feat(course): audit fixes and twelve extensions (Modules 7b, 12, 13)

Audit pass:
- Frame rule counts frames since the last move, so every step lasts
  exactly movesEvery(score) frames, also right after a speed-up
- clampTime + MAX_GAP_MS: a hidden tab no longer replays the whole gap
- Pong Restart keeps held keys
- Fact corrections (mulberry32 output coverage, Haskell IO, uniform)
- New fixture cases from a mutation analysis

Extensions (all three tracks, shared fixtures, course text):
- Turn queue (Exercise 10.6), persisted high score (10.7),
  replay viewer (10.8), play() end-to-end helper (9.5)
- Optional State (4.7) and optional Module 7b, stack-safe IO
- Module 12, property-based testing with generators, shrinking,
  invariants and a model-based oracle
- Module 13, Conway's Life with a Click event
- JS type checking via jsconfig, JSDoc and types.d.ts
- Maintainer tools: scripts/mutate.mjs, scripts/typecheck-js.sh,
  Playwright end-to-end suite (e2e/, scripts/e2e.sh)

Docs: docs/contract.md is the canonical contract; specs and build log
under docs/superpowers.

Verified: JS 728, Scala 689, F# 691 unit tests; stubs and progression
M01-M13 and M07b; 176 end-to-end tests on Chromium and WebKit.

Co-Authored-By: Claude Fable 5.2 <noreply@anthropic.com>

fsharp/solutions/App.fs

@@ -2,11 +2,14 @@
 module Snake.App
 
 open Snake.Model
+open Snake.View
 open Snake.IO
 open Snake.Stream
 
 let TICK_MS = 150L
 let FRAME_MS = 25L
+// m10: the longest step of time the game accepts at once; a longer silence (a background tab) counts as this much.
+let MAX_GAP_MS = 250L
 
 // Everything that can happen to the game: a Tick every TICK_MS, and an action for each bound key.
 let classicActions: Stream<Action> =
@@ -22,7 +25,47 @@ let states (seed: uint32) (actions: Stream<Action>) : Stream<GameState> =
 
 let render (state: GameState) : IO<unit> = IO.drawScene (View.view state)
 
-// The whole program as a value: get a seed, then turn the stream of states into a stream of drawings.
+// m10: a session wraps the game with the best score so far. GameState, update and view do not know about it.
+// `Improved` says whether the action that produced this session raised the best, which is when it must be saved.
+type Session = { Game: GameState; Best: int; Improved: bool }
+
+let sessionUpdate (session: Session) (action: Action) : Session =
+    let game = Game.update session.Game action
+    let best = max session.Best game.Score
+    { Game = game; Best = best; Improved = best > session.Best }
+
+// The same running fold as `states`, one level up.
+let sessions (seed: uint32) (best: int) (actions: Stream<Action>) : Stream<Session> =
+    let s0 = { Game = Spawn.newGame seed; Best = best; Improved = false }
+    actions |> Stream.scan sessionUpdate s0 |> Stream.startWith s0
+
+// The game's picture with the best score at the top right, level with the score on the left.
+let sessionView (session: Session) : Scene =
+    View.view session.Game
+    @ [ Text(session.Game.Width * CELL - 6, 18, $"Best: {session.Best}", "#ffffff", 16, "right") ]
+
+// Draw, and save the best only on the action that raised it.
+let renderSession (session: Session) : IO<unit> =
+    if session.Improved then
+        IO.sequence [ IO.drawScene (sessionView session); IO.saveBest session.Best ]
+    else
+        IO.drawScene (sessionView session)
+
+// The whole program as a value: get a seed, then the stored best, then turn the stream of sessions into a stream
+// of drawings. clampTime wraps the whole stream, so everything inside it sees the clamped clock.
 // Nothing runs here; Runtime interprets it.
 let main () : IO<Stream<IO<unit>>> =
-    IO.randomSeed () |> IO.map (fun seed -> states seed speedActions |> Stream.map render)
+    IO.randomSeed ()
+    |> IO.bind (fun seed ->
+        IO.loadBest ()
+        |> IO.map (fun best -> sessions seed best speedActions |> Stream.map renderSession |> Stream.clampTime MAX_GAP_MS))
+
+// m09: the whole program in one equality. For each drawing that `main` emits while fed `events`, the ops it
+// performs, answered by the pure interpreter (`seed` for RandomSeed, and from m10 `best` for LoadBest).
+let playWith (seed: uint32) (best: int) (events: Event list) (main: IO<Stream<IO<unit>>>) : Op list list =
+    (IO.traceWith seed best main).Result
+    |> Stream.run events
+    |> List.map (fun io -> (IO.traceWith seed best io).Ops)
+
+// The Module 9 helper is the "no stored best" case.
+let play (seed: uint32) (events: Event list) (main: IO<Stream<IO<unit>>>) : Op list list = playWith seed 0 events main

