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 -> Imandra_surface.Document.t = <fun>
val doc_of_grid : grid -> Imandra_surface.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 -> Imandra_surface.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, 0.025s):
 let (g : game_state) =
   {grid =
     {a = (Some X); b = (Some X); c = (Some X); d = (Some X); e = (Some X);
      f = (Some X); g = (Some X); h = None; i = None};
    last_player = None}
Instance
proof attempt
ground_instances0
definitions0
inductions0
search_time
0.025s
details
Expand
smt_stats
num checks1
rlimit count1594
mk clause40
datatype occurs check1
mk bool var149
datatype splits41
decisions39
propagations32
conflicts7
datatype accessor ax7
datatype constructor ax58
num allocs3255253
final checks1
added eqs275
memory7.500000
max memory8.060000
Expand
  • start[0.025s]
      (:var_0:.grid.a, :var_0:.grid.b, :var_0:.grid.c) = (Some X, Some X, Some X)
      || (:var_0:.grid.d, :var_0:.grid.e, :var_0:.grid.f) =
         (Some X, Some X, Some X)
         || (:var_0:.grid.g, :var_0:.grid.h, :var_0:.grid.i) =
            (Some X, Some X, Some X)
            || (:var_0:.grid.a, :var_0:.grid.d, :var_0:.grid.g) =
               (Some X, Some X, Some X)
               || (:var_0:.grid.b, :var_0:.grid.e, :var_0:.grid.h) =
                  (Some X, Some X, Some X)
                  || (:var_0:.grid.c, :var_0:.grid.f, :var_0:.grid.i) =
                     (Some X, Some X, Some X)
                     || (:var_0:.grid.a, :var_0:.grid.e, :var_0:.grid.i) =
                        (Some X, Some X, Some X)
                        || (:var_0:.grid.c, :var_0:.grid.e, :var_0:.grid.g) =
                           (Some X, Some X, Some X)
  • simplify

    into
    (((((((:var_0:.grid.a = Some X && :var_0:.grid.b = Some X)
          && :var_0:.grid.c = Some X
          || (:var_0:.grid.d = Some X && :var_0:.grid.e = Some X)
             && :var_0:.grid.f = Some X)
         || (:var_0:.grid.g = Some X && :var_0:.grid.h = Some X)
            && :var_0:.grid.i = Some X)
        || (:var_0:.grid.a = Some X && :var_0:.grid.d = Some X)
           && :var_0:.grid.g = Some X)
       || (:var_0:.grid.b = Some X && :var_0:.grid.e = Some X)
          && :var_0:.grid.h = Some X)
      || (:var_0:.grid.c = Some X && :var_0:.grid.f = Some X)
         && :var_0:.grid.i = Some X)
     || (:var_0:.grid.a = Some X && :var_0:.grid.e = Some X)
        && :var_0:.grid.i = Some X)
    || (:var_0:.grid.c = Some X && :var_0:.grid.e = Some X)
       && :var_0:.grid.g = Some X
    expansions
    []
    rewrite_steps
      forward_chaining
      • Sat (Some let (g : game_state) = {grid = {a = (Some X); b = (Some X); c = (Some X); d = (Some X); e = (Some X); f = (Some X); g = (Some X); h = None; 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 */
      /* with these rules, either player can go first so we check that one player is never too far ahead */
      let is_valid_grid = (grid, last_player) => {
        let (x, o) = move_counts(grid);
        if (x > o) {
          last_player == Some(X) && x - o == 1;
        } else if (x < o) {
          last_player == Some(O) && o - x == 1;
        } else {
          true;
        };
      };
      
      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, 0.046s):
       let (game : game_state) =
         {grid =
           {a = (Some O); b = (Some X); c = (Some O); d = None; e = (Some O);
            f = None; g = (Some X); h = (Some X); i = (Some X)};
          last_player = (Some X)}
      
      Instance
      proof attempt
      ground_instances10
      definitions0
      inductions0
      search_time
      0.046s
      details
      Expand
      smt_stats
      arith offset eqs342
      num checks21
      arith assert lower291
      arith pivots147
      rlimit count43310
      mk clause311
      datatype occurs check225
      mk bool var1624
      arith assert upper279
      datatype splits130
      decisions528
      arith add rows694
      arith bound prop27
      propagations1055
      interface eqs9
      conflicts43
      arith fixed eqs310
      datatype accessor ax122
      arith assert diseq48
      datatype constructor ax720
      num allocs7432788
      final checks20
      added eqs4340
      del clause158
      arith eq adapter211
      memory12.140000
      max memory12.140000
      Expand
      • start[0.046s]
          if (List.fold_right
              (fun el _x
               -> let (x : int) = _x.0 and (o : int) = _x.1 in
                  if el = None then (x, o)
                  else
                  if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                  else (x, o + 1))
              (0, 0) (….a :: (… :: …))).0
             >
             (List.fold_right
              (fun el _x
               -> let (x : int) = _x.0 and (o : int) = _x.1 in
                  if el = None then (x, o)
                  else
                  if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                  else (x, o + 1))
              (0, 0) (….a :: (… :: …))).1
          then
            :var_0:.last_player = Some X
            && (List.fold_right
                (fun el _x
                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                    if el = None then (x, o)
                    else
                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                    else (x, o + 1))
                (0, 0) (….a :: (… :: …))).0
               -
               (List.fold_right
                (fun el _x
                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                    if el = None then (x, o)
                    else
                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                    else (x, o + 1))
                (0, 0) (….a :: (… :: …))).1
               = 1
          else
          if (List.fold_right
              (fun el _x
               -> let (x : int) = _x.0 and (o : int) = _x.1 in
                  if el = None then (x, o)
                  else
                  if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                  else (x, o + 1))
              (0, 0) (….a :: (… :: …))).0
             <
             (List.fold_right
              (fun el _x
               -> let (x : int) = _x.0 and (o : int) = _x.1 in
                  if el = None then (x, o)
                  else
                  if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                  else (x, o + 1))
              (0, 0) (….a :: (… :: …))).1
          then
            :var_0:.last_player = Some O
            && (List.fold_right
                (fun el _x
                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                    if el = None then (x, o)
                    else
                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                    else (x, o + 1))
                (0, 0) (….a :: (… :: …))).1
               -
               (List.fold_right
                (fun el _x
                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                    if el = None then (x, o)
                    else
                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                    else (x, o + 1))
                (0, 0) (….a :: (… :: …))).0
               = 1
          else true
      • simplify

        into
        if (List.fold_right
            (fun el _x
             -> let (x : int) = _x.0 and (o : int) = _x.1 in
                if el = None then (x, o)
                else
                if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                else (x, o + 1))
            (0, 0) (….a :: (… :: …))).0
           <=
           (List.fold_right
            (fun el _x
             -> let (x : int) = _x.0 and (o : int) = _x.1 in
                if el = None then (x, o)
                else
                if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                else (x, o + 1))
            (0, 0) (….a :: (… :: …))).1
        then
          (List.fold_right
           (fun el _x
            -> let (x : int) = _x.0 and (o : int) = _x.1 in
               if el = None then (x, o)
               else
               if Is_a(Some, el) && Option.get el = X then (x + 1, o) else (x, o + 1))
           (0, 0) (….a :: (… :: …))).1
          <=
          (List.fold_right
           (fun el _x
            -> let (x : int) = _x.0 and (o : int) = _x.1 in
               if el = None then (x, o)
               else
               if Is_a(Some, el) && Option.get el = X then (x + 1, o) else (x, o + 1))
           (0, 0) (….a :: (… :: …))).0
          || :var_0:.last_player = Some O
             && (List.fold_right
                 (fun el _x
                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                     if el = None then (x, o)
                     else
                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                     else (x, o + 1))
                 (0, 0) (….a :: (… :: …))).1
                + -1
                  * (List.fold_right
                     (fun el _x
                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                         if el = None then (x, o)
                         else
                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                         else (x, o + 1))
                     (0, 0) (….a :: (… :: …))).0
                = 1
        else
          :var_0:.last_player = Some X
          && (List.fold_right
              (fun el _x
               -> let (x : int) = _x.0 and (o : int) = _x.1 in
                  if el = None then (x, o)
                  else
                  if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                  else (x, o + 1))
              (0, 0) (….a :: (… :: …))).0
             + -1
               * (List.fold_right
                  (fun el _x
                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                      if el = None then (x, o)
                      else
                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                      else (x, o + 1))
                  (0, 0) (….a :: (… :: …))).1
             = 1
        expansions
        []
        rewrite_steps
          forward_chaining
          • unroll
            expr
            (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                               (|::_3| (i_27 (grid_40 game_99)) |[]_…
            expansions
            • unroll
              expr
              (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                 (|::_3| (i_27 (grid_40 game_99)) |[]_…
              expansions
              • unroll
                expr
                (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                   (|::_3| (i_27 (grid_40 game_99)) |[]_…
                expansions
                • unroll
                  expr
                  (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                     (|::_3| (i_27 (grid_40 game_99)) |[]_…
                  expansions
                  • unroll
                    expr
                    (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                       (|::_3| (i_27 (grid_40 game_99)) |[]_…
                    expansions
                    • unroll
                      expr
                      (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                         (|::_3| (i_27 (grid_40 game_99)) |[]_…
                      expansions
                      • unroll
                        expr
                        (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                           (|::_3| (i_27 (grid_40 game_99)) |[]_…
                        expansions
                        • unroll
                          expr
                          (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                             (|::_3| (i_27 (grid_40 game_99)) |[]_…
                          expansions
                          • unroll
                            expr
                            (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                               (|::_3| (i_27 (grid_40 game_99)) |[]_…
                            expansions
                            • unroll
                              expr
                              (let ((a!1 (|::_3| (h_26 (grid_40 game_99))
                                                 (|::_3| (i_27 (grid_40 game_99)) |[]_…
                              expansions
                              • Sat (Some let (game : game_state) = {grid = {a = (Some O); b = (Some X); c = (Some O); d = None; e = (Some O); f = None; g = (Some X); h = (Some X); i = (Some X)}; last_player = (Some X)} )
                              In [13]:
                              CX.game;
                              
