nng levels
This commit is contained in:
Jon Eugster
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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--import Mathlib.Tactic.Ring
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Game "NNG"
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World "Inequality"
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Level 1
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Title ""
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Title "the `use` tactic"
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open MyNat
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Introduction
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"
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The goal below is to prove $x\\le 1+x$ for any natural number $x$.
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First let's turn the goal explicitly into an existence problem with
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`rw le_iff_exists_add,`
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and now the goal has become `∃ c : mynat, 1 + x = x + c`. Clearly
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this statement is true, and the proof is that $c=1$ will work (we also
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need the fact that addition is commutative, but we proved that a long
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time ago). How do we make progress with this goal?
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The `use` tactic can be used on goals of the form `∃ c, ...`. The idea
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is that we choose which natural number we want to use, and then we use it.
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So try
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`use 1,`
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and now the goal becomes `⊢ 1 + x = x + 1`. You can solve this by
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`exact add_comm 1 x`, or if you are lazy you can just use the `ring` tactic,
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which is a powerful AI which will solve any equality in algebra which can
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be proved using the standard rules of addition and multiplication. Now
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look at your proof. We're going to remove a line.
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## Important
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An important time-saver here is to note that because `a ≤ b` is *defined*
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as `∃ c : mynat, b = a + c`, you *do not need to write* `rw le_iff_exists_add`.
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The `use` tactic will work directly on a goal of the form `a ≤ b`. Just
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use the difference `b - a` (note that we have not defined subtraction so
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this does not formally make sense, but you can do the calculation in your head).
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If you have written `rw le_iff_exists_add` below, then just put two minus signs `--`
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before it and comment it out. See that the proof still compiles.
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"
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Statement
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""
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: true := by
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trivial
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axiom add_comm (a b : ℕ) : a + b = b + a
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Statement --one_add_le_self
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"If $x$ is a natural number, then $x\\le 1+x$.
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"
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(x : ℕ) : x ≤ 1 + x := by
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rw [le_iff_exists_add]
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use 1
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rw [add_comm]
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rfl
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Conclusion
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"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,22 +1,26 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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Level 2
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Title ""
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Title "le_rfl"
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open MyNat
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Introduction
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"
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Here's a nice easy one.
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"
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Statement
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""
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: true := by
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trivial
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"The $\\le$ relation is reflexive. In other words, if $x$ is a natural number,
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then $x\\le x$."
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(x : ℕ) : x ≤ x := by
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use 0
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rw [add_zero]
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rfl
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Conclusion
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"
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@@ -1,22 +1,57 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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import Std.Tactic.RCases
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Game "NNG"
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World "Inequality"
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Level 3
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Title ""
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Title "le_succ_of_le"
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open MyNat
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Introduction
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"
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We have seen how the `use` tactic makes progress on goals of the form `⊢ ∃ c, ...`.
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But what do we do when we have a *hypothesis* of the form `h : ∃ c, ...`?
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The hypothesis claims that there exists some natural number `c` with some
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property. How are we going to get to that natural number `c`? It turns out
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that the `cases` tactic can be used (just like it was used to extract
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information from `∧` and `∨` and `↔` hypotheses). Let me talk you through
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the proof of $a\\le b\\implies a\\le\\operatorname{succ}(b)$.
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The goal is an implication so we clearly want to start with
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`intro h,`
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. After this, if you *want*, you can do something like
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`rw le_iff_exists_add at h ⊢,`
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(get the sideways T with `\\|-` then space). This changes the `≤` into
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its `∃` form in `h` and the goal -- but if you are happy with just
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*imagining* the `∃` whenever you read a `≤` then you don't need to do this line.
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Our hypothesis `h` is now `∃ (c : mynat), b = a + c` (or `a ≤ b` if you
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elected not to do the definitional rewriting) so
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`cases h with c hc,`
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gives you the natural number `c` and the hypothesis `hc : b = a + c`.
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Now use `use` wisely and you're home.
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"
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Statement
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""
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: true := by
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trivial
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"For all naturals $a$, $b$, if $a\\leq b$ then $a\\leq \\operatorname{succ}(b)$.
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"
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(a b : ℕ) : a ≤ b → a ≤ (succ b) := by
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intro h
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rcases h with ⟨c, hc⟩
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rw [hc]
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use c + 1
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sorry
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--rfl
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Conclusion
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"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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@@ -1,5 +1,6 @@
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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.MyNat.LE
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import Mathlib.Tactic.Use
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Game "NNG"
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World "Inequality"
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