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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) 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:
*' < prod

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_#:13_2(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)

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

Induction Step:
*'(gen_#:13_2(+(n137824_2, 1)), gen_#:13_2(0)) →_R^Ω(1)
+'(0(*'(gen_#:13_2(n137824_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).

(12) Complex Obligation (BEST)

(13) 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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons4_2(n142939_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n142939₂)

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

Induction Step:
sum(gen_nil:cons4_2(+(n142939_2, 1))) →_R^Ω(1)
+'(#, sum(gen_nil:cons4_2(n142939_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).

(15) Complex Obligation (BEST)

(16) 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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)
sum(gen_nil:cons4_2(n142939_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n142939₂)

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) 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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)
sum(gen_nil:cons4_2(n142939_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n142939₂)

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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)

(20) BOUNDS(n^1, INF)

(21) 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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)
sum(gen_nil:cons4_2(n142939_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n142939₂)

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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)

(23) BOUNDS(n^1, INF)

(24) 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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)

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(n137824_2), gen_#:13_2(0)) → gen_#:13_2(0), rt ∈ Ω(1 + n137824₂)

(26) BOUNDS(n^1, INF)