df46ab90 - mathematics_in

Authored by: Patrick Massot
Authored at: 2024-05-02 16:46:44 -0400
Committed by: Patrick Massot
Committed at: 2024-05-02 16:46:44 -0400
Object name: df46ab9048edb89df1491db0e4db16155c4c735f
Parent: a04873bc

Update Thu May 2 04:46:44 PM EDT 2024

Changes

37 changed files (+121/-97)

MIL/C01_Introduction/S02_Overview.lean
- @@ -1,4 +1,3 @@
  - import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Parity import MIL.Common
MIL/C01_Introduction/solutions/Solutions_S02_Overview.lean
- @@ -1,4 +1,3 @@
  - import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Parity import MIL.Common
MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean
- @@ -59,10 +59,10 @@ section
  variable {α : Type*} [DistribLattice α] variable (x y z : α) - #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) - #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) - #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) - #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + #check (inf_sup_left x y z : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right x y z : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left x y z : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right x y z : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section
MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
- @@ -62,7 +62,7 @@ example : |a| - |b| ≤ |a - b| :=
  _ ≤ |a - b| + |b| - |b| := by apply sub_le_sub_right apply abs_add - _ ≤ |a - b| := by rw [add_sub_cancel] + _ ≤ |a - b| := by rw [add_sub_cancel_right] -- alternatively
MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean
- @@ -81,10 +81,10 @@ section
  variable {α : Type*} [DistribLattice α] variable (x y z : α) - #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) - #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) - #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) - #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + #check (inf_sup_left x y z : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right x y z : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left x y z : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right x y z : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section
MIL/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
- @@ -62,12 +62,12 @@ open Function
  example {c : ℝ} (h : c ≠ 0) : Surjective fun x ↦ c * x := by intro x use x / c - dsimp; rw [mul_div_cancel' _ h] + dsimp; rw [mul_div_cancel₀ _ h] example {c : ℝ} (h : c ≠ 0) : Surjective fun x ↦ c * x := by intro x use x / c - field_simp [h] ; ring + field_simp end
MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean
- @@ -51,7 +51,7 @@ theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : Conver
 calc |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub] _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos) - _ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm + _ = ε := mul_div_cancel₀ _ (ne_of_lt acpos).symm theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → |s n| < b := by
- @@ -80,7 +80,7 @@ theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : Converges
 calc |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero] _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _)) - _ = ε := mul_div_cancel' _ (ne_of_lt Bpos).symm + _ = ε := mul_div_cancel₀ _ (ne_of_lt Bpos).symm theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) :
- @@ -125,7 +125,7 @@ theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ}
 _ ≤ |(-(s N - a))| + |s N - b| := (abs_add _ _) _ = |s N - a| + |s N - b| := by rw [abs_neg] _ < ε + ε := (add_lt_add absa absb) - _ = |a - b| := by norm_num + _ = |a - b| := by norm_num [ε] exact lt_irrefl _ this
MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean
- @@ -45,7 +45,7 @@ theorem sb_injective (hf : Injective f) : Injective (sbFun f g) := by
  · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by - apply not_imp_self.mp + apply _root_.not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A
MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
- @@ -46,7 +46,7 @@ theorem sb_injective (hf : Injective f) : Injective (sbFun f g) := by
  · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by - apply not_imp_self.mp + apply _root_.not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A
MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean
- @@ -96,7 +96,7 @@ theorem sum_id (n : ℕ) : ∑ i in range (n + 1), i = n * (n + 1) / 2 := by
 symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) induction' n with n ih · simp - rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] + rw [Finset.sum_range_succ, mul_add 2, ← ih] ring theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n + 1) / 6 := by
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
- @@ -1,5 +1,5 @@
 import Mathlib.Data.Nat.Prime - import Mathlib.Algebra.BigOperators.Order + import Mathlib.Algebra.BigOperators.Basic import MIL.Common open BigOperators
- @@ -156,15 +156,14 @@ example : 27 % 4 = 3 := by norm_num
 example (n : ℕ) : (4 * n + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] - norm_num theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) - interval_cases hm : m % 4 <;> simp [hm] + interval_cases m % 4 <;> simp [-Nat.mul_mod_mod] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) - interval_cases hn : n % 4 <;> simp [hn] + interval_cases n % 4 <;> simp theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;>
MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean
- @@ -12,7 +12,7 @@ theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by
 induction' n with n ih · simp [fac] simp at * - rw [pow_succ, fac] + rw [pow_succ', fac] apply Nat.mul_le_mul _ ih repeat' apply Nat.succ_le_succ apply zero_le
- @@ -29,7 +29,7 @@ theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n
 apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) induction' n with n ih · simp - rw [Finset.sum_range_succ, mul_add 6, ← ih, Nat.succ_eq_add_one] + rw [Finset.sum_range_succ, mul_add 6, ← ih] ring end
MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
- @@ -1,5 +1,5 @@
 import Mathlib.Data.Nat.Prime - import Mathlib.Algebra.BigOperators.Order + import Mathlib.Algebra.BigOperators.Basic import MIL.Common open BigOperators
- @@ -148,9 +148,9 @@ theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨
 revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) - interval_cases hm : m % 4 <;> simp [hm] + interval_cases m % 4 <;> simp [-Nat.mul_mod_mod] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) - interval_cases hn : n % 4 <;> simp [hn] + interval_cases n % 4 <;> simp theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;>
- @@ -206,7 +206,6 @@ theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3
 rcases this with ⟨s, hs⟩ have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] - norm_num rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ have ps : p ∈ s := by rw [← hs p]
MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean
- @@ -1,4 +1,3 @@
 - import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import MIL.Common
- @@ -87,6 +86,8 @@ instance instCommRing : CommRing gaussInt where
 add := (· + ·) neg x := -x mul := (· * ·) + nsmul := nsmulRec + zsmul := zsmulRec add_assoc := by intros ext <;> simp <;> ring
- @@ -120,8 +121,12 @@ instance instCommRing : CommRing gaussInt where
 mul_comm := by intros ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry + zero_mul := by + intros + ext <;> simp + mul_zero := by + intros + ext <;> simp @[simp] theorem sub_re (x y : gaussInt) : (x - y).re = x.re - y.re :=
- @@ -258,7 +263,7 @@ instance : EuclideanDomain gaussInt :=
 quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := - fun x y ↦ by simp only; rw [mod_def, add_comm, sub_add_cancel] + fun x y ↦ by simp only; rw [mod_def, add_comm] ; ring quotient_zero := fun x ↦ by simp [div_def, norm, Int.div'] rfl
MIL/C06_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean
- @@ -1,4 +1,3 @@
 - import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import MIL.Common
- @@ -87,6 +86,8 @@ instance instCommRing : CommRing gaussInt where
 add := (· + ·) neg x := -x mul := (· * ·) + nsmul := nsmulRec + zsmul := zsmulRec add_assoc := by intros ext <;> simp <;> ring
- @@ -120,8 +121,12 @@ instance instCommRing : CommRing gaussInt where
 mul_comm := by intros ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry + zero_mul := by + intros + ext <;> simp + mul_zero := by + intros + ext <;> simp @[simp] theorem sub_re (x y : gaussInt) : (x - y).re = x.re - y.re :=
- @@ -272,7 +277,7 @@ instance : EuclideanDomain gaussInt :=
 quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := - fun x y ↦ by simp only; rw [mod_def, add_comm, sub_add_cancel] + fun x y ↦ by simp only; rw [mod_def, add_comm] ; ring quotient_zero := fun x ↦ by simp [div_def, norm, Int.div'] rfl
MIL/C08_Groups_and_Rings/S01_Groups.lean
- @@ -273,6 +273,9 @@ open MonoidHom
  #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right + -- The following line is working around a Lean bug that will be fixed very soon. + attribute [-instance] Subtype.instInhabited + lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G ⧸ H) = card K := by sorry variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K)
MIL/C08_Groups_and_Rings/S02_Rings.lean
- @@ -70,7 +70,7 @@ example {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] (f : ι → Ideal R)
  open BigOperators PiNotation example {ι : Type*} [Fintype ι] (a : ι → ℕ) (coprime : ∀ i j, i ≠ j → (a i).Coprime (a j)) : - ZMod (∏ i, a i) ≃+* ∀ i, ZMod (a i) := + ZMod (∏ i, a i) ≃+* Π i, ZMod (a i) := ZMod.prodEquivPi a coprime section
MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean
- @@ -128,6 +128,9 @@ open MonoidHom
  #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right + -- The following line is working around a Lean bug that will be fixed very soon. + attribute [-instance] Subtype.instInhabited + lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G ⧸ H) = card K := by have := calc card (G ⧸ H) * card H = card G := by rw [← H.index_eq_card, H.index_mul_card]
MIL/C08_Groups_and_Rings/solutions/Solutions_S02_Rings.lean
- @@ -64,7 +64,6 @@ lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}
  replace he : ∀ j, j ≠ i → e ∈ I j := by simpa using he refine ⟨e, ?_, ?_⟩ · simp [eq_sub_of_add_eq' hue, map_sub, eq_zero_iff_mem.mpr hu] - rfl · exact fun j hj ↦ eq_zero_iff_mem.mpr (he j hj) choose e he using key use mk _ (∑ i, f i * e i)
MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean
- @@ -152,14 +152,14 @@ example {X : Type*} [MetricSpace X] [CompactSpace X] {Y : Type*} [MetricSpace Y]
 · use 1, by norm_num intro x y _ have : (x, y) ∉ K := by simp [hK] - simpa using this + simpa [K] using this · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ use dist x₀ x₁ constructor · change _ < _ rw [dist_pos] intro h - have : ε ≤ 0 := by simpa [*] using xx_in + have : ε ≤ 0 := by simpa [K, φ, *] using xx_in linarith · intro x x' contrapose!
MIL/C10_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -2,7 +2,7 @@ import MIL.Common
 import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv - import Mathlib.Analysis.Calculus.ContDiff.IsROrC + import Mathlib.Analysis.Calculus.ContDiff.RCLike import Mathlib.Analysis.Calculus.FDeriv.Prod
- @@ -106,7 +106,7 @@ example {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x,
 have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m sorry have εk_pos : 0 < ε / ‖k‖ := sorry - refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.opNorm_le_of_shell ε_pos _ hk _⟩ sorry sorry
- @@ -153,7 +153,7 @@ example (n : WithTop ℕ) {f : E → F} :
 ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x ↦ iteratedFDeriv 𝕜 m f x := contDiff_iff_continuous_differentiable - example {𝕂 : Type*} [IsROrC 𝕂] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type*} + example {𝕂 : Type*} [RCLike 𝕂] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := hf.hasStrictFDerivAt hn
MIL/C10_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -2,7 +2,7 @@ import MIL.Common
 import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv - import Mathlib.Analysis.Calculus.ContDiff.IsROrC + import Mathlib.Analysis.Calculus.ContDiff.RCLike import Mathlib.Analysis.Calculus.FDeriv.Prod
- @@ -43,11 +43,11 @@ example {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x,
 replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i apply interior_subset hz have εk_pos : 0 < ε / ‖k‖ := div_pos ε_pos (zero_lt_one.trans hk) - refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.opNorm_le_of_shell ε_pos _ hk _⟩ · exact div_nonneg (Nat.cast_nonneg _) εk_pos.le intro y le_y y_lt calc - ‖g i y‖ = ‖g i (y + x) - g i x‖ := by rw [(g i).map_add, add_sub_cancel] + ‖g i y‖ = ‖g i (y + x) - g i x‖ := by rw [(g i).map_add, add_sub_cancel_right] _ ≤ ‖g i (y + x)‖ + ‖g i x‖ := (norm_sub_le _ _) _ ≤ m + m := (add_le_add (real_norm_le (y + x) (by rwa [add_comm, add_mem_ball_iff_norm]) i)
MIL/C11_Integration_and_Measure_Theory/S03_Integration.lean