                              Out[13]:
                              - : game_state = <document>
                              
                              ··
                              Last:

                              This is a bit better - the grid looks more reasonable now. Interestingly, we notice that the valid grid Imandra gave us is also 'winning'. We can see this by computing with the CX:

                              In [14]:
                              is_winning(CX.game, O);
                              
                              Out[14]:
                              - : bool = false
                              

                              This begs the question - have we inadvertantly specified that every valid game is a winning game? Let's ask Imandra for an instance where this isn't the case to make sure:

                              In [15]:
                              instance((game, player) => is_valid_grid(game.grid, game.last_player) && !is_winning(game, player));
                              
                              Out[15]:
                              - : game_state -> player -> bool = <fun>
                              module CX : sig val game : game_state val player : player end
                              
                              Instance (after 10 steps, 0.052s):
                               let (game : game_state) =
                                 {grid =
                                   {a = (Some O); b = None; c = (Some X); d = None; e = (Some X); f = None;
                                    g = (Some X); h = (Some O); i = None};
                                  last_player = (Some X)}
                               let (player : player) = O
                              
                              Instance
                              proof attempt
                              ground_instances10
                              definitions0
                              inductions0
                              search_time
                              0.052s
                              details
                              Expand
                              smt_stats
                              arith offset eqs607
                              num checks21
                              arith assert lower469
                              arith pivots247
                              rlimit count89593
                              mk clause574
                              datatype occurs check212
                              mk bool var2036
                              arith assert upper439
                              datatype splits152
                              decisions858
                              arith add rows1779
                              arith bound prop90
                              propagations1879
                              interface eqs10
                              conflicts83
                              arith fixed eqs551
                              datatype accessor ax141
                              minimized lits25
                              arith conflicts11
                              arith assert diseq69
                              datatype constructor ax665
                              num allocs14998412
                              final checks21
                              added eqs6179
                              del clause272
                              arith eq adapter355
                              memory16.520000
                              max memory16.520000
                              Expand
                              • start[0.052s]
                                  (if (List.fold_right
                                       (fun el _x
                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                           if el = None then (x, o)
                                           else
                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                           else (x, o + 1))
                                       (0, 0) (….a :: (… :: …))).0
                                      >
                                      (List.fold_right
                                       (fun el _x
                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                           if el = None then (x, o)
                                           else
                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                           else (x, o + 1))
                                       (0, 0) (….a :: (… :: …))).1
                                   then
                                     :var_0:.last_player = Some X
                                     && (List.fold_right
                                         (fun el _x
                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                             if el = None then (x, o)
                                             else
                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                             else (x, o + 1))
                                         (0, 0) (….a :: (… :: …))).0
                                        -
                                        (List.fold_right
                                         (fun el _x
                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                             if el = None then (x, o)
                                             else
                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                             else (x, o + 1))
                                         (0, 0) (….a :: (… :: …))).1
                                        = 1
                                   else
                                   if (List.fold_right
                                       (fun el _x
                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                           if el = None then (x, o)
                                           else
                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                           else (x, o + 1))
                                       (0, 0) (….a :: (… :: …))).0
                                      <
                                      (List.fold_right
                                       (fun el _x
                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                           if el = None then (x, o)
                                           else
                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                           else (x, o + 1))
                                       (0, 0) (….a :: (… :: …))).1
                                   then
                                     :var_0:.last_player = Some O
                                     && (List.fold_right
                                         (fun el _x
                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                             if el = None then (x, o)
                                             else
                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                             else (x, o + 1))
                                         (0, 0) (….a :: (… :: …))).1
                                        -
                                        (List.fold_right
                                         (fun el _x
                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                             if el = None then (x, o)
                                             else
                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                             else (x, o + 1))
                                         (0, 0) (….a :: (… :: …))).0
                                        = 1
                                   else true)
                                  && not
                                     ((….a, …, ….c) = (Some :var_1:, Some :var_1:, Some :var_1:)
                                      || (….d, ….e, ….f) = (Some :var_1:, Some :var_1:, Some :var_1:)
                                         || (….g, ….h, ….i) =
                                            (Some :var_1:, Some :var_1:, Some :var_1:)
                                            || (….a, ….d, ….g) =
                                               (Some :var_1:, Some :var_1:, Some :var_1:)
                                               || (…, ….e, ….h) =
                                                  (Some :var_1:, Some :var_1:, Some :var_1:)
                                                  || (….c, ….f, ….i) =
                                                     (Some :var_1:, Some :var_1:, Some :var_1:)
                                                     || (….a, ….e, ….i) =
                                                        (Some :var_1:, Some :var_1:, Some :var_1:)
                                                        || (….c, ….e, ….g) =
                                                           (Some :var_1:, Some :var_1:, Some :var_1:))
                              • simplify

