NNG progress
This commit is contained in:
Jon Eugster
@@ -1,7 +1,5 @@
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import NNG.Metadata
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import NNG.MyNat.Multiplication
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import Std.Tactic.RCases
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import Mathlib.Tactic.LeftRight
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import NNG.Levels.Multiplication
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import NNG.Levels.AdvAddition
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Game "NNG"
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World "AdvMultiplication"
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@@ -12,28 +10,24 @@ open MyNat
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Introduction
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"
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Welcome to Advanced Multiplication World! Before attempting this
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world you should have completed seven other worlds, including
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Multiplication World and Advanced Addition World. There are four
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levels in this world.
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Recall that if `b : mynat` is a hypothesis and you do `cases b with n`,
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your one goal will split into two goals,
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namely the cases `b = 0` and `b = succ(n)`. So `cases` here is like
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a weaker version of induction (you don't get the inductive hypothesis).
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## Tricks
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1) if your goal is `⊢ X ≠ Y` then `intro h` will give you `h : X = Y` and
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a goal of `⊢ false`. This is because `X ≠ Y` *means* `(X = Y) → false`.
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Conversely if your goal is `false` and you have `h : X ≠ Y` as a hypothesis
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1) if your goal is `X ≠ Y` then `intro h` will give you `h : X = Y` and
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a goal of `False`. This is because `X ≠ Y` *means* `(X = Y) → False`.
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Conversely if your goal is `False` and you have `h : X ≠ Y` as a hypothesis
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then `apply h` will turn the goal into `X = Y`.
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2) if `hab : succ (3 * x + 2 * y + 1) = 0` is a hypothesis and your goal is `⊢ false`,
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2) if `hab : succ (3 * x + 2 * y + 1) = 0` is a hypothesis and your goal is `False`,
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then `exact succ_ne_zero _ hab` will solve the goal, because Lean will figure
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out that `_` is supposed to be `3 * x + 2 * y + 1`.
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"
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-- TODO: cases
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-- Recall that if `b : ℕ` is a hypothesis and you do `cases b with n`,
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-- your one goal will split into two goals,
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-- namely the cases `b = 0` and `b = succ n`. So `cases` here is like
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-- a weaker version of induction (you don't get the inductive hypothesis).
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Statement
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"The product of two non-zero natural numbers is non-zero."
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@@ -1,7 +1,4 @@
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import NNG.Metadata
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import NNG.MyNat.Multiplication
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import Std.Tactic.RCases
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import Mathlib.Tactic.LeftRight
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import NNG.Levels.AdvMultiplication.Level_1
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Game "NNG"
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World "AdvMultiplication"
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@@ -15,14 +12,14 @@ Introduction
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A variant on the previous level.
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"
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Statement -- eq_zero_or_eq_zero_of_mul_eq_zero
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Statement MyNat.eq_zero_or_eq_zero_of_mul_eq_zero
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"If $ab = 0$, then at least one of $a$ or $b$ is equal to zero."
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(a b : ℕ) (h : a * b = 0) :
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a = 0 ∨ b = 0 := by
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induction a with d
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left
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rfl
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induction b with e he
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induction b with e
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right
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rfl
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exfalso
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@@ -1,7 +1,5 @@
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import NNG.Metadata
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import NNG.MyNat.Multiplication
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import Std.Tactic.RCases
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import Mathlib.Tactic.LeftRight
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import NNG.Levels.AdvMultiplication.Level_2
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Game "NNG"
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World "AdvMultiplication"
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@@ -15,13 +13,10 @@ Introduction
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Now you have `eq_zero_or_eq_zero_of_mul_eq_zero` this is pretty straightforward.
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"
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axiom eq_zero_or_eq_zero_of_mul_eq_zero (a b : ℕ) (h : a * b = 0) : a = 0 ∨ b = 0
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axiom zero_mul (a : ℕ) : 0 * a = 0
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Statement
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"$ab = 0$, if and only if at least one of $a$ or $b$ is equal to zero.