fsharp/solutions/Check.fs

@@ -0,0 +1,170 @@
+/// m12 check: property-based testing built from the course's own parts.
+/// A generator of test data is nothing new: it is a `Random<'a>` from Random.fs (`seed -> (value, nextSeed)`).
+module Snake.Check
+
+open Snake.Model
+open Snake.Random
+
+/// What `forAll` reports: every value passed, or the first one that did not.
+type CheckResult<'a> =
+    | Passed of runs: int
+    | Failed of run: int * seed: uint32 * value: 'a
+
+/// One of the elements of the non-empty list `xs`: the one at index `intBelow (List.length xs)`.
+let elementOf (xs: 'a list) : Random<'a> =
+    Random.intBelow (List.length xs) |> Random.map (fun i -> List.item i xs)
+
+/// `n` values drawn from `gen`, one after another; the seed is threaded from each draw to the next.
+let rec listOf (n: int) (gen: Random<'a>) : Random<'a list> =
+    if n <= 0 then
+        Random.pure []
+    else
+        gen |> Random.bind (fun a -> listOf (n - 1) gen |> Random.map (fun rest -> a :: rest))
+
+// Ticks appear three times so that the snake moves more often than it turns.
+let private actionPool: Action list =
+    [ Tick; Tick; Tick; Turn Up; Turn Down; Turn Left; Turn Right ]
+
+/// A generator of `n` random actions (given).
+let genActions (n: int) : Random<Action list> = listOf n (elementOf actionPool)
+
+/// Check `prop` on `runs` values drawn from `gen`, starting from `seed`; stop at the first value that fails.
+/// Recursion carries the seed: no counter, no loop variable. `go` calls itself last, so it runs in constant stack.
+let forAll (runs: int) (seed: uint32) (prop: 'a -> bool) (gen: Random<'a>) : CheckResult<'a> =
+    let rec go (i: int) (s: uint32) : CheckResult<'a> =
+        if i >= runs then
+            Passed runs
+        else
+            let value, next = Random.run s gen
+            if prop value then go (i + 1) next else Failed(i, s, value)
+
+    go 0 seed
+
+/// Every list that is `xs` with exactly one element removed, in the order of the removed index.
+let removals (xs: 'a list) : 'a list list = xs |> List.mapi (fun i _ -> List.removeAt i xs)
+
+/// Keep removing one element for as long as the list still fails; the result is 1-minimal.
+let rec minimize (fails: 'a list -> bool) (xs: 'a list) : 'a list =
+    match List.tryFind fails (removals xs) with
+    | Some smaller -> minimize fails smaller
+    | None -> xs
+
+let private neighbours (a: Point, b: Point) : bool = abs (a.X - b.X) + abs (a.Y - b.Y) = 1
+
+let rec private allDifferent (ps: Point list) : bool =
+    match ps with
+    | [] -> true
+    | p :: rest -> not (Util.contains rest p (=)) && allDifferent rest
+
+// Each invariant: its name, and a predicate that is true when the state is fine.
+// `Option.forall` is true for `None`: no apple breaks nothing.
+let private invariants: (string * (GameState -> bool)) list =
+    [ "snake-not-empty", (fun s -> not (List.isEmpty s.Snake))
+      "snake-in-bounds", (fun s -> s.Snake |> List.forall (Rules.inBounds s))
+      "snake-no-overlap", (fun s -> allDifferent s.Snake)
+      "snake-connected", (fun s -> s.Snake |> List.pairwise |> List.forall neighbours)
+      "apple-in-bounds", (fun s -> s.Apple |> Option.forall (Rules.inBounds s))
+      "apple-not-on-snake", (fun s -> s.Apple |> Option.forall (fun a -> not (Util.contains s.Snake a (=))))
+      "turn-not-reversed", (fun s -> s.NextDir <> opposite s.Dir)
+      "queued-not-reversed", (fun s -> s.Queued |> Option.forall (fun q -> q <> opposite s.NextDir)) ]
+
+/// The names of the invariants that `state` breaks, in the order of the table above.
+let violations (state: GameState) : string list =
+    invariants |> List.filter (fun (_, holds) -> not (holds state)) |> List.map fst
+
+/// The numbers of states between consecutive moves of the snake's head.
+let moveGaps (states: GameState list) : int list =
+    let moved =
+        states
+        |> List.pairwise
+        |> List.indexed
+        |> List.filter (fun (_, (before, after)) -> List.tryHead after.Snake <> List.tryHead before.Snake)
+        |> List.map (fun (i, _) -> i + 1)
+
+    moved |> List.pairwise |> List.map (fun (a, b) -> b - a)
+
+// ---- random states (Exercise 12.6) and a second implementation of the step as an oracle (Exercise 12.7)
+
+let private directions: Direction list = [ Up; Down; Left; Right ]
+
+// The direction the head moved in to get from the second cell to the first (`Right` for a one-cell snake).
+let private headingOf (snake: Point list) : Direction =
+    match snake with
+    | head :: second :: _ -> directions |> List.find (fun d -> Rules.move second d = head)
+    | _ -> Right
+
+/// A random snake, head first: a random head, then a random walk that never leaves the board or crosses itself.
+let genSnake (width: int) (height: int) (maxLen: int) : Random<Point list> =
+    // Extend `snake` at its tail by up to `left` more cells, each a random free in-bounds neighbour of the last cell.
+    let rec grow (snake: Point list) (left: int) : Random<Point list> =
+        if left <= 0 then
+            Random.pure snake
+        else
+            let last = List.last snake
+
+            let options =
+                directions
+                |> List.map (Rules.move last)
+                |> List.filter (fun p ->
+                    p.X >= 0 && p.X < width && p.Y >= 0 && p.Y < height && not (Util.contains snake p (=)))
+
+            if List.isEmpty options then
+                Random.pure snake
+            else
+                elementOf options |> Random.bind (fun p -> grow (snake @ [ p ]) (left - 1))
+
+    Random.intBelow width
+    |> Random.bind (fun x -> Random.intBelow height |> Random.bind (fun y -> grow [ { X = x; Y = y } ] (maxLen - 1)))
+
+/// A random sane Playing state on a `width` x `height` board (given).
+let genState (width: int) (height: int) : Random<GameState> =
+    Random.intBelow (width * height)
+    |> Random.bind (fun n ->
+        genSnake width height (n + 1)
+        |> Random.bind (fun snake ->
+            let dir = headingOf snake
+
+            let state =
+                { Width = width
+                  Height = height
+                  Snake = snake
+                  Dir = dir
+                  NextDir = dir
+                  Apple = None
+                  Seed = 0u
+                  Score = 0
+                  Status = Playing
+                  Frame = 0
+                  Queued = None }
+
+            let free = Spawn.freeCells state
+
+            elementOf (directions |> List.filter (fun d -> d <> opposite dir))
+            |> Random.bind (fun nextDir ->
+                Random.intBelow 2
+                |> Random.bind (fun coin ->
+                    (if coin = 0 || List.isEmpty free then Random.pure None else elementOf free |> Random.map Some)
+                    |> Random.map (fun apple -> { state with NextDir = nextDir; Apple = apple })))))
+
+/// The step of m03, written from the rules instead of from `advance`: build the snake the step would give,
+/// then ask `violations` whether that snake is on the board and does not overlap itself.
+let advanceSpec (state: GameState) : GameState =
+    if state.Status <> Playing then
+        state
+    else
+        let head = Rules.move (List.head state.Snake) state.NextDir
+        let eating = state.Apple = Some head
+        let snake = head :: (if eating then state.Snake else Util.dropLast state.Snake)
+
+        let broken =
+            violations { state with Snake = snake }
+            |> List.exists (fun v -> v = "snake-in-bounds" || v = "snake-no-overlap")
+
+        if broken then
+            { state with Dir = state.NextDir; Status = GameOver }
+        else
+            { state with
+                Snake = snake
+                Dir = state.NextDir
+                Apple = if eating then None else state.Apple
+                Score = state.Score + (if eating then 1 else 0) }