                                into
                                (if (List.fold_right
                                     (fun el _x
                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                         if el = None then (x, o)
                                         else
                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                         else (x, o + 1))
                                     (0, 0) (….a :: (… :: …))).0
                                    <=
                                    (List.fold_right
                                     (fun el _x
                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                         if el = None then (x, o)
                                         else
                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                         else (x, o + 1))
                                     (0, 0) (….a :: (… :: …))).1
                                 then
                                   (List.fold_right
                                    (fun el _x
                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                        if el = None then (x, o)
                                        else
                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                        else (x, o + 1))
                                    (0, 0) (….a :: (… :: …))).1
                                   <=
                                   (List.fold_right
                                    (fun el _x
                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                        if el = None then (x, o)
                                        else
                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                        else (x, o + 1))
                                    (0, 0) (….a :: (… :: …))).0
                                   || :var_0:.last_player = Some O
                                      && (List.fold_right
                                          (fun el _x
                                           -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                              if el = None then (x, o)
                                              else
                                              if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                              else (x, o + 1))
                                          (0, 0) (….a :: (… :: …))).1
                                         + -1
                                           * (List.fold_right
                                              (fun el _x
                                               -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                  if el = None then (x, o)
                                                  else
                                                  if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                  else (x, o + 1))
                                              (0, 0) (….a :: (… :: …))).0
                                         = 1
                                 else
                                   (:var_0:.last_player = Some X
                                    && (List.fold_right
                                        (fun el _x
                                         -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                            if el = None then (x, o)
                                            else
                                            if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                            else (x, o + 1))
                                        (0, 0) (….a :: (… :: …))).0
                                       + -1
                                         * (List.fold_right
                                            (fun el _x
                                             -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                if el = None then (x, o)
                                                else
                                                if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                else (x, o + 1))
                                            (0, 0) (….a :: (… :: …))).1
                                       = 1))
                                && not
                                   ((((((((….a = Some :var_1: && … = Some :var_1:)
                                          && ….c = Some :var_1:
                                          || (….d = Some :var_1: && ….e = Some :var_1:)
                                             && ….f = Some :var_1:)
                                         || (….g = Some :var_1: && ….h = Some :var_1:)
                                            && ….i = Some :var_1:)
                                        || (….a = Some :var_1: && ….d = Some :var_1:)
                                           && ….g = Some :var_1:)
                                       || (… = Some :var_1: && ….e = Some :var_1:)
                                          && ….h = Some :var_1:)
                                      || (….c = Some :var_1: && ….f = Some :var_1:)
                                         && ….i = Some :var_1:)
                                     || (….a = Some :var_1: && ….e = Some :var_1:)
                                        && ….i = Some :var_1:)
                                    || (….c = Some :var_1: && ….e = Some :var_1:) && ….g = Some :var_1:)
                                expansions
                                []
                                rewrite_steps
                                  forward_chaining
                                  • unroll
                                    expr
                                    (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                       (|::_3| (i_27 (grid_40 game_101)) |[…
                                    expansions
                                    • unroll
                                      expr
                                      (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                         (|::_3| (i_27 (grid_40 game_101)) |[…
                                      expansions
                                      • unroll
                                        expr
                                        (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                           (|::_3| (i_27 (grid_40 game_101)) |[…
                                        expansions
                                        • unroll
                                          expr
                                          (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                             (|::_3| (i_27 (grid_40 game_101)) |[…
                                          expansions
                                          • unroll
                                            expr
                                            (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                               (|::_3| (i_27 (grid_40 game_101)) |[…
                                            expansions
                                            • unroll
                                              expr
                                              (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                                 (|::_3| (i_27 (grid_40 game_101)) |[…
                                              expansions
                                              • unroll
                                                expr
                                                (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                                   (|::_3| (i_27 (grid_40 game_101)) |[…
                                                expansions
                                                • unroll
                                                  expr
                                                  (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                                     (|::_3| (i_27 (grid_40 game_101)) |[…
                                                  expansions
                                                  • unroll
                                                    expr
                                                    (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                                       (|::_3| (i_27 (grid_40 game_101)) |[…
                                                    expansions
                                                    • unroll
                                                      expr
                                                      (let ((a!1 (|::_3| (h_26 (grid_40 game_101))
                                                                         (|::_3| (i_27 (grid_40 game_101)) |[…
                                                      expansions
                                                      • Sat (Some let (game : game_state) = {grid = {a = (Some O); b = None; c = (Some X); d = None; e = (Some X); f = None; g = (Some X); h = (Some O); i = None}; last_player = (Some X)} let (player : player) = O )
                                                      In [16]:
                                                      CX.game
                                                      
                                                      Out[16]:
                                                      - : game_state = <document>
                                                      
                                                      ·
                                                      ··
                                                      ·
                                                      Last:

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

                                                      In [17]:
                                                      instance((game) => {
                                                         let (x, o) = move_counts(game.grid);
                                                         is_valid_grid(game.grid, game.last_player) && ((x - o) >= 2)
                                                      });
                                                      
                                                      Out[17]:
                                                      - : game_state -> bool = <fun>
                                                      
                                                      Unsatisfiable
                                                      proof
                                                      ground_instances0
                                                      definitions0
                                                      inductions0
                                                      search_time
                                                      0.015s
                                                      details
                                                      Expand
                                                      smt_stats
                                                      num checks1
                                                      arith assert lower1
                                                      arith pivots1
                                                      rlimit count1027
                                                      mk clause29
                                                      mk bool var113
                                                      datatype splits49
                                                      propagations7
                                                      conflicts1
                                                      datatype accessor ax19
                                                      datatype constructor ax4
                                                      num allocs24188350
                                                      added eqs94
                                                      arith eq adapter2
                                                      memory19.760000
                                                      max memory19.760000
                                                      Expand
                                                      • start[0.015s]
                                                          (if (List.fold_right
                                                               (fun el _x
                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                   if el = None then (x, o)
                                                                   else
                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                   else (x, o + 1))
                                                               (0, 0) (….a :: (… :: …))).0
                                                              >
                                                              (List.fold_right
                                                               (fun el _x
                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                   if el = None then (x, o)
                                                                   else
                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                   else (x, o + 1))
                                                               (0, 0) (….a :: (… :: …))).1
                                                           then
                                                             :var_0:.last_player = Some X
                                                             && (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).0
                                                                -
                                                                (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).1
                                                                = 1
                                                           else
                                                           if (List.fold_right
                                                               (fun el _x
                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                   if el = None then (x, o)
                                                                   else
                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                   else (x, o + 1))
                                                               (0, 0) (….a :: (… :: …))).0
                                                              <
                                                              (List.fold_right
                                                               (fun el _x
                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                   if el = None then (x, o)
                                                                   else
                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                   else (x, o + 1))
                                                               (0, 0) (….a :: (… :: …))).1
                                                           then
                                                             :var_0:.last_player = Some O
                                                             && (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).1
                                                                -
                                                                (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).0
                                                                = 1
                                                           else true)
                                                          && ((List.fold_right
                                                               (fun el _x
                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                   if el = None then (x, o)
                                                                   else
                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                   else (x, o + 1))
                                                               (0, 0) (….a :: (… :: …))).0
                                                              -
                                                              (List.fold_right
                                                               (fun el _x
                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                   if el = None then (x, o)
                                                                   else
                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                   else (x, o + 1))
                                                               (0, 0) (….a :: (… :: …))).1)
                                                             >= 2
                                                      • simplify

                                                        into
                                                        (if (List.fold_right
                                                             (fun el _x
                                                              -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                 if el = None then (x, o)
                                                                 else
                                                                 if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                 else (x, o + 1))
                                                             (0, 0) (….a :: (… :: …))).0
                                                            <=
                                                            (List.fold_right
                                                             (fun el _x
                                                              -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                 if el = None then (x, o)
                                                                 else
                                                                 if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                 else (x, o + 1))
                                                             (0, 0) (….a :: (… :: …))).1
                                                         then
                                                           (List.fold_right
                                                            (fun el _x
                                                             -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                if el = None then (x, o)
                                                                else
                                                                if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                else (x, o + 1))
                                                            (0, 0) (….a :: (… :: …))).1
                                                           <=
                                                           (List.fold_right
                                                            (fun el _x
                                                             -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                if el = None then (x, o)
                                                                else
                                                                if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                else (x, o + 1))
                                                            (0, 0) (….a :: (… :: …))).0
                                                           || :var_0:.last_player = Some O
                                                              && (List.fold_right
                                                                  (fun el _x
                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                      if el = None then (x, o)
                                                                      else
                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                      else (x, o + 1))
                                                                  (0, 0) (….a :: (… :: …))).1
                                                                 + -1
                                                                   * (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).0
                                                                 = 1
                                                         else
                                                           (:var_0:.last_player = Some X
                                                            && (List.fold_right
                                                                (fun el _x
                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                    if el = None then (x, o)
                                                                    else
                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                    else (x, o + 1))
                                                                (0, 0) (….a :: (… :: …))).0
                                                               + -1
                                                                 * (List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).1
                                                               = 1))
                                                        && ((List.fold_right
                                                             (fun el _x
                                                              -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                 if el = None then (x, o)
                                                                 else
                                                                 if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                 else (x, o + 1))
                                                             (0, 0) (….a :: (… :: …))).0
                                                            + -1
                                                              * (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).1)
                                                           >= 2
                                                        expansions
                                                        []
                                                        rewrite_steps
                                                          forward_chaining
                                                          • unsat

                                                            (let ((a!1 (|::_3| (h_26 (grid_40 game_104))
                                                                               (|::_3| (i_27 (grid_40 game_104)) |[…

                                                          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 like a test.

