a7420e27 - mathematics_in

Authored by: Patrick Massot
Authored at: 2023-07-04 15:08:22 +0200
Committed by: Patrick Massot
Committed at: 2023-07-04 15:08:22 +0200
Object name: a7420e274eeae12110cdd7db38e42a69f40b5aed
Parent: 9b0beb48

Update mar. 04 juil. 2023 15:08:22 CEST

Changes

64 changed files (+186/-143)

MIL/C02_Basics/S01_Calculating.lean
- @@ -1,7 +1,7 @@
  - import Mathlib.Data.Real.Basic + -- An example. + import Mathlib.Tactic import Mathlib.Data.Real.Basic - -- An example. example (a b c : ℝ) : a * b * c = b * (a * c) := by rw [mul_comm a b] rw [mul_assoc b a c]
- @@ -46,7 +46,7 @@ example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c *
  section - variable (a b c d e f g : ℝ) + variable (a b c d e f : ℝ) example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc]
- @@ -64,7 +64,6 @@ variable (a b c : ℝ)
  #check mul_assoc c a b #check mul_comm a #check mul_comm - #check @mul_comm end
MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean
- @@ -40,8 +40,8 @@ theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add]
  theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] - #check @MyRing.add_zero - #check @add_zero + #check MyRing.add_zero + #check add_zero end MyRing
MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean
- @@ -74,7 +74,7 @@ example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by
 #check (add_pos : 0 < a → 0 < b → 0 < a + b) #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) #check (exp_pos : ∀ a, 0 < exp a) - #check @add_le_add_left + #check add_le_add_left example (h : a ≤ b) : exp a ≤ exp b := by rw [exp_le_exp]
MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C02S04
MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.MetricSpace.Basic section
MIL/C02_Basics/solutions/Solutions_S01_Calculating.lean
- @@ -1,6 +1,3 @@
  - import Mathlib.Data.Real.Basic - import Mathlib.Data.Real.Basic - example (a b c : ℝ) : c * b * a = b * (a * c) := by rw [mul_comm c b] rw [mul_assoc b c a]
MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C02S04
MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.MetricSpace.Basic section
MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S01
- @@ -76,7 +77,7 @@ end
  section variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] - #check @add_le_add + #check add_le_add def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a
MIL/C03_Logic/S02_The_Existential_Quantifier.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S02
MIL/C03_Logic/S03_Negation.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S03
MIL/C03_Logic/S04_Conjunction_and_Bi-implication.lean
- @@ -1,6 +1,6 @@
 + import Mathlib.Tactic import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum namespace C03S04 example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by
- @@ -60,7 +60,6 @@ example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prim
 use 5 use 7 norm_num - sorry example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by rintro ⟨h₀, h₁⟩
MIL/C03_Logic/S05_Disjunction.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S05
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
- @@ -1,3 +1,4 @@
 + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S06
- @@ -13,7 +14,7 @@ example (a b : ℝ) : |a| = |a - b + b| := by
 ring example {a : ℝ} (h : 1 < a) : a < a * a := by - convert(mul_lt_mul_right _).2 h + convert (mul_lt_mul_right _).2 h · rw [one_mul] exact lt_trans zero_lt_one h
MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S01
MIL/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S02
MIL/C03_Logic/solutions/Solutions_S03_Negation.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S03
MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean
- @@ -1,6 +1,6 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum namespace C03S04 example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by
MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S05
MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S06
MIL/C04_Sets_and_Functions/S02_Functions.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic
MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic
MIL/C05_Number_Theory/S01_Irrational_Roots.lean
- @@ -1,7 +1,6 @@
 + import Mathlib.Tactic import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum.GCD - import Mathlib.Tactic.NormNum.Prime #print Nat.coprime
- @@ -16,7 +15,7 @@ example : Nat.coprime 12 7 := by norm_num
 example : Nat.gcd 12 8 = 4 := by norm_num - #check @Nat.prime_def_lt + #check Nat.prime_def_lt example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by rwa [Nat.prime_def_lt] at prime_p
- @@ -35,7 +34,7 @@ example : Nat.Prime 2 :=
 example : Nat.Prime 3 := Nat.prime_three - #check @Nat.Prime.dvd_mul + #check Nat.Prime.dvd_mul #check Nat.Prime.dvd_mul Nat.prime_two #check Nat.prime_two.dvd_mul
MIL/C05_Number_Theory/S03_Infinitely_Many_Primes.lean
- @@ -1,7 +1,6 @@
  import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic - import Mathlib.Tactic.IntervalCases open BigOperators
MIL/C05_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean
- @@ -1,7 +1,6 @@
  + import Mathlib.Tactic import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum.GCD - import Mathlib.Tactic.NormNum.Prime theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by rw [pow_two, Nat.prime_two.dvd_mul] at h
MIL/C05_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
- @@ -1,7 +1,6 @@
  import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic - import Mathlib.Tactic.IntervalCases open BigOperators
MIL/C06_Structures/S01_Structures.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
MIL/C06_Structures/S02_Algebraic_Structures.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C06S02
- @@ -110,12 +111,12 @@ instance {α : Type _} : Group₂ (Equiv.Perm α) where
  mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm - #check @Group₂.mul + #check Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x - #check @mySquare + #check mySquare section variable {β : Type _} (f g : Equiv.Perm β)
MIL/C06_Structures/solutions/Solutions_S01_Structures.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
MIL/C06_Structures/solutions/Solutions_S02_Algebraic_Structures.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C06S02
MIL/C07_Hierarchies/S01_Basics.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic.LibrarySearch import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
MIL/C07_Hierarchies/S02_Morphisms.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real
MIL/C07_Hierarchies/S03_Subobjects.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.GroupTheory.QuotientGroup
MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic.LibrarySearch import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
MIL/C07_Hierarchies/solutions/Solutions_S02_Morphisms.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real
MIL/C07_Hierarchies/solutions/Solutions_S03_Subobjects.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.GroupTheory.QuotientGroup
MIL/C08_Topology/S01_Filters.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real open Set Filter Topology
- @@ -79,9 +80,9 @@ example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) :
  Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := tendsto_congr' h - #check @eventually_of_forall - #check @Eventually.mono - #check @Eventually.and + #check eventually_of_forall + #check Eventually.mono + #check Eventually.and example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by
MIL/C08_Topology/S02_Metric_Spaces.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
MIL/C08_Topology/S03_Topological_Spaces.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
- @@ -112,7 +113,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X}
  ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry - #check @HasBasis.tendsto_right_iff + #check HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} :
MIL/C08_Topology/solutions/Solutions_S01_Filters.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real open Set Filter Topology
MIL/C08_Topology/solutions/Solutions_S02_Metric_Spaces.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
MIL/C08_Topology/solutions/Solutions_S03_Topological_Spaces.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
- @@ -126,7 +127,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X}
  ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry - #check @HasBasis.tendsto_right_iff + #check HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)
MIL/C09_Differential_Calculus/S01_Elementary_Differential_Calculus.lean
- @@ -1,3 +1,4 @@
 + import Mathlib.Tactic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.Calculus.MeanValue
- @@ -27,6 +28,8 @@ example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : Diff
 example {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := h.deriv_eq_zero + open Set + example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : ∃ c ∈ Ioo a b, deriv f c = 0 := exists_deriv_eq_zero f hab hfc hfI
MIL/C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.Inverse
- @@ -113,8 +114,6 @@ end
  open Asymptotics - open Asymptotics - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff
- @@ -133,6 +132,8 @@ example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g :
  section + open Topology + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
- @@ -171,7 +172,7 @@ example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
  ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := hf.eventually_right_inverse - example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + example {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := HasStrictFDerivAt.to_localInverse hf
MIL/C09_Differential_Calculus/solutions/Solutions_S01_Elementary_Differential_Calculus.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.Calculus.MeanValue
MIL/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.Inverse
MIL/C10_Integration_and_Measure_Theory/S01_Elementary_Integration.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.MeasureTheory.Integral.IntervalIntegral import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.Analysis.Convolution
MIL/C10_Integration_and_Measure_Theory/S02_Measure_Theory.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
MIL/C10_Integration_and_Measure_Theory/S03_Integration.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
- @@ -24,6 +25,8 @@ example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) :
  example {s : Set α} (c : E) : (∫ x in s, c ∂μ) = (μ s).toReal • c := set_integral_const c + open Filter + example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) :
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S01_Elementary_Integration.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.MeasureTheory.Integral.IntervalIntegral import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.Analysis.Convolution
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S02_Measure_Theory.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S03_Integration.lean
- @@ -1,3 +1,4 @@
  + import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
html/C01_Introduction.html
- @@ -203,7 +203,7 @@ Such an expression is a proof of the proposition <cite>P</cite>.
 #check hard </pre></div> </div> - If you manage to construct an expression of type <cite>fermat_last_theorem</cite> and + If you manage to construct an expression of type <code class="docutils literal notranslate">FermatLastTheorem</code> and Lean accepts it as a term of that type, you have done something very impressive. (Using <code class="docutils literal notranslate">sorry</code> is cheating, and Lean knows it.)
- @@ -269,7 +269,7 @@ declares that we are going to show that <code class="docutils literal notranslat
 showing <code class="docutils literal notranslate">m * n = 2 * (m * k)</code>. The next line uses the <code class="docutils literal notranslate">rewrite</code> tactic to replace <code class="docutils literal notranslate">n</code> by <code class="docutils literal notranslate">2 * k</code> in the goal, - and the <cite>ring</cite> tactic solves the resulting goal <code class="docutils literal notranslate">m * (2 * k) = 2 * (m * k)</code>. + and the <code class="docutils literal notranslate">ring</code> tactic solves the resulting goal <code class="docutils literal notranslate">m * (2 * k) = 2 * (m * k)</code>. The ability to build a proof in small steps with incremental feedback is extremely powerful. For that reason, tactic proofs are often easier and quicker to write than
html/C02_Basics.html
- @@ -121,7 +121,10 @@ so the left-hand side of <code class="docutils literal notranslate"> Let’s try out <code class="docutils literal notranslate">rw</code>. - <div class="highlight-lean notranslate" id="index-1"><div class="highlight"><pre>example (a b c : ℝ) : a * b * c = b * (a * c) := by + <div class="highlight-lean notranslate" id="index-1"><div class="highlight"><pre>import Mathlib.Tactic + import Mathlib.Data.Real.Basic + + example (a b c : ℝ) : a * b * c = b * (a * c) := by rw [mul_comm a b] rw [mul_assoc b a c] </pre></div>
- @@ -141,7 +144,7 @@ If you are curious to see all available abbreviations, you can hit Ctrl-Shift-p
 and then type abbreviations to get access to the <code class="docutils literal notranslate">Lean 4: Show all abbreviations</code> command. If your keyboard does not have an easily accessible backslash, you can change the leading character by changing the - <code class="docutils literal notranslate">lean.input.leader</code> setting. + <code class="docutils literal notranslate">lean4.input.leader</code> setting. When a cursor is in the middle of a tactic proof, Lean reports on the current proof state in the Lean infoview window.
- @@ -176,7 +179,7 @@ In practice, the intended meaning is usually clear.
 in each case replacing <code class="docutils literal notranslate">sorry</code> by a tactic proof. With the <code class="docutils literal notranslate">rw</code> tactic, you can use a left arrow (<code class="docutils literal notranslate">\l</code>) to reverse an identity. - For example, <code class="docutils literal notranslate">rw ← mul_assoc a b c</code> + For example, <code class="docutils literal notranslate">rw [← mul_assoc a b c]</code> replaces <code class="docutils literal notranslate">a * (b * c)</code> by <code class="docutils literal notranslate">a * b * c</code> in the current goal. Note that the left-pointing arrow refers to going from right to left in the identity provided by <code class="docutils literal notranslate">mul_assoc</code>, it has nothing to do with the left or right side of the goal.
