add NNG

server/nng/NNG.lean

@@ -1,9 +1,35 @@
 import GameServer.Commands
 
+import NNG.Levels.Tutorial
+import NNG.Levels.Addition
+import NNG.Levels.Multiplication
+import NNG.Levels.Power
+import NNG.Levels.Function
+import NNG.Levels.Proposition
+import NNG.Levels.AdvProposition
+import NNG.Levels.AdvAddition
+import NNG.Levels.AdvMultiplication
+import NNG.Levels.Inequality
+
 Game "NNG"
-World "HelloWorld"
+Title "Natural Number Game"
-Level 1
+Introduction
+"
+[intro text missing]
+
+## Credits
+* Content and Lean3-version: Kevin Buzzard, Mohammad Pedramfar
+* Game Engine: Alexander Bentkamp, Jon Eugster, Patrick Massot
+* Port to Lean 4: Chris Lovett
+
+## Resources
+* [Original Lean3 version](https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/)
+* [Chris' translation to lean4](https://lovettsoftware.com/NaturalNumbers/TutorialWorld/Level1.lean.html)
+"
 
-Statement : 1 + 1 = 2 := rfl
+Path Tutorial → Addition → Function → Proposition → AdvProposition → AdvAddition
+Path AdvAddition → AdvMultiplication → Inequality
+Path Addition → Multiplication → AdvMultiplication
+Path Multiplication → Power
 
 MakeGame

server/nng/NNG/Doc/Definitions.lean

@@ -0,0 +1,6 @@
+import GameServer.Commands
+
+DefinitionDoc MyNat as "ℕ"
+"
+The Natural Numbers.
+"

server/nng/NNG/Doc/Lemmas.lean

@@ -0,0 +1,31 @@
+import GameServer.Commands
+
+LemmaDoc MyNat.add_zero as "add_zero" in "Nat"
+""
+
+LemmaDoc MyNat.add_succ as "add_succ" in "Nat"
+""
+
+LemmaDoc MyNat.zero_add as "zero_add" in "Nat"
+""
+
+LemmaDoc MyNat.add_assoc as "add_assoc" in "Nat"
+""
+
+LemmaDoc MyNat.succ_add as "succ_add" in "Nat"
+""
+
+LemmaDoc MyNat.add_comm as "add_comm" in "Nat"
+""
+
+LemmaDoc MyNat.one_eq_succ_zero as "one_eq_succ_zero" in "Nat"
+""
+
+LemmaDoc not_iff_imp_false as "not_iff_imp_false" in "Prop"
+""
+
+LemmaDoc MyNat.succ_inj as "succ_inj" in "Nat"
+""
+
+LemmaDoc MyNat.zero_ne_succ as "zero_ne_succ" in "Nat"
+""

server/nng/NNG/Doc/Tactics.lean

@@ -0,0 +1,57 @@
+import GameServer.Commands
+
+TacticDoc rfl
+"
+"
+
+TacticDoc rewrite
+"
+"
+
+TacticDoc rw
+"
+"
+
+TacticDoc induction
+"
+"
+
+TacticDoc exact
+"
+"
+
+TacticDoc apply
+"
+"
+
+TacticDoc intro
+"
+"
+
+TacticDoc «have»
+"
+"
+
+TacticDoc constructor
+"
+"
+
+TacticDoc rcases
+"
+"
+
+TacticDoc left
+"
+"
+
+TacticDoc right
+"
+"
+
+TacticDoc contradiction
+"
+"
+
+TacticDoc exfalso
+"
+"

server/nng/NNG/Levels/Addition.lean

@@ -0,0 +1,42 @@
+import NNG.Levels.Addition.Level_1
+import NNG.Levels.Addition.Level_2
+import NNG.Levels.Addition.Level_3
+import NNG.Levels.Addition.Level_4
+import NNG.Levels.Addition.Level_5
+import NNG.Levels.Addition.Level_6
+
+
+Game "NNG"
+World "Addition"
+Title "Addition World"
+
+Introduction
+"
+Welcome to Addition World. If you've done all four levels in tutorial world
+and know about `rewrite` and `rfl`, then you're in the right place. Here's
+a reminder of the things you're now equipped with which we'll need in this world.
+
+## Data:
+
+  * a type called `ℕ` or `MyNat`.
+  * a term `0 : ℕ`, interpreted as the number zero.
+  * a function `succ : ℕ → ℕ`, with `succ n` interpreted as \"the number after `n`\".
+  * Usual numerical notation `0,1,2` etc. (although `2` onwards will be of no use to us until much later ;-) ).
+  * Addition (with notation `a + b`).
+
+## Theorems:
+
+  * `add_zero (a : ℕ) : a + 0 = a`. Use with `rewrite [add_zero]`.
+  * `add_succ (a b : ℕ) : a + succ(b) = succ(a + b)`. Use with `rewrite [add_succ]`.
+  * The principle of mathematical induction. Use with `induction` (which we learn about in this chapter).
+
+
+## Tactics:
+
+  * `rfl` :  proves goals of the form `X = X`.
+  * `rewrite [h]` : if `h` is a proof of `A = B`, changes all `A`'s in the goal to `B`'s.
+  * `induction n with d hd` : we're going to learn this right now.
+
+
+You will also find all this information in your Inventory to read the documentation.
+"