                                                          In [18]:
                                                          verify((game) => {
                                                             let (x, o) = move_counts(game.grid);
                                                             not(is_valid_grid(game.grid, game.last_player) && ((x - o) >= 2))
                                                          });
                                                          
                                                          Out[18]:
                                                          - : game_state -> bool = <fun>
                                                          
                                                          Proved
                                                          proof
                                                          ground_instances0
                                                          definitions0
                                                          inductions0
                                                          search_time
                                                          0.011s
                                                          details
                                                          Expand
                                                          smt_stats
                                                          num checks1
                                                          arith assert lower1
                                                          arith pivots1
                                                          rlimit count1035
                                                          mk clause29
                                                          mk bool var113
                                                          datatype splits49
                                                          propagations7
                                                          conflicts1
                                                          datatype accessor ax19
                                                          datatype constructor ax4
                                                          num allocs36237559
                                                          added eqs94
                                                          arith eq adapter2
                                                          memory23.230000
                                                          max memory23.230000
                                                          Expand
                                                          • start[0.011s]
                                                              not
                                                              ((if (List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).0
                                                                   >
                                                                   (List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).1
                                                                then
                                                                  :var_0:.last_player = Some X
                                                                  && (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).0
                                                                     -
                                                                     (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).1
                                                                     = 1
                                                                else
                                                                if (List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).0
                                                                   <
                                                                   (List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).1
                                                                then
                                                                  :var_0:.last_player = Some O
                                                                  && (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).1
                                                                     -
                                                                     (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).0
                                                                     = 1
                                                                else true)
                                                               && ((List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).0
                                                                   -
                                                                   (List.fold_right
                                                                    (fun el _x
                                                                     -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                        if el = None then (x, o)
                                                                        else
                                                                        if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                        else (x, o + 1))
                                                                    (0, 0) (….a :: (… :: …))).1)
                                                                  >= 2)
                                                          • simplify

                                                            into
                                                            not
                                                            ((if (List.fold_right
                                                                  (fun el _x
                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                      if el = None then (x, o)
                                                                      else
                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                      else (x, o + 1))
                                                                  (0, 0) (….a :: (… :: …))).0
                                                                 <=
                                                                 (List.fold_right
                                                                  (fun el _x
                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                      if el = None then (x, o)
                                                                      else
                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                      else (x, o + 1))
                                                                  (0, 0) (….a :: (… :: …))).1
                                                              then
                                                                (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).1
                                                                <=
                                                                (List.fold_right
                                                                 (fun el _x
                                                                  -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                     if el = None then (x, o)
                                                                     else
                                                                     if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                     else (x, o + 1))
                                                                 (0, 0) (….a :: (… :: …))).0
                                                                || :var_0:.last_player = Some O
                                                                   && (List.fold_right
                                                                       (fun el _x
                                                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                           if el = None then (x, o)
                                                                           else
                                                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                           else (x, o + 1))
                                                                       (0, 0) (….a :: (… :: …))).1
                                                                      + -1
                                                                        * (List.fold_right
                                                                           (fun el _x
                                                                            -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                               if el = None then (x, o)
                                                                               else
                                                                               if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                               else (x, o + 1))
                                                                           (0, 0) (….a :: (… :: …))).0
                                                                      = 1
                                                              else
                                                                (:var_0:.last_player = Some X
                                                                 && (List.fold_right
                                                                     (fun el _x
                                                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                         if el = None then (x, o)
                                                                         else
                                                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                         else (x, o + 1))
                                                                     (0, 0) (….a :: (… :: …))).0
                                                                    + -1
                                                                      * (List.fold_right
                                                                         (fun el _x
                                                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                             if el = None then (x, o)
                                                                             else
                                                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                             else (x, o + 1))
                                                                         (0, 0) (….a :: (… :: …))).1
                                                                    = 1))
                                                             && ((List.fold_right
                                                                  (fun el _x
                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                      if el = None then (x, o)
                                                                      else
                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                      else (x, o + 1))
                                                                  (0, 0) (….a :: (… :: …))).0
                                                                 + -1
                                                                   * (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).1)
                                                                >= 2)
                                                            expansions
                                                            []
                                                            rewrite_steps
                                                              forward_chaining
                                                              • unsat

                                                                (let ((a!1 (|::_3| (h_26 (grid_40 game_108))
                                                                                   (|::_3| (i_27 (grid_40 game_108)) |[…

                                                              Valid games

                                                              Let's continue adding some more definitions - we havent really covered winning criteria as part of is_valid_grid, so let's include a few:

                                                              In [19]:
                                                              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[19]:
                                                              val is_tie : game_state -> bool = <fun>
                                                              
                                                              In [20]:
                                                              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[20]:
                                                              val is_valid_game : game_state -> bool = <fun>
                                                              

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

                                                              In [21]:
                                                              instance((game) => is_valid_game(game) && is_winning(game, X));
                                                              
                                                              Out[21]:
                                                              - : game_state -> bool = <fun>
                                                              module CX : sig val game : game_state end
                                                              
                                                              Instance (after 10 steps, 0.052s):
                                                               let (game : game_state) =
                                                                 {grid =
                                                                   {a = (Some X); b = None; c = (Some O); d = (Some O); e = (Some X);
                                                                    f = None; g = (Some O); h = None; i = (Some X)};
                                                                  last_player = None}
                                                              
