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