fsharp/solutions/Game.fs

@@ -1,19 +1,40 @@
 /// m05 game: one `update` turns every action into a new state; replaying a game is a fold over its actions.
-/// m10 adds pause, restart and a frame clock that speeds up as the score grows.
+/// m10 adds pause, restart, a frame counter that makes the snake speed up as the score grows, and a turn queue.
 module Snake.Game
 
 open Snake.Model
 
+/// What `update` does with `Turn dir` from m10 on. With no turn pending (`NextDir = Dir`) it is the m03 `Rules.turn`.
+/// With a turn pending, a key that is neither `NextDir` nor its opposite is remembered in `Queued`
+/// (replacing an earlier one) and taken on the step after; any other key changes nothing.
+let steer (state: GameState) (dir: Direction) : GameState =
+    if state.Status <> Playing then state
+    elif state.NextDir = state.Dir then Rules.turn state dir
+    elif dir <> state.NextDir && dir <> opposite state.NextDir then { state with Queued = Some dir }
+    else state
+
+// After a step: a game still in play takes the queued turn, if there is one. It is always a legal turn, because the
+// step just made `Dir` equal to the old `NextDir`, and `Queued` is neither that nor its opposite.
+let private takeQueued (state: GameState) : GameState =
+    match state.Queued with
+    | Some d when state.Status = Playing -> { state with NextDir = d; Queued = None }
+    | _ -> state
+
+// One game step as `update` takes it: `Spawn.tick`, then the queued turn.
+let private step (state: GameState) : GameState = takeQueued (Spawn.tick state)
+
 /// How many frames pass between game steps: 8 at first, one fewer every 3 points, never fewer than 2.
 let movesEvery (score: int) : int = max 2 (8 - score / 3)
 
-// One animation frame: count it, and every `movesEvery score`-th frame run a game step.
+// One animation frame. `Frame` counts the frames since the snake last moved; when it reaches
+// `movesEvery Score` the count starts again and the game takes a step. Counting from the last move
+// (not from the start of the game) keeps every step exactly `movesEvery` frames long, also right after a speed-up.
 let private frame (state: GameState) : GameState =
     if state.Status <> Playing then
         state
     else
-        let counted = { state with Frame = state.Frame + 1 }
-        if counted.Frame % movesEvery counted.Score = 0 then Spawn.tick counted else counted
+        let n = state.Frame + 1
+        if n >= movesEvery state.Score then step { state with Frame = 0 } else { state with Frame = n }
 
 // Pause and resume only make sense in those two statuses; a finished game stays finished.
 let private togglePause (state: GameState) : GameState =
@@ -26,8 +47,8 @@ let private togglePause (state: GameState) : GameState =
 // Every case is listed (no `| _ ->`): adding an Action later makes the compiler point here.
 let update (state: GameState) (action: Action) : GameState =
     match action with
-    | Tick -> Spawn.tick state
-    | Turn dir -> Rules.turn state dir
+    | Tick -> step state
+    | Turn dir -> steer state dir
     | TogglePause -> togglePause state
     | Restart -> Spawn.newGame state.Seed
     | Frame -> frame state

fsharp/solutions/IO.fs

@@ -9,11 +9,16 @@ open Snake.View
 type Op =
     | Draw of Shape
     | RandomSeed
+    // m10: the stored best score, read and written.
+    | LoadBest
+    | SaveBest of int
 
