988d4dff - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2023-06-05 16:04:24 -0700
Committed by: Jeremy Avigad
Committed at: 2023-06-05 16:04:24 -0700
Object name: 988d4dff7c2a58a56883078db6b5f22037fa2160
Parent: 6c0e47b2

Update Mon Jun 5 16:04:24 PDT 2023

Changes

10 changed files (+30/-38)

html/C02_Basics.html
- @@ -303,7 +303,7 @@ an expression that begins with <code class="docutils literal notranslate">calc</code> expression can also be used inside a tactic proof, but Lean interprets it as the instruction to use the resulting proof term to solve the goal. - The <code class="docutils literal notranslate">calc</code> syntax is finicky: the dots and underscires and justification + The <code class="docutils literal notranslate">calc</code> syntax is finicky: the dots and underscores and justification have to be in the format indicated above. Lean uses indentation to determine things like where a block of tactics or a <code class="docutils literal notranslate">calc</code> block begins and ends;
- @@ -381,8 +381,8 @@ It can be imported explicitly with the command
 We will see there are similar tactics for other common kind of algebraic structures. There is a variation of <code class="docutils literal notranslate">rw</code> called <code class="docutils literal notranslate">nth_rewrite</code> that allows you to replace only particular instances of an expression in the goal. - Possible matches are enumerated starting with zero, - so in the following example, <code class="docutils literal notranslate">nth_rewrite 1 h</code> replaces the second + Possible matches are enumerated starting with 1, + so in the following example, <code class="docutils literal notranslate">nth_rewrite 2 h</code> replaces the second occurrence of <code class="docutils literal notranslate">a + b</code> with <code class="docutils literal notranslate">c</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by nth_rw 2 [h]
- @@ -626,7 +626,7 @@ addition of the additive inverse.
 The proof term <code class="docutils literal notranslate">rfl</code> is short for <code class="docutils literal notranslate">reflexivity</code>. Presenting it as a proof of <code class="docutils literal notranslate">a - b = a + -b</code> forces Lean to unfold the definition and recognize both sides as being the same. - The <code class="docutils literal notranslate">reflexivity</code> tactic, which can be abbreviated as <code class="docutils literal notranslate">refl</code>, + The <code class="docutils literal notranslate">reflexivity</code> tactic, which can be abbreviated as <code class="docutils literal notranslate">rfl</code>, does the same. This is an instance of what is known as a definitional equality in Lean’s underlying logic.
- @@ -910,10 +910,6 @@ which tries to find the relevant theorem in the library.</li>
 </div> To try out <code class="docutils literal notranslate">library_search</code> in this example, delete the <code class="docutils literal notranslate">exact</code> command and uncomment the previous line. - If you replace <code class="docutils literal notranslate">library_search</code> with <code class="docutils literal notranslate">suggest</code>, - you’ll see a long list of suggestions. - In this case, the suggestions are not helpful, but in other cases - it does better. Using these tricks, see if you can find what you need to do the next example:
html/C03_Logic.html
- @@ -213,7 +213,7 @@ of an if-and-only-if statement.
 Universal quantifiers are often hidden in definitions, and Lean will unfold definitions to expose them when necessary. For example, let’s define two predicates, - <code class="docutils literal notranslate">fn_ub f a</code> and <code class="docutils literal notranslate">fn_lb f a</code>, + <code class="docutils literal notranslate">FnUb f a</code> and <code class="docutils literal notranslate">FnLb f a</code>, where <code class="docutils literal notranslate">f</code> is a function from the real numbers to the real numbers and <code class="docutils literal notranslate">a</code> is a real number. The first says that <code class="docutils literal notranslate">a</code> is an upper bound on the
- @@ -237,8 +237,8 @@ function that maps <code class="docutils literal notranslate">
 apply hgb </pre></div> </div> - Applying <code class="docutils literal notranslate">intro</code> to the goal <code class="docutils literal notranslate">fn_ub (fun x ↦ f x + g x) (a + b)</code> - forces Lean to unfold the definition of <code class="docutils literal notranslate">fn_ub</code> + Applying <code class="docutils literal notranslate">intro</code> to the goal <code class="docutils literal notranslate">FnUb (fun x ↦ f x + g x) (a + b)</code> + forces Lean to unfold the definition of <code class="docutils literal notranslate">FnUb</code> and introduce <code class="docutils literal notranslate">x</code> for the universal quantifier. The goal is then <code class="docutils literal notranslate">(fun (x : ℝ) ↦ f x + g x) x ≤ a + b</code>. But applying <code class="docutils literal notranslate">(fun x ↦ f x + g x)</code> to <code class="docutils literal notranslate">x</code> should result in <code class="docutils literal notranslate">f x + g x</code>,
- @@ -255,7 +255,7 @@ This helps make the proof more readable,
 and gives you more control over how the goal is transformed. The rest of the proof is routine. The last two <code class="docutils literal notranslate">apply</code> commands force Lean to unfold the definitions - of <code class="docutils literal notranslate">fn_ub</code> in the hypotheses. + of <code class="docutils literal notranslate">FnUb</code> in the hypotheses. Try carrying out similar proofs of these: <div class="highlight-lean notranslate"><div class="highlight"><pre>example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := sorry
- @@ -268,7 +268,7 @@ Try carrying out similar proofs of these:
