patience.mlw 27.4 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18


(** {1 The Patience Solitaire Game}

Problem 1 from the {h <a href="http://vscomp.org/">Verified Software
Competition 2014</a>}

Patience Solitaire is played by taking cards one-by-one from a deck of
cards and arranging them face up in a sequence of stacks arranged from
left to right as follows. The very first card from the deck is kept
face up to form a singleton stack. Each subsequent card is placed on
the leftmost stack where its card value is no greater than the topmost
card on that stack. If there is no such stack, then a new stack is
started to right of the other stacks. We can do this with positive
numbers instead of cards. If the input sequence is 9, 7, 10, 9, 5, 4,
and 10, then the stacks develop as

{h <pre>}
19 20 21 22 23 24 25
<[[9]]>
<[[7, 9]]>
<[[7, 9]], [[10]]>
<[[7, 9]], [[9, 10]]>
<[[5, 7, 9]], [[9, 10]]>
<[[4, 5, 7, 9]], [[9, 10]]>
<[[4, 5, 7, 9]], [[9, 10]], [[10]]>
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
{h </pre>}

Verify the claim is that the number of stacks at the end of the game
is the length of the longest (strictly) increasing subsequence in the
input sequence.


*)



(** {2 Preliminary: pigeon-hole lemma} *)


module PigeonHole

(** The Why standard library provides a lemma
43 44
    `map.MapInjection.injective_surjective` stating that a map from
    `(0..n-1)` to `(0..n-1)` that is an injection is also a
45 46 47 48 49 50 51 52
    surjection.

    This is more or less equivalent to the pigeon-hole lemma. However, we need such a lemma more generally on functions instead of maps.

    Thus we restate the pigeon-hole lemma here. Proof is left as an exercise.

*)

53
  use int.Int
54 55 56

  predicate range (f: int -> int) (n: int) (m:int) =
    forall i: int. 0 <= i < n -> 0 <= f i < m
57 58
  (** `range f n m` is true when `f` maps the domain
      `(0..n-1)` into `(0..m-1)` *)
59 60 61

  predicate injective (f: int -> int) (n: int) (m:int) =
    forall i j: int. 0 <= i < j < n -> f i <> f j
62 63
  (** `injective f n m` is true when `f` is an injection
      from `(0..n-1)` to `(0..m-1)` *)
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80




(*
  lemma pigeon_hole2:
    forall n m:int, f: int -> int.
      range f n m /\ n > m >= 0 ->
        not (injective f n m)
*)




  exception Found

  function shift (f: int -> int) (i:int) : int -> int =
81
    fun k -> if k < i then f k else f (k+1)
82 83 84 85 86 87

  let rec lemma pigeon_hole (n m:int) (f: int -> int)
    requires { range f n m }
    requires { n > m >= 0 }
    variant  { m }
    ensures  { not (injective f n m) }
88
  =
89 90
      for i = 0 to n-1 do
        invariant { forall k. 0 <= k < i -> f k <> m-1 }
91
        if f i = m-1 then begin
92 93 94
          (* we have found index i such that f i = m-1 *)
          for j = i+1 to n-1 do
            invariant { forall k. i < k < j -> f k <> m-1 }
95 96
          (* we know that f i = f j = m-1 hence we are done *)
            if f j = m-1 then return
97 98
          done;
          (* we know that for all k <> i, f k <> m-1 *)
99
          let g = shift f i in
100 101
          assert { range g (n-1) (m-1) };
          pigeon_hole (n-1) (m-1) g;
102 103
          return
        end
104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
      done;
      (* we know that for all k, f k <> m-1 *)
      assert { range f n (m-1) };
      pigeon_hole n (m-1) f

end




(** {2 Patience idiomatic code} *)


module PatienceCode

119 120 121
  use int.Int
  use list.List
  use list.RevAppend
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

  (** this code was the one written initially, without any
      specification, except for termination, ans unreachability
      of the 'absurd' branch'.