- @@ -23,7 +23,7 @@ example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) :
  integral_add hf hg example {s : Set α} (c : E) : ∫ x in s, c ∂μ = (μ s).toReal • c := - set_integral_const c + setIntegral_const c open Filter
html/C01_Introduction.html
- @@ -290,7 +290,7 @@ tactic proofs are often easier and quicker to write than
 proof terms. There isn’t a sharp distinction between the two: tactic proofs can be inserted in proof terms, - as we did with the phrase <code class="docutils literal notranslate">by rw [hk, mul_left_comm]</code> in the example above. + as we did with the phrase <code class="docutils literal notranslate">by rw [hk, mul_add]</code> in the example above. We will also see that, conversely, it is often useful to insert a short proof term in the middle of a tactic proof. That said, in this book, our emphasis will be on the use of tactics.
html/C02_Basics.html
- @@ -380,9 +380,9 @@ It can be imported explicitly with the command
 <code class="docutils literal notranslate">import Mathlib.Tactic</code>. 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. + There is a variation of <code class="docutils literal notranslate">rw</code> called <code class="docutils literal notranslate">nth_rw</code> that allows you to replace only particular instances of an expression in the goal. 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 + so in the following example, <code class="docutils literal notranslate">nth_rw 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]
- @@ -1113,7 +1113,7 @@ 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> </div> - Use it to prove the following variant: + Use it to prove the following variant, using also <code class="docutils literal notranslate">add_sub_cancel_right</code>: <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a| - |b| ≤ |a - b| := sorry end
- @@ -1314,10 +1314,10 @@ is called a distributive lattice. Lean knows about these too:
 <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [DistribLattice α] variable (x y z : α) - #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) - #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) - #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) - #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + #check (inf_sup_left x y z : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right x y z : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left x y z : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right x y z : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) </pre></div> </div> The left and right versions are easily shown to be
html/C03_Logic.html
- @@ -782,7 +782,7 @@ why the next example makes use of both <code class="docutils literal notranslate
 dsimp; ring </pre></div> </div> - Try this example yourself using the theorem <code class="docutils literal notranslate">mul_div_cancel'</code>.: + Try this example yourself using the theorem <code class="docutils literal notranslate">mul_div_cancel₀</code>.: <div class="highlight-lean notranslate"><div class="highlight"><pre>example {c : ℝ} (h : c ≠ 0) : Surjective fun x ↦ c * x := by sorry </pre></div>
html/C04_Sets_and_Functions.html
- @@ -1044,7 +1044,7 @@ See if you can finish off the proof using <code class="docutils literal notransl
 · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by - apply not_imp_self.mp + apply _root_.not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A
html/C05_Elementary_Number_Theory.html
- @@ -514,7 +514,8 @@ function.
 It turns out to be easier to start with a proof by cases, so that the remainder of the proof starts with the case \(n = 1\). - See if you can complete the argument with a proof by induction. + See if you can complete the argument with a proof by induction using <code class="docutils literal notranslate">pow_succ</code> + or <code class="docutils literal notranslate">pow_succ'</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by rcases n with _ | n · simp [fac]
- @@ -613,7 +614,7 @@ because calculations with division generally have side conditions.
 symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) induction' n with n ih · simp - rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] + rw [Finset.sum_range_succ, mul_add 2, ← ih] ring </pre></div> </div>
- @@ -1042,19 +1043,18 @@ the facts about this function that we will need to use below.
