61f99a62 - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2023-06-05 09:16:18 -0700
Committed by: Jeremy Avigad
Committed at: 2023-06-05 09:16:18 -0700
Object name: 61f99a62d9c2e80043c75f73d5de3da6b543d207
Parent: e3465109

Update Mon Jun 5 09:16:18 PDT 2023

Changes

109 changed files (+1677/-11495)

html/C01_Introduction.html
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html/C02_Basics.html
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html/C03_Logic.html
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html/C04_Sets_and_Functions.html
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html/C05_Number_Theory.html
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html/C06_Structures.html
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html/C07_Hierarchies.html
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html/C08_Topology.html
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html/C09_Differential_Calculus.html
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html/C10_Integration_and_Measure_Theory.html
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html/index.html
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html/searchindex.js
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lake-manifest.json
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lake-packages/Qq (deleted)
- @@ -1,1 +0,0 @@
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lake-packages/aesop (deleted)
- @@ -1,1 +0,0 @@
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lake-packages/mathlib (deleted)
- @@ -1,1 +0,0 @@
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lake-packages/std (deleted)
- @@ -1,1 +0,0 @@
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mathematics_in_lean.pdf
src/02_Basics/01_Calculating.lean
- @@ -184,3 +184,9 @@ end
  end + example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := + begin + nth_rewrite 1 h, + rw add_mul + end +
src/06_Abstract_Algebra/03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,194 @@
 + import data.int.basic + import ring_theory.principal_ideal_domain + import tactic + + @[ext] structure gaussint := (re : ℤ) (im : ℤ) + + namespace gaussint + + instance : has_zero gaussint := ⟨⟨0, 0⟩⟩ + instance : has_one gaussint := ⟨⟨1, 0⟩⟩ + instance : has_add gaussint := ⟨λ x y, ⟨x.re + y.re, x.im + y.im⟩⟩ + instance : has_neg gaussint := ⟨λ x, ⟨-x.re, -x.im⟩⟩ + instance : has_mul gaussint := + ⟨λ x y, ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussint) = ⟨0, 0⟩ := rfl + theorem one_def : (1 : gaussint) = ⟨1, 0⟩ := rfl + theorem add_def (x y : gaussint) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl + theorem neg_def (x : gaussint) : -x = ⟨-x.re, -x.im⟩ := rfl + theorem mul_def (x y : gaussint) : + x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl + + @[simp] theorem zero_re : (0 : gaussint).re = 0 := rfl + @[simp] theorem zero_im : (0 : gaussint).im = 0 := rfl + @[simp] theorem one_re : (1 : gaussint).re = 1 := rfl + @[simp] theorem one_im : (1 : gaussint).im = 0 := rfl + @[simp] theorem add_re (x y : gaussint) : (x + y).re = x.re + y.re := rfl + @[simp] theorem add_im (x y : gaussint) : (x + y).im = x.im + y.im := rfl + @[simp] theorem neg_re (x : gaussint) : (-x).re = - x.re := rfl + @[simp] theorem neg_im (x : gaussint) : (-x).im = - x.im := rfl + @[simp] theorem mul_re (x y : gaussint) : (x * y).re = x.re * y.re - x.im * y.im := rfl + @[simp] theorem mul_im (x y : gaussint) : (x * y).im = x.re * y.im + x.im * y.re := rfl + + instance : comm_ring gaussint := + { zero := 0, + one := 1, + add := (+), + neg := λ x, -x, + mul := (*), + + add_assoc := by { intros, ext; simp; ring }, + zero_add := by { intros, ext; simp }, + add_zero := by { intros, ext; simp }, + add_left_neg := by { intros, ext; simp }, + add_comm := by { intros, ext; simp; ring }, + mul_assoc := by { intros, ext; simp; ring }, + one_mul := by { intros, ext; simp }, + mul_one := by { intros, ext; simp }, + left_distrib := by { intros, ext; simp; ring }, + right_distrib := by { intros, ext; simp; ring }, + mul_comm := by { intros, ext; simp; ring } } + + instance : nontrivial gaussint := + by { use [0, 1], rw [ne, gaussint.ext_iff], simp } + + end gaussint + + example (a b : ℤ) : a = b * (a / b) + a % b := eq.symm $ int.div_add_mod a b + + example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := int.mod_nonneg a + + example (a b : ℤ) : b ≠ 0 → a % b < abs b := int.mod_lt a + + namespace int + + def div' (a b : ℤ) := (a + b / 2) / b + + def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := + begin + rw [div', mod'], + linarith [int.div_add_mod (a + b / 2) b], + end + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b): abs (mod' a b) ≤ b / 2 := + begin + rw [mod', abs_le], + split, + { linarith [int.mod_nonneg (a + b / 2) h.ne'] }, + have := int.mod_lt_of_pos (a + b / 2) h, + have := int.div_add_mod b 2, + have := int.mod_lt_of_pos b zero_lt_two, + linarith + end + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := + by linarith [div'_add_mod' a b] + + end int + + theorem sq_add_sq_eq_zero {α : Type*} [linear_ordered_ring α] (x y : α) : + x^2 + y^2 = 0 ↔ x = 0 ∧ y = 0 := + sorry + + namespace gaussint + + def norm (x : gaussint) := x.re^2 + x.im^2 + + @[simp] theorem norm_nonneg (x : gaussint) : 0 ≤ norm x := + sorry + + theorem norm_eq_zero (x : gaussint) : norm x = 0 ↔ x = 0 := + sorry + + theorem norm_pos (x : gaussint) : 0 < norm x ↔ x ≠ 0 := + sorry + + theorem norm_mul (x y : gaussint) : norm (x * y) = norm x * norm y := + sorry + + def conj (x : gaussint) : gaussint := ⟨x.re, -x.im⟩ + + @[simp] theorem conj_re (x : gaussint) : (conj x).re = x.re := rfl + @[simp] theorem conj_im (x : gaussint) : (conj x).im = - x.im := rfl + + theorem norm_conj (x : gaussint) : norm (conj x) = norm x := + by { simp [norm] } + + instance : has_div gaussint := + ⟨λ x y, ⟨int.div' (x * conj y).re (norm y), int.div' (x * conj y).im (norm y)⟩⟩ + + instance : has_mod gaussint := ⟨λ x y, x - y * (x / y)⟩ + + theorem div_def (x y : gaussint) : + x / y = ⟨int.div' (x * conj y).re (norm y), int.div' (x * conj y).im (norm y)⟩ := rfl + + theorem mod_def (x y : gaussint) : x % y = x - y * (x / y) := rfl + + lemma norm_mod_lt (x : gaussint) {y : gaussint} (hy : y ≠ 0) : (x % y).norm < y.norm := + begin + have norm_y_pos : 0 < norm y, + by rwa [norm_pos], + have : (x % y) * conj y = + ⟨int.mod' (x * conj y).re (norm y), int.mod' (x * conj y).im (norm y)⟩, + { rw [mod_def, sub_mul, int.mod'_eq, int.mod'_eq, sub_eq_add_neg, norm, div_def], + ext; simp; ring }, + have : norm (x % y) * norm y ≤ (norm y / 2) * norm y, + { conv { to_lhs, rw [←norm_conj y, ←norm_mul, this, norm] }, + simp, + transitivity 2 * (y.norm / 2)^2, + { rw [two_mul], + apply add_le_add; + { rw [sq_le_sq], + apply le_trans (int.abs_mod'_le _ _ norm_y_pos), + apply le_abs_self } }, + rw [pow_two, ←mul_assoc, mul_comm, mul_comm (2 : ℤ)], + apply mul_le_mul_of_nonneg_left _ _, + { apply int.div_mul_le, norm_num }, + apply int.div_nonneg (norm_nonneg y), norm_num }, + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos, + apply lt_of_le_of_lt this, + apply int.div_lt_of_lt_mul, { norm_num }, + linarith + end + + lemma coe_nat_abs_norm (x : gaussint) : (x.norm.nat_abs : ℤ) = x.norm := + int.nat_abs_of_nonneg (norm_nonneg _) + + lemma nat_abs_norm_mod_lt (x y : gaussint) (hy : y ≠ 0) : + (x % y).norm.nat_abs < y.norm.nat_abs := + begin + apply int.coe_nat_lt.1, simp, + exact int.nat_abs_lt_nat_abs_of_nonneg_of_lt (norm_nonneg _) (norm_mod_lt x hy) + end + + lemma not_norm_mul_left_lt_norm (x : gaussint) {y : gaussint} (hy : y ≠ 0) : + ¬ (norm (x * y)).nat_abs < (norm x).nat_abs := + begin + apply not_lt_of_ge, + rw [norm_mul, int.nat_abs_mul], + apply le_mul_of_one_le_right (nat.zero_le _), + apply int.coe_nat_le.1, + rw [coe_nat_abs_norm], + exact int.add_one_le_of_lt ((norm_pos _).mpr hy) + end + + instance : euclidean_domain gaussint := + { quotient := (/), + remainder := (%), + quotient_mul_add_remainder_eq := + λ x y, by {rw [mod_def, add_comm, sub_add_cancel] }, + quotient_zero := λ x, by { simp [div_def, norm, int.div'], refl }, + r := measure (int.nat_abs ∘ norm), + r_well_founded := measure_wf (int.nat_abs ∘ norm), + remainder_lt := nat_abs_norm_mod_lt, + mul_left_not_lt := not_norm_mul_left_lt_norm, + .. gaussint.comm_ring } + + example (x : gaussint) : irreducible x ↔ prime x := + principal_ideal_ring.irreducible_iff_prime + + end gaussint +
src/06_Abstract_Algebra/solutions/solutions_03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,206 @@
 + import data.int.basic + import ring_theory.principal_ideal_domain + import tactic + + @[ext] structure gaussint := (re : ℤ) (im : ℤ) + + namespace gaussint + + instance : has_zero gaussint := ⟨⟨0, 0⟩⟩ + instance : has_one gaussint := ⟨⟨1, 0⟩⟩ + instance : has_add gaussint := ⟨λ x y, ⟨x.re + y.re, x.im + y.im⟩⟩ + instance : has_neg gaussint := ⟨λ x, ⟨-x.re, -x.im⟩⟩ + instance : has_mul gaussint := + ⟨λ x y, ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussint) = ⟨0, 0⟩ := rfl + theorem one_def : (1 : gaussint) = ⟨1, 0⟩ := rfl + theorem add_def (x y : gaussint) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl + theorem neg_def (x : gaussint) : -x = ⟨-x.re, -x.im⟩ := rfl + theorem mul_def (x y : gaussint) : + x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl + + @[simp] theorem zero_re : (0 : gaussint).re = 0 := rfl + @[simp] theorem zero_im : (0 : gaussint).im = 0 := rfl + @[simp] theorem one_re : (1 : gaussint).re = 1 := rfl + @[simp] theorem one_im : (1 : gaussint).im = 0 := rfl + @[simp] theorem add_re (x y : gaussint) : (x + y).re = x.re + y.re := rfl + @[simp] theorem add_im (x y : gaussint) : (x + y).im = x.im + y.im := rfl + @[simp] theorem neg_re (x : gaussint) : (-x).re = - x.re := rfl + @[simp] theorem neg_im (x : gaussint) : (-x).im = - x.im := rfl + @[simp] theorem mul_re (x y : gaussint) : (x * y).re = x.re * y.re - x.im * y.im := rfl + @[simp] theorem mul_im (x y : gaussint) : (x * y).im = x.re * y.im + x.im * y.re := rfl + + instance : comm_ring gaussint := + { zero := 0, + one := 1, + add := (+), + neg := λ x, -x, + mul := (*), + + add_assoc := by { intros, ext; simp; ring }, + zero_add := by { intros, ext; simp }, + add_zero := by { intros, ext; simp }, + add_left_neg := by { intros, ext; simp }, + add_comm := by { intros, ext; simp; ring }, + mul_assoc := by { intros, ext; simp; ring }, + one_mul := by { intros, ext; simp }, + mul_one := by { intros, ext; simp }, + left_distrib := by { intros, ext; simp; ring }, + right_distrib := by { intros, ext; simp; ring }, + mul_comm := by { intros, ext; simp; ring } } + + instance : nontrivial gaussint := + by { use [0, 1], rw [ne, gaussint.ext_iff], simp } + + end gaussint + + namespace int + + def div' (a b : ℤ) := (a + b / 2) / b + + def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := + begin + rw [div', mod'], + linarith [int.div_add_mod (a + b / 2) b], + end + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b): abs (mod' a b) ≤ b / 2 := + begin + rw [mod', abs_le], + split, + { linarith [int.mod_nonneg (a + b / 2) h.ne'] }, + have := int.mod_lt_of_pos (a + b / 2) h, + have := int.div_add_mod b 2, + have := int.mod_lt_of_pos b zero_lt_two, + linarith + end + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := + by linarith [div'_add_mod' a b] + + end int + + private theorem aux {α : Type*} [linear_ordered_ring α] {x y : α} (h : x^2 + y^2 = 0) : x = 0 := + begin + have h' : x^2 = 0, + { apply le_antisymm _ (sq_nonneg x), + rw ←h, + apply le_add_of_nonneg_right (sq_nonneg y) }, + exact pow_eq_zero h' + end + + theorem sq_add_sq_eq_zero {α : Type*} [linear_ordered_ring α] (x y : α) : + x^2 + y^2 = 0 ↔ x = 0 ∧ y = 0 := + begin + split, + { intro h, + split, + { exact aux h }, + rw add_comm at h, + exact aux h }, + rintros ⟨rfl, rfl⟩, + norm_num + end + + namespace gaussint + + def norm (x : gaussint) := x.re^2 + x.im^2 + + @[simp] theorem norm_nonneg (x : gaussint) : 0 ≤ norm x := + by { apply add_nonneg; apply sq_nonneg } + + theorem norm_eq_zero (x : gaussint) : norm x = 0 ↔ x = 0 := + by { rw [norm, sq_add_sq_eq_zero, gaussint.ext_iff], refl } + + theorem norm_pos (x : gaussint) : 0 < norm x ↔ x ≠ 0 := + by { rw [lt_iff_le_and_ne, ne_comm, ne, norm_eq_zero], simp [norm_nonneg] } + + theorem norm_mul (x y : gaussint) : norm (x * y) = norm x * norm y := + by { simp [norm], ring } + + def conj (x : gaussint) : gaussint := ⟨x.re, -x.im⟩ + + @[simp] theorem conj_re (x : gaussint) : (conj x).re = x.re := rfl + @[simp] theorem conj_im (x : gaussint) : (conj x).im = - x.im := rfl + + theorem norm_conj (x : gaussint) : norm (conj x) = norm x := + by { simp [norm] } + + instance : has_div gaussint := + ⟨λ x y, ⟨int.div' (x * conj y).re (norm y), int.div' (x * conj y).im (norm y)⟩⟩ + + instance : has_mod gaussint := ⟨λ x y, x - y * (x / y)⟩ + + theorem div_def (x y : gaussint) : + x / y = ⟨int.div' (x * conj y).re (norm y), int.div' (x * conj y).im (norm y)⟩ := rfl + + theorem mod_def (x y : gaussint) : x % y = x - y * (x / y) := rfl + + lemma norm_mod_lt (x : gaussint) {y : gaussint} (hy : y ≠ 0) : (x % y).norm < y.norm := + begin + have norm_y_pos : 0 < norm y, + by rwa [norm_pos], + have : (x % y) * conj y = + ⟨int.mod' (x * conj y).re (norm y), int.mod' (x * conj y).im (norm y)⟩, + { rw [mod_def, sub_mul, int.mod'_eq, int.mod'_eq, sub_eq_add_neg, norm, div_def], + ext; simp; ring }, + have : norm (x % y) * norm y ≤ (norm y / 2) * norm y, + { conv { to_lhs, rw [←norm_conj y, ←norm_mul, this, norm] }, + simp, + transitivity 2 * (y.norm / 2)^2, + { rw [two_mul], + apply add_le_add; + { rw [sq_le_sq], + apply le_trans (int.abs_mod'_le _ _ norm_y_pos), + apply le_abs_self } }, + rw [pow_two, ←mul_assoc, mul_comm, mul_comm (2 : ℤ)], + apply mul_le_mul_of_nonneg_left _ _, + { apply int.div_mul_le, norm_num }, + apply int.div_nonneg (norm_nonneg y), norm_num }, + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos, + apply lt_of_le_of_lt this, + apply int.div_lt_of_lt_mul, { norm_num }, + linarith + end + + lemma coe_nat_abs_norm (x : gaussint) : (x.norm.nat_abs : ℤ) = x.norm := + int.nat_abs_of_nonneg (norm_nonneg _) + + lemma nat_abs_norm_mod_lt (x y : gaussint) (hy : y ≠ 0) : + (x % y).norm.nat_abs < y.norm.nat_abs := + begin + apply int.coe_nat_lt.1, simp, + exact int.nat_abs_lt_nat_abs_of_nonneg_of_lt (norm_nonneg _) (norm_mod_lt x hy) + end + + lemma not_norm_mul_left_lt_norm (x : gaussint) {y : gaussint} (hy : y ≠ 0) : + ¬ (norm (x * y)).nat_abs < (norm x).nat_abs := + begin + apply not_lt_of_ge, + rw [norm_mul, int.nat_abs_mul], + apply le_mul_of_one_le_right (nat.zero_le _), + apply int.coe_nat_le.1, + rw [coe_nat_abs_norm], + exact int.add_one_le_of_lt ((norm_pos _).mpr hy) + end + + instance : euclidean_domain gaussint := + { quotient := (/), + remainder := (%), + quotient_mul_add_remainder_eq := + λ x y, by {rw [mod_def, add_comm, sub_add_cancel] }, + quotient_zero := λ x, by { simp [div_def, norm, int.div'], refl }, + r := measure (int.nat_abs ∘ norm), + r_well_founded := measure_wf (int.nat_abs ∘ norm), + remainder_lt := nat_abs_norm_mod_lt, + mul_left_not_lt := not_norm_mul_left_lt_norm, + .. gaussint.comm_ring } + + example (x : gaussint) : irreducible x ↔ prime x := + principal_ideal_ring.irreducible_iff_prime + + end gaussint +
src/07_Topology/02_Metric_Spaces.lean
- @@ -164,36 +164,6 @@ example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space
 sorry - -- SOLUTIONs: - example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] - {f : X → Y} (hf : continuous f) : uniform_continuous f := - begin - rw metric.uniform_continuous_iff, - intros ε ε_pos, - let φ : X × X → ℝ := λ p, dist (f p.1) (f p.2), - have φ_cont : continuous φ := hf.fst'.dist hf.snd', - let K := { p : X × X | ε ≤ φ p }, - have K_closed : is_closed K := is_closed_le continuous_const φ_cont, - have K_cpct : is_compact K := K_closed.is_compact, - cases eq_empty_or_nonempty K with hK hK, - { use [1, by norm_num], - intros x y hxy, - have : (x, y) ∉ K, by simp [hK], - simpa [K] }, - { rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩, - use dist x₀ x₁, - split, - { change _ < _, - rw dist_pos, - intro h, - have : ε ≤ 0, by simpa [*] using xx_in, - linarith }, - { intros x x', - contrapose!, - intros hxx', - exact H (x, x') hxx' } }, - end - example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε :=
src/07_Topology/03_Topological_Spaces.lean (new)
- @@ -0,0 +1,200 @@
  + import topology.instances.real + import analysis.normed_space.banach_steinhaus + + open set filter + open_locale topological_space filter + + + section + + variables {X : Type*} [topological_space X] + + example : is_open (univ : set X) := is_open_univ + + example : is_open (∅ : set X) := is_open_empty + + example {ι : Type*} {s : ι → set X} (hs : ∀ i, is_open $ s i) : + is_open (⋃ i, s i) := + is_open_Union hs + + example {ι : Type*} [fintype ι] {s : ι → set X} (hs : ∀ i, is_open $ s i) : + is_open (⋂ i, s i) := + is_open_Inter hs + + + + + variables {Y : Type*} [topological_space Y] + + example {f : X → Y} : continuous f ↔ ∀ s, is_open s → is_open (f ⁻¹' s) := + continuous_def + + + + + example {f : X → Y} {x : X} : continuous_at f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + iff.rfl + + + + example {f : X → Y} {x : X} : continuous_at f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + iff.rfl + + + example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ t ⊆ s, is_open t ∧ x ∈ t := + mem_nhds_iff + + + example (x : X) : pure x ≤ 𝓝 x := pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + + + #check topological_space.mk_of_nhds + #check topological_space.nhds_mk_of_nhds. + + + example {α : Type*} (n : α → filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → (∀ᶠ y in n a, ∀ᶠ x in n y, p x)) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + + end + -- BOTH. + + variables {X Y : Type*} + + example (f : X → Y) : topological_space X → topological_space Y := + topological_space.coinduced f + + example (f : X → Y) : topological_space Y → topological_space X := + topological_space.induced f + + example (f : X → Y) (T_X : topological_space X) (T_Y : topological_space Y) : + topological_space.coinduced f T_X ≤ T_Y ↔ T_X ≤ topological_space.induced f T_Y := + coinduced_le_iff_le_induced + + + + #check coinduced_compose + #check induced_compose. + + + example {T T' : topological_space X} : + T ≤ T' ↔ ∀ s, T'.is_open s → T.is_open s := + iff.rfl + + + + example (T_X : topological_space X) (T_Y : topological_space Y) (f : X → Y) : + continuous f ↔ topological_space.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + + + + example {Z : Type*} (f : X → Y) + (T_X : topological_space X) (T_Z : topological_space Z) (g : Y → Z) : + @continuous Y Z (topological_space.coinduced f T_X) T_Z g ↔ @continuous X Z T_X T_Z (g ∘ f) := + by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + + + example (ι : Type*) (X : ι → Type*) (T_X : Π i, topological_space $ X i) : + (Pi.topological_space : topological_space (Π i, X i)) = ⨅ i, topological_space.induced (λ x, x i) (T_X i) := + rfl + + + + + example [topological_space X] [t2_space X] {u : ℕ → X} {a b : X} + (ha : tendsto u at_top (𝓝 a)) (hb : tendsto u at_top (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [topological_space X] [regular_space X] (a : X) : + (𝓝 a).has_basis (λ (s : set X), s ∈ 𝓝 a ∧ is_closed s) id := + closed_nhds_basis a + + + example [topological_space X] {x : X} : (𝓝 x).has_basis (λ t : set X, t ∈ 𝓝 x ∧ is_open t) id := + nhds_basis_opens' x + + + lemma aux {X Y A : Type*} [topological_space X] {c : A → X} {f : A → Y} {x : X} {F : filter Y} + (h : tendsto f (comap c (𝓝 x)) F) {V' : set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, is_open V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + + + + example [topological_space X] [topological_space Y] [regular_space Y] + {A : set X} (hA : ∀ x, x ∈ closure A) + {f : A → Y} (f_cont : continuous f) + (hf : ∀ x : X, ∃ c : Y, tendsto f (comap coe $ 𝓝 x) $ 𝓝 c) : + ∃ φ : X → Y, continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + + + example [topological_space X] [topological_space.first_countable_topology X] {s : set X} {a : X} : + a ∈ closure s ↔ ∃ (u : ℕ → X), (∀ n, u n ∈ s) ∧ tendsto u at_top (𝓝 a) := + mem_closure_iff_seq_limit + + + + variables [topological_space X] + + example {F : filter X} {x : X} : cluster_pt x F ↔ ne_bot (𝓝 x ⊓ F) := + iff.rfl + + example {s : set X} : + is_compact s ↔ ∀ (F : filter X) [ne_bot F], F ≤ 𝓟 s → ∃ a ∈ s, cluster_pt a F := + iff.rfl + + + example [topological_space.first_countable_topology X] + {s : set X} {u : ℕ → X} (hs : is_compact s) (hu : ∀ n, u n ∈ s) : + ∃ (a ∈ s) (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (u ∘ φ) at_top (𝓝 a) := + hs.tendsto_subseq hu + + + + variables [topological_space Y] + + example {x : X} {F : filter X} {G : filter Y} (H : cluster_pt x F) + {f : X → Y} (hfx : continuous_at f x) (hf : tendsto f F G) : + cluster_pt (f x) G := + cluster_pt.map H hfx hf + + + example [topological_space Y] {f : X → Y} (hf : continuous f) + {s : set X} (hs : is_compact s) : is_compact (f '' s) := + begin + intros F F_ne F_le, + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F, + { sorry }, + haveI Hne : (𝓟 s ⊓ comap f F).ne_bot, + { sorry }, + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s, from inf_le_left, + sorry + end + + + example {ι : Type*} {s : set X} (hs : is_compact s) + (U : ι → set X) (hUo : ∀ i, is_open (U i)) (hsU : s ⊆ ⋃ i, U i) : + ∃ t : finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + + + example [compact_space X] : is_compact (univ : set X) := + compact_univ
src/07_Topology/solutions/solutions_02_Metric_Spaces.lean
- @@ -4,10 +4,120 @@ import analysis.normed_space.banach_steinhaus
 open set filter open_locale topological_space filter + + variables {X : Type*} [metric_space X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check emetric_space + #check pseudo_metric_space + #check pseudo_emetric_space + + + + example {u : ℕ → X} {a : X} : + tendsto u at_top (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + metric.tendsto_at_top + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} : + continuous f ↔ + ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + metric.continuous_iff + + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + by continuity + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + begin + apply continuous.dist, + exact hf.comp continuous_fst, + exact hf.comp continuous_snd + end + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + hf.fst'.dist hf.snd' + + + + example {f : ℝ → X} (hf : continuous f) : continuous (λ x : ℝ, f (x^2 + x)) := + sorry + + example {f : ℝ → X} (hf : continuous f) : continuous (λ x : ℝ, f (x^2 + x)) := hf.comp $ (continuous_pow 2).add continuous_id + + + example {X Y : Type*} [metric_space X] [metric_space Y] (f : X → Y) (a : X) : + continuous_at f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + metric.continuous_at_iff + + + + variables r : ℝ + + example : metric.ball a r = {b | dist b a < r} := rfl + + example : metric.closed_ball a r = {b | dist b a ≤ r} := rfl + + + + example (hr : 0 < r) : a ∈ metric.ball a r := metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ metric.closed_ball a r := metric.mem_closed_ball_self hr + + + example (s : set X) : is_open s ↔ ∀ x ∈ s, ∃ ε > 0, metric.ball x ε ⊆ s := + metric.is_open_iff + + + + example {s : set X} : is_closed s ↔ is_open sᶜ := + is_open_compl_iff.symm + + example {s : set X} (hs : is_closed s) {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ metric.ball b ε := + metric.mem_closure_iff + + + example {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) {s : set X} (hs : ∀ n, u n ∈ s) : + a ∈ closure s := + sorry + + example {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) {s : set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := begin