- @@ -217,7 +220,7 @@ and the second with only one argument.
 rw [mul_assoc] </pre></div> </div> - Try these, using the theorem <cite>sub_self</cite> for the second one: + Try these, using the theorem <code class="docutils literal notranslate">sub_self</code> for the second one: <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by sorry
- @@ -235,7 +238,7 @@ by listing the relevant identities separated by commas inside the square bracket
 a comma in any list of rewrites. Another trick is that we can declare variables once and for all outside an example or theorem. Lean then includes them automatically. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (a b c d e f g : ℝ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (a b c d e f : ℝ) example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc]
- @@ -258,7 +261,6 @@ command to determine the type of an expression:
 #check mul_assoc c a b #check mul_comm a #check mul_comm - #check @mul_comm end </pre></div>
- @@ -304,7 +306,7 @@ 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 underscores and justification + The <code class="docutils literal notranslate">calc</code> syntax is finicky: the 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;
- @@ -498,8 +500,8 @@ as ring axioms, because they follow from the other axioms.
 theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] - #check @MyRing.add_zero - #check @add_zero + #check MyRing.add_zero + #check add_zero end MyRing </pre></div>
- @@ -579,16 +581,15 @@ that was introduced by the <code class="docutils literal notranslate">h</code> has been added: having proved it, we are now free to use it. - At this point, the goal is exactly the result of <code class="docutils literal notranslate">add_left_cancel h</code>. - .. index:: apply, tactics ; apply, exact, tactics ; exact - We could equally well have closed the proof with + At this point, the goal is exactly the result of <code class="docutils literal notranslate">add_left_cancel h</code>. + We could equally well have closed the proof with <code class="docutils literal notranslate">apply add_left_cancel h</code> or <code class="docutils literal notranslate">exact add_left_cancel h</code>. The <code class="docutils literal notranslate">exact</code> tactic takes as argument a proof term which completely proves the current goal, without creating any new goal. The <code class="docutils literal notranslate">apply</code> tactic is a variant whose argument is not necessarily a complete proof. The missing pieces are either inferred automatically by Lean or become new goals to prove. While the <code class="docutils literal notranslate">exact</code> tactic is technically redundant since it is strictly less powerful - than <code class="docutils literal notranslate">apply</code>, it makes proof scripts slightly clearer tho + than <code class="docutils literal notranslate">apply</code>, it makes proof scripts slightly clearer to human readers and easier to maintain when the library evolves. Remember that multiplication is not assumed to be commutative, so the following theorem also requires some work.
- @@ -632,7 +633,7 @@ addition of the additive inverse.
 rfl </pre></div> </div> - The proof term <code class="docutils literal notranslate">rfl</code> is short for “reflexivity”. + The proof term <code class="docutils literal notranslate">rfl</code> is short for “reflexivity”. 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">rfl</code> tactic does the same.
- @@ -663,7 +664,7 @@ the last section:
 sorry </pre></div> </div> - We close this section by noting that some of the facts about + We close this section by noting that some of the facts about addition and negation that we established above do not need the full strength of the ring axioms, or even commutativity of addition. The weaker notion of a group
- @@ -702,10 +703,10 @@ The proofs we have carried out in this section provide some hints.
 sorry </pre></div> </div> - Explicitly invoking those lemmas is tedious, so mathlib provides + Explicitly invoking those lemmas is tedious, so mathlib provides tactics similar to <cite>ring</cite> in order to cover most uses: <cite>group</cite> is for non-commutative multiplicative groups, <cite>abel</cite> for abelian - additive groups, and <cite>noncomm_ring</cite> for non-commutative groups. + additive groups, and <cite>noncomm_ring</cite> for non-commutative rings. It may seem odd that the algebraic structures are called <cite>Ring</cite> and <cite>CommRing</cite> while the tactics are named <cite>noncomm_ring</cite> and <cite>ring</cite>. This is partly for historical reasons,
- @@ -714,7 +715,7 @@ tactic that deals with commutative rings, since it is used more often.
 </section> <section id="using-theorems-and-lemmas"> <h2>2.3. Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> - Rewriting is great for proving equations, + Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality, like the fact that \(a + e^b \le a + e^c\) holds whenever \(b \le c\)?
- @@ -745,7 +746,7 @@ all the following work:
 #check (le_trans h h' : a ≤ c) </pre></div> </div> - The <code class="docutils literal notranslate">apply</code> tactic takes a proof of a general statement or implication, + The <code class="docutils literal notranslate">apply</code> tactic takes a proof of a general statement or implication, tries to match the conclusion with the current goal, and leaves the hypotheses, if any, as new goals. If the given proof matches the goal exactly
- @@ -795,7 +796,7 @@ we avoid going into tactic mode entirely:
 sorry </pre></div> </div> - In fact, Lean has a tactic that does this sort of thing automatically: + In fact, Lean has a tactic that does this sort of thing automatically: <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by linarith </pre></div>
- @@ -820,7 +821,7 @@ would be parsed as <code class="docutils literal notranslate">
 linarith [exp_le_exp.mpr h'] </pre></div> </div> - Here are some more theorems in the library that can be used to establish + Here are some more theorems in the library that can be used to establish inequalities on the real numbers. <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) #check (exp_lt_exp : exp a < exp b ↔ a < b)
- @@ -837,7 +838,7 @@ inequalities on the real numbers.
 #check (add_pos : 0 < a → 0 < b → 0 < a + b) #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) #check (exp_pos : ∀ a, 0 < exp a) - #check @add_le_add_left + #check add_le_add_left </pre></div> </div> Some of the theorems, <code class="docutils literal notranslate">exp_le_exp</code>, <code class="docutils literal notranslate">exp_lt_exp</code>, and <code class="docutils literal notranslate">log_le_log</code>
- @@ -872,7 +873,7 @@ Thus the following proof works:
 creates two goals, and once again we use a dot to separate the proof of the first from the proof of the second. - Try the following examples on your own. + Try the following examples on your own. The example in the middle shows you that the <code class="docutils literal notranslate">norm_num</code> tactic can be used to solve concrete numeric goals. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry
- @@ -981,7 +982,7 @@ You are well on your way to becoming a master formalizer.
