### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

length(nil) → 0
length(cons(x, l)) → s(length(l))
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0, l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
length(cons(x, l)) →+ s(length(l))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [l / cons(x, 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:

length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
length, lt, rev

They will be analysed ascendingly in the following order:
length < rev
lt < rev

### (8) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(undefined, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
length, lt, rev

They will be analysed ascendingly in the following order:
length < rev
lt < rev

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

Proved the following rewrite lemma:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)

Induction Base:
length(gen_nil:cons6_0(0)) →RΩ(1)
0'

Induction Step:
length(gen_nil:cons6_0(+(n8_0, 1))) →RΩ(1)
s(length(gen_nil:cons6_0(n8_0))) →IH
s(gen_0':s5_0(c9_0))

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

### (11) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(undefined, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
lt, rev

They will be analysed ascendingly in the following order:
lt < rev

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

Proved the following rewrite lemma:
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) → false, rt ∈ Ω(1 + n2320)

Induction Base:
lt(gen_0':s5_0(0), gen_0':s5_0(0)) →RΩ(1)
false

Induction Step:
lt(gen_0':s5_0(+(n232_0, 1)), gen_0':s5_0(+(n232_0, 1))) →RΩ(1)
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) →IH
false

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

### (14) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) → false, rt ∈ Ω(1 + n2320)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(undefined, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
rev

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

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

### (16) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) → false, rt ∈ Ω(1 + n2320)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(undefined, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)

### (19) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) → false, rt ∈ Ω(1 + n2320)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(undefined, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)

### (22) Obligation:

TRS:
Rules:
length(nil) → 0'
length(cons(x, l)) → s(length(l))
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
tail(nil) → nil
tail(cons(x, l)) → l
reverse(l) → rev(0', l, nil, l)
rev(x, l, accu, orig) → if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) → rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) → accu

Types:
length :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined → nil:cons → nil:cons
s :: 0':s → 0':s
lt :: 0':s → 0':s → false:true
false :: false:true
true :: false:true
undefined :: undefined
tail :: nil:cons → nil:cons
reverse :: nil:cons → nil:cons
rev :: 0':s → nil:cons → nil:cons → nil:cons → nil:cons
if :: false:true → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(undefined, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
length(gen_nil:cons6_0(n8_0)) → gen_0':s5_0(n8_0), rt ∈ Ω(1 + n80)