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:

norm(nil) → 0
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0) → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

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

Heuristically decided to analyse the following defined symbols:
norm, f, rem

(8) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

The following defined symbols remain to be analysed:
norm, f, rem

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

Proved the following rewrite lemma:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

Induction Base:
norm(gen_nil:g5_0(0)) →_R^Ω(1)
0'

Induction Step:
norm(gen_nil:g5_0(+(n7_0, 1))) →_R^Ω(1)
s(norm(gen_nil:g5_0(n7_0))) →_IH
s(gen_0':s4_0(c8_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Lemmas:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(hole_a3_0, gen_nil:g5_0(n207_0)) → gen_nil:g5_0(+(1, n207_0)), rt ∈ Ω(1 + n207₀)

Induction Base:
f(hole_a3_0, gen_nil:g5_0(0)) →_R^Ω(1)
g(nil, hole_a3_0)

Induction Step:
f(hole_a3_0, gen_nil:g5_0(+(n207_0, 1))) →_R^Ω(1)
g(f(hole_a3_0, gen_nil:g5_0(n207_0)), hole_a3_0) →_IH
g(gen_nil:g5_0(+(1, c208_0)), hole_a3_0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Lemmas:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)
f(hole_a3_0, gen_nil:g5_0(n207_0)) → gen_nil:g5_0(+(1, n207_0)), rt ∈ Ω(1 + n207₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

The following defined symbols remain to be analysed:
rem

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
rem(gen_nil:g5_0(n488_0), gen_0':s4_0(n488_0)) → gen_nil:g5_0(0), rt ∈ Ω(1 + n488₀)

Induction Base:
rem(gen_nil:g5_0(0), gen_0':s4_0(0)) →_R^Ω(1)
nil

Induction Step:
rem(gen_nil:g5_0(+(n488_0, 1)), gen_0':s4_0(+(n488_0, 1))) →_R^Ω(1)
rem(gen_nil:g5_0(n488_0), gen_0':s4_0(n488_0)) →_IH
gen_nil:g5_0(0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Lemmas:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)
f(hole_a3_0, gen_nil:g5_0(n207_0)) → gen_nil:g5_0(+(1, n207_0)), rt ∈ Ω(1 + n207₀)
rem(gen_nil:g5_0(n488_0), gen_0':s4_0(n488_0)) → gen_nil:g5_0(0), rt ∈ Ω(1 + n488₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Lemmas:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)
f(hole_a3_0, gen_nil:g5_0(n207_0)) → gen_nil:g5_0(+(1, n207_0)), rt ∈ Ω(1 + n207₀)
rem(gen_nil:g5_0(n488_0), gen_0':s4_0(n488_0)) → gen_nil:g5_0(0), rt ∈ Ω(1 + n488₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Lemmas:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)
f(hole_a3_0, gen_nil:g5_0(n207_0)) → gen_nil:g5_0(+(1, n207_0)), rt ∈ Ω(1 + n207₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
Rules:
norm(nil) → 0'
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0') → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

Types:
norm :: nil:g → 0':s
nil :: nil:g
0' :: 0':s
g :: nil:g → a → nil:g
s :: 0':s → 0':s
f :: a → nil:g → nil:g
rem :: nil:g → 0':s → nil:g
hole_0':s1_0 :: 0':s
hole_nil:g2_0 :: nil:g
hole_a3_0 :: a
gen_0':s4_0 :: Nat → 0':s
gen_nil:g5_0 :: Nat → nil:g

Lemmas:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:g5_0(0) ⇔ nil
gen_nil:g5_0(+(x, 1)) ⇔ g(gen_nil:g5_0(x), hole_a3_0)

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
norm(gen_nil:g5_0(n7_0)) → gen_0':s4_0(n7_0), rt ∈ Ω(1 + n7₀)

(28) BOUNDS(n^1, INF)