- @@ -21,6 +131,121 @@ begin
 end + + example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, metric.ball x ε ⊆ s := + metric.nhds_basis_ball.mem_iff + + example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, metric.closed_ball x ε ⊆ s := + metric.nhds_basis_closed_ball.mem_iff + + + + + example : is_compact (set.Icc 0 1 : set ℝ) := + is_compact_Icc + + example {s : set X} (hs : is_compact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, strict_mono φ ∧ tendsto (u ∘ φ) at_top (𝓝 a) := + hs.tendsto_subseq hu + + example {s : set X} (hs : is_compact s) (hs' : s.nonempty) + {f : X → ℝ} (hfs : continuous_on f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : set X} (hs : is_compact s) (hs' : s.nonempty) + {f : X → ℝ} (hfs : continuous_on f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : set X} (hs : is_compact s) : is_closed s := + hs.is_closed + + + + example {X : Type*} [metric_space X] [compact_space X] : is_compact (univ : set X) := + compact_univ + + + #check is_compact.is_closed + + + example {X : Type*} [metric_space X] {Y : Type*} [metric_space Y] {f : X → Y} : + uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + metric.uniform_continuous_iff + + + + example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] + {f : X → Y} (hf : continuous f) : uniform_continuous f := + sorry + + + example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] + {f : X → Y} (hf : continuous f) : uniform_continuous f := + begin + rw metric.uniform_continuous_iff, + intros ε ε_pos, + let φ : X × X → ℝ := λ p, dist (f p.1) (f p.2), + have φ_cont : continuous φ := hf.fst'.dist hf.snd', + let K := { p : X × X | ε ≤ φ p }, + have K_closed : is_closed K := is_closed_le continuous_const φ_cont, + have K_cpct : is_compact K := K_closed.is_compact, + cases eq_empty_or_nonempty K with hK hK, + { use [1, by norm_num], + intros x y hxy, + have : (x, y) ∉ K, by simp [hK], + simpa [K] }, + { rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩, + use dist x₀ x₁, + split, + { change _ < _, + rw dist_pos, + intro h, + have : ε ≤ 0, by simpa [*] using xx_in, + linarith }, + { intros x x', + contrapose!, + intros hxx', + exact H (x, x') hxx' } }, + end + + + + example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + metric.cauchy_seq_iff + + example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + metric.cauchy_seq_iff' + + + example [complete_space X] (u : ℕ → X) (hu : cauchy_seq u) : ∃ x, tendsto u at_top (𝓝 x) := + cauchy_seq_tendsto_of_complete hu + + + + open_locale big_operators + open finset + + + lemma cauchy_seq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : + cauchy_seq u := + begin + rw metric.cauchy_seq_iff', + intros ε ε_pos, + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε, + { sorry }, + use N, + intros n hn, + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn, + calc dist (u (N + k)) (u N) = dist (u (N+0)) (u (N + k)) : sorry + ... ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) : sorry + ... ≤ ∑ i in range k, (1/2 : ℝ)^(N+i) : sorry + ... = 1/2^N*∑ i in range k, (1 / 2) ^ i : sorry + ... ≤ 1/2^N*2 : sorry + ... < ε : sorry + end + example {u : ℕ → X} (hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : cauchy_seq u := begin
- @@ -46,6 +271,55 @@ begin
 ... < ε : hN end + + open metric + + example [complete_space X] (f : ℕ → set X) (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) : dense (⋂n, f n) := + begin + let B : ℕ → ℝ := λ n, (1/2)^n, + have Bpos : ∀ n, 0 < B n, sorry, + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : ∀ (n : ℕ) (x : X) (δ > 0), ∃ (y : X) (r > 0), r ≤ B (n+1) ∧ closed_ball y r ⊆ (closed_ball x δ) ∩ f n, + { sorry }, + choose! center radius Hpos HB Hball using this, + intros x, + rw mem_closure_iff_nhds_basis nhds_basis_closed_ball, + intros ε εpos, + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → (X × ℝ) := λn, nat.rec_on n (prod.mk x (min ε (B 0))) + (λn p, prod.mk (center n p.1 p.2) (radius n p.1 p.2)), + let c : ℕ → X := λ n, (F n).1, + let r : ℕ → ℝ := λ n, (F n).2, + have rpos : ∀ n, 0 < r n, + { sorry }, + + have rB : ∀n, r n ≤ B n, + { sorry }, + have incl : ∀n, closed_ball (c (n+1)) (r (n+1)) ⊆ (closed_ball (c n) (r n)) ∩ (f n), + { sorry }, + have cdist : ∀ n, dist (c n) (c (n+1)) ≤ B n, + { sorry }, + have : cauchy_seq c, from cauchy_seq_of_le_geometric_two' cdist, + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchy_seq_tendsto_of_complete this with ⟨y, ylim⟩, + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y, + have I : ∀n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n), + { sorry }, + have yball : ∀n, y ∈ closed_ball (c n) (r n), + { sorry }, + sorry + end + + example [complete_space X] (f : ℕ → set X) (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) : dense (⋂n, f n) := begin let B : ℕ → ℝ := λ n, (1/2)^n,
- @@ -133,3 +407,4 @@ begin
 ... ≤ ε : min_le_left _ _, end +
src/07_Topology/solutions/solutions_03_Topological_Spaces.lean (new)
- @@ -0,0 +1,263 @@
  + import topology.instances.real + import analysis.normed_space.banach_steinhaus + + open set filter + open_locale topological_space filter + + + section + + variables {X : Type*} [topological_space X] + + example : is_open (univ : set X) := is_open_univ + + example : is_open (∅ : set X) := is_open_empty + + example {ι : Type*} {s : ι → set X} (hs : ∀ i, is_open $ s i) : + is_open (⋃ i, s i) := + is_open_Union hs + + example {ι : Type*} [fintype ι] {s : ι → set X} (hs : ∀ i, is_open $ s i) : + is_open (⋂ i, s i) := + is_open_Inter hs + + + + + variables {Y : Type*} [topological_space Y] + + example {f : X → Y} : continuous f ↔ ∀ s, is_open s → is_open (f ⁻¹' s) := + continuous_def + + + + + example {f : X → Y} {x : X} : continuous_at f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + iff.rfl + + + + example {f : X → Y} {x : X} : continuous_at f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + iff.rfl + + + example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ t ⊆ s, is_open t ∧ x ∈ t := + mem_nhds_iff + + + example (x : X) : pure x ≤ 𝓝 x := pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + + + #check topological_space.mk_of_nhds + #check topological_space.nhds_mk_of_nhds. + + + example {α : Type*} (n : α → filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → (∀ᶠ y in n a, ∀ᶠ x in n y, p x)) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + + example {α : Type*} (n : α → filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → (∀ᶠ y in n a, ∀ᶠ x in n y, p x)) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + begin + intros a s s_in, + refine ⟨{y | s ∈ n y}, H a (λ x, x ∈ s) s_in, _, by tauto⟩, + rintros y (hy : s ∈ n y), + exact H₀ y hy + end + end + -- BOTH. + + variables {X Y : Type*} + + example (f : X → Y) : topological_space X → topological_space Y := + topological_space.coinduced f + + example (f : X → Y) : topological_space Y → topological_space X := + topological_space.induced f + + example (f : X → Y) (T_X : topological_space X) (T_Y : topological_space Y) : + topological_space.coinduced f T_X ≤ T_Y ↔ T_X ≤ topological_space.induced f T_Y := + coinduced_le_iff_le_induced + + + + #check coinduced_compose + #check induced_compose. + + + example {T T' : topological_space X} : + T ≤ T' ↔ ∀ s, T'.is_open s → T.is_open s := + iff.rfl + + + + example (T_X : topological_space X) (T_Y : topological_space Y) (f : X → Y) : + continuous f ↔ topological_space.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + + + + example {Z : Type*} (f : X → Y) + (T_X : topological_space X) (T_Z : topological_space Z) (g : Y → Z) : + @continuous Y Z (topological_space.coinduced f T_X) T_Z g ↔ @continuous X Z T_X T_Z (g ∘ f) := + by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + + + example (ι : Type*) (X : ι → Type*) (T_X : Π i, topological_space $ X i) : + (Pi.topological_space : topological_space (Π i, X i)) = ⨅ i, topological_space.induced (λ x, x i) (T_X i) := + rfl + + + + + example [topological_space X] [t2_space X] {u : ℕ → X} {a b : X} + (ha : tendsto u at_top (𝓝 a)) (hb : tendsto u at_top (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [topological_space X] [regular_space X] (a : X) : + (𝓝 a).has_basis (λ (s : set X), s ∈ 𝓝 a ∧ is_closed s) id := + closed_nhds_basis a + + + example [topological_space X] {x : X} : (𝓝 x).has_basis (λ t : set X, t ∈ 𝓝 x ∧ is_open t) id := + nhds_basis_opens' x + + + lemma aux {X Y A : Type*} [topological_space X] {c : A → X} {f : A → Y} {x : X} {F : filter Y} + (h : tendsto f (comap c (𝓝 x)) F) {V' : set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, is_open V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + + example {X Y A : Type*} [topological_space X] {c : A → X} {f : A → Y} {x : X} {F : filter Y} + (h : tendsto f (comap c (𝓝 x)) F) {V' : set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, is_open V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + begin + simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in + end + + + + example [topological_space X] [topological_space Y] [regular_space Y] + {A : set X} (hA : ∀ x, x ∈ closure A) + {f : A → Y} (f_cont : continuous f) + (hf : ∀ x : X, ∃ c : Y, tendsto f (comap coe $ 𝓝 x) $ 𝓝 c) : + ∃ φ : X → Y, continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + + example [topological_space X] [topological_space Y] [regular_space Y] {A : set X} (hA : ∀ x, x ∈ closure A) + {f : A → Y} (f_cont : continuous f) + (hf : ∀ x : X, ∃ c : Y, tendsto f (comap coe $ 𝓝 x) $ 𝓝 c) : + ∃ φ : X → Y, continuous φ ∧ ∀ a : A, φ a = f a := + begin + choose φ hφ using hf, + use φ, + split, + { rw continuous_iff_continuous_at, + intros x, + suffices : ∀ V' ∈ 𝓝 (φ x), is_closed V' → φ ⁻¹' V' ∈ 𝓝 x, + by simpa [continuous_at, (closed_nhds_basis _).tendsto_right_iff], + intros V' V'_in V'_closed, + obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, is_open V ∧ coe ⁻¹' V ⊆ f ⁻¹' V', + { exact aux (hφ x) V'_in }, + suffices : ∀ y ∈ V, φ y ∈ V', + from mem_of_superset V_in this, + intros y y_in, + have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in, + haveI : (comap (coe : A → X) (𝓝 y)).ne_bot := by simpa [mem_closure_iff_comap_ne_bot] using hA y, + apply V'_closed.mem_of_tendsto (hφ y), + exact mem_of_superset (preimage_mem_comap hVx) hV }, + { intros a, + have lim : tendsto f (𝓝 a) (𝓝 $ φ a), + by simpa [nhds_induced] using hφ a, + exact tendsto_nhds_unique lim f_cont.continuous_at }, + end + + + + + example [topological_space X] [topological_space.first_countable_topology X] {s : set X} {a : X} : + a ∈ closure s ↔ ∃ (u : ℕ → X), (∀ n, u n ∈ s) ∧ tendsto u at_top (𝓝 a) := + mem_closure_iff_seq_limit + + + + variables [topological_space X] + + example {F : filter X} {x : X} : cluster_pt x F ↔ ne_bot (𝓝 x ⊓ F) := + iff.rfl + + example {s : set X} : + is_compact s ↔ ∀ (F : filter X) [ne_bot F], F ≤ 𝓟 s → ∃ a ∈ s, cluster_pt a F := + iff.rfl + + + example [topological_space.first_countable_topology X] + {s : set X} {u : ℕ → X} (hs : is_compact s) (hu : ∀ n, u n ∈ s) : + ∃ (a ∈ s) (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (u ∘ φ) at_top (𝓝 a) := + hs.tendsto_subseq hu + + + + variables [topological_space Y] + + example {x : X} {F : filter X} {G : filter Y} (H : cluster_pt x F) + {f : X → Y} (hfx : continuous_at f x) (hf : tendsto f F G) : + cluster_pt (f x) G := + cluster_pt.map H hfx hf + + + example [topological_space Y] {f : X → Y} (hf : continuous f) + {s : set X} (hs : is_compact s) : is_compact (f '' s) := + begin + intros F F_ne F_le, + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F, + { sorry }, + haveI Hne : (𝓟 s ⊓ comap f F).ne_bot, + { sorry }, + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s, from inf_le_left, + sorry + end + + example [topological_space Y] {f : X → Y} (hf : continuous f) + {s : set X} (hs : is_compact s) : is_compact (f '' s) := + begin + intros F F_ne F_le, + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F, + { rw [filter.push_pull, map_principal] }, + haveI Hne : (𝓟 s ⊓ comap f F).ne_bot, + { apply ne_bot.of_map, + rwa [map_eq, inf_of_le_right F_le] }, + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s, from inf_le_left, + rcases hs Hle with ⟨x, x_in, hx⟩, + refine ⟨f x, mem_image_of_mem f x_in, _⟩, + apply hx.map hf.continuous_at, + rw [tendsto, map_eq], + exact inf_le_right + end + + + example {ι : Type*} {s : set X} (hs : is_compact s) + (U : ι → set X) (hUo : ∀ i, is_open (U i)) (hsU : s ⊆ ⋃ i, U i) : + ∃ t : finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + + + example [compact_space X] : is_compact (univ : set X) := + compact_univ
src/08_Differential_Calculus/01_Elementary_Differential_Calculus.lean (new)
- @@ -0,0 +1,61 @@
 + import analysis.special_functions.trigonometric.deriv + import analysis.calculus.mean_value + + open set filter + open_locale topological_space filter classical real + + noncomputable theory + + open real + + /-- The sin function has derivative 1 at 0. -/ + example : has_deriv_at sin 1 0 := + by simpa using has_deriv_at_sin 0 + + example (x : ℝ) : differentiable_at ℝ sin x := + (has_deriv_at_sin x).differentiable_at + + example {f : ℝ → ℝ} {x a : ℝ} (h : has_deriv_at f a x) : deriv f x = a := + h.deriv + + example {f : ℝ → ℝ} {x : ℝ} (h : ¬ differentiable_at ℝ f x) : deriv f x = 0 := + deriv_zero_of_not_differentiable_at h + + example {f g : ℝ → ℝ} {x : ℝ} (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) : + deriv (f + g) x = deriv f x + deriv g x := + deriv_add hf hg + + example {f : ℝ → ℝ} {a : ℝ} (h : is_local_min f a) : deriv f a = 0 := + h.deriv_eq_zero + + example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) + (hfc : continuous_on 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 + + example (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hf : continuous_on f (Icc a b)) + (hf' : differentiable_on ℝ f (Ioo a b)) : + ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) := + exists_deriv_eq_slope f hab hf hf' + + example : deriv (λ x : ℝ, x^5) 6 = 5 * 6^4 := by simp + + example (x₀ : ℝ) (h₀ : x₀ ≠ 0) : deriv (λ x : ℝ, 1 / x) x₀ = -(x₀^2)⁻¹ := by simp + + example : deriv sin π = -1 := by simp + + example (x₀ : ℝ) (h : x₀ ≠ 0) : + deriv (λ x : ℝ, exp(x^2) / x^5) x₀ = (2 * x₀^2 - 5) * exp (x₀^2) / x₀^6 := + begin + have : x₀^5 ≠ 0, { exact pow_ne_zero 5 h, }, + field_simp, + ring, + end + + example (y : ℝ) : has_deriv_at (λ x : ℝ, 2 * x + 5) 2 y := + begin + have := ((has_deriv_at_id y).const_mul 2).add_const 5, + rwa [mul_one] at this, + end + + example (y : ℝ) : deriv (λ x : ℝ, 2 * x + 5) y = 2 := by simp
src/08_Differential_Calculus/02_Differential_Calculus_in_Normed_Spaces.lean (new)
- @@ -0,0 +1,193 @@
 + import analysis.normed_space.banach_steinhaus + import analysis.normed_space.finite_dimension + + import analysis.calculus.inverse + + open set filter + open_locale topological_space filter + + noncomputable theory + + section + variables {E : Type*} [normed_group E] + + example (x : E) : 0 ≤ ∥x∥ := + norm_nonneg x + + example {x : E} : ∥x∥ = 0 ↔ x = 0 := + norm_eq_zero + + example (x y : E) : ∥x + y∥ ≤ ∥x∥ + ∥y∥ := + norm_add_le x y + + example : metric_space E := by apply_instance + + example {X : Type*} [topological_space X] {f : X → E} (hf : continuous f) : + continuous (λ x, ∥f x∥) := + hf.norm + + variables [normed_space ℝ E] + + example (a : ℝ) (x : E) : ∥a • x∥ = |a| * ∥x∥ := + norm_smul a x + + example [finite_dimensional ℝ E] : complete_space E := + by apply_instance + + example (𝕜 : Type*) [nondiscrete_normed_field 𝕜] (x y : 𝕜) : ∥x * y∥ = ∥x∥ * ∥y∥ := + norm_mul x y + + example (𝕜 : Type*) [nondiscrete_normed_field 𝕜] : ∃ x : 𝕜, 1 < ∥x∥ := + normed_field.exists_one_lt_norm 𝕜 + + example (𝕜 : Type*) [nondiscrete_normed_field 𝕜] (E : Type*) [normed_group E] + [normed_space 𝕜 E] [complete_space 𝕜] [finite_dimensional 𝕜 E] : complete_space E := + finite_dimensional.complete 𝕜 E + + end + + section + variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] + {E : Type*} [normed_group E] [normed_space 𝕜 E] + {F : Type*} [normed_group F] [normed_space 𝕜 F] + + example : E →L[𝕜] E := continuous_linear_map.id 𝕜 E + + example (f : E →L[𝕜] F) : E → F := + f + + example (f : E →L[𝕜] F) : continuous f := + f.cont + + example (f : E →L[𝕜] F) (x y : E) : f (x + y) = f x + f y := + f.map_add x y + + example (f : E →L[𝕜] F) (a : 𝕜) (x : E) : f (a • x) = a • f x := + f.map_smul a x + + variables (f : E →L[𝕜] F) + + example (x : E) : ∥f x∥ ≤ ∥f∥ * ∥x∥ := + f.le_op_norm x + + example {M : ℝ} (hMp: 0 ≤ M) (hM : ∀ x, ∥f x∥ ≤ M * ∥x∥) : + ∥f∥ ≤ M := + f.op_norm_le_bound hMp hM + + end + + section + variables + {𝕜 : Type*} [nondiscrete_normed_field 𝕜] + {E : Type*} [normed_group E] [normed_space 𝕜 E] + {F : Type*} [normed_group F] [normed_space 𝕜 F] + + open metric + + example {ι : Type*} [complete_space E] {g : ι → E →L[𝕜] F} + (h : ∀ x, ∃ C, ∀ i, ∥g i x∥ ≤ C) : + ∃ C', ∀ i, ∥g i∥ ≤ C' := + begin + /- sequence of subsets consisting of those `x : E` with norms `∥g i x∥` bounded by `n` -/ + let e : ℕ → set E := λ n, ⋂ i : ι, { x : E | ∥g i x∥ ≤ n }, + /- each of these sets is closed -/ + have hc : ∀ n : ℕ, is_closed (e n), + sorry, + /- the union is the entire space; this is where we use `h` -/ + have hU : (⋃ n : ℕ, e n) = univ, + sorry, + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry, + obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry, + obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ∥k∥ := sorry, + /- show all elements in the ball have norm bounded by `m` after applying any `g i` -/ + 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∥), + λ i, continuous_linear_map.op_norm_le_of_shell ε_pos _ hk _⟩, + sorry, + sorry + end + + end + + open asymptotics + open_locale asymptotics + + example {α : Type*} {E : Type*} [normed_group E] {F : Type*} [normed_group F] + (c : ℝ) (l : filter α) (f : α → E) (g : α → F) : + is_O_with c l f g ↔ ∀ᶠ x in l, ∥ f x ∥ ≤ c * ∥ g x ∥ := + is_O_with_iff + + example {α : Type*} {E : Type*} [normed_group E] {F : Type*} [normed_group F] + (c : ℝ) (l : filter α) (f : α → E) (g : α → F) : + f =O[l] g ↔ ∃ C, is_O_with C l f g := + is_O_iff_is_O_with + + example {α : Type*} {E : Type*} [normed_group E] {F : Type*} [normed_group F] + (c : ℝ) (l : filter α) (f : α → E) (g : α → F) : + f =o[l] g ↔ ∀ C > 0, is_O_with C l f g := + is_o_iff_forall_is_O_with + + example {α : Type*} {E : Type*} [normed_group E] (c : ℝ) (l : filter α) (f g : α → E) : + f ~[l] g ↔ (f - g) =o[l] g := + iff.rfl + + section + variables + {𝕜 : Type*} [nondiscrete_normed_field 𝕜] + {E : Type*} [normed_group E] [normed_space 𝕜 E] + {F : Type*} [normed_group F] [normed_space 𝕜 F] + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : + has_fderiv_at f f' x₀ ↔ (λ x, f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] (λ x, x - x₀) := + iff.rfl + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : has_fderiv_at f f' x₀) : + fderiv 𝕜 f x₀ = f' := + hff'.fderiv + + example (n : ℕ) (f : E → F) : E → (E [×n]→L[𝕜] F) := + iterated_fderiv 𝕜 n f + + example (n : with_top ℕ) {f : E → F} : + cont_diff 𝕜 n f ↔ + (∀ (m : ℕ), (m : with_top ℕ) ≤ n → continuous (λ x, iterated_fderiv 𝕜 m f x)) + ∧ (∀ (m : ℕ), (m : with_top ℕ) < n → differentiable 𝕜 (λ x, iterated_fderiv 𝕜 m f x)) := + cont_diff_iff_continuous_differentiable + + example {𝕂 : Type*} [is_R_or_C 𝕂] {E : Type*} [normed_group E] [normed_space 𝕂 E] + {F : Type*} [normed_group F] [normed_space 𝕂 F] + {f : E → F} {x : E} {n : with_top ℕ} + (hf : cont_diff_at 𝕂 n f x) (hn : 1 ≤ n) : + has_strict_fderiv_at f (fderiv 𝕂 f x) x := + hf.has_strict_fderiv_at hn + + section local_inverse + variables [complete_space E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + + example (hf : has_strict_fderiv_at f ↑f' a) : F → E := + has_strict_fderiv_at.local_inverse f f' a hf + + example (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 a, hf.local_inverse f f' a (f x) = x := + hf.eventually_left_inverse + + example (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 (f a), f (hf.local_inverse f f' a x) = x := + hf.eventually_right_inverse + + example [complete_space E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + (hf : has_strict_fderiv_at f ↑f' a) : + has_strict_fderiv_at (has_strict_fderiv_at.local_inverse f f' a hf) + (f'.symm : F →L[𝕜] E) (f a) := + has_strict_fderiv_at.to_local_inverse hf + + end local_inverse + + #check has_fderiv_within_at + #check has_fderiv_at_filter + + end
src/08_Differential_Calculus/solutions/solutions_01_Elementary_Differential_Calculus.lean (new)
- @@ -0,0 +1,8 @@
  + import analysis.special_functions.trigonometric.deriv + import analysis.calculus.mean_value + + open set filter + open_locale topological_space filter classical real + + noncomputable theory +
src/08_Differential_Calculus/solutions/solutions_02_Differential_Calculus_in_Normed_Spaces.lean (new)
- @@ -0,0 +1,61 @@
 + import analysis.normed_space.banach_steinhaus + import analysis.normed_space.finite_dimension + + import analysis.calculus.inverse + + open set filter + open_locale topological_space filter + + noncomputable theory + + section + variables + {𝕜 : Type*} [nondiscrete_normed_field 𝕜] + {E : Type*} [normed_group E] [normed_space 𝕜 E] + {F : Type*} [normed_group F] [normed_space 𝕜 F] + + open metric + + example {ι : Type*} [complete_space E] {g : ι → E →L[𝕜] F} + (h : ∀ x, ∃ C, ∀ i, ∥g i x∥ ≤ C) : + ∃ C', ∀ i, ∥g i∥ ≤ C' := + begin + /- sequence of subsets consisting of those `x : E` with norms `∥g i x∥` bounded by `n` -/ + let e : ℕ → set E := λ n, ⋂ i : ι, { x : E | ∥g i x∥ ≤ n }, + /- each of these sets is closed -/ + have hc : ∀ n : ℕ, is_closed (e n), + from λ i, is_closed_Inter (λ i, is_closed_le (g i).cont.norm continuous_const), + /- the union is the entire space; this is where we use `h` -/ + have hU : (⋃ n : ℕ, e n) = univ, + { refine eq_univ_of_forall (λ x, _), + cases h x with C hC, + obtain ⟨m, hm⟩ := exists_nat_ge C, + exact ⟨e m, mem_range_self m, mem_Inter.mpr (λ i, le_trans (hC i) hm)⟩ }, + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_Union_of_closed hc hU, + obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := is_open_iff.mp is_open_interior x hx, + obtain ⟨k : 𝕜, hk : 1 < ∥k∥⟩ := normed_field.exists_one_lt_norm 𝕜, + /- show all elements in the ball have norm bounded by `m` after applying any `g i` -/ + have real_norm_le : ∀ (z ∈ ball x ε) (i : ι), ∥g i z∥ ≤ m, + { intros z hz i, + replace hz := mem_Inter.mp (interior_Inter_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∥), λ i, continuous_linear_map.op_norm_le_of_shell ε_pos _ hk _⟩, + { exact div_nonneg (nat.cast_nonneg _) εk_pos.le }, + intros 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 + 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) + (real_norm_le x (mem_ball_self ε_pos) i) + ... = (m + m : ℕ) : by norm_cast + ... ≤ (m + m : ℕ) * (∥y∥ / (ε / ∥k∥)) + : le_mul_of_one_le_right (nat.cast_nonneg _) + ((one_le_div $ div_pos ε_pos (zero_lt_one.trans hk)).2 le_y) + ... = (m + m : ℕ) / (ε / ∥k∥) * ∥y∥ : (mul_comm_div _ _ _).symm, + end + + end +