      It can be tested, see below. *)

  type card = int

  (** stacks are well-formed if they are non-empty *)
  predicate wf_stacks (stacks: list (list card)) =
    match stacks with
    | Nil -> true
    | Cons Nil _ -> false
    | Cons (Cons _ _) rem -> wf_stacks rem
    end

  (** concatenation of well-formed stacks is well-formed *)
  let rec lemma wf_rev_append_stacks (s1 s2: list (list int))
    requires { wf_stacks s1 }
    requires { wf_stacks s2 }
    variant { s1 }
    ensures { wf_stacks (rev_append s1 s2) }
  = match s1 with
    | Nil -> ()
    | Cons Nil _ -> absurd
    | Cons s rem -> wf_rev_append_stacks rem (Cons s s2)
    end

151 152 153
  (** `push_card c stacks acc` pushes card `c` on stacks `stacks`,
      assuming `acc` is an accumulator (in reverse order) of stacks
      where `c` could not be pushed.
154 155 156 157 158 159 160 161 162 163
  *)
  let rec push_card (c:card) (stacks : list (list card))
     (acc : list (list card)) : list (list card)
    requires { wf_stacks stacks }
    requires { wf_stacks acc }
    variant  { stacks }
    ensures  { wf_stacks result }
  =
    match stacks with
    | Nil ->
164
      (* we put card `c` in a new stack *)
165 166 167
      rev_append (Cons (Cons c Nil) acc) Nil
    | Cons stack remaining_stacks ->
        match stack with
168
        | Nil -> absurd (* because `wf_stacks stacks` *)
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
        | Cons c' _ ->
           if c <= c' then
             (* card is placed on the leftmost stack where its card
                value is no greater than the topmost card on that
                stack *)
             rev_append (Cons (Cons c stack) acc) remaining_stacks
           else
             (* try next stack *)
             push_card c remaining_stacks (Cons stack acc)
        end
     end

  let rec play_cards (input: list card) (stacks: list (list card))
    : list (list card)
    requires { wf_stacks stacks }
    variant { input }
    ensures  { wf_stacks result }
  =
    match input with
    | Nil -> stacks
    | Cons c rem ->
        let stacks' = push_card c stacks Nil in
        play_cards rem stacks'
    end


  let play_game (input: list card) : list (list card) =
    play_cards input Nil


199
  (** test, can be run using `why3 patience.mlw --exec PatienceCode.test` *)
200 201 202 203 204 205 206 207 208 209 210 211 212
  let test () =
    (** the list given in the problem description
       9, 7, 10, 9, 5, 4, and 10 *)
    play_game
      (Cons 9 (Cons 7 (Cons 10 (Cons 9 (Cons 5 (Cons 4 (Cons 10 Nil)))))))

end


(** {2 Abstract version of Patience game} *)

module PatienceAbstract

213
  use int.Int
214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230

(** To specify the expected property of the Patience game, we first
    provide an abstract version, working on a abstract state that
    includes a lot of information regarding the positions of the cards
    in the stack and so on.

    This abstract state should then be including in the real code as a
    ghost state, with a gluing invariant that matches the ghost state
    and the concrete stacks of cards.

*)


  type card = int

  (** {3 Abstract state} *)

231 232
  use map.Map
  use map.Const
233 234

  type state = {
235 236 237
    ghost mutable num_stacks : int;
    (** number of stacks built so far *)
    ghost mutable num_elts : int;
238
    (** number of cards already seen *)
239
    ghost mutable values : map int card;
240
    (** cards values seen, indexed in the order they have been seen,
241
        from `0` to `num_elts-1` *)
242
    ghost mutable stack_sizes : map int int;
243
    (** sizes of these stacks, numbered from `0` to `num_stacks - 1` *)
244
    ghost mutable stacks : map int (map int int);
245
    (** indexes of the cards in respective stacks *)
246
    ghost mutable positions : map int (int,int);
247 248
    (** table that given a card index, provides its position, i.e. in
        which stack it is and at which height *)
249
    ghost mutable preds : map int int;
250
    (** predecessors of cards, i.e. for each card index `i`, `preds[i]`
251
        provides an index of a card in the stack on the immediate left,
252
        whose value is smaller. Defaults to `-1` if the card is on the
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269
        leftmost stack. *)
  }