server/nng/NNG/Levels/Addition/Level_1.lean

@@ -0,0 +1,93 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Addition"
+Level 1
+Title "the induction tactic."
+
+open MyNat
+
+set_option tactic.hygienic false
+
+Introduction
+"
+OK so let's see induction in action. We're going to prove
+
+```
+zero_add (n : ℕ) : 0 + n = n
+```
+
+Wait… what is going on here? Didn't we already prove that adding zero to $n$ gave us $n$?
+
+No we didn't! We proved $n + 0 = n$, and that proof was called `add_zero`. We're now
+trying to establish `zero_add`, the proof that $0 + n = n$.
+
+But aren't these two theorems the same?
+
+No they're not! It is *true* that `x + y = y + x`, but we haven't *proved* it yet,
+and in fact we will need both `add_zero` and `zero_add` in order
+to prove this. In fact `x + y = y + x` is the boss level for addition world,
+and `induction` is the only other tactic you'll need to beat it.
+
+Now `add_zero` is one of Peano's axioms, so we don't need to prove it, we already have it.
+To prove `0 + n = n` we need to use induction on $n$. While we're here,
+note that `zero_add` is about zero add something, and `add_zero` is about something add zero.
+The names of the proofs tell you what the theorems are. Anyway, let's prove `0 + n = n`.
+"
+
+Statement zero_add
+"For all natural numbers $n$, we have $0 + n = n$."
+    (n : ℕ) : 0 + n = n := by
+  Hint "You can start a proof by induction over `n` by typing:
+  `induction n with d hd`.
+
+  If you use the `with` part, you can name your variable and induction hypothesis, otherwise
+  they get default names."
+  induction n with n hn
+  · Hint "Now you have two goals. Once you proved the first, you will jump to the second one.
+    This first goal is the base case $n = 0$.
+
+    Recall that you can use all lemmas that are visible in your inventory."
+    Hint (hidden := true) "try using `add_zero`."
+    rw [add_zero]
+    rfl
+  · Hint "Now you jumped to the second goal. Here you have the induction hypothesis
+    `{hn} : 0 + {n} = {n}` and you need to prove the statement for `succ {n}`."
+    Hint (hidden := true) "look at `add_succ`."
+    rw [add_succ]
+    Hint (hidden := true) "At this point you see the term `0 + {n}`, so you can use the
+    induction hypothesis with `rw [{hn}]`."
+    rw [hn]
+    rfl
+
+NewTactic induction
+
+Conclusion
+"
+## Now venture off on your own.
+
+Those three tactics: 
+
+* `induction n with d hd` 
+* `rw [h]`
+* `rfl`
+
+will get you quite a long way through this game. Using only these tactics
+you can beat Addition World level 4 (the boss level of Addition World),
+all of Multiplication World including the boss level `a * b = b * a`,
+and even all of Power World including the fiendish final boss. This route will
+give you a good grounding in these three basic tactics; after that, if you
+are still interested, there are other worlds to master, where you can learn
+more tactics.
+
+But we're getting ahead of ourselves, you still have to beat the rest of Addition World. 
+We're going to stop explaining stuff carefully now. If you get stuck or want
+to know more about Lean (e.g. how to do much harder maths in Lean),
+ask in `#new members` at
+[the Lean chat](https://leanprover.zulipchat.com)
+(login required, real name preferred). Kevin or Mohammad or one of the other
+people there might be able to help.
+
+Good luck! Click on \"Next\" to solve some levels on your own.
+"