src/09_Integration_and_Measure_Theory/01_Elementary_Integration.lean (new)
- @@ -0,0 +1,35 @@
  + import measure_theory.integral.interval_integral + import analysis.special_functions.integrals + import analysis.convolution + + open set filter + open_locale topological_space filter + + noncomputable theory + + open measure_theory interval_integral + open_locale interval -- this introduces the notation [a, b] + + example (a b : ℝ): ∫ x in a..b, x = (b ^ 2 - a ^ 2) / 2 := + integral_id + + example {a b : ℝ} (h : (0:ℝ) ∉ [a, b]) : ∫ x in a..b, 1/x = real.log (b / a) := + integral_one_div h + + example (f : ℝ → ℝ) (hf : continuous f) (a b : ℝ) : + deriv (λ u, ∫ (x : ℝ) in a..u, f x) b = f b := + (integral_has_strict_deriv_at_right + (hf.interval_integrable _ _) (hf.strongly_measurable_at_filter _ _) + hf.continuous_at).has_deriv_at.deriv + + example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} + (h : ∀ x ∈ [a, b], has_deriv_at f (f' x) x) (h' : interval_integrable f' volume a b) : + ∫ y in a..b, f' y = f b - f a := + integral_eq_sub_of_has_deriv_at h h' + + + open_locale convolution + + example (f : ℝ → ℝ) (g : ℝ → ℝ) : + f ⋆ g = λ x, ∫ t, (f t) * (g (x - t)) := + rfl
src/09_Integration_and_Measure_Theory/02_Measure_Theory.lean (new)
- @@ -0,0 +1,52 @@
  + import analysis.normed_space.finite_dimension + import analysis.convolution + import measure_theory.function.jacobian + import measure_theory.integral.bochner + import measure_theory.measure.lebesgue + + open set filter + open_locale topological_space filter ennreal + noncomputable theory + + variables {α : Type*} [measurable_space α] + + example : measurable_set (∅ : set α) := measurable_set.empty + + example : measurable_set (univ : set α) := measurable_set.univ + + example {s : set α} (hs : measurable_set s) : measurable_set sᶜ := + hs.compl + + example : encodable ℕ := + by apply_instance + + example (n : ℕ) : encodable (fin n) := + by apply_instance + + variables {ι : Type*} [encodable ι] + + example {f : ι → set α} (h : ∀ b, measurable_set (f b)) : + measurable_set (⋃ b, f b) := + measurable_set.Union h + + example {f : ι → set α} (h : ∀ b, measurable_set (f b)) : + measurable_set (⋂ b, f b) := + measurable_set.Inter h + + open measure_theory + + variables {μ : measure α} + + example (s : set α) : μ s = ⨅ t (st : s ⊆ t) (ht : measurable_set t), μ t := + measure_eq_infi s + + example (s : ι → set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) := + measure_Union_le s + + example {f : ℕ → set α} + (hmeas : ∀ i, measurable_set (f i)) (hdis : pairwise (disjoint on f)) : + μ (⋃ i, f i) = ∑' i, μ (f i) := + μ.m_Union hmeas hdis + + example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in μ.ae, P x := + iff.rfl
src/09_Integration_and_Measure_Theory/03_Integration.lean (new)
- @@ -0,0 +1,68 @@
  + import analysis.normed_space.finite_dimension + import analysis.convolution + import measure_theory.function.jacobian + import measure_theory.integral.bochner + import measure_theory.measure.lebesgue + + open set filter + open_locale topological_space filter ennreal + open measure_theory + + noncomputable theory + + variables {α : Type*} [measurable_space α] + variables {μ : measure α} + + section + + variables {E : Type*} [normed_group E] [normed_space ℝ E] [complete_space E] + {f : α → E} + + example {f g : α → E} (hf : integrable f μ) (hg : integrable g μ) : + ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ := + integral_add hf hg + + example {s : set α} (c : E) : + ∫ x in s, c ∂μ = (μ s).to_real • c := + set_integral_const c + + example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) + (hmeas : ∀ n, ae_strongly_measurable (F n) μ) + (hint : integrable bound μ) + (hbound : ∀ n, ∀ᵐ a ∂μ, ∥F n a∥ ≤ bound a) + (hlim : ∀ᵐ a ∂μ, tendsto (λ (n : ℕ), F n a) at_top (𝓝 (f a))) : + tendsto (λ n, ∫ a, F n a ∂μ) at_top (𝓝 (∫ a, f a ∂μ)) := + tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim + + example + {α : Type*} [measurable_space α] + {μ : measure α} [sigma_finite μ] + {β : Type*} [measurable_space β] {ν : measure β} [sigma_finite ν] + (f : α × β → E) (hf : integrable f (μ.prod ν)) : + ∫ z, f z ∂μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := + integral_prod f hf + + end + section + open_locale convolution + + variables {𝕜 : Type*} {G : Type*} {E : Type*} {E' : Type*} {F : Type*} [normed_group E] + [normed_group E'] [normed_group F] [nondiscrete_normed_field 𝕜] + [normed_space 𝕜 E] [normed_space 𝕜 E'] [normed_space 𝕜 F] + [measurable_space G] [normed_space ℝ F] [complete_space F] [has_sub G] + + example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : measure G) : + f ⋆[L, μ] g = λ x, ∫ t, L (f t) (g (x - t)) ∂μ := + rfl + + end + example {E : Type*} [normed_group E] [normed_space ℝ E] [finite_dimensional ℝ E] + [measurable_space E] [borel_space E] (μ : measure E) [μ.is_add_haar_measure] + {F : Type*}[normed_group F] [normed_space ℝ F] [complete_space F] + {s : set E} {f : E → E} {f' : E → (E →L[ℝ] E)} + (hs : measurable_set s) + (hf : ∀ (x : E), x ∈ s → has_fderiv_within_at f (f' x) s x) + (h_inj : inj_on f s) + (g : E → F) : + ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ := + integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g
src/09_Integration_and_Measure_Theory/solutions/solutions_01_Elementary_Integration.lean (new)
- @@ -0,0 +1,9 @@
  + import measure_theory.integral.interval_integral + import analysis.special_functions.integrals + import analysis.convolution + + open set filter + open_locale topological_space filter + + noncomputable theory +
src/09_Integration_and_Measure_Theory/solutions/solutions_02_Measure_Theory.lean (new)
- @@ -0,0 +1,18 @@
  + import analysis.normed_space.finite_dimension + import analysis.convolution + import measure_theory.function.jacobian + import measure_theory.integral.bochner + import measure_theory.measure.lebesgue + + open set filter + open_locale topological_space filter ennreal + noncomputable theory + + variables {α : Type*} [measurable_space α] + + variables {ι : Type*} [encodable ι] + + open measure_theory + + variables {μ : measure α} +
src/09_Integration_and_Measure_Theory/solutions/solutions_03_Integration.lean (new)
- @@ -0,0 +1,15 @@
  + import analysis.normed_space.finite_dimension + import analysis.convolution + import measure_theory.function.jacobian + import measure_theory.integral.bochner + import measure_theory.measure.lebesgue + + open set filter + open_locale topological_space filter ennreal + open measure_theory + + noncomputable theory + + variables {α : Type*} [measurable_space α] + variables {μ : measure α} +
src/C06_Abstract_Algebra/S01_Structures.lean (deleted)
- @@ -1,224 +0,0 @@
 - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - namespace C06S01 - noncomputable section - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - #check Point.ext - - example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by - ext - repeat' assumption - - def myPoint1 : Point where - x := 2 - y := -1 - z := 4 - - def myPoint2 : Point := - ⟨2, -1, 4⟩ - - def myPoint3 := - Point.mk 2 (-1) 4 - - structure Point' where build :: - x : ℝ - y : ℝ - z : ℝ - - #check Point'.build 2 (-1) 4 - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def add' (a b : Point) : Point where - x := a.x + b.x - y := a.y + b.y - z := a.z + b.z - - #check add myPoint1 myPoint2 - #check myPoint1.add myPoint2 - - end Point - - #check Point.add myPoint1 myPoint2 - #check myPoint1.add myPoint2 - - namespace Point - - protected theorem add_comm (a b : Point) : add a b = add b a := by - rw [add, add] - ext <;> dsimp - repeat' apply add_comm - - example (a b : Point) : add a b = add b a := by simp [add, add_comm] - - theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := - rfl - - def addAlt : Point → Point → Point - | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ - - def addAlt' : Point → Point → Point - | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ - - theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by - cases a - cases b - rfl - - theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by - rcases a with ⟨xa, ya, za⟩ - rcases b with ⟨xb, yb, zb⟩ - rw [addAlt, addAlt] - ext <;> dsimp - apply add_comm - repeat' apply add_comm - - example (a b : Point) : addAlt a b = addAlt b a := by - rcases a with ⟨xa, ya, za⟩ - rcases b with ⟨xb, yb, zb⟩ - simp [addAlt, add_comm] - - example : ∀ a b : Point, addAlt a b = addAlt b a := by - rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ - simp [addAlt, add_comm] - - example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by - simp [add, add_comm] - - protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by - sorry - - def smul (r : ℝ) (a : Point) : Point := - sorry - - theorem smul_distrib (r : ℝ) (a b : Point) : - (smul r a).add (smul r b) = smul r (a.add b) := by - sorry - - end Point - - structure StandardTwoSimplex where - x : ℝ - y : ℝ - z : ℝ - x_nonneg : 0 ≤ x - y_nonneg : 0 ≤ y - z_nonneg : 0 ≤ z - sum_eq : x + y + z = 1 - - namespace StandardTwoSimplex - - def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex - where - x := a.y - y := a.x - z := a.z - x_nonneg := a.y_nonneg - y_nonneg := a.x_nonneg - z_nonneg := a.z_nonneg - sum_eq := by rw [add_comm a.y a.x, a.sum_eq] - - noncomputable section - - def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex - where - x := (a.x + b.x) / 2 - y := (a.y + b.y) / 2 - z := (a.z + b.z) / 2 - x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) - y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) - z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) - sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] - - def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex := - sorry - - end - - end StandardTwoSimplex - - open BigOperators - - structure StandardSimplex (n : ℕ) where - V : Fin n → ℝ - NonNeg : ∀ i : Fin n, 0 ≤ V i - sum_eq_one : (∑ i, V i) = 1 - - namespace StandardSimplex - - def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n - where - V i := (a.V i + b.V i) / 2 - NonNeg := by - intro i - apply div_nonneg - · linarith [a.NonNeg i, b.NonNeg i] - norm_num - sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, - a.sum_eq_one, b.sum_eq_one] - field_simp - - end StandardSimplex - - structure IsLinear (f : ℝ → ℝ) where - is_additive : ∀ x y, f (x + y) = f x + f y - preserves_mul : ∀ x c, f (c * x) = c * f x - - section - variable (f : ℝ → ℝ) (linf : IsLinear f) - - #check linf.is_additive - #check linf.preserves_mul - - end - - def Point'' := - ℝ × ℝ × ℝ - - def IsLinear' (f : ℝ → ℝ) := - (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x - - def PReal := - { y : ℝ // 0 < y } - - section - variable (x : PReal) - - #check x.val - #check x.property - #check x.1 - #check x.2 - - end - - def StandardTwoSimplex' := - { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } - - def StandardSimplex' (n : ℕ) := - { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } - - def StdSimplex := Σ n : ℕ, StandardSimplex n - - section - variable (s : StdSimplex) - - #check s.fst - #check s.snd - - #check s.1 - #check s.2 - - end -
src/C06_Abstract_Algebra/S02_Algebraic_Structures.lean (deleted)
- @@ -1,171 +0,0 @@
  - import Mathlib.Data.Real.Basic - - namespace C06S02 - structure Group₁ (α : Type _) where - mul : α → α → α - one : α - inv : α → α - mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) - mul_one : ∀ x : α, mul x one = x - one_mul : ∀ x : α, mul one x = x - mul_left_inv : ∀ x : α, mul (inv x) x = one - - structure Group₁Cat where - α : Type _ - str : Group₁ α - - section - variable (α β γ : Type _) - variable (f : α ≃ β) (g : β ≃ γ) - - #check Equiv α β - #check (f.toFun : α → β) - #check (f.invFun : β → α) - #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) - #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) - #check (Equiv.refl α : α ≃ α) - #check (f.symm : β ≃ α) - #check (f.trans g : α ≃ γ) - - example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := - rfl - - example (x : α) : (f.trans g) x = g (f x) := - rfl - - example : (f.trans g : α → γ) = g ∘ f := - rfl - - end - - example (α : Type _) : Equiv.Perm α = (α ≃ α) := - rfl - - def permGroup {α : Type _} : Group₁ (Equiv.Perm α) - where - mul f g := Equiv.trans g f - one := Equiv.refl α - inv := Equiv.symm - mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm - one_mul := Equiv.trans_refl - mul_one := Equiv.refl_trans - mul_left_inv := Equiv.self_trans_symm - - structure AddGroup₁ (α : Type _) where - (add : α → α → α) - -- fill in the rest - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def neg (a : Point) : Point := sorry - - def zero : Point := sorry - - def add_group_point : AddGroup₁ point := sorry - - end Point - - section - variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) - - #check f * g - #check mul_assoc f g g⁻¹ - - -- group power, defined for any group - #check g ^ n - - example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] - - example : f * g * g⁻¹ = f := - mul_inv_cancel_right f g - - example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := - mul_inv_cancel_right f g - - end - - class Group₂ (α : Type _) where - mul : α → α → α - one : α - inv : α → α - mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) - mul_one : ∀ x : α, mul x one = x - one_mul : ∀ x : α, mul one x = x - mul_left_inv : ∀ x : α, mul (inv x) x = one - - instance {α : Type _} : Group₂ (Equiv.Perm α) where - mul f g := Equiv.trans g f - one := Equiv.refl α - inv := Equiv.symm - mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm - one_mul := Equiv.trans_refl - mul_one := Equiv.refl_trans - mul_left_inv := Equiv.self_trans_symm - - #check @Group₂.mul - - def mySquare {α : Type _} [Group₂ α] (x : α) := - Group₂.mul x x - - #check @mySquare - - section - variable {β : Type _} (f g : Equiv.Perm β) - - example : Group₂.mul f g = g.trans f := - rfl - - example : mySquare f = f.trans f := - rfl - - end - - instance : Inhabited Point where default := ⟨0, 0, 0⟩ - - #check (default : Point) - - example : ([] : List Point).headI = default := - rfl - - instance : Add Point where add := Point.add - - section - variable (x y : Point) - - #check x + y - - example : x + y = Point.add x y := - rfl - - end - - instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := - ⟨Group₂.mul⟩ - - instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := - ⟨Group₂.one⟩ - - instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := - ⟨Group₂.inv⟩ - - section - variable {α : Type _} (f g : Equiv.Perm α) - - #check f * 1 * g⁻¹ - - def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := - rfl - - end - - class AddGroup₂ (α : Type _) where - add : α → α → α - -- fill in the rest
src/C06_Abstract_Algebra/S03_Building_the_Gaussian_Integers.lean (deleted)
- @@ -1,276 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : - x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intro - ext <;> simp - add_zero := by - intro - ext <;> simp - add_left_neg := by - intro - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intro - ext <;> simp - mul_one := by - intro - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - example (a b : ℤ) : a = b * (a / b) + a % b := - Eq.symm <| Int.ediv_add_emod a b - - example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := - Int.emod_nonneg a - - example (a b : ℤ) : b ≠ 0 → a % b < abs b := - Int.emod_lt a - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - sorry - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - sorry - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - sorry - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - sorry - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - sorry - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C06_Abstract_Algebra/solutions/Solutions_S01_Structures.lean (deleted)
- @@ -1,99 +0,0 @@
  - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - namespace C06S01 - noncomputable section - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by - simp [add, add_assoc] - - def smul (r : ℝ) (a : Point) : Point := - ⟨r * a.x, r * a.y, r * a.z⟩ - - theorem smul_distrib (r : ℝ) (a b : Point) : - (smul r a).add (smul r b) = smul r (a.add b) := by - simp [add, smul, mul_add] - - end Point - - structure StandardTwoSimplex where - x : ℝ - y : ℝ - z : ℝ - x_nonneg : 0 ≤ x - y_nonneg : 0 ≤ y - z_nonneg : 0 ≤ z - sum_eq : x + y + z = 1 - - namespace StandardTwoSimplex - - noncomputable section - - def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where - x := lambda * a.x + (1 - lambda) * b.x - y := lambda * a.y + (1 - lambda) * b.y - z := lambda * a.z + (1 - lambda) * b.z - x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) - y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) - z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) - sum_eq := by - trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) - · ring - simp [a.sum_eq, b.sum_eq] - - end - - end StandardTwoSimplex - - open BigOperators - - structure StandardSimplex (n : ℕ) where - V : Fin n → ℝ - NonNeg : ∀ i : Fin n, 0 ≤ V i - sum_eq_one : (∑ i, V i) = 1 - - namespace StandardSimplex - - def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n - where - V i := (a.V i + b.V i) / 2 - NonNeg := by - intro i - apply div_nonneg - · linarith [a.NonNeg i, b.NonNeg i] - norm_num - sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, - a.sum_eq_one, b.sum_eq_one] - field_simp - - end StandardSimplex - - namespace StandardSimplex - - def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardSimplex n) : StandardSimplex n - where - V i := lambda * a.V i + (1 - lambda) * b.V i - NonNeg i := - add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) - sum_eq_one := by - trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i - · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] - simp [a.sum_eq_one, b.sum_eq_one] - - end StandardSimplex -
src/C06_Abstract_Algebra/solutions/Solutions_S02_Algebraic_Structures.lean (deleted)
- @@ -1,74 +0,0 @@
  - import Mathlib.Data.Real.Basic - - namespace C06S02 - structure AddGroup₁ (α : Type _) where - add : α → α → α - zero : α - neg : α → α - add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) - add_zero : ∀ x : α, add x zero = x - zero_add : ∀ x : α, add x zero = x - add_left_neg : ∀ x : α, add (neg x) x = zero - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def neg (a : Point) : Point := - ⟨-a.x, -a.y, -a.z⟩ - - def zero : Point := - ⟨0, 0, 0⟩ - - def addGroupPoint : AddGroup₁ Point where - add := Point.add - zero := Point.zero - neg := Point.neg - add_assoc := by simp [Point.add, add_assoc] - add_zero := by simp [Point.add, Point.zero] - zero_add := by simp [Point.add, Point.zero] - add_left_neg := by simp [Point.add, Point.neg, Point.zero] - - end Point - - class AddGroup₂ (α : Type _) where - add : α → α → α - zero : α - neg : α → α - add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) - add_zero : ∀ x : α, add x zero = x - zero_add : ∀ x : α, add x zero = x - add_left_neg : ∀ x : α, add (neg x) x = zero - - instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := - ⟨AddGroup₂.add⟩ - - instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := - ⟨AddGroup₂.zero⟩ - - instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := - ⟨AddGroup₂.neg⟩ - - instance : AddGroup₂ Point where - add := Point.add - zero := Point.zero - neg := Point.neg - add_assoc := by simp [Point.add, add_assoc] - add_zero := by simp [Point.add, Point.zero] - zero_add := by simp [Point.add, Point.zero] - add_left_neg := by simp [Point.add, Point.neg, Point.zero] - - section - variable (x y : Point) - - #check x + -y + 0 - - end -
src/C06_Abstract_Algebra/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (deleted)
- @@ -1,290 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : - x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intro - ext <;> simp - add_zero := by - intro - ext <;> simp - add_left_neg := by - intro - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intro - ext <;> simp - mul_one := by - intro - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := - haveI h' : x ^ 2 = 0 := by - apply le_antisymm _ (sq_nonneg x) - rw [← h] - apply le_add_of_nonneg_right (sq_nonneg y) - pow_eq_zero h' - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - constructor - · intro h - constructor - · exact aux h - rw [add_comm] at h - exact aux h - rintro ⟨rfl, rfl⟩ - norm_num - - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - apply add_nonneg <;> - apply sq_nonneg - - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] - rfl - - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] - simp [norm_nonneg] - - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - simp [norm] - ring - - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C07_Topology/S01_Filters.lean (deleted)
- @@ -1,105 +0,0 @@
 - import Mathlib.Topology.Instances.Real - - open Set Filter Topology - - def principal {α : Type _} (s : Set α) : Filter α - where - sets := { t | s ⊆ t } - univ_sets := sorry - sets_of_superset := sorry - inter_sets := sorry - - example : Filter ℕ := - { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } - univ_sets := sorry - sets_of_superset := sorry - inter_sets := sorry } - - def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := - ∀ V ∈ G, f ⁻¹' V ∈ F - - def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := - map f F ≤ G - - example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : - Tendsto₂ f F G ↔ Tendsto₁ f F G := - Iff.rfl - - #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) - - #check - (@Filter.map_map : - ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) - - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} - (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := - sorry - - variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) - - #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) - - #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) - - section - variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} - - #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) - - end - - example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ˢ 𝓝 y₀ := - nhds_prod_eq - - #check le_inf_iff - - example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ - Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := - sorry - - example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := - nhds_basis_Ioo_pos x₀ - - example (u : ℕ → ℝ) (x₀ : ℝ) : - Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by - have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis - rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] - simp - - example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : - ∀ᶠ n in atTop, P n ∧ Q n := - hP.and hQ - - example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : - Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := - tendsto_congr' h - - 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 - - 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 - apply (hP.and (hQ.and hR)).mono - rintro n ⟨h, h', h''⟩ - exact h'' ⟨h, h'⟩ - - 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 - filter_upwards [hP, hQ, hR] - intro n h h' h'' - exact h'' ⟨h, h'⟩ - - #check mem_closure_iff_clusterPt - #check le_principal_iff - #check neBot_of_le - - example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) - (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := - sorry -