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"
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(a b : ℕ): a * b = 0 ↔ a = 0 ∨ b = 0 := by
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{a b : ℕ} : a * b = 0 ↔ a = 0 ∨ b = 0 := by
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constructor
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intro h
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exact eq_zero_or_eq_zero_of_mul_eq_zero a b h
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@@ -1,7 +1,4 @@
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import NNG.Metadata
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import NNG.MyNat.Multiplication
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import Std.Tactic.RCases
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import Mathlib.Tactic.LeftRight
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import NNG.Levels.AdvMultiplication.Level_3
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Game "NNG"
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World "AdvMultiplication"
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@@ -19,14 +16,17 @@ People find this level hard. I have probably had more questions about this
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level than all the other levels put together, in fact. Many levels in this
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game can just be solved by \"running at it\" -- do induction on one of the
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variables, keep your head, and you're done. In fact, if you like a challenge,
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it might be instructive if you stop reading after the end of this paragraph and try solving this level now by induction,
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seeing the trouble you run into, and reading the rest of these comments afterwards. This level
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it might be instructive if you stop reading after the end of this paragraph
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and try solving this level now by induction,
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seeing the trouble you run into, and reading the rest of these comments afterwards.
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This level
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has a sting in the tail. If you are a competent mathematician, try
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and figure out what is going on. Write down a maths proof of the
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theorem in this level. Exactly what statement do you want to prove
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by induction? It is subtle.
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Ok so here are some spoilers. The problem with naively running at it, is that if you try induction on,
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Ok so here are some spoilers. The problem with naively running at it,
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is that if you try induction on,
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say, $c$, then you are imagining a and b as fixed, and your inductive
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hypothesis $P(c)$ is $ab=ac \\implies b=c$. So for your inductive step
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you will be able to assume $ab=ad \\implies b=d$ and your goal will
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@@ -36,30 +36,64 @@ $ab=ad$ is not true! The statement $P(c)$ (with $a$ and $b$ regarded
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as constants) is not provable by induction.
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What you *can* prove by induction is the following *stronger* statement.
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Imagine $a\not=0$ as fixed, and then prove \"for all $b$, if $ab=ac$ then $b=c$\"
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Imagine $a\\ne 0$ as fixed, and then prove \"for all $b$, if $ab=ac$ then $b=c$\"
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by induction on $c$. This gives us the extra flexibility we require.
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Note that we are quantifying over all $b$ in the inductive hypothesis -- it
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is essential that $b$ is not fixed.
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is essential that $b$ is not fixed.
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You can do this in two ways in Lean -- before you start the induction
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you can write `revert b,`. The `revert` tactic is the opposite of the `intro`
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you can write `revert b`. The `revert` tactic is the opposite of the `intro`
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tactic; it replaces the `b` in the hypotheses with \"for all $b$\" in the goal.
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Alternatively, you can write `induction c with d hd
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generalizing b` as the first line of the proof.
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[TODO: The second way does not work yet in this game]
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If you do not modify your technique in this way, then this level seems
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to be impossible (judging by the comments I've had about it!)
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"
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-- Alternatively, you can write `induction c with d hd
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-- generalizing b` as the first line of the proof.
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Statement
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Statement MyNat.mul_left_cancel
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"If $a \\neq 0$, $b$ and $c$ are natural numbers such that
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$ ab = ac, $
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then $b = c$."
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(a b c : ℕ) (ha : a ≠ 0) : a * b = a * c → b = c := by
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sorry
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Hint "NOTE: As is, this level is probably too hard and contains no hints yet.
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Good luck!
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Your first step should be `revert b`!"
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revert b
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induction c with c hc
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· intro b hb
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rw [mul_zero] at hb
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rcases eq_zero_or_eq_zero_of_mul_eq_zero _ _ hb
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exfalso
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apply ha
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exact h
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exact h
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· intro b h
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induction b with b hb
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exfalso
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apply ha
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rw [mul_zero] at h
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rcases eq_zero_or_eq_zero_of_mul_eq_zero _ _ h.symm with h | h
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exact h
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exfalso
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exact succ_ne_zero _ h
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rw [mul_succ, mul_succ] at h
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have j : b = c
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apply hc
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exact add_right_cancel _ _ _ h
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rw [j]
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rfl
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-- TODO: generalizing b.
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NewTactic revert
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Conclusion
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"
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"
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