(** {3 Invariants on the abstract state} *)

  predicate inv (s:state) =
     0 <= s.num_stacks <= s.num_elts
     (** the number of stacks is less or equal the number of cards *)
  /\ (s.num_elts > 0 -> s.num_stacks > 0)
     (** when there is at least one card, there is at least one stack *)
  /\ (forall i. 0 <= i < s.num_stacks ->
         s.stack_sizes[i] >= 1
         (** stacks are non-empty *)
      /\ forall j. 0 <= j < s.stack_sizes[i] ->
           0 <= s.stacks[i][j] < s.num_elts)
         (** contents of stacks are valid card indexes *)
  /\ (forall i. 0 <= i < s.num_elts ->
270
       let is,ip = s.positions[i] in
271 272 273 274 275
       0 <= is < s.num_stacks &&
       let st = s.stacks[is] in
         0 <= ip < s.stack_sizes[is] &&
         st[ip] = i)
     (** the position table of cards is correct, i.e. when
276 277
        `(is,ip) = s.positions[i]` then card `i` indeed
        occurs in stack `is` at height `ip` *)
278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295
  /\ (forall is. 0 <= is < s.num_stacks ->
        forall ip. 0 <= ip < s.stack_sizes[is] ->
        let idx = s.stacks[is][ip] in
        (is,ip) = s.positions[idx])
     (** positions is the proper inverse of stacks *)
  /\ (forall i. 0 <= i < s.num_stacks ->
        let stack_i = s.stacks[i] in
        forall j,k. 0 <= j < k < s.stack_sizes[i] ->
           stack_i[j] < stack_i[k])
     (** in a given stack, indexes are increasing from bottom to top *)
  /\ (forall i. 0 <= i < s.num_stacks ->
        let stack_i = s.stacks[i] in
        forall j,k. 0 <= j <= k < s.stack_sizes[i] ->
           s.values[stack_i[j]] >= s.values[stack_i[k]])
     (** in a given stack, card values are decreasing from bottom to top *)
  /\ (forall i. 0 <= i < s.num_elts ->
       let pred = s.preds[i] in
       -1 <= pred < s.num_elts &&
296
       (** the predecessor is a valid index or `-1` *)
297 298
       pred < i /\
       (** predecessor is always a smaller index *)
299
       let is,_ip = s.positions[i] in
300
       if pred < 0 then is = 0
301
         (** if predecessor is `-1` then `i` is in leftmost stack *)
302 303
       else
         s.values[pred] < s.values[i] /\
304
         (** if predecessor is not `-1`, it denotes a card with smaller value... *)
305 306
         is > 0 &&
         (** ...the card is not on the leftmost stack... *)
307
         let ps,_pp = s.positions[pred] in
308 309 310 311 312 313 314
         ps = is - 1)
         (** ...and predecessor is in the stack on the immediate left *)



  (** {2 Programs} *)