 </section> <section id="more-examples-using-apply-and-rw"> <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this heading"></a></h2> - The <code class="docutils literal notranslate">min</code> function on the real numbers is uniquely characterized + The <code class="docutils literal notranslate">min</code> function on the real numbers is uniquely characterized by the following three facts: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b)
- @@ -1022,7 +1023,7 @@ we can show that <code class="docutils literal notranslate">mi
 apply min_le_left </pre></div> </div> - Here we have used dots to separate proofs of + Here we have used dots to separate proofs of different goals. Our usage is inconsistent: at the outer level,
- @@ -1035,7 +1036,7 @@ the proof
 and indicate what is being proved in each block. The proof still works without the <code class="docutils literal notranslate">show</code> commands, but using them makes the proof easier to read and maintain. - It may bother you that the the proof is repetitive. + It may bother you that the proof is repetitive. To foreshadow skills you will learn later on, we note that one way to avoid the repetition is to state a local lemma and then use it:
- @@ -1059,7 +1060,7 @@ and the <code class="docutils literal notranslate">introx</code> and <code class="docutils literal notranslate">y</code> to establish the conclusion. The first <code class="docutils literal notranslate">apply</code> after <code class="docutils literal notranslate">le_antisymm</code> implicitly uses <code class="docutils literal notranslate">h a b</code>, whereas the second one uses <code class="docutils literal notranslate">h b a</code>. - Another solution is to use the <code class="docutils literal notranslate">repeat</code> tactic, + Another solution is to use the <code class="docutils literal notranslate">repeat</code> tactic, which applies a tactic (or a block) as many times as it can. <div class="highlight-lean notranslate"><div class="highlight"><pre>example : min a b = min b a := by
- @@ -1113,7 +1114,7 @@ but applying it to suitable values yields the other direction
 as well. As a hint, you can use the theorem <code class="docutils literal notranslate">add_neg_cancel_right</code> and the <code class="docutils literal notranslate">linarith</code> tactic. - Lean’s naming convention is made manifest + Lean’s naming convention is made manifest in the library’s name for the triangle inequality: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (abs_add : ∀ a b : ℝ, |a + b| ≤ |a| + |b|) </pre></div>
- @@ -1125,7 +1126,7 @@ in the library’s name for the triangle inequality:
 </div> See if you can do this in three lines or less. You can use the theorem <code class="docutils literal notranslate">sub_add_cancel</code>. - Another important relation that we will make use of + Another important relation that we will make use of in the sections to come is the divisibility relation on the natural numbers, <code class="docutils literal notranslate">x ∣ y</code>. Be careful: the divisibility symbol is not the
- @@ -1155,7 +1156,7 @@ you need to prove the following:
 sorry </pre></div> </div> - With respect to divisibility, the greatest common divisor, + With respect to divisibility, the greatest common divisor, <code class="docutils literal notranslate">gcd</code>, and least common multiple, <code class="docutils literal notranslate">lcm</code>, are analogous to <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code>. Since every number divides <code class="docutils literal notranslate">0</code>,
- @@ -1190,7 +1191,7 @@ either one will work.
 </section> <section id="proving-facts-about-algebraic-structures"> <h2>2.5. Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> - In <a class="reference internal" href="#proving-identities-in-algebraic-structures">Section 2.2</a>, + In <a class="reference internal" href="#proving-identities-in-algebraic-structures">Section 2.2</a>, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures, such as commutative rings.
- @@ -1240,7 +1241,7 @@ the symbol <code class="docutils literal notranslate">¬</
 In <a class="reference internal" href="C03_Logic.html#logic">Chapter 3</a>, you will learn how to use these logical connectives to prove that <code class="docutils literal notranslate"><</code> has the properties indicated. - A lattice is a structure that extends a partial + A lattice is a structure that extends a partial order with operations <code class="docutils literal notranslate">⊓</code> and <code class="docutils literal notranslate">⊔</code> that are analogous to <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code> on the real numbers: <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Lattice α]
- @@ -1380,7 +1381,7 @@ enumerated in the last two examples:
 sorry </pre></div> </div> - Finally, here is one last example. + Finally, here is one last example. A metric space consists of a set equipped with a notion of distance, <code class="docutils literal notranslate">dist x y</code>, mapping any pair of elements to a real number.
html/C03_Logic.html
- @@ -287,7 +287,7 @@ So if we prove the theorem <code class="docutils literal notranslate"> <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] - #check @add_le_add + #check add_le_add def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a
- @@ -410,7 +410,7 @@ direction, but not much harder).
 once you know how to spot them. Mathlib includes a good library for manipulating sets. Recall that Lean does not use foundations based on set theory, so here the word set has its mundane meaning - of a collection of mathematical objets of some given type <code class="docutils literal notranslate">α</code>. + of a collection of mathematical objects of some given type <code class="docutils literal notranslate">α</code>. If <code class="docutils literal notranslate">x</code> has type <code class="docutils literal notranslate">α</code> and <code class="docutils literal notranslate">s</code> has type <code class="docutils literal notranslate">Set α</code>, then <code class="docutils literal notranslate">x ∈ s</code> is a proposition that asserts that <code class="docutils literal notranslate">x</code> is an element of <code class="docutils literal notranslate">s</code>. If <code class="docutils literal notranslate">y</code> has some different type <code class="docutils literal notranslate">β</code> then the expression <code class="docutils literal notranslate">y ∈ s</code> makes no sense. Here “makes no sense” means “has no type hence Lean does not
- @@ -735,7 +735,7 @@ why the next example makes use of both <code class="docutils literal notranslate
 sorry </pre></div> </div> - At this point, it is worth mentioning that there is a tactic, <cite>field_simp</cite>, + At this point, it is worth mentioning that there is a tactic, <code class="docutils literal notranslate">field_simp</code>, that will often clear denominators in a useful way. It can be used in conjunction with the <code class="docutils literal notranslate">ring</code> tactic. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by
- @@ -1126,7 +1126,6 @@ with anonymous constructors, <code class="docutils literal notranslate">use 5 use 7 norm_num - sorry example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by rintro ⟨h₀, h₁⟩
- @@ -1205,7 +1204,7 @@ to replace an expression of the form <code class="docutils literal notranslate">
 See if you can use <code class="docutils literal notranslate">rw</code> with the theorem below to provide a short proof that negation is not a nondecreasing function. (Note that <code class="docutils literal notranslate">push_neg</code> won’t - unfold definitions for you, so the <code class="docutils literal notranslate">rw monotone</code> in + unfold definitions for you, so the <code class="docutils literal notranslate">rw [Monotone]</code> in the proof of the theorem is needed.) <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by rw [Monotone]
- @@ -1521,7 +1520,7 @@ as it is,
 and leaves us with the task of proving the equations that are needed to make the goal match. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {a : ℝ} (h : 1 < a) : a < a * a := by - convert(mul_lt_mul_right _).2 h + convert (mul_lt_mul_right _).2 h · rw [one_mul] exact lt_trans zero_lt_one h </pre></div>
html/C05_Number_Theory.html
- @@ -151,7 +151,7 @@ to the natural numbers.