 The first named theorem is another illustration of reasoning by a small number of cases. In the second named theorem, remember that the semicolon means that - the subsequent tactic block is applied to both of the goals - that result from the application of <code class="docutils literal notranslate">two_le</code>. + the subsequent tactic block is applied to all the goals created by the + preceding tactic. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) : (4 * n + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] - norm_num theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) - interval_cases hm : m % 4 <;> simp [hm] + interval_cases m % 4 <;> simp [-Nat.mul_mod_mod] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) - interval_cases hn : n % 4 <;> simp [hn] + interval_cases n % 4 <;> simp theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;>
html/C06_Structures.html
- @@ -479,7 +479,7 @@ it will help to consider some examples.
 <ol class="arabic simple"> <li>A partially ordered set consists of a set \(P\) and a binary relation \(\le\) on \(P\) that is transitive - and antireflexive.</li> + and reflexive.</li> <li>A group consists of a set \(G\) with an associative binary operation, an identity element \(1\), and a function \(g \mapsto g^{-1}\) that returns
- @@ -1146,17 +1146,23 @@ that appears in VS Code, and then ask Lean to fill in a skeleton for the
 structure definition, you will see a scary number of entries. Jumping to the definition of the structure, however, shows that many of the fields have default definitions that Lean will fill in for you automatically. - The essential ones appear in the definition below. In each case, the relevant + The essential ones appear in the definition below. + A special case are <code class="docutils literal notranslate">nsmul</code> and <code class="docutils literal notranslate">zsmul</code> which should be ignored for now + and will be explained in the next chapter. + In each case, the relevant identity is proved by unfolding definitions, using the <code class="docutils literal notranslate">ext</code> tactic to reduce the identities to their real and imaginary components, simplifying, and, if necessary, carrying out the relevant ring calculation in - the integers. + the integers. Note that we could easily avoid repeating all this code, but + this is not the topic of the current discussion. <div class="highlight-lean notranslate"><div class="highlight"><pre>instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) + nsmul := nsmulRec + zsmul := zsmulRec add_assoc := by intros ext <;> simp <;> ring
- @@ -1190,8 +1196,12 @@ the integers.
 mul_comm := by intros ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry + zero_mul := by + intros + ext <;> simp + mul_zero := by + intros + ext <;> simp </pre></div> </div> Lean’s library defines the class of nontrivial types to be types with at
- @@ -1268,7 +1278,7 @@ numbers, carry out the division
 \frac{ac + bd}{c^2 + d^2} + \frac{bc -ad}{c^2+d^2} i.\]</div> The real and imaginary parts might not be integers, but we can round them to the nearest integers \(u\) and \(v\). We can then express the - right-hand size as \((u + vi) + (u' + v'i)\), where + right-hand side as \((u + vi) + (u' + v'i)\), where \(u' + v'i\) is the part left over. Note that we have \(|u'| \le 1/2\) and \(|v'| \le 1/2\), and hence <div class="math notranslate nohighlight">
- @@ -1419,7 +1429,7 @@ less than the norm of <code class="docutils literal notranslate"> <blockquote> - <div><code class="docutils literal notranslate">(x % y) * conj y = (x - x / y * y) * conj y = x * conj y - x * (y * conj j)</code> + <div><code class="docutils literal notranslate">(x % y) * conj y = (x - x / y * y) * conj y = x * conj y - x / y * (y * conj y)</code> </div></blockquote> The real and imaginary parts of the right-hand side are exactly <code class="docutils literal notranslate">mod' (x * conj y).re (norm y)</code> and <code class="docutils literal notranslate">mod' (x * conj y).im (norm y)</code>. By the properties of <code class="docutils literal notranslate">div'</code> and <code class="docutils literal notranslate">mod'</code>,
- @@ -1500,7 +1510,7 @@ and in that case, the required properties are the theorems
 quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := - fun x y ↦ by simp only; rw [mod_def, add_comm, sub_add_cancel] + fun x y ↦ by simp only; rw [mod_def, add_comm] ; ring quotient_zero := fun x ↦ by simp [div_def, norm, Int.div'] rfl
html/C08_Groups_and_Rings.html
- @@ -667,6 +667,9 @@ or disjoint.