315
  use ref.Ref
316 317
  exception Return int

318
  (** `play_card c i s` pushes the card `c` on state `s` *)
319
  let ghost play_card (c:card) (s:state) : unit
320 321 322 323 324 325
    requires { inv s }
    writes   { s }
    ensures  { inv s }
    ensures  { s.num_elts = (old s).num_elts + 1 }
    ensures  { s.values = (old s).values[(old s).num_elts <- c] }
  =
326
    let ghost pred = ref (-1) in
327
  try
328 329 330 331 332 333 334 335
    for i = 0 to s.num_stacks - 1 do
      invariant { if i=0 then !pred = -1 else
        let stack_im1 = s.stacks[i-1] in
        let stack_im1_size = s.stack_sizes[i-1] in
        let top_stack_im1 = stack_im1[stack_im1_size - 1] in
        !pred = top_stack_im1 /\
        c > s.values[!pred]  /\
        0 <= !pred < s.num_elts /\
336
        let ps,_pp = s.positions[!pred] in
337 338 339 340 341
        ps = i - 1
      }
      let stack_i = s.stacks[i] in
      let stack_i_size = s.stack_sizes[i] in
      let top_stack_i = stack_i[stack_i_size - 1] in
342 343
      if c <= s.values[top_stack_i] then raise (Return i);
      assert { 0 <= top_stack_i < s.num_elts };
344
      assert { let is,ip = s.positions[top_stack_i] in
345 346 347 348 349 350
        0 <= is < s.num_stacks &&
        0 <= ip < s.stack_sizes[is] &&
        s.stacks[is][ip] = top_stack_i &&
        is = i /\ ip = stack_i_size - 1
      };
      pred := top_stack_i
351 352 353 354 355 356 357 358 359 360 361
    done;
    (* we add a new stack *)
    let idx = s.num_elts in
    let i = s.num_stacks in
    let stack_i = s.stacks[i] in
    let new_stack_i = stack_i[0 <- idx] in
    s.num_elts <- idx + 1;
    s.values <- s.values[idx <- c];
    s.num_stacks <- s.num_stacks + 1;
    s.stack_sizes <- s.stack_sizes[i <- 1];
    s.stacks <- s.stacks[i <- new_stack_i];
362
    s.positions <- s.positions[idx <- i,0];
363 364 365 366 367 368 369 370 371 372 373 374
    s.preds <- s.preds[idx <- !pred]
  with Return i ->
         let stack_i = s.stacks[i] in
         let stack_i_size = s.stack_sizes[i] in
         (* we put c on top of stack i *)
         let idx = s.num_elts in
         let new_stack_i = stack_i[stack_i_size <- idx] in
         s.num_elts <- idx + 1;
         s.values <- s.values[idx <- c];
         (* s.num_stacks unchanged *)
         s.stack_sizes <- s.stack_sizes[i <- stack_i_size + 1];
         s.stacks <- s.stacks[i <- new_stack_i];
375
         s.positions <- s.positions[idx <- i,stack_i_size];
376 377 378 379 380 381 382
         s.preds <- s.preds[idx <- !pred];
  end





383 384 385
  use list.List
  use list.Length
  use list.NthNoOpt
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



  let rec play_cards (input: list int) (s: state) : unit
    requires { inv s }
    variant  { input }
    writes   { s }
    ensures  { inv s }
    ensures  { s.num_elts = (old s).num_elts + length input }
    ensures  { forall i. 0 <= i < (old s).num_elts ->
                 s.values[i] = (old s).values[i] }
    ensures  { forall i. (old s).num_elts <= i < s.num_elts ->
                 s.values[i] = nth (i - (old s).num_elts) input }
  =
    match input with
    | Nil -> ()
    | Cons c rem -> play_card c s; play_cards rem s
    end







  type seq 'a = { seqlen: int; seqval: map int 'a }

  predicate increasing_subsequence (s:seq int) (l:list int) =
    0 <= s.seqlen <= length l &&
    (* subsequence *)
    ((forall i. 0 <= i < s.seqlen -> 0 <= s.seqval[i] < length l)
    /\ (forall i,j. 0 <= i < j < s.seqlen -> s.seqval[i] < s.seqval[j]))
    (* increasing *)
    && (forall i,j. 0 <= i < j < s.seqlen ->
          nth s.seqval[i] l < nth s.seqval[j] l)






427
  use PigeonHole
428 429 430 431 432 433 434 435








436
  let ghost play_game (input: list int) : state
437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473
    ensures { exists s: seq int.  s.seqlen = result.num_stacks /\
        increasing_subsequence s input
      }
    ensures { forall s: seq int.
        increasing_subsequence s input -> s.seqlen <= result.num_stacks
      }
  = let s = {
      num_elts = 0;
      values = Const.const (-1) ;
      num_stacks = 0;
      stack_sizes = Const.const 0;
      stacks = Const.const (Const.const (-1));
      positions = Const.const (-1,-1);
      preds = Const.const (-1);
    }
    in
    play_cards input s;
    (**

