Tic Tac Toe with ReasonML

ReasonML provides an alternative syntax for OCaml, which Imandra can read. Let's walk through an example ReasonML model of Tic Tac Toe and use Imandra to understand and verify some properties of it.

This model is wired into some ReasonReact components in order to produce this final, playable version. You can see the model code and components as a ReasonReact project in this github repo.

Setting up the model

Let's start by adding some definitions for the player, state of a square on the game grid, and the grid itself.

In [1]:
type player =
  | X
  | O;

/* the state of a particular square - None = (no move yet), Some(X/O) = either X or O has moved there */
type square_state = option(player);

/* each letter represents a square on the grid, starting in the top left and going along each row */
type grid = {
  a: square_state,
  b: square_state,
  c: square_state,
  d: square_state,
  e: square_state,
  f: square_state,
  g: square_state,
  h: square_state,
  i: square_state,
};
Out[1]:
type player = X | O
type square_state = player option
type grid = {
  a : square_state;
  b : square_state;
  c : square_state;
  d : square_state;
  e : square_state;
  f : square_state;
  g : square_state;
  h : square_state;
  i : square_state;
}

(NB: we're inputting in Reason syntax, but Imandra currently outputs OCaml syntax - this is something we'll have resolved very soon!).

Grid document printer

Here's an example grid:

In [2]:
let a_grid = {
  a: Some(O),
  b: Some(X),
  c: Some(O),
  d: None,
  e: None,
  f: Some(X),
  g: None,
  h: None,
  i: None,
};
Out[2]:
val a_grid : grid =
  {a = Some O; b = Some X; c = Some O; d = None; e = None; f = Some X;
   g = None; h = None; i = None}

As you can tell, it's not the easiest thing to interpret from Imandra's output given the way we've described the grid. Let's setup a custom document mapping for the grid type. Imandra will render the Document data structures as richer output to display our types in a clearer way:

In [3]:
[@program] let doc_of_square_state = (s : square_state) =>
  Document.s(switch (s) {
  | Some(X) => "❌" /* emojis! */
  | Some(O) => "⭕"
  | None => "·"
});

[@program] let doc_of_grid = (g: grid) => {
  let f = doc_of_square_state;
  Document.tbl([[f(g.a), f(g.b), f(g.c)], [f(g.d), f(g.e), f(g.f)], [f(g.g), f(g.h), f(g.i)]])
};
[@install_doc doc_of_grid];
Out[3]:
val doc_of_square_state : square_state -> Document.t = <fun>
val doc_of_grid : grid -> Document.t = <fun>
In [4]:
a_grid
Out[4]:
- : grid = <document>
··
···

Much better! The [@program] annotation tells Imandra that we're only going to be using these functions for non-analytical (and potentially side-effecting) operations, and that it doesn't have to worry about analysing them. See Logic and program modes for more information.

More definitions

Let's continue adding some more definitions:

In [5]:
/* maps to the grid entries */
type move =
  | A
  | B
  | C
  | D
  | E
  | F
  | G
  | H
  | I;

/* the full game state - the current grid, and who went last */
type game_state = {
  grid,
  last_player: option(player),
};

type game_status =
  | Won(player)
  | InProgress
  | InvalidMove(move)
  | Tied;

let initial_game = {
  grid: {
    a: None,
    b: None,
    c: None,
    d: None,
    e: None,
    f: None,
    g: None,
    h: None,
    i: None,
  },
  last_player: None,
};

let initial_player = X;

/* convenience function from our move variant to the value in that grid position */
let value = ({grid: {a, b, c, d, e, f, g, h, i}, _}) =>
  fun
  | A => a
  | B => b
  | C => c
  | D => d
  | E => e
  | F => f
  | G => g
  | H => h
  | I => i;
Out[5]:
type move = A | B | C | D | E | F | G | H | I
type game_state = { grid : grid; last_player : player option; }
type game_status = Won of player | InProgress | InvalidMove of move | Tied
val initial_game : game_state = {grid = <document>; last_player = None}
val initial_player : player = X
val value : game_state -> move -> square_state = <fun>
···
···
···

Game state document printer

Let's also add another document for printing game_states, showing both the grid and who went last:

In [6]:
[@program] let doc_of_game_state = (gs: game_state) => {
  Document.tbl([[doc_of_grid(gs.grid), Document.s("Last:"), doc_of_square_state(gs.last_player)]])
};

[@install_doc doc_of_game_state];

initial_game;
Out[6]:
val doc_of_game_state : game_state -> Document.t = <fun>
- : game_state = <document>
···
···
···
Last:·

Some logic and our first instance

Now for our first bit of proper logic - whether or not a game is won for a particular player:

In [7]:
let is_winning = ({grid: {a, b, c, d, e, f, g, h, i}, _}, player) => {
  let winning_state = (Some(player), Some(player), Some(player));
  (a, b, c) == winning_state
  || (d, e, f) == winning_state
  || (g, h, i) == winning_state
  || (a, d, g) == winning_state
  || (b, e, h) == winning_state
  || (c, f, i) == winning_state
  || (a, e, i) == winning_state
  || (c, e, g) == winning_state;
};
Out[7]:
val is_winning : game_state -> player -> bool = <fun>

Let's use Imandra to get a feel for this function - let's ask for an instance of a game where the game is being won by the player X. We use Imandra's instance mechanism, and ask for a game g where the predicate is_winning(g, X) is true:

In [8]:
instance((g : game_state) => is_winning(g, X));
Out[8]:
- : game_state -> bool = <fun>
module CX : sig val g : game_state end
Instance (after 0 steps, 0.018s):
let g : game_state =
  let (_x_0 : player option) = Some X in
  {grid =
   {a = None; b = _x_0; c = _x_0; d = _x_0; e = _x_0; f = None; g = None;
    h = _x_0; i = None};
   last_player = None}
Instance
proof attempt
ground_instances:0
definitions:0
inductions:0
search_time:
0.018s
details:
Expand
smt_stats:
num checks:1
arith-make-feasible:2
arith-max-columns:4
rlimit count:2616
mk clause:40
datatype occurs check:1
mk bool var:97
datatype splits:23
decisions:38
propagations:34
eliminated applications:1
conflicts:7
datatype accessor ax:7
minimized lits:3
datatype constructor ax:41
num allocs:988774
final checks:1
added eqs:168
time:0.008000
memory:5.190000
max memory:5.630000
Expand
  • start[0.018s]
      let (_x_0 : grid) = ( :var_0: ).grid in
      let (_x_1 : player option) = _x_0.a in
      let (_x_2 : player option) = _x_0.b in
      let (_x_3 : player option) = _x_0.c in
      let (_x_4 : player option) = Some X in
      let (_x_5 : (player option * player option * player option))
          = (_x_4, _x_4, _x_4)
      in
      let (_x_6 : player option) = _x_0.d in
      let (_x_7 : player option) = _x_0.e in
      let (_x_8 : player option) = _x_0.f in
      let (_x_9 : player option) = _x_0.g in
      let (_x_10 : player option) = _x_0.h in
      let (_x_11 : player option) = _x_0.i in
      (_x_1, _x_2, _x_3) = _x_5
      || (_x_6, _x_7, _x_8) = _x_5
         || (_x_9, _x_10, _x_11) = _x_5
            || (_x_1, _x_6, _x_9) = _x_5
               || (_x_2, _x_7, _x_10) = _x_5
                  || (_x_3, _x_8, _x_11) = _x_5
                     || (_x_1, _x_7, _x_11) = _x_5 || (_x_3, _x_7, _x_9) = _x_5
  • simplify