 We also need the notion of a prime number, <code class="docutils literal notranslate">Nat.Prime</code>. The theorem <code class="docutils literal notranslate">Nat.prime_def_lt</code> provides one familiar characterization, and <code class="docutils literal notranslate">Nat.Prime.eq_one_or_self_of_dvd</code> provides another. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @Nat.prime_def_lt + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.prime_def_lt example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by rwa [Nat.prime_def_lt] at prime_p
- @@ -186,7 +186,7 @@ if the square of a number is even, then that number is even as well.
 Mathlib defines the predicate <code class="docutils literal notranslate">Even</code> in <code class="docutils literal notranslate">Data.Nat.Parity</code>, but for reasons that will become clear below, we will simply use <code class="docutils literal notranslate">2 ∣ m</code> to express that <code class="docutils literal notranslate">m</code> is even. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @Nat.Prime.dvd_mul + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.Prime.dvd_mul #check Nat.Prime.dvd_mul Nat.prime_two #check Nat.prime_two.dvd_mul
- @@ -478,7 +478,7 @@ that the simplifier uses by default.
 about the natural numbers by proving that the statement holds of 0 and that whenever it holds of a natural number \(n\), it also holds of \(n + 1\). - The line <code class="docutils literal notranslate">induction n with n ih</code> in the proof + The line <code class="docutils literal notranslate">induction' n with n ih</code> in the proof below therefore results in two goals: in the first we need to prove <code class="docutils literal notranslate">0 < fac 0</code>, and in the second we have the added assumption <code class="docutils literal notranslate">ih : 0 < fac n</code>
html/C06_Structures.html
- @@ -360,7 +360,7 @@ function above.
 Structures can depend on parameters. For example, we can generalize the standard 2-simplex to the standard \(n\)-simplex for any \(n\). - At this stage, you don’t have to know anything about the type <cite>Fin n</cite> + At this stage, you don’t have to know anything about the type <code class="docutils literal notranslate">Fin n</code> except that it has \(n\) elements, and that Lean knows how to sum over it. <div class="highlight-lean notranslate"><div class="highlight"><pre>open BigOperators
- @@ -742,7 +742,7 @@ In Lean, the components of any <code class="docutils literal notranslate"><span
 and in a moment we will see how multiplicative notation is set up to refer to them. If we want to use additive notation, we instead use an isomorphic structure - <code class="docutils literal notranslate">AdditiveGroup</code>. Its components are named <code class="docutils literal notranslate">add</code>, <code class="docutils literal notranslate">zero</code>, + <code class="docutils literal notranslate">AddGroup</code> (the structure underlying additive groups). Its components are named <code class="docutils literal notranslate">add</code>, <code class="docutils literal notranslate">zero</code>, and <code class="docutils literal notranslate">neg</code>, and the associated notation is what you would expect it to be. Recall the type <code class="docutils literal notranslate">Point</code> that we defined in <a class="reference internal" href="#section-structures">Section 6.1</a>, and the addition function that we defined there.
- @@ -885,12 +885,12 @@ without troubling us with the details.
 </pre></div> </div> The following illustrates their use. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @Group₂.mul + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x - #check @mySquare + #check mySquare section variable {β : Type _} (f g : Equiv.Perm β)
- @@ -1213,7 +1213,7 @@ the integers.
 </div> Lean’s library defines the class of nontrivial types to be types with at least two distinct elements. In the context of a ring, this is equivalent - to saying the the zero is not equal to the one. Since some common theorems + to saying that the zero is not equal to the one. Since some common theorems depend on that fact, we may as well establish it now. <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Nontrivial gaussInt := by use 0, 1
- @@ -1307,7 +1307,7 @@ between these number systems appropriately.
 In fact, this is exactly the approach that is followed in mathlib, where the Gaussian integers themselves are constructed as a special case of a ring of quadratic integers. - See the file <a class="reference external" href="https://github.com/leanprover-community/mathlib/blob/master/src/number_theory/zsqrtd/gaussian_int.lean">gaussian_int.lean</a>. + See the file <a class="reference external" href="https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean">GaussianInt.lean</a>. Here we will instead carry out an argument that stays in the integers. This illustrates an choice one commonly faces when formalizing mathematics. Given an argument that requires concepts or machinery that is not already
html/C07_Hierarchies.html
- @@ -225,7 +225,7 @@ property that uses both a <cite>Dia₁</cite> instance and a <cite>One&#8321
 those instances we set a tracing option whose result can be seen in the info view. This result is rather terse by default but can be expended by clicking one lines ending with black arrows. It includes failed attempts where Lean tried to find instances before having enough type - information to succceed. The successful attempts do involve the instances generated by the + information to succeed. The successful attempts do involve the instances generated by the <code class="docutils literal notranslate">extends</code> syntax. <div class="highlight-lean notranslate"><div class="highlight"><pre>set_option trace.Meta.synthInstance true in example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙
- @@ -490,7 +490,7 @@ type <code class="docutils literal notranslate">β<
 </pre></div> </div> There is something interesting going on here. While it isn’t too surprising that the - ring structure on <code class="docutils literal notranslate">R</code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate">AddCommGroup3 M</code> + ring structure on <code class="docutils literal notranslate">R</code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate">AddCommGroup₃ M</code> to be part of the <code class="docutils literal notranslate">extends</code> clause just as <code class="docutils literal notranslate">SMul₃ R M</code> is. Trying to do that would lead to a mysterious sounding error message: <code class="docutils literal notranslate">cannot find synthesization order for instance Module₁.toAddCommGroup₃ with type (R : Type) → [inst : Ring₃ R] → {M : Type} → [self : Module₁ R M] → AddCommGroup₃ M
- @@ -944,7 +944,9 @@ convenient notation are built and code duplication is avoided in Mathlib. Here t
 is the <code class="docutils literal notranslate">HasQuotient</code> class which allows notations like <code class="docutils literal notranslate">M ⧸ N</code>. Beware the quotient symbol <code class="docutils literal notranslate">⧸</code> is a special unicode character, not a regular ASCII division symbol. As an example, we will build the quotient of a commutative monoid by a submonoid, leave proofs - to you. + to you. In the last example, you can use <code class="docutils literal notranslate">Setoid.refl</code> but it won’t automatically pick up + the relevant <code class="docutils literal notranslate">Setoid</code> structure. You can fix this issue by providing all arguments using + the <code class="docutils literal notranslate">@</code> syntax, as in <code class="docutils literal notranslate">@Setoid.refl M N.Setoid</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z iseqv := {
html/C08_Topology.html
- @@ -343,7 +343,7 @@ which is to say, it reverses the order of the arguments.