                                                              Instance
                                                              proof attempt
                                                              ground_instances10
                                                              definitions0
                                                              inductions0
                                                              search_time
                                                              0.052s
                                                              details
                                                              Expand
                                                              smt_stats
                                                              arith offset eqs470
                                                              num checks21
                                                              arith assert lower418
                                                              arith pivots269
                                                              rlimit count87823
                                                              mk clause609
                                                              datatype occurs check224
                                                              mk bool var1976
                                                              arith assert upper371
                                                              datatype splits176
                                                              decisions844
                                                              arith add rows1930
                                                              arith bound prop65
                                                              propagations2009
                                                              interface eqs7
                                                              conflicts106
                                                              arith fixed eqs427
                                                              datatype accessor ax136
                                                              minimized lits50
                                                              arith conflicts12
                                                              arith assert diseq63
                                                              datatype constructor ax642
                                                              num allocs56921594
                                                              final checks18
                                                              added eqs6359
                                                              del clause290
                                                              arith eq adapter318
                                                              memory28.400000
                                                              max memory28.460000
                                                              Expand
                                                              • start[0.052s]
                                                                  ((if (List.fold_right
                                                                        (fun el _x
                                                                         -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                            if el = None then (x, o)
                                                                            else
                                                                            if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                            else (x, o + 1))
                                                                        (0, 0) (….a :: (… :: …))).0
                                                                       >
                                                                       (List.fold_right
                                                                        (fun el _x
                                                                         -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                            if el = None then (x, o)
                                                                            else
                                                                            if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                            else (x, o + 1))
                                                                        (0, 0) (….a :: (… :: …))).1
                                                                    then
                                                                      :var_0:.last_player = Some X
                                                                      && (List.fold_right
                                                                          (fun el _x
                                                                           -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                              if el = None then (x, o)
                                                                              else
                                                                              if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                              else (x, o + 1))
                                                                          (0, 0) (….a :: (… :: …))).0
                                                                         -
                                                                         (List.fold_right
                                                                          (fun el _x
                                                                           -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                              if el = None then (x, o)
                                                                              else
                                                                              if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                              else (x, o + 1))
                                                                          (0, 0) (….a :: (… :: …))).1
                                                                         = 1
                                                                    else
                                                                    if (List.fold_right
                                                                        (fun el _x
                                                                         -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                            if el = None then (x, o)
                                                                            else
                                                                            if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                            else (x, o + 1))
                                                                        (0, 0) (….a :: (… :: …))).0
                                                                       <
                                                                       (List.fold_right
                                                                        (fun el _x
                                                                         -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                            if el = None then (x, o)
                                                                            else
                                                                            if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                            else (x, o + 1))
                                                                        (0, 0) (….a :: (… :: …))).1
                                                                    then
                                                                      :var_0:.last_player = Some O
                                                                      && (List.fold_right
                                                                          (fun el _x
                                                                           -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                              if el = None then (x, o)
                                                                              else
                                                                              if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                              else (x, o + 1))
                                                                          (0, 0) (….a :: (… :: …))).1
                                                                         -
                                                                         (List.fold_right
                                                                          (fun el _x
                                                                           -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                              if el = None then (x, o)
                                                                              else
                                                                              if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                              else (x, o + 1))
                                                                          (0, 0) (….a :: (… :: …))).0
                                                                         = 1
                                                                    else true)
                                                                   && (not
                                                                       ((….a, …, ….c) = (Some X, Some X, Some X)
                                                                        || (….d, ….e, ….f) = (Some X, Some X, Some X)
                                                                           || (….g, ….h, ….i) = (Some X, Some X, Some X)
                                                                              || (….a, ….d, ….g) = (Some X, Some X, Some X)
                                                                                 || (…, ….e, ….h) = (Some X, Some X, Some X)
                                                                                    || (….c, ….f, ….i) = (Some X, Some X, Some X)
                                                                                       || (….a, ….e, ….i) = (Some X, Some X, Some X)
                                                                                          || (….c, ….e, ….g) = (Some X, Some X, Some X))
                                                                       && not
                                                                          ((….a, …, ….c) = (Some O, Some O, Some O)
                                                                           || (….d, ….e, ….f) = (Some O, Some O, Some O)
                                                                              || (….g, ….h, ….i) = (Some O, Some O, Some O)
                                                                                 || (….a, ….d, ….g) = (Some O, Some O, Some O)
                                                                                    || (…, ….e, ….h) = (Some O, Some O, Some O)
                                                                                       || (….c, ….f, ….i) = (Some O, Some O, Some O)
                                                                                          || (….a, ….e, ….i) = (Some O, Some O, Some O)
                                                                                             || (….c, ….e, ….g) =
                                                                                                (Some O, Some O, Some O))
                                                                       || not
                                                                          (((….a, …, ….c) = (Some X, Some X, Some X)
                                                                            || (….d, ….e, ….f) = (Some X, Some X, Some X)
                                                                               || (….g, ….h, ….i) = (Some X, Some X, Some X)
                                                                                  || (….a, ….d, ….g) = (Some X, Some X, Some X)
                                                                                     || (…, ….e, ….h) = (Some X, Some X, Some X)
                                                                                        || (….c, ….f, ….i) = (Some X, Some X, Some X)
                                                                                           || (….a, ….e, ….i) =
                                                                                              (Some X, Some X, Some X)
                                                                                              || (….c, ….e, ….g) =
                                                                                                 (Some X, Some X, Some X))
                                                                           =
                                                                           ((….a, …, ….c) = (Some O, Some O, Some O)
                                                                            || (….d, ….e, ….f) = (Some O, Some O, Some O)
                                                                               || (….g, ….h, ….i) = (Some O, Some O, Some O)
                                                                                  || (….a, ….d, ….g) = (Some O, Some O, Some O)
                                                                                     || (…, ….e, ….h) = (Some O, Some O, Some O)
                                                                                        || (….c, ….f, ….i) = (Some O, Some O, Some O)
                                                                                           || (….a, ….e, ….i) =
                                                                                              (Some O, Some O, Some O)
                                                                                              || (….c, ….e, ….g) =
                                                                                                 (Some O, Some O, Some O)))))
                                                                  && ((….a, …, ….c) = (Some X, Some X, Some X)
                                                                      || (….d, ….e, ….f) = (Some X, Some X, Some X)
                                                                         || (….g, ….h, ….i) = (Some X, Some X, Some X)
                                                                            || (….a, ….d, ….g) = (Some X, Some X, Some X)
                                                                               || (…, ….e, ….h) = (Some X, Some X, Some X)
                                                                                  || (….c, ….f, ….i) = (Some X, Some X, Some X)
                                                                                     || (….a, ….e, ….i) = (Some X, Some X, Some X)
                                                                                        || (….c, ….e, ….g) = (Some X, Some X, Some X))
                                                              • simplify

                                                                into
                                                                ((if (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).0
                                                                     <=
                                                                     (List.fold_right
                                                                      (fun el _x
                                                                       -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                          if el = None then (x, o)
                                                                          else
                                                                          if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                          else (x, o + 1))
                                                                      (0, 0) (….a :: (… :: …))).1
                                                                  then
                                                                    (List.fold_right
                                                                     (fun el _x
                                                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                         if el = None then (x, o)
                                                                         else
                                                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                         else (x, o + 1))
                                                                     (0, 0) (….a :: (… :: …))).1
                                                                    <=
                                                                    (List.fold_right
                                                                     (fun el _x
                                                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                         if el = None then (x, o)
                                                                         else
                                                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                         else (x, o + 1))
                                                                     (0, 0) (….a :: (… :: …))).0
                                                                    || :var_0:.last_player = Some O
                                                                       && (List.fold_right
                                                                           (fun el _x
                                                                            -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                               if el = None then (x, o)
                                                                               else
                                                                               if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                               else (x, o + 1))
                                                                           (0, 0) (….a :: (… :: …))).1
                                                                          + -1
                                                                            * (List.fold_right
                                                                               (fun el _x
                                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                   if el = None then (x, o)
                                                                                   else
                                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                   else (x, o + 1))
                                                                               (0, 0) (….a :: (… :: …))).0
                                                                          = 1
                                                                  else
                                                                    (:var_0:.last_player = Some X
                                                                     && (List.fold_right
                                                                         (fun el _x
                                                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                             if el = None then (x, o)
                                                                             else
                                                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                             else (x, o + 1))
                                                                         (0, 0) (….a :: (… :: …))).0
                                                                        + -1
                                                                          * (List.fold_right
                                                                             (fun el _x
                                                                              -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                 if el = None then (x, o)
                                                                                 else
                                                                                 if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                 else (x, o + 1))
                                                                             (0, 0) (….a :: (… :: …))).1
                                                                        = 1))
                                                                 && (not
                                                                     ((((((((….a = Some X && … = Some X) && ….c = Some X
                                                                            || (….d = Some X && ….e = Some X) && ….f = Some X)
                                                                           || (….g = Some X && ….h = Some X) && ….i = Some X)
                                                                          || (….a = Some X && ….d = Some X) && ….g = Some X)
                                                                         || (… = Some X && ….e = Some X) && ….h = Some X)
                                                                        || (….c = Some X && ….f = Some X) && ….i = Some X)
                                                                       || (….a = Some X && ….e = Some X) && ….i = Some X)
                                                                      || (….c = Some X && ….e = Some X) && ….g = Some X)
                                                                     && not
                                                                        ((((((((….a = Some O && … = Some O) && ….c = Some O
                                                                               || (….d = Some O && ….e = Some O) && ….f = Some O)
                                                                              || (….g = Some O && ….h = Some O) && ….i = Some O)
                                                                             || (….a = Some O && ….d = Some O) && ….g = Some O)
                                                                            || (… = Some O && ….e = Some O) && ….h = Some O)
                                                                           || (….c = Some O && ….f = Some O) && ….i = Some O)
                                                                          || (….a = Some O && ….e = Some O) && ….i = Some O)
                                                                         || (….c = Some O && ….e = Some O) && ….g = Some O)
                                                                     || not
                                                                        ((((((((….a = Some X && … = Some X) && ….c = Some X
                                                                               || (….d = Some X && ….e = Some X) && ….f = Some X)
                                                                              || (….g = Some X && ….h = Some X) && ….i = Some X)
                                                                             || (….a = Some X && ….d = Some X) && ….g = Some X)
                                                                            || (… = Some X && ….e = Some X) && ….h = Some X)
                                                                           || (….c = Some X && ….f = Some X) && ….i = Some X)
                                                                          || (….a = Some X && ….e = Some X) && ….i = Some X)
                                                                         || (….c = Some X && ….e = Some X) && ….g = Some X)
                                                                        =
                                                                        ((((((((….a = Some O && … = Some O) && ….c = Some O
                                                                               || (….d = Some O && ….e = Some O) && ….f = Some O)
                                                                              || (….g = Some O && ….h = Some O) && ….i = Some O)
                                                                             || (….a = Some O && ….d = Some O) && ….g = Some O)
                                                                            || (… = Some O && ….e = Some O) && ….h = Some O)
                                                                           || (….c = Some O && ….f = Some O) && ….i = Some O)
                                                                          || (….a = Some O && ….e = Some O) && ….i = Some O)
                                                                         || (….c = Some O && ….e = Some O) && ….g = Some O)))
                                                                && ((((((((….a = Some X && … = Some X) && ….c = Some X
                                                                          || (….d = Some X && ….e = Some X) && ….f = Some X)
                                                                         || (….g = Some X && ….h = Some X) && ….i = Some X)
                                                                        || (….a = Some X && ….d = Some X) && ….g = Some X)
                                                                       || (… = Some X && ….e = Some X) && ….h = Some X)
                                                                      || (….c = Some X && ….f = Some X) && ….i = Some X)
                                                                     || (….a = Some X && ….e = Some X) && ….i = Some X)
                                                                    || (….c = Some X && ….e = Some X) && ….g = Some X)
                                                                expansions
                                                                []
                                                                rewrite_steps
                                                                  forward_chaining
                                                                  • unroll
                                                                    expr
                                                                    (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                       (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                    expansions
                                                                    • unroll
                                                                      expr
                                                                      (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                         (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                      expansions
                                                                      • unroll
                                                                        expr
                                                                        (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                           (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                        expansions
                                                                        • unroll
                                                                          expr
                                                                          (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                             (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                          expansions
                                                                          • unroll
                                                                            expr
                                                                            (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                               (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                            expansions
                                                                            • unroll
                                                                              expr
                                                                              (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                                 (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                              expansions
                                                                              • unroll
                                                                                expr
                                                                                (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                                   (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                                expansions
                                                                                • unroll
                                                                                  expr
                                                                                  (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                                     (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                                  expansions
                                                                                  • unroll
                                                                                    expr
                                                                                    (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                                       (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                                    expansions
                                                                                    • unroll
                                                                                      expr
                                                                                      (let ((a!1 (|::_3| (h_26 (grid_40 game_127))
                                                                                                         (|::_3| (i_27 (grid_40 game_127)) |[…
                                                                                      expansions
                                                                                      • Sat (Some let (game : game_state) = {grid = {a = (Some X); b = None; c = (Some O); d = (Some O); e = (Some X); f = None; g = (Some O); h = None; i = (Some X)}; last_player = None} )
                                                                                      In [22]:
                                                                                      CX.game;
                                                                                      
