:0':cons
hole_h3_0 :: h
gen_s:0':cons4_0 :: Nat → s:0':cons

Lemmas:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_s:0':cons4_0(0) ⇔ 0'
gen_s:0':cons4_0(+(x, 1)) ⇔ s(gen_s:0':cons4_0(x))

The following defined symbols remain to be analysed:
g, h

They will be analysed ascendingly in the following order:
g < h

### (12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (13) Obligation:

TRS:
Rules:
f(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Types:
f :: s:0':cons → f
s :: s:0':cons → s:0':cons
g :: s:0':cons → s:0':cons
cons :: s:0':cons → s:0':cons → s:0':cons
0' :: s:0':cons
h :: s:0':cons → h
hole_f1_0 :: f
hole_s:0':cons2_0 :: s:0':cons
hole_h3_0 :: h
gen_s:0':cons4_0 :: Nat → s:0':cons

Lemmas:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_s:0':cons4_0(0) ⇔ 0'
gen_s:0':cons4_0(+(x, 1)) ⇔ s(gen_s:0':cons4_0(x))

The following defined symbols remain to be analysed:
h

### (14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (15) Obligation:

TRS:
Rules:
f(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Types:
f :: s:0':cons → f
s :: s:0':cons → s:0':cons
g :: s:0':cons → s:0':cons
cons :: s:0':cons → s:0':cons → s:0':cons
0' :: s:0':cons
h :: s:0':cons → h
hole_f1_0 :: f
hole_s:0':cons2_0 :: s:0':cons
hole_h3_0 :: h
gen_s:0':cons4_0 :: Nat → s:0':cons

Lemmas:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_s:0':cons4_0(0) ⇔ 0'
gen_s:0':cons4_0(+(x, 1)) ⇔ s(gen_s:0':cons4_0(x))

No more defined symbols left to analyse.

### (16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

### (18) Obligation:

TRS:
Rules:
f(s(X)) → f(X)
g(cons(0', Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Types:
f :: s:0':cons → f
s :: s:0':cons → s:0':cons
g :: s:0':cons → s:0':cons
cons :: s:0':cons → s:0':cons → s:0':cons
0' :: s:0':cons
h :: s:0':cons → h
hole_f1_0 :: f
hole_s:0':cons2_0 :: s:0':cons
hole_h3_0 :: h
gen_s:0':cons4_0 :: Nat → s:0':cons

Lemmas:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_s:0':cons4_0(0) ⇔ 0'
gen_s:0':cons4_0(+(x, 1)) ⇔ s(gen_s:0':cons4_0(x))

No more defined symbols left to analyse.

### (19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_s:0':cons4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)