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import Hashmap.Funs
open Primitives
open Result
namespace List
-- TODO: we don't want to use the original List.lookup because it uses BEq
-- TODO: rewrite rule: match x == y with ... -> if x = y then ... else ... ? (actually doesn't work because of sugar)
-- TODO: move?
@[simp]
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
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
@[simp] theorem v_nil (α : Type) : (Nil : List α).v = [] := by rfl
@[simp] theorem v_cons {α : Type} k x (tl: List α) : (Cons k x tl).v = (k, x) :: v tl := by rfl
@[simp]
abbrev lookup {α : Type} (ls: List α) (key: Usize) : Option α :=
ls.v.lookup' key
@[simp]
abbrev len {α : Type} (ls : List α) : Int := ls.v.len
end List
namespace HashMap
abbrev Core.List := _root_.List
namespace List
end List
-- TODO: move
@[simp] theorem neq_imp_nbeq [BEq α] [LawfulBEq α] (x y : α) (heq : ¬ x = y) : ¬ x == y := by simp [*]
@[simp] theorem neq_imp_nbeq_rev [BEq α] [LawfulBEq α] (x y : α) (heq : ¬ x = y) : ¬ y == x := by simp [*]
-- TODO: move
-- TODO: this doesn't work because of sugar
theorem match_lawful_beq [BEq α] [LawfulBEq α] [DecidableEq α] (x y : α) :
(x == y) = (if x = y then true else false) := by
split <;> simp_all
theorem insert_in_list_spec0 {α : Type} (key: Usize) (value: α) (ls: List α) :
∃ b,
insert_in_list α key value ls = ret b ∧
(b ↔ ls.lookup key = none)
:= match ls with
| .Nil => by simp [insert_in_list, insert_in_list_loop]
| .Cons k v tl =>
if h: k = key then -- TODO: The order of k/key matters
by
simp [insert_in_list]
rw [insert_in_list_loop]
simp [h]
else
have ⟨ b, hi ⟩ := insert_in_list_spec0 key value tl
by
exists b
simp [insert_in_list]
rw [insert_in_list_loop] -- TODO: Using simp leads to infinite recursion
simp only [insert_in_list] at hi
simp [*]
-- Variation: use progress
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 ↔ ls.lookup key = none)
:= match ls with
| .Nil => by simp [insert_in_list, insert_in_list_loop]
| .Cons k v tl =>
if h: k = key then -- TODO: The order of k/key matters
by
simp [insert_in_list]
rw [insert_in_list_loop]
simp [h]
else
by
simp only [insert_in_list]
rw [insert_in_list_loop]
conv => rhs; ext; simp [*]
progress keep as heq as ⟨ b, hi ⟩
simp only [insert_in_list] at heq
exists b
-- Variation: use tactics from the beginning
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 ↔ (ls.lookup key = none))
:= by
induction ls
case Nil => simp [insert_in_list, insert_in_list_loop]
case Cons k v tl ih =>
simp only [insert_in_list]
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 [*]
def distinct_keys (ls : Core.List (Usize × α)) := ls.pairwise_rel (λ x y => x.fst ≠ y.fst)
def hash_mod_key (k : Usize) (l : Int) : Int :=
match hash_key k with
| .ret k => k.val % l
| _ => 0
def slot_s_inv_hash (l i : Int) (ls : Core.List (Usize × α)) : Prop :=
ls.allP (λ (k, _) => hash_mod_key k l = i)
@[simp]
def slot_s_inv (l i : Int) (ls : Core.List (Usize × α)) : Prop :=
distinct_keys ls ∧
slot_s_inv_hash l i ls
def slot_t_inv (l i : Int) (s : List α) : Prop := slot_s_inv l i s.v
theorem insert_in_list_back_spec_aux {α : Type} (l : Int) (key: Usize) (value: α) (l0: List α)
(hinv : slot_s_inv_hash l (hash_mod_key key l) l0.v)
(hdk : distinct_keys l0.v) :
∃ l1,
insert_in_list_back α key value l0 = ret l1 ∧
-- We update the binding
l1.lookup key = value ∧
(∀ k, k ≠ key → l1.lookup k = l0.lookup k) ∧
-- We preserve part of the key invariant
slot_s_inv_hash l (hash_mod_key key l) l1.v ∧
-- Reasoning about the length
(match l0.lookup key with
| none => l1.len = l0.len + 1
| some _ => l1.len = l0.len) ∧
-- The keys are distinct
distinct_keys l1.v ∧
-- We need this auxiliary property to prove that the keys distinct properties is preserved
(∀ k, k ≠ key → l0.v.allP (λ (k1, _) => k ≠ k1) → l1.v.allP (λ (k1, _) => k ≠ k1))
:= match l0 with
| .Nil => by checkpoint
simp (config := {contextual := true})
[insert_in_list_back, insert_in_list_loop_back,
List.v, slot_s_inv_hash, distinct_keys, List.pairwise_rel]
| .Cons k v tl0 =>
if h: k = key then by checkpoint
simp [insert_in_list_back]
rw [insert_in_list_loop_back]
simp [h]
split_conjs
. intros; simp [*]
. simp [List.v, slot_s_inv_hash] at *
simp [*]
. simp [*, distinct_keys] at *
apply hdk
. tauto
else by checkpoint
simp [insert_in_list_back]
rw [insert_in_list_loop_back]
simp [h]
have : slot_s_inv_hash l (hash_mod_key key l) (List.v tl0) := by checkpoint
simp_all [List.v, slot_s_inv_hash]
have : distinct_keys (List.v tl0) := by checkpoint
simp [distinct_keys] at hdk
simp [hdk, distinct_keys]
progress keep as heq as ⟨ tl1 .. ⟩
simp only [insert_in_list_back] at heq
have : slot_s_inv_hash l (hash_mod_key key l) (List.v (List.Cons k v tl1)) := by checkpoint
simp [List.v, slot_s_inv_hash] at *
simp [*]
have : distinct_keys ((k, v) :: List.v tl1) := by checkpoint
simp [distinct_keys] at *
simp [*]
-- TODO: canonize addition by default?
simp_all [Int.add_assoc, Int.add_comm, Int.add_left_comm]
end HashMap
end hashmap
|