    into:
    let (_x_0 : grid) = ( :var_0: ).grid in
    let (_x_1 : player option) = Some X in
    let (_x_2 : bool) = _x_0.a = _x_1 in
    let (_x_3 : bool) = _x_0.b = _x_1 in
    let (_x_4 : bool) = _x_0.c = _x_1 in
    let (_x_5 : bool) = _x_0.d = _x_1 in
    let (_x_6 : bool) = _x_0.e = _x_1 in
    let (_x_7 : bool) = _x_0.f = _x_1 in
    let (_x_8 : bool) = _x_0.g = _x_1 in
    let (_x_9 : bool) = _x_0.h = _x_1 in
    let (_x_10 : bool) = _x_0.i = _x_1 in
    (((((((_x_2 && _x_3) && _x_4 || (_x_5 && _x_6) && _x_7)
         || (_x_8 && _x_9) && _x_10)
        || (_x_2 && _x_5) && _x_8)
       || (_x_3 && _x_6) && _x_9)
      || (_x_4 && _x_7) && _x_10)
     || (_x_2 && _x_6) && _x_10)
    || (_x_4 && _x_6) && _x_8
    expansions:
    []
    rewrite_steps:
      forward_chaining:
      • Sat (Some let g : game_state = let (_x_0 : player option) = Some X in {grid = {a = None; b = _x_0; c = _x_0; d = _x_0; e = _x_0; f = None; g = None; h = _x_0; i = None}; last_player = None} )

      Rather than executing our function with random inputs as you might imagine, Imandra has examined our definitions mathematically, and generated an instance that matches what we asked for based on the code itself, without executing it directly. This means we don't have to write generators as we would if we were using a library like QuickCheck for example. We can also use Imandra's other tools, such as state space decomposition, to identify edge cases we may not have thought about. See the other documentation pages/notebooks for more detail.

      In this case we explicitly annotated the type of g as game_state, but if this can be inferred using the usual ReasonML/OCaml inference rules, it can be omitted.

      instance has also reflected the value it found back into our Imandra session, so we can take a look at it:

      In [9]:
      CX.g
      
      Out[9]:
      - : game_state = <document>
      
      ·
      ·
      ··
      Last:·

      and compute with it:

      In [10]:
      is_winning(CX.g, X)
      
      Out[10]:
      - : bool = true
      

      We can see here that X is definitely 'winning', although the game doesn't look particularly valid. O hasn't made any moves, and the game state hasn't got a last_player set. That's fine for now though, we'll build up additional layers in our logic to handle these other cases. It can be very useful to see a concrete example of our function in action!

      Valid grids

      Let's continue:

      In [11]:
      let other_player =
        fun
        | X => O
        | O => X;
      
      /* count up the number of moves for each player across the whole grid - sounds useful! */
      let move_counts = ({a, b, c, d, e, f, g, h, i}) =>
        List.fold_right(
          (el, (x, o)) =>
            switch (el) {
            | None => (x, o)
            | Some(X) => (x + 1, o)
            | Some(O) => (x, o + 1)
            },
          ~base=(0, 0),
          [a, b, c, d, e, f, g, h, i],
        );
      
      /* whether a grid is 'valid' - the difference between move totals should never be more than 1 */
      let is_valid_grid = (grid, last_player) => {
        let (x, o) = move_counts(grid);
      
        (x + o == 0 && last_player == None) ||
        (x - o == 1 && last_player == Some(X)) ||
        (x == o && last_player == Some(O));
      };
      
      Out[11]:
      val other_player : player -> player = <fun>
      val move_counts : grid -> Z.t * Z.t = <fun>
      val is_valid_grid : grid -> player option -> bool = <fun>
      

      Let's grab a game where the grid is valid:

      In [12]:
      instance((game) => is_valid_grid(game.grid, game.last_player));
      
      Out[12]:
      - : game_state -> bool = <fun>
      module CX : sig val game : game_state end
      
      Instance (after 10 steps, 0.060s):
      let game : game_state =
        {grid =
         {a = None; b = None; c = None; d = None; e = None; f = None; g = None;
          h = None; i = None};
         last_player = None}
      
      Instance
      proof attempt
      ground_instances:10
      definitions:0
      inductions:0
      search_time:
      0.060s
      details:
      Expand
      smt_stats:
      num checks:21
      arith-assume-eqs:4
      arith-make-feasible:108
      arith-max-columns:66
      arith-conflicts:2
      rlimit count:20194
      arith-cheap-eqs:141
      mk clause:276
      datatype occurs check:227
      mk bool var:1431
      arith-lower:140
      arith-diseq:54
      datatype splits:61
      decisions:465
      arith-propagations:57
      propagations:426
      interface eqs:4
      arith-bound-propagations-cheap:57
      arith-max-rows:42
      conflicts:38
      datatype accessor ax:53
      minimized lits:3
      arith-bound-propagations-lp:1
      datatype constructor ax:841
      num allocs:4863531
      final checks:15
      added eqs:3465
      del clause:100
      arith eq adapter:126
      arith-upper:205
      time:0.001000
      memory:6.290000
      max memory:8.050000
      Expand
      • start[0.060s]
          let (_x_0 : (int * int))
              = List.fold_right
                (fun el _y
                 -> let (x, o) = _y in
                    if el = None then (x, o)
                    else if el = Some X then (x + 1, o) else (x, o + 1))
                (0, 0) (….a :: (… :: …))
          in
          let (_x_1 : int) = _x_0.0 in
          let (_x_2 : int) = _x_0.1 in
          let (_x_3 : player option) = ( :var_0: ).last_player in
          _x_1 + _x_2 = 0 && _x_3 = None
          || _x_1 - _x_2 = 1 && _x_3 = Some X || _x_1 = _x_2 && _x_3 = Some O
      • simplify

        into:
        let (_x_0 : (int * int))
            = List.fold_right
              (fun el _y
               -> let (x, o) = _y in
                  if el = None then (x, o)
                  else if el = Some X then (x + 1, o) else (x, o + 1))
              (0, 0) (….a :: (… :: …))
        in
        let (_x_1 : int) = _x_0.0 in
        let (_x_2 : int) = _x_0.1 in
        let (_x_3 : player option) = ( :var_0: ).last_player in
        (_x_1 + _x_2 = 0 && _x_3 = None || _x_1 = _x_2 && _x_3 = Some O)
        || _x_1 + (-1) * _x_2 = 1 && _x_3 = Some X
        expansions:
        []
        rewrite_steps:
          forward_chaining:
          • unroll
            expr:
            (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                             (|::| (i_1358/…
            expansions:
            • unroll
              expr:
              (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                               (|::| (i_1358/…
              expansions:
              • unroll
                expr:
                (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                                 (|::| (i_1358/…
                expansions:
                • unroll
                  expr:
                  (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                                   (|::| (i_1358/…
                  expansions:
                  • unroll
                    expr:
                    (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                                     (|::| (i_1358/…
                    expansions:
                    • unroll
                      expr:
                      (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                                       (|::| (i_1358/…
                      expansions:
                      • unroll
                        expr:
                        (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                                         (|::| (i_1358/…
                        expansions:
                        • unroll
                          expr:
                          (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1442/client))
                                           (|::| (i_1358/…
                          expansions:
                          • unroll
                            expr:
                            (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                            expansions:
                            • unroll
                              expr:
                              (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                              expansions:
                              • Sat (Some let game : game_state = {grid = {a = None; b = None; c = None; d = None; e = None; f = None; g = None; h = None; i = None}; last_player = None} )
                              In [13]:
                              CX.game;
                              
                              Out[13]:
                              - : game_state = <document>
                              
                              ···
                              ···
                              ···
                              Last:·

                              This is a bit better - the grid looks more like a real game of Tic Tac Toe now.

