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))
Renamed function symbols to avoid clashes with predefined symbol.
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))
Sliced the following arguments:
add'/0
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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))
Infered types.
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
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'
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(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'
Proved the following rewrite lemma:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
Induction Base:
minus'(_gen_0':s'5(0), _gen_0':s'5(0)) →RΩ(1)
_gen_0':s'5(0)
Induction Step:
minus'(_gen_0':s'5(+(_$n10, 1)), _gen_0':s'5(+(_$n10, 1))) →RΩ(1)
minus'(_gen_0':s'5(_$n10), _gen_0':s'5(_$n10)) →IH
_gen_0':s'5(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(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'
Could not prove a rewrite lemma for the defined symbol quot'.
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(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'
Proved the following rewrite lemma:
app'(_gen_nil':add'6(_n1172), _gen_nil':add'6(b)) → _gen_nil':add'6(+(_n1172, b)), rt ∈ Ω(1 + n1172)
Induction Base:
app'(_gen_nil':add'6(0), _gen_nil':add'6(b)) →RΩ(1)
_gen_nil':add'6(b)
Induction Step:
app'(_gen_nil':add'6(+(_$n1173, 1)), _gen_nil':add'6(_b1431)) →RΩ(1)
add'(app'(_gen_nil':add'6(_$n1173), _gen_nil':add'6(_b1431))) →IH
add'(_gen_nil':add'6(+(_$n1173, _b1431)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
app'(_gen_nil':add'6(_n1172), _gen_nil':add'6(b)) → _gen_nil':add'6(+(_n1172, b)), rt ∈ Ω(1 + n1172)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(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'
Proved the following rewrite lemma:
reverse'(_gen_nil':add'6(_n2235)) → _gen_nil':add'6(_n2235), rt ∈ Ω(1 + n2235 + n22352)
Induction Base:
reverse'(_gen_nil':add'6(0)) →RΩ(1)
nil'
Induction Step:
reverse'(_gen_nil':add'6(+(_$n2236, 1))) →RΩ(1)
app'(reverse'(_gen_nil':add'6(_$n2236)), add'(nil')) →IH
app'(_gen_nil':add'6(_$n2236), add'(nil')) →LΩ(1 + $n2236)
_gen_nil':add'6(+(_$n2236, +(0, 1)))
We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
app'(_gen_nil':add'6(_n1172), _gen_nil':add'6(b)) → _gen_nil':add'6(+(_n1172, b)), rt ∈ Ω(1 + n1172)
reverse'(_gen_nil':add'6(_n2235)) → _gen_nil':add'6(_n2235), rt ∈ Ω(1 + n2235 + n22352)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(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'
Proved the following rewrite lemma:
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)
Induction Base:
shuffle'(_gen_nil':add'6(0)) →RΩ(1)
nil'
Induction Step:
shuffle'(_gen_nil':add'6(+(_$n3045, 1))) →RΩ(1)
add'(shuffle'(reverse'(_gen_nil':add'6(_$n3045)))) →LΩ(1 + $n3045 + $n30452)
add'(shuffle'(_gen_nil':add'6(_$n3045))) →IH
add'(_gen_nil':add'6(_$n3045))
We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
app'(_gen_nil':add'6(_n1172), _gen_nil':add'6(b)) → _gen_nil':add'6(+(_n1172, b)), rt ∈ Ω(1 + n1172)
reverse'(_gen_nil':add'6(_n2235)) → _gen_nil':add'6(_n2235), rt ∈ Ω(1 + n2235 + n22352)
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))
The following defined symbols remain to be analysed:
concat', less_leaves'
They will be analysed ascendingly in the following order:
concat' < less_leaves'
Proved the following rewrite lemma:
concat'(_gen_leaf':cons'7(_n3755), _gen_leaf':cons'7(b)) → _gen_leaf':cons'7(+(_n3755, b)), rt ∈ Ω(1 + n3755)
Induction Base:
concat'(_gen_leaf':cons'7(0), _gen_leaf':cons'7(b)) →RΩ(1)
_gen_leaf':cons'7(b)
Induction Step:
concat'(_gen_leaf':cons'7(+(_$n3756, 1)), _gen_leaf':cons'7(_b4076)) →RΩ(1)
cons'(leaf', concat'(_gen_leaf':cons'7(_$n3756), _gen_leaf':cons'7(_b4076))) →IH
cons'(leaf', _gen_leaf':cons'7(+(_$n3756, _b4076)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
app'(_gen_nil':add'6(_n1172), _gen_nil':add'6(b)) → _gen_nil':add'6(+(_n1172, b)), rt ∈ Ω(1 + n1172)
reverse'(_gen_nil':add'6(_n2235)) → _gen_nil':add'6(_n2235), rt ∈ Ω(1 + n2235 + n22352)
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)
concat'(_gen_leaf':cons'7(_n3755), _gen_leaf':cons'7(b)) → _gen_leaf':cons'7(+(_n3755, b)), rt ∈ Ω(1 + n3755)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))
The following defined symbols remain to be analysed:
less_leaves'
Proved the following rewrite lemma:
less_leaves'(_gen_leaf':cons'7(_n5027), _gen_leaf':cons'7(_n5027)) → false', rt ∈ Ω(1 + n5027)
Induction Base:
less_leaves'(_gen_leaf':cons'7(0), _gen_leaf':cons'7(0)) →RΩ(1)
false'
Induction Step:
less_leaves'(_gen_leaf':cons'7(+(_$n5028, 1)), _gen_leaf':cons'7(+(_$n5028, 1))) →RΩ(1)
less_leaves'(concat'(leaf', _gen_leaf':cons'7(_$n5028)), concat'(leaf', _gen_leaf':cons'7(_$n5028))) →LΩ(1)
less_leaves'(_gen_leaf':cons'7(+(0, _$n5028)), concat'(leaf', _gen_leaf':cons'7(_$n5028))) →LΩ(1)
less_leaves'(_gen_leaf':cons'7(_$n5028), _gen_leaf':cons'7(+(0, _$n5028))) →IH
false'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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'(x), y) → add'(app'(x, y))
reverse'(nil') → nil'
reverse'(add'(x)) → app'(reverse'(x), add'(nil'))
shuffle'(nil') → nil'
shuffle'(add'(x)) → add'(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' :: 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':s'1 :: 0':s'
_hole_nil':add'2 :: nil':add'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_nil':add'6 :: Nat → nil':add'
_gen_leaf':cons'7 :: Nat → leaf':cons'
Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
app'(_gen_nil':add'6(_n1172), _gen_nil':add'6(b)) → _gen_nil':add'6(+(_n1172, b)), rt ∈ Ω(1 + n1172)
reverse'(_gen_nil':add'6(_n2235)) → _gen_nil':add'6(_n2235), rt ∈ Ω(1 + n2235 + n22352)
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)
concat'(_gen_leaf':cons'7(_n3755), _gen_leaf':cons'7(b)) → _gen_leaf':cons'7(+(_n3755, b)), rt ∈ Ω(1 + n3755)
less_leaves'(_gen_leaf':cons'7(_n5027), _gen_leaf':cons'7(_n5027)) → false', rt ∈ Ω(1 + n5027)
Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_nil':add'6(0) ⇔ nil'
_gen_nil':add'6(+(x, 1)) ⇔ add'(_gen_nil':add'6(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))
No more defined symbols left to analyse.
The lowerbound Ω(n3) was proven with the following lemma:
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)