-/// What the outside world answers: `Done` for a `Draw`, `Seed value` for a `RandomSeed`.
+/// What the outside world answers: `Done` for a `Draw` or a `SaveBest`, `Seed value` for a `RandomSeed`,
+/// `Best value` for a `LoadBest`.
 type OpResult =
     | Done
     | Seed of uint32
+    | Best of int
 
 type IO<'a> =
     | Pure of 'a
@@ -47,13 +52,24 @@ let randomSeed () : IO<uint32> =
     |> map (fun result ->
         match result with
         | Seed value -> value
-        | Done -> failwith "RandomSeed must be answered with a Seed")
+        | _ -> failwith "RandomSeed must be answered with a Seed")
 
 let drawScene (scene: Scene) : IO<unit> = sequence (List.map draw scene)
 
+// m10: the stored best score. Reading it answers a number (built like randomSeed); saving answers Done, like a draw.
+let loadBest () : IO<int> =
+    effect LoadBest
+    |> map (fun result ->
+        match result with
+        | Best value -> value
+        | _ -> failwith "LoadBest must be answered with a Best")
+
+let saveBest (score: int) : IO<unit> = effect (SaveBest score) |> map ignore
+
 // Answer each op with a canned result, collecting the ops as we go. Tail recursion: the compiler turns it
 // into a loop, so long programs cannot overflow the stack, and still nothing is mutated.
-let trace (seed: uint32) (io: IO<'a>) : Traced<'a> =
+// `best` (m10) answers LoadBest, the way `seed` answers RandomSeed.
+let traceWith (seed: uint32) (best: int) (io: IO<'a>) : Traced<'a> =
     let rec go io performed =
         match io with
         | Pure a -> { Result = a; Ops = List.rev performed }
@@ -61,8 +77,13 @@ let trace (seed: uint32) (io: IO<'a>) : Traced<'a> =
             let answer =
                 match op with
                 | RandomSeed -> Seed seed
-                | Draw _ -> Done
+                | LoadBest -> Best best
+                | Draw _
+                | SaveBest _ -> Done
 
             go (k answer) (op :: performed)
 
     go io []
+
+// The Module 7 interpreter is the "no stored best" case.
+let trace (seed: uint32) (io: IO<'a>) : Traced<'a> = traceWith seed 0 io

fsharp/solutions/Life/App.fs

@@ -0,0 +1,44 @@
+/// m13 life app: the same wiring as Snake and Pong, with a third kind of input, the mouse.
+module Snake.Life.App
+
+open Snake
+open Snake.Model
+open Snake.IO
+open Snake.Stream
+open Snake.Life.Model
+open Snake.Life.Rules
+open Snake.Life.View
+
+let GEN_MS = 150L
+
+let lifeKey (key: string) : LifeAction option =
+    match key with
+    | " " -> Some LifeAction.TogglePause
+    | "n" | "N" -> Some LifeAction.Step
+    | "c" | "C" -> Some LifeAction.Clear
+    | "r" | "R" -> Some LifeAction.Randomize
+    | _ -> None
+
+// Integer division rounds towards zero (-1 / 16 = 0), so the pixel is checked before dividing.
+let lifeClick (x: int) (y: int) : LifeAction option =
+    if x >= 0 && y >= 0 && x / LCELL < LW && y / LCELL < LH then
+        Some(LifeAction.Toggle { X = x / LCELL; Y = y / LCELL })
+    else
+        None
+
+let lifeActions: Stream<LifeAction> =
+    Stream.merge
+        (Stream.merge
+            (Stream.interval GEN_MS |> Stream.map (fun _ -> LifeAction.Frame))
+            (Stream.keyDowns |> Stream.choose lifeKey))
+        (Stream.clicks |> Stream.choose (fun (x, y) -> lifeClick x y))
+
+let lifeStates (seed: uint32) (actions: Stream<LifeAction>) : Stream<LifeState> =
+    actions |> Stream.scan lifeUpdate (initialLife seed) |> Stream.startWith (initialLife seed)
+
+let lifeMain () : IO<Stream<IO<unit>>> =
+    IO.randomSeed ()
+    |> IO.map (fun seed ->
+        lifeStates seed lifeActions
+        |> Stream.map (fun state -> IO.drawScene (lifeView state))
+        |> Stream.clampTime App.MAX_GAP_MS)

fsharp/solutions/Life/Model.fs

@@ -0,0 +1,32 @@
+/// m13 life model: the whole world is one value, like the snake and the match.
+module Snake.Life.Model
+
+open Snake.Model
+
+/// `Cells` is canonical: sorted row-major (by `Y`, then `X`), no duplicates, all on the board.
+type LifeState =
+    { Cells: Point list
+      Running: bool
+      Seed: uint32
+      Generation: int }
+
+[<RequireQualifiedAccess>]
+type LifeAction =
+    | Frame
+    | Toggle of Point
+    | TogglePause
+    | Step
+    | Clear
+    | Randomize
+
+// The board, in cells: LW columns, LH rows, LCELL pixels per cell. Below it the page shows a status strip.
+let LW = 40
+let LH = 30
+let LCELL = 16
+let STRIP = 24
+
+let initialLife (seed: uint32) : LifeState =
+    { Cells = []
+      Running = false
+      Seed = seed
+      Generation = 0 }