server/nng/NNG/Levels/Addition/Level_2.lean

@@ -0,0 +1,70 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Addition"
+Level 2
+Title "add_assoc (associativity of addition)"
+
+open MyNat
+
+theorem MyNat.zero_add (n : ℕ) : 0 + n = n := by
+  induction n with n hn
+  · rw [add_zero]
+    rfl
+  · rw [add_succ]
+    rw [hn]
+    rfl
+
+Introduction
+"
+It's well-known that $(1 + 2) + 3 = 1 + (2 + 3)$; if we have three numbers
+to add up, it doesn't matter which of the additions we do first. This fact
+is called *associativity of addition* by mathematicians, and it is *not*
+obvious. For example, subtraction really is not associative: $(6 - 2) - 1$
+is really not equal to $6 - (2 - 1)$. We are going to have to prove
+that addition, as defined the way we've defined it, is associative.
+
+See if you can prove associativity of addition.
+"
+
+Statement add_assoc
+"On the set of natural numbers, addition is associative.
+In other words, for all natural numbers $a, b$ and $c$, we have
+$ (a + b) + c = a + (b + c). $"
+    (a b c : ℕ) : (a + b) + c = a + (b + c)  := by
+  Hint "Because addition was defined by recursion on the right-most variable,
+  use induction on the right-most variable (try other variables at your peril!).
+
+  Note that when Lean writes `a + b + c`, it means `(a + b) + c`. If it wants to talk
+  about `a + (b + c)` it will put the brackets in explictly."
+  Branch
+    induction a
+    Hint "Good luck with that…"
+    rw [zero_add, zero_add]
+    rfl
+  Branch
+    induction b
+    Hint "Good luck with that…"
+  induction c with c hc
+  Hint (hidden := true) "look at the lemma `add_zero`."
+  rw [add_zero]
+  Hint "`rw [add_zero]` only rewrites one term of the form `… + 0`, so you might to
+  use it multiple times."
+  rw [add_zero]
+  rfl
+  Hint (hidden := true) "`add_succ` might help here."
+  rw [add_succ]
+  rw [add_succ]
+  rw [add_succ]
+  Hint (hidden := true) "Now you can use the induction hypothesis."
+  rw [hc]
+  rfl
+
+
+Conclusion
+"
+
+"
+
+NewLemma MyNat.zero_add

server/nng/NNG/Levels/Addition/Level_3.lean

@@ -0,0 +1,66 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import NNG.Levels.Addition.Level_2
+
+Game "NNG"
+World "Addition"
+Level 3
+Title "succ_add"
+
+open MyNat
+
+theorem MyNat.add_assoc (a b c : ℕ) : (a + b) + c = a + (b + c)  := by
+  induction c with c hc
+  · rw [add_zero]
+    rw [add_zero]
+    rfl
+  · rw [add_succ]
+    rw [add_succ]
+    rw [add_succ]
+    rw [hc]
+    rfl
+
+
+Introduction
+"
+Oh no! On the way to `add_comm`, a wild `succ_add` appears. `succ_add`
+is the proof that `succ(a) + b = succ(a + b)` for `a` and `b` in your
+natural number type. We need to prove this now, because we will need
+to use this result in our proof that `a + b = b + a` in the next level.
+
+NB: think about why computer scientists called this result `succ_add` .
+There is a logic to all the names.
+
+Note that if you want to be more precise about exactly where you want
+to rewrite something like `add_succ` (the proof you already have),
+you can do things like `rw [add_succ (succ a)]` or
+`rw [add_succ (succ a) d]`, telling Lean explicitly what to use for
+the input variables for the function `add_succ`. Indeed, `add_succ`
+is a function: it takes as input two variables `a` and `b` and outputs a proof
+that `a + succ(b) = succ(a + b)`. The tactic `rw [add_succ]` just says to Lean \"guess
+what the variables are\".
+"
+
+Statement succ_add
+"For all natural numbers $a, b$, we have
+$ \\operatorname{succ}(a) + b = \\operatorname{succ}(a + b)$."
+    (a b : ℕ) : succ a + b = succ (a + b)  := by
+  Hint (hidden := true) "You might again want to start by induction
+  on the right-most variable."
+  Branch
+    induction a
+    Hint "Induction on `a` will not work."
+  induction b with d hd
+  · rw [add_zero]
+    rfl
+  · rw [add_succ]
+    rw [hd]
+    rw [add_succ]
+    rfl
+
+NewLemma MyNat.add_assoc
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Addition/Level_4.lean

@@ -0,0 +1,60 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import NNG.Levels.Addition.Level_3
+
+Game "NNG"
+World "Addition"
+Level 4
+Title "`add_comm` (boss level)"
+
+open MyNat
+
+theorem MyNat.succ_add (a b : ℕ) : succ a + b = succ (a + b)  := by
+  induction b with d hd
+  · rw [add_zero]
+    rfl
+  · rw [add_succ]
+    rw [hd]
+    rw [add_succ]
+    rfl
+
+Introduction
+"
+[boss battle music]
+
+Look in your inventory to see the proofs you have available.
+These should be enough.
+"
+
+Statement add_comm
+"On the set of natural numbers, addition is commutative.
+In other words, for all natural numbers $a$ and $b$, we have
+$a + b = b + a$."
+    (a b : ℕ) : a + b = b + a := by
+  Branch
+    induction a with d hd
+    · rw [zero_add]
+      rw [add_zero]
+      rfl
+    · rw [succ_add]
+      rw [hd]
+      rw [add_succ]
+      rfl
+  induction b with d hd
+  · rw [zero_add]
+    rw [add_zero]
+    rfl
+  · rw [add_succ]
+    rw [hd]
+    rw [succ_add]
+    rfl
+
+NewLemma MyNat.succ_add
+
+Conclusion
+"
+If you got this far -- nice! You're nearly ready to make a choice:
+Multiplication World or Function World. But there are just a couple
+more useful lemmas in Addition World which you should prove first.
+Press on to level 5.
+"