 <blockquote> <div><code class="docutils literal notranslate">F ×ˢ G = (comap Prod.fst F) ⊓ (comap Prod.snd G)</code>. </div></blockquote> - Here the <code class="docutils literal notranslate">inf</code> operation refers to the lattice structure on <code class="docutils literal notranslate">filter X</code> for any type <code class="docutils literal notranslate">X</code>, whereby + Here the <code class="docutils literal notranslate">inf</code> operation refers to the lattice structure on <code class="docutils literal notranslate">Filter X</code> for any type <code class="docutils literal notranslate">X</code>, whereby <code class="docutils literal notranslate">F ⊓ G</code> is the greatest filter that is smaller than both <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. Thus the <code class="docutils literal notranslate">inf</code> operation generalizes the notion of the intersection of sets. A lot of proofs in mathlib use all of the aforementioned structure (<code class="docutils literal notranslate">map</code>, <code class="docutils literal notranslate">comap</code>, <code class="docutils literal notranslate">inf</code>, <code class="docutils literal notranslate">sup</code>, and <code class="docutils literal notranslate">prod</code>)
- @@ -400,7 +400,7 @@ this for the standard neighborhoods <code class="docutils literal notranslate"><
 The key is that, by definition, every neighborhood contains such a standard one. This observation leads to the notion of a filter basis. Given <code class="docutils literal notranslate">F : Filter X</code>, - a family of sets <cite>s : ι → Set X</cite> is a basis for <code class="docutils literal notranslate">F</code> if for every set <code class="docutils literal notranslate">U</code>, + a family of sets <code class="docutils literal notranslate">s : ι → Set X</code> is a basis for <code class="docutils literal notranslate">F</code> if for every set <code class="docutils literal notranslate">U</code>, we have <code class="docutils literal notranslate">U ∈ F</code> if and only if it contains some <code class="docutils literal notranslate">s i</code>. In other words, formally speaking, <code class="docutils literal notranslate">s</code> is a basis if it satisfies <code class="docutils literal notranslate">∀ U : Set X, U ∈ F ↔ ∃ i, s i ⊆ U</code>. It is even more flexible to consider
- @@ -471,9 +471,9 @@ Given <code class="docutils literal notranslate">F <spa
 <li>the condition <code class="docutils literal notranslate">U ∈ F → U ⊆ V → V ∈ F</code> ensures <code class="docutils literal notranslate">(∀ᶠ x in F, P x) → (∀ x, P x → Q x) → ∀ᶠ x in F, Q x</code>, and</li> <li>the condition <code class="docutils literal notranslate">U ∈ F → V ∈ F → U ∩ V ∈ F</code> ensures <code class="docutils literal notranslate">(∀ᶠ x in F, P x) → (∀ᶠ x in F, Q x) → ∀ᶠ x in F, P x ∧ Q x</code>.</li> </ul> - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @eventually_of_forall - #check @Eventually.mono - #check @Eventually.and + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check eventually_of_forall + #check Eventually.mono + #check Eventually.and </pre></div> </div> The second item, corresponding to <code class="docutils literal notranslate">eventually.mono</code>, supports nice ways
- @@ -961,9 +961,9 @@ precise sense the neighborhood filters is all there is in a topological space st
 of formal properties of <code class="docutils literal notranslate">topological_space : Type u → Type u</code>. From a purely mathematical point of view, those formal properties are a very clean way to explain how topological spaces solve issues that metric spaces have. From this point of view, the issues solved by topological spaces is that metric spaces enjoy very - little fonctoriality, and have very bad categorical properties in general. This comes on top of the fact + little functoriality, and have very bad categorical properties in general. This comes on top of the fact already discussed that metric spaces contain a lot of geometrical information that is not topologically relevant. - Let us focus on fonctoriality first. A metric space structure can be induced on a subset or, + Let us focus on functoriality first. A metric space structure can be induced on a subset or, equivalently, it can be pulled back by an injective map. But that’s pretty much everything. They cannot be pulled back by general map or pushed forward, even by surjective maps. In particular there is no sensible distance to put on a quotient of a metric space or on an uncountable
- @@ -1123,7 +1123,7 @@ together with the fact that <code class="docutils literal notranslate">∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry - #check @HasBasis.tendsto_right_iff + #check HasBasis.tendsto_right_iff </pre></div> </div> In addition to separation property, the main kind of assumption you can make on a topological
html/C09_Differential_Calculus.html
- @@ -160,7 +160,9 @@ definition of <code class="docutils literal notranslate">deriv
 </div> We can eve state Rolle’s theorem without any differentiability assumptions, which seems even weirder. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Set + + example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : ∃ c ∈ Ioo a b, deriv f c = 0 := exists_deriv_eq_zero f hab hfc hfI </pre></div>
- @@ -337,8 +339,6 @@ notation.
 Here we will only use little o to define differentiability. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Asymptotics - open Asymptotics - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff
- @@ -361,10 +361,12 @@ Here we will only use little o to define differentiability.