fsharp/solutions/Life/Rules.fs

@@ -0,0 +1,71 @@
+/// m13 life rules: the next generation is a function of the whole old board, and nothing else.
+module Snake.Life.Rules
+
+open Snake
+open Snake.Model
+open Snake.Life.Model
+
+// Every cell of the board, row-major (y, then x). Filtering this list keeps any result in canonical order for free.
+let private board: Point list =
+    [ for y in 0 .. LH - 1 do
+          for x in 0 .. LW - 1 -> { X = x; Y = y } ]
+
+let private onBoard (p: Point) : bool = p.X >= 0 && p.X < LW && p.Y >= 0 && p.Y < LH
+
+// A cell's position in row-major order: the order the canonical list is sorted by.
+let private index (p: Point) : int = p.Y * LW + p.X
+
+let private offsets = [ (-1, -1); (0, -1); (1, -1); (-1, 0); (1, 0); (-1, 1); (0, 1); (1, 1) ]
+
+// The up to eight neighbours of `p` that are on the board (the edges are dead: nothing wraps around).
+let private neighbours (p: Point) : Point list =
+    offsets |> List.map (fun (dx, dy) -> { X = p.X + dx; Y = p.Y + dy }) |> List.filter onBoard
+
+// B3/S23, every cell judged on the *old* list only. A Set of the old cells makes "is this neighbour alive?" one
+// lookup, so a generation is 1200 x 8 lookups however many cells live. The set is built once and only read.
+let lifeStep (cells: Point list) : Point list =
+    let alive = Set.ofList cells
+    let count p = neighbours p |> List.filter alive.Contains |> List.length
+
+    board
+    |> List.filter (fun p ->
+        let n = count p
+        n = 3 || (n = 2 && alive.Contains p))
+
+// Add a dead cell or remove a live one, keeping the list sorted row-major without duplicates. Off the board: unchanged.
+let private toggle (cells: Point list) (p: Point) : Point list =
+    if not (onBoard p) then
+        cells
+    elif List.contains p cells then
+        List.filter (fun c -> c <> p) cells
+    else
+        List.filter (fun c -> index c < index p) cells
+        @ [ p ]
+        @ List.filter (fun c -> index c > index p) cells
+
+// A random board: one `intBelow 4` per cell in row-major order, the cell alive when the draw is 0 (about one cell in
+// four). mapFold threads the seed through the 1200 draws; the result is the cells and the seed after the last draw.
+let private randomCells (seed: uint32) : Point list * uint32 =
+    let draws, next = List.mapFold (fun s _ -> Random.run s (Random.intBelow 4)) seed board
+    List.zip board draws |> List.filter (fun (_, d) -> d = 0) |> List.map fst, next
+
+let private nextGeneration (state: LifeState) : LifeState =
+    { state with
+        Cells = lifeStep state.Cells
+        Generation = state.Generation + 1 }
+
+let lifeUpdate (state: LifeState) (action: LifeAction) : LifeState =
+    match action with
+    | LifeAction.Frame -> if state.Running then nextGeneration state else state
+    | LifeAction.Step -> if state.Running then state else nextGeneration state
+    | LifeAction.TogglePause -> { state with Running = not state.Running }
+    // Clicking edits the world whether it runs or not: the click is one action between two generations.
+    | LifeAction.Toggle p -> { state with Cells = toggle state.Cells p }
+    | LifeAction.Clear -> { state with Cells = []; Generation = 0 }
+    | LifeAction.Randomize ->
+        let cells, seed = randomCells state.Seed
+
+        { state with
+            Cells = cells
+            Seed = seed
+            Generation = 0 }

fsharp/solutions/Life/View.fs

@@ -0,0 +1,17 @@
+/// m13 life view: live cells as squares, and a status strip under the board.
+module Snake.Life.View
+
+open Snake.View
+open Snake.Life.Model
+
+let private text x y str align = Text(x, y, str, "#ffffff", 14, align)
+
+let private W = LW * LCELL
+let private H = LH * LCELL
+
+let lifeView (state: LifeState) : Scene =
+    [ Rect(0, 0, W, H, "#111111") ]
+    @ (state.Cells |> List.map (fun p -> Rect(p.X * LCELL + 1, p.Y * LCELL + 1, LCELL - 2, LCELL - 2, "#7cb342")))
+    @ [ Rect(0, H, W, STRIP, "#222222")
+        text 6 (H + 17) $"Generation {state.Generation}" "left"
+        text (W - 6) (H + 17) (if state.Running then "Running: Space pauses" else "Paused: Space runs, N steps") "right" ]

fsharp/solutions/Model.fs

@@ -25,6 +25,7 @@ type Action =
     | Frame
 
 /// The whole game. `Snake` is head first; `Seed` is the unsigned 32-bit random seed.
+/// `Frame` and `Queued` stay unused until m10 (`Queued` holds a turn waiting for the next step).
 type GameState =
     { Width: int
       Height: int
@@ -35,7 +36,8 @@ type GameState =
       Seed: uint32
       Score: int
       Status: Status
-      Frame: int }
+      Frame: int
+      Queued: Direction option }
 
 let delta (dir: Direction) : Point =
     match dir with
@@ -61,4 +63,5 @@ let initialState (seed: uint32) : GameState =
       Seed = seed
       Score = 0
       Status = Playing
-      Frame = 0 }
+      Frame = 0
+      Queued = None }

fsharp/solutions/Pong/App.fs

@@ -33,4 +33,7 @@ let pongStates (seed: uint32) (actions: Stream<PongAction>) : Stream<PongState>
 
