The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

f(0) → 0
f(s(0)) → s(0)
f(s(s(x))) → p(h(g(x)))
g(0) → pair(s(0), s(0))
g(s(x)) → h(g(x))
h(x) → pair(+(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
f(s(s(x))) → +(p(g(x)), q(g(x)))
g(s(x)) → pair(+(p(g(x)), q(g(x))), p(g(x)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
g(s(s(x5076_1))) →⁺ h(pair(+(p(g(x5076_1)), q(g(x5076_1))), p(g(x5076_1))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0].
The pumping substitution is [x5076_1 / s(s(x5076_1))].
The result substitution is [ ].

The rewrite sequence
g(s(s(x5076_1))) →⁺ h(pair(+(p(g(x5076_1)), q(g(x5076_1))), p(g(x5076_1))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1,0].
The pumping substitution is [x5076_1 / s(s(x5076_1))].
The result substitution is [ ].

(2) BOUNDS(2^n, 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') → 0'
f(s(0')) → s(0')
f(s(s(x))) → p(h(g(x)))
g(0') → pair(s(0'), s(0'))
g(s(x)) → h(g(x))
h(x) → pair(+'(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
f(s(s(x))) → +'(p(g(x)), q(g(x)))
g(s(x)) → pair(+'(p(g(x)), q(g(x))), p(g(x)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(0') → 0'
f(s(0')) → s(0')
f(s(s(x))) → p(h(g(x)))
g(0') → pair(s(0'), s(0'))
g(s(x)) → h(g(x))
h(x) → pair(+'(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
f(s(s(x))) → +'(p(g(x)), q(g(x)))
g(s(x)) → pair(+'(p(g(x)), q(g(x))), p(g(x)))

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: pair → 0':s
h :: pair → pair
g :: 0':s → pair
pair :: 0':s → 0':s → pair
+' :: 0':s → 0':s → 0':s
q :: pair → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
g, +'

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

(8) Obligation:

TRS:
Rules:
f(0') → 0'
f(s(0')) → s(0')
f(s(s(x))) → p(h(g(x)))
g(0') → pair(s(0'), s(0'))
g(s(x)) → h(g(x))
h(x) → pair(+'(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
f(s(s(x))) → +'(p(g(x)), q(g(x)))
g(s(x)) → pair(+'(p(g(x)), q(g(x))), p(g(x)))

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: pair → 0':s
h :: pair → pair
g :: 0':s → pair
pair :: 0':s → 0':s → pair
+' :: 0':s → 0':s → 0':s
q :: pair → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
+', g

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

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n5_0))) →_IH
s(gen_0':s3_0(+(a, c6_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:
f(0') → 0'
f(s(0')) → s(0')
f(s(s(x))) → p(h(g(x)))
g(0') → pair(s(0'), s(0'))
g(s(x)) → h(g(x))
h(x) → pair(+'(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
f(s(s(x))) → +'(p(g(x)), q(g(x)))
g(s(x)) → pair(+'(p(g(x)), q(g(x))), p(g(x)))

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: pair → 0':s
h :: pair → pair
g :: 0':s → pair
pair :: 0':s → 0':s → pair
+' :: 0':s → 0':s → 0':s
q :: pair → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
g

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

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

(13) Obligation:

TRS:
Rules:
f(0') → 0'
f(s(0')) → s(0')
f(s(s(x))) → p(h(g(x)))
g(0') → pair(s(0'), s(0'))
g(s(x)) → h(g(x))
h(x) → pair(+'(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
f(s(s(x))) → +'(p(g(x)), q(g(x)))
g(s(x)) → pair(+'(p(g(x)), q(g(x))), p(g(x)))

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: pair → 0':s
h :: pair → pair
g :: 0':s → pair
pair :: 0':s → 0':s → pair
+' :: 0':s → 0':s → 0':s
q :: pair → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(0') → 0'
f(s(0')) → s(0')
f(s(s(x))) → p(h(g(x)))
g(0') → pair(s(0'), s(0'))
g(s(x)) → h(g(x))
h(x) → pair(+'(p(x), q(x)), p(x))
p(pair(x, y)) → x
q(pair(x, y)) → y
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
f(s(s(x))) → +'(p(g(x)), q(g(x)))
g(s(x)) → pair(+'(p(g(x)), q(g(x))), p(g(x)))

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: pair → 0':s
h :: pair → pair
g :: 0':s → pair
pair :: 0':s → 0':s → pair
+' :: 0':s → 0':s → 0':s
q :: pair → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)