### (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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
prod(cons(1(x51065_3), l)) →+ +(0(*(x51065_3, 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(x51065_3), l)].
The result substitution is [ ].

The rewrite sequence
prod(cons(1(x51065_3), l)) →+ +(0(*(x51065_3, 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(x51065_3), 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))
*'(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: 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))
*'(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

### (7) 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

### (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))
*'(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

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

Proved the following rewrite lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

Induction Base:
+'(gen_#:13_3(0), gen_#:13_3(0))

Induction Step:
+'(gen_#:13_3(+(n6_3, 1)), gen_#:13_3(+(n6_3, 1))) →RΩ(1)
0(+'(+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)), 1(#))) →IH
0(+'(*5_3, 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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

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

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

Proved the following rewrite lemma:
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)

Induction Base:
*'(gen_#:13_3(0), gen_#:13_3(0)) →RΩ(1)
#

Induction Step:
*'(gen_#:13_3(+(n104928_3, 1)), gen_#:13_3(0)) →RΩ(1)
+'(0(*'(gen_#:13_3(n104928_3), gen_#:13_3(0))), gen_#:13_3(0)) →IH
+'(0(gen_#:13_3(0)), gen_#:13_3(0)) →RΩ(1)
+'(#, gen_#:13_3(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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)

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

### (15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)

Induction Base:
app(gen_nil:cons4_3(0), gen_nil:cons4_3(b)) →RΩ(1)
gen_nil:cons4_3(b)

Induction Step:
app(gen_nil:cons4_3(+(n119121_3, 1)), gen_nil:cons4_3(b)) →RΩ(1)
cons(#, app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b))) →IH
cons(#, gen_nil:cons4_3(+(b, c119122_3)))

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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)

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

### (18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons4_3(n120223_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n1202233)

Induction Base:
sum(gen_nil:cons4_3(0)) →RΩ(1)
0(#) →RΩ(1)
#

Induction Step:
sum(gen_nil:cons4_3(+(n120223_3, 1))) →RΩ(1)
+'(#, sum(gen_nil:cons4_3(n120223_3))) →IH
+'(#, gen_#:13_3(0)) →RΩ(1)
#

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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)
sum(gen_nil:cons4_3(n120223_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n1202233)

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

### (21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
prod(gen_nil:cons4_3(n126303_3)) → *5_3, rt ∈ Ω(n1263033)

Induction Base:
prod(gen_nil:cons4_3(0))

Induction Step:
prod(gen_nil:cons4_3(+(n126303_3, 1))) →RΩ(1)
*'(#, prod(gen_nil:cons4_3(n126303_3))) →IH
*'(#, *5_3)

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

### (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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)
sum(gen_nil:cons4_3(n120223_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n1202233)
prod(gen_nil:cons4_3(n126303_3)) → *5_3, rt ∈ Ω(n1263033)

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.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

### (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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)
sum(gen_nil:cons4_3(n120223_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n1202233)
prod(gen_nil:cons4_3(n126303_3)) → *5_3, rt ∈ Ω(n1263033)

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.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

### (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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)
sum(gen_nil:cons4_3(n120223_3)) → gen_#:13_3(0), rt ∈ Ω(1 + n1202233)

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.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

### (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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)
app(gen_nil:cons4_3(n119121_3), gen_nil:cons4_3(b)) → gen_nil:cons4_3(+(n119121_3, b)), rt ∈ Ω(1 + n1191213)

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.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

### (35) 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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)
*'(gen_#:13_3(n104928_3), gen_#:13_3(0)) → gen_#:13_3(0), rt ∈ Ω(1 + n1049283)

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.

### (36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

### (38) 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))
*'(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(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)

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.

### (39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_3(n6_3), gen_#:13_3(n6_3)) → *5_3, rt ∈ Ω(n63)