 #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right + -- The following line is working around a Lean bug that will be fixed very soon. + attribute [-instance] Subtype.instInhabited + lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G ⧸ H) = card K := by sorry </pre></div>
- @@ -838,7 +841,7 @@ deduced from the previous one:
 <div class="highlight-lean notranslate"><div class="highlight"><pre>open BigOperators PiNotation example {ι : Type*} [Fintype ι] (a : ι → ℕ) (coprime : ∀ i j, i ≠ j → (a i).Coprime (a j)) : - ZMod (∏ i, a i) ≃+* ∀ i, ZMod (a i) := + ZMod (∏ i, a i) ≃+* Π i, ZMod (a i) := ZMod.prodEquivPi a coprime </pre></div> </div>
html/C09_Topology.html
- @@ -178,7 +178,7 @@ conditions that we will spell out below. The notion
 supports two related ideas: <ul class="simple"> <li>limits, including all the kinds of limits discussed above: finite and infinite limits of sequences, finite and infinite limits of functions at a point or at infinity, and so on.</li> - <li>things happening eventually, including things happening for large enough <code class="docutils literal notranslate">n : ℕ</code>, or sufficiently near a point <code class="docutils literal notranslate">x</code>, or for sufficiently close pairs of points, or almost everywhere in the sense of measure theory. Dually, filters can also express the idea of things happening often: for arbitrarily large <code class="docutils literal notranslate">n</code>, at a point in any neighborhood of given a point, etc.</li> + <li>things happening eventually, including things happening for large enough <code class="docutils literal notranslate">n : ℕ</code>, or sufficiently near a point <code class="docutils literal notranslate">x</code>, or for sufficiently close pairs of points, or almost everywhere in the sense of measure theory. Dually, filters can also express the idea of things happening often: for arbitrarily large <code class="docutils literal notranslate">n</code>, at a point in any neighborhood of a given point, etc.</li> </ul> The filters that correspond to these descriptions will be defined later in this section, but we can already name them: <ul class="simple">
- @@ -548,7 +548,7 @@ will be explained as part of the calculus chapter.
 <h3>9.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Link to this heading"></a></h3> Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, - but we have lemmas recasting the definition is terms of distances. + but we have lemmas recasting the definition in terms of distances. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop
- @@ -654,7 +654,7 @@ and get our final proof, now bordering obfuscation.
 </pre></div> </div> Once we have balls, we can define open sets. They are actually defined in a more general setting covered in the next section, - but we have lemmas recasting the definition is terms of balls. + but we have lemmas recasting the definition in terms of balls. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.isOpen_iff </pre></div>
- @@ -971,7 +971,7 @@ products of metric spaces. Consider for instance the type <code class="docutils
 a product of copies of <code class="docutils literal notranslate">ℝ</code> indexed by <code class="docutils literal notranslate">ℝ</code>. We would like to say that pointwise convergence of sequences of functions is a respectable notion of convergence. But there is no distance on <code class="docutils literal notranslate">ℝ → ℝ</code> that gives this notion of convergence. Relatedly, there is no distance ensuring that - a map <code class="docutils literal notranslate">f : X → (ℝ → ℝ)</code> is continuous if and only <code class="docutils literal notranslate">fun x ↦ f x t</code> is continuous for every <code class="docutils literal notranslate">t : ℝ</code>. + a map <code class="docutils literal notranslate">f : X → (ℝ → ℝ)</code> is continuous if and only if <code class="docutils literal notranslate">fun x ↦ f x t</code> is continuous for every <code class="docutils literal notranslate">t : ℝ</code>. We now review the data used to solve all those issues. First we can use any map <code class="docutils literal notranslate">f : X → Y</code> to push or pull topologies from one side to the other. Those two operations form a Galois connection. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X Y : Type*}
