## VSCOMP 2014, problem 1

Auteurs: Claude Marché

Catégories: Ghost code

Outils: Why3

see also the index (by topic, by tool, by reference, by year)

# The Patience Solitaire Game

Problem 1 from the Verified Software Competition 2014

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

```<[]>
<[[7, 9]]>
<[[7, 9]], []>
<[[7, 9]], [[9, 10]]>
<[[5, 7, 9]], [[9, 10]]>
<[[4, 5, 7, 9]], [[9, 10]]>
<[[4, 5, 7, 9]], [[9, 10]], []>
```

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.

## Preliminary: pigeon-hole lemma

```module PigeonHole

```

The Why standard library provides a lemma `map.MapInjection.injective_surjective` stating that a map from `(0..n-1)` to `(0..n-1)` that is an injection is also a 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.

```  use int.Int

predicate range (f: int -> int) (n: int) (m:int) =
forall i: int. 0 <= i < n -> 0 <= f i < m
```

`range f n m` is true when `f` maps the domain `(0..n-1)` into `(0..m-1)`

```  predicate injective (f: int -> int) (n: int) (m:int) =
forall i j: int. 0 <= i < j < n -> f i <> f j
```

`injective f n m` is true when `f` is an injection from `(0..n-1)` to `(0..m-1)`

```(*
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 =
fun k -> if k < i then f k else f (k+1)

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) }
=
for i = 0 to n-1 do
invariant { forall k. 0 <= k < i -> f k <> m-1 }
if f i = m-1 then begin
(* 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 }
(* we know that f i = f j = m-1 hence we are done *)
if f j = m-1 then return
done;
(* we know that for all k <> i, f k <> m-1 *)
let g = shift f i in
assert { range g (n-1) (m-1) };
pigeon_hole (n-1) (m-1) g;
return
end
done;
(* we know that for all k, f k <> m-1 *)
assert { range f n (m-1) };
pigeon_hole n (m-1) f

end

```

## Patience idiomatic code

```module PatienceCode

use int.Int
use list.List
use list.RevAppend

```

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

predicate wf_stacks (stacks: list (list card)) =
match stacks with
| Nil -> true
| Cons Nil _ -> false
| Cons (Cons _ _) rem -> wf_stacks rem
end
```

stacks are well-formed if they are non-empty

```  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
```

concatenation of well-formed stacks is well-formed

```  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 ->
(* we put card `c` in a new stack *)
rev_append (Cons (Cons c Nil) acc) Nil
| Cons stack remaining_stacks ->
match stack with
| Nil -> absurd (* because `wf_stacks stacks` *)
| 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
```

`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.

```  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

let test () =
```

test, can be run using `why3 patience.mlw --exec PatienceCode.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

```

## Abstract version of Patience game

```module PatienceAbstract

use int.Int

```

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

```

### Abstract state

```  use map.Map
use map.Const

type state = {
ghost mutable num_stacks : int;
```

number of stacks built so far

```    ghost mutable num_elts : int;
```

number of cards already seen

```    ghost mutable values : map int card;
```

cards values seen, indexed in the order they have been seen, from `0` to `num_elts-1`

```    ghost mutable stack_sizes : map int int;
```

sizes of these stacks, numbered from `0` to `num_stacks - 1`

```    ghost mutable stacks : map int (map int int);
```

indexes of the cards in respective stacks

```    ghost mutable positions : map int (int,int);
```

table that given a card index, provides its position, i.e. in which stack it is and at which height

```    ghost mutable preds : map int int;
```

predecessors of cards, i.e. for each card index `i`, `preds[i]` provides an index of a card in the stack on the immediate left, whose value is smaller. Defaults to `-1` if the card is on the leftmost stack.

```  }

```

### 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 ->
let is,ip = s.positions[i] in
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 `(is,ip) = s.positions[i]` then card `i` indeed occurs in stack `is` at height `ip`

```  /\ (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 &&
```

the predecessor is a valid index or `-1`

```       pred < i /\
```

predecessor is always a smaller index

```       let is,_ip = s.positions[i] in
if pred < 0 then is = 0
```

if predecessor is `-1` then `i` is in leftmost stack

```       else
s.values[pred] < s.values[i] /\
```

if predecessor is not `-1`, it denotes a card with smaller value...

```         is > 0 &&
```

...the card is not on the leftmost stack...

```         let ps,_pp = s.positions[pred] in
ps = is - 1)
```

...and predecessor is in the stack on the immediate left

## Programs

```  use ref.Ref
exception Return int

let ghost play_card (c:card) (s:state) : unit
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] }
=
let ghost pred = ref (-1) in
try
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 /\
let ps,_pp = s.positions[!pred] in
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
if c <= s.values[top_stack_i] then raise (Return i);
assert { 0 <= top_stack_i < s.num_elts };
assert { let is,ip = s.positions[top_stack_i] in
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
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];
s.positions <- s.positions[idx <- i,0];
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];
s.positions <- s.positions[idx <- i,stack_i_size];
s.preds <- s.preds[idx <- !pred];
end
```

`play_card c i s` pushes the card `c` on state `s`

```  use list.List
use list.Length
use list.NthNoOpt

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)

use PigeonHole

let ghost play_game (input: list int) : state
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 &&
let is,_ = s.positions[!idx] in is = i }
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 ->
let f = fun i ->
let si = sigma.seqval[i] in
let stack_i,_ = s.positions[si] in
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
let stack_i,_pi = s.positions[si] in
let stack_j,_pj = s.positions[sj] in
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
let stack_i,pi = s.positions[si] in
let stack_j,pj = s.positions[sj] in
stack_i = stack_j ->
si < sj && pi < pj && s.values[si] < s.values[sj]
};
s

let ghost 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

```

## Gluing abstract version with the original idiomatic code