 let pongMain () : IO<Stream<IO<unit>>> =
     IO.randomSeed ()
-    |> IO.map (fun seed -> pongStates seed pongActions |> Stream.map (fun state -> IO.drawScene (pongView state)))
+    |> IO.map (fun seed ->
+        pongStates seed pongActions
+        |> Stream.map (fun state -> IO.drawScene (pongView state))
+        |> Stream.clampTime App.MAX_GAP_MS)

fsharp/solutions/Pong/Rules.fs

@@ -89,5 +89,6 @@ let pongUpdate (state: PongState) (action: PongAction) : PongState =
     match action with
     | PongAction.Press key -> setHeld state key true
     | PongAction.Release key -> setHeld state key false
-    | PongAction.Restart -> initialPong state.Seed
+    // A new match, but the keys are still where the player's fingers are.
+    | PongAction.Restart -> { initialPong state.Seed with Held = state.Held }
     | PongAction.Frame -> frame state

fsharp/solutions/Replay.fs

@@ -0,0 +1,96 @@
+/// m10 replay viewer: a recorded game is a seed and a list of actions; watching it is a second, tiny program
+/// whose state is only a position in the list of the game's states.
+module Snake.Replay
+
+open Snake.Model
+open Snake.View
+open Snake.IO
+open Snake.Stream
+
+type Recording = { Seed: uint32; Actions: Action list }
+
+type Viewer = { Index: int; Last: int; Playing: bool }
+
+[<RequireQualifiedAccess>]
+type ViewerAction =
+    | Next
+    | Prev
+    | First
+    | Last
+    | TogglePlay
+    | Beat
+
+let REPLAY_MS = 100L
+
+// Every state of the recorded game, from the new game to the end: the recording replayed with `history`.
+let frames (recording: Recording) : GameState list =
+    Game.history (Spawn.newGame recording.Seed) recording.Actions
+
+let viewerStart (last: int) : Viewer = { Index = 0; Last = last; Playing = false }
+
+// Where each action moves the viewer, and whether it plays afterwards (before the end-of-recording rule).
+let private move (v: Viewer) (action: ViewerAction) : Viewer =
+    let clamp i = min v.Last (max 0 i)
+
+    match action with
+    | ViewerAction.Next -> { v with Index = clamp (v.Index + 1) }
+    | ViewerAction.Prev -> { v with Index = clamp (v.Index - 1) }
+    | ViewerAction.First -> { v with Index = 0 }
+    | ViewerAction.Last -> { v with Index = v.Last }
+    // Play from the end means: watch it again from the start.
+    | ViewerAction.TogglePlay ->
+        if v.Playing then
+            { v with Playing = false }
+        else
+            { v with
+                Index = (if v.Index = v.Last then 0 else v.Index)
+                Playing = true }
+    | ViewerAction.Beat -> if v.Playing then { v with Index = clamp (v.Index + 1) } else v
+
+// The index never leaves [0, Last], and the viewer never plays on the last frame, so autoplay stops by itself.
+let viewerUpdate (viewer: Viewer) (action: ViewerAction) : Viewer =
+    let next = move viewer action
+    { next with Playing = next.Playing && next.Index < next.Last }
+
+let viewerKey (key: string) : ViewerAction option =
+    match key with
+    | "ArrowRight" | "d" | "D" -> Some ViewerAction.Next
+    | "ArrowLeft" | "a" | "A" -> Some ViewerAction.Prev
+    | "Home" -> Some ViewerAction.First
+    | "End" -> Some ViewerAction.Last
+    | " " -> Some ViewerAction.TogglePlay
+    | _ -> None
+
+// The game's own picture of the current frame, then the position at the top right and a hint at the bottom.
+let viewerView (frames: GameState list) (viewer: Viewer) : Scene =
+    let game = List.item viewer.Index frames
+    let w = game.Width * CELL
+
+    let hint =
+        if viewer.Playing then
+            "Space: pause"
+        else
+            "Left/Right: step   Home/End: jump   Space: play"
+
+    View.view game
+    @ [ Text(w - 6, 18, $"step {viewer.Index} / {viewer.Last}", "#ffffff", 16, "right")
+        Text(w / 2, game.Height * CELL - 8, hint, "#aaaaaa", 12, "center") ]
+
+let replayActions: Stream<ViewerAction> =
+    Stream.merge
+        (Stream.interval REPLAY_MS |> Stream.map (fun _ -> ViewerAction.Beat))
+        (Stream.keyDowns |> Stream.choose viewerKey)
+
+let viewerStates (last: int) (actions: Stream<ViewerAction>) : Stream<Viewer> =
+    actions |> Stream.scan viewerUpdate (viewerStart last) |> Stream.startWith (viewerStart last)
+
+// It needs no seed from the world (the recording has one), so it starts with `pure`.
+// The frames are computed once, here, and every drawing picks one of them.
+let replayMain (recording: Recording) : IO<Stream<IO<unit>>> =
+    let fs = frames recording
+
+    IO.pure (
+        viewerStates (List.length fs - 1) replayActions
+        |> Stream.map (fun v -> IO.drawScene (viewerView fs v))
+        |> Stream.clampTime App.MAX_GAP_MS
+    )

fsharp/solutions/Runtime.fs

@@ -11,14 +11,23 @@ type Platform =
     { FillRect: int -> int -> int -> int -> string -> unit
       FillText: int -> int -> string -> string -> int -> string -> unit
       RandomSeed: unit -> uint32
+      // m10: the stored best score; reading must never fail (anything unreadable is 0).
+      LoadBest: unit -> int
+      SaveBest: int -> unit
       StartClock: (int64 -> unit) -> unit
       OnKeyDown: (string -> unit) -> unit
-      OnKeyUp: (string -> unit) -> unit }
+      OnKeyUp: (string -> unit) -> unit
+      // m13: clicks on the canvas, in canvas pixels.
+      OnClick: (int -> int -> unit) -> unit }
 