- @@ -992,7 +992,7 @@ As usual, pushing forward is covariant and pulling back is contravariant, see <c
 On paper we will use notations \(f_*T\) for <code class="docutils literal notranslate">TopologicalSpace.coinduced f T</code> and \(f^*T\) for <code class="docutils literal notranslate">TopologicalSpace.induced f T</code>. Then the next big piece is a complete lattice structure on <code class="docutils literal notranslate">TopologicalSpace X</code> - for any given structure. If you think of topologies are being primarily the data of open sets then you expect + for any given structure. If you think of topologies as being primarily the data of open sets then you expect the order relation on <code class="docutils literal notranslate">TopologicalSpace X</code> to come from <code class="docutils literal notranslate">Set (Set X)</code>, ie you expect <code class="docutils literal notranslate">t ≤ t'</code> if a set <code class="docutils literal notranslate">u</code> is open for <code class="docutils literal notranslate">t'</code> as soon as it is open for <code class="docutils literal notranslate">t</code>. However we already know that Mathlib focuses on neighborhoods more than open sets so, for any <code class="docutils literal notranslate">x : X</code> we want the map from topological spaces to neighborhoods
- @@ -1085,7 +1085,7 @@ but we’ll stick to Bourbaki’s version here.
 insert that funny up arrow when needed. And the (inclusion) coercion map is <code class="docutils literal notranslate">(↑) : A → X</code>. The assumption “tends to \(x\) while remaining in \(A\)” corresponds to the pull-back filter <code class="docutils literal notranslate">comap (↑) (𝓝 x)</code>. - Let’s prove first an auxiliary lemma, extracted to simplify the context + Let’s first prove an auxiliary lemma, extracted to simplify the context (in particular we don’t need Y to be a topological space here). <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux {X Y A : Type*} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y}
- @@ -1115,8 +1115,8 @@ In particular <code class="docutils literal notranslate">(&#85
 In addition <code class="docutils literal notranslate">comap (↑) (𝓝 y) ≠ ⊥</code> because <code class="docutils literal notranslate">A</code> is dense. Because we know <code class="docutils literal notranslate">Tendsto f (comap (↑) (𝓝 y)) (𝓝 (φ y))</code> this implies <code class="docutils literal notranslate">φ y ∈ closure V'</code> and, since <code class="docutils literal notranslate">V'</code> is closed, we have proved <code class="docutils literal notranslate">φ y ∈ V'</code>. - It remains to prove that <code class="docutils literal notranslate">φ</code> extends <code class="docutils literal notranslate">f</code>. This is were continuity of <code class="docutils literal notranslate">f</code> enters the discussion, - together with the fact that <code class="docutils literal notranslate">Y</code> is Hausdorff. + It remains to prove that <code class="docutils literal notranslate">φ</code> extends <code class="docutils literal notranslate">f</code>. This is where the continuity of <code class="docutils literal notranslate">f</code> enters the + discussion, together with the fact that <code class="docutils literal notranslate">Y</code> is Hausdorff. <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :
- @@ -1128,7 +1128,7 @@ together with the fact that <code class="docutils literal notranslate"> In addition to separation property, the main kind of assumption you can make on a topological space to bring it closer to metric spaces is countability assumption. The main one is first countability - asking that every point has a countable neighborhood basic. In particular this ensures that closure + asking that every point has a countable neighborhood basis. In particular this ensures that closure of sets can be understood using sequences. <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} {a : X} :