                              Let's also see what happens if we ask for an instance that we know shouldn't be possible - if player X is more than 1 move ahead:

                              In [14]:
                              instance((game) => {
                                 let (x, o) = move_counts(game.grid);
                                 is_valid_grid(game.grid, game.last_player) && ((x - o) >= 2)
                              });
                              
                              Out[14]:
                              - : game_state -> bool = <fun>
                              
                              Unsatisfiable
                              proof
                              ground_instances:10
                              definitions:0
                              inductions:0
                              search_time:
                              0.061s
                              details:
                              Expand
                              smt_stats:
                              num checks:21
                              arith-assume-eqs:20
                              arith-make-feasible:588
                              arith-max-columns:116
                              arith-conflicts:38
                              rlimit count:57031
                              arith-cheap-eqs:2350
                              mk clause:2009
                              datatype occurs check:457
                              restarts:1
                              mk bool var:3244
                              arith-lower:1430
                              arith-diseq:628
                              datatype splits:275
                              decisions:968
                              arith-propagations:458
                              propagations:4290
                              interface eqs:20
                              arith-bound-propagations-cheap:458
                              arith-max-rows:92
                              conflicts:203
                              datatype accessor ax:160
                              minimized lits:158
                              arith-bound-propagations-lp:346
                              datatype constructor ax:881
                              num allocs:13579415
                              final checks:30
                              added eqs:8177
                              del clause:886
                              arith eq adapter:547
                              arith-upper:1197
                              time:0.026000
                              memory:7.530000
                              max memory:9.170000
                              Expand
                              • start[0.061s]
                                  let (_x_0 : (int * int))
                                      = List.fold_right
                                        (fun el _y
                                         -> let (x, o) = _y in
                                            if el = None then (x, o)
                                            else if el = Some X then (x + 1, o) else (x, o + 1))
                                        (0, 0) (….a :: (… :: …))
                                  in
                                  let (_x_1 : int) = _x_0.0 in
                                  let (_x_2 : int) = _x_0.1 in
                                  let (_x_3 : player option) = ( :var_0: ).last_player in
                                  let (_x_4 : int) = _x_1 - _x_2 in
                                  (_x_1 + _x_2 = 0 && _x_3 = None
                                   || _x_4 = 1 && _x_3 = Some X || _x_1 = _x_2 && _x_3 = Some O)
                                  && _x_4 >= 2
                              • simplify

                                into:
                                let (_x_0 : (int * int))
                                    = List.fold_right
                                      (fun el _y
                                       -> let (x, o) = _y in
                                          if el = None then (x, o)
                                          else if el = Some X then (x + 1, o) else (x, o + 1))
                                      (0, 0) (….a :: (… :: …))
                                in
                                let (_x_1 : int) = _x_0.0 in
                                let (_x_2 : int) = _x_0.1 in
                                let (_x_3 : player option) = ( :var_0: ).last_player in
                                let (_x_4 : int) = _x_1 + (-1) * _x_2 in
                                ((_x_1 + _x_2 = 0 && _x_3 = None || _x_1 = _x_2 && _x_3 = Some O)
                                 || _x_4 = 1 && _x_3 = Some X)
                                && _x_4 >= 2
                                expansions:
                                []
                                rewrite_steps:
                                  forward_chaining:
                                  • unroll
                                    expr:
                                    (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                     (|::| (i_1358/…
                                    expansions:
                                    • unroll
                                      expr:
                                      (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                       (|::| (i_1358/…
                                      expansions:
                                      • unroll
                                        expr:
                                        (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                         (|::| (i_1358/…
                                        expansions:
                                        • unroll
                                          expr:
                                          (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                           (|::| (i_1358/…
                                          expansions:
                                          • unroll
                                            expr:
                                            (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                             (|::| (i_1358/…
                                            expansions:
                                            • unroll
                                              expr:
                                              (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                               (|::| (i_1358/…
                                              expansions:
                                              • unroll
                                                expr:
                                                (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                                 (|::| (i_1358/…
                                                expansions:
                                                • unroll
                                                  expr:
                                                  (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1444/client))
                                                                   (|::| (i_1358/…
                                                  expansions:
                                                  • unroll
                                                    expr:
                                                    (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                                                    expansions:
                                                    • unroll
                                                      expr:
                                                      (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                                                      expansions:
                                                      • Unsat

                                                      As this is invalid according to the description we've provided, Imandra can't find an instance. In fact, this Unsatisfiable judgment tells us that Imandra has actually constructed a mathematical proof that no such instance exists.

                                                      Verifying

                                                      We can invert this question, and turn it into a verification statement and ask Imandra to check it for us - similar to a test, but mathematically checked for all possible games (as game is the free argument to the verification function).

                                                      In [15]:
                                                      verify((game) => {
                                                         let (x, o) = move_counts(game.grid);
                                                         not(is_valid_grid(game.grid, game.last_player) && ((x - o) >= 2))
                                                      });
                                                      
                                                      Out[15]:
                                                      At jupyter cell 15:3,3--6
                                                      3 |    not(is_valid_grid(game.grid, game.last_player) && ((x - o) >= 2))
                                                             ^^^
                                                      
                                                      Error:
                                                        File "jupyter cell 15", line 3, characters 3-6:
                                                        Error: Unbound value not_
                                                        Hint: Did you mean not?
                                                      

                                                      Valid games

                                                      Let's continue adding some more definitions - we haven't really covered winning criteria as part of is_valid_grid, so let's include a few:

                                                      In [16]:
                                                      let is_tie = ({grid: {a, b, c, d, e, f, g, h, i}, _}) =>
                                                        List.for_all((!=)(None), [a, b, c, d, e, f, g, h, i]);
                                                      
                                                      Out[16]:
                                                      val is_tie : game_state -> bool = <fun>
                                                      
                                                      In [17]:
                                                      let is_valid_game = game => {
                                                        let winning_X = is_winning(game, X);
                                                        let winning_O = is_winning(game, O);
                                                        is_valid_grid(game.grid, game.last_player)
                                                        && (! winning_X && ! winning_O || winning_X != winning_O);
                                                      };
                                                      
                                                      Out[17]:
                                                      val is_valid_game : game_state -> bool = <fun>
                                                      

                                                      ... and let's ask for a valid game where O is winning:

                                                      In [18]:
                                                      instance((game) => is_valid_game(game) && is_winning(game, O));
                                                      
                                                      Out[18]:
                                                      - : game_state -> bool = <fun>
                                                      module CX : sig val game : game_state end
                                                      
                                                      Instance (after 10 steps, 0.073s):
                                                      let game : game_state =
                                                        let (_x_0 : player option) = Some X in
                                                        let (_x_1 : player option) = Some O in
                                                        {grid =
                                                         {a = _x_0; b = _x_1; c = None; d = _x_1; e = _x_1; f = _x_0; g = _x_0;
                                                          h = _x_1; i = _x_0};
                                                         last_player = _x_1}
                                                      
