(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
shuffle(nil) → nil
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) DecreasingLoopProof (EQUIVALENT transformation)

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

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
shuffle(nil) → nil
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

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:

(6) 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
reverse(nil) → nil
shuffle(nil) → nil
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

Infered types.

(8) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

(9) 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

(10) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_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

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

Proved the following rewrite lemma:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)

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

Induction Step:
minus(gen_0':s5_0(+(n9_0, 1)), gen_0':s5_0(+(n9_0, 1))) →RΩ(1)
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) →IH
gen_0':s5_0(0)

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

(13) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_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

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)

Induction Base:

Induction Step:

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

(18) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_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

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)

Induction Base:
nil

Induction Step:

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

(21) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_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

(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

Induction Base:
nil

Induction Step:

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

(24) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_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

(25) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
concat(gen_leaf:cons7_0(n2005_0), gen_leaf:cons7_0(b)) → gen_leaf:cons7_0(+(n2005_0, b)), rt ∈ Ω(1 + n20050)

Induction Base:
concat(gen_leaf:cons7_0(0), gen_leaf:cons7_0(b)) →RΩ(1)
gen_leaf:cons7_0(b)

Induction Step:
concat(gen_leaf:cons7_0(+(n2005_0, 1)), gen_leaf:cons7_0(b)) →RΩ(1)
cons(leaf, concat(gen_leaf:cons7_0(n2005_0), gen_leaf:cons7_0(b))) →IH
cons(leaf, gen_leaf:cons7_0(+(b, c2006_0)))

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

(27) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)
concat(gen_leaf:cons7_0(n2005_0), gen_leaf:cons7_0(b)) → gen_leaf:cons7_0(+(n2005_0, b)), rt ∈ Ω(1 + n20050)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

The following defined symbols remain to be analysed:
less_leaves

(28) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
less_leaves(gen_leaf:cons7_0(n3096_0), gen_leaf:cons7_0(n3096_0)) → false, rt ∈ Ω(1 + n30960)

Induction Base:
less_leaves(gen_leaf:cons7_0(0), gen_leaf:cons7_0(0)) →RΩ(1)
false

Induction Step:
less_leaves(gen_leaf:cons7_0(+(n3096_0, 1)), gen_leaf:cons7_0(+(n3096_0, 1))) →RΩ(1)
less_leaves(concat(leaf, gen_leaf:cons7_0(n3096_0)), concat(leaf, gen_leaf:cons7_0(n3096_0))) →LΩ(1)
less_leaves(gen_leaf:cons7_0(+(0, n3096_0)), concat(leaf, gen_leaf:cons7_0(n3096_0))) →LΩ(1)
less_leaves(gen_leaf:cons7_0(n3096_0), gen_leaf:cons7_0(+(0, n3096_0))) →IH
false

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

(30) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)
concat(gen_leaf:cons7_0(n2005_0), gen_leaf:cons7_0(b)) → gen_leaf:cons7_0(+(n2005_0, b)), rt ∈ Ω(1 + n20050)
less_leaves(gen_leaf:cons7_0(n3096_0), gen_leaf:cons7_0(n3096_0)) → false, rt ∈ Ω(1 + n30960)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

(33) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)
concat(gen_leaf:cons7_0(n2005_0), gen_leaf:cons7_0(b)) → gen_leaf:cons7_0(+(n2005_0, b)), rt ∈ Ω(1 + n20050)
less_leaves(gen_leaf:cons7_0(n3096_0), gen_leaf:cons7_0(n3096_0)) → false, rt ∈ Ω(1 + n30960)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

(36) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)
concat(gen_leaf:cons7_0(n2005_0), gen_leaf:cons7_0(b)) → gen_leaf:cons7_0(+(n2005_0, b)), rt ∈ Ω(1 + n20050)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

(39) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

No more defined symbols left to analyse.

(40) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuffle(gen_nil:add6_0(n1804_0)) → gen_nil:add6_0(n1804_0), rt ∈ Ω(1 + n18040 + n180402 + n180403)

(42) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

No more defined symbols left to analyse.

(43) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
reverse(gen_nil:add6_0(n1522_0)) → gen_nil:add6_0(n1522_0), rt ∈ Ω(1 + n15220 + n152202)

(45) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)
app(gen_nil:add6_0(n603_0), gen_nil:add6_0(b)) → gen_nil:add6_0(+(n603_0, b)), rt ∈ Ω(1 + n6030)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_leaf:cons7_0(0) ⇔ leaf
gen_leaf:cons7_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons7_0(x))

No more defined symbols left to analyse.

(46) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)

(48) 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
reverse(nil) → nil
shuffle(nil) → nil
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
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_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat → 0':s
gen_leaf:cons7_0 :: Nat → leaf:cons

Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) → gen_0':s5_0(0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))