### (0) Obligation:

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

+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
f(0, s(0), X) → f(X, +(X, X), X)
g(X, Y) → X
g(X, Y) → Y

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

+'(X, 0') → X
+'(X, s(Y)) → s(+'(X, Y))
f(0', s(0'), X) → f(X, +'(X, X), X)
g(X, Y) → X
g(X, Y) → Y

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → f
g :: g → g → g
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

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

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

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

### (8) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → f
g :: g → g → g
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
+'(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Induction Base:
+'(gen_0':s4_0(a), gen_0':s4_0(0)) →RΩ(1)
gen_0':s4_0(a)

Induction Step:
+'(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) →RΩ(1)
s(+'(gen_0':s4_0(a), gen_0':s4_0(n6_0))) →IH
s(gen_0':s4_0(+(a, c7_0)))

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

### (11) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → f
g :: g → g → g
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
f

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

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

### (13) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → f
g :: g → g → g
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_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':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

### (16) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → f
g :: g → g → g
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_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':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)