                                                      Instance
                                                      proof attempt
                                                      ground_instances:10
                                                      definitions:0
                                                      inductions:0
                                                      search_time:
                                                      0.073s
                                                      details:
                                                      Expand
                                                      smt_stats:
                                                      arith-assume-eqs:20
                                                      arith-make-feasible:612
                                                      arith-max-columns:116
                                                      arith-conflicts:23
                                                      arith-gcd-calls:1
                                                      datatype occurs check:587
                                                      arith-lower:1241
                                                      arith-diseq:565
                                                      datatype splits:213
                                                      arith-propagations:504
                                                      propagations:5381
                                                      interface eqs:20
                                                      arith-bound-propagations-cheap:504
                                                      conflicts:213
                                                      datatype constructor ax:709
                                                      num allocs:30126414
                                                      final checks:32
                                                      added eqs:9182
                                                      del clause:551
                                                      arith-gcd-conflict:1
                                                      num checks:21
                                                      rlimit count:66918
                                                      arith-cheap-eqs:1865
                                                      mk clause:1901
                                                      restarts:1
                                                      mk bool var:2657
                                                      decisions:1091
                                                      arith-max-rows:92
                                                      datatype accessor ax:129
                                                      minimized lits:202
                                                      arith-bound-propagations-lp:345
                                                      arith eq adapter:541
                                                      arith-upper:1328
                                                      time:0.025000
                                                      memory:8.820000
                                                      max memory:10.740000
                                                      Expand
                                                      • start[0.073s]
                                                          let (_x_0 : player option) = ….a in
                                                          let (_x_1 : (int * int))
                                                              = List.fold_right
                                                                (fun el _y
                                                                 -> let (x, o) = _y in
                                                                    if el = None then (x, o)
                                                                    else if el = Some X then (x + 1, o) else (x, o + 1))
                                                                (0, 0) (_x_0 :: (… :: …))
                                                          in
                                                          let (_x_2 : int) = _x_1.0 in
                                                          let (_x_3 : int) = _x_1.1 in
                                                          let (_x_4 : player option) = ( :var_0: ).last_player in
                                                          let (_x_5 : player option) = Some X in
                                                          let (_x_6 : player option) = Some O in
                                                          let (_x_7 : player option) = ….c in
                                                          let (_x_8 : (player option * player option * player option))
                                                              = (_x_0, …, _x_7)
                                                          in
                                                          let (_x_9 : (player option * player option * player option))
                                                              = (_x_5, _x_5, _x_5)
                                                          in
                                                          let (_x_10 : player option) = ….d in
                                                          let (_x_11 : player option) = ….e in
                                                          let (_x_12 : player option) = ….f in
                                                          let (_x_13 : (player option * player option * player option))
                                                              = (_x_10, _x_11, _x_12)
                                                          in
                                                          let (_x_14 : player option) = ….g in
                                                          let (_x_15 : player option) = ….h in
                                                          let (_x_16 : player option) = ….i in
                                                          let (_x_17 : (player option * player option * player option))
                                                              = (_x_14, _x_15, _x_16)
                                                          in
                                                          let (_x_18 : (player option * player option * player option))
                                                              = (_x_0, _x_10, _x_14)
                                                          in
                                                          let (_x_19 : (player option * player option * player option))
                                                              = (…, _x_11, _x_15)
                                                          in
                                                          let (_x_20 : (player option * player option * player option))
                                                              = (_x_7, _x_12, _x_16)
                                                          in
                                                          let (_x_21 : (player option * player option * player option))
                                                              = (_x_0, _x_11, _x_16)
                                                          in
                                                          let (_x_22 : (player option * player option * player option))
                                                              = (_x_7, _x_11, _x_14)
                                                          in
                                                          let (_x_23 : bool)
                                                              = _x_8 = _x_9
                                                                || _x_13 = _x_9
                                                                   || _x_17 = _x_9
                                                                      || _x_18 = _x_9
                                                                         || _x_19 = _x_9
                                                                            || _x_20 = _x_9 || _x_21 = _x_9 || _x_22 = _x_9
                                                          in
                                                          let (_x_24 : (player option * player option * player option))
                                                              = (_x_6, _x_6, _x_6)
                                                          in
                                                          let (_x_25 : bool)
                                                              = _x_8 = _x_24
                                                                || _x_13 = _x_24
                                                                   || _x_17 = _x_24
                                                                      || _x_18 = _x_24
                                                                         || _x_19 = _x_24
                                                                            || _x_20 = _x_24 || _x_21 = _x_24 || _x_22 = _x_24
                                                          in
                                                          ((_x_2 + _x_3 = 0 && _x_4 = None
                                                            || _x_2 - _x_3 = 1 && _x_4 = _x_5 || _x_2 = _x_3 && _x_4 = _x_6)
                                                           && (not _x_23 && not _x_25 || not (_x_23 = _x_25)))
                                                          && _x_25
                                                      • simplify

                                                        into:
                                                        let (_x_0 : player option) = ….a in
                                                        let (_x_1 : (int * int))
                                                            = List.fold_right
                                                              (fun el _y
                                                               -> let (x, o) = _y in
                                                                  if el = None then (x, o)
                                                                  else if el = Some X then (x + 1, o) else (x, o + 1))
                                                              (0, 0) (_x_0 :: (… :: …))
                                                        in
                                                        let (_x_2 : int) = _x_1.0 in
                                                        let (_x_3 : int) = _x_1.1 in
                                                        let (_x_4 : player option) = ( :var_0: ).last_player in
                                                        let (_x_5 : player option) = Some O in
                                                        let (_x_6 : player option) = Some X in
                                                        let (_x_7 : bool) = _x_0 = _x_6 in
                                                        let (_x_8 : bool) = … = _x_6 in
                                                        let (_x_9 : player option) = ….c in
                                                        let (_x_10 : bool) = _x_9 = _x_6 in
                                                        let (_x_11 : player option) = ….d in
                                                        let (_x_12 : bool) = _x_11 = _x_6 in
                                                        let (_x_13 : player option) = ….e in
                                                        let (_x_14 : bool) = _x_13 = _x_6 in
                                                        let (_x_15 : player option) = ….f in
                                                        let (_x_16 : bool) = _x_15 = _x_6 in
                                                        let (_x_17 : player option) = ….g in
                                                        let (_x_18 : bool) = _x_17 = _x_6 in
                                                        let (_x_19 : player option) = ….h in
                                                        let (_x_20 : bool) = _x_19 = _x_6 in
                                                        let (_x_21 : player option) = ….i in
                                                        let (_x_22 : bool) = _x_21 = _x_6 in
                                                        let (_x_23 : bool)
                                                            = not
                                                              ((((((((_x_7 && _x_8) && _x_10 || (_x_12 && _x_14) && _x_16)
                                                                    || (_x_18 && _x_20) && _x_22)
                                                                   || (_x_7 && _x_12) && _x_18)
                                                                  || (_x_8 && _x_14) && _x_20)
                                                                 || (_x_10 && _x_16) && _x_22)
                                                                || (_x_7 && _x_14) && _x_22)
                                                               || (_x_10 && _x_14) && _x_18)
                                                        in
                                                        let (_x_24 : bool) = _x_0 = _x_5 in
                                                        let (_x_25 : bool) = … = _x_5 in
                                                        let (_x_26 : bool) = _x_9 = _x_5 in
                                                        let (_x_27 : bool) = _x_11 = _x_5 in
                                                        let (_x_28 : bool) = _x_13 = _x_5 in
                                                        let (_x_29 : bool) = _x_15 = _x_5 in
                                                        let (_x_30 : bool) = _x_17 = _x_5 in
                                                        let (_x_31 : bool) = _x_19 = _x_5 in
                                                        let (_x_32 : bool) = _x_21 = _x_5 in
                                                        let (_x_33 : bool)
                                                            = (((((((_x_24 && _x_25) && _x_26 || (_x_27 && _x_28) && _x_29)
                                                                   || (_x_30 && _x_31) && _x_32)
                                                                  || (_x_24 && _x_27) && _x_30)
                                                                 || (_x_25 && _x_28) && _x_31)
                                                                || (_x_26 && _x_29) && _x_32)
                                                               || (_x_24 && _x_28) && _x_32)
                                                              || (_x_26 && _x_28) && _x_30
                                                        in
                                                        (((_x_2 + _x_3 = 0 && _x_4 = None || _x_2 = _x_3 && _x_4 = _x_5)
                                                          || _x_2 + (-1) * _x_3 = 1 && _x_4 = _x_6)
                                                         && (_x_23 && not _x_33 || _x_23 = _x_33))
                                                        && _x_33
                                                        expansions:
                                                        []
                                                        rewrite_steps:
                                                          forward_chaining:
                                                          • unroll
                                                            expr:
                                                            (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                             (|::| (i_1358/…
                                                            expansions:
                                                            • unroll
                                                              expr:
                                                              (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                               (|::| (i_1358/…
                                                              expansions:
                                                              • unroll
                                                                expr:
                                                                (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                                 (|::| (i_1358/…
                                                                expansions:
                                                                • unroll
                                                                  expr:
                                                                  (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                                   (|::| (i_1358/…
                                                                  expansions:
                                                                  • unroll
                                                                    expr:
                                                                    (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                                     (|::| (i_1358/…
                                                                    expansions:
                                                                    • unroll
                                                                      expr:
                                                                      (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                                       (|::| (i_1358/…
                                                                      expansions:
                                                                      • unroll
                                                                        expr:
                                                                        (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                                         (|::| (i_1358/…
                                                                        expansions:
                                                                        • unroll
                                                                          expr:
                                                                          (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1464/client))
                                                                                           (|::| (i_1358/…
                                                                          expansions:
                                                                          • unroll
                                                                            expr:
                                                                            (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                                                                            expansions:
                                                                            • unroll
                                                                              expr:
                                                                              (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                                                                              expansions:
                                                                              • Sat (Some let game : game_state = let (_x_0 : player option) = Some X in let (_x_1 : player option) = Some O in {grid = {a = _x_0; b = _x_1; c = None; d = _x_1; e = _x_1; f = _x_0; g = _x_0; h = _x_1; i = _x_0}; last_player = _x_1} )
                                                                              In [19]:
                                                                              CX.game;
                                                                              