server/nng/NNG/Levels/Addition/Level_5.lean

@@ -0,0 +1,54 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import NNG.Levels.Addition.Level_4
+
+Game "NNG"
+World "Addition"
+Level 5
+Title "succ_eq_add_one"
+
+open MyNat
+
+theorem MyNat.add_comm (a b : ℕ) : a + b = b + a := by
+  induction b with d hd
+  · rw [zero_add]
+    rw [add_zero]
+    rfl
+  · rw [add_succ]
+    rw [hd]
+    rw [succ_add]
+    rfl
+
+theorem MyNat.one_eq_succ_zero : (1 : ℕ) = succ 0 := by
+  rfl
+
+NewLemma MyNat.add_comm MyNat.one_eq_succ_zero
+
+Introduction
+"
+I've just added `one_eq_succ_zero` (a proof of $1 = \\operatorname{succ}(0)$)
+to your list of theorems; this is true
+by definition of $1$, but we didn't need it until now.
+
+Levels 5 and 6 are the two last levels in Addition World.
+Level 5 involves the number $1$. When you see a $1$ in your goal,
+you can write `rw [one_eq_succ_zero]` to get back
+to something which only mentions `0`. This is a good move because $0$ is easier for us to
+manipulate than $1$ right now, because we have
+some theorems about $0$ (`zero_add`, `add_zero`), but, other than `1 = succ 0`,
+no theorems at all which mention $1$. Let's prove one now.
+"
+
+Statement succ_eq_add_one
+"For any natural number $n$, we have
+$ \\operatorname{succ}(n) = n+1$ ."
+    (n : ℕ) : succ n = n + 1 := by
+  rw [one_eq_succ_zero]
+  rw [add_succ]
+  rw [add_zero]
+  rfl
+
+Conclusion
+"
+Well done! On to the last level!
+"

server/nng/NNG/Levels/Addition/Level_6.lean

@@ -0,0 +1,47 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import NNG.Levels.Addition.Level_5
+
+Game "NNG"
+World "Addition"
+Level 6
+Title "add_right_comm"
+
+open MyNat
+
+Introduction
+"
+Lean sometimes writes `a + b + c`. What does it mean? The convention is
+that if there are no brackets displayed in an addition formula, the brackets
+are around the left most `+` (Lean's addition is \"left associative\"). 
+So the goal in this level is `(a + b) + c = (a + c) + b`. This isn't
+quite `add_assoc` or `add_comm`, it's something you'll have to prove
+by putting these two theorems together.
+
+If you hadn't picked up on this already, `rw [add_assoc]` will
+change `(x + y) + z` to `x + (y + z)`, but to change it back
+you will need `rw [← add_assoc]`. Get the left arrow by typing `\\l`
+then the space bar (note that this is L for left, not a number 1).
+Similarly, if `h : a = b` then `rw [h]` will change `a`'s to `b`'s
+and `rw [← h]` will change `b`'s to `a`'s.
+
+Also, you can be (and will need to be, in this level) more precise
+about where to rewrite theorems. `rw add_comm,` will just find the
+first `? + ?` it sees and swap it around. You can target more specific
+additions like this: `rw add_comm a` will swap around
+additions of the form `a + ?`, and `rw add_comm a b,` will only
+swap additions of the form `a + b`.
+"
+
+Statement add_right_comm
+"For all natural numbers $a, b$ and $c$, we have
+$a + b + c = a + c + b$."
+    (a b c : ℕ) : a + b + c = a + c + b := by
+  rw [add_assoc]
+  rw [add_comm b c]
+  rw [←add_assoc]
+  rfl
+
+Conclusion
+"
+"

server/nng/NNG/Levels/AdvAddition.lean

@@ -0,0 +1,22 @@
+import NNG.Levels.AdvAddition.Level_1
+import NNG.Levels.AdvAddition.Level_2
+import NNG.Levels.AdvAddition.Level_3
+import NNG.Levels.AdvAddition.Level_4
+import NNG.Levels.AdvAddition.Level_5
+import NNG.Levels.AdvAddition.Level_6
+import NNG.Levels.AdvAddition.Level_7
+import NNG.Levels.AdvAddition.Level_8
+import NNG.Levels.AdvAddition.Level_9
+import NNG.Levels.AdvAddition.Level_10
+import NNG.Levels.AdvAddition.Level_11
+import NNG.Levels.AdvAddition.Level_12
+import NNG.Levels.AdvAddition.Level_13
+
+
+Game "NNG"
+World "AdvAddition"
+Title "Advanced Addition World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/AdvAddition/Level_1.lean