                                                                                      Out[22]:
                                                                                      - : 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 [23]:
                                                                                      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[23]:
                                                                                      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 [24]:
                                                                                      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[24]:
                                                                                      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 [25]:
                                                                                      (==>)
                                                                                      
                                                                                      Out[25]:
                                                                                      - : bool -> bool -> bool = <fun>
                                                                                      

                                                                                      Anyway! Let's actually run the statement:

                                                                                      In [26]:
                                                                                      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>
                                                                                      module CX : sig val game : game_state val player : player val move : move end
                                                                                      
                                                                                      Counterexample (after 24 steps, 0.415s):
                                                                                       let (game : game_state) =
                                                                                         {grid =
                                                                                           {a = (Some X); b = (Some O); c = (Some X); d = None; e = (Some O);
                                                                                            f = None; g = (Some X); h = None; i = None};
                                                                                          last_player = (Some X)}
                                                                                       let (player : player) = O
                                                                                       let (move : move) = B
                                                                                      
                                                                                      Refuted
                                                                                      proof attempt
                                                                                      ground_instances24
                                                                                      definitions0
                                                                                      inductions0
                                                                                      search_time
                                                                                      0.415s
                                                                                      details
                                                                                      Expand
                                                                                      smt_stats
                                                                                      arith offset eqs8659
                                                                                      num checks49
                                                                                      arith assert lower7297
                                                                                      arith pivots1022
                                                                                      rlimit count863200
                                                                                      mk clause4224
                                                                                      datatype occurs check3181
                                                                                      restarts3
                                                                                      mk bool var10593
                                                                                      arith assert upper7165
                                                                                      datatype splits1350
                                                                                      decisions7472
                                                                                      arith add rows13084
                                                                                      arith bound prop1776
                                                                                      propagations38574
                                                                                      interface eqs63
                                                                                      conflicts578
                                                                                      arith fixed eqs7084
                                                                                      datatype accessor ax552
                                                                                      minimized lits508
                                                                                      arith conflicts41
                                                                                      arith assert diseq3399
                                                                                      datatype constructor ax5066
                                                                                      num allocs148916452
                                                                                      final checks88
                                                                                      added eqs76568
                                                                                      del clause1371
                                                                                      arith eq adapter1911
                                                                                      memory34.160000
                                                                                      max memory34.160000
                                                                                      Expand
                                                                                      • start[0.415s]
                                                                                          ((if (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).0
                                                                                               >
                                                                                               (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).1
                                                                                            then
                                                                                              :var_0:.last_player = Some X
                                                                                              && (List.fold_right
                                                                                                  (fun el _x
                                                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                      if el = None then (x, o)
                                                                                                      else
                                                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                      else (x, o + 1))
                                                                                                  (0, 0) (….a :: (… :: …))).0
                                                                                                 -
                                                                                                 (List.fold_right
                                                                                                  (fun el _x
                                                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                      if el = None then (x, o)
                                                                                                      else
                                                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                      else (x, o + 1))
                                                                                                  (0, 0) (….a :: (… :: …))).1
                                                                                                 = 1
                                                                                            else
                                                                                            if (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).0
                                                                                               <
                                                                                               (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).1
                                                                                            then
                                                                                              :var_0:.last_player = Some O
                                                                                              && (List.fold_right
                                                                                                  (fun el _x
                                                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                      if el = None then (x, o)
                                                                                                      else
                                                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                      else (x, o + 1))
                                                                                                  (0, 0) (….a :: (… :: …))).1
                                                                                                 -
                                                                                                 (List.fold_right
                                                                                                  (fun el _x
                                                                                                   -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                      if el = None then (x, o)
                                                                                                      else
                                                                                                      if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                      else (x, o + 1))
                                                                                                  (0, 0) (….a :: (… :: …))).0
                                                                                                 = 1
                                                                                            else true)
                                                                                           && (not
                                                                                               ((….a, …, ….c) = (Some X, Some X, Some X)
                                                                                                || (….d, ….e, ….f) = (Some X, Some X, Some X)
                                                                                                   || (….g, ….h, ….i) = (Some X, Some X, Some X)
                                                                                                      || (….a, ….d, ….g) = (Some X, Some X, Some X)
                                                                                                         || (…, ….e, ….h) = (Some X, Some X, Some X)
                                                                                                            || (….c, ….f, ….i) = (Some X, Some X, Some X)
                                                                                                               || (….a, ….e, ….i) = (Some X, Some X, Some X)
                                                                                                                  || (….c, ….e, ….g) = (Some X, Some X, Some X))
                                                                                               && not
                                                                                                  ((….a, …, ….c) = (Some O, Some O, Some O)
                                                                                                   || (….d, ….e, ….f) = (Some O, Some O, Some O)
                                                                                                      || (….g, ….h, ….i) = (Some O, Some O, Some O)
                                                                                                         || (….a, ….d, ….g) = (Some O, Some O, Some O)
                                                                                                            || (…, ….e, ….h) = (Some O, Some O, Some O)
                                                                                                               || (….c, ….f, ….i) = (Some O, Some O, Some O)
                                                                                                                  || (….a, ….e, ….i) = (Some O, Some O, Some O)
                                                                                                                     || (….c, ….e, ….g) =
                                                                                                                        (Some O, Some O, Some O))
                                                                                               || not
                                                                                                  (((….a, …, ….c) = (Some X, Some X, Some X)
                                                                                                    || (….d, ….e, ….f) = (Some X, Some X, Some X)
                                                                                                       || (….g, ….h, ….i) = (Some X, Some X, Some X)
                                                                                                          || (….a, ….d, ….g) = (Some X, Some X, Some X)
                                                                                                             || (…, ….e, ….h) = (Some X, Some X, Some X)
                                                                                                                || (….c, ….f, ….i) = (Some X, Some X, Some X)
                                                                                                                   || (….a, ….e, ….i) =
                                                                                                                      (Some X, Some X, Some X)
                                                                                                                      || (….c, ….e, ….g) =
                                                                                                                         (Some X, Some X, Some X))
                                                                                                   =
                                                                                                   ((….a, …, ….c) = (Some O, Some O, Some O)
                                                                                                    || (….d, ….e, ….f) = (Some O, Some O, Some O)
                                                                                                       || (….g, ….h, ….i) = (Some O, Some O, Some O)
                                                                                                          || (….a, ….d, ….g) = (Some O, Some O, Some O)
                                                                                                             || (…, ….e, ….h) = (Some O, Some O, Some O)
                                                                                                                || (….c, ….f, ….i) = (Some O, Some O, Some O)
                                                                                                                   || (….a, ….e, ….i) =
                                                                                                                      (Some O, Some O, Some O)
                                                                                                                      || (….c, ….e, ….g) =
                                                                                                                         (Some O, Some O, Some O)))))
                                                                                          && not
                                                                                             (((….a, …, ….c) = (Some X, Some X, Some X)
                                                                                               || (….d, ….e, ….f) = (Some X, Some X, Some X)
                                                                                                  || (….g, ….h, ….i) = (Some X, Some X, Some X)
                                                                                                     || (….a, ….d, ….g) = (Some X, Some X, Some X)
                                                                                                        || (…, ….e, ….h) = (Some X, Some X, Some X)
                                                                                                           || (….c, ….f, ….i) = (Some X, Some X, Some X)
                                                                                                              || (….a, ….e, ….i) = (Some X, Some X, Some X)
                                                                                                                 || (….c, ….e, ….g) = (Some X, Some X, Some X))
                                                                                              || ((….a, …, ….c) = (Some O, Some O, Some O)
                                                                                                  || (….d, ….e, ….f) = (Some O, Some O, Some O)
                                                                                                     || (….g, ….h, ….i) = (Some O, Some O, Some O)
                                                                                                        || (….a, ….d, ….g) = (Some O, Some O, Some O)
                                                                                                           || (…, ….e, ….h) = (Some O, Some O, Some O)
                                                                                                              || (….c, ….f, ….i) = (Some O, Some O, Some O)
                                                                                                                 || (….a, ….e, ….i) = (Some O, Some O, Some O)
                                                                                                                    || (….c, ….e, ….g) =
                                                                                                                       (Some O, Some O, Some O))
                                                                                                 || List.for_all ((<>) None) (….a :: (… :: …)))
                                                                                             && …
                                                                                          ==> …
                                                                                      • simplify