                                                                              Out[19]:
                                                                              - : game_state = <document>
                                                                              
                                                                              ·
                                                                              Last:

                                                                              Valid moves

                                                                              Looks good! We're on the home stretch now - let's add the last few helper functions around playing moves:

                                                                              In [20]:
                                                                              let is_valid_move = (game, player, move) =>
                                                                                ! (is_winning(game, X) || is_winning(game, O) || is_tie(game))
                                                                                && is_valid_game(game)
                                                                                && (
                                                                                  game.last_player == None
                                                                                  && player == initial_player
                                                                                  || game.last_player == Some(other_player(player))
                                                                                );
                                                                              
                                                                              let play_move = ({grid, _}, player, move) => {
                                                                                let play = Some(player);
                                                                                let grid =
                                                                                  switch (move) {
                                                                                  | A => {...grid, a: play}
                                                                                  | B => {...grid, b: play}
                                                                                  | C => {...grid, c: play}
                                                                                  | D => {...grid, d: play}
                                                                                  | E => {...grid, e: play}
                                                                                  | F => {...grid, f: play}
                                                                                  | G => {...grid, g: play}
                                                                                  | H => {...grid, h: play}
                                                                                  | I => {...grid, i: play}
                                                                                  };
                                                                                {grid, last_player: play};
                                                                              };
                                                                              
                                                                              let status = game =>
                                                                                if (is_winning(game, X)) {
                                                                                  Won(X);
                                                                                } else if (is_winning(game, O)) {
                                                                                  Won(O);
                                                                                } else if (is_tie(game)) {
                                                                                  Tied;
                                                                                } else {
                                                                                  InProgress;
                                                                                };
                                                                              
                                                                              Out[20]:
                                                                              val is_valid_move : game_state -> player -> 'a -> bool = <fun>
                                                                              val play_move : game_state -> player -> move -> game_state = <fun>
                                                                              val status : game_state -> game_status = <fun>
                                                                              

                                                                              And finally, the function we call from the ReasonReact frontend to actually play the game!

                                                                              In [21]:
                                                                              let play = ({last_player, _} as game, move) => {
                                                                                let player =
                                                                                  switch (last_player) {
                                                                                  | None => initial_player
                                                                                  | Some(player) => other_player(player)
                                                                                  };
                                                                                if (is_valid_move(game, player, move)) {
                                                                                  let game = play_move(game, player, move);
                                                                                  (game, status(game));
                                                                                } else {
                                                                                  (game, InvalidMove(move));
                                                                                };
                                                                              };
                                                                              
                                                                              Out[21]:
                                                                              val play : game_state -> move -> game_state * game_status = <fun>
                                                                              

                                                                              ...and we're done! Let's add one more verification statement to check that these functions work as we expect. If we've got a valid game, and a valid move, then if we play the move on the game, we should still have a valid game.

                                                                              verify((game, player, move) =>
                                                                                (is_valid_game(game) && is_valid_move(game, player, move))
                                                                                ==> is_valid_game(play_move(game, player, move))
                                                                              );

                                                                              The a ==> b operator here is the logical "implies" - "if a is true, then b is also true". It's defined as a regular infix operator in the Imandra prelude:

                                                                              In [22]:
                                                                              (==>)
                                                                              
                                                                              Out[22]:
                                                                              - : bool -> bool -> bool = <fun>
                                                                              

                                                                              Anyway! Let's actually run the statement:

                                                                              In [23]:
                                                                              verify((game, player, move) =>
                                                                                (is_valid_game(game) && is_valid_move(game, player, move))
                                                                                ==> is_valid_game(play_move(game, player, move))
                                                                              );
                                                                              
                                                                              Out[23]:
                                                                              - : game_state -> player -> move -> bool = <fun>
                                                                              module CX : sig val game : game_state val player : player val move : move end
                                                                              
                                                                              Counterexample (after 23 steps, 0.169s):
                                                                              let game : game_state =
                                                                                let (_x_0 : player option) = Some X in
                                                                                let (_x_1 : player option) = Some O in
                                                                                {grid =
                                                                                 {a = None; b = _x_0; c = None; d = _x_0; e = None; f = _x_0; g = _x_1;
                                                                                  h = _x_1; i = None};
                                                                                 last_player = _x_0}
                                                                              let player : player = O
                                                                              let move : move = B
                                                                              