      This is ghost code to build an increasing subsequence of maximal length

    *)
    let ns = s.num_stacks in
    if ns = 0 then
      begin
        assert { input = Nil };
        let seq = { seqlen = 0 ; seqval = Const.const (-1) } in
        assert { increasing_subsequence seq input };
        s
      end
    else
    let last_stack = s.stacks[ns-1] in
    let idx = ref (last_stack[s.stack_sizes[ns-1]-1]) in
    let seq = ref (Const.const (-1)) in
    for i = ns-1 downto 0 do
       invariant { -1 <= !idx < s.num_elts }
       invariant { i >= 0 -> !idx >= 0 &&
474
         let is,_ = s.positions[!idx] in is = i }
475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496
       invariant { i+1 < ns -> !idx < !seq[i+1] }
       invariant { 0 <= i < ns-1 -> s.values[!idx] < s.values[!seq[i+1]] }
       invariant { forall j. i < j < ns -> 0 <= !seq[j] < s.num_elts }
       invariant { forall j,k. i < j < k < ns -> !seq[j] < !seq[k] }
       invariant { forall j,k. i < j < k < ns ->
         s.values[!seq[j]] < s.values[!seq[k]]
       }
       seq := !seq[i <- !idx];
       idx := s.preds[!idx];
    done;
    let sigma = { seqlen = ns ; seqval = !seq } in
    assert { forall i. 0 <= i < length input -> nth i input = s.values[i] };
    assert { increasing_subsequence sigma input };
    (**

      These are assertions to prove that no increasing subsequence of
      length larger than the number of stacks may exists

    *)
    assert {  (* non-injectivity *)
      forall sigma: seq int.
        increasing_subsequence sigma input /\ sigma.seqlen > s.num_stacks ->
497
        let f = fun i ->
498
          let si = sigma.seqval[i] in
499
          let stack_i,_ = s.positions[si] in
500 501 502 503 504 505 506 507 508 509 510 511
          stack_i
        in range f sigma.seqlen s.num_stacks &&
           not (injective f sigma.seqlen s.num_stacks)

    };
    assert {  (* non-injectivity *)
      forall sigma: seq int.
        increasing_subsequence sigma input /\ sigma.seqlen > s.num_stacks ->
        exists i,j.
          0 <= i < j < sigma.seqlen &&
          let si = sigma.seqval[i] in
          let sj = sigma.seqval[j] in
512 513
          let stack_i,_pi = s.positions[si] in
          let stack_j,_pj = s.positions[sj] in
514 515 516 517 518 519 520 521 522
          stack_i = stack_j
    };
    assert { (* contradiction from non-injectivity *)
      forall sigma: seq int.
        increasing_subsequence sigma input /\ sigma.seqlen > s.num_stacks ->
        forall i,j.
          0 <= i < j < sigma.seqlen ->
          let si = sigma.seqval[i] in
          let sj = sigma.seqval[j] in
523 524
          let stack_i,pi = s.positions[si] in
          let stack_j,pj = s.positions[sj] in
525 526 527 528 529
          stack_i = stack_j ->
          si < sj && pi < pj && s.values[si] < s.values[sj]
    };
    s

530
  let ghost test () =
531 532 533 534 535 536 537 538 539 540 541
    (* the list given in the problem description
       9, 7, 10, 9, 5, 4, and 10 *)
    play_game
      (Cons 9 (Cons 7 (Cons 10 (Cons 9 (Cons 5 (Cons 4 (Cons 10 Nil)))))))

end

(** {2 Gluing abstract version with the original idiomatic code} *)

module PatienceFull

542 543
  use int.Int
  use PatienceAbstract
544 545 546 547


(** glue between the ghost state and the stacks of cards *)

548 549 550 551
  use list.List
  use list.Length
  use list.NthNoOpt
  use map.Map
552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568

  predicate glue_stack (s:state) (i:int) (st:list card) =
      length st = s.stack_sizes[i] /\
      let stack_i = s.stacks[i] in
      forall j. 0 <= i < length st ->
        nth j st = s.values[stack_i[j]]

  predicate glue (s:state) (st:list (list card)) =
    length st = s.num_stacks /\
    forall i. 0 <= i < length st ->
      glue_stack s i (nth i st)




