da0d309d65
Write `mergeSort xs ys cmp` to provide an explicit comparator, or otherwise `mergeSort xs ys` falls back to `LE` and `DecidablePred` via an autoparam.
50 lines
1.7 KiB
Text
50 lines
1.7 KiB
Text
def List.insert' (p : α → α → Bool) (a : α) (bs : List α) : List α :=
|
||
match bs with
|
||
| [] => [a]
|
||
| b :: bs' => if p a b then a :: bs else b :: bs'.insert' p a
|
||
|
||
def List.merge' (p : α → α → Bool) (as bs : List α) : List α :=
|
||
match as with
|
||
| [] => bs
|
||
| a :: as' => insert' p a (merge' p as' bs)
|
||
|
||
def List.split (as : List α) : List α × List α :=
|
||
match as with
|
||
| [] => ([], [])
|
||
| [a] => ([a], [])
|
||
| a :: b :: as =>
|
||
let (as, bs) := split as
|
||
(a :: as, b :: bs)
|
||
|
||
@[simp] def List.atLeast2 (as : List α) : Bool :=
|
||
match as with
|
||
| [] => false
|
||
| [_] => false
|
||
| _::_::_ => true
|
||
|
||
theorem List.length_split_of_atLeast2 {as : List α} (h : as.atLeast2) : as.split.1.length < as.length ∧ as.split.2.length < as.length := by
|
||
match as with
|
||
| [] => simp at h
|
||
| [_] => simp at h
|
||
| [_, _] => simp (config := { decide := true }) [split]
|
||
| [_, _, _] => simp (config := { decide := true }) [split]
|
||
| a::b::c::d::as =>
|
||
-- TODO: simplify using linear arith and more automation
|
||
have : (c::d::as).atLeast2 := by simp_arith
|
||
have ih := length_split_of_atLeast2 this
|
||
simp_arith [split] at ih |-
|
||
have ⟨ih₁, ih₂⟩ := ih
|
||
exact ⟨Nat.le_trans ih₁ (by simp_arith), Nat.le_trans ih₂ (by simp_arith)⟩
|
||
|
||
def List.mergeSort' (p : α → α → Bool) (as : List α) : List α :=
|
||
if h : as.atLeast2 then
|
||
match he:as.split with
|
||
| (as', bs') =>
|
||
-- TODO: simplify using more automation
|
||
have ⟨h₁, h₂⟩ := length_split_of_atLeast2 h
|
||
have : as'.length < as.length := by simp [he] at h₁; assumption
|
||
have : bs'.length < as.length := by simp [he] at h₂; assumption
|
||
merge' p (mergeSort' p as') (mergeSort' p bs')
|
||
else
|
||
as
|
||
termination_by as.length
|