 sorry </pre></div> </div> - Even though we have defined <code class="docutils literal notranslate">fn_ub</code> and <code class="docutils literal notranslate">fn_lb</code> for functions + Even though we have defined <code class="docutils literal notranslate">FnUb</code> and <code class="docutils literal notranslate">FnLb</code> for functions from the reals to the reals, you should recognize that the definitions and proofs are much more general.
- @@ -279,7 +279,7 @@ of any structure that is an “ordered additive commutative monoid”;
 the details of what that means don’t matter now, but it is worth knowing that the natural numbers, integers, rationals, and real numbers are all instances. - So if we prove the theorem <code class="docutils literal notranslate">fn_ub_add</code> at that level of generality, + So if we prove the theorem <code class="docutils literal notranslate">fnUb_add</code> at that level of generality, it will apply in all these instances. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R]
- @@ -288,7 +288,7 @@ it will apply in all these instances.
 def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a - theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : + theorem fnUb_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) </pre></div> </div>
- @@ -453,7 +453,7 @@ then <code class="docutils literal notranslate">b</code
 A function \(f\) is said to be injective if for every \(x_1\) and \(x_2\), if \(f(x_1) = f(x_2)\) then \(x_1 = x_2\). - Mathlib defines <code class="docutils literal notranslate">function.injective f</code> with + Mathlib defines <code class="docutils literal notranslate">Function.Injective f</code> with <code class="docutils literal notranslate">x₁</code> and <code class="docutils literal notranslate">x₂</code> implicit. The next example shows that, on the real numbers, any function that adds a constant is injective.
- @@ -543,14 +543,14 @@ without specifying the bound:
 </div> We can use the theorem <code class="docutils literal notranslate">FnUb_add</code> from the last section to prove that if <code class="docutils literal notranslate">f</code> and <code class="docutils literal notranslate">g</code> have upper bounds, - then so does `` gun x => f x + g x``. + then so does <code class="docutils literal notranslate">fun x => f x + g x</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {f g : ℝ → ℝ} example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by cases' ubf with a ubfa - cases' ubg with b ubfb + cases' ubg with b ubgb use a + b - apply fnUb_add ubfa ubfb + apply fnUb_add ubfa ubgb </pre></div> </div> The <code class="docutils literal notranslate">cases</code> tactic unpacks the information
- @@ -597,18 +597,18 @@ is a combination of <code class="docutils literal notranslate"> <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by rcases ubf with ⟨a, ubfa⟩ - rcases ubg with ⟨b, ubfb⟩ - exact ⟨a + b, fnUb_add ubfa ubfb⟩ + rcases ubg with ⟨b, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by - rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ - exact ⟨a + b, fnUb_add ubfa ubfb⟩ + rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ </pre></div> </div> In fact, Lean also supports a pattern-matching lambda in expressions and proof terms: <div class="highlight-lean notranslate"><div class="highlight"><pre>example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := - fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ + fun ⟨a, ubfa⟩ ⟨b, ubgb⟩ => ⟨a + b, fnUb_add ubfa ubgb⟩ </pre></div> </div> These are power-user moves, and there is no harm
html/C04_Sets_and_Functions.html
- @@ -612,7 +612,7 @@ for this sort of situation.
 use x, Or.inr xt </pre></div> </div> - Notice also that the <code class="docutils literal notranslate">use</code> tactic applies <code class="docutils literal notranslate">refl</code> + Notice also that the <code class="docutils literal notranslate">use</code> tactic applies <code class="docutils literal notranslate">rfl</code> to close goals when it can. Here is another example: <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ⊆ f ⁻¹' (f '' s) := by
html/searchindex.js
- @@ -1,1 +1,1 @@
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mathematics_in_lean.pdf
src/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean
- @@ -81,7 +81,7 @@ variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R]
  def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a - theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : + theorem fnUb_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) end
src/C03_Logic/S02_The_Existential_Quantifier.lean
- @@ -34,9 +34,9 @@ variable {f g : ℝ → ℝ}
  example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by cases' ubf with a ubfa - cases' ubg with b ubfb + cases' ubg with b ubgb use a + b - apply fnUb_add ubfa ubfb + apply fnUb_add ubfa ubgb example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by sorry
- @@ -46,15 +46,15 @@ example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x :=
  example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by rcases ubf with ⟨a, ubfa⟩ - rcases ubg with ⟨b, ubfb⟩ - exact ⟨a + b, fnUb_add ubfa ubfb⟩ + rcases ubg with ⟨b, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by - rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ - exact ⟨a + b, fnUb_add ubfa ubfb⟩ + rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := - fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ + fun ⟨a, ubfa⟩ ⟨b, ubgb⟩ => ⟨a + b, fnUb_add ubfa ubgb⟩ end
- @@ -129,4 +129,3 @@ example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f
  sorry end -
src/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
- @@ -79,4 +79,3 @@ example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f
  use x end -
src/C04_Sets_and_Functions/S02_Functions.lean
- @@ -211,4 +211,3 @@ theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by
  -- COMMENTS: TODO: improve this end -
src/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
- @@ -250,4 +250,3 @@ theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by
  contradiction end -