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

(0) Obligation:

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

f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Types:
f :: a:c:b:d → a:c:b:d
a :: a:c:b:d
c :: a:c:b:d → a:c:b:d
d :: a:c:b:d → a:c:b:d
b :: a:c:b:d
e :: g → e
g :: g → g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g

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

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

(8) Obligation:

TRS:
Rules:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Types:
f :: a:c:b:d → a:c:b:d
a :: a:c:b:d
c :: a:c:b:d → a:c:b:d
d :: a:c:b:d → a:c:b:d
b :: a:c:b:d
e :: g → e
g :: g → g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g

Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))

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

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

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

(10) Obligation:

TRS:
Rules:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Types:
f :: a:c:b:d → a:c:b:d
a :: a:c:b:d
c :: a:c:b:d → a:c:b:d
d :: a:c:b:d → a:c:b:d
b :: a:c:b:d
e :: g → e
g :: g → g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g

Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))

The following defined symbols remain to be analysed:
e

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n43₀)

Induction Base:
e(gen_g5_0(+(1, 0)))

Induction Step:
e(gen_g5_0(+(1, +(n43_0, 1)))) →_R^Ω(1)
e(gen_g5_0(+(1, n43_0))) →_IH
*6_0

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Types:
f :: a:c:b:d → a:c:b:d
a :: a:c:b:d
c :: a:c:b:d → a:c:b:d
d :: a:c:b:d → a:c:b:d
b :: a:c:b:d
e :: g → e
g :: g → g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g

Lemmas:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n43₀)

Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n43₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Types:
f :: a:c:b:d → a:c:b:d
a :: a:c:b:d
c :: a:c:b:d → a:c:b:d
d :: a:c:b:d → a:c:b:d
b :: a:c:b:d
e :: g → e
g :: g → g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g

Lemmas:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n43₀)

Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n43₀)

(18) BOUNDS(n^1, INF)