src/C07_Topology/S02_Metric_Spaces.lean (deleted)
- @@ -1,206 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter - open Topology Filter - - variable {X : Type _} [MetricSpace X] (a b c : X) - - #check (dist a b : ℝ) - #check (dist_nonneg : 0 ≤ dist a b) - #check (dist_eq_zero : dist a b = 0 ↔ a = b) - #check (dist_comm a b : dist a b = dist b a) - #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) - - -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. - #check EMetricSpace - #check PseudoMetricSpace - #check PseudoEMetricSpace - - example {u : ℕ → X} {a : X} : - Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := - Metric.tendsto_atTop - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : - Continuous f ↔ - ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := - Metric.continuous_iff - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by - apply Continuous.dist - exact hf.comp continuous_fst - exact hf.comp continuous_snd - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - (hf.comp continuous_fst).dist (hf.comp continuous_snd) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - hf.fst'.dist hf.snd' - - example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := - sorry - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : - ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := - Metric.continuousAt_iff - - variable (r : ℝ) - - example : Metric.ball a r = { b | dist b a < r } := - rfl - - example : Metric.closedBall a r = { b | dist b a ≤ r } := - rfl - - example (hr : 0 < r) : a ∈ Metric.ball a r := - Metric.mem_ball_self hr - - example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := - Metric.mem_closedBall_self hr - - example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.isOpen_iff - - example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := - isOpen_compl_iff.symm - - example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) - (hus : ∀ n, u n ∈ s) : a ∈ s := - hs.mem_of_tendsto hu (eventually_of_forall hus) - - example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := - Metric.mem_closure_iff - - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : - a ∈ closure s := - sorry - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.nhds_basis_ball.mem_iff - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := - Metric.nhds_basis_closedBall.mem_iff - - example : IsCompact (Set.Icc 0 1 : Set ℝ) := - isCompact_Icc - - example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : - ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := - hs.exists_forall_le hs' hfs - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := - hs.exists_forall_ge hs' hfs - - example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed - - example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ - - #check IsCompact.isClosed - - example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : - UniformContinuous f ↔ - ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := - Metric.uniformContinuous_iff - - example {X : Type _} [MetricSpace X] [CompactSpace X] - {Y : Type _} [MetricSpace Y] {f : X → Y} - (hf : Continuous f) : UniformContinuous f := - sorry - - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := - Metric.cauchySeq_iff - - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := - Metric.cauchySeq_iff' - - example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : - ∃ x, Tendsto u atTop (𝓝 x) := - cauchySeq_tendsto_of_complete hu - - open BigOperators - - open Finset - - theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} - (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by - rw [Metric.cauchySeq_iff'] - intro ε ε_pos - obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry - use N - intro n hn - obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn - calc - dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry - _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry - _ ≤ 1 / 2 ^ N * 2 := sorry - _ < ε := sorry - - - open Metric - - example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : - Dense (⋂ n, f n) := by - let B : ℕ → ℝ := fun n => (1 / 2) ^ n - have Bpos : ∀ n, 0 < B n - sorry - /- Translate the density assumption into two functions `center` and `radius` associating - to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. - We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ - have : - ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := - by sorry - choose! center radius Hpos HB Hball using this - intro x - rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] - intro ε εpos - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` - belonging to all `f n`. For this, we construct inductively a sequence - `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included - in the previous ball and in `f n`, and such that `r n` is small enough to ensure - that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs - to all the `f n`. -/ - let F : ℕ → X × ℝ := fun n => - Nat.recOn n (Prod.mk x (min ε (B 0))) - fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) - let c : ℕ → X := fun n => (F n).1 - let r : ℕ → ℝ := fun n => (F n).2 - have rpos : ∀ n, 0 < r n := by sorry - have rB : ∀ n, r n ≤ B n := by sorry - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by - sorry - have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry - have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist - -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. - rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ - -- this point `y` will be the desired point. We will check that it belongs to all - -- `f n` and to `ball x ε`. - use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry - sorry -
src/C07_Topology/S03_Topological_Spaces.lean (deleted)
- @@ -1,155 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter Topology - - section - variable {X : Type _} [TopologicalSpace X] - - example : IsOpen (univ : Set X) := - isOpen_univ - - example : IsOpen (∅ : Set X) := - isOpen_empty - - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := - isOpen_iUnion hs - - example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : - IsOpen (⋂ i, s i) := - isOpen_iInter hs - - variable {Y : Type _} [TopologicalSpace Y] - - example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := - continuous_def - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := - Iff.rfl - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := - Iff.rfl - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := - mem_nhds_iff - - example (x : X) : pure x ≤ 𝓝 x := - pure_le_nhds x - - example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := - pure_le_nhds x h - - example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := - eventually_eventually_nhds.mpr h - - #check TopologicalSpace.mkOfNhds - - #check TopologicalSpace.nhds_mkOfNhds - - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) - (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : - ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := - sorry - - end - - -- BOTH. - variable {X Y : Type _} - - example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := - TopologicalSpace.coinduced f - - example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := - TopologicalSpace.induced f - - example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : - TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := - coinduced_le_iff_le_induced - - #check coinduced_compose - - #check induced_compose - - example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := - Iff.rfl - - example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : - Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := - continuous_iff_coinduced_le - - example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) - (g : Y → Z) : - @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ - @Continuous X Z T_X T_Z (g ∘ f) := by - rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] - - example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : - (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = - ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := - rfl - - example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) - (hb : Tendsto u atTop (𝓝 b)) : a = b := - tendsto_nhds_unique ha hb - - example [TopologicalSpace X] [RegularSpace X] (a : X) : - (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := - closed_nhds_basis a - - example [TopologicalSpace X] {x : X} : - (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := - nhds_basis_opens' x - - theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} - {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : - ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := - sorry - - example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace 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) : - ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := - sorry - - #check @HasBasis.tendsto_right_iff - - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] - {s : Set X} {a : X} : - a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := - mem_closure_iff_seq_limit - - variable [TopologicalSpace X] - - example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := - Iff.rfl - - example {s : Set X} : - IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := - Iff.rfl - - example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) - (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - variable [TopologicalSpace Y] - - example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} - (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := - ClusterPt.map H hfx hf - - example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : - IsCompact (f '' s) := by - intro F F_ne F_le - have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry - have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry - have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left - sorry - - example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) - (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := - hs.elim_finite_subcover U hUo hsU - - example [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ -
src/C07_Topology/solutions/Solutions_S01_Filters.lean (deleted)
- @@ -1,71 +0,0 @@
 - import Mathlib.Topology.Instances.Real - - open Set Filter Topology - - -- In the next example we could use `tauto` in each proof instead of knowing the lemmas - example {α : Type _} (s : Set α) : Filter α := - { sets := { t | s ⊆ t } - univ_sets := subset_univ s - sets_of_superset := fun hU hUV => Subset.trans hU hUV - inter_sets := fun hU hV => subset_inter hU hV } - - example : Filter ℕ := - { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } - univ_sets := by - use 42 - simp - sets_of_superset := by - rintro U V ⟨N, hN⟩ hUV - use N - tauto - inter_sets := by - rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ - use max N N' - intro b hb - rw [max_le_iff] at hb - constructor <;> tauto } - - def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := - ∀ V ∈ G, f ⁻¹' V ∈ F - - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} - (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := - calc - map (g ∘ f) F = map g (map f F) := by rw [map_map] - _ ≤ map g G := (map_mono hf) - _ ≤ H := hg - - - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} - (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by - intro V hV - rw [preimage_comp] - apply hf - apply hg - exact hV - - example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ - Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := - calc - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl - _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq] - _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl - _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff - _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by - rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] - _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by - rw [map_map, map_map] - - - -- an alternative solution - example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ - Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by - rw [nhds_prod_eq] - unfold Tendsto SProd.sprod Filter.instSProd Filter.prod - erw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] - - example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) - (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := - mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
src/C07_Topology/solutions/Solutions_S02_Metric_Spaces.lean (deleted)
- @@ -1,371 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter - open Topology Filter - - variable {X : Type _} [MetricSpace X] (a b c : X) - - #check (dist a b : ℝ) - #check (dist_nonneg : 0 ≤ dist a b) - #check (dist_eq_zero : dist a b = 0 ↔ a = b) - #check (dist_comm a b : dist a b = dist b a) - #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) - - -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. - #check EMetricSpace - #check PseudoMetricSpace - #check PseudoEMetricSpace - - example {u : ℕ → X} {a : X} : - Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := - Metric.tendsto_atTop - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : - Continuous f ↔ - ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := - Metric.continuous_iff - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by - apply Continuous.dist - exact hf.comp continuous_fst - exact hf.comp continuous_snd - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - (hf.comp continuous_fst).dist (hf.comp continuous_snd) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - hf.fst'.dist hf.snd' - - example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := - sorry - - example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := - hf.comp <| (continuous_pow 2).add continuous_id - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : - ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := - Metric.continuousAt_iff - - variable (r : ℝ) - - example : Metric.ball a r = { b | dist b a < r } := - rfl - - example : Metric.closedBall a r = { b | dist b a ≤ r } := - rfl - - example (hr : 0 < r) : a ∈ Metric.ball a r := - Metric.mem_ball_self hr - - example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := - Metric.mem_closedBall_self hr - - example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.isOpen_iff - - example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := - isOpen_compl_iff.symm - - example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) - (hus : ∀ n, u n ∈ s) : a ∈ s := - hs.mem_of_tendsto hu (eventually_of_forall hus) - - example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := - Metric.mem_closure_iff - - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : - a ∈ closure s := - sorry - - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by - rw [Metric.tendsto_atTop] at hu - rw [Metric.mem_closure_iff] - intro ε ε_pos - rcases hu ε ε_pos with ⟨N, hN⟩ - refine' ⟨u N, hs _, _⟩ - rw [dist_comm] - exact hN N le_rfl - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.nhds_basis_ball.mem_iff - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := - Metric.nhds_basis_closedBall.mem_iff - - example : IsCompact (Set.Icc 0 1 : Set ℝ) := - isCompact_Icc - - example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : - ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := - hs.exists_forall_le hs' hfs - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := - hs.exists_forall_ge hs' hfs - - example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed - - example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ - - #check IsCompact.isClosed - - example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : - UniformContinuous f ↔ - ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := - Metric.uniformContinuous_iff - - example {X : Type _} [MetricSpace X] [CompactSpace X] - {Y : Type _} [MetricSpace Y] {f : X → Y} - (hf : Continuous f) : UniformContinuous f := - sorry - - example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} - (hf : Continuous f) : UniformContinuous f := by - rw [Metric.uniformContinuous_iff] - intro ε ε_pos - let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) - have φ_cont : Continuous φ := hf.fst'.dist hf.snd' - let K := { p : X × X | ε ≤ φ p } - have K_closed : IsClosed K := isClosed_le continuous_const φ_cont - have K_cpct : IsCompact K := K_closed.isCompact - cases' eq_empty_or_nonempty K with hK hK - · use 1, by norm_num - intro x y _ - have : (x, y) ∉ K := by simp [hK] - simpa 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 - linarith - · intro x x' - contrapose! - intro hxx' - exact H (x, x') hxx' - - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := - Metric.cauchySeq_iff - - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := - Metric.cauchySeq_iff' - - example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : - ∃ x, Tendsto u atTop (𝓝 x) := - cauchySeq_tendsto_of_complete hu - - open BigOperators - - open Finset - - theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} - (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by - rw [Metric.cauchySeq_iff'] - intro ε ε_pos - obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry - use N - intro n hn - obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn - calc - dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry - _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry - _ ≤ 1 / 2 ^ N * 2 := sorry - _ < ε := sorry - - - example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by - rw [Metric.cauchySeq_iff'] - intro ε ε_pos - obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by - have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by - rw [← MulZeroClass.zero_mul (2 : ℝ)] - apply Tendsto.mul - simp_rw [← one_div_pow (2 : ℝ)] - apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith - exact tendsto_const_nhds - rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ - exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ - use N - intro n hn - obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn - calc - dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] - _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := - (dist_le_range_sum_dist (fun i => u (N + i)) k) - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] - _ ≤ 1 / 2 ^ N * 2 := - (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) - (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) - _ < ε := hN - - - open Metric - - example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : - Dense (⋂ n, f n) := by - let B : ℕ → ℝ := fun n => (1 / 2) ^ n - have Bpos : ∀ n, 0 < B n - sorry - /- Translate the density assumption into two functions `center` and `radius` associating - to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. - We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ - have : - ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := - by sorry - choose! center radius Hpos HB Hball using this - intro x - rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] - intro ε εpos - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` - belonging to all `f n`. For this, we construct inductively a sequence - `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included - in the previous ball and in `f n`, and such that `r n` is small enough to ensure - that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs - to all the `f n`. -/ - let F : ℕ → X × ℝ := fun n => - Nat.recOn n (Prod.mk x (min ε (B 0))) - fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) - let c : ℕ → X := fun n => (F n).1 - let r : ℕ → ℝ := fun n => (F n).2 - have rpos : ∀ n, 0 < r n := by sorry - have rB : ∀ n, r n ≤ B n := by sorry - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by - sorry - have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry - have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist - -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. - rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ - -- this point `y` will be the desired point. We will check that it belongs to all - -- `f n` and to `ball x ε`. - use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry - sorry - - example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : - Dense (⋂ n, f n) := by - let B : ℕ → ℝ := fun n => (1 / 2) ^ n - have Bpos : ∀ n, 0 pow_pos sorry n - /- Translate the density assumption into two functions `center` and `radius` associating - to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. - We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ - have : - ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by - intro n x δ δpos - have : x ∈ closure (f n) := hd n x - rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ - rw [dist_comm] at xy - obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := - nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) - refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ - show 0 < min (min (δ / 2) r) (B (n + 1)) - exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) - show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) - exact min_le_right _ _ - show z ∈ closedBall x δ - exact - calc - dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ - _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) - _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) - _ = δ := add_halves δ - - show z ∈ f n - exact - hr - (calc - dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz - _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) - ) - choose! center radius Hpos HB Hball using this - refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all - `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball - `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that - `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a - limit which belongs to all the `f n`. -/ - let F : ℕ → X × ℝ := fun n => - Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) - let c : ℕ → X := fun n => (F n).1 - let r : ℕ → ℝ := fun n => (F n).2 - have rpos : ∀ n, 0 < r n := by - intro n - induction' n with n hn - exact lt_min εpos (Bpos 0) - exact Hpos n (c n) (r n) hn - have rB : ∀ n, r n ≤ B n := by - intro n - induction' n with n hn - exact min_le_right _ _ - exact HB n (c n) (r n) (rpos n) - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => - Hball n (c n) (r n) (rpos n) - have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by - intro n - rw [dist_comm] - have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := - mem_closedBall_self (rpos <| n + 1).le - have I := - calc - closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := - (incl n).trans (inter_subset_left _ _) - _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) - - exact I A - have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist - -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. - rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ - -- this point `y` will be the desired point. We will check that it belongs to all - -- `f n` and to `ball x ε`. - use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by - intro n - refine' Nat.le_induction _ fun m hnm h => _ - · exact Subset.rfl - · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by - intro n - refine' isClosed_ball.mem_of_tendsto ylim _ - refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ - exact I n m hm (mem_closedBall_self (rpos _).le) - constructor - · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] - intro n - have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := - Subset.trans (incl n) (inter_subset_right _ _) - exact this (yball (n + 1)) - calc - dist y x ≤ r 0 := yball 0 - _ ≤ ε := min_le_left _ _ - -
