### (0) Obligation:

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

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
minus, min

They will be analysed ascendingly in the following order:
min < minus

### (8) Obligation:

TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
min, minus

They will be analysed ascendingly in the following order:
min < minus

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

Proved the following rewrite lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n40)

Induction Base:
min(gen_s:0'2_0(0), gen_s:0'2_0(0)) →RΩ(1)
0'

Induction Step:
min(gen_s:0'2_0(+(n4_0, 1)), gen_s:0'2_0(+(n4_0, 1))) →RΩ(1)
s(min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0))) →IH
s(gen_s:0'2_0(c5_0))

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

### (11) Obligation:

TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
minus

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

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

### (13) Obligation:

TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n40)

### (16) Obligation:

TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n40)