@@ -0,0 +1,30 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+theorem MyNat.succ_inj {a b : ℕ} : succ a = succ b → a = b := by simp only [succ.injEq, imp_self]
+
+theorem MyNat.zero_ne_succ (a : ℕ) : zero ≠ succ a := by simp only [ne_eq, not_false_iff]
+
+Statement succ_inj'
+""
+    {a b : ℕ} (hs : succ a = succ b) :  a = b := by
+    exact succ_inj hs
+
+NewLemma MyNat.succ_inj MyNat.zero_ne_succ
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_10.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 10
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_11.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 11
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_12.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 12
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_13.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 13
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_2.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_3.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_4.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_5.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_6.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_7.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_8.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvAddition/Level_9.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvAddition"
+Level 9
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvMultiplication.lean

@@ -0,0 +1,13 @@
+import NNG.Levels.AdvMultiplication.Level_1
+import NNG.Levels.AdvMultiplication.Level_2
+import NNG.Levels.AdvMultiplication.Level_3
+import NNG.Levels.AdvMultiplication.Level_4
+
+
+Game "NNG"
+World "AdvMultiplication"
+Title "Advanced Multiplication World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/AdvMultiplication/Level_1.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvMultiplication"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvMultiplication/Level_2.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvMultiplication"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvMultiplication/Level_3.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvMultiplication"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvMultiplication/Level_4.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvMultiplication"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition.lean

@@ -0,0 +1,19 @@
+import NNG.Levels.AdvProposition.Level_1
+import NNG.Levels.AdvProposition.Level_2
+import NNG.Levels.AdvProposition.Level_3
+import NNG.Levels.AdvProposition.Level_4
+import NNG.Levels.AdvProposition.Level_5
+import NNG.Levels.AdvProposition.Level_6
+import NNG.Levels.AdvProposition.Level_7
+import NNG.Levels.AdvProposition.Level_8
+import NNG.Levels.AdvProposition.Level_9
+import NNG.Levels.AdvProposition.Level_10
+
+
+Game "NNG"
+World "AdvProposition"
+Title "Advanced Proposition World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/AdvProposition/Level_1.lean

@@ -0,0 +1,28 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "AdvProposition"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Prop) (p : P) (q : Q) : P ∧ Q := by
+  constructor
+  exact p
+  exact q
+
+NewTactic constructor
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_10.lean

@@ -0,0 +1,36 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+
+Game "NNG"
+World "AdvProposition"
+Level 10
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Prop) : (¬ Q → ¬ P) → (P → Q) := by
+  by_cases p : P
+  · by_cases q : Q
+    intro h p' -- cc
+    assumption
+    intro h p'
+    have g : ¬ P := h q
+    contradiction
+  · by_cases q : Q
+    intro h p
+    assumption
+    intro h p
+    contradiction
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_2.lean

@@ -0,0 +1,33 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+
+Game "NNG"
+World "AdvProposition"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+set_option tactic.hygienic false
+
+Statement and_symm
+""
+    (P Q : Prop) : P ∧ Q → Q ∧ P :=  by
+  intro h
+  rcases h with ⟨p, q⟩
+  constructor
+  exact q
+  exact p
+
+NewTactic rcases
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_3.lean

@@ -0,0 +1,31 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+
+Game "NNG"
+World "AdvProposition"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement and_trans
+""
+    (P Q R : Prop) : P ∧ Q → Q ∧ R → P ∧ R := by
+  intro hpq
+  intro hqr
+  rcases hpq with ⟨p, q⟩
+  rcases hqr with ⟨q', r⟩
+  constructor
+  assumption
+  assumption
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_4.lean

@@ -0,0 +1,31 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+
+Game "NNG"
+World "AdvProposition"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement iff_trans
+""
+    (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) := by
+  intro hpq
+  intro hqr
+  rcases hpq with ⟨hpq, hqp⟩
+  rcases hqr with ⟨hqr, hrq⟩
+  constructor
+  exact fun x => hqr (hpq x) -- cc
+  exact fun x => hqp (hrq x) -- cc
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_5.lean

@@ -0,0 +1,34 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+
+Game "NNG"
+World "AdvProposition"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement iff_trans
+""
+    (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) := by
+  intro hpq hqr
+  constructor
+  intro p
+  apply hqr.1
+  apply hpq.1
+  assumption
+  intro r
+  apply hpq.2
+  apply hqr.2
+  assumption
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_6.lean

