more realistic claims about prime number world

.i18n/en/Game.pot

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

Game.lean

@@ -41,10 +41,10 @@ those who read the help texts like this one.
 To start, click on \"Tutorial World\".
 
 Note: this is a new Lean 4 version of the game containing several
-worlds which were not present in the old Lean 3 version. A new version
-of Advanced Multiplication World is in preparation, and worlds
-such as Prime Number World and more will be appearing during October and
-November 2023.
+worlds which were not present in the old Lean 3 version. More new worlds
+such as Strong Induction World, Even/Odd World and Prime Number World
+are in development; if you want to see their state or even help out, checkout
+out the [issues in the github repo](https://github.com/leanprover-community/NNG4/issues).
 
 ## More
 
@@ -53,9 +53,9 @@ links, and ways to interact with the Lean community.
 "
 
 Info "
-*Game version: 4.2*
+*Game version: 4.3*
 
-*Recent additions: Inequality world, algorithm world*
+*Recent additions: bug fixes*
 
 ## Progress saving
 

Game/Levels/AdvMultiplication/L05le_mul_right.lean

@@ -18,7 +18,8 @@ TheoremDoc MyNat.le_mul_right as "le_mul_right" in "≤"
 
 Introduction
 "
-In Prime Number World we will be proving that $2$ is prime.
+One day this game will have a Prime Number World, with a final boss
+of proving that $2$ is prime.
 To do this, we will have to rule out things like $2 = 37 × 42.$
 We will do this by proving that any factor of $2$ is at most $2$,
 which we will do using this lemma. The proof I have in mind manipulates the hypothesis