src/C07_Topology/solutions/Solutions_S03_Topological_Spaces.lean (deleted)
- @@ -1,207 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter Topology - - section - variable {X : Type _} [TopologicalSpace X] - - example : IsOpen (univ : Set X) := - isOpen_univ - - example : IsOpen (∅ : Set X) := - isOpen_empty - - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := - isOpen_iUnion hs - - example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : - IsOpen (⋂ i, s i) := - isOpen_iInter hs - - variable {Y : Type _} [TopologicalSpace Y] - - example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := - continuous_def - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := - Iff.rfl - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := - Iff.rfl - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := - mem_nhds_iff - - example (x : X) : pure x ≤ 𝓝 x := - pure_le_nhds x - - example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := - pure_le_nhds x h - - example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := - eventually_eventually_nhds.mpr h - - #check TopologicalSpace.mkOfNhds - - #check TopologicalSpace.nhds_mkOfNhds - - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) - (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : - ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := - sorry - - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) - (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : - ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by - intro a s s_in - refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ - rintro y (hy : s ∈ n y) - exact H₀ y hy - - end - - -- BOTH. - variable {X Y : Type _} - - example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := - TopologicalSpace.coinduced f - - example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := - TopologicalSpace.induced f - - example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : - TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := - coinduced_le_iff_le_induced - - #check coinduced_compose - - #check induced_compose - - example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := - Iff.rfl - - example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : - Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := - continuous_iff_coinduced_le - - example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) - (g : Y → Z) : - @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ - @Continuous X Z T_X T_Z (g ∘ f) := by - rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] - - example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : - (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = - ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := - rfl - - example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) - (hb : Tendsto u atTop (𝓝 b)) : a = b := - tendsto_nhds_unique ha hb - - example [TopologicalSpace X] [RegularSpace X] (a : X) : - (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := - closed_nhds_basis a - - example [TopologicalSpace X] {x : X} : - (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := - nhds_basis_opens' x - - theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} - {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : - ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := - sorry - - example {X Y A : Type _} [TopologicalSpace X] {c : A → X} - {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : - ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by - simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in - - example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace 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) : - ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := - sorry - - #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) - (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : - ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by - choose φ hφ using hf - use φ - constructor - · rw [continuous_iff_continuousAt] - intro x - suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by - simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] - intro V' V'_in V'_closed - obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in - suffices : ∀ y ∈ V, φ y ∈ V' - exact mem_of_superset V_in this - intro y y_in - have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in - haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y - apply V'_closed.mem_of_tendsto (hφ y) - exact mem_of_superset (preimage_mem_comap hVx) hV - · intro a - have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a - exact tendsto_nhds_unique lim f_cont.continuousAt - - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] - {s : Set X} {a : X} : - a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := - mem_closure_iff_seq_limit - - variable [TopologicalSpace X] - - example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := - Iff.rfl - - example {s : Set X} : - IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := - Iff.rfl - - example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) - (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - variable [TopologicalSpace Y] - - example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} - (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := - ClusterPt.map H hfx hf - - example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : - IsCompact (f '' s) := by - intro F F_ne F_le - have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry - have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry - have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left - sorry - - example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : - IsCompact (f '' s) := by - intro F F_ne F_le - have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] - have Hne : (𝓟 s ⊓ comap f F).NeBot := by - apply NeBot.of_map - rwa [map_eq, inf_of_le_right F_le] - have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left - rcases hs Hle with ⟨x, x_in, hx⟩ - refine' ⟨f x, mem_image_of_mem f x_in, _⟩ - apply hx.map hf.continuousAt - rw [Tendsto, map_eq] - exact inf_le_right - - example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) - (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := - hs.elim_finite_subcover U hUo hsU - - example [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ -
src/C_Abstract_Algebra/S01_Structures.lean (deleted)
- @@ -1,227 +0,0 @@
 - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - noncomputable section - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - #check Point.ext - - example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by - ext - repeat' assumption - - def myPoint1 : Point where - x := 2 - y := -1 - z := 4 - - def myPoint2 : Point := - ⟨2, -1, 4⟩ - - def myPoint3 := - Point.mk 2 (-1) 4 - - structure Point' where build :: - x : ℝ - y : ℝ - z : ℝ - - #check Point'.build 2 (-1) 4 - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def add' (a b : Point) : Point where - x := a.x + b.x - y := a.y + b.y - z := a.z + b.z - - #check add myPoint1 myPoint2 - - #check myPoint1.add myPoint2 - - end Point - - #check Point.add myPoint1 myPoint2 - - #check myPoint1.add myPoint2 - - namespace Point - - protected theorem add_comm (a b : Point) : add a b = add b a := by - rw [add, add] - ext <;> dsimp - repeat' apply add_comm - - example (a b : Point) : add a b = add b a := by simp [add, add_comm] - - theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := - rfl - - def addAlt : Point → Point → Point - | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ - - def addAlt' : Point → Point → Point - | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ - - theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by - cases a - cases b - rfl - - theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by - rcases a with ⟨xa, ya, za⟩ - rcases b with ⟨xb, yb, zb⟩ - rw [addAlt, addAlt] - ext <;> dsimp - apply add_comm - repeat' apply add_comm - - example (a b : Point) : addAlt a b = addAlt b a := by - rcases a with ⟨xa, ya, za⟩ - rcases b with ⟨xb, yb, zb⟩ - simp [addAlt, add_comm] - - example : ∀ a b : Point, addAlt a b = addAlt b a := by - rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ - simp [addAlt, add_comm] - - example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by - simp [add, add_comm] - - protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by - sorry - def smul (r : ℝ) (a : Point) : Point := - sorry - theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by - sorry - end Point - - structure StandardTwoSimplex where - x : ℝ - y : ℝ - z : ℝ - x_nonneg : 0 ≤ x - y_nonneg : 0 ≤ y - z_nonneg : 0 ≤ z - sum_eq : x + y + z = 1 - - namespace StandardTwoSimplex - - def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex - where - x := a.y - y := a.x - z := a.z - x_nonneg := a.y_nonneg - y_nonneg := a.x_nonneg - z_nonneg := a.z_nonneg - sum_eq := by rw [add_comm a.y a.x, a.sum_eq] - - noncomputable section - - def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex - where - x := (a.x + b.x) / 2 - y := (a.y + b.y) / 2 - z := (a.z + b.z) / 2 - x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) - y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) - z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) - sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] - - def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where - sorry - end - - end StandardTwoSimplex - - open BigOperators - - structure StandardSimplex (n : ℕ) where - V : Fin n → ℝ - NonNeg : ∀ i : Fin n, 0 ≤ V i - sum_eq_one : (∑ i, V i) = 1 - - namespace StandardSimplex - - def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n - where - V i := (a.V i + b.V i) / 2 - NonNeg := by - intro i - apply div_nonneg - · linarith [a.NonNeg i, b.NonNeg i] - norm_num - sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] - field_simp - - end StandardSimplex - - structure IsLinear (f : ℝ → ℝ) where - is_additive : ∀ x y, f (x + y) = f x + f y - preserves_mul : ∀ x c, f (c * x) = c * f x - - section - - variable (f : ℝ → ℝ) (linf : IsLinear f) - - #check linf.is_additive - - #check linf.preserves_mul - - end - - def Point'' := - ℝ × ℝ × ℝ - - def IsLinear' (f : ℝ → ℝ) := - (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x - - def PReal := - { y : ℝ // 0 < y } - - section - - variable (x : PReal) - - #check x.val - - #check x.property - - #check x.1 - - #check x.2 - - end - - def StandardTwoSimplex' := - { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } - - def StandardSimplex' (n : ℕ) := - { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } - - def StdSimplex := Σ n : ℕ, StandardSimplex n - - section - - variable (s : StdSimplex) - - #check s.fst - #check s.snd - - #check s.1 - #check s.2 - - end -
src/C_Abstract_Algebra/S02_Algebraic_Structures.lean (deleted)
- @@ -1,172 +0,0 @@
  - import Mathlib.Data.Real.Basic - - structure Group₁ (α : Type _) where - mul : α → α → α - one : α - inv : α → α - mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) - mul_one : ∀ x : α, mul x one = x - one_mul : ∀ x : α, mul one x = x - mul_left_inv : ∀ x : α, mul (inv x) x = one - - structure Group₁Cat where - α : Type _ - str : Group₁ α - - section - - variable (α β γ : Type _) - variable (f : α ≃ β) (g : β ≃ γ) - - #check Equiv α β - #check (f.toFun : α → β) - #check (f.invFun : β → α) - #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) - #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) - #check (Equiv.refl α : α ≃ α) - #check (f.symm : β ≃ α) - #check (f.trans g : α ≃ γ) - - example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := - rfl - - example (x : α) : (f.trans g) x = g (f x) := - rfl - - example : (f.trans g : α → γ) = g ∘ f := - rfl - - end - - example (α : Type _) : Equiv.Perm α = (α ≃ α) := - rfl - - def permGroup {α : Type _} : Group₁ (Equiv.Perm α) - where - mul f g := Equiv.trans g f - one := Equiv.refl α - inv := Equiv.symm - mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm - one_mul := Equiv.trans_refl - mul_one := Equiv.refl_trans - mul_left_inv := Equiv.self_trans_symm - - structure AddGroup₁ (α : Type _) where - (add : α → α → α) - -- fill in the rest - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def neg (a : point) : point := sorry - - def zero : point := sorry - - def add_group_point : add_group₁ point := sorry - end Point - - section - - variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) - - #check f * g - #check mul_assoc f g g⁻¹ - - -- group power, defined for any group - #check g ^ n - - example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] - - example : f * g * g⁻¹ = f := - mul_inv_cancel_right f g - - example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := - mul_inv_cancel_right f g - - end - - class Group₂ (α : Type _) where - mul : α → α → α - one : α - inv : α → α - mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) - mul_one : ∀ x : α, mul x one = x - one_mul : ∀ x : α, mul one x = x - mul_left_inv : ∀ x : α, mul (inv x) x = one - - instance {α : Type _} : Group₂ (Equiv.Perm α) where - mul f g := Equiv.trans g f - one := Equiv.refl α - inv := Equiv.symm - mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm - one_mul := Equiv.trans_refl - mul_one := Equiv.refl_trans - mul_left_inv := Equiv.self_trans_symm - - #check @Group₂.mul - - def mySquare {α : Type _} [Group₂ α] (x : α) := - Group₂.mul x x - - #check @mySquare - - section - variable {β : Type _} (f g : Equiv.Perm β) - - example : Group₂.mul f g = g.trans f := - rfl - - example : mySquare f = f.trans f := - rfl - - end - - instance : Inhabited Point where default := ⟨0, 0, 0⟩ - - #check (default : Point) - - example : ([] : List Point).headI = default := - rfl - - instance : Add Point where add := Point.add - - section - variable (x y : Point) - - #check x + y - - example : x + y = Point.add x y := - rfl - - end - - instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := - ⟨Group₂.mul⟩ - - instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := - ⟨Group₂.one⟩ - - instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := - ⟨Group₂.inv⟩ - - section - - variable {α : Type _} (f g : Equiv.Perm α) - - #check f * 1 * g⁻¹ - - def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := - rfl - - end - - class AddGroup₂ (α : Type _) where - add : α → α → α - -- fill in the rest
src/C_Abstract_Algebra/S03_Building_the_Gaussian_Integers.lean (deleted)
- @@ -1,271 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intros - ext <;> simp - add_zero := by - intros - ext <;> simp - add_left_neg := by - intros - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intros - ext <;> simp - mul_one := by - intros - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - example (a b : ℤ) : a = b * (a / b) + a % b := - Eq.symm <| Int.ediv_add_emod a b - - example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := - Int.emod_nonneg a - - example (a b : ℤ) : b ≠ 0 → a % b < abs b := - Int.emod_lt a - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - sorry - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - sorry - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - sorry - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - sorry - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - sorry - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C_Abstract_Algebra/solutions/Solutions_S01_Structures.lean (deleted)
- @@ -1,96 +0,0 @@
  - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - noncomputable section - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by - simp [add, add_assoc] - - def smul (r : ℝ) (a : Point) : Point := - ⟨r * a.x, r * a.y, r * a.z⟩ - - theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by - simp [add, smul, mul_add] - - end Point - - structure StandardTwoSimplex where - x : ℝ - y : ℝ - z : ℝ - x_nonneg : 0 ≤ x - y_nonneg : 0 ≤ y - z_nonneg : 0 ≤ z - sum_eq : x + y + z = 1 - - namespace StandardTwoSimplex - - noncomputable section - - def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where - x := lambda * a.x + (1 - lambda) * b.x - y := lambda * a.y + (1 - lambda) * b.y - z := lambda * a.z + (1 - lambda) * b.z - x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) - y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) - z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) - sum_eq := by - trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) - · ring - simp [a.sum_eq, b.sum_eq] - - end - - end StandardTwoSimplex - - open BigOperators - - structure StandardSimplex (n : ℕ) where - V : Fin n → ℝ - NonNeg : ∀ i : Fin n, 0 ≤ V i - sum_eq_one : (∑ i, V i) = 1 - - namespace StandardSimplex - - def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n - where - V i := (a.V i + b.V i) / 2 - NonNeg := by - intro i - apply div_nonneg - · linarith [a.NonNeg i, b.NonNeg i] - norm_num - sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] - field_simp - - end StandardSimplex - - namespace StandardSimplex - - def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardSimplex n) : StandardSimplex n - where - V i := lambda * a.V i + (1 - lambda) * b.V i - NonNeg i := - add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) - sum_eq_one := by - trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i - · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] - simp [a.sum_eq_one, b.sum_eq_one] - - end StandardSimplex -
src/C_Abstract_Algebra/solutions/Solutions_S02_Algebraic_Structures.lean (deleted)
- @@ -1,73 +0,0 @@
  - import Mathlib.Data.Real.Basic - - structure AddGroup₁ (α : Type _) where - add : α → α → α - zero : α - neg : α → α - add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) - add_zero : ∀ x : α, add x zero = x - zero_add : ∀ x : α, add x zero = x - add_left_neg : ∀ x : α, add (neg x) x = zero - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def neg (a : Point) : Point := - ⟨-a.x, -a.y, -a.z⟩ - - def zero : Point := - ⟨0, 0, 0⟩ - - def addGroupPoint : AddGroup₁ Point where - add := Point.add - zero := Point.zero - neg := Point.neg - add_assoc := by simp [Point.add, add_assoc] - add_zero := by simp [Point.add, Point.zero] - zero_add := by simp [Point.add, Point.zero] - add_left_neg := by simp [Point.add, Point.neg, Point.zero] - - end Point - - class AddGroup₂ (α : Type _) where - add : α → α → α - zero : α - neg : α → α - add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) - add_zero : ∀ x : α, add x zero = x - zero_add : ∀ x : α, add x zero = x - add_left_neg : ∀ x : α, add (neg x) x = zero - - instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := - ⟨AddGroup₂.add⟩ - - instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := - ⟨AddGroup₂.zero⟩ - - instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := - ⟨AddGroup₂.neg⟩ - - instance : AddGroup₂ Point where - add := Point.add - zero := Point.zero - neg := Point.neg - add_assoc := by simp [Point.add, add_assoc] - add_zero := by simp [Point.add, Point.zero] - zero_add := by simp [Point.add, Point.zero] - add_left_neg := by simp [Point.add, Point.neg, Point.zero] - - section - variable (x y : Point) - - #check x + -y + 0 - - end -
src/C_Abstract_Algebra/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (deleted)
- @@ -1,285 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intros - ext <;> simp - add_zero := by - intros - ext <;> simp - add_left_neg := by - intros - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intros - ext <;> simp - mul_one := by - intros - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := - haveI h' : x ^ 2 = 0 := by - apply le_antisymm _ (sq_nonneg x) - rw [← h] - apply le_add_of_nonneg_right (sq_nonneg y) - pow_eq_zero h' - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - constructor - · intro h - constructor - · exact aux h - rw [add_comm] at h - exact aux h - rintro ⟨rfl, rfl⟩ - norm_num - - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - apply add_nonneg <;> - apply sq_nonneg - - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] - rfl - - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] - simp [norm_nonneg] - - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - simp [norm] - ring - - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C_Basics/S01_Calculating.lean (deleted)
- @@ -1,164 +0,0 @@
  - import Mathlib.Data.Real.Basic - 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] - - -- Try these. - example (a b c : ℝ) : c * b * a = b * (a * c) := by - sorry - - example (a b c : ℝ) : a * (b * c) = b * (a * c) := by - sorry - - -- An example. - example (a b c : ℝ) : a * b * c = b * c * a := by - rw [mul_assoc] - rw [mul_comm] - - /- Try doing the first of these without providing any arguments at all, - and the second with only one argument. -/ - example (a b c : ℝ) : a * (b * c) = b * (c * a) := by - sorry - - example (a b c : ℝ) : a * (b * c) = b * (a * c) := by - sorry - - -- Using facts from the local context. - example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by - rw [h'] - rw [← mul_assoc] - rw [h] - rw [mul_assoc] - - -- Try these. For the second one, use the theorem `sub_self`. - example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by - sorry - - example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by - sorry - - -- Examples. - example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by - rw [h', ← mul_assoc, h, mul_assoc] - - section - - variable (a b c d e f g : ℝ) - - example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by - rw [h', ← mul_assoc, h, mul_assoc] - - end - - section - - variable (a b c : ℝ) - - #check a - - #check a + b - - #check (a : ℝ) - - #check mul_comm a b - - #check (mul_comm a b : a * b = b * a) - - #check mul_assoc c a b - - #check mul_comm a - - #check mul_comm - - #check @mul_comm - - end - - section - - variable (a b : ℝ) - - example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by - rw [mul_add, add_mul, add_mul] - rw [← add_assoc, add_assoc (a * a)] - rw [mul_comm b a, ← two_mul] - - example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := - calc - (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by - rw [mul_add, add_mul, add_mul] - _ = a * a + (b * a + a * b) + b * b := by - rw [← add_assoc, add_assoc (a * a)] - _ = a * a + 2 * (a * b) + b * b := by - rw [mul_comm b a, ← two_mul] - - example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := - calc - (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by - sorry - _ = a * a + (b * a + a * b) + b * b := by - sorry - _ = a * a + 2 * (a * b) + b * b := by - sorry - - end - - -- Try these. For the second, use the theorems listed underneath. - section - - variable (a b c d : ℝ) - - example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by - sorry - - example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by - sorry - - #check pow_two a - - #check mul_sub a b c - - #check add_mul a b c - - #check add_sub a b c - - #check sub_sub a b c - - #check add_zero a - - end - - -- Examples. - section - - variable (a b c d : ℝ) - - example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by - rw [hyp'] at hyp - rw [mul_comm d a] at hyp - rw [← two_mul (a * d)] at hyp - rw [← mul_assoc 2 a d] at hyp - exact hyp - - example : c * b * a = b * (a * c) := by - ring - - example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by - ring - - example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by - ring - - example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by - rw [hyp, hyp'] - ring - - end - - example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by - nth_rw 2 [h] - rw [add_mul] -