(** {3 playing a card} *)

569 570
  use list.RevAppend
  use ref.Ref
571 572 573
  exception Return


574
(*** FIXME: not proved
575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598

  let play_card (c:card) (old_stacks : list (list card)) (ghost state:state) : list (list card)
    requires { inv state }
    requires { glue state old_stacks }
    writes   { state }
    ensures  { inv state }
    ensures  { state.num_elts = (old state).num_elts + 1 }
    ensures  { state.values = (old state).values[(old state).num_elts <- c] }
    ensures  { glue state result }
  =
    let acc = ref Nil in
    let rem_stacks = ref old_stacks in
    let ghost pred = ref (-1) in
    let ghost i = ref 0 in
    try
    while !rem_stacks <> Nil do
      invariant { 0 <= !i <= state.num_stacks }
      invariant { if !i = 0 then !pred = -1 else
        let stack_im1 = state.stacks[!i-1] in
        let stack_im1_size = state.stack_sizes[!i-1] in
        let top_stack_im1 = stack_im1[stack_im1_size - 1] in
        !pred = top_stack_im1 /\
        c > state.values[!pred]  /\
        0 <= !pred < state.num_elts /\
599
        let ps,_pp = state.positions[!pred] in
600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628
        ps = !i - 1
      }
      invariant { old_stacks = rev_append !acc !rem_stacks }
      invariant {
        forall j. 0 <= j < !i -> glue_stack state j (nth (!i - j) !acc)
      }
      invariant {
        forall j. !i <= j < state.num_stacks ->
          glue_stack state j (nth (j - !i) !rem_stacks)
      }
      variant { !rem_stacks }
      match !rem_stacks with
      | Nil -> absurd
      | Cons stack remaining_stacks ->
          rem_stacks := remaining_stacks;
          match stack with
          | Nil ->
            assert { glue_stack state !i stack };
            absurd
          | Cons c' _ ->
             if c <= c' then
               begin
                 acc := Cons (Cons c stack) !acc;
                 raise Return;
               end;
             let ghost stack_i = state.stacks[!i] in
             let ghost stack_i_size = state.stack_sizes[!i] in
             let ghost top_stack_i = stack_i[stack_i_size - 1] in
             assert { 0 <= top_stack_i < state.num_elts };
629
             assert { let is,ip = state.positions[top_stack_i] in
630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650
               0 <= is < state.num_stacks &&
               0 <= ip < state.stack_sizes[is] &&
               state.stacks[is][ip] = top_stack_i &&
               is = !i /\ ip = stack_i_size - 1
             };
             i := !i + 1;
             acc := Cons stack !acc;
             pred := top_stack_i
         end
      end
    done;
    (* we add a new stack *)
    let ghost idx = state.num_elts in
    let ghost i = state.num_stacks in
    let ghost stack_i = state.stacks[i] in
    let ghost new_stack_i = stack_i[0 <- idx] in
    state.num_elts <- idx + 1;
    state.values <- state.values[idx <- c];
    state.num_stacks <- state.num_stacks + 1;
    state.stack_sizes <- state.stack_sizes[i <- 1];
    state.stacks <- state.stacks[i <- new_stack_i];
651
    state.positions <- state.positions[idx <- i,0];
652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667
    state.preds <- state.preds[idx <- !pred];
    (* we put card [c] in a new stack *)
    rev_append (Cons (Cons c Nil) !acc) Nil
  with Return ->
         let ghost stack_i = state.stacks[!i] in
         let ghost stack_i_size = state.stack_sizes[!i] in
         let ghost top_stack_i = stack_i[stack_i_size - 1] in
         assert { c <= state.values[top_stack_i] };
         (* we put c on top of stack i *)
         let ghost idx = state.num_elts in
         let ghost new_stack_i = stack_i[stack_i_size <- idx] in
         state.num_elts <- idx + 1;
         state.values <- state.values[idx <- c];
         (* state.num_stacks unchanged *)
         state.stack_sizes <- state.stack_sizes[!i <- stack_i_size + 1];
         state.stacks <- state.stacks[!i <- new_stack_i];
668
         state.positions <- state.positions[idx <- !i,stack_i_size];
669 670 671 672 673 674 675 676 677 678
         state.preds <- state.preds[idx <- !pred];
         (* card is placed on the leftmost stack where its card
            value is no greater than the topmost card on that
            stack *)
         rev_append !acc !rem_stacks
  end