 <h3>9.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> We are now ready to discuss differentiable functions between normed spaces. In analogy the elementary one-dimensional, - mathlib defines a predicate <code class="docutils literal notranslate">has_fderiv_at</code> and a function <code class="docutils literal notranslate">fderiv</code>. + mathlib defines a predicate <code class="docutils literal notranslate">HasFDerivAt</code> and a function <code class="docutils literal notranslate">fderiv</code>. Here the letter “f” stands for Fréchet. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Topology + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) :
- @@ -393,7 +395,7 @@ So \(\mathcal{C}^\infty\) func
 </pre></div> </div> There is a stricter notion of differentiability called - <code class="docutils literal notranslate">has_strict_fderiv_at</code>, which is used in the statement + <code class="docutils literal notranslate">HasStrictFDerivAt</code>, which is used in the statement of the inverse function theorem and the statement of the implicit function theorem, both of which are in mathlib. Over <code class="docutils literal notranslate">ℝ</code> or <code class="docutils literal notranslate">ℂ</code>, continuously differentiable
- @@ -425,7 +427,7 @@ from the left and from the right, and that it is strictly differentiable.
 ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := hf.eventually_right_inverse - example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + example {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := HasStrictFDerivAt.to_localInverse hf
html/C10_Integration_and_Measure_Theory.html
- @@ -140,14 +140,14 @@ not necessarily finite dimensional.
 The first component in the development of measure theory is the notion of a \(\sigma\)-algebra of sets, which are called the measurable sets. - The type class <code class="docutils literal notranslate">measurable_space</code> serves to equip a type with such a structure. + The type class <code class="docutils literal notranslate">MeasurableSpace</code> serves to equip a type with such a structure. The sets <code class="docutils literal notranslate">empty</code> and <code class="docutils literal notranslate">univ</code> are measurable, the complement of a measurable set is measurable, and a countable union or intersection of measurable sets is measurable. - Note that these axioms are redundant; if you <code class="docutils literal notranslate">#print measurable_space</code>, + Note that these axioms are redundant; if you <code class="docutils literal notranslate">#print MeasurableSpace</code>, you will see the ones that mathlib uses. As the examples below show, countability assumptions can be expressed using the - <code class="docutils literal notranslate">encodable</code> type class. + <code class="docutils literal notranslate">Encodable</code> type class. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [MeasurableSpace α] example : MeasurableSet (∅ : Set α) :=
- @@ -228,7 +228,7 @@ Most lemmas having to do with integrals have integrability assumptions.
 </div> As an example of the complex interactions between our various conventions, let us see how to integrate constant functions. Recall that a measure <code class="docutils literal notranslate">μ</code> takes values in <code class="docutils literal notranslate">ℝ≥0∞</code>, the type of extended non-negative reals. - There is a function <code class="docutils literal notranslate">ennreal.to_real : ℝ≥0∞ → ℝ</code> which sends <code class="docutils literal notranslate">⊤</code>, + There is a function <code class="docutils literal notranslate">ENNReal.toReal : ℝ≥0∞ → ℝ</code> which sends <code class="docutils literal notranslate">⊤</code>, the point at infinity, to zero. For any <code class="docutils literal notranslate">s : Set α</code>, if <code class="docutils literal notranslate">μ s = ⊤</code>, then nonzero constant functions are not integrable on <code class="docutils literal notranslate">s</code>. In that case, their integrals are equal to zero by definition, as is <code class="docutils literal notranslate">(μ s).to_real</code>.
- @@ -240,7 +240,9 @@ So in all cases we have the following lemma.
 We now quickly explain how to access the most important theorems in integration theory, starting with the dominated convergence theorem. There are several versions in mathlib, and here we only show the most basic one. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Filter + + example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) : Tendsto (fun n ↦ ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) :=
- @@ -269,9 +271,9 @@ continuous bilinear form.
 </pre></div> </div> Finally, mathlib has a very general version of the change-of-variables formula. - In the statement below, <code class="docutils literal notranslate">borel_space E</code> means the + In the statement below, <code class="docutils literal notranslate">BorelSpace E</code> means the \(\sigma\)-algebra on <code class="docutils literal notranslate">E</code> is generated by the open sets of <code class="docutils literal notranslate">E</code>, - and <code class="docutils literal notranslate">is_add_haar_measure μ</code> means that the measure <code class="docutils literal notranslate">μ</code> is left-invariant, + and <code class="docutils literal notranslate">IsAddHaarMeasure μ</code> means that the measure <code class="docutils literal notranslate">μ</code> is left-invariant, gives finite mass to compact sets, and give positive mass to open sets. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type _}
html/genindex.html
- @@ -107,7 +107,7 @@
 <h2 id="A">A</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-23">absolute value</a> + <li><a href="C02_Basics.html#index-24">absolute value</a> </li> <li><a href="C03_Logic.html#index-17">absurd</a> </li>
- @@ -115,7 +115,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-7">anonymous constructor</a> </li> - <li><a href="C02_Basics.html#index-16">apply</a> + <li><a href="C02_Basics.html#index-12">apply</a>, <a href="C02_Basics.html#index-17">[1]</a> </li> <li><a href="C03_Logic.html#index-19">assumption</a> </li>
- @@ -183,13 +183,13 @@
 <h2 id="D">D</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-12">definitional equality</a> + <li><a href="C02_Basics.html#index-13">definitional equality</a> </li> <li><a href="C09_Differential_Calculus.html#index-0">differential calculus</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-24">divisibility</a> + <li><a href="C02_Basics.html#index-25">divisibility</a> </li> <li><a href="C03_Logic.html#index-2">dsimp</a> </li>
