### (0) Obligation:

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

0(#) → #
+(x, #) → x
+(#, x) → x
+(0(x), 0(y)) → 0(+(x, y))
+(0(x), 1(y)) → 1(+(x, y))
+(1(x), 0(y)) → 1(+(x, y))
+(1(x), 1(y)) → 0(+(+(x, y), 1(#)))
+(+(x, y), z) → +(x, +(y, z))
*(#, x) → #
*(0(x), y) → 0(*(x, y))
*(1(x), y) → +(0(*(x, y)), y)
*(*(x, y), z) → *(x, *(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +(x, sum(l))
prod(nil) → 1(#)
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(2n):
The rewrite sequence
prod(cons(1(x27767_2), l)) →+ +(0(*(x27767_2, prod(l))), prod(l))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1].
The pumping substitution is [l / cons(1(x27767_2), l)].
The result substitution is [ ].

The rewrite sequence
prod(cons(1(x27767_2), l)) →+ +(0(*(x27767_2, prod(l))), prod(l))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [l / cons(1(x27767_2), l)].
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:

0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: 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:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(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_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

Induction Base:
+'(gen_#:13_2(0), gen_#:13_2(0))

Induction Step:
+'(gen_#:13_2(+(n6_2, 1)), gen_#:13_2(+(n6_2, 1))) →RΩ(1)
0(+'(+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)), 1(#))) →IH
0(+'(*5_2, 1(#)))

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

### (11) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(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_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)

Induction Base:
*'(gen_#:13_2(0), gen_#:13_2(0)) →RΩ(1)
#

Induction Step:
*'(gen_#:13_2(+(n104413_2, 1)), gen_#:13_2(0)) →RΩ(1)
+'(0(*'(gen_#:13_2(n104413_2), gen_#:13_2(0))), gen_#:13_2(0)) →IH
+'(0(gen_#:13_2(0)), gen_#:13_2(0)) →RΩ(1)
+'(#, gen_#:13_2(0)) →RΩ(1)
#

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

### (14) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)
*'(gen_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(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_2(n117917_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1179172)

Induction Base:
sum(gen_nil:cons4_2(0)) →RΩ(1)
0(#) →RΩ(1)
#

Induction Step:
sum(gen_nil:cons4_2(+(n117917_2, 1))) →RΩ(1)
+'(#, sum(gen_nil:cons4_2(n117917_2))) →IH
+'(#, gen_#:13_2(0)) →RΩ(1)
#

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

### (17) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)
*'(gen_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)
sum(gen_nil:cons4_2(n117917_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1179172)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(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_2(n123858_2)) → *5_2, rt ∈ Ω(n1238582)

Induction Base:
prod(gen_nil:cons4_2(0))

Induction Step:
prod(gen_nil:cons4_2(+(n123858_2, 1))) →RΩ(1)
*'(#, prod(gen_nil:cons4_2(n123858_2))) →IH
*'(#, *5_2)

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

### (20) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)
*'(gen_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)
sum(gen_nil:cons4_2(n117917_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1179172)
prod(gen_nil:cons4_2(n123858_2)) → *5_2, rt ∈ Ω(n1238582)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

### (23) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)
*'(gen_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)
sum(gen_nil:cons4_2(n117917_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1179172)
prod(gen_nil:cons4_2(n123858_2)) → *5_2, rt ∈ Ω(n1238582)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

### (26) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)
*'(gen_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)
sum(gen_nil:cons4_2(n117917_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1179172)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

### (29) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)
*'(gen_#:13_2(n104413_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n1044132)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

### (32) Obligation:

TRS:
Rules:
0(#) → #
+'(x, #) → x
+'(#, x) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
sum :: nil:cons → #:1
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
prod :: nil:cons → #:1
hole_#:11_2 :: #:1
hole_nil:cons2_2 :: nil:cons
gen_#:13_2 :: Nat → #:1
gen_nil:cons4_2 :: Nat → nil:cons

Lemmas:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))
gen_nil:cons4_2(0) ⇔ nil
gen_nil:cons4_2(+(x, 1)) ⇔ cons(#, gen_nil:cons4_2(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n6_2), gen_#:13_2(n6_2)) → *5_2, rt ∈ Ω(n62)