*)


679
(*** a version closer to the original code
680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723
  let play_card (c:card) (old_stacks : list (list card)) (ghost state:state) : list (list card)
    requires { inv state }
    requires { glue state old_stacks }
    writes   { state }
    ensures  { inv state }
    ensures  { state.num_elts = (old state).num_elts + 1 }
    ensures  { state.values = (old state).values[(old state).num_elts <- c] }
    ensures  { glue state result }
  = let i = ref 0 in
    let pred = ref (-1) in
    let rec push_card (c:card) (st : list (list card))
                      (acc : list (list card)) : list (list card)
      requires { old_stacks = rev_append acc st }
      variant { st }
    =
    match st with
    | Nil ->
        (* we put card [c] in a new stack *)
        rev_append (Cons (Cons c Nil) acc) Nil
    | Cons stack remaining_stacks ->
        match stack with
        | Nil ->
          assert { glue_stack state !i stack };
          absurd
        | Cons c' _ ->
           if c <= c' then
             (* card is placed on the leftmost stack where its card
                value is no greater than the topmost card on that
                stack *)
             rev_append (Cons (Cons c stack) acc) remaining_stacks
           else
             (* try next stack *)
             push_card c remaining_stacks (Cons stack acc)
        end
    end
    in
   let new_stacks = push_card c old_stacks Nil in
   let idx = state.num_elts in
   state.num_elts <- idx + 1;
   state.values <- state.values[idx <- c];
   new_stacks
*)


724
(*** {3 playing cards} *)
725 726 727



728
(***
729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758


  let rec play_cards (input: list card) (stacks: list (list card))
    (ghost state:state) : list (list card)
    requires { inv state }
    requires { glue state stacks }
    variant  { input }
    (* writes   { state } *)
    ensures  { inv state }
    ensures  { state.num_elts = (old state).num_elts + length input }
    ensures  { forall i. 0 <= i < (old state).num_elts ->
                 state.values[i] = (old state).values[i] }
    ensures  { forall i. (old state).num_elts <= i < state.num_elts ->
                 state.values[i] = nth (i - (old state).num_elts) input }
    ensures  { glue state result }
  =
    match input with
    | Nil -> stacks
    | Cons c rem ->
        let stacks' = play_card c stacks state in
        play_cards rem stacks' state
    end

*)






759
(*** {3 playing a whole game} *)
760

761
(***
762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777

  type seq 'a = { seqlen: int; seqval: map int 'a }
  (** a sequence is defined by a length and a mapping *)

  (** definition of an increasing sub-sequence of a list of card *)
  predicate increasing_subsequence (sigma:seq int) (l:list card) =
       0 <= sigma.seqlen <= length l
       (** the length of [sigma] is at most the number of cards *)
    && (forall i. 0 <= i < sigma.seqlen -> 0 <= sigma.seqval[i] < length l)
       (** [sigma] maps indexes to valid indexes in the card list *)
    && (forall i,j. 0 <= i < j < sigma.seqlen -> sigma.seqval[i] < sigma.seqval[j])
       (** [sigma] is an increasing sequence of indexes *)
    && (forall i,j. 0 <= i < j < sigma.seqlen ->
          nth sigma.seqval[i] l < nth sigma.seqval[j] l)
       (** the card values denoted by [sigma] are increasing *)