@@ -0,0 +1,31 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+import Mathlib.Tactic.LeftRight
+--import Mathlib.Logic.Basic
+
+Game "NNG"
+World "AdvProposition"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Prop) : Q → (P ∨ Q) := by
+  intro q
+  right
+  assumption
+
+NewTactic left right
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_7.lean

@@ -0,0 +1,31 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+import Mathlib.Tactic.LeftRight
+
+Game "NNG"
+World "AdvProposition"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement or_symm
+""
+    (P Q : Prop) : P ∨ Q → Q ∨ P := by
+  intro h
+  rcases h with p | q
+  right
+  exact p
+  left
+  exact q
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_8.lean

@@ -0,0 +1,49 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+import Mathlib.Tactic.LeftRight
+
+Game "NNG"
+World "AdvProposition"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement and_or_distrib_left
+""
+    (P Q R : Prop) : P ∧ (Q ∨ R) ↔ (P ∧ Q) ∨ (P ∧ R) := by
+  constructor
+  intro h
+  rcases h with ⟨hp, hqr⟩
+  rcases hqr with q | r
+  left
+  constructor
+  assumption
+  assumption
+  right
+  constructor
+  assumption
+  assumption
+  intro h
+  rcases h with hpq | hpr
+  rcases hpq with ⟨p, q⟩
+  constructor
+  assumption
+  left
+  assumption
+  rcases hpr with ⟨hp, hr⟩
+  constructor
+  assumption
+  right
+  assumption
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/AdvProposition/Level_9.lean

@@ -0,0 +1,36 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import Std.Tactic.RCases
+import NNG.MyNat.Theorems.Proposition
+
+
+
+Game "NNG"
+World "AdvProposition"
+Level 9
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement contra
+""
+  (P Q : Prop) : (P ∧ ¬ P) → Q := by
+  intro h
+  rcases h with ⟨p, np ⟩
+  contradiction
+  -- rw [not_iff_imp_false] at np
+  -- exfalso
+  -- apply np
+  -- exact p
+
+NewTactic exfalso contradiction
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function.lean

@@ -0,0 +1,18 @@
+import NNG.Levels.Function.Level_1
+import NNG.Levels.Function.Level_2
+import NNG.Levels.Function.Level_3
+import NNG.Levels.Function.Level_4
+import NNG.Levels.Function.Level_5
+import NNG.Levels.Function.Level_6
+import NNG.Levels.Function.Level_7
+import NNG.Levels.Function.Level_8
+import NNG.Levels.Function.Level_9
+
+
+Game "NNG"
+World "Function"
+Title "Function World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/Function/Level_1.lean

@@ -0,0 +1,27 @@
+import NNG.Metadata
+import NNG.MyNat.Theorems.Addition
+import NNG.MyNat.Multiplication
+
+Game "NNG"
+World "Function"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+"If $P$ is true, and $P\\implies Q$ is also true, then $Q$ is true."
+    (P Q : Prop) (p : P) (h : P → Q) : Q := by
+  exact h p
+
+NewTactic exact
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_2.lean

@@ -0,0 +1,28 @@
+import NNG.Metadata
+import NNG.MyNat.Theorems.Addition
+import NNG.MyNat.Multiplication
+
+Game "NNG"
+World "Function"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : ℕ → ℕ := by
+  intro n
+  exact 3 * n + 2
+
+NewTactic intro
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_3.lean

@@ -0,0 +1,31 @@
+import NNG.Metadata
+import NNG.MyNat.Theorems.Addition
+import NNG.MyNat.Multiplication
+
+Game "NNG"
+World "Function"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R S T U: Type) (p : P) (h : P → Q) (i : Q → R) (j : Q → T) (k : S → T) (l : T → U) :
+    U := by
+  have q := h p
+  have t : T := j q
+  have u : U := l t
+  exact u
+
+NewTactic «have»
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_4.lean

@@ -0,0 +1,37 @@
+import NNG.Metadata
+import NNG.MyNat.Theorems.Addition
+import NNG.MyNat.Multiplication
+
+Game "NNG"
+World "Function"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R S T U: Type)
+(p : P)
+(h : P → Q)
+(i : Q → R)
+(j : Q → T)
+(k : S → T)
+(l : T → U) : U :=
+by
+  apply l
+  apply j
+  apply h
+  exact p
+
+NewTactic apply
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_5.lean

@@ -0,0 +1,27 @@
+import NNG.Metadata
+import NNG.MyNat.Theorems.Addition
+import NNG.MyNat.Multiplication
+
+Game "NNG"
+World "Function"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Type) : P → (Q → P) := by
+  intro p
+  intro q
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_6.lean

@@ -0,0 +1,30 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Function"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R : Type) : (P → (Q → R)) → ((P → Q) → (P → R)) := by
+  intro f
+  intro h
+  intro p
+  have j : Q → R := f p
+  apply j
+  apply h
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_7.lean