                                                                                        into
                                                                                        not
                                                                                        ((((if (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).0
                                                                                               <=
                                                                                               (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).1
                                                                                            then
                                                                                              (List.fold_right
                                                                                               (fun el _x
                                                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                   if el = None then (x, o)
                                                                                                   else
                                                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                   else (x, o + 1))
                                                                                               (0, 0) (….a :: (… :: …))).1
                                                                                              <=
                                                                                              (List.fold_right
                                                                                               (fun el _x
                                                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                   if el = None then (x, o)
                                                                                                   else
                                                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                   else (x, o + 1))
                                                                                               (0, 0) (….a :: (… :: …))).0
                                                                                              || :var_0:.last_player = Some O
                                                                                                 && (List.fold_right
                                                                                                     (fun el _x
                                                                                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                         if el = None then (x, o)
                                                                                                         else
                                                                                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                         else (x, o + 1))
                                                                                                     (0, 0) (….a :: (… :: …))).1
                                                                                                    + -1
                                                                                                      * (List.fold_right
                                                                                                         (fun el _x
                                                                                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                             if el = None then (x, o)
                                                                                                             else
                                                                                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                             else (x, o + 1))
                                                                                                         (0, 0) (….a :: (… :: …))).0
                                                                                                    = 1
                                                                                            else
                                                                                              (:var_0:.last_player = Some X
                                                                                               && (List.fold_right
                                                                                                   (fun el _x
                                                                                                    -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                       if el = None then (x, o)
                                                                                                       else
                                                                                                       if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                       else (x, o + 1))
                                                                                                   (0, 0) (….a :: (… :: …))).0
                                                                                                  + -1
                                                                                                    * (List.fold_right
                                                                                                       (fun el _x
                                                                                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                           if el = None then (x, o)
                                                                                                           else
                                                                                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                           else (x, o + 1))
                                                                                                       (0, 0) (….a :: (… :: …))).1
                                                                                                  = 1))
                                                                                           && (not
                                                                                               ((((((((….a = Some X && … = Some X) && ….c = Some X
                                                                                                      || (….d = Some X && ….e = Some X) && ….f = Some X)
                                                                                                     || (….g = Some X && ….h = Some X) && ….i = Some X)
                                                                                                    || (….a = Some X && ….d = Some X) && ….g = Some X)
                                                                                                   || (… = Some X && ….e = Some X) && ….h = Some X)
                                                                                                  || (….c = Some X && ….f = Some X) && ….i = Some X)
                                                                                                 || (….a = Some X && ….e = Some X) && ….i = Some X)
                                                                                                || (….c = Some X && ….e = Some X) && ….g = Some X)
                                                                                               && not
                                                                                                  ((((((((….a = Some O && … = Some O) && ….c = Some O
                                                                                                         || (….d = Some O && ….e = Some O) && ….f = Some O)
                                                                                                        || (….g = Some O && ….h = Some O) && ….i = Some O)
                                                                                                       || (….a = Some O && ….d = Some O) && ….g = Some O)
                                                                                                      || (… = Some O && ….e = Some O) && ….h = Some O)
                                                                                                     || (….c = Some O && ….f = Some O) && ….i = Some O)
                                                                                                    || (….a = Some O && ….e = Some O) && ….i = Some O)
                                                                                                   || (….c = Some O && ….e = Some O) && ….g = Some O)
                                                                                               || not
                                                                                                  ((((((((….a = Some X && … = Some X) && ….c = Some X
                                                                                                         || (….d = Some X && ….e = Some X) && ….f = Some X)
                                                                                                        || (….g = Some X && ….h = Some X) && ….i = Some X)
                                                                                                       || (….a = Some X && ….d = Some X) && ….g = Some X)
                                                                                                      || (… = Some X && ….e = Some X) && ….h = Some X)
                                                                                                     || (….c = Some X && ….f = Some X) && ….i = Some X)
                                                                                                    || (….a = Some X && ….e = Some X) && ….i = Some X)
                                                                                                   || (….c = Some X && ….e = Some X) && ….g = Some X)
                                                                                                  =
                                                                                                  ((((((((….a = Some O && … = Some O) && ….c = Some O
                                                                                                         || (….d = Some O && ….e = Some O) && ….f = Some O)
                                                                                                        || (….g = Some O && ….h = Some O) && ….i = Some O)
                                                                                                       || (….a = Some O && ….d = Some O) && ….g = Some O)
                                                                                                      || (… = Some O && ….e = Some O) && ….h = Some O)
                                                                                                     || (….c = Some O && ….f = Some O) && ….i = Some O)
                                                                                                    || (….a = Some O && ….e = Some O) && ….i = Some O)
                                                                                                   || (….c = Some O && ….e = Some O) && ….g = Some O)))
                                                                                          && not
                                                                                             (((((((((((((((((….a = Some X && … = Some X) && ….c = Some X
                                                                                                             || (….d = Some X && ….e = Some X) && ….f = Some X)
                                                                                                            || (….g = Some X && ….h = Some X) && ….i = Some X)
                                                                                                           || (….a = Some X && ….d = Some X) && ….g = Some X)
                                                                                                          || (… = Some X && ….e = Some X) && ….h = Some X)
                                                                                                         || (….c = Some X && ….f = Some X) && ….i = Some X)
                                                                                                        || (….a = Some X && ….e = Some X) && ….i = Some X)
                                                                                                       || (….c = Some X && ….e = Some X) && ….g = Some X)
                                                                                                      || (….a = Some O && … = Some O) && ….c = Some O)
                                                                                                     || (….d = Some O && ….e = Some O) && ….f = Some O)
                                                                                                    || (….g = Some O && ….h = Some O) && ….i = Some O)
                                                                                                   || (….a = Some O && ….d = Some O) && ….g = Some O)
                                                                                                  || (… = Some O && ….e = Some O) && ….h = Some O)
                                                                                                 || (….c = Some O && ….f = Some O) && ….i = Some O)
                                                                                                || (….a = Some O && ….e = Some O) && ….i = Some O)
                                                                                               || (….c = Some O && ….e = Some O) && ….g = Some O)
                                                                                              || List.for_all ((<>) None) (….a :: (… :: …))))
                                                                                         && (:var_0:.last_player = None && :var_1: = X
                                                                                             || :var_0:.last_player = Some (if :var_1: = X then O else X)))
                                                                                        || (if (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).0
                                                                                               <=
                                                                                               (List.fold_right
                                                                                                (fun el _x
                                                                                                 -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                    if el = None then (x, o)
                                                                                                    else
                                                                                                    if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                    else (x, o + 1))
                                                                                                (0, 0) (….a :: (… :: …))).1
                                                                                            then
                                                                                              (List.fold_right
                                                                                               (fun el _x
                                                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                   if el = None then (x, o)
                                                                                                   else
                                                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                   else (x, o + 1))
                                                                                               (0, 0) (….a :: (… :: …))).1
                                                                                              <=
                                                                                              (List.fold_right
                                                                                               (fun el _x
                                                                                                -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                   if el = None then (x, o)
                                                                                                   else
                                                                                                   if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                   else (x, o + 1))
                                                                                               (0, 0) (….a :: (… :: …))).0
                                                                                              || :var_1: = O
                                                                                                 && (List.fold_right
                                                                                                     (fun el _x
                                                                                                      -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                         if el = None then (x, o)
                                                                                                         else
                                                                                                         if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                         else (x, o + 1))
                                                                                                     (0, 0) (….a :: (… :: …))).1
                                                                                                    + -1
                                                                                                      * (List.fold_right
                                                                                                         (fun el _x
                                                                                                          -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                             if el = None then (x, o)
                                                                                                             else
                                                                                                             if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                             else (x, o + 1))
                                                                                                         (0, 0) (….a :: (… :: …))).0
                                                                                                    = 1
                                                                                            else
                                                                                              (:var_1: = X
                                                                                               && (List.fold_right
                                                                                                   (fun el _x
                                                                                                    -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                       if el = None then (x, o)
                                                                                                       else
                                                                                                       if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                       else (x, o + 1))
                                                                                                   (0, 0) (….a :: (… :: …))).0
                                                                                                  + -1
                                                                                                    * (List.fold_right
                                                                                                       (fun el _x
                                                                                                        -> let (x : int) = _x.0 and (o : int) = _x.1 in
                                                                                                           if el = None then (x, o)
                                                                                                           else
                                                                                                           if Is_a(Some, el) && Option.get el = X then (x + 1, o)
                                                                                                           else (x, o + 1))
                                                                                                       (0, 0) (….a :: (… :: …))).1
                                                                                                  = 1))
                                                                                           && (not
                                                                                               ((((((((….a = Some X && … = Some X)
