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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
+
+@[pspec]
+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 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
+
+@[simp]
+theorem hash_mod_key_eq : hash_mod_key k l = k.val % l := by
+ simp [hash_mod_key, hash_key]
+
+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
+
+-- Interpret the hashmap as a list of lists
+def v (hm : HashMap α) : Core.List (Core.List (Usize × α)) :=
+ hm.slots.val.map List.v
+
+-- Interpret the hashmap as an associative list
+def al_v (hm : HashMap α) : Core.List (Usize × α) :=
+ hm.v.flatten
+
+-- TODO: automatic derivation
+instance : Inhabited (List α) where
+ default := .Nil
+
+@[simp]
+def slots_s_inv (s : Core.List (List α)) : Prop :=
+ ∀ (i : Int), 0 ≤ i → i < s.len → slot_t_inv s.len i (s.index i)
+
+def slots_t_inv (s : Vec (List α)) : Prop :=
+ slots_s_inv s.v
+
+@[simp]
+def base_inv (hm : HashMap α) : Prop :=
+ -- [num_entries] correctly tracks the number of entries
+ hm.num_entries.val = hm.al_v.len ∧
+ -- Slots invariant
+ slots_t_inv hm.slots ∧
+ -- The capacity must be > 0 (otherwise we can't resize)
+ 0 < hm.slots.length
+ -- TODO: load computation
+
+def inv (hm : HashMap α) : Prop :=
+ -- Base invariant
+ base_inv hm
+ -- TODO: either the hashmap is not overloaded, or we can't resize it
+
+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 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]
+
+@[pspec]
+theorem insert_in_list_back_spec {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
+ := by
+ progress with insert_in_list_back_spec_aux as ⟨ l1 .. ⟩
+ exists l1
+
+@[simp]
+def slots_t_lookup (s : Core.List (List α)) (k : Usize) : Option α :=
+ let i := hash_mod_key k s.len
+ let slot := s.index i
+ slot.lookup k
+
+def lookup (hm : HashMap α) (k : Usize) : Option α :=
+ slots_t_lookup hm.slots.val k
+
+@[simp]
+abbrev len_s (hm : HashMap α) : Int := hm.al_v.len
+
+-- Remark: α and β must live in the same universe, otherwise the
+-- bind doesn't work
+theorem if_update_eq
+ {α β : Type u} (b : Bool) (y : α) (e : Result α) (f : α → Result β) :
+ (if b then Bind.bind e f else f y) = Bind.bind (if b then e else pure y) f
+ := by
+ split <;> simp [Pure.pure]
+
+-- Small helper
+-- TODO: move, and introduce a better solution with nice syntax
+def mk_opaque {α : Sort u} (x : α) : { y : α // y = x} :=
+ ⟨ x, by simp ⟩
+
+--set_option profiler true
+--set_option profiler.threshold 10
+--set_option trace.profiler true
+
+-- For pretty printing (useful when copy-pasting goals)
+attribute [pp_dot] List.length -- use the dot notation when printing
+set_option pp.coercions false -- do not print coercions with ↑ (this doesn't parse)
+
+theorem insert_no_resize_spec {α : Type} (hm : HashMap α) (key : Usize) (value : α)
+ (hinv : hm.inv) (hnsat : hm.lookup key = none → hm.len_s < Usize.max) :
+ ∃ nhm, hm.insert_no_resize α key value = ret nhm ∧
+ -- We preserve the invariant
+ nhm.inv ∧
+ -- We updated the binding for key
+ nhm.lookup key = some value ∧
+ -- We left the other bindings unchanged
+ (∀ k, ¬ k = key → nhm.lookup k = hm.lookup k) ∧
+ -- Reasoning about the length
+ (match hm.lookup key with
+ | none => nhm.len_s = hm.len_s + 1
+ | some _ => nhm.len_s = hm.len_s) := by
+ rw [insert_no_resize]
+ simp only [hash_key, bind_tc_ret] -- TODO: annoying
+ have _ : (Vec.len (List α) hm.slots).val ≠ 0 := by checkpoint
+ intro
+ simp_all [inv]
+ progress keep _ as ⟨ hash_mod, hhm ⟩
+ have _ : 0 ≤ hash_mod.val := by checkpoint scalar_tac
+ have _ : hash_mod.val < Vec.length hm.slots := by
+ have : 0 < hm.slots.val.len := by
+ simp [inv] at hinv
+ simp [hinv]
+ -- TODO: we want to automate that
+ simp [*, Int.emod_lt_of_pos]
+ -- TODO: change the spec of Vec.index_mut to introduce a let-binding.
+ -- or: make progress introduce the let-binding by itself (this is clearer)
+ progress as ⟨ l, h_leq ⟩
+ -- TODO: make progress use the names written in the goal
+ progress as ⟨ inserted ⟩
+ rw [if_update_eq] -- TODO: necessary because we don't have a join
+ -- TODO: progress to ...
+ have hipost :
+ ∃ i0, (if inserted = true then hm.num_entries + Usize.ofInt 1 else pure hm.num_entries) = ret i0 ∧
+ i0.val = if inserted then hm.num_entries.val + 1 else hm.num_entries.val
+ := by
+ if inserted then
+ simp [*]
+ have hbounds : hm.num_entries.val + (Usize.ofInt 1).val ≤ Usize.max := by
+ simp [lookup] at hnsat
+ simp_all
+ simp [inv] at hinv
+ int_tac
+ -- TODO: progress fails in command line mode with "index out of bounds"
+ -- and I have no idea how to fix this. The error happens after progress
+ -- introduced the new goals. It must be when we exit the "withApp", etc.