```module PatienceFull

use int.Int
use PatienceAbstract

```

glue between the ghost state and the stacks of cards

```  use list.List
use list.Length
use list.NthNoOpt
use map.Map

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)

```

### playing a card

```  use list.RevAppend
use ref.Ref
exception Return

end
```

# Why3 Proof Results for Project "patience"

## Theory "patience.PigeonHole": fully verified

 Obligations Alt-Ergo 2.0.0 Z3 4.5.0 VC pigeon_hole --- --- split_goal_right VC pigeon_hole.0 0.00 --- VC pigeon_hole.1 0.00 --- VC pigeon_hole.2 0.00 --- VC pigeon_hole.3 0.00 --- VC pigeon_hole.4 0.01 --- VC pigeon_hole.5 0.00 --- VC pigeon_hole.6 0.00 --- VC pigeon_hole.7 0.00 --- VC pigeon_hole.8 0.01 --- VC pigeon_hole.9 0.00 --- VC pigeon_hole.10 0.00 --- VC pigeon_hole.11 0.00 --- VC pigeon_hole.12 0.00 --- VC pigeon_hole.13 0.00 --- VC pigeon_hole.14 --- 0.02 VC pigeon_hole.15 0.00 --- VC pigeon_hole.16 0.00 ---

## Theory "patience.PatienceCode": fully verified

 Obligations Alt-Ergo 2.0.0 VC wf_rev_append_stacks 0.03 VC push_card 0.07 VC play_cards 0.01 VC play_game 0.00

## Theory "patience.PatienceAbstract": fully verified

 Obligations Alt-Ergo 2.0.0 Z3 4.5.0 VC play_card --- --- split_goal_right VC play_card.0 0.00 --- VC play_card.1 --- --- introduce_premises VC play_card.1.0 --- --- inline_goal VC play_card.1.0.0 --- --- split_goal_right VC play_card.1.0.0.0 0.05 --- VC play_card.1.0.0.1 0.05 --- VC play_card.1.0.0.2 0.05 --- VC play_card.1.0.0.3 0.05 --- VC play_card.1.0.0.4 --- 0.06 VC play_card.1.0.0.5 --- 0.04 VC play_card.1.0.0.6 0.09 --- VC play_card.1.0.0.7 0.09 --- VC play_card.1.0.0.8 0.09 --- VC play_card.1.0.0.9 0.16 --- VC play_card.1.0.0.10 0.09 --- VC play_card.1.0.0.11 0.23 --- VC play_card.1.0.0.12 --- 0.04 VC play_card.1.0.0.13 --- 0.06 VC play_card.1.0.0.14 --- 0.04 VC play_card.1.0.0.15 --- 0.04 VC play_card.1.0.0.16 --- 0.04 VC play_card.1.0.0.17 --- 0.02 VC play_card.1.0.0.18 0.21 --- VC play_card.1.0.0.19 --- 0.03 VC play_card.1.0.0.20 --- 0.04 VC play_card.2 0.02 --- VC play_card.3 0.02 --- VC play_card.4 0.01 --- VC play_card.5 0.04 --- VC play_card.6 0.00 --- VC play_card.7 --- --- introduce_premises VC play_card.7.0 --- --- inline_goal VC play_card.7.0.0 --- --- split_goal_right VC play_card.7.0.0.0 0.04 --- VC play_card.7.0.0.1 0.04 --- VC play_card.7.0.0.2 0.04 --- VC play_card.7.0.0.3 0.05 --- VC play_card.7.0.0.4 --- 0.04 VC play_card.7.0.0.5 --- 0.03 VC play_card.7.0.0.6 0.06 --- VC play_card.7.0.0.7 0.06 --- VC play_card.7.0.0.8 0.06 --- VC play_card.7.0.0.9 0.08 --- VC play_card.7.0.0.10 0.08 --- VC play_card.7.0.0.11 0.12 --- VC play_card.7.0.0.12 --- 0.04 VC play_card.7.0.0.13 --- 0.07 VC play_card.7.0.0.14 --- 0.03 VC play_card.7.0.0.15 --- 0.02 VC play_card.7.0.0.16 --- 0.03 VC play_card.7.0.0.17 --- 0.02 VC play_card.7.0.0.18 0.08 --- VC play_card.7.0.0.19 --- 0.03 VC play_card.7.0.0.20 --- 0.03 VC play_card.8 0.02 --- VC play_card.9 0.02 --- VC play_card.10 0.00 --- VC play_cards 0.04 --- VC play_game --- --- split_goal_right VC play_game.0 0.01 --- VC play_game.1 0.01 --- VC play_game.2 0.00 --- VC play_game.3 0.01 --- VC play_game.4 0.01 --- VC play_game.5 0.01 --- VC play_game.6 0.01 --- VC play_game.7 0.00 --- VC play_game.8 0.01 --- VC play_game.9 0.00 --- VC play_game.10 0.01 --- VC play_game.11 0.00 --- VC play_game.12 0.01 --- VC play_game.13 0.02 --- VC play_game.14 0.01 --- VC play_game.15 0.04 --- VC play_game.16 0.02 --- VC play_game.17 0.02 --- VC play_game.18 --- 0.03 VC play_game.19 0.01 --- VC play_game.20 0.08 --- VC play_game.21 --- 0.03 VC play_game.22 --- 0.02 VC play_game.23 0.12 --- VC play_game.24 0.00 --- VC play_game.25 0.08 --- VC play_game.26 0.00 ---