 // Perform one op on the platform and return its answer.
 let private perform (p: Platform) (op: Op) : OpResult =
     match op with
     | RandomSeed -> Seed(p.RandomSeed())
+    | LoadBest -> Best(p.LoadBest())
+    | SaveBest score ->
+        p.SaveBest score
+        Done
     | Draw(Rect(x, y, w, h, color)) ->
         p.FillRect x y w h color
         Done
@@ -43,4 +52,6 @@ let runApp (p: Platform) (main: IO<Stream<IO<unit>>>) : unit =
 
     p.OnKeyDown(fun key -> feed (KeyDown key))
     p.OnKeyUp(fun key -> feed (KeyUp key))
+    // m13: a fourth kind of event. Programs that do not listen for clicks simply ignore them.
+    p.OnClick(fun x y -> feed (Click(x, y)))
     p.StartClock(fun t -> feed (Time t))

fsharp/solutions/SafeIO.fs

@@ -0,0 +1,65 @@
+/// m07b safeio (optional): a stack-safe IO. Same idea as m07's IO, plus one more case, so that `bind` only builds
+/// data and no program shape can overflow the stack, however deeply its `bind`s are nested.
+///   - `Pure a`            done, with a value
+///   - `Effect (op, k)`    perform `op`, then continue with `k result`
+///   - `FlatMap (io, f)`   run `io`, then continue with `f result`
+/// The ops, their answers and the result of `trace` are m07's `IO.Op`, `IO.OpResult` and `IO.Traced`.
+/// (Not `open Snake.IO`: its cases are also called Pure and Effect.)
+module Snake.SafeIO
+
+/// `FlatMap`'s inner program produces some type `X` that only its continuation knows. A union case cannot have a
+/// type parameter of its own, so `X` is erased to `obj`; only `bind` builds a `FlatMap`, and it keeps the two in step.
+type SafeIO<'a> =
+    | Pure of 'a
+    | Effect of IO.Op * (IO.OpResult -> SafeIO<'a>)
+    | FlatMap of SafeIO<obj> * (obj -> SafeIO<'a>)
+
+// Given: the same program, seen as a SafeIO<obj>. Constant work: continuations are wrapped, not called.
+let rec private erase (io: SafeIO<'a>) : SafeIO<obj> =
+    match io with
+    | Pure a -> Pure(box a)
+    | Effect(op, k) -> Effect(op, fun r -> erase (k r))
+    | FlatMap(m, g) -> FlatMap(m, fun x -> erase (g x))
+
+let pure (a: 'a) : SafeIO<'a> = Pure a
+
+let effect (op: IO.Op) : SafeIO<IO.OpResult> = Effect(op, pure)
+
+// Only builds a node: never calls f, not even when io is Pure. The unbox cannot fail: x is the result of `io`.
+let bind (f: 'a -> SafeIO<'b>) (io: SafeIO<'a>) : SafeIO<'b> = FlatMap(erase io, fun x -> f (unbox<'a> x))
+
+let map (f: 'a -> 'b) (io: SafeIO<'a>) : SafeIO<'b> = io |> bind (fun a -> Pure(f a))
+
+// A left fold on purpose: with SafeIO the nesting no longer matters.
+let sequence (ios: SafeIO<unit> list) : SafeIO<unit> = List.fold (fun acc io -> acc |> bind (fun () -> io)) (pure ()) ios
+
+// One rewrite towards a program whose top is a Pure or an Effect. Constant work, no recursion.
+//   FlatMap(Pure a, f)          -> f a
+//   FlatMap(Effect(op, k), f)   -> Effect(op, fun r -> FlatMap(k r, f))
+//   FlatMap(FlatMap(m, g), f)   -> FlatMap(m, fun x -> FlatMap(g x, f))     (re-association)
+// Pure and Effect are already in that shape and come back unchanged.
+let step (io: SafeIO<'a>) : SafeIO<'a> =
+    match io with
+    | FlatMap(Pure x, f) -> f x
+    | FlatMap(Effect(op, k), f) -> Effect(op, fun r -> FlatMap(k r, f))
+    | FlatMap(FlatMap(m, g), f) -> FlatMap(m, fun x -> FlatMap(g x, f))
+    | other -> other
+
+// Given: the canned answer of the pure interpreter, as in m07. SafeIO keeps Module 7's two operations; the stored
+// best score of Module 10 is not part of it, so its operations are refused loudly rather than given a meaning here.
+let private answer (seed: uint32) (op: IO.Op) : IO.OpResult =
+    match op with
+    | IO.RandomSeed -> IO.Seed seed
+    | IO.Draw _ -> IO.Done
+    | IO.LoadBest
+    | IO.SaveBest _ -> failwith "SafeIO.trace answers only Module 7's operations (Draw and RandomSeed)"
+
+// `loop` calls itself only in tail position, so the compiler turns it into a loop: the stack stays flat.
+let trace (seed: uint32) (io: SafeIO<'a>) : IO.Traced<'a> =
+    let rec loop (current: SafeIO<'a>) (performed: IO.Op list) : IO.Traced<'a> =
+        match current with
+        | Pure a -> { IO.Result = a; IO.Ops = List.rev performed }
+        | Effect(op, k) -> loop (k (answer seed op)) (op :: performed)
+        | FlatMap _ -> loop (step current) performed
+
+    loop io []