+ -- helpers.
+ -- progress as ⟨ z, hp ⟩
+ have ⟨ z, hp ⟩ := Usize.add_spec hbounds
+ simp [hp]
+ else
+ simp [*, Pure.pure]
+ progress as ⟨ i0 ⟩
+ have h_slot : slot_s_inv_hash hm.slots.length (hash_mod_key key hm.slots.length) l.v
+ := by
+ simp [inv] at hinv
+ have h := (hinv.right.left hash_mod.val (by assumption) (by assumption)).right
+ simp [slot_t_inv, hhm] at h
+ simp [h, hhm, h_leq]
+ have hd : distinct_keys l.v := by checkpoint
+ simp [inv, slots_t_inv, slot_t_inv] at hinv
+ have h := hinv.right.left hash_mod.val (by assumption) (by assumption)
+ simp [h, h_leq]
+ -- TODO: hide the variables and only keep the props
+ -- TODO: allow providing terms to progress to instantiate the meta variables
+ -- which are not propositions
+ progress as ⟨ l0, _, _, _, hlen .. ⟩
+ progress keep hv as ⟨ v, h_veq ⟩
+ -- TODO: update progress to automate that
+ -- TODO: later I don't want to inline nhm - we need to control simp: deactivate
+ -- zeta reduction? For now I have to do this peculiar manipulation
+ have ⟨ nhm, nhm_eq ⟩ := @mk_opaque (HashMap α) { num_entries := i0, max_load_factor := hm.max_load_factor, max_load := hm.max_load, slots := v }
+ exists nhm
+ have hupdt : lookup nhm key = some value := by checkpoint
+ simp [lookup, List.lookup] at *
+ simp_all
+ have hlkp : ∀ k, ¬ k = key → nhm.lookup k = hm.lookup k := by
+ simp [lookup, List.lookup] at *
+ intro k hk
+ -- We have to make a case disjunction: either the hashes are different,
+ -- in which case we don't even lookup the same slots, or the hashes
+ -- are the same, in which case we have to reason about what happens
+ -- in one slot
+ let k_hash_mod := k.val % v.val.len
+ have : 0 < hm.slots.val.len := by simp_all [inv]
+ have hvpos : 0 < v.val.len := by simp_all
+ have hvnz: v.val.len ≠ 0 := by
+ simp_all
+ have _ : 0 ≤ k_hash_mod := by
+ -- TODO: we want to automate this
+ simp
+ apply Int.emod_nonneg k.val hvnz
+ have _ : k_hash_mod < Vec.length hm.slots := by
+ -- TODO: we want to automate this
+ simp
+ have h := Int.emod_lt_of_pos k.val hvpos
+ simp_all
+ if h_hm : k_hash_mod = hash_mod.val then
+ simp_all
+ else
+ simp_all
+ have _ :
+ match hm.lookup key with
+ | none => nhm.len_s = hm.len_s + 1
+ | some _ => nhm.len_s = hm.len_s := by checkpoint
+ simp only [lookup, List.lookup, len_s, al_v, HashMap.v, slots_t_lookup] at *
+ -- We have to do a case disjunction
+ simp_all
+ simp [_root_.List.update_map_eq]
+ -- TODO: dependent rewrites
+ have _ : key.val % hm.slots.val.len < (List.map List.v hm.slots.val).len := by
+ simp [*]
+ simp [_root_.List.len_flatten_update_eq, *]
+ split <;>
+ rename_i heq <;>
+ simp [heq] at hlen <;>
+ -- TODO: canonize addition by default? We need a tactic to simplify arithmetic equalities
+ -- with addition and substractions ((ℤ, +) is a group or something - there should exist a tactic
+ -- somewhere in mathlib?)
+ simp [Int.add_assoc, Int.add_comm, Int.add_left_comm] <;>
+ int_tac
+ have hinv : inv nhm := by
+ simp [inv] at *
+ split_conjs
+ . match h: lookup hm key with
+ | none =>
+ simp [h, lookup] at *
+ simp_all
+ | some _ =>
+ simp_all [lookup]
+ . simp [slots_t_inv, slot_t_inv] at *
+ intro i hipos _
+ have _ := hinv.right.left i hipos (by simp_all)
+ simp [hhm, h_veq, nhm_eq] at * -- TODO: annoying, we do that because simp_all fails below
+ -- We need a case disjunction
+ if h_ieq : i = key.val % _root_.List.len hm.slots.val then
+ -- TODO: simp_all fails: "(deterministic) timeout at 'whnf'"
+ -- Also, it is annoying to do this kind
+ -- of rewritings by hand. We could have a different simp
+ -- which safely substitutes variables when we have an
+ -- equality `x = ...` and `x` doesn't appear in the rhs
+ simp [h_ieq] at *
+ simp [*]
+ else
+ simp [*]
+ . -- TODO: simp[*] fails: "(deterministic) timeout at 'whnf'"
+ simp [hinv, h_veq, nhm_eq]
+ simp_all
+
+end HashMap
+
+end hashmap