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_0(x) → a
f_1(x) → g_1(x, x)
g_1(s(x), y) → b(f_0(y), g_1(x, y))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(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
g_1(s(x), y) →⁺ b(f_0(y), g_1(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / s(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_0(x) → a
f_1(x) → g_1(x, x)
g_1(s(x), y) → b(f_0(y), g_1(x, y))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
f_0/0
g_1/1

(6) Obligation:

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

f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
g_1, g_2, g_3, g_4, g_5

(10) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

The following defined symbols remain to be analysed:
g_1, g_2, g_3, g_4, g_5

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

Proved the following rewrite lemma:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Induction Base:
g_1(gen_s4_6(+(1, 0)))

Induction Step:
g_1(gen_s4_6(+(1, +(n6_6, 1)))) →_R^Ω(1)
b(f_0, g_1(gen_s4_6(+(1, n6_6)))) →_R^Ω(1)
b(a, g_1(gen_s4_6(+(1, n6_6)))) →_IH
b(a, *5_6)

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_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Lemmas:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

The following defined symbols remain to be analysed:
g_2, g_3, g_4, g_5

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

Could not prove a rewrite lemma for the defined symbol g_2.

(15) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Lemmas:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

The following defined symbols remain to be analysed:
g_3, g_4, g_5

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

Could not prove a rewrite lemma for the defined symbol g_3.

(17) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Lemmas:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

The following defined symbols remain to be analysed:
g_4, g_5

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

Could not prove a rewrite lemma for the defined symbol g_4.

(19) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Lemmas:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

The following defined symbols remain to be analysed:
g_5

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

Could not prove a rewrite lemma for the defined symbol g_5.

(21) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Lemmas:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))

Types:
f_0 :: a:b
a :: a:b
f_1 :: s → a:b
g_1 :: s → a:b
s :: s → s
b :: a:b → a:b → a:b
f_2 :: s → a:b
g_2 :: s → s → a:b
f_3 :: s → a:b
g_3 :: s → s → a:b
f_4 :: s → a:b
g_4 :: s → s → a:b
f_5 :: s → a:b
g_5 :: s → s → a:b
hole_a:b1_6 :: a:b
hole_s2_6 :: s
gen_a:b3_6 :: Nat → a:b
gen_s4_6 :: Nat → s

Lemmas:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

Generator Equations:
gen_a:b3_6(0) ⇔ a
gen_a:b3_6(+(x, 1)) ⇔ b(a, gen_a:b3_6(x))
gen_s4_6(0) ⇔ hole_s2_6
gen_s4_6(+(x, 1)) ⇔ s(gen_s4_6(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g_1(gen_s4_6(+(1, n6_6))) → *5_6, rt ∈ Ω(n6₆)

(26) BOUNDS(n^1, INF)