fsharp/solutions/Snake.fsproj

@@ -10,17 +10,25 @@
     <Compile Include="Model.fs" />
     <Compile Include="Rules.fs" />
     <Compile Include="Random.fs" />
+    <Compile Include="State.fs" />
     <Compile Include="Spawn.fs" />
     <Compile Include="Game.fs" />
     <Compile Include="View.fs" />
     <Compile Include="IO.fs" />
+    <Compile Include="SafeIO.fs" />
     <Compile Include="Stream.fs" />
     <Compile Include="Input.fs" />
     <Compile Include="App.fs" />
     <Compile Include="Runtime.fs" />
+    <Compile Include="Replay.fs" />
     <Compile Include="Pong/Model.fs" />
     <Compile Include="Pong/Rules.fs" />
     <Compile Include="Pong/View.fs" />
     <Compile Include="Pong/App.fs" />
+    <Compile Include="Life/Model.fs" />
+    <Compile Include="Life/Rules.fs" />
+    <Compile Include="Life/View.fs" />
+    <Compile Include="Life/App.fs" />
+    <Compile Include="Check.fs" />
   </ItemGroup>
 </Project>

fsharp/solutions/State.fs

@@ -0,0 +1,27 @@
+/// m04 state (optional, Exercise 4.7): the general pattern behind Random. A `State<'s, 'a>` wraps a function
+/// `s -> (value, nextS)`: it reads a state, produces a value and hands on a new state. `Random<'a>` is
+/// `State<uint32, 'a>`. Nothing in the game uses this file; it is here to show that Random was one example of a bigger idea.
+module Snake.State
+
+type State<'s, 'a> = State of ('s -> 'a * 's)
+
+let run (s: 's) (State f: State<'s, 'a>) : 'a * 's = f s
+
+let pure (a: 'a) : State<'s, 'a> = State(fun s -> (a, s))
+
+let map (f: 'a -> 'b) (st: State<'s, 'a>) : State<'s, 'b> =
+    State(fun s ->
+        let a, next = run s st
+        (f a, next))
+
+let bind (f: 'a -> State<'s, 'b>) (st: State<'s, 'a>) : State<'s, 'b> =
+    State(fun s ->
+        let a, next = run s st
+        run next (f a))
+
+// A function of unit: a generic module-level value is not allowed in F#.
+let get () : State<'s, 's> = State(fun s -> (s, s))
+
+let put (s: 's) : State<'s, unit> = State(fun _ -> ((), s))
+
+let modify (f: 's -> 's) : State<'s, unit> = State(fun s -> ((), f s))

fsharp/solutions/Stream.fs

@@ -2,11 +2,13 @@
 /// There is no hidden state: whatever a stream "remembers" lives in the next stream it returns.
 module Snake.Stream
 
-/// Something that happens to the program: time passing (ms since start) or a key going down or up.
+/// Something that happens to the program: time passing (ms since start) or a key going down or up, and from
+/// Module 13 a mouse click at `(x, y)` in canvas pixels. Every stream ignores the kinds of event it does not use.
 type Event =
     | Time of int64
     | KeyDown of string
     | KeyUp of string
+    | Click of int * int
 
 type Stream<'a> = Stream of (Event -> 'a list * Stream<'a>)
 
@@ -31,6 +33,16 @@ let keyUps: Stream<string> =
 
     source ()
 
+// (m13, given in src/) The position of every Click event, in canvas pixels. Like the key sources it remembers nothing.
+let clicks: Stream<int * int> =
+    let rec source () =
+        Stream(fun e ->
+            match e with
+            | Click(x, y) -> [ (x, y) ], source ()
+            | _ -> [], source ())
+
+    source ()
+
 // `last` is the time of the previous Time event. Multiples of ms in (last, t] fire, ascending.
 let rec private intervalFrom (ms: int64) (last: int64) : Stream<int64> =
     Stream(fun e ->
@@ -76,6 +88,23 @@ let startWith (v: 'a) (s: Stream<'a>) : Stream<'a> =
         let out, next = step e s
         v :: out, next)
 
+// (m10) `last` is the previous real time, `clock` the time the inner stream has been told. A long silence
+// (a background tab) moves `clock` by at most `maxGap`, so the inner stream never has to catch up on minutes of beats.
+let rec private clampFrom (maxGap: int64) (last: int64) (clock: int64) (s: Stream<'a>) : Stream<'a> =
+    Stream(fun e ->
+        match e with
+        | Time t ->
+            let now = clock + min (max (t - last) 0L) maxGap
+            let out, next = step (Time now) s
+            out, clampFrom maxGap t now next
+        | KeyDown _
+        | KeyUp _
+        | Click _ ->
+            let out, next = step e s
+            out, clampFrom maxGap last clock next)
+
+let clampTime (maxGap: int64) (s: Stream<'a>) : Stream<'a> = clampFrom maxGap 0L 0L s
+
 // mapFold threads the stream through the events like fold, and keeps each step's outputs.
 let run (events: Event list) (s: Stream<'a>) : 'a list =
     List.mapFold (fun stream e -> step e stream) s events |> fst |> List.concat