more realistic claims about prime number world
This commit is contained in:
Kevin Buzzard
@@ -1,7 +1,7 @@
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msgid ""
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msgstr "Project-Id-Version: Game v4.7.0\n"
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"Report-Msgid-Bugs-To: \n"
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"POT-Creation-Date: Mon Apr 29 13:18:35 2024\n"
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"POT-Creation-Date: Sat Feb 1 23:22:17 2025\n"
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"Last-Translator: \n"
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"Language-Team: none\n"
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"Language: en\n"
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@@ -166,7 +166,7 @@ msgid "## Summary\n"
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"\n"
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"1) Basic usage: if `h : A = B` is an assumption or\n"
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"the proof of a theorem, and if the goal contains one or more `A`s, then `rw [h]`\n"
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"will change them all to `B`'s. The tactic will error\n"
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"will change them all to `B`s. The tactic will error\n"
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"if there are no `A`s in the goal.\n"
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"\n"
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"2) Advanced usage: Assumptions coming from theorem proofs\n"
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@@ -1343,7 +1343,7 @@ msgstr ""
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msgid "The music gets ever more dramatic, as we explore\n"
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"the interplay between exponentiation and multiplication.\n"
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"\n"
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"If you're having trouble exchanging the right `x * y`\n"
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"If you're having trouble exchanging the right `a * b`\n"
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"because `rw [mul_comm]` swaps the wrong multiplication,\n"
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"then read the documentation of `rw` for tips on how to fix this."
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msgstr ""
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@@ -1378,7 +1378,7 @@ msgstr ""
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msgid "The music dies down. Is that it?\n"
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"\n"
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"Course it isn't, you can\n"
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"clearly see that there are two worlds left.\n"
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"clearly see that there are two levels left.\n"
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"\n"
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"A passing mathematician says that mathematicians don't have a name\n"
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"for the structure you just constructed. You feel cheated.\n"
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@@ -1855,7 +1855,7 @@ msgid "`a ≠ b` is *notation* for `(a = b) → False`.\n"
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"is the logical opposite of `P`. Indeed `True → False` is false,\n"
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"and `False → False` is true!\n"
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"\n"
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"The upshot of this is that use can treat `a ≠ b` in exactly\n"
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"The upshot of this is that you can treat `a ≠ b` in exactly\n"
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"the same way as you treat any implication `P → Q`. For example,\n"
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"if your *goal* is of the form `a ≠ b` then you can make progress\n"
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"with `intro h`, and if you have a hypothesis `h` of the\n"
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@@ -2301,7 +2301,7 @@ msgstr ""
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#: Game.Levels.Algorithm.L07succ_ne_succ
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msgid "Start with `contrapose! h`, to change the goal into its\n"
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"contrapositive, namely a hypothesis of `succ m = succ m` and a goal of `m = n`."
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"contrapositive, namely a hypothesis of `succ m = succ n` and a goal of `m = n`."
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msgstr ""
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#: Game.Levels.Algorithm.L07succ_ne_succ
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@@ -2518,20 +2518,18 @@ msgid "Here's a proof using `add_left_eq_self`:\n"
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"exact add_left_eq_self y x\n"
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"```\n"
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"\n"
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"Alternatively you can just prove it by induction on `x`\n"
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"(the dots in the proof just indicate the two goals and\n"
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"can be omitted):\n"
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"Alternatively you can just prove it by induction on `x`:\n"
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"\n"
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"```\n"
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" induction x with d hd\n"
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" · intro h\n"
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" rw [zero_add] at h\n"
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" assumption\n"
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" · intro h\n"
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" rw [succ_add] at h\n"
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" apply succ_inj at h\n"
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" apply hd at h\n"
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" assumption\n"
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"induction x with d hd\n"
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"intro h\n"
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"rw [zero_add] at h\n"
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"exact h\n"
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"intro h\n"
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"rw [succ_add] at h\n"
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"apply succ_inj at h\n"
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"apply hd at h\n"
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"exact h\n"
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"```"
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msgstr ""
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@@ -2666,7 +2664,13 @@ msgid "## Summary\n"
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"\n"
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"Because `a ≤ b` is notation for \\\"there exists `c` such that `b = a + c`\\\",\n"
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"you can make progress on goals of the form `a ≤ b` by `use`ing the\n"
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"number which is morally `b - a`."
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"number which is morally `b - a` (i.e. `use b - a`)\n"
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"\n"
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"Any of the following examples is possible assuming the type of the argument passed to the `use` function is accurate:\n"
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"\n"
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"- `use 37`\n"
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"- `use a`\n"
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"- `use a * a + 1`"
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msgstr ""
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#: Game.Levels.LessOrEqual.L01le_refl
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@@ -3180,7 +3184,7 @@ msgstr ""
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#: Game.Levels.AdvMultiplication.L03eq_succ_of_ne_zero
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msgid "Multiplication usually makes a number bigger, but multiplication by zero can make\n"
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"it smaller. Thus many lemmas about inequalities and multiplication need the\n"
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"hypothesis `a ≠ 0`. Here is a key lemma enables us to use this hypothesis.\n"
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"hypothesis `a ≠ 0`. Here is a key lemma that enables us to use this hypothesis.\n"
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"To help us with the proof, we can use the `tauto` tactic. Click on the tactic's name\n"
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"on the right to see what it does."
