summaryrefslogtreecommitdiff
path: root/tests/lean/Hashmap/Properties.lean
blob: 6bc821d39c5be8916b83337b4cb3f08c73143776 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
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)

-- The proof below is a bit expensive, so we need to increase the maximum number
-- of heart beats
set_option maxHeartbeats 400000

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
   intro
   simp_all [inv]
  progress as  hash_mod, hhm 
  have _ : 0  hash_mod.val := by 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]
  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