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50 lines
1.7 KiB
Plaintext
50 lines
1.7 KiB
Plaintext
import NNG.Levels.Inequality.Level_1
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-- import NNG.Levels.Inequality.Level_2
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-- import NNG.Levels.Inequality.Level_3
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-- import NNG.Levels.Inequality.Level_4
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-- import NNG.Levels.Inequality.Level_5
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-- import NNG.Levels.Inequality.Level_6
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-- import NNG.Levels.Inequality.Level_7
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-- import NNG.Levels.Inequality.Level_8
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-- import NNG.Levels.Inequality.Level_9
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-- import NNG.Levels.Inequality.Level_10
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-- import NNG.Levels.Inequality.Level_11
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-- import NNG.Levels.Inequality.Level_12
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-- import NNG.Levels.Inequality.Level_13
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-- import NNG.Levels.Inequality.Level_14
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-- import NNG.Levels.Inequality.Level_15
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-- import NNG.Levels.Inequality.Level_16
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-- import NNG.Levels.Inequality.Level_17
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Game "NNG"
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World "Inequality"
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Title "Inequality World"
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Introduction
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"
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A new import, giving us a new definition. If `a` and `b` are naturals,
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`a ≤ b` is *defined* to mean
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`∃ (c : mynat), b = a + c`
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The upside-down E means \"there exists\". So in words, $a\\le b$
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if and only if there exists a natural $c$ such that $b=a+c$.
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If you really want to change an `a ≤ b` to `∃ c, b = a + c` then
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you can do so with `rw le_iff_exists_add`:
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```
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le_iff_exists_add (a b : mynat) :
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a ≤ b ↔ ∃ (c : mynat), b = a + c
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```
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But because `a ≤ b` is *defined as* `∃ (c : mynat), b = a + c`, you
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do not need to `rw le_iff_exists_add`, you can just pretend when you see `a ≤ b`
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that it says `∃ (c : mynat), b = a + c`. You will see a concrete
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example of this below.
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A new construction like `∃` means that we need to learn how to manipulate it.
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There are two situations. Firstly we need to know how to solve a goal
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of the form `⊢ ∃ c, ...`, and secondly we need to know how to use a hypothesis
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of the form `∃ c, ...`.
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" |