The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, 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, 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: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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

(5) 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

(6) Obligation:

Innermost 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

(7) 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))) →_R^Ω(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 irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_#:13_2(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)

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

Induction Step:
*'(gen_#:13_2(+(n106070_2, 1)), gen_#:13_2(0)) →_R^Ω(1)
+'(0(*'(gen_#:13_2(n106070_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).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost 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(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)

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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons4_2(n119750_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n119750₂)

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

Induction Step:
sum(gen_nil:cons4_2(+(n119750_2, 1))) →_R^Ω(1)
+'(#, sum(gen_nil:cons4_2(n119750_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).

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost 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(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)
sum(gen_nil:cons4_2(n119750_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n119750₂)

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

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
prod(gen_nil:cons4_2(n125723_2)) → *5_2, rt ∈ Ω(n125723₂)

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

Induction Step:
prod(gen_nil:cons4_2(+(n125723_2, 1))) →_R^Ω(1)
*'(#, prod(gen_nil:cons4_2(n125723_2))) →_IH
*'(#, *5_2)

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

(17) Complex Obligation (BEST)

(18) Obligation:

Innermost 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(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)
sum(gen_nil:cons4_2(n119750_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n119750₂)
prod(gen_nil:cons4_2(n125723_2)) → *5_2, rt ∈ Ω(n125723₂)

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.

(19) 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)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost 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(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)
sum(gen_nil:cons4_2(n119750_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n119750₂)
prod(gen_nil:cons4_2(n125723_2)) → *5_2, rt ∈ Ω(n125723₂)

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.

(22) 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) BOUNDS(n^1, INF)

(24) Obligation:

Innermost 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(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)
sum(gen_nil:cons4_2(n119750_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n119750₂)

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.

(25) 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) BOUNDS(n^1, INF)

(27) Obligation:

Innermost 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(n106070_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n106070₂)

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.

(28) 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) BOUNDS(n^1, INF)

(30) Obligation:

Innermost 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.

(31) 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) BOUNDS(n^1, INF)