The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).

(0) Obligation:

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

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Rewrite Strategy: FULL

(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:

minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

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

Heuristically decided to analyse the following defined symbols:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

(6) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)

Induction Base:
minus(gen_0':s6_0(0), gen_0':s6_0(0)) →_R^Ω(1)
gen_0':s6_0(0)

Induction Step:
minus(gen_0':s6_0(+(n10_0, 1)), gen_0':s6_0(+(n10_0, 1))) →_R^Ω(1)
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) →_IH
gen_0':s6_0(0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
app < reverse
reverse < shuffle
concat < less_leaves

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
app < reverse
reverse < shuffle
concat < less_leaves

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

Proved the following rewrite lemma:
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)

Induction Base:
app(gen_nil:add7_0(0), gen_nil:add7_0(b)) →_R^Ω(1)
gen_nil:add7_0(b)

Induction Step:
app(gen_nil:add7_0(+(n485_0, 1)), gen_nil:add7_0(b)) →_R^Ω(1)
add(hole_a3_0, app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b))) →_IH
add(hole_a3_0, gen_nil:add7_0(+(b, c486_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
reverse < shuffle
concat < less_leaves

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

Proved the following rewrite lemma:
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)

Induction Base:
reverse(gen_nil:add7_0(0)) →_R^Ω(1)
nil

Induction Step:
reverse(gen_nil:add7_0(+(n1422_0, 1))) →_R^Ω(1)
app(reverse(gen_nil:add7_0(n1422_0)), add(hole_a3_0, nil)) →_IH
app(gen_nil:add7_0(c1423_0), add(hole_a3_0, nil)) →_L^{Ω(1 + n1422₀)}
gen_nil:add7_0(+(n1422_0, +(0, 1)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

Proved the following rewrite lemma:
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

Induction Base:
shuffle(gen_nil:add7_0(0)) →_R^Ω(1)
nil

Induction Step:
shuffle(gen_nil:add7_0(+(n1722_0, 1))) →_R^Ω(1)
add(hole_a3_0, shuffle(reverse(gen_nil:add7_0(n1722_0)))) →_L^{Ω(1 + n1722₀ + n1722₀2)}
add(hole_a3_0, shuffle(gen_nil:add7_0(n1722_0))) →_IH
add(hole_a3_0, gen_nil:add7_0(c1723_0))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

Proved the following rewrite lemma:
concat(gen_leaf:cons8_0(n1933_0), gen_leaf:cons8_0(b)) → gen_leaf:cons8_0(+(n1933_0, b)), rt ∈ Ω(1 + n1933₀)

Induction Base:
concat(gen_leaf:cons8_0(0), gen_leaf:cons8_0(b)) →_R^Ω(1)
gen_leaf:cons8_0(b)

Induction Step:
concat(gen_leaf:cons8_0(+(n1933_0, 1)), gen_leaf:cons8_0(b)) →_R^Ω(1)
cons(leaf, concat(gen_leaf:cons8_0(n1933_0), gen_leaf:cons8_0(b))) →_IH
cons(leaf, gen_leaf:cons8_0(+(b, c1934_0)))

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

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)
concat(gen_leaf:cons8_0(n1933_0), gen_leaf:cons8_0(b)) → gen_leaf:cons8_0(+(n1933_0, b)), rt ∈ Ω(1 + n1933₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

The following defined symbols remain to be analysed:
less_leaves

(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
less_leaves(gen_leaf:cons8_0(n3024_0), gen_leaf:cons8_0(n3024_0)) → false, rt ∈ Ω(1 + n3024₀)

Induction Base:
less_leaves(gen_leaf:cons8_0(0), gen_leaf:cons8_0(0)) →_R^Ω(1)
false

Induction Step:
less_leaves(gen_leaf:cons8_0(+(n3024_0, 1)), gen_leaf:cons8_0(+(n3024_0, 1))) →_R^Ω(1)
less_leaves(concat(leaf, gen_leaf:cons8_0(n3024_0)), concat(leaf, gen_leaf:cons8_0(n3024_0))) →_L^Ω(1)
less_leaves(gen_leaf:cons8_0(+(0, n3024_0)), concat(leaf, gen_leaf:cons8_0(n3024_0))) →_L^Ω(1)
less_leaves(gen_leaf:cons8_0(n3024_0), gen_leaf:cons8_0(+(0, n3024_0))) →_IH
false

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

(25) Complex Obligation (BEST)

(26) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)
concat(gen_leaf:cons8_0(n1933_0), gen_leaf:cons8_0(b)) → gen_leaf:cons8_0(+(n1933_0, b)), rt ∈ Ω(1 + n1933₀)
less_leaves(gen_leaf:cons8_0(n3024_0), gen_leaf:cons8_0(n3024_0)) → false, rt ∈ Ω(1 + n3024₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

(28) BOUNDS(n^3, INF)

(29) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)
concat(gen_leaf:cons8_0(n1933_0), gen_leaf:cons8_0(b)) → gen_leaf:cons8_0(+(n1933_0, b)), rt ∈ Ω(1 + n1933₀)
less_leaves(gen_leaf:cons8_0(n3024_0), gen_leaf:cons8_0(n3024_0)) → false, rt ∈ Ω(1 + n3024₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

(31) BOUNDS(n^3, INF)

(32) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)
concat(gen_leaf:cons8_0(n1933_0), gen_leaf:cons8_0(b)) → gen_leaf:cons8_0(+(n1933_0, b)), rt ∈ Ω(1 + n1933₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

(34) BOUNDS(n^3, INF)

(35) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
shuffle(gen_nil:add7_0(n1722_0)) → gen_nil:add7_0(n1722_0), rt ∈ Ω(1 + n1722₀ + n1722₀² + n1722₀³)

(37) BOUNDS(n^3, INF)

(38) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
reverse(gen_nil:add7_0(n1422_0)) → gen_nil:add7_0(n1422_0), rt ∈ Ω(1 + n1422₀ + n1422₀²)

(40) BOUNDS(n^2, INF)

(41) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)
app(gen_nil:add7_0(n485_0), gen_nil:add7_0(b)) → gen_nil:add7_0(+(n485_0, b)), rt ∈ Ω(1 + n485₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(42) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)

(43) BOUNDS(n^1, INF)

(44) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
app :: nil:add → nil:add → nil:add
nil :: nil:add
add :: a → nil:add → nil:add
reverse :: nil:add → nil:add
shuffle :: nil:add → nil:add
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
less_leaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_a3_0 :: a
hole_leaf:cons4_0 :: leaf:cons
hole_false:true5_0 :: false:true
gen_0':s6_0 :: Nat → 0':s
gen_nil:add7_0 :: Nat → nil:add
gen_leaf:cons8_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))
gen_nil:add7_0(0) ⇔ nil
gen_nil:add7_0(+(x, 1)) ⇔ add(hole_a3_0, gen_nil:add7_0(x))
gen_leaf:cons8_0(0) ⇔ leaf
gen_leaf:cons8_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons8_0(x))

No more defined symbols left to analyse.

(45) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s6_0(n10_0), gen_0':s6_0(n10_0)) → gen_0':s6_0(0), rt ∈ Ω(1 + n10₀)

(46) BOUNDS(n^1, INF)