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import NNG.Levels.Multiplication.Level_1
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import NNG.Levels.Multiplication.Level_2
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import NNG.Levels.Multiplication.Level_3
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import NNG.Levels.Multiplication.Level_4
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import NNG.Levels.Multiplication.Level_5
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import NNG.Levels.Multiplication.Level_6
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import NNG.Levels.Multiplication.Level_7
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import NNG.Levels.Multiplication.Level_8
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import NNG.Levels.Multiplication.Level_9
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Game "NNG"
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World "Multiplication"
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Title "Multiplication World"
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Introduction
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"
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In this world you start with the definition of multiplication on `ℕ`. It is
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defined by recursion, just like addition was. So you get two new axioms:
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* `mul_zero (a : ℕ) : a * 0 = 0`
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* `mul_succ (a b : ℕ) : a * succ b = a * b + a`
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In words, we define multiplication by \"induction on the second variable\",
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with `a * 0` defined to be `0` and, if we know `a * b`, then `a` times
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the number after `b` is defined to be `a * b + a`.
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You can keep all the theorems you proved about addition, but
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for multiplication, those two results above are what you've got right now.
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So what's going on in multiplication world? Like addition, we need to go
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for the proofs that multiplication
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is commutative and associative, but as well as that we will
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need to prove facts about the relationship between multiplication
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and addition, for example `a * (b + c) = a * b + a * c`, so now
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there is a lot more to do. Good luck!
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" |