html/C10_Differential_Calculus.html
- @@ -298,7 +298,7 @@ into a normed space is pointwise
 bounded, then the norms of these linear maps are uniformly bounded. The main ingredient is Baire’s theorem <code class="docutils literal notranslate">nonempty_interior_of_iUnion_of_closed</code>. (You proved a version of this in the topology chapter.) - Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.op_norm_le_of_shell</code>, + Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.opNorm_le_of_shell</code>, <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_iInter_subset</code> and <code class="docutils literal notranslate">is_closed_le</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
- @@ -324,7 +324,7 @@ Minor ingredients include <code class="docutils literal notranslate">have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m sorry have εk_pos : 0 < ε / ‖k‖ := sorry - refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.opNorm_le_of_shell ε_pos _ hk _⟩ sorry sorry </pre></div>
- @@ -401,7 +401,7 @@ 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 functions are strictly differentiable. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type*} [IsROrC 𝕂] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type*} + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type*} [RCLike 𝕂] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := hf.hasStrictFDerivAt hn
html/C11_Integration_and_Measure_Theory.html
- @@ -235,7 +235,7 @@ For any <code class="docutils literal notranslate">s <s
 In that case, their integrals are equal to zero by definition, as is <code class="docutils literal notranslate">(μ s).toReal</code>. So in all cases we have the following lemma. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set α} (c : E) : ∫ x in s, c ∂μ = (μ s).toReal • c := - set_integral_const c + setIntegral_const c </pre></div> </div> We now quickly explain how to access the most important theorems in integration theory, starting
html/searchindex.js
- @@ -1,1 +1,1 @@
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lake-manifest.json
- @@ -4,7 +4,7 @@
  [{"url": "https://github.com/leanprover/std4", "type": "git", "subDir": null, - "rev": "276953b13323ca151939eafaaec9129bf7970306", + "rev": "3025cb124492b423070f20cf0a70636f757d117f", "name": "std", "manifestFile": "lake-manifest.json", "inputRev": "main",
- @@ -13,7 +13,7 @@
  {"url": "https://github.com/leanprover-community/quote4", "type": "git", "subDir": null, - "rev": "1c88406514a636d241903e2e288d21dc6d861e01", + "rev": "64365c656d5e1bffa127d2a1795f471529ee0178", "name": "Qq", "manifestFile": "lake-manifest.json", "inputRev": "master",
- @@ -22,7 +22,7 @@
  {"url": "https://github.com/leanprover-community/aesop", "type": "git", "subDir": null, - "rev": "6beed82dcfbb7731d173cd517675df27d62ad0f4", + "rev": "0a21a48c286c4a4703c0be6ad2045f601f31b1d0", "name": "aesop", "manifestFile": "lake-manifest.json", "inputRev": "master",
- @@ -31,16 +31,16 @@
  {"url": "https://github.com/leanprover-community/ProofWidgets4", "type": "git", "subDir": null, - "rev": "af1e86cf7a37389632a02f4a111e6b501b2b818f", + "rev": "fe1eff53bd0838c657aa6126fe4dd75ad9939d9a", "name": "proofwidgets", "manifestFile": "lake-manifest.json", - "inputRev": "v0.0.27", + "inputRev": "v0.0.35", "inherited": true, "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover/lean4-cli", "type": "git", "subDir": null, - "rev": "a751d21d4b68c999accb6fc5d960538af26ad5ec", + "rev": "a11566029bd9ec4f68a65394e8c3ff1af74c1a29", "name": "Cli", "manifestFile": "lake-manifest.json", "inputRev": "main",
- @@ -49,7 +49,7 @@
  {"url": "https://github.com/leanprover-community/import-graph.git", "type": "git", "subDir": null, - "rev": "8079d2d1d0e073bde42eab159c24f4c2d0d3a871", + "rev": "188eb34fcf1125e89d651ad462d02598219718ca", "name": "importGraph", "manifestFile": "lake-manifest.json", "inputRev": "main",
- @@ -58,7 +58,7 @@
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lean-toolchain
- @@ -1,1 +1,1 @@
  - leanprover/lean4:v4.6.0-rc1 + leanprover/lean4:v4.8.0-rc1
mathematics_in_lean.pdf