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))
*(x, +(y, z)) → +(*(x, y), *(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +(x, sum(l))
sum(app(l1, l2)) → +(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *(x, prod(l))
prod(app(l1, l2)) → *(prod(l1), prod(l2))

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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
+', *', app, 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

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

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

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol +'.

(8) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

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

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

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

Proved the following rewrite lemma:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

Induction Base:
*'(gen_#:13_3(0), gen_#:13_3(0)) →_R^Ω(1)
#

Induction Step:
*'(gen_#:13_3(+(n106700_3, 1)), gen_#:13_3(0)) →_R^Ω(1)
+'(0(*'(gen_#:13_3(n106700_3), gen_#:13_3(0))), gen_#:13_3(0)) →_IH
+'(0(gen_#:13_3(0)), gen_#:13_3(0)) →_R^Ω(1)
+'(#, gen_#:13_3(0)) →_R^Ω(1)
#

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

(10) Complex Obligation (BEST)

(11) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

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

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

Proved the following rewrite lemma:
app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n120873_3, b)), rt ∈ Ω(1 + n120873₃)

Induction Base:
app(gen_nil:cons4_3(0), gen_nil:cons4_3(b)) →_R^Ω(1)
gen_nil:cons4_3(b)

Induction Step:
app(gen_nil:cons4_3(+(n120873_3, 1)), gen_nil:cons4_3(b)) →_R^Ω(1)
cons(#, app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b))) →_IH
cons(#, gen_nil:cons4_3(+(b, c120874_3)))

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

(13) Complex Obligation (BEST)

(14) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)
app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n120873_3, b)), rt ∈ Ω(1 + n120873₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(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_3(n121836_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n121836₃)

Induction Base:
sum(gen_nil:cons4_3(0)) →_R^Ω(1)
0(#) →_R^Ω(1)
#

Induction Step:
sum(gen_nil:cons4_3(+(n121836_3, 1))) →_R^Ω(1)
+'(#, sum(gen_nil:cons4_3(n121836_3))) →_IH
+'(#, gen_#:13_3(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:

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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)
app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n120873_3, b)), rt ∈ Ω(1 + n120873₃)
sum(gen_nil:cons4_3(n121836_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n121836₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

The following defined symbols remain to be analysed:
prod

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)
app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n120873_3, b)), rt ∈ Ω(1 + n120873₃)
sum(gen_nil:cons4_3(n121836_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n121836₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

(21) BOUNDS(n^1, INF)

(22) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)
app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n120873_3, b)), rt ∈ Ω(1 + n120873₃)
sum(gen_nil:cons4_3(n121836_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n121836₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

(24) BOUNDS(n^1, INF)

(25) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)
app(gen_nil:cons4_3(n120873_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n120873_3, b)), rt ∈ Ω(1 + n120873₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

(27) BOUNDS(n^1, INF)

(28) 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))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
app(nil, l) → l
app(cons(x, l1), l2) → cons(x, app(l1, l2))
sum(nil) → 0(#)
sum(cons(x, l)) → +'(x, sum(l))
sum(app(l1, l2)) → +'(sum(l1), sum(l2))
prod(nil) → 1(#)
prod(cons(x, l)) → *'(x, prod(l))
prod(app(l1, l2)) → *'(prod(l1), prod(l2))

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
*' :: #:1 → #:1 → #:1
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: #:1 → nil:cons → nil:cons
sum :: nil:cons → #:1
prod :: nil:cons → #:1
hole_#:11_3 :: #:1
hole_nil:cons2_3 :: nil:cons
gen_#:13_3 :: Nat → #:1
gen_nil:cons4_3 :: Nat → nil:cons

Lemmas:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

Generator Equations:
gen_#:13_3(0) ⇔ #
gen_#:13_3(+(x, 1)) ⇔ 1(gen_#:13_3(x))
gen_nil:cons4_3(0) ⇔ nil
gen_nil:cons4_3(+(x, 1)) ⇔ cons(#, gen_nil:cons4_3(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_#:13_3(n106700_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n106700₃)

(30) BOUNDS(n^1, INF)