src/C_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean (deleted)
- @@ -1,168 +0,0 @@
  - import Mathlib.Algebra.Ring.Defs - import Mathlib.Data.Real.Basic - import Mathlib.Tactic - - section - - variable (R : Type _) [Ring R] - - #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) - - #check (add_comm : ∀ a b : R, a + b = b + a) - - #check (zero_add : ∀ a : R, 0 + a = a) - - #check (add_left_neg : ∀ a : R, -a + a = 0) - - #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) - - #check (mul_one : ∀ a : R, a * 1 = a) - - #check (one_mul : ∀ a : R, 1 * a = a) - - #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) - - #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) - - end - - section - - variable (R : Type _) [CommRing R] - - variable (a b c d : R) - - example : c * b * a = b * (a * c) := by ring - - example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring - - example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring - - example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by - rw [hyp, hyp'] - ring - - end - - namespace MyRing - - variable {R : Type _} [Ring R] - - 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 - - end MyRing - - namespace MyRing - - variable {R : Type _} [Ring R] - - theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by - rw [← add_assoc, add_left_neg, zero_add] - - -- Prove these: - theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by - sorry - - theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by - sorry - - theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by - sorry - - theorem mul_zero (a : R) : a * 0 = 0 := by - have h : a * 0 + a * 0 = a * 0 + 0 := by - rw [← mul_add, add_zero, add_zero] - rw [add_left_cancel h] - - theorem zero_mul (a : R) : 0 * a = 0 := by - sorry - - theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by - sorry - - theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by - sorry - - theorem neg_zero : (-0 : R) = 0 := by - apply neg_eq_of_add_eq_zero - rw [add_zero] - - theorem neg_neg (a : R) : - -a = a := by - sorry - - end MyRing - - -- Examples. - section - - variable {R : Type _} [Ring R] - - example (a b : R) : a - b = a + -b := - sub_eq_add_neg a b - - end - - example (a b : ℝ) : a - b = a + -b := - rfl - - example (a b : ℝ) : a - b = a + -b := by - rfl - - namespace MyRing - - variable {R : Type _} [Ring R] - - theorem self_sub (a : R) : a - a = 0 := - sorry - - theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by - norm_num - - theorem two_mul (a : R) : 2 * a = a + a := - sorry - - end MyRing - - section - - variable (A : Type _) [AddGroup A] - - #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) - - #check (zero_add : ∀ a : A, 0 + a = a) - - #check (add_left_neg : ∀ a : A, -a + a = 0) - - end - - section - - variable {G : Type _} [Group G] - - #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) - - #check (one_mul : ∀ a : G, 1 * a = a) - - #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) - - namespace MyGroup - - theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by - sorry - - theorem mul_one (a : G) : a * 1 = a := by - sorry - - theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by - sorry - - end MyGroup - - end -
src/C_Basics/S03_Using_Theorems_and_Lemmas.lean (deleted)
- @@ -1,156 +0,0 @@
 - import Mathlib.Analysis.SpecialFunctions.Log.Basic - import Mathlib.Tactic - - variable (a b c d e : ℝ) - - open Real - - #check (le_refl : ∀ a : ℝ, a ≤ a) - - #check (le_trans : a ≤ b → b ≤ c → a ≤ c) - - section - - variable (h : a ≤ b) (h' : b ≤ c) - - #check (le_refl : ∀ a : Real, a ≤ a) - - #check (le_refl a : a ≤ a) - - #check (le_trans : a ≤ b → b ≤ c → a ≤ c) - - #check (le_trans h : b ≤ c → a ≤ c) - - #check (le_trans h h' : a ≤ c) - - end - - example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by - apply le_trans - · apply h₀ - . apply h₁ - - example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by - apply le_trans h₀ - apply h₁ - - example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := - le_trans h₀ h₁ - - example (x : ℝ) : x ≤ x := by - apply le_refl - - example (x : ℝ) : x ≤ x := - le_refl x - - #check (le_refl : ∀ a, a ≤ a) - - #check (le_trans : a ≤ b → b ≤ c → a ≤ c) - - #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) - - #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) - - #check (lt_trans : a < b → b < c → a < c) - - -- Try this. - example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by - sorry - - example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by - linarith - - section - - example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by - linarith - - end - - example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by - linarith [exp_le_exp.mpr h'] - - #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) - - #check (exp_lt_exp : exp a < exp b ↔ a < b) - - #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) - - #check (log_lt_log : 0 < a → a < b → log a < log b) - - #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) - - #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) - - #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) - - #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) - - #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) - - #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) - - #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) - - #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) - - #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 - - example (h : a ≤ b) : exp a ≤ exp b := by - rw [exp_le_exp] - exact h - - example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by - apply add_lt_add_of_lt_of_le - · apply add_lt_add_of_le_of_lt h₀ - apply exp_lt_exp.mpr h₁ - apply le_refl - - example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry - - example : (0 : ℝ) < 1 := by norm_num - - example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by - have h₀ : 0 < 1 + exp a := by sorry - have h₁ : 0 < 1 + exp b := by sorry - apply (log_le_log h₀ h₁).mpr - sorry - - example : 0 ≤ a ^ 2 := by - -- library_search - exact sq_nonneg a - - example (h : a ≤ b) : c - exp b ≤ c - exp a := by - sorry - - example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 - calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring - _ ≥ 0 := by apply pow_two_nonneg - - calc - 2 * a * b = 2 * a * b + 0 := by ring - _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := - add_le_add (le_refl _) h - _ = a ^ 2 + b ^ 2 := by ring - - - example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 - calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring - _ ≥ 0 := by apply pow_two_nonneg - linarith - - example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by - sorry - - #check abs_le'.mpr -
src/C_Basics/S04_More_on_Order_and_Divisibility.lean (deleted)
- @@ -1,98 +0,0 @@
  - import Mathlib.Data.Real.Basic - - section - - variable (a b c d : ℝ) - - #check (min_le_left a b : min a b ≤ a) - - #check (min_le_right a b : min a b ≤ b) - - #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) - - example : min a b = min b a := by - apply le_antisymm - · show min a b ≤ min b a - apply le_min - · apply min_le_right - apply min_le_left - · show min b a ≤ min a b - apply le_min - · apply min_le_right - apply min_le_left - - example : min a b = min b a := by - have h : ∀ x y : ℝ, min x y ≤ min y x := by - intro x y - apply le_min - apply min_le_right - apply min_le_left - apply le_antisymm - apply h - apply h - - example : min a b = min b a := by - apply le_antisymm - repeat - apply le_min - apply min_le_right - apply min_le_left - - example : max a b = max b a := by - sorry - - example : min (min a b) c = min a (min b c) := by - sorry - - theorem aux : min a b + c ≤ min (a + c) (b + c) := by - sorry - - example : min a b + c = min (a + c) (b + c) := by - sorry - - #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) - - example : abs a - abs b ≤ abs (a - b) := - sorry - - end - - section - - variable (w x y z : ℕ) - - example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := - dvd_trans h₀ h₁ - - example : x ∣ y * x * z := by - apply dvd_mul_of_dvd_left - apply dvd_mul_left - - example : x ∣ x ^ 2 := by - apply dvd_mul_left - - example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by - sorry - - end - - section - - variable (m n : ℕ) - - open Nat - - #check (gcd_zero_right n : gcd n 0 = n) - - #check (gcd_zero_left n : gcd 0 n = n) - - #check (lcm_zero_right n : lcm n 0 = 0) - - #check (lcm_zero_left n : lcm 0 n = 0) - - example : gcd m n = gcd n m := by - sorry - - end - -
src/C_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean (deleted)
- @@ -1,141 +0,0 @@
 - import Mathlib.Topology.MetricSpace.Basic - - section - - variable {α : Type _} [PartialOrder α] - - variable (x y z : α) - - #check x ≤ y - - #check (le_refl x : x ≤ x) - - #check (le_trans : x ≤ y → y ≤ z → x ≤ z) - - #check x < y - - #check (lt_irrefl x : ¬x < x) - - #check (lt_trans : x < y → y < z → x < z) - - #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) - - #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) - - example : x < y ↔ x ≤ y ∧ x ≠ y := - lt_iff_le_and_ne - - end - - section - - variable {α : Type _} [Lattice α] - - variable (x y z : α) - - #check x ⊓ y - - #check (inf_le_left : x ⊓ y ≤ x) - - #check (inf_le_right : x ⊓ y ≤ y) - - #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) - - #check x ⊔ y - - #check (le_sup_left : x ≤ x ⊔ y) - - #check (le_sup_right : y ≤ x ⊔ y) - - #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) - - example : x ⊓ y = y ⊓ x := by - sorry - - example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by - sorry - - example : x ⊔ y = y ⊔ x := by - sorry - - example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by - sorry - - theorem absorb1 : x ⊓ (x ⊔ y) = x := by - sorry - - theorem absorb2 : x ⊔ x ⊓ y = x := by - sorry - - end - - 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)) - - end - - section - - variable {α : Type _} [Lattice α] - - variable (a b c : α) - - example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by - sorry - - example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by - sorry - - end - - section - - variable {R : Type _} [StrictOrderedRing R] - - variable (a b c : R) - - #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) - - #check (mul_pos : 0 < a → 0 < b → 0 < a * b) - - #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) - - example : a ≤ b → 0 ≤ b - a := by - sorry - - example : 0 ≤ b - a → a ≤ b := by - sorry - - example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by - sorry - - end - - section - - variable {X : Type _} [MetricSpace X] - - variable (x y z : X) - - #check (dist_self x : dist x x = 0) - - #check (dist_comm x y : dist x y = dist y x) - - #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) - - example (x y : X) : 0 ≤ dist x y := by - sorry - - end -
src/C_Basics/solutions/Solutions_S01_Calculating.lean (deleted)
- @@ -1,33 +0,0 @@
  - 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] - rw [mul_comm c a] - - example (a b c : ℝ) : a * (b * c) = b * (a * c) := by - rw [← mul_assoc a b c] - rw [mul_comm a b] - rw [mul_assoc b a c] - - example (a b c : ℝ) : a * (b * c) = b * (c * a) := by - rw [mul_comm] - rw [mul_assoc] - - example (a b c : ℝ) : a * (b * c) = b * (a * c) := by - rw [← mul_assoc] - rw [mul_comm a] - rw [mul_assoc] - - example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by - rw [mul_assoc a] - rw [h] - rw [← mul_assoc] - - example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by - rw [hyp] - rw [hyp'] - rw [mul_comm] - rw [sub_self] -
src/C_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean (deleted)
- @@ -1,76 +0,0 @@
  - import Mathlib.Algebra.Ring.Defs - import Mathlib.Data.Real.Basic - import Mathlib.Tactic - - namespace MyRing - - variable {R : Type _} [Ring R] - - theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by - rw [add_assoc, add_right_neg, add_zero] - - theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by - rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] - - theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by - rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] - - theorem zero_mul (a : R) : 0 * a = 0 := by - have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] - rw [add_left_cancel h] - - theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by - rw [← neg_add_cancel_left a b, h, add_zero] - - theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by - symm - apply neg_eq_of_add_eq_zero - rw [add_comm, h] - - theorem neg_zero : (-0 : R) = 0 := by - apply neg_eq_of_add_eq_zero - rw [add_zero] - - theorem neg_neg (a : R) : - -a = a := by - apply neg_eq_of_add_eq_zero - rw [add_left_neg] - - end MyRing - - namespace MyRing - - variable {R : Type _} [Ring R] - - theorem self_sub (a : R) : a - a = 0 := by - rw [sub_eq_add_neg, add_right_neg] - - theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by - norm_num - - theorem two_mul (a : R) : 2 * a = a + a := by - rw [← one_add_one_eq_two, add_mul, one_mul] - - end MyRing - - section - - variable {G : Type _} [Group G] - - namespace MyGroup - - theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by - have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by - rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] - rw [← h, ← mul_assoc, mul_left_inv, one_mul] - - theorem mul_one (a : G) : a * 1 = a := by - rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul] - - theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by - rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹, - mul_right_inv, one_mul, mul_right_inv, mul_one] - - end MyGroup - - end -
src/C_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean (deleted)
- @@ -1,62 +0,0 @@
 - import Mathlib.Analysis.SpecialFunctions.Log.Basic - import Mathlib.Tactic - - variable (a b c d e : ℝ) - - open Real - - example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by - apply lt_of_le_of_lt h₀ - apply lt_trans h₁ - exact lt_of_le_of_lt h₂ h₃ - - example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by - apply add_le_add_left - rw [exp_le_exp] - apply add_le_add_left h₀ - - -- an alternative using `linarith`. - example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by - have : exp (a + d) ≤ exp (a + e) := by - rw [exp_le_exp] - linarith - linarith [this] - - example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by - have h₀ : 0 < 1 + exp a := by linarith [exp_pos a] - have h₁ : 0 < 1 + exp b := by linarith [exp_pos b] - apply (log_le_log h₀ h₁).mpr - apply add_le_add_left (exp_le_exp.mpr h) - - -- SOLUTION. - example (h : a ≤ b) : c - exp b ≤ c - exp a := by - apply sub_le_sub_left - exact exp_le_exp.mpr h - - -- alternatively: - example (h : a ≤ b) : c - exp b ≤ c - exp a := by - linarith [exp_le_exp.mpr h] - - theorem fact1 : a * b * 2 ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 - calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring - _ ≥ 0 := by apply pow_two_nonneg - linarith - - theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 - calc - a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring - _ ≥ 0 := by apply pow_two_nonneg - linarith - - example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by - have h : (0 : ℝ) < 2 := by norm_num - apply abs_le'.mpr - constructor - · rw [le_div_iff h] - apply fact1 - rw [le_div_iff h] - apply fact2 -
src/C_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean (deleted)
- @@ -1,124 +0,0 @@
  - import Mathlib.Data.Real.Basic - - section - - variable (a b c d : ℝ) - - #check (min_le_left a b : min a b ≤ a) - - #check (min_le_right a b : min a b ≤ b) - - #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) - - example : max a b = max b a := by - apply le_antisymm - repeat' - apply max_le - apply le_max_right - apply le_max_left - - example : min (min a b) c = min a (min b c) := by - apply le_antisymm - · apply le_min - · apply le_trans - apply min_le_left - apply min_le_left - apply le_min - · apply le_trans - apply min_le_left - apply min_le_right - apply min_le_right - apply le_min - · apply le_min - · apply min_le_left - apply le_trans - apply min_le_right - apply min_le_left - apply le_trans - apply min_le_right - apply min_le_right - - theorem aux : min a b + c ≤ min (a + c) (b + c) := by - apply le_min - · apply add_le_add_right - apply min_le_left - apply add_le_add_right - apply min_le_right - - example : min a b + c = min (a + c) (b + c) := by - apply le_antisymm - · apply aux - have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel] - rw [h] - apply add_le_add_right - rw [sub_eq_add_neg] - apply le_trans - apply aux - rw [add_neg_cancel_right, add_neg_cancel_right] - - example : abs a - abs b ≤ abs (a - b) := - calc - abs a - abs b = abs (a - b + b) - abs b := by rw [sub_add_cancel] - _ ≤ abs (a - b) + abs b - abs b := by - apply sub_le_sub_right - apply abs_add - _ ≤ abs (a - b) := by rw [add_sub_cancel] - - - -- alternatively - example : abs a - abs b ≤ abs (a - b) := by - have h := abs_add (a - b) b - rw [sub_add_cancel] at h - linarith - - end - - section - - variable (w x y z : ℕ) - - example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := - dvd_trans h₀ h₁ - - example : x ∣ y * x * z := by - apply dvd_mul_of_dvd_left - apply dvd_mul_left - - example : x ∣ x ^ 2 := by - apply dvd_mul_left - - example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by - apply dvd_add - · apply dvd_add - · apply dvd_mul_of_dvd_right - apply dvd_mul_right - apply dvd_mul_left - rw [pow_two] - apply dvd_mul_of_dvd_right - exact h - - end - - section - - variable (m n : ℕ) - - open Nat - - #check (gcd_zero_right n : gcd n 0 = n) - - #check (gcd_zero_left n : gcd 0 n = n) - - #check (lcm_zero_right n : lcm n 0 = 0) - - #check (lcm_zero_left n : lcm 0 n = 0) - - example : gcd m n = gcd n m := by - apply _root_.dvd_antisymm - repeat' - apply dvd_gcd - apply gcd_dvd_right - apply gcd_dvd_left - - end -
src/C_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean (deleted)
- @@ -1,149 +0,0 @@
  - import Mathlib.Topology.MetricSpace.Basic - - section - - variable {α : Type _} [Lattice α] - - variable (x y z : α) - - example : x ⊓ y = y ⊓ x := by - apply le_antisymm - repeat' - apply le_inf - · apply inf_le_right - apply inf_le_left - - example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by - apply le_antisymm - · apply le_inf - · apply le_trans - apply inf_le_left - apply inf_le_left - apply le_inf - · apply le_trans - apply inf_le_left - apply inf_le_right - apply inf_le_right - apply le_inf - · apply le_inf - · apply inf_le_left - apply le_trans - apply inf_le_right - apply inf_le_left - apply le_trans - apply inf_le_right - apply inf_le_right - - example : x ⊔ y = y ⊔ x := by - apply le_antisymm - repeat' - apply sup_le - · apply le_sup_right - apply le_sup_left - - example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by - apply le_antisymm - · apply sup_le - · apply sup_le - apply le_sup_left - · apply le_trans - apply @le_sup_left _ _ y z - apply le_sup_right - apply le_trans - apply @le_sup_right _ _ y z - apply le_sup_right - apply sup_le - · apply le_trans - apply @le_sup_left _ _ x y - apply le_sup_left - apply sup_le - · apply le_trans - apply @le_sup_right _ _ x y - apply le_sup_left - apply le_sup_right - - theorem absorb1 : x ⊓ (x ⊔ y) = x := by - apply le_antisymm - · apply inf_le_left - apply le_inf - · apply le_refl - apply le_sup_left - - theorem absorb2 : x ⊔ x ⊓ y = x := by - apply le_antisymm - · apply sup_le - · apply le_refl - apply inf_le_left - apply le_sup_left - - end - - 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)) - - end - - section - - variable {α : Type _} [Lattice α] - - variable (a b c : α) - - example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by - rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, ← sup_assoc, @inf_comm _ _ c a, - absorb2, inf_comm] - - example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by - rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, ← inf_assoc, @sup_comm _ _ c a, - absorb1, sup_comm] - - end - - section - - variable {R : Type _} [StrictOrderedRing R] - - variable (a b c : R) - - theorem aux1 : a ≤ b → 0 ≤ b - a := by - intro h - rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] - apply add_le_add_left h - - theorem aux2 : 0 ≤ b - a → a ≤ b := by - intro h - rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] - apply add_le_add_left h - - example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by - have h1 : 0 ≤ (b - a) * c := mul_nonneg (aux1 _ _ h) h' - rw [sub_mul] at h1 - exact aux2 _ _ h1 - - end - - section - - variable {X : Type _} [MetricSpace X] - - variable (x y z : X) - - example (x y : X) : 0 ≤ dist x y :=by - have : 0 ≤ dist x y + dist y x := by - rw [← dist_self x] - apply dist_triangle - linarith [dist_comm x y] - - end -
src/C_Hierarchies/S01_Basics.lean (deleted)
- @@ -1,308 +0,0 @@
 - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - - class One₁ (α : Type) where - /-- The element one -/ - one : α - - - #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α - - @[class] structure One₂ (α : Type) where - /-- The element one -/ - one : α - - #check One₂.one - - - example (α : Type) [One₁ α] : α := One₁.one - - example (α : Type) [One₁ α] := (One₁.one : α) - - @[inherit_doc] - notation "𝟙" => One₁.one - - example {α : Type} [One₁ α] : α := 𝟙 - - example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl - - - class Dia₁ (α : Type) where - dia : α → α → α - - infixl:70 " ⋄ " => Dia₁.dia - - - class Semigroup₁ (α : Type) where - toDia₁ : Dia₁ α - /-- Diamond is associative -/ - dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) - - - attribute [instance] Semigroup₁.toDia₁ - - example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b - - - class Semigroup₂ (α : Type) extends Dia₁ α where - /-- Diamond is associative -/ - dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) - - example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b - - class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where - /-- One is a left neutral element for diamond. -/ - one_dia : ∀ a : α, 𝟙 ⋄ a = a - /-- One is a right neutral element for diamond -/ - dia_one : ∀ a : α, a ⋄ 𝟙 = a - - - - set_option trace.Meta.synthInstance true in - example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 - - - class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α - - - class Monoid₂ (α : Type) where - toSemigroup₁ : Semigroup₁ α - toDiaOneClass₁ : DiaOneClass₁ α - - - example {α : Type} [Monoid₁ α] : - (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl - - - /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ - #check Monoid₂.mk - - /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ - #check Monoid₁.mk - - - #check Monoid₁.toSemigroup₁ - #check Monoid₁.toDiaOneClass₁ - - - class Inv₁ (α : Type) where - /-- The inversion function -/ - inv : α → α - - @[inherit_doc] - postfix:max "⁻¹" => Inv₁.inv - - class Group₁ (G : Type) extends Monoid₁ G, Inv G where - inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 - - - lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by - rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] - - - export DiaOneClass₁ (one_dia dia_one) - export Semigroup₁ (dia_assoc) - export Group₁ (inv_dia) - - - example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by - rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] - - - lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := - sorry - - lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := - sorry - - - - - class AddSemigroup₃ (α : Type) extends Add α where - /-- Multiplication is associative -/ - add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) - - @[to_additive AddSemigroup₃] - class Semigroup₃ (α : Type) extends Mul α where - /-- Multiplication is associative -/ - mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) - - class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α - - @[to_additive AddMonoid₃] - class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α - - attribute [to_additive existing] Monoid₃.toMulOneClass - - export Semigroup₃ (mul_assoc₃) - export AddSemigroup₃ (add_assoc₃) - - whatsnew in - @[to_additive] - lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by - rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] - - #check left_neg_eq_right_neg' - - class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where - add_comm : ∀ a b : α, a + b = b + a - - @[to_additive AddCommSemigroup₃] - class CommSemigroup₃ (α : Type) extends Semigroup₃ α where - mul_comm : ∀ a b : α, a * b = b * a - - class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α - - @[to_additive AddCommMonoid₃] - class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α - - class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where - neg_add : ∀ a : G, -a + a = 0 - - @[to_additive AddGroup₃] - class Group₃ (G : Type) extends Monoid₃ G, Inv G where - inv_mul : ∀ a : G, a⁻¹ * a = 1 - - - attribute [simp] Group₃.inv_mul AddGroup₃.neg_add - - - - @[to_additive] - lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := - sorry - - - @[to_additive (attr := simp)] - lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by - sorry - - @[to_additive] - lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by - sorry - - @[to_additive] - lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by - sorry - - class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G - - @[to_additive AddCommGroup₃] - class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G - - - - class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where - /-- Multiplication is left distributive over addition -/ - left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c - /-- Multiplication is right distributive over addition -/ - right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c - - instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := - { Ring₃.toAddGroup₃ with - add_comm := by - sorry } - - instance : Ring₃ ℤ where - add := (· + ·) - add_assoc₃ := add_assoc - zero := 0 - zero_add := by simp - add_zero := by simp - neg := (- ·) - neg_add := by simp - mul := (· * ·) - mul_assoc₃ := mul_assoc - one := 1 - one_mul := by simp - mul_one := by simp - zero_mul := by simp - mul_zero := by simp - left_distrib := Int.mul_add - right_distrib := Int.add_mul - - class LE₁ (α : Type) where - /-- The Less-or-Equal relation. -/ - le : α → α → Prop - - @[inherit_doc] infix:50 " ≤₁ " => LE₁.le - - class Preorder₁ (α : Type) - - class PartialOrder₁ (α : Type) - - class OrderedCommMonoid₁ (α : Type) - - instance : OrderedCommMonoid₁ ℕ where - - class SMul₃ (α : Type) (β : Type) where - /-- Scalar multiplication -/ - smul : α → β → β - - infixr:73 " • " => SMul₃.smul - - - class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where - zero_smul : ∀ m : M, (0 : R) • m = 0 - one_smul : ∀ m : M, (1 : R) • m = m - mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m - add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m - smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n - - instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where - smul := fun r s ↦ r*s - zero_smul := zero_mul - one_smul := one_mul - mul_smul := mul_assoc₃ - add_smul := Ring₃.right_distrib - smul_add := Ring₃.left_distrib - - def nsmul₁ [Zero M] [Add M] : ℕ → M → M - | 0, _ => 0 - | n + 1, a => a + nsmul₁ n a - - def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M - | Int.ofNat n, a => nsmul₁ n a - | Int.negSucc n, a => -nsmul₁ n.succ a - - instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where - smul := zsmul₁ - zero_smul := sorry - one_smul := sorry - mul_smul := sorry - add_smul := sorry - smul_add := sorry - - #synth Module₁ ℤ ℤ -- abGrpModule ℤ - - - class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where - /-- Multiplication by a natural number. -/ - nsmul : ℕ → M → M := nsmul₁ - /-- Multiplication by `(0 : ℕ)` gives `0`. -/ - nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl - /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ - nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl - - instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ - - instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where - add := fun p q ↦ (p.1 + q.1, p.2 + q.2) - add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ - zero := (0, 0) - zero_add := fun a ↦ by ext <;> apply zero_add - add_zero := fun a ↦ by ext <;> apply add_zero - - instance : AddMonoid₄ ℤ where - add := (· + ·) - add_assoc₃ := Int.add_assoc - zero := 0 - zero_add := Int.zero_add - add_zero := Int.add_zero - nsmul := fun n m ↦ (n : ℤ) * m - nsmul_zero := Int.zero_mul - nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m - by rw [Int.add_mul, Int.add_comm, Int.one_mul] - - example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl
src/C_Hierarchies/S02_Morphisms.lean (deleted)
- @@ -1,114 +0,0 @@
  - import Mathlib.Topology.Instances.Real - - - def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := - f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' - structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where - map_one : f 1 = 1 - map_mul : ∀ g g', f (g * g') = f g * f g' - example : Continuous (id : ℝ → ℝ) := continuous_id - @[ext] - structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where - toFun : G → H - map_one : toFun 1 = 1 - map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' - - instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where - coe := MonoidHom₁.toFun - - attribute [coe] MonoidHom₁.toFun - - - example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one - - @[ext] - structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where - toFun : G → H - map_zero : toFun 0 = 0 - map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' - - instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where - coe := AddMonoidHom₁.toFun - - attribute [coe] AddMonoidHom₁.toFun - - @[ext] - structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S - - - - class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where - toFun : F → M → N - map_one : ∀ f : F, toFun f 1 = 1 - map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' - - - def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where - coe := MonoidHomClass₁.toFun - - - class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where - toFun : F → M → N - map_one : ∀ f : F, toFun f 1 = 1 - map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' - - instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where - coe := MonoidHomClass₂.toFun - - attribute [coe] MonoidHomClass₂.toFun - - - instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where - toFun := MonoidHom₁.toFun - map_one := fun f ↦ f.map_one - map_mul := fun f ↦ f.map_mul - - instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where - toFun := fun f ↦ f.toMonoidHom₁.toFun - map_one := fun f ↦ f.toMonoidHom₁.map_one - map_mul := fun f ↦ f.toMonoidHom₁.map_mul - - - lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : - f m * f m' = 1 := by - rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] - - example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := - map_inv_of_inv f h - - example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := - map_inv_of_inv f h - - - - class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends - FunLike F M (fun _ ↦ N) where - map_one : ∀ f : F, f 1 = 1 - map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' - - instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where - coe := MonoidHom₁.toFun - coe_injective' := MonoidHom₁.ext - map_one := MonoidHom₁.map_one - map_mul := MonoidHom₁.map_mul - - - @[ext] - structure OrderPresHom (α β : Type) [LE α] [LE β] where - toFun : α → β - le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' - - @[ext] - structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends - MonoidHom₁ M N, OrderPresHom M N - - class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] - - instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where - - instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : - OrderPresHomClass (OrderPresMonoidHom α β) α β where - - instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : - MonoidHomClass₃ (OrderPresMonoidHom α β) α β - := sorry
src/C_Hierarchies/S03_Subobjects.lean (deleted)
- @@ -1,88 +0,0 @@
  - import Mathlib.GroupTheory.QuotientGroup - - - - @[ext] - structure Submonoid₁ (M : Type) [Monoid M] where - /-- The carrier of a submonoid. -/ - carrier : Set M - /-- The product of two elements of a submonoid belongs to the submonoid. -/ - mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier - /-- The unit element belongs to the submonoid. -/ - one_mem : 1 ∈ carrier - - /-- Submonoids in `M` can be seen as sets in `M`. -/ - instance [Monoid M] : SetLike (Submonoid₁ M) M where - coe := Submonoid₁.carrier - coe_injective' := Submonoid₁.ext - - - - example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem - - example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N - - - example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property - - - instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where - mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ - mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) - one := ⟨1, N.one_mem⟩ - one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) - mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) - - - example [Monoid M] (N : Submonoid₁ M) : Monoid N where - mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ - mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) - one := ⟨1, N.one_mem⟩ - one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) - mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) - - - class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where - mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s - one_mem : ∀ s : S, 1 ∈ s - - instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where - mul_mem := Submonoid₁.mul_mem - one_mem := Submonoid₁.one_mem - - - instance [Monoid M] : Inf (Submonoid₁ M) := - ⟨fun S₁ S₂ => - { carrier := S₁ ∩ S₂ - one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ - mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ - - - example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P - - - def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where - r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z - iseqv := { - refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ - symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ - trans := by - sorry - } - - instance [CommMonoid M] : HasQuotient M (Submonoid M) where - quotient' := fun N ↦ Quotient N.Setoid - - def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid - - instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where - mul := Quotient.map₂' (· * ·) (by - sorry - ) - mul_assoc := by - sorry - one := QuotientMonoid.mk N 1 - one_mul := by - sorry - mul_one := by - sorry
src/C_Hierarchies/solutions/Solutions_S01_Basics.lean (deleted)
- @@ -1,348 +0,0 @@
 - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - - class One₁ (α : Type) where - /-- The element one -/ - one : α - - - #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α - - @[class] structure One₂ (α : Type) where - /-- The element one -/ - one : α - - #check One₂.one - - - example (α : Type) [One₁ α] : α := One₁.one - - example (α : Type) [One₁ α] := (One₁.one : α) - - @[inherit_doc] - notation "𝟙" => One₁.one - - example {α : Type} [One₁ α] : α := 𝟙 - - example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl - - - class Dia₁ (α : Type) where - dia : α → α → α - - infixl:70 " ⋄ " => Dia₁.dia - - - class Semigroup₁ (α : Type) where - toDia₁ : Dia₁ α - /-- Diamond is associative -/ - dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) - - - attribute [instance] Semigroup₁.toDia₁ - - example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b - - - class Semigroup₂ (α : Type) extends Dia₁ α where - /-- Diamond is associative -/ - dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) - - example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b - - class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where - /-- One is a left neutral element for diamond. -/ - one_dia : ∀ a : α, 𝟙 ⋄ a = a - /-- One is a right neutral element for diamond -/ - dia_one : ∀ a : α, a ⋄ 𝟙 = a - - - - set_option trace.Meta.synthInstance true in - example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 - - - class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α - - - class Monoid₂ (α : Type) where - toSemigroup₁ : Semigroup₁ α - toDiaOneClass₁ : DiaOneClass₁ α - - - example {α : Type} [Monoid₁ α] : - (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl - - - /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ - #check Monoid₂.mk - - /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ - #check Monoid₁.mk - - - #check Monoid₁.toSemigroup₁ - #check Monoid₁.toDiaOneClass₁ - - - class Inv₁ (α : Type) where - /-- The inversion function -/ - inv : α → α - - @[inherit_doc] - postfix:max "⁻¹" => Inv₁.inv - - class Group₁ (G : Type) extends Monoid₁ G, Inv G where - inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 - - - lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by - rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] - - - export DiaOneClass₁ (one_dia dia_one) - export Semigroup₁ (dia_assoc) - export Group₁ (inv_dia) - - - example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by - rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] - - - lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := - left_inv_eq_right_inv₁ (inv_dia a) h - - lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := - by rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)] - - - - - - class AddSemigroup₃ (α : Type) extends Add α where - /-- Multiplication is associative -/ - add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) - - @[to_additive AddSemigroup₃] - class Semigroup₃ (α : Type) extends Mul α where - /-- Multiplication is associative -/ - mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) - - class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α - - @[to_additive AddMonoid₃] - class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α - - attribute [to_additive existing] Monoid₃.toMulOneClass - - export Semigroup₃ (mul_assoc₃) - export AddSemigroup₃ (add_assoc₃) - - whatsnew in - @[to_additive] - lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by - rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] - - #check left_neg_eq_right_neg' - - class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where - add_comm : ∀ a b : α, a + b = b + a - - @[to_additive AddCommSemigroup₃] - class CommSemigroup₃ (α : Type) extends Semigroup₃ α where - mul_comm : ∀ a b : α, a * b = b * a - - class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α - - @[to_additive AddCommMonoid₃] - class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α - - class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where - neg_add : ∀ a : G, -a + a = 0 - - @[to_additive AddGroup₃] - class Group₃ (G : Type) extends Monoid₃ G, Inv G where - inv_mul : ∀ a : G, a⁻¹ * a = 1 - - - attribute [simp] Group₃.inv_mul AddGroup₃.neg_add - - - - @[to_additive] - lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := - left_inv_eq_right_inv' (Group₃.inv_mul a) h - - - @[to_additive (attr := simp)] - lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by - rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)] - - @[to_additive] - lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by - simpa [← mul_assoc₃] using congr_arg (a⁻¹ * ·) h - - @[to_additive] - lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by - simpa [mul_assoc₃] using congr_arg (· * a⁻¹) h - - class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G - - @[to_additive AddCommGroup₃] - class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G - - - - class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where - /-- Multiplication is left distributive over addition -/ - left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c - /-- Multiplication is right distributive over addition -/ - right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c - - instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := - { Ring₃.toAddGroup₃ with - add_comm := by - intro a b - have : a + (a + b + b) = a + (b + a + b) := calc - a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃] - _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp - _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib] - _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib] - _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib] - _ = (a + b) + (a + b) := by simp - _ = a + (b + a + b) := by simp [add_assoc₃] - exact add_right_cancel₃ (add_left_cancel₃ this) } - - instance : Ring₃ ℤ where - add := (· + ·) - add_assoc₃ := add_assoc - zero := 0 - zero_add := by simp - add_zero := by simp - neg := (- ·) - neg_add := by simp - mul := (· * ·) - mul_assoc₃ := mul_assoc - one := 1 - one_mul := by simp - mul_one := by simp - zero_mul := by simp - mul_zero := by simp - left_distrib := Int.mul_add - right_distrib := Int.add_mul - - class LE₁ (α : Type) where - /-- The Less-or-Equal relation. -/ - le : α → α → Prop - - @[inherit_doc] infix:50 " ≤₁ " => LE₁.le - - class Preorder₁ (α : Type) - extends LE₁ α where - le_refl : ∀ a : α, a ≤₁ a - le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c - - class PartialOrder₁ (α : Type) - extends Preorder₁ α where - le_antisymm : ∀ a b : α, a ≤₁ b → b ≤₁ a → a = b - - class OrderedCommMonoid₁ (α : Type) - extends PartialOrder₁ α, CommMonoid₃ α where - mul_of_le : ∀ a b : α, a ≤₁ b → ∀ c : α, c * a ≤₁ c * b - - instance : OrderedCommMonoid₁ ℕ where - le := (· ≤ ·) - le_refl := fun _ ↦ le_rfl - le_trans := fun _ _ _ ↦ le_trans - le_antisymm := fun _ _ ↦ le_antisymm - mul := (· * ·) - mul_assoc₃ := mul_assoc - one := 1 - one_mul := one_mul - mul_one := mul_one - mul_comm := mul_comm - mul_of_le := fun _ _ h c ↦ Nat.mul_le_mul_left c h - - class SMul₃ (α : Type) (β : Type) where - /-- Scalar multiplication -/ - smul : α → β → β - - infixr:73 " • " => SMul₃.smul - - - class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where - zero_smul : ∀ m : M, (0 : R) • m = 0 - one_smul : ∀ m : M, (1 : R) • m = m - mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m - add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m - smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n - - instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where - smul := fun r s ↦ r*s - zero_smul := zero_mul - one_smul := one_mul - mul_smul := mul_assoc₃ - add_smul := Ring₃.right_distrib - smul_add := Ring₃.left_distrib - - def nsmul₁ [Zero M] [Add M] : ℕ → M → M - | 0, _ => 0 - | n + 1, a => a + nsmul₁ n a - - def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M - | Int.ofNat n, a => nsmul₁ n a - | Int.negSucc n, a => -nsmul₁ n.succ a - - instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where - smul := zsmul₁ - zero_smul := sorry - one_smul := sorry - mul_smul := sorry - add_smul := sorry - smul_add := sorry - - #synth Module₁ ℤ ℤ -- abGrpModule ℤ - - - class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where - /-- Multiplication by a natural number. -/ - nsmul : ℕ → M → M := nsmul₁ - /-- Multiplication by `(0 : ℕ)` gives `0`. -/ - nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl - /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ - nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl - - instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ - - instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where - add := fun p q ↦ (p.1 + q.1, p.2 + q.2) - add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ - zero := (0, 0) - zero_add := fun a ↦ by ext <;> apply zero_add - add_zero := fun a ↦ by ext <;> apply add_zero - - instance : AddMonoid₄ ℤ where - add := (· + ·) - add_assoc₃ := Int.add_assoc - zero := 0 - zero_add := Int.zero_add - add_zero := Int.add_zero - nsmul := fun n m ↦ (n : ℤ) * m - nsmul_zero := Int.zero_mul - nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m - by rw [Int.add_mul, Int.add_comm, Int.one_mul] - - example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl - - class LT₁ (α : Type) where - /-- The Less-Than relation -/ - lt : α → α → Prop - - @[inherit_doc] infix:50 " <₁ " => LT₁.lt - - class PreOrder₂ (α : Type) extends LE₁ α, LT₁ α where - le_refl : ∀ a : α, a ≤₁ a - le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c - lt := fun a b => a ≤₁ b ∧ ¬b ≤₁ a - lt_iff_le_not_le : ∀ a b : α, a <₁ b ↔ a ≤₁ b ∧ ¬b ≤₁ a := by intros; rfl