                                                                              Refuted
                                                                              proof attempt
                                                                              ground_instances:23
                                                                              definitions:0
                                                                              inductions:0
                                                                              search_time:
                                                                              0.169s
                                                                              details:
                                                                              Expand
                                                                              smt_stats:
                                                                              num checks:47
                                                                              arith-assume-eqs:151
                                                                              arith-make-feasible:1541
                                                                              arith-max-columns:223
                                                                              arith-conflicts:50
                                                                              rlimit count:231652
                                                                              arith-cheap-eqs:5177
                                                                              mk clause:4070
                                                                              datatype occurs check:5368
                                                                              restarts:2
                                                                              mk bool var:8730
                                                                              arith-lower:3695
                                                                              arith-diseq:1355
                                                                              datatype splits:574
                                                                              decisions:4464
                                                                              arith-propagations:1149
                                                                              propagations:14290
                                                                              interface eqs:151
                                                                              arith-bound-propagations-cheap:1149
                                                                              arith-max-rows:181
                                                                              conflicts:349
                                                                              datatype accessor ax:271
                                                                              minimized lits:378
                                                                              arith-bound-propagations-lp:680
                                                                              datatype constructor ax:4164
                                                                              num allocs:62385248
                                                                              final checks:175
                                                                              added eqs:37988
                                                                              del clause:1335
                                                                              arith eq adapter:1337
                                                                              arith-upper:2978
                                                                              time:0.058000
                                                                              memory:11.460000
                                                                              max memory:12.610000
                                                                              Expand
                                                                              • start[0.169s]
                                                                                  let (_x_0 : (int * int)) = (0, 0) in
                                                                                  let (_x_1 : player option) = ….a in
                                                                                  let (_x_2 : player option list) = _x_1 :: (… :: …) in
                                                                                  let (_x_3 : (int * int))
                                                                                      = List.fold_right
                                                                                        (fun el _y
                                                                                         -> let (x, o) = _y in
                                                                                            if el = None then (x, o)
                                                                                            else if el = Some X then (x + 1, o) else (x, o + 1))
                                                                                        _x_0 _x_2
                                                                                  in
                                                                                  let (_x_4 : int) = _x_3.0 in
                                                                                  let (_x_5 : int) = _x_3.1 in
                                                                                  let (_x_6 : player option) = ( :var_0: ).last_player in
                                                                                  let (_x_7 : bool) = _x_6 = None in
                                                                                  let (_x_8 : player option) = Some X in
                                                                                  let (_x_9 : player option) = Some O in
                                                                                  let (_x_10 : player option) = ….c in
                                                                                  let (_x_11 : (player option * player option * player option))
                                                                                      = (_x_1, …, _x_10)
                                                                                  in
                                                                                  let (_x_12 : (player option * player option * player option))
                                                                                      = (_x_8, _x_8, _x_8)
                                                                                  in
                                                                                  let (_x_13 : player option) = ….d in
                                                                                  let (_x_14 : player option) = ….e in
                                                                                  let (_x_15 : player option) = ….f in
                                                                                  let (_x_16 : (player option * player option * player option))
                                                                                      = (_x_13, _x_14, _x_15)
                                                                                  in
                                                                                  let (_x_17 : player option) = ….g in
                                                                                  let (_x_18 : player option) = ….h in
                                                                                  let (_x_19 : player option) = ….i in
                                                                                  let (_x_20 : (player option * player option * player option))
                                                                                      = (_x_17, _x_18, _x_19)
                                                                                  in
                                                                                  let (_x_21 : (player option * player option * player option))
                                                                                      = (_x_1, _x_13, _x_17)
                                                                                  in
                                                                                  let (_x_22 : (player option * player option * player option))
                                                                                      = (…, _x_14, _x_18)
                                                                                  in
                                                                                  let (_x_23 : (player option * player option * player option))
                                                                                      = (_x_10, _x_15, _x_19)
                                                                                  in
                                                                                  let (_x_24 : (player option * player option * player option))
                                                                                      = (_x_1, _x_14, _x_19)
                                                                                  in
                                                                                  let (_x_25 : (player option * player option * player option))
                                                                                      = (_x_10, _x_14, _x_17)
                                                                                  in
                                                                                  let (_x_26 : bool)
                                                                                      = _x_11 = _x_12
                                                                                        || _x_16 = _x_12
                                                                                           || _x_20 = _x_12
                                                                                              || _x_21 = _x_12
                                                                                                 || _x_22 = _x_12
                                                                                                    || _x_23 = _x_12 || _x_24 = _x_12 || _x_25 = _x_12
                                                                                  in
                                                                                  let (_x_27 : (player option * player option * player option))
                                                                                      = (_x_9, _x_9, _x_9)
                                                                                  in
                                                                                  let (_x_28 : bool)
                                                                                      = _x_11 = _x_27
                                                                                        || _x_16 = _x_27
                                                                                           || _x_20 = _x_27
                                                                                              || _x_21 = _x_27
                                                                                                 || _x_22 = _x_27
                                                                                                    || _x_23 = _x_27 || _x_24 = _x_27 || _x_25 = _x_27
                                                                                  in
                                                                                  let (_x_29 : bool)
                                                                                      = (_x_4 + _x_5 = 0 && _x_7
                                                                                         || _x_4 - _x_5 = 1 && _x_6 = _x_8 || _x_4 = _x_5 && _x_6 = _x_9)
                                                                                        && (not _x_26 && not _x_28 || not (_x_26 = _x_28))
                                                                                  in
                                                                                  let (_x_30 : bool) = ( :var_1: ) = X in
                                                                                  let (_x_31 : player option) = ….a in
                                                                                  let (_x_32 : (int * int))
                                                                                      = List.fold_right
                                                                                        (fun el _y
                                                                                         -> let (x, o) = _y in
                                                                                            if el = None then (x, o)
                                                                                            else if el = Some X then (x + 1, o) else (x, o + 1))
                                                                                        _x_0 (_x_31 :: (… :: …))
                                                                                  in
                                                                                  let (_x_33 : int) = _x_32.0 in
                                                                                  let (_x_34 : int) = _x_32.1 in
                                                                                  let (_x_35 : player option) = ….c in
                                                                                  let (_x_36 : (player option * player option * player option))
                                                                                      = (_x_31, …, _x_35)
                                                                                  in
                                                                                  let (_x_37 : player option) = ….d in
                                                                                  let (_x_38 : player option) = ….e in
                                                                                  let (_x_39 : player option) = ….f in
                                                                                  let (_x_40 : (player option * player option * player option))
                                                                                      = (_x_37, _x_38, _x_39)
                                                                                  in
                                                                                  let (_x_41 : player option) = ….g in
                                                                                  let (_x_42 : player option) = ….h in
                                                                                  let (_x_43 : player option) = ….i in
                                                                                  let (_x_44 : (player option * player option * player option))
                                                                                      = (_x_41, _x_42, _x_43)
                                                                                  in
                                                                                  let (_x_45 : (player option * player option * player option))
                                                                                      = (_x_31, _x_37, _x_41)
                                                                                  in
                                                                                  let (_x_46 : (player option * player option * player option))
                                                                                      = (…, _x_38, _x_42)
                                                                                  in
                                                                                  let (_x_47 : (player option * player option * player option))
                                                                                      = (_x_35, _x_39, _x_43)
                                                                                  in
                                                                                  let (_x_48 : (player option * player option * player option))
                                                                                      = (_x_31, _x_38, _x_43)
                                                                                  in
                                                                                  let (_x_49 : (player option * player option * player option))
                                                                                      = (_x_35, _x_38, _x_41)
                                                                                  in
                                                                                  let (_x_50 : bool)
                                                                                      = _x_36 = _x_12
                                                                                        || _x_40 = _x_12
                                                                                           || _x_44 = _x_12
                                                                                              || _x_45 = _x_12
                                                                                                 || _x_46 = _x_12
                                                                                                    || _x_47 = _x_12 || _x_48 = _x_12 || _x_49 = _x_12
                                                                                  in
                                                                                  let (_x_51 : bool)
                                                                                      = _x_36 = _x_27
                                                                                        || _x_40 = _x_27
                                                                                           || _x_44 = _x_27
                                                                                              || _x_45 = _x_27
                                                                                                 || _x_46 = _x_27
                                                                                                    || _x_47 = _x_27 || _x_48 = _x_27 || _x_49 = _x_27
                                                                                  in
                                                                                  _x_29
                                                                                  && not (_x_26 || _x_28 || List.for_all ((<>) None) _x_2)
                                                                                     && _x_29 && (_x_7 && _x_30 || _x_6 = Some (if _x_30 then O else X))
                                                                                  ==> (_x_33 + _x_34 = 0 && … = None
                                                                                       || _x_33 - _x_34 = 1 && … = _x_8 || _x_33 = _x_34 && … = _x_9)
                                                                                      && (not _x_50 && not _x_51 || not (_x_50 = _x_51))
                                                                              • simplify

