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|
import Hashmap.Funs
open Primitives
open Result
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
abbrev Core.List := _root_.List
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 ↔ 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
have ⟨ b, hi ⟩ := insert_in_list_spec0 key value tl
by
exists b
simp [insert_in_list, List.lookup]
rw [insert_in_list_loop] -- TODO: Using simp leads to infinite recursion
simp [h]
simp [insert_in_list] at hi
exact hi
-- 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 ↔ 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 keep as heq as ⟨ b hi ⟩
simp only [insert_in_list] at heq
exists b
simp only [heq, hi]
simp [*, List.lookup]
-- 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 ↔ (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 [*]
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 ∧
-- We update the binding
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]
progress keep as heq as ⟨ tl hp1 hp2 ⟩
simp [insert_in_list_back] at heq
simp (config := {contextual := true}) [*, List.lookup]
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
@[pspec]
theorem insert_in_list_back_spec1 {α : Type} (l : Int) (key: Usize) (value: α) (l0: List α)
(hinv : slot_s_inv_hash l (hash_mod_key key l) l0.v) :
∃ l1,
insert_in_list_back α key value l0 = ret l1 ∧
-- We update the binding
List.lookup l1.v key = value ∧
(∀ k, k ≠ key → List.lookup l1.v k = List.lookup l0.v k) ∧
-- We preserve part of the key invariant
slot_s_inv_hash l (hash_mod_key key l) l1.v
:= match l0 with
| .Nil => by
simp [insert_in_list_back, insert_in_list_loop_back, List.lookup, List.v, slot_s_inv_hash]
tauto
| .Cons k v tl0 =>
if h: k = key then
by
simp [insert_in_list_back, List.lookup]
rw [insert_in_list_loop_back]
simp [h, List.lookup]
constructor
. intros; simp [*]
. simp [List.v, slot_s_inv_hash] at *
simp [*]
else
by
simp [insert_in_list_back, List.lookup]
rw [insert_in_list_loop_back]
simp [h, List.lookup]
have : slot_s_inv_hash l (hash_mod_key key l) (List.v tl0) := by
simp_all [List.v, slot_s_inv_hash]
progress keep as heq as ⟨ tl1 hp1 hp2 hp3 ⟩
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
simp [List.v, slot_s_inv_hash] at *
simp [*]
simp (config := {contextual := true}) [*, List.lookup]
end HashMap
end hashmap
|
