add NNG
This commit is contained in:
Jon Eugster
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import NNG.Metadata
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import NNG.MyNat.Multiplication
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Game "NNG"
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World "Tutorial"
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Level 1
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Title "The rfl tactic"
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Introduction
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"
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Each level in this game involves proving a mathematical statement. In this first level
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you have three natural numbers $x, y, z$ (listed under \"Objects\") and you want to prove
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$x \\cdot y + z = x \\cdot y + z$ (displayed under \"Goal\").
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You can modify the Goal using *Tactics* until you can close (i.e. prove) it.
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The first tactic is called `rfl`, which stands for \"reflexivity\",
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a fancy way of saying that it will prove any goal of the form `A = A`. It doesn't matter how
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complicated `A` is, all that matters is that the left hand side is exactly equal to the right hand
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side (a computer scientist would say \"definitionally equal\"). I really mean \"press the same buttons
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on your computer in the same order\" equal. For example, `x * y + z = x * y + z` can be proved by `rfl`,
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but `x + y = y + x` cannot.
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"
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Statement
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"For all natural numbers $x, y$ and $z$, we have $xy + z = xy + z$."
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(x y z : ℕ) : x * y + z = x * y + z := by
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Hint "In order to use the tactic `rfl` you can enter it above and hit \"Execute\"."
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rfl
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NewTactic rfl
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NewDefinition MyNat
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Conclusion
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"
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Congratulations! You completed your first verified proof!
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If you want to be reminded about the `rfl` tactic, your inventory on the right contains useful
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information about things you've learned.
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Now click on \"Next\" to continue the journey.
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"
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import NNG.Metadata
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import NNG.MyNat.Multiplication
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Game "NNG"
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World "Tutorial"
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Level 2
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Title "the rewrite (rw) tactic"
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Introduction
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"
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In this level, you also get \"Assumptions\" about your objects. These are hypotheses of which
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you assume (or know) that they are true.
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The `rewrite` tactic is the way to \"substitute in\" the value of a variable.
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If you have a hypothesis of the form `A = B`, and your goal mentions
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the left hand side `A` somewhere,
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then the rewrite tactic will replace the `A` in your goal with a `B`.
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*(Note: For this game, `rw` is a shorthand for `rewrite`. Out in the real world, `rw` tries to call
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`rfl` automatically afterwards.)*
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"
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Statement
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"If $x$ and $y$ are natural numbers, and $y = x + 7$, then $2y = 2(x + 7)$."
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(x y : ℕ) (h : y = x + 7) : 2 * y = 2 * (x + 7) := by
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Hint "You can use `rewrite [h]` to replace the `{y}` with `x + 7`.
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Note that the assumption `h` is written
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inside square brackets: `[h]`."
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rw [h]
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Hint "In this game not all hints are directly shown. If you need help finishing the proof, click
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on \"More Help\" below!"
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Hint (hidden := true)
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"Now both sides are identical, so you can use `rfl` to close the goal."
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rfl
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NewTactic rewrite rw
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Conclusion
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"
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If you want to see the entire proof you created, toggle \"Editor mode\" above.
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There you can also move your cursor around the proof to see the \"state\" of the proof at this point.
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Each tactic is written on a new line and Lean is sensitive to indentation (i.e. there must be no
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spaces before any of the tactics)
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"
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import NNG.Metadata
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import NNG.MyNat.Definition
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Game "NNG"
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World "Tutorial"
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Level 3
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Title "Peano axioms"
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open MyNat
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Introduction
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"
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Now we start from the beginning, where we don't know about addition or multiplication on `ℕ`.
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All we get is the following data:
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* a term `(0 : ℕ)`, interpreted as the zero number.
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* a function `succ : ℕ → ℕ`, with `succ n` interpreted as \"the number after $n$\".
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* the principle of mathematical induction.
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These axioms are essentially the axioms isolated by Peano which uniquely characterise the natural
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numbers (we also need recursion, but we can ignore it for now).
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The first axiom says that $0$ is a natural number.
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The second says that there is a $\\operatorname{succ}$ function which eats a number and spits out
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the number after it, so $\\operatorname{succ}(0)=1$, $\\operatorname{succ}(1)=2$ and so on.
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Peano's last axiom is the principle of mathematical induction. This is a deeper fact.
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It says that if we have infinitely many true/false statements $P(0)$, $P(1)$, $P(2)$ and so on,
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and if $P(0)$ is true, and if for every natural number $d$ we know that $P(d)$ implies
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$P(\\operatorname{succ}(d))$, then $P(n)$ must be true for every natural number $n$.