@@ -0,0 +1,29 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Function"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q F : Type) : (P → Q) → ((Q → F) → (P → F)) := by
+  intro f
+  intro h
+  intro p
+  apply h
+  apply f
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_8.lean

@@ -0,0 +1,27 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Function"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Type) : (P → Q) → ((Q → empty) → (P → empty)) := by
+  intros f h p
+  apply h
+  apply f
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Function/Level_9.lean

@@ -0,0 +1,36 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Function"
+Level 9
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (A B C D E F G H I J K L : Type)
+    (f1 : A → B) (f2 : B → E) (f3 : E → D) (f4 : D → A) (f5 : E → F)
+    (f6 : F → C) (f7 : B → C) (f8 : F → G) (f9 : G → J) (f10 : I → J)
+    (f11 : J → I) (f12 : I → H) (f13 : E → H) (f14 : H → K) (f15 : I → L) : A → L := by
+  intro a
+  apply f15
+  apply f11
+  apply f9
+  apply f8
+  apply f5
+  apply f2
+  apply f1
+  exact a
+
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality.lean

@@ -0,0 +1,25 @@
+import NNG.Levels.Inequality.Level_1
+import NNG.Levels.Inequality.Level_2
+import NNG.Levels.Inequality.Level_3
+import NNG.Levels.Inequality.Level_4
+import NNG.Levels.Inequality.Level_5
+import NNG.Levels.Inequality.Level_6
+import NNG.Levels.Inequality.Level_7
+import NNG.Levels.Inequality.Level_8
+import NNG.Levels.Inequality.Level_9
+import NNG.Levels.Inequality.Level_10
+import NNG.Levels.Inequality.Level_11
+import NNG.Levels.Inequality.Level_12
+import NNG.Levels.Inequality.Level_13
+import NNG.Levels.Inequality.Level_14
+import NNG.Levels.Inequality.Level_15
+import NNG.Levels.Inequality.Level_16
+import NNG.Levels.Inequality.Level_17
+
+Game "NNG"
+World "Inequality"
+Title "Inequality World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/Inequality/Level_1.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_10.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 10
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_11.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 11
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_12.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 12
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_13.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 13
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_14.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 14
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_15.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 15
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_16.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 16
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_17.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 17
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_2.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_3.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_4.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_5.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_6.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_7.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_8.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Inequality/Level_9.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Inequality"
+Level 9
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication.lean

@@ -0,0 +1,18 @@
+import NNG.Levels.Multiplication.Level_1
+import NNG.Levels.Multiplication.Level_2
+import NNG.Levels.Multiplication.Level_3
+import NNG.Levels.Multiplication.Level_4
+import NNG.Levels.Multiplication.Level_5
+import NNG.Levels.Multiplication.Level_6
+import NNG.Levels.Multiplication.Level_7
+import NNG.Levels.Multiplication.Level_8
+import NNG.Levels.Multiplication.Level_9
+
+
+Game "NNG"
+World "Multiplication"
+Title "Multiplication World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/Multiplication/Level_1.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_2.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_3.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_4.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_5.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_6.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_7.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_8.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Multiplication/Level_9.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Multiplication"
+Level 9
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power.lean

@@ -0,0 +1,16 @@
+import NNG.Levels.Power.Level_1
+import NNG.Levels.Power.Level_2
+import NNG.Levels.Power.Level_3
+import NNG.Levels.Power.Level_4
+import NNG.Levels.Power.Level_5
+import NNG.Levels.Power.Level_6
+import NNG.Levels.Power.Level_7
+import NNG.Levels.Power.Level_8
+
+Game "NNG"
+World "Power"
+Title "Power World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/Power/Level_1.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_2.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_3.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_4.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_5.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_6.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_7.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Power/Level_8.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Power"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : true := by
+  trivial
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition.lean

@@ -0,0 +1,18 @@
+import NNG.Levels.Proposition.Level_1
+import NNG.Levels.Proposition.Level_2
+import NNG.Levels.Proposition.Level_3
+import NNG.Levels.Proposition.Level_4
+import NNG.Levels.Proposition.Level_5
+import NNG.Levels.Proposition.Level_6
+import NNG.Levels.Proposition.Level_7
+import NNG.Levels.Proposition.Level_8
+-- import NNG.Levels.Proposition.Level_9 -- `cc` is not ported
+
+
+Game "NNG"
+World "Proposition"
+Title "Proposition World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/Proposition/Level_1.lean

@@ -0,0 +1,24 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 1
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Prop) (p : P) (h : P → Q) : Q := by
+exact h p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_2.lean

@@ -0,0 +1,25 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 2
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    : P → P := by
+  intro p
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_3.lean