                                                                                into:
                                                                                let (_x_0 : (int * int)) = (0, 0) in
                                                                                let (_x_1 : player option) = ….a in
                                                                                let (_x_2 : player option list) = _x_1 :: (… :: …) in
                                                                                let (_x_3 : (int * int))
                                                                                    = List.fold_right
                                                                                      (fun el _y
                                                                                       -> let (x, o) = _y in
                                                                                          if el = None then (x, o)
                                                                                          else if el = Some X then (x + 1, o) else (x, o + 1))
                                                                                      _x_0 _x_2
                                                                                in
                                                                                let (_x_4 : int) = _x_3.0 in
                                                                                let (_x_5 : int) = _x_3.1 in
                                                                                let (_x_6 : player option) = ( :var_0: ).last_player in
                                                                                let (_x_7 : bool) = _x_6 = None in
                                                                                let (_x_8 : player option) = Some O in
                                                                                let (_x_9 : player option) = Some X in
                                                                                let (_x_10 : bool) = _x_1 = _x_9 in
                                                                                let (_x_11 : bool) = … = _x_9 in
                                                                                let (_x_12 : player option) = ….c in
                                                                                let (_x_13 : bool) = _x_12 = _x_9 in
                                                                                let (_x_14 : bool) = (_x_10 && _x_11) && _x_13 in
                                                                                let (_x_15 : player option) = ….d in
                                                                                let (_x_16 : bool) = _x_15 = _x_9 in
                                                                                let (_x_17 : player option) = ….e in
                                                                                let (_x_18 : bool) = _x_17 = _x_9 in
                                                                                let (_x_19 : player option) = ….f in
                                                                                let (_x_20 : bool) = _x_19 = _x_9 in
                                                                                let (_x_21 : bool) = (_x_16 && _x_18) && _x_20 in
                                                                                let (_x_22 : player option) = ….g in
                                                                                let (_x_23 : bool) = _x_22 = _x_9 in
                                                                                let (_x_24 : player option) = ….h in
                                                                                let (_x_25 : bool) = _x_24 = _x_9 in
                                                                                let (_x_26 : player option) = ….i in
                                                                                let (_x_27 : bool) = _x_26 = _x_9 in
                                                                                let (_x_28 : bool) = (_x_23 && _x_25) && _x_27 in
                                                                                let (_x_29 : bool) = (_x_10 && _x_16) && _x_23 in
                                                                                let (_x_30 : bool) = (_x_11 && _x_18) && _x_25 in
                                                                                let (_x_31 : bool) = (_x_13 && _x_20) && _x_27 in
                                                                                let (_x_32 : bool) = (_x_10 && _x_18) && _x_27 in
                                                                                let (_x_33 : bool) = (_x_13 && _x_18) && _x_23 in
                                                                                let (_x_34 : bool)
                                                                                    = not
                                                                                      (((((((_x_14 || _x_21) || _x_28) || _x_29) || _x_30) || _x_31) || _x_32)
                                                                                       || _x_33)
                                                                                in
                                                                                let (_x_35 : bool) = _x_1 = _x_8 in
                                                                                let (_x_36 : bool) = … = _x_8 in
                                                                                let (_x_37 : bool) = _x_12 = _x_8 in
                                                                                let (_x_38 : bool) = (_x_35 && _x_36) && _x_37 in
                                                                                let (_x_39 : bool) = _x_15 = _x_8 in
                                                                                let (_x_40 : bool) = _x_17 = _x_8 in
                                                                                let (_x_41 : bool) = _x_19 = _x_8 in
                                                                                let (_x_42 : bool) = (_x_39 && _x_40) && _x_41 in
                                                                                let (_x_43 : bool) = _x_22 = _x_8 in
                                                                                let (_x_44 : bool) = _x_24 = _x_8 in
                                                                                let (_x_45 : bool) = _x_26 = _x_8 in
                                                                                let (_x_46 : bool) = (_x_43 && _x_44) && _x_45 in
                                                                                let (_x_47 : bool) = (_x_35 && _x_39) && _x_43 in
                                                                                let (_x_48 : bool) = (_x_36 && _x_40) && _x_44 in
                                                                                let (_x_49 : bool) = (_x_37 && _x_41) && _x_45 in
                                                                                let (_x_50 : bool) = (_x_35 && _x_40) && _x_45 in
                                                                                let (_x_51 : bool) = (_x_37 && _x_40) && _x_43 in
                                                                                let (_x_52 : bool)
                                                                                    = ((((((_x_38 || _x_42) || _x_46) || _x_47) || _x_48) || _x_49) || _x_50)
                                                                                      || _x_51
                                                                                in
                                                                                let (_x_53 : bool) = ( :var_1: ) = X in
                                                                                let (_x_54 : player option) = ….a in
                                                                                let (_x_55 : (int * int))
                                                                                    = List.fold_right
                                                                                      (fun el _y
                                                                                       -> let (x, o) = _y in
                                                                                          if el = None then (x, o)
                                                                                          else if el = Some X then (x + 1, o) else (x, o + 1))
                                                                                      _x_0 (_x_54 :: (… :: …))
                                                                                in
                                                                                let (_x_56 : int) = _x_55.0 in
                                                                                let (_x_57 : int) = _x_55.1 in
                                                                                let (_x_58 : bool) = _x_54 = _x_9 in
                                                                                let (_x_59 : bool) = … = _x_9 in
                                                                                let (_x_60 : bool) = ( :var_2: ) = C in
                                                                                let (_x_61 : player option) = Some ( :var_1: ) in
                                                                                let (_x_62 : grid)
                                                                                    = {a = _x_1; b = _x_61; c = _x_12; d = _x_15; e = _x_17; f = _x_19;
                                                                                       g = _x_22; h = _x_24; i = _x_26}
                                                                                in
                                                                                let (_x_63 : bool) = ( :var_2: ) = B in
                                                                                let (_x_64 : grid)
                                                                                    = {a = _x_61; b = …; c = _x_12; d = _x_15; e = _x_17; f = _x_19;
                                                                                       g = _x_22; h = _x_24; i = _x_26}
                                                                                in
                                                                                let (_x_65 : bool) = ( :var_2: ) = A in
                                                                                let (_x_66 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).c
                                                                                in
                                                                                let (_x_67 : bool) = _x_66 = _x_9 in
                                                                                let (_x_68 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).d
                                                                                in
                                                                                let (_x_69 : bool) = _x_68 = _x_9 in
                                                                                let (_x_70 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).e
                                                                                in
                                                                                let (_x_71 : bool) = _x_70 = _x_9 in
                                                                                let (_x_72 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).f
                                                                                in
                                                                                let (_x_73 : bool) = _x_72 = _x_9 in
                                                                                let (_x_74 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).g
                                                                                in
                                                                                let (_x_75 : bool) = _x_74 = _x_9 in
                                                                                let (_x_76 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).h
                                                                                in
                                                                                let (_x_77 : bool) = _x_76 = _x_9 in
                                                                                let (_x_78 : player option)
                                                                                    = (if _x_65 then _x_64
                                                                                       else if _x_63 then _x_62 else if _x_60 then … else …).i
                                                                                in
                                                                                let (_x_79 : bool) = _x_78 = _x_9 in
                                                                                let (_x_80 : bool)
                                                                                    = not
                                                                                      ((((((((_x_58 && _x_59) && _x_67 || (_x_69 && _x_71) && _x_73)
                                                                                            || (_x_75 && _x_77) && _x_79)
                                                                                           || (_x_58 && _x_69) && _x_75)
                                                                                          || (_x_59 && _x_71) && _x_77)
                                                                                         || (_x_67 && _x_73) && _x_79)
                                                                                        || (_x_58 && _x_71) && _x_79)
                                                                                       || (_x_67 && _x_71) && _x_75)
                                                                                in
                                                                                let (_x_81 : bool) = _x_54 = _x_8 in
                                                                                let (_x_82 : bool) = … = _x_8 in
                                                                                let (_x_83 : bool) = _x_66 = _x_8 in
                                                                                let (_x_84 : bool) = _x_68 = _x_8 in
                                                                                let (_x_85 : bool) = _x_70 = _x_8 in
                                                                                let (_x_86 : bool) = _x_72 = _x_8 in
                                                                                let (_x_87 : bool) = _x_74 = _x_8 in
                                                                                let (_x_88 : bool) = _x_76 = _x_8 in
                                                                                let (_x_89 : bool) = _x_78 = _x_8 in
                                                                                let (_x_90 : bool)
                                                                                    = (((((((_x_81 && _x_82) && _x_83 || (_x_84 && _x_85) && _x_86)
                                                                                           || (_x_87 && _x_88) && _x_89)
                                                                                          || (_x_81 && _x_84) && _x_87)
                                                                                         || (_x_82 && _x_85) && _x_88)
                                                                                        || (_x_83 && _x_86) && _x_89)
                                                                                       || (_x_81 && _x_85) && _x_89)
                                                                                      || (_x_83 && _x_85) && _x_87
                                                                                in
                                                                                not
                                                                                (((((_x_4 + _x_5 = 0 && _x_7 || _x_4 = _x_5 && _x_6 = _x_8)
                                                                                    || _x_4 + (-1) * _x_5 = 1 && _x_6 = _x_9)
                                                                                   && (_x_34 && not _x_52 || _x_34 = _x_52))
                                                                                  && not
                                                                                     ((((((((((((((((List.for_all ((<>) None) _x_2 || _x_14) || _x_21)
                                                                                                   || _x_28)
                                                                                                  || _x_29)
                                                                                                 || _x_30)
                                                                                                || _x_31)
                                                                                               || _x_32)
                                                                                              || _x_33)
                                                                                             || _x_38)
                                                                                            || _x_42)
                                                                                           || _x_46)
                                                                                          || _x_47)
                                                                                         || _x_48)
                                                                                        || _x_49)
                                                                                       || _x_50)
                                                                                      || _x_51))
                                                                                 && (_x_7 && _x_53 || _x_6 = Some (if _x_53 then O else X)))
                                                                                || (_x_56 + (-1) * _x_57 = 1 && _x_53 || _x_56 = _x_57 && ( :var_1: ) = O)
                                                                                   && (_x_80 && not _x_90 || _x_80 = _x_90)
                                                                                expansions:
                                                                                []
                                                                                rewrite_steps:
                                                                                  forward_chaining:
                                                                                  • unroll
                                                                                    expr:
                                                                                    (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                    (|rec_mk.grid_2265/server|
                                                                                          …
                                                                                    expansions:
                                                                                    • unroll
                                                                                      expr:
                                                                                      (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                       (|::| (i_1358/…
                                                                                      expansions:
                                                                                      • unroll
                                                                                        expr:
                                                                                        (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                         (|::| (i_1358/…
                                                                                        expansions:
                                                                                        • unroll
                                                                                          expr:
                                                                                          (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                          (|rec_mk.grid_2265/server|
                                                                                                …
                                                                                          expansions:
                                                                                          • unroll
                                                                                            expr:
                                                                                            (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                             (|::| (i_1358/…
                                                                                            expansions:
                                                                                            • unroll
                                                                                              expr:
                                                                                              (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                              (|rec_mk.grid_2265/server|
                                                                                                    …
                                                                                              expansions:
                                                                                              • unroll
                                                                                                expr:
                                                                                                (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                 (|::| (i_1358/…
                                                                                                expansions:
                                                                                                • unroll
                                                                                                  expr:
                                                                                                  (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                                  (|rec_mk.grid_2265/server|
                                                                                                        …
                                                                                                  expansions:
                                                                                                  • unroll
                                                                                                    expr:
                                                                                                    (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                     (|::| (i_1358/…
                                                                                                    expansions:
                                                                                                    • unroll
                                                                                                      expr:
                                                                                                      (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                       (|::| (i_1358/…
                                                                                                      expansions:
                                                                                                      • unroll
                                                                                                        expr:
                                                                                                        (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                                        (|rec_mk.grid_2265/server|
                                                                                                              …
                                                                                                        expansions:
                                                                                                        • unroll
                                                                                                          expr:
                                                                                                          (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                           (|::| (i_1358/…
                                                                                                          expansions:
                                                                                                          • unroll
                                                                                                            expr:
                                                                                                            (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                                            (|rec_mk.grid_2265/server|
                                                                                                                  …
                                                                                                            expansions:
                                                                                                            • unroll
                                                                                                              expr:
                                                                                                              (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                               (|::| (i_1358/…
                                                                                                              expansions:
                                                                                                              • unroll
                                                                                                                expr:
                                                                                                                (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                                 (|::| (i_1358/…
                                                                                                                expansions:
                                                                                                                • unroll
                                                                                                                  expr:
                                                                                                                  (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                                                  (|rec_mk.grid_2265/server|
                                                                                                                        …
                                                                                                                  expansions:
                                                                                                                  • unroll
                                                                                                                    expr:
                                                                                                                    (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                                     (|::| (i_1358/…
                                                                                                                    expansions:
                                                                                                                    • unroll
                                                                                                                      expr:
                                                                                                                      (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                                                      (|rec_mk.grid_2265/server|
                                                                                                                            …
                                                                                                                      expansions:
                                                                                                                      • unroll
                                                                                                                        expr:
                                                                                                                        (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                                         (|::| (i_1358/…
                                                                                                                        expansions:
                                                                                                                        • unroll
                                                                                                                          expr:
                                                                                                                          (let ((a!1 (|::| (h_1357/client (grid_1375/client game_1488/client))
                                                                                                                                           (|::| (i_1358/…
                                                                                                                          expansions:
                                                                                                                          • unroll
                                                                                                                            expr:
                                                                                                                            (let ((a!1 (ite (= move_1490/client H_1370/client)
                                                                                                                                            (|rec_mk.grid_2265/server|
                                                                                                                                  …
                                                                                                                            expansions:
                                                                                                                            • unroll
                                                                                                                              expr:
                                                                                                                              (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                                                                                                                              expansions:
                                                                                                                              • unroll
                                                                                                                                expr:
                                                                                                                                (|`List.fold_right { Resolved_term.spec_idx = 0; spec_arg = (v.cls (cls_ty: player option -> (int * …
                                                                                                                                expansions:
                                                                                                                                • Sat (Some let game : game_state = let (_x_0 : player option) = Some X in let (_x_1 : player option) = Some O in {grid = {a = None; b = _x_0; c = None; d = _x_0; e = None; f = _x_0; g = _x_1; h = _x_1; i = None}; last_player = _x_0} let player : player = O let move : move = B )