- @@ -203,7 +203,7 @@
 </li> <li><a href="C03_Logic.html#index-4">erw</a> </li> - <li><a href="C02_Basics.html#index-16">exact</a>, <a href="C02_Basics.html#index-5">[1]</a> + <li><a href="C02_Basics.html#index-12">exact</a>, <a href="C02_Basics.html#index-17">[1]</a>, <a href="C02_Basics.html#index-5">[2]</a> </li> <li><a href="C03_Logic.html#index-22">excluded middle</a> </li>
- @@ -211,7 +211,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-17">exfalso</a> </li> - <li><a href="C02_Basics.html#index-18">exponential</a> + <li><a href="C02_Basics.html#index-19">exponential</a> </li> <li><a href="C03_Logic.html#index-24">ext</a> </li>
- @@ -237,16 +237,16 @@
 <h2 id="G">G</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-25">gcd</a> + <li><a href="C02_Basics.html#index-26">gcd</a> </li> <li><a href="C02_Basics.html#index-2">goal</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-13">group (algebraic structure)</a> + <li><a href="C02_Basics.html#index-14">group (algebraic structure)</a> <ul> - <li><a href="C02_Basics.html#index-14">(tactic)</a> + <li><a href="C02_Basics.html#index-15">(tactic)</a> </li> </ul></li> </ul></td>
- @@ -265,7 +265,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-10">implicit argument</a> </li> - <li><a href="C02_Basics.html#index-15">inequalities</a> + <li><a href="C02_Basics.html#index-16">inequalities</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul>
- @@ -283,9 +283,9 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-1">lambda abstraction</a> </li> - <li><a href="C02_Basics.html#index-27">lattice</a> + <li><a href="C02_Basics.html#index-28">lattice</a> </li> - <li><a href="C02_Basics.html#index-25">lcm</a> + <li><a href="C02_Basics.html#index-26">lcm</a> </li> <li><a href="C03_Logic.html#index-21">left</a> </li>
- @@ -293,11 +293,11 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-13">let</a> </li> - <li><a href="C02_Basics.html#index-17">linarith</a> + <li><a href="C02_Basics.html#index-18">linarith</a> </li> <li><a href="C02_Basics.html#index-2">local context</a> </li> - <li><a href="C02_Basics.html#index-18">logarithm</a> + <li><a href="C02_Basics.html#index-19">logarithm</a> </li> </ul></td> </tr></table>
- @@ -305,15 +305,15 @@
 <h2 id="M">M</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-20">max</a> + <li><a href="C02_Basics.html#index-21">max</a> </li> <li><a href="C10_Integration_and_Measure_Theory.html#index-2">measure theory</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-28">metric space</a>, <a href="C08_Topology.html#index-2">[1]</a> + <li><a href="C02_Basics.html#index-29">metric space</a>, <a href="C08_Topology.html#index-2">[1]</a> </li> - <li><a href="C02_Basics.html#index-20">min</a> + <li><a href="C02_Basics.html#index-21">min</a> </li> <li><a href="C03_Logic.html#index-3">monotone function</a> </li>
- @@ -327,7 +327,7 @@
 </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-19">norm_num</a> + <li><a href="C02_Basics.html#index-20">norm_num</a> </li> <li><a href="C09_Differential_Calculus.html#index-2">normed space</a> </li>
- @@ -341,7 +341,7 @@
 </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-26">order relation</a> + <li><a href="C02_Basics.html#index-27">order relation</a> </li> </ul></td> </tr></table>
- @@ -349,7 +349,7 @@
 <h2 id="P">P</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> - <li><a href="C02_Basics.html#index-26">partial order</a> + <li><a href="C02_Basics.html#index-27">partial order</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul>
- @@ -367,13 +367,13 @@
 </li> <li><a href="C02_Basics.html#index-1">real numbers</a> </li> - <li><a href="C02_Basics.html#index-12">reflexivity</a> + <li><a href="C02_Basics.html#index-13">reflexivity</a> </li> - <li><a href="C02_Basics.html#index-22">repeat</a> + <li><a href="C02_Basics.html#index-23">repeat</a> </li> <li><a href="C02_Basics.html#index-0">rewrite</a> </li> - <li><a href="C02_Basics.html#index-12">rfl</a> + <li><a href="C02_Basics.html#index-13">rfl</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul>
- @@ -399,7 +399,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C04_Sets_and_Functions.html#index-0">set operations</a> </li> - <li><a href="C02_Basics.html#index-21">show</a> + <li><a href="C02_Basics.html#index-22">show</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul>
- @@ -424,9 +424,9 @@
 tactics <ul> - <li><a href="C02_Basics.html#index-14">abel</a> + <li><a href="C02_Basics.html#index-15">abel</a> </li> - <li><a href="C02_Basics.html#index-16">apply</a> + <li><a href="C02_Basics.html#index-12">apply</a>, <a href="C02_Basics.html#index-17">[1]</a> </li> <li><a href="C03_Logic.html#index-19">assumption</a> </li>
- @@ -456,7 +456,7 @@
 </li> <li><a href="C03_Logic.html#index-4">erw</a> </li> - <li><a href="C02_Basics.html#index-16">exact</a>, <a href="C02_Basics.html#index-5">[1]</a> + <li><a href="C02_Basics.html#index-12">exact</a>, <a href="C02_Basics.html#index-17">[1]</a>, <a href="C02_Basics.html#index-5">[2]</a> </li> <li><a href="C03_Logic.html#index-17">exfalso</a> </li>
- @@ -464,7 +464,7 @@
 </li> <li><a href="C03_Logic.html#index-12">from</a> </li> - <li><a href="C02_Basics.html#index-14">group</a> + <li><a href="C02_Basics.html#index-15">group</a> </li> <li><a href="C02_Basics.html#index-11">have</a>, <a href="C03_Logic.html#index-12">[1]</a> </li>
- @@ -474,19 +474,19 @@
 </li> <li><a href="C03_Logic.html#index-13">let</a> </li> - <li><a href="C02_Basics.html#index-17">linarith</a> + <li><a href="C02_Basics.html#index-18">linarith</a> </li> - <li><a href="C02_Basics.html#index-14">noncomm_ring</a> + <li><a href="C02_Basics.html#index-15">noncomm_ring</a> </li> - <li><a href="C02_Basics.html#index-19">norm_num</a> + <li><a href="C02_Basics.html#index-20">norm_num</a> </li> <li><a href="C03_Logic.html#index-15">push_neg</a> </li> <li><a href="C03_Logic.html#index-9">rcases</a> </li> - <li><a href="C02_Basics.html#index-12">refl and reflexivity</a> + <li><a href="C02_Basics.html#index-13">refl and reflexivity</a> </li> - <li><a href="C02_Basics.html#index-22">repeat</a> + <li><a href="C02_Basics.html#index-23">repeat</a> </li> <li><a href="C03_Logic.html#index-21">right</a> </li>
- @@ -498,7 +498,7 @@
 </li> <li><a href="C04_Sets_and_Functions.html#index-2">rwa</a> </li> - <li><a href="C02_Basics.html#index-21">show</a> + <li><a href="C02_Basics.html#index-22">show</a> </li> <li><a href="C03_Logic.html#index-20">simp</a>, <a href="C04_Sets_and_Functions.html#index-1">[1]</a> </li>
html/searchindex.js
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