@@ -0,0 +1,30 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 3
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R S T U: Prop) (p : P) (h : P → Q) (i : Q → R)
+    (j : Q → T) (k : S → T) (l : T → U) : U := by
+  have q := h p
+  have t := j q
+  have u := l t
+  exact u
+
+DisabledTactic apply
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_4.lean

@@ -0,0 +1,30 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 4
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R S T U: Prop) (p : P) (h : P → Q) (i : Q → R)
+    (j : Q → T) (k : S → T) (l : T → U) : U := by
+  apply l
+  apply j
+  apply h
+  exact p
+
+DisabledTactic «have» 
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_5.lean

@@ -0,0 +1,27 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 5
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Prop) : P → (Q → P) := by
+  intro p
+  intro q
+  exact p
+  rfl
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_6.lean

@@ -0,0 +1,30 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 6
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R : Prop) : (P → (Q → R)) → ((P → Q) → (P → R)) := by
+  intro f
+  intro h
+  intro p
+  have j : Q → R := f p
+  apply j
+  apply h
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_7.lean

@@ -0,0 +1,28 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+
+Game "NNG"
+World "Proposition"
+Level 7
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q R : Prop) : (P → Q) → ((Q → R) → (P → R)) := by
+  intro hpq hqr
+  intro p
+  apply hqr
+  apply hpq
+  exact p
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_8.lean

@@ -0,0 +1,35 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import NNG.MyNat.Theorems.Proposition
+
+
+Game "NNG"
+World "Proposition"
+Level 8
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (P Q : Prop) : (P → Q) → (¬ Q → ¬ P) := by
+  rw [not_iff_imp_false]
+  rw [not_iff_imp_false]
+  intro f
+  intro h
+  intro p
+  apply h
+  apply f
+  exact p
+
+NewLemma not_iff_imp_false
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Proposition/Level_9.lean

@@ -0,0 +1,29 @@
+import NNG.Metadata
+import NNG.MyNat.Addition
+import NNG.MyNat.Theorems.Proposition
+
+Game "NNG"
+World "Proposition"
+Level 9
+Title ""
+
+open MyNat
+
+Introduction
+"
+
+"
+
+Statement
+""
+    (A B C D E F G H I J K L : Prop)
+    (f1 : A → B) (f2 : B → E) (f3 : E → D) (f4 : D → A) (f5 : E → F)
+    (f6 : F → C) (f7 : B → C) (f8 : F → G) (f9 : G → J) (f10 : I → J)
+    (f11 : J → I) (f12 : I → H) (f13 : E → H) (f14 : H → K) (f15 : I → L) : A → L := by
+  -- cc -- TODO: `cc` is not ported yet.
+  sorry
+
+Conclusion
+"
+
+"

server/nng/NNG/Levels/Tutorial.lean

@@ -0,0 +1,12 @@
+import NNG.Levels.Tutorial.Level_1
+import NNG.Levels.Tutorial.Level_2
+import NNG.Levels.Tutorial.Level_3
+import NNG.Levels.Tutorial.Level_4
+
+Game "NNG"
+World "Tutorial"
+Title "Tutorial World"
+
+Introduction
+"
+"

server/nng/NNG/Levels/Tutorial/Level_1.lean

@@ -0,0 +1,42 @@
+import NNG.Metadata
+import NNG.MyNat.Multiplication
+
+Game "NNG"
+World "Tutorial"
+Level 1
+Title "The rfl tactic"
+
+Introduction
+"
+Each level in this game involves proving a mathematical statement. In this first level
+you have three natural numbers $x, y, z$ (listed under \"Objects\") and you want to prove
+$x \\cdot y + z = x \\cdot y + z$ (displayed under \"Goal\").
+
+You can modify the Goal using *Tactics* until you can close (i.e. prove) it. 
+
+The first tactic is called `rfl`, which stands for \"reflexivity\",
+a fancy way of saying that it will prove any goal of the form `A = A`. It doesn't matter how
+complicated `A` is, all that matters is that the left hand side is exactly equal to the right hand
+side (a computer scientist would say \"definitionally equal\"). I really mean \"press the same buttons
+on your computer in the same order\" equal. For example, `x * y + z = x * y + z` can be proved by `rfl`,
+but `x + y = y + x` cannot.
+"
+
+Statement
+"For all natural numbers $x, y$ and $z$, we have $xy + z = xy + z$."
+    (x y z : ℕ) : x * y + z = x * y + z := by
+  Hint "In order to use the tactic `rfl` you can enter it above and hit \"Execute\"."
+  rfl
+
+NewTactic rfl
+NewDefinition MyNat
+
+Conclusion
+"
+Congratulations! You completed your first verified proof!
+
+If you want to be reminded about the `rfl` tactic, your inventory on the right contains useful
+information about things you've learned.
+
+Now click on \"Next\" to continue the journey.
+"