                                                                                                                                Uh oh! What's gone wrong?

                                                                                                                                In this case, the problem is clear - the player is trying to play a move in a square that's already occupied:

                                                                                                                                In [24]:
                                                                                                                                value(CX.game, CX.move)
                                                                                                                                
                                                                                                                                Out[24]:
                                                                                                                                - : square_state = Some X
                                                                                                                                

                                                                                                                                If you try out the initial version of the game based on the code so far, you'll see the bug if you try and make a move in a square that's already filled.

                                                                                                                                Imandra has spotted the issue for us from a straightforward statement about the logic of the game!

                                                                                                                                The culprit is the is_valid_move function - we don't check that the square is empty. Let's add && value(game, move) == None to our check:

                                                                                                                                In [25]:
                                                                                                                                let is_valid_move = (game, player, move) =>
                                                                                                                                  ! (is_winning(game, X) || is_winning(game, O) || is_tie(game))
                                                                                                                                  && is_valid_game(game)
                                                                                                                                  && (
                                                                                                                                    game.last_player == None
                                                                                                                                    && player == initial_player
                                                                                                                                    || game.last_player == Some(other_player(player))
                                                                                                                                  )
                                                                                                                                  && value(game, move) == None
                                                                                                                                ;
                                                                                                                                
                                                                                                                                Out[25]:
                                                                                                                                val is_valid_move : game_state -> player -> move -> bool = <fun>
                                                                                                                                

                                                                                                                                ... and verify once more (using [@blast] to speed it up):

                                                                                                                                In [26]:
                                                                                                                                [@blast]
                                                                                                                                verify((game, player, move) =>
                                                                                                                                  (is_valid_game(game) && is_valid_move(game, player, move))
                                                                                                                                  ==> is_valid_game(play_move(game, player, move))
                                                                                                                                );
                                                                                                                                
                                                                                                                                Out[26]:
                                                                                                                                - : game_state -> player -> move -> bool = <fun>
                                                                                                                                
                                                                                                                                proof from "blast" (after 1.22s)!
                                                                                                                                Proved

                                                                                                                                Much better! You can play the final version here, and you'll see that the bug is fixed. Hopefully this gives you a feel for an interactive development workflow with Imandra, in particular how seeing concrete examples can be really useful to understanding what's going on.