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msgstr ""
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@@ -3227,7 +3231,7 @@ msgstr ""
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#: Game.Levels.AdvMultiplication.L05le_mul_right
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msgid "In Prime Number World we will be proving that $2$ is prime.\n"
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"To do this, we will have to rule out things like $2 ≠ 37 × 42.$\n"
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"To do this, we will have to rule out things like $2 = 37 × 42.$\n"
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"We will do this by proving that any factor of $2$ is at most $2$,\n"
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"which we will do using this lemma. The proof I have in mind manipulates the hypothesis\n"
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"until it becomes the goal, using pretty much everything which we've proved in this world so far."
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@@ -3278,7 +3282,7 @@ msgid "# Summary\n"
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"## Example\n"
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"\n"
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"If you have a proof to hand, then you don't even need to state what you\n"
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"are proving. example\n"
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"are proving. For example\n"
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"\n"
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"`have h2 := succ_inj a b`\n"
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"\n"
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@@ -3310,7 +3314,7 @@ msgid "Now you can `apply le_mul_right at h2`."
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msgstr ""
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#: Game.Levels.AdvMultiplication.L06mul_right_eq_one
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msgid "Now `rw [h] at h2` so you can `apply le_one at hx`."
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msgid "Now `rw [«{h}»] at «{h2}»` so you can `apply le_one at «{h2}»`."
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msgstr ""
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#: Game.Levels.AdvMultiplication.L06mul_right_eq_one
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@@ -3381,8 +3385,8 @@ msgid "In this level we prove that if `a * b = a * c` and `a ≠ 0` then `b = c`
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"so the induction hypothesis does not apply!\n"
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"\n"
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"Assume `a ≠ 0` is fixed. The actual statement we want to prove by induction on `b` is\n"
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"\"for all `c`, if `a * b = a * c` then `b = c`. This *can* be proved by induction,\n"
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"because we now have the flexibility to change `c`.\""
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"\"for all `c`, if `a * b = a * c` then `b = c`\". This *can* be proved by induction,\n"
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"because we now have the flexibility to change `c`."
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msgstr ""
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#: Game.Levels.AdvMultiplication.L09mul_left_cancel
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@@ -3479,10 +3483,10 @@ msgid "# Welcome to the Natural Number Game\n"
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"To start, click on \"Tutorial World\".\n"
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"\n"
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"Note: this is a new Lean 4 version of the game containing several\n"
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"worlds which were not present in the old Lean 3 version. A new version\n"
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"of Advanced Multiplication World is in preparation, and worlds\n"
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"such as Prime Number World and more will be appearing during October and\n"
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"November 2023.\n"
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"worlds which were not present in the old Lean 3 version. More new worlds\n"
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"such as Strong Induction World, Even/Odd World and Prime Number World\n"
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"are in development; if you want to see their state or even help out, checkout\n"
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"out the [issues in the github repo](https://github.com/leanprover-community/NNG4/issues).\n"
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"\n"
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"## More\n"
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"\n"
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@@ -3491,9 +3495,9 @@ msgid "# Welcome to the Natural Number Game\n"
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msgstr ""
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#: Game
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msgid "*Game version: 4.2*\n"
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msgid "*Game version: 4.3*\n"
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"\n"
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"*Recent additions: Inequality world, algorithm world*\n"
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"*Recent additions: bug fixes*\n"
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"\n"
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"## Progress saving\n"
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"\n"
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12
Game.lean
12
Game.lean
@@ -41,10 +41,10 @@ those who read the help texts like this one.
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To start, click on \"Tutorial World\".
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Note: this is a new Lean 4 version of the game containing several
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worlds which were not present in the old Lean 3 version. A new version
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of Advanced Multiplication World is in preparation, and worlds
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such as Prime Number World and more will be appearing during October and
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November 2023.
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worlds which were not present in the old Lean 3 version. More new worlds
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such as Strong Induction World, Even/Odd World and Prime Number World
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are in development; if you want to see their state or even help out, checkout
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out the [issues in the github repo](https://github.com/leanprover-community/NNG4/issues).
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## More
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@@ -53,9 +53,9 @@ links, and ways to interact with the Lean community.
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"
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Info "
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*Game version: 4.2*
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*Game version: 4.3*
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*Recent additions: Inequality world, algorithm world*
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*Recent additions: bug fixes*
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## Progress saving
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@@ -18,7 +18,8 @@ TheoremDoc MyNat.le_mul_right as "le_mul_right" in "≤"
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Introduction
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"
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In Prime Number World we will be proving that $2$ is prime.
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One day this game will have a Prime Number World, with a final boss
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of proving that $2$ is prime.
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To do this, we will have to rule out things like $2 = 37 × 42.$
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We will do this by proving that any factor of $2$ is at most $2$,
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which we will do using this lemma. The proof I have in mind manipulates the hypothesis
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