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It's like saying that if you have a long line of dominoes, and if you knock the first
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one down, and if you know that if a domino falls down then the one after it will fall
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down too, then you can deduce that all the dominos will fall down. One can also think
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of it as saying that every natural number can be built by starting at $0$ and then applying
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$\\operatorname{succ}$ a finite number of times.
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Peano's insights were firstly that these axioms completely characterise the natural numbers,
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and secondly that these axioms alone can be used to build a whole bunch of other structure
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on the natural numbers, for example addition, multiplication and so on.
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This game is all about seeing how far these axioms of Peano can take us.
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Now let us practise the use of `rewrite` with this new function `succ`:
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"
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Statement
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"If $\\operatorname{succ}(a) = b$, then $\\operatorname{succ}(\\operatorname{succ}(a)) = \\operatorname{succ}(a)$."
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(a b : ℕ) (h : (succ a) = b) : succ (succ a) = succ b := by
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Hint "You can use `rewrite` and your assumption `{h}` to substitute `succ a` with `b`.
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Notes:
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1) We do not need brackets for function application the way we would write
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them in mathematics: `succ b` means $\\operatorname\{succ}(b)$.
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2) If you would want to substitute instead `b` with `succ a`, you can do that
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writing a small `←` (`\\l`, i.e. backslash + small letter L + space)
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before `h` like this: `rewrite [← h]`."
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Branch
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rewrite [← h]
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Hint (hidden := true) "Now both sides are identical…"
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rewrite [h]
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Hint (hidden := true) "Now both sides are identical…"
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rfl
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Conclusion
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"
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You may also be wondering why we keep writing `succ b` instead of `b + 1`.
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This is because we haven't defined addition yet!
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On the next level, the final level of Tutorial World,
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we will introduce addition, and then we'll be ready to enter Addition World.
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"
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import NNG.Metadata
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import NNG.MyNat.Addition
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Game "NNG"
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World "Tutorial"
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Level 4
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Title "addition"
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open MyNat
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Introduction
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"
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Peano defined addition $a + b$ by induction on $b$, or, more precisely, by *recursion* on $b$.
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He first explained how to add $0$ to a number: this is the base case.
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* `add_zero (a : ℕ) : a + 0 = a`
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We will call this theorem `add_zero`.
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More precisely, `add_zero` is the name of the *proof* of the
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theorem. Note the name of this proof. Mathematicians sometimes call it
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\"Lemma 2.1\" or \"Hypothesis P6\" or something.
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But computer scientists call it `add_zero` because it tells you what the answer to
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\"x add zero\" is. It's a much better name than \"Lemma 2.1\".
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Even better, you can use the `rewrite` tactic
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with `add_zero`. If you ever see `x + 0` in your goal,
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`rewrite [add_zero]` should simplify it to `x`. This is
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because `add_zero` is a proof that `x + 0 = x`
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(more precisely, `add_zero x` is a proof that `x + 0 = x` but
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Lean can figure out the `x` from the context).
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Now here's the inductive step. If you know how to add $d$ to $a$, then Peano
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tells you how to add $\\operatorname{succ} (d)$ to $a$. It looks like this:
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- `add_succ (a d : ℕ) : a + (succ d) = succ (a + d)`
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What's going on here is that we assume `a + d` is already defined, and we define
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`a + (succ d)` to be the number after it.
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Note the name of this proof too: `add_succ` tells you how to add a successor
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to something.
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If you ever see `… + succ …` in your goal, you should be able to use
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`rewrite [add_succ]`, to make progress.
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"
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Statement
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"For all natural numbers $a$, we have $a + \\operatorname{succ}(0) = a$."
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(a : ℕ) : a + succ 0 = succ a := by
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Hint "You find `{a} + succ …` in the goal, so you can use `rewrite` and `add_succ`
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to make progress."
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Hint (hidden := true) "Explicitely, type `rewrite [add_succ]`!"
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rewrite [add_succ]
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Hint "Now you see a term of the form `… + 0`, so you can use `add_zero`."
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Hint (hidden := true) "Explicitely, type `rewrite [add_zero]`!"
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rewrite [add_zero]
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Hint (hidden := true) "Finally both sides are identical."
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rfl
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NewLemma MyNat.add_succ MyNat.add_zero
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Conclusion
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"
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You have finished tutorial world! If you're happy, let's move onto Addition World,
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and learn about proof by induction.
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## Inspection time
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If you want to examine the proof, toggle \"Editor mode\" and click somewhere
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inside the proof to see the state at that point!
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"
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