NNG progress
This commit is contained in:
Jon Eugster
@@ -1,5 +1,5 @@
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import NNG.Metadata
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--import NNG.MyNat.Power
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import NNG.Levels.Multiplication
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import NNG.MyNat.Power
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Game "NNG"
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World "Power"
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@@ -8,18 +8,12 @@ Title "zero_pow_zero"
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open MyNat
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Introduction
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"
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"
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Statement
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Statement MyNat.zero_pow_zero
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"$0 ^ 0 = 1$"
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: (0 : ℕ) ^ 0 = (1 : ℕ) := by -- (0 : ℕ) ^ (0 : ℕ) = 1 := by
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trivial
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Conclusion
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"
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"
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: (0 : ℕ) ^ 0 = 1 := by
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rw [pow_zero]
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rfl
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NewLemma MyNat.pow_zero MyNat.pow_succ
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NewDefinition Pow
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LemmaTab "Pow"
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@@ -1,24 +1,17 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_1
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Game "NNG"
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World "Power"
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Level 2
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Title ""
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Title "zero_pow_succ"
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open MyNat
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Introduction
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"
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Statement MyNat.zero_pow_succ
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"For all naturals $m$, $0 ^{\\operatorname{succ} (m)} = 0$."
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(m : ℕ) : (0 : ℕ) ^ (succ m) = 0 := by
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rw [pow_succ]
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rw [mul_zero]
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rfl
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"
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Statement
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""
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: true := by
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trivial
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Conclusion
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"
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"
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LemmaTab "Pow"
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@@ -1,24 +1,19 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_2
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Game "NNG"
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World "Power"
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Level 3
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Title ""
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Title "pow_one"
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open MyNat
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Introduction
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"
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Statement MyNat.pow_one
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"For all naturals $a$, $a ^ 1 = a$."
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(a : ℕ) : a ^ 1 = a := by
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rw [one_eq_succ_zero]
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rw [pow_succ]
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rw [pow_zero]
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rw [one_mul]
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rfl
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"
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Statement
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""
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: true := by
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trivial
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Conclusion
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"
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"
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LemmaTab "Pow"
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@@ -1,25 +1,22 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_3
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Game "NNG"
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World "Power"
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Level 4
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Title ""
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Title "one_pow"
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open MyNat
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Introduction
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"
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Statement MyNat.one_pow
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"For all naturals $m$, $1 ^ m = 1$."
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(m : ℕ) : (1 : ℕ) ^ m = 1 := by
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induction m with t ht
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· rw [pow_zero]
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rfl
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· rw [pow_succ]
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rw [ht]
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rw [mul_one]
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rfl
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"
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Statement
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""
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: true := by
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trivial
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Conclusion
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"
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"
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LemmaTab "Pow"
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@@ -1,24 +1,20 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_4
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Game "NNG"
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World "Power"
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Level 5
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Title ""
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Title "pow_add"
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open MyNat
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Introduction
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"
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Statement MyNat.pow_add
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"For all naturals $a$, $m$, $n$, we have $a^{m + n} = a ^ m a ^ n$."
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(a m n : ℕ) : a ^ (m + n) = a ^ m * a ^ n := by
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induction n with t ht
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· rw [add_zero, pow_zero, mul_one]
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rfl
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· rw [add_succ, pow_succ, pow_succ, ht, mul_assoc]
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rfl
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"
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Statement
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""
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: true := by
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trivial
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Conclusion
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"
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"
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LemmaTab "Pow"
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@@ -1,24 +1,30 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_5
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Game "NNG"
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World "Power"
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Level 6
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Title ""
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Title "mul_pow"
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open MyNat
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Introduction
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"
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You might find the tip at the end of level 9 of Multiplication World
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useful in this one. You can go to the main menu and pop back into
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Multiplication World and take a look -- you won't lose any of your
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proofs.
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"
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Statement
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""
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: true := by
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trivial
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Statement MyNat.mul_pow
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"For all naturals $a$, $b$, $n$, we have $(ab) ^ n = a ^ nb ^ n$."
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(a b n : ℕ) : (a * b) ^ n = a ^ n * b ^ n := by
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induction n with t Ht
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· rw [pow_zero, pow_zero, pow_zero, mul_one]
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rfl
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· rw [pow_succ, pow_succ, pow_succ, Ht]
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-- simp
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repeat rw [mul_assoc]
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rw [mul_comm a (_ * b), mul_assoc, mul_comm b a]
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rfl
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Conclusion
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"
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"
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LemmaTab "Pow"
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@@ -1,24 +1,35 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_6
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Game "NNG"
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World "Power"
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Level 7
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Title ""
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Title "pow_pow"
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open MyNat
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Introduction
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"
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Boss level! What will the collectible be?
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"
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Statement
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""
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: true := by
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trivial
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Statement MyNat.pow_pow
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"For all naturals $a$, $m$, $n$, we have $(a ^ m) ^ n = a ^ {mn}$."
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(a m n : ℕ) : (a ^ m) ^ n = a ^ (m * n) := by
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induction n with t Ht
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· rw [mul_zero, pow_zero, pow_zero]
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rfl
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· rw [pow_succ, Ht, mul_succ, pow_add]
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rfl
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LemmaTab "Pow"
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Conclusion
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"
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Apparently Lean can't find a collectible, even though you feel like you
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just finished power world so you must have proved *something*. What should the
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collectible for this level be called?
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But what is this? It's one of those twists where there's another
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boss after the boss you thought was the final boss! Go to the next
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level!
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"
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@@ -1,22 +1,49 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Power.Level_7
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-- import Mathlib.Tactic.Ring
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Game "NNG"
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World "Power"
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Level 8
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Title ""
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Title "add_squared"
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open MyNat
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Introduction
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"
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[final boss music]
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You see something written on the stone dungeon wall:
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```
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by
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rw [two_eq_succ_one]
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rw [one_eq_succ_zero]
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repeat rw [pow_succ]
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…
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```
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and you can't make out the last two lines because there's a kind
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of thing in the way that will magically disappear
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but only when you've beaten the boss.
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"
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Statement
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""
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: true := by
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trivial
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Statement MyNat.add_squared
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"For all naturals $a$ and $b$, we have
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$$(a+b)^2=a^2+b^2+2ab.$$"
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(a b : ℕ) : (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b := by
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rw [two_eq_succ_one]
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rw [one_eq_succ_zero]
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repeat rw [pow_succ]
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repeat rw [pow_zero]
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--ring
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repeat rw [one_mul]
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rw [add_mul, mul_add, mul_add, mul_comm b a]
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rw [succ_mul, succ_mul, zero_mul, zero_add, add_mul]
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repeat rw [add_assoc]
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rw [add_comm _ (b * b), ← add_assoc _ (b*b), add_comm _ (b*b), add_assoc]
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rfl
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NewLemma MyNat.two_eq_succ_one
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LemmaTab "Pow"
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Conclusion
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"
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