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:

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) → #
*(0(x), y) → 0(*(x, y))
*(1(x), y) → +(0(*(x, y)), y)
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(x13789_2), l)) →⁺ +(0(*(x13789_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(x13789_2), l)].
The result substitution is [ ].

The rewrite sequence
prod(cons(1(x13789_2), l)) →⁺ +(0(*(x13789_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(x13789_2), l)].
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:

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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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

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

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

Induction Base:
+'(gen_#:13_2(+(1, 0)), gen_#:13_2(+(1, 0)))

Induction Step:
+'(gen_#:13_2(+(1, +(n6_2, 1))), gen_#:13_2(+(1, +(n6_2, 1)))) →_R^Ω(1)
0(+'(+'(gen_#:13_2(+(1, n6_2)), gen_#:13_2(+(1, n6_2))), 1(#))) →_IH
0(+'(*5_2, 1(#)))

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

(10) Complex Obligation (BEST)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, 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(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)

Induction Base:
*'(gen_#:13_2(0), gen_#:13_2(0)) →_R^Ω(1)
#

Induction Step:
*'(gen_#:13_2(+(n138054_2, 1)), gen_#:13_2(0)) →_R^Ω(1)
+'(0(*'(gen_#:13_2(n138054_2), gen_#:13_2(0))), gen_#:13_2(0)) →_IH
+'(0(gen_#:13_2(0)), gen_#:13_2(0)) →_R^Ω(1)
+'(#, gen_#:13_2(0)) →_R^Ω(1)
#

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

(13) Complex Obligation (BEST)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)
*'(gen_#:13_2(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)

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(n143364_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n143364₂)

Induction Base:
sum(gen_nil:cons4_2(0)) →_R^Ω(1)
0(#) →_R^Ω(1)
#

Induction Step:
sum(gen_nil:cons4_2(+(n143364_2, 1))) →_R^Ω(1)
+'(#, sum(gen_nil:cons4_2(n143364_2))) →_IH
+'(#, gen_#:13_2(0)) →_R^Ω(1)
#

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

(16) Complex Obligation (BEST)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)
*'(gen_#:13_2(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)
sum(gen_nil:cons4_2(n143364_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n143364₂)

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(n145801_2)) → *5_2, rt ∈ Ω(n145801₂)

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

Induction Step:
prod(gen_nil:cons4_2(+(n145801_2, 1))) →_R^Ω(1)
*'(#, prod(gen_nil:cons4_2(n145801_2))) →_IH
*'(#, *5_2)

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

(19) Complex Obligation (BEST)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)
*'(gen_#:13_2(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)
sum(gen_nil:cons4_2(n143364_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n143364₂)
prod(gen_nil:cons4_2(n145801_2)) → *5_2, rt ∈ Ω(n145801₂)

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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)

(22) BOUNDS(n^1, INF)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)
*'(gen_#:13_2(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)
sum(gen_nil:cons4_2(n143364_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n143364₂)
prod(gen_nil:cons4_2(n145801_2)) → *5_2, rt ∈ Ω(n145801₂)

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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)

(25) BOUNDS(n^1, INF)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)
*'(gen_#:13_2(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)
sum(gen_nil:cons4_2(n143364_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n143364₂)

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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)

(28) BOUNDS(n^1, INF)

(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) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)
*'(gen_#:13_2(n138054_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n138054₂)

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(+(1, n6_2)), gen_#:13_2(+(1, n6_2))) → *5_2, rt ∈ Ω(n6₂)

(31) BOUNDS(n^1, INF)

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

(34) BOUNDS(n^1, INF)