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import Hashmap.Funs
open Primitives
open Result
namespace List
theorem index_ne
{α : Type u} [Inhabited α] (l: List α) (i: ℤ) (j: ℤ) (x: α) :
0 ≤ i → i < l.len → 0 ≤ j → j < l.len → j ≠ i →
(l.update i x).index j = l.index j
:=
λ _ _ _ _ _ => match l with
| [] => by simp at *
| hd :: tl =>
if h: i = 0 then
have : j ≠ 0 := by scalar_tac
by simp [*]
else if h : j = 0 then
have : i ≠ 0 := by scalar_tac
by simp [*]
else
by
simp [*]
simp at *
apply index_ne <;> scalar_tac
theorem index_eq
{α : Type u} [Inhabited α] (l: List α) (i: ℤ) (x: α) :
0 ≤ i → i < l.len →
(l.update i x).index i = x
:=
fun _ _ => match l with
| [] => by simp at *; exfalso; scalar_tac -- TODO: exfalso needed. Son FIXME
| hd :: tl =>
if h: i = 0 then
by
simp [*]
else
by
simp [*]
simp at *
apply index_eq <;> scalar_tac
def allP {α : Type u} (l : List α) (p: α → Prop) : Prop :=
foldr (fun a r => p a ∧ r) True l
@[simp]
theorem allP_nil {α : Type u} (p: α → Prop) : allP [] p :=
by simp [allP, foldr]
@[simp]
theorem allP_cons {α : Type u} (hd: α) (tl : List α) (p: α → Prop) :
allP (hd :: tl) p ↔ p hd ∧ allP tl p
:= by simp [allP, foldr]
def pairwise_rel
{α : Type u} (rel : α → α → Prop) (l: List α) : Prop
:= match l with
| [] => True
| hd :: tl => allP tl (rel hd) ∧ pairwise_rel rel tl
@[simp]
theorem pairwise_rel_nil {α : Type u} (rel : α → α → Prop) :
pairwise_rel rel []
:= by simp [pairwise_rel]
@[simp]
theorem pairwise_rel_cons {α : Type u} (rel : α → α → Prop) (hd: α) (tl: List α) :
pairwise_rel rel (hd :: tl) ↔ allP tl (rel hd) ∧ pairwise_rel rel tl
:= by simp [pairwise_rel]
end List
namespace hashmap
namespace List
def v {α : Type} (ls: List α) : _root_.List (Usize × α) :=
match ls with
| Nil => []
| Cons k x tl => (k, x) :: v tl
def lookup {α : Type} (ls: _root_.List (Usize × α)) (key: Usize) : Option α :=
match ls with
| [] => none
| (k, x) :: tl => if k = key then some x else lookup tl key
end List
namespace HashMap
@[pspec]
theorem insert_in_list_spec {α : Type} (key: Usize) (value: α) (ls: List α) :
∃ b,
insert_in_list α key value ls = ret b ∧
(b ↔ List.lookup ls.v key = none)
:= match ls with
| .Nil => by simp [insert_in_list, insert_in_list_loop, List.lookup]
| .Cons k v tl =>
if h: k = key then -- TODO: The order of k/key matters
by
simp [insert_in_list, List.lookup]
rw [insert_in_list_loop]
simp [h]
else
by
-- TODO: use progress: detect that this is a recursive call, or give
-- the possibility of specifying an identifier
have ⟨ b, hi ⟩ := insert_in_list_spec key value tl
exists b
simp [insert_in_list, List.lookup]
rw [insert_in_list_loop] -- TODO: Using it in simp leads to infinite recursion
simp [h]
simp [insert_in_list] at hi
exact hi
set_option trace.Progress true
@[pspec]
theorem insert_in_list_spec1 {α : Type} (key: Usize) (value: α) (ls: List α) :
∃ b,
insert_in_list α key value ls = ret b ∧
(b ↔ List.lookup ls.v key = none)
:= match ls with
| .Nil => by simp [insert_in_list, insert_in_list_loop, List.lookup]
| .Cons k v tl =>
if h: k = key then -- TODO: The order of k/key matters
by
simp [insert_in_list, List.lookup]
rw [insert_in_list_loop]
simp [h]
else
by
simp only [insert_in_list]
rw [insert_in_list_loop]
conv => rhs; ext; simp [*]
progress as ⟨ b hi ⟩
-- TODO: use progress: detect that this is a recursive call, or give
-- the possibility of specifying an identifier
have ⟨ b, hi ⟩ := insert_in_list_spec key value tl
exists b
simp [insert_in_list, List.lookup]
rw [insert_in_list_loop] -- TODO: Using it in simp leads to infinite recursion
simp [h]
simp [insert_in_list] at hi
exact hi
@[pspec]
theorem insert_in_list_spec2 {α : Type} (key: Usize) (value: α) (ls: List α) :
∃ b,
insert_in_list α key value ls = ret b ∧
(b ↔ (List.lookup ls.v key = none))
:= by
induction ls
case Nil => simp [insert_in_list, insert_in_list_loop, List.lookup]
case Cons k v tl ih =>
simp only [insert_in_list, List.lookup]
rw [insert_in_list_loop]
simp only
if h: k = key then
simp [h]
else
conv => rhs; ext; left; simp [h] -- TODO: Simplify
simp only [insert_in_list] at ih;
-- TODO: give the possibility of using underscores
progress as ⟨ b h ⟩
simp [*]
@[pspec]
theorem insert_in_list_back_spec {α : Type} (key: Usize) (value: α) (l0: List α) :
∃ l1,
insert_in_list_back α key value l0 = ret l1 ∧
List.lookup l1.v key = value ∧
(∀ k, k ≠ key → List.lookup l1.v k = List.lookup l0.v k)
:= match l0 with
| .Nil => by simp [insert_in_list_back, insert_in_list_loop_back, List.lookup]; tauto
| .Cons k v tl =>
if h: k = key then
by
simp [insert_in_list_back, List.lookup]
rw [insert_in_list_loop_back]
simp [h, List.lookup]
intro k1 h1
simp [*]
else
by
simp [insert_in_list_back, List.lookup]
rw [insert_in_list_loop_back]
simp [h, List.lookup]
have ⟨tl0 , _, _ ⟩ := insert_in_list_back_spec key value tl -- TODO: Use progress
simp [insert_in_list_back] at *
simp [List.lookup, *]
simp (config := {contextual := true}) [*]
end HashMap
end hashmap
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