778
  use PigeonHole
779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826

  let play_game (input: list card) : list (list card)
    requires { length input > 0 }
    ensures  { exists sigma: seq int.
                 sigma.seqlen = length result /\
                 increasing_subsequence sigma input
             }
    ensures  { forall sigma: seq int.
                increasing_subsequence sigma input ->
                  sigma.seqlen <= length result
             }
  = let ghost state = {
      num_elts = 0;
      values = Const.const (-1) ;
      num_stacks = 0;
      stack_sizes = Const.const 0;
      stacks = Const.const (Const.const (-1));
      positions = Const.const (-1,-1);
      preds = Const.const (-1);
    }
    in
    let final_stacks = play_cards input Nil state in
    assert { forall i. 0 <= i < length input -> nth i input = state.values[i] };
    (**

      This is ghost code to build an increasing subsequence of maximal length

    *)
    let ghost ns = state.num_stacks in
    let ghost _sigma =
      if ns = 0 then
      begin
        assert { input = Nil };
        absurd
(*
        TODO: if input is empty, we may be able to prove that:
        let sigma = { seqlen = 0 ; seqval = Const.const (-1) } in
        assert { increasing_subsequence sigma input };
        sigma
*)
      end
    else
    let ghost last_stack = state.stacks[ns-1] in
    let ghost idx = ref (last_stack[state.stack_sizes[ns-1]-1]) in
    let ghost seq = ref (Const.const (-1)) in
    for i = ns-1 downto 0 do
       invariant { -1 <= !idx < state.num_elts }
       invariant { i >= 0 -> !idx >= 0 &&
827
         let is,_ = state.positions[!idx] in is = i }
828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848
       invariant { i+1 < ns -> !idx < !seq[i+1] }
       invariant { 0 <= i < ns-1 -> state.values[!idx] < state.values[!seq[i+1]] }
       invariant { forall j. i < j < ns -> 0 <= !seq[j] < state.num_elts }
       invariant { forall j,k. i < j < k < ns -> !seq[j] < !seq[k] }
       invariant { forall j,k. i < j < k < ns ->
         state.values[!seq[j]] < state.values[!seq[k]]
       }
       seq := !seq[i <- !idx];
       idx := state.preds[!idx];
    done;
    let ghost sigma = { seqlen = ns ; seqval = !seq } in
    assert { increasing_subsequence sigma input };
    (**

      These are assertions to prove that no increasing subsequence of
      length larger than the number of stacks may exists

    *)
    assert {  (* non-injectivity *)
      forall sigma: seq int.
        increasing_subsequence sigma input /\ sigma.seqlen > state.num_stacks ->
849
        let f = fun i ->
850
          let si = sigma.seqval[i] in
851
          let stack_i,_ = state.positions[si] in
852 853 854 855 856 857 858 859 860 861 862
          stack_i
        in range f sigma.seqlen state.num_stacks &&
           not (injective f sigma.seqlen state.num_stacks)
    };
    assert {  (* non-injectivity *)
      forall sigma: seq int.
        increasing_subsequence sigma input /\ sigma.seqlen > state.num_stacks ->
        exists i,j.
          0 <= i < j < sigma.seqlen &&
          let si = sigma.seqval[i] in
          let sj = sigma.seqval[j] in
863 864
          let stack_i,_pi = state.positions[si] in
          let stack_j,_pj = state.positions[sj] in
865 866 867 868 869 870 871 872 873
          stack_i = stack_j
    };
    assert { (* contradiction from non-injectivity *)
      forall sigma: seq int.
        increasing_subsequence sigma input /\ sigma.seqlen > state.num_stacks ->
        forall i,j.
          0 <= i < j < sigma.seqlen ->
          let si = sigma.seqval[i] in
          let sj = sigma.seqval[j] in
874 875
          let stack_i,pi = state.positions[si] in
          let stack_j,pj = state.positions[sj] in
876 877 878 879 880 881 882 883 884 885
          stack_i = stack_j ->
          si < sj && pi < pj && state.values[si] < state.values[sj]
    };
    sigma
  in
  final_stacks

*)

end