                                                                                                      && (if :var_2: = A
                                                                                                          then
                                                                                                            {a = Some :var_1:; b = …; c = ….c; d = ….d;
                                                                                                             e = ….e; f = ….f; g = ….g; h = ….h; i = ….i}
                                                                                                          else …).c
                                                                                                         = Some X
                                                                                                      || …)
                                                                                                     || …)
                                                                                                    || …)
                                                                                                   || …)
                                                                                                  || …)
                                                                                                 || …)
                                                                                                || …)
                                                                                               && … || …)
                                                                                        expansions
                                                                                        []
                                                                                        rewrite_steps
                                                                                          forward_chaining
                                                                                          • unroll
                                                                                            expr
                                                                                            (let ((a!1 (ite (= move_153 H_37)
                                                                                                            (|rec_mk.grid_1866|
                                                                                                              (a_19 (grid_…
                                                                                            expansions
                                                                                            • unroll
                                                                                              expr
                                                                                              (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                 (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                              expansions
                                                                                              • unroll
                                                                                                expr
                                                                                                (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                   (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                expansions
                                                                                                • unroll
                                                                                                  expr
                                                                                                  (let ((a!1 (ite (= move_153 H_37)
                                                                                                                  (|rec_mk.grid_1866|
                                                                                                                    (a_19 (grid_…
                                                                                                  expansions
                                                                                                  • unroll
                                                                                                    expr
                                                                                                    (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                       (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                    expansions
                                                                                                    • unroll
                                                                                                      expr
                                                                                                      (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                         (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                      expansions
                                                                                                      • unroll
                                                                                                        expr
                                                                                                        (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                           (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                        expansions
                                                                                                        • unroll
                                                                                                          expr
                                                                                                          (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                             (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                          expansions
                                                                                                          • unroll
                                                                                                            expr
                                                                                                            (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                               (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                            expansions
                                                                                                            • unroll
                                                                                                              expr
                                                                                                              (let ((a!1 (ite (= move_153 H_37)
                                                                                                                              (|rec_mk.grid_1866|
                                                                                                                                (a_19 (grid_…
                                                                                                              expansions
                                                                                                              • unroll
                                                                                                                expr
                                                                                                                (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                   (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                expansions
                                                                                                                • unroll
                                                                                                                  expr
                                                                                                                  (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                     (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                  expansions
                                                                                                                  • unroll
                                                                                                                    expr
                                                                                                                    (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                       (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                    expansions
                                                                                                                    • unroll
                                                                                                                      expr
                                                                                                                      (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                         (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                      expansions
                                                                                                                      • unroll
                                                                                                                        expr
                                                                                                                        (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                           (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                        expansions
                                                                                                                        • unroll
                                                                                                                          expr
                                                                                                                          (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                             (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                          expansions
                                                                                                                          • unroll
                                                                                                                            expr
                                                                                                                            (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                            (|rec_mk.grid_1866|
                                                                                                                                              (a_19 (grid_…
                                                                                                                            expansions
                                                                                                                            • unroll
                                                                                                                              expr
                                                                                                                              (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                              (|rec_mk.grid_1866|
                                                                                                                                                (a_19 (grid_…
                                                                                                                              expansions
                                                                                                                              • unroll
                                                                                                                                expr
                                                                                                                                (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                                (|rec_mk.grid_1866|
                                                                                                                                                  (a_19 (grid_…
                                                                                                                                expansions
                                                                                                                                • unroll
                                                                                                                                  expr
                                                                                                                                  (let ((a!1 (|::_3| (h_26 (grid_40 game_151))
                                                                                                                                                     (|::_3| (i_27 (grid_40 game_151)) |[…
                                                                                                                                  expansions
                                                                                                                                  • unroll
                                                                                                                                    expr
                                                                                                                                    (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                                    (|rec_mk.grid_1866|
                                                                                                                                                      (a_19 (grid_…
                                                                                                                                    expansions
                                                                                                                                    • unroll
                                                                                                                                      expr
                                                                                                                                      (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                                      (|rec_mk.grid_1866|
                                                                                                                                                        (a_19 (grid_…
                                                                                                                                      expansions
                                                                                                                                      • unroll
                                                                                                                                        expr
                                                                                                                                        (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                                        (|rec_mk.grid_1866|
                                                                                                                                                          (a_19 (grid_…
                                                                                                                                        expansions
                                                                                                                                        • unroll
                                                                                                                                          expr
                                                                                                                                          (let ((a!1 (ite (= move_153 H_37)
                                                                                                                                                          (|rec_mk.grid_1866|
                                                                                                                                                            (a_19 (grid_…
                                                                                                                                          expansions
                                                                                                                                          • Sat (Some let (game : game_state) = {grid = {a = (Some X); b = (Some O); c = (Some X); d = None; e = (Some O); f = None; g = (Some X); h = None; i = None}; last_player = (Some X)} let (player : player) = O let (move : move) = B )
                                                                                                                                          In [27]:
                                                                                                                                          CX.game;
                                                                                                                                          
                                                                                                                                          Out[27]:
                                                                                                                                          - : game_state = <document>
                                                                                                                                          
                                                                                                                                          ··
                                                                                                                                          ··
                                                                                                                                          Last:

                                                                                                                                          Uh oh! What's gone wrong?

                                                                                                                                          In this case, the problem is clear - player O is moving in square B, but there's already a move there. 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 [28]:
                                                                                                                                          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[28]:
                                                                                                                                          val is_valid_move : game_state -> player -> move -> bool = <fun>
                                                                                                                                          

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

                                                                                                                                          In [29]:
                                                                                                                                          [@blast]
                                                                                                                                          verify((game, player, move) =>
                                                                                                                                            (is_valid_game(game) && is_valid_move(game, player, move))
                                                                                                                                            ==> is_valid_game(play_move(game, player, move))
                                                                                                                                          );
                                                                                                                                          
                                                                                                                                          Out[29]:
                                                                                                                                          - : game_state -> player -> move -> bool = <fun>
                                                                                                                                          
                                                                                                                                          proof from "blast" (after 2.13s)!
                                                                                                                                          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.