### (0) Obligation:

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

+(x, 0) → x
+(0, x) → x
+(s(x), s(y)) → s(s(+(x, y)))
*(x, 0) → 0
*(0, x) → 0
*(s(x), s(y)) → s(+(*(x, y), +(x, y)))
sum(nil) → 0
sum(cons(x, l)) → +(x, sum(l))
prod(nil) → s(0)
prod(cons(x, l)) → *(x, prod(l))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
+', *', sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

### (8) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
+', *', sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
gen_0':s3_0(0)

Induction Step:
+'(gen_0':s3_0(+(n6_0, 1)), gen_0':s3_0(+(n6_0, 1))) →RΩ(1)
s(s(+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)))) →IH
s(s(gen_0':s3_0(*(2, c7_0))))

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

### (11) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
*', sum, prod

They will be analysed ascendingly in the following order:
*' < prod

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

Proved the following rewrite lemma:
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

Induction Base:
*'(gen_0':s3_0(0), gen_0':s3_0(0))

Induction Step:
*'(gen_0':s3_0(+(n476_0, 1)), gen_0':s3_0(+(n476_0, 1))) →RΩ(1)
s(+'(*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)), +'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)))) →IH
s(+'(*5_0, +'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)))) →LΩ(1 + n4760)
s(+'(*5_0, gen_0':s3_0(*(2, n476_0))))

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (14) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
sum, prod

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

Proved the following rewrite lemma:
sum(gen_nil:cons4_0(n10181_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n101810)

Induction Base:
sum(gen_nil:cons4_0(0)) →RΩ(1)
0'

Induction Step:
sum(gen_nil:cons4_0(+(n10181_0, 1))) →RΩ(1)
+'(0', sum(gen_nil:cons4_0(n10181_0))) →IH
+'(0', gen_0':s3_0(0)) →LΩ(1)
gen_0':s3_0(*(2, 0))

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

### (17) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)
sum(gen_nil:cons4_0(n10181_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n101810)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
prod

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

Proved the following rewrite lemma:
prod(gen_nil:cons4_0(n10615_0)) → *5_0, rt ∈ Ω(n106150)

Induction Base:
prod(gen_nil:cons4_0(0))

Induction Step:
prod(gen_nil:cons4_0(+(n10615_0, 1))) →RΩ(1)
*'(0', prod(gen_nil:cons4_0(n10615_0))) →IH
*'(0', *5_0)

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

### (20) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)
sum(gen_nil:cons4_0(n10181_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n101810)
prod(gen_nil:cons4_0(n10615_0)) → *5_0, rt ∈ Ω(n106150)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

### (23) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)
sum(gen_nil:cons4_0(n10181_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n101810)
prod(gen_nil:cons4_0(n10615_0)) → *5_0, rt ∈ Ω(n106150)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

### (26) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)
sum(gen_nil:cons4_0(n10181_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n101810)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

### (29) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) → *5_0, rt ∈ Ω(n4760 + n47602)

### (32) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n60)