modify Statement

server/nng/NNG/Doc/Definitions.lean

@@ -3,4 +3,12 @@ import GameServer.Commands
 DefinitionDoc MyNat as "ℕ"
 "
 The Natural Numbers.
+
+## Game Modifications
+
+This notation is for our own version of the natural numbers, called `MyNat`.
+The natural numbers implemented in Lean's core are called `Nat`.
+
+If you end up getting someting of type `Nat` in this game, you probably
+need to write `(4 : ℕ)` to force it to be of type `MyNat`.
 "

server/nng/NNG/Doc/Tactics.lean

@@ -2,6 +2,34 @@ import GameServer.Commands
 
 TacticDoc rfl
 "
+## Summary
+
+`rfl` proves goals of the form `X = X`.
+
+## Details
+
+The `rfl` tactic will close any goal of the form `A = B`
+where `A` and `B` are *exactly the same thing*.
+
+## Example:
+If it looks like this in the top right hand box:
+```
+Objects:
+  a b c d : ℕ
+Goal:
+  (a + b) * (c + d) = (a + b) * (c + d)
+```
+
+then `rfl` will close the goal and solve the level.
+
+## Game modifications
+`rfl` in this game is weakened.
+
+The real `rfl` could also proof goals of the form `A = B` where the
+two sides are not *exactly identical* but merely
+*\"definitionally equal\"*. That means the real `rfl`
+could prove things like `a + 0 = a`.
+
 "
 
 TacticDoc rewrite
@@ -10,6 +38,83 @@ TacticDoc rewrite
 
 TacticDoc rw
 "
+## Summary
+
+If `h` is a proof of `X = Y`, then `rw h,` will change
+all `X`s in the goal to `Y`s. Variants: `rw ← h` (changes
+`Y` to `X`) and
+`rw h at h2` (changes `X` to `Y` in hypothesis `h2` instead
+of the goal).
+
+## Details
+
+The `rw` tactic is a way to do \"substituting in\". There
+are two distinct situations where use this tactics.
+
+1) If `h : A = B` is a hypothesis (i.e., a proof of `A = B`)
+in your local context (the box in the top right)
+and if your goal contains one or more `A`s, then `rw h`
+will change them all to `B`'s.
+
+2) The `rw` tactic will also work with proofs of theorems
+which are equalities (look for them in the drop down
+menu on the left, within Theorem Statements).
+For example, in world 1 level 4
+we learn about `add_zero x : x + 0 = x`, and `rw add_zero`
+will change `x + 0` into `x` in your goal (or fail with
+an error if Lean cannot find `x + 0` in the goal).
+
+Important note: if `h` is not a proof of the form `A = B`
+or `A ↔ B` (for example if `h` is a function, an implication,
+or perhaps even a proposition itself rather than its proof),
+then `rw` is not the tactic you want to use. For example,
+`rw (P = Q)` is never correct: `P = Q` is the true-false
+statement itself, not the proof.
+If `h : P = Q` is its proof, then `rw h` will work.
+
+Pro tip 1: If `h : A = B` and you want to change
+`B`s to `A`s instead, try `rw ←h` (get the arrow with `\\l` and
+note that this is a small letter L, not a number 1).
+
+### Example:
+If it looks like this in the top right hand box:
+
+```
+  x y : ℕ
+  h : x = y + y
+Goal:
+  succ (x + 0) = succ (y + y)
+```
+
+then
+
+`rw add_zero,`
+
+will change the goal into `⊢ succ x = succ (y + y)`, and then
+
+`rw h,`
+
+will change the goal into `⊢ succ (y + y) = succ (y + y)`, which
+can be solved with `refl,`.
+
+### Example:
+
+You can use `rw` to change a hypothesis as well.
+For example, if your local context looks like this:
+
+```
+  x y : ℕ
+  h1 : x = y + 3
+  h2 : 2 * y = x
+Goal:
+  y = 3
+```
+then `rw h1 at h2` will turn `h2` into `h2 : 2 * y = y + 3`.
+
+## Game modifications
+
+The real `rw` tactic does automatically try to call `rfl` afterwards. We disabled this
+feature for the game.
 "
 
 TacticDoc induction