src/C_Hierarchies/solutions/Solutions_S02_Morphisms.lean (deleted)
- @@ -1,126 +0,0 @@
  - import Mathlib.Topology.Instances.Real - - - def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := - f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' - structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where - map_one : f 1 = 1 - map_mul : ∀ g g', f (g * g') = f g * f g' - example : Continuous (id : ℝ → ℝ) := continuous_id - @[ext] - structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where - toFun : G → H - map_one : toFun 1 = 1 - map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' - - instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where - coe := MonoidHom₁.toFun - - attribute [coe] MonoidHom₁.toFun - - - example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one - - @[ext] - structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where - toFun : G → H - map_zero : toFun 0 = 0 - map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' - - instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where - coe := AddMonoidHom₁.toFun - - attribute [coe] AddMonoidHom₁.toFun - - @[ext] - structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S - - - - class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where - toFun : F → M → N - map_one : ∀ f : F, toFun f 1 = 1 - map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' - - - def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where - coe := MonoidHomClass₁.toFun - - - class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where - toFun : F → M → N - map_one : ∀ f : F, toFun f 1 = 1 - map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' - - instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where - coe := MonoidHomClass₂.toFun - - attribute [coe] MonoidHomClass₂.toFun - - - instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where - toFun := MonoidHom₁.toFun - map_one := fun f ↦ f.map_one - map_mul := fun f ↦ f.map_mul - - instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where - toFun := fun f ↦ f.toMonoidHom₁.toFun - map_one := fun f ↦ f.toMonoidHom₁.map_one - map_mul := fun f ↦ f.toMonoidHom₁.map_mul - - - lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : - f m * f m' = 1 := by - rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] - - example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := - map_inv_of_inv f h - - example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := - map_inv_of_inv f h - - - - class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends - FunLike F M (fun _ ↦ N) where - map_one : ∀ f : F, f 1 = 1 - map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' - - instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where - coe := MonoidHom₁.toFun - coe_injective' := MonoidHom₁.ext - map_one := MonoidHom₁.map_one - map_mul := MonoidHom₁.map_mul - - - @[ext] - structure OrderPresHom (α β : Type) [LE α] [LE β] where - toFun : α → β - le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' - - @[ext] - structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends - MonoidHom₁ M N, OrderPresHom M N - - class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] - extends FunLike F α (fun _ ↦ β) where - le_of_le : ∀ (f : F) a a', a ≤ a' → f a ≤ f a' - - instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where - coe := OrderPresHom.toFun - coe_injective' := OrderPresHom.ext - le_of_le := OrderPresHom.le_of_le - - instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : - OrderPresHomClass (OrderPresMonoidHom α β) α β where - coe := fun f ↦ f.toOrderPresHom.toFun - coe_injective' := OrderPresMonoidHom.ext - le_of_le := fun f ↦ f.toOrderPresHom.le_of_le - - instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : - MonoidHomClass₃ (OrderPresMonoidHom α β) α β - where - coe := fun f ↦ f.toOrderPresHom.toFun - coe_injective' := OrderPresMonoidHom.ext - map_one := fun f ↦ f.toMonoidHom₁.map_one - map_mul := fun f ↦ f.toMonoidHom₁.map_mul
src/C_Hierarchies/solutions/Solutions_S03_Subobjects.lean (deleted)
- @@ -1,126 +0,0 @@
  - import Mathlib.GroupTheory.QuotientGroup - - - - @[ext] - structure Submonoid₁ (M : Type) [Monoid M] where - /-- The carrier of a submonoid. -/ - carrier : Set M - /-- The product of two elements of a submonoid belongs to the submonoid. -/ - mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier - /-- The unit element belongs to the submonoid. -/ - one_mem : 1 ∈ carrier - - /-- Submonoids in `M` can be seen as sets in `M`. -/ - instance [Monoid M] : SetLike (Submonoid₁ M) M where - coe := Submonoid₁.carrier - coe_injective' := Submonoid₁.ext - - - - example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem - - example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N - - - example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property - - - instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where - mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ - mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) - one := ⟨1, N.one_mem⟩ - one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) - mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) - - - example [Monoid M] (N : Submonoid₁ M) : Monoid N where - mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ - mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) - one := ⟨1, N.one_mem⟩ - one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) - mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) - - - class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where - mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s - one_mem : ∀ s : S, 1 ∈ s - - instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where - mul_mem := Submonoid₁.mul_mem - one_mem := Submonoid₁.one_mem - - @[ext] - structure Subgroup₁ (G : Type) [Group G] extends Submonoid₁ G where - /-- The inverse of an element of a subgroup belongs to the subgroup. -/ - inv_mem {a} : a ∈ carrier → a⁻¹ ∈ carrier - - - /-- Subgroups in `M` can be seen as sets in `M`. -/ - instance [Group G] : SetLike (Subgroup₁ G) G where - coe := fun H ↦ H.toSubmonoid₁.carrier - coe_injective' := Subgroup₁.ext - - instance [Group G] (H : Subgroup₁ G) : Group H := - { SubMonoid₁Monoid H.toSubmonoid₁ with - inv := fun x ↦ ⟨x⁻¹, H.inv_mem x.property⟩ - mul_left_inv := fun x ↦ SetCoe.ext (mul_left_inv (x : G)) } - - class SubgroupClass₁ (S : Type _) (G : Type) [Group G] [SetLike S G] - extends SubmonoidClass₁ S G : Prop where - inv_mem : ∀ (s : S) {a : G}, a ∈ s → a⁻¹ ∈ s - - instance [Group G] : SubmonoidClass₁ (Subgroup₁ G) G where - mul_mem := fun H ↦ H.toSubmonoid₁.mul_mem - one_mem := fun H ↦ H.toSubmonoid₁.one_mem - - instance [Group G] : SubgroupClass₁ (Subgroup₁ G) G := - { (inferInstance : SubmonoidClass₁ (Subgroup₁ G) G) with - inv_mem := Subgroup₁.inv_mem } - - - instance [Monoid M] : Inf (Submonoid₁ M) := - ⟨fun S₁ S₂ => - { carrier := S₁ ∩ S₂ - one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ - mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ - - - example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P - - - def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where - r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z - iseqv := { - refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ - symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ - trans := by - rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩ - refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ - rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc] - } - - instance [CommMonoid M] : HasQuotient M (Submonoid M) where - quotient' := fun N ↦ Quotient N.Setoid - - def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid - - instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where - mul := Quotient.map₂' (· * ·) (by - rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩ - refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ - rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha, - mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc] - ) - mul_assoc := by - rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ - apply Quotient.sound - dsimp only - rw [mul_assoc] - apply @Setoid.refl M N.Setoid - one := QuotientMonoid.mk N 1 - one_mul := by - rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul] ; apply @Setoid.refl M N.Setoid - mul_one := by - rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one] ; apply @Setoid.refl M N.Setoid -
src/C_Introduction/S01_Getting_Started.lean (deleted)
- @@ -1,2 +0,0 @@
  - #eval "Hello, World!" -
src/C_Introduction/S02_Overview.lean (deleted)
- @@ -1,56 +0,0 @@
  - import Mathlib.Data.Nat.Basic - import Mathlib.Data.Nat.Parity - import Mathlib.Tactic - - open Nat - - -- These are pieces of data. - #check 2 + 2 - - def f (x : ℕ) := - x + 3 - - #check f - - -- These are propositions, of type `Prop`. - #check 2 + 2 = 4 - - def FermatLastTheorem := - ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n - - #check FermatLastTheorem - - -- These are proofs of propositions. - theorem easy : 2 + 2 = 4 := - rfl - - #check easy - - theorem hard : FermatLastTheorem := - sorry - - #check hard - - -- Here are some proofs. - example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => - have hmn : m * n = m * k + m * k := by rw [hk, mul_add] - show ∃ l, m * n = l + l from ⟨_, hmn⟩ - - example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ - - example : ∀ m n : Nat, Even n → Even (m * n) := by - -- say m and n are natural numbers, and assume n=2*k - rintro m n ⟨k, hk⟩ - -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. - use m * k - -- substitute in for n - rw [hk] - -- and now it's obvious - ring - - example : ∀ m n : Nat, Even n → Even (m * n) := by - rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring - - example : ∀ m n : Nat, Even n → Even (m * n) := by - intros; simp [*, parity_simps] -
src/C_Introduction/solutions/Solutions_S01_Getting_Started.lean (deleted)
src/C_Introduction/solutions/Solutions_S02_Overview.lean (deleted)
- @@ -1,7 +0,0 @@
  - import Mathlib.Data.Nat.Basic - import Mathlib.Data.Nat.Parity - import Mathlib.Tactic - - open Nat - - -- There are no exercises in this section.
src/C_Logic/S01_Implication_and_the_Universal_Quantifier.lean (deleted)
- @@ -1,178 +0,0 @@
 - import Mathlib.Data.Real.Basic - - #check ∀ x : ℝ, 0 ≤ x → abs x = x - - #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε - - theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := - sorry - - section - variable (a b δ : ℝ) - variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) - variable (ha : abs a < δ) (hb : abs b < δ) - - #check my_lemma a b δ - #check my_lemma a b δ h₀ h₁ - #check my_lemma a b δ h₀ h₁ ha hb - - end - - theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := - sorry - - section - variable (a b δ : ℝ) - variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) - variable (ha : abs a < δ) (hb : abs b < δ) - - #check my_lemma2 h₀ h₁ ha hb - - end - - theorem my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by - intro x y ε epos ele1 xlt ylt - sorry - - theorem my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by - intro x y ε epos ele1 xlt ylt - calc - abs (x * y) = abs x * abs y := sorry - _ ≤ abs x * ε := sorry - _ < 1 * ε := sorry - _ = ε := sorry - - def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, f x ≤ a - - def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, a ≤ f x - - section - variable (f g : ℝ → ℝ) (a b : ℝ) - - example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by - intro x - dsimp - apply add_le_add - apply hfa - apply hgb - - example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := - sorry - - example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := - sorry - - example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : - FnUb (fun x => f x * g x) (a * b) := - sorry - - end - - section - variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] - - #check @add_le_add - - def FnUb' (f : α → R) (a : R) : Prop := - ∀ x, f x ≤ a - - theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : - FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) - - end - - example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := - @h - - section - variable (f g : ℝ → ℝ) - - example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by - intro a b aleb - apply add_le_add - apply mf aleb - apply mg aleb - - example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := - fun a b aleb => add_le_add (mf aleb) (mg aleb) - - example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := - sorry - - example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := - sorry - - def FnEven (f : ℝ → ℝ) : Prop := - ∀ x, f x = f (-x) - - def FnOdd (f : ℝ → ℝ) : Prop := - ∀ x, f x = -f (-x) - - example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by - intro x - calc - (fun x => f x + g x) x = f x + g x := rfl - _ = f (-x) + g (-x) := by rw [ef, eg] - - - example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by - sorry - - example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by - sorry - - example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by - sorry - - end - - section - - variable {α : Type _} (r s t : Set α) - - example : s ⊆ s := by - intro x xs - exact xs - - theorem Subset.refl : s ⊆ s := fun x xs => xs - - theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by - sorry - - end - - section - - variable {α : Type _} [PartialOrder α] - - variable (s : Set α) (a b : α) - - def SetUb (s : Set α) (a : α) := - ∀ x, x ∈ s → x ≤ a - - example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := - sorry - - end - - section - - open Function - - example (c : ℝ) : Injective fun x => x + c := by - intro x₁ x₂ h' - exact (add_left_inj c).mp h' - - example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by - sorry - - variable {α : Type _} {β : Type _} {γ : Type _} - - variable {g : β → γ} {f : α → β} - - example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by - sorry - - end
src/C_Logic/S02_The_Existential_Quantifier.lean (deleted)
- @@ -1,135 +0,0 @@
 - import Mathlib.Data.Real.Basic - - example : ∃ x : ℝ, 2 < x ∧ x < 3 := by - use 5 / 2 - norm_num - - example : ∃ x : ℝ, 2 < x ∧ x < 3 := - have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num - ⟨5 / 2, h⟩ - - example : ∃ x : ℝ, 2 < x ∧ x < 3 := - ⟨5 / 2, by norm_num⟩ - - def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, f x ≤ a - - def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, a ≤ f x - - def FnHasUb (f : ℝ → ℝ) := - ∃ a, FnUb f a - - def FnHasLb (f : ℝ → ℝ) := - ∃ a, FnLb f a - - theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : - FnUb (fun x => f x + g x) (a + b) := - fun x => add_le_add (hfa x) (hgb x) - - section - - variable {f g : ℝ → ℝ} - - example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by - cases' ubf with a ubfa - cases' ubg with b ubfb - use a + b - apply fnUb_add ubfa ubfb - - example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by - sorry - - example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by - sorry - - example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by - rcases ubf with ⟨a, ubfa⟩ - rcases ubg with ⟨b, ubfb⟩ - exact ⟨a + b, fnUb_add ubfa ubfb⟩ - - example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by - rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ - exact ⟨a + b, fnUb_add ubfa ubfb⟩ - - example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := - fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ - - end - - section - - variable {α : Type _} [CommRing α] - - def SumOfSquares (x : α) := - ∃ a b, x = a ^ 2 + b ^ 2 - - theorem sumOfSquares_mul {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : - SumOfSquares (x * y) := by - rcases sosx with ⟨a, b, xeq⟩ - rcases sosy with ⟨c, d, yeq⟩ - rw [xeq, yeq] - use a * c - b * d, a * d + b * c - ring - - theorem sumOfSquares_mul' {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : - SumOfSquares (x * y) := by - rcases sosx with ⟨a, b, rfl⟩ - rcases sosy with ⟨c, d, rfl⟩ - use a * c - b * d, a * d + b * c - ring - - end - - section - - variable {a b c : ℕ} - - example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by - cases' divab with d beq - cases' divbc with e ceq - rw [ceq, beq] - use d * e; ring - - example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by - sorry - - end - - section - - open Function - - example {c : ℝ} : Surjective fun x => x + c := by - intro x - use x - c - dsimp; ring - - example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by - sorry - - example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by - field_simp [h] - ring - - example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by - cases' h 2 with x hx - use x - rw [hx] - norm_num - - end - - section - - open Function - - variable {α : Type _} {β : Type _} {γ : Type _} - - variable {g : β → γ} {f : α → β} - - example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by - sorry - - end -
src/C_Logic/S03_Negation.lean (deleted)
- @@ -1,143 +0,0 @@
 - import Mathlib.Data.Real.Basic - - section - - variable (a b : ℝ) - - example (h : a < b) : ¬b < a := by - intro h' - have : a < a := lt_trans h h' - apply lt_irrefl a this - - def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, f x ≤ a - - def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, a ≤ f x - - def FnHasUb (f : ℝ → ℝ) := - ∃ a, FnUb f a - - def FnHasLb (f : ℝ → ℝ) := - ∃ a, FnLb f a - - variable (f : ℝ → ℝ) - - example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by - intro fnub - cases' fnub with a fnuba - cases' h a with x hx - have : f x ≤ a := fnuba x - linarith - - example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := - sorry - - example : ¬FnHasUb fun x => x := - sorry - - #check (not_le_of_gt : a > b → ¬a ≤ b) - - #check (not_lt_of_ge : a ≥ b → ¬a < b) - - #check (lt_of_not_ge : ¬a ≥ b → a < b) - - #check (le_of_not_gt : ¬a > b → a ≤ b) - - example (h : Monotone f) (h' : f a < f b) : a (0 : ℝ) - have monof : Monotone f := by sorry - have h' : f 1 ≤ f 0 := le_refl _ - sorry - - example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by - sorry - - end - - section - - variable {α : Type _} (P : α → Prop) (Q : Prop) - - example (h : ¬∃ x, P x) : ∀ x, ¬P x := by - sorry - - example (h : ∀ x, ¬P x) : ¬∃ x, P x := by - sorry - - example (h : ¬∀ x, P x) : ∃ x, ¬P x := by - sorry - - example (h : ∃ x, ¬P x) : ¬∀ x, P x := by - sorry - - example (h : ¬∀ x, P x) : ∃ x, ¬P x := by - by_contra h' - apply h - intro x - show P x - by_contra h'' - exact h' ⟨x, h''⟩ - - example (h : ¬¬Q) : Q := by - sorry - - example (h : Q) : ¬¬Q := by - sorry - - end - - section - - variable (f : ℝ → ℝ) - - example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by - sorry - - example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by - push_neg at h - exact h - - example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by - simp only [FnHasUb, FnUb] at h - push_neg at h - exact h - - example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by - sorry - - example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by - contrapose! h - exact h - - example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by - contrapose! h - use x / 2 - constructor <;> linarith - - end - - section - - variable (a : ℕ) - - example (h : 0 < 0) : a > 37 := by - exfalso - apply lt_irrefl 0 h - - example (h : 0 < 0) : a > 37 := - absurd h (lt_irrefl 0) - - example (h : 0 < 0) : a > 37 := by - have h' : ¬0 < 0 := lt_irrefl 0 - contradiction - - end -
src/C_Logic/S04_Conjunction_and_Bi-implication.lean (deleted)
- @@ -1,138 +0,0 @@
 - import Mathlib.Data.Real.Basic - import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum - - example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by - constructor - · assumption - intro h - apply h₁ - rw [h] - - example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := - ⟨h₀, fun h => h₁ (by rw [h])⟩ - - example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := - have h : x ≠ y := by - contrapose! h₁ - rw [h₁] - ⟨h₀, h⟩ - - example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - cases' h with h₀ h₁ - contrapose! h₁ - exact le_antisymm h₀ h₁ - - example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by - rintro ⟨h₀, h₁⟩ h' - exact h₁ (le_antisymm h₀ h') - - example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := - fun ⟨h₀, h₁⟩ h' => h₁ (le_antisymm h₀ h') - - example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - intro h' - apply h.right - exact le_antisymm h.left h' - - example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := - fun h' => h.right (le_antisymm h.left h') - - example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := - sorry - - example : ∃ x : ℝ, 2 < x ∧ x < 4 := - ⟨5 / 2, by norm_num, by norm_num⟩ - - example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by - rintro ⟨z, xltz, zlty⟩ - exact lt_trans xltz zlty - - example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := - fun ⟨z, xltz, zlty⟩ => lt_trans xltz zlty - - example : ∃ x : ℝ, 2 < x ∧ x < 4 := by - use 5 / 2 - constructor <;> norm_num - - example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by - use 5 - use 7 - norm_num - sorry - - example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by - rintro ⟨h₀, h₁⟩ - use h₀ - exact fun h' => h₁ (le_antisymm h₀ h') - - example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by - constructor - · contrapose! - rintro rfl - rfl - contrapose! - exact le_antisymm h - - example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := - ⟨fun h₀ h₁ => h₀ (by rw [h₁]), fun h₀ h₁ => h₀ (le_antisymm h h₁)⟩ - - example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := - sorry - - theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := - have h' : x ^ 2 = 0 := by sorry - pow_eq_zero h' - - example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := - sorry - - section - - example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by - rw [abs_lt] - intro h - constructor <;> linarith - - example : 3 ∣ Nat.gcd 6 15 := by - rw [Nat.dvd_gcd_iff] - constructor <;> norm_num - - end - - theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by - rw [Monotone] - push_neg - rfl - - example : ¬Monotone fun x : ℝ => -x := by - sorry - - section - - variable {α : Type _} [PartialOrder α] - - variable (a b : α) - - example : a < b ↔ a ≤ b ∧ a ≠ b := by - rw [lt_iff_le_not_le] - sorry - - end - - section - - variable {α : Type _} [Preorder α] - - variable (a b c : α) - - example : ¬a < a := by - rw [lt_iff_le_not_le] - sorry - - example : a < b → b < c → a < c := by - simp only [lt_iff_le_not_le] - sorry - - end -
src/C_Logic/S05_Disjunction.lean (deleted)
- @@ -1,102 +0,0 @@
 - import Mathlib.Data.Real.Basic - - section - - variable {x y : ℝ} - - example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by - left - linarith [pow_two_nonneg x] - - example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by - right - linarith [pow_two_nonneg x] - - example (h : y > 0) : y > 0 ∨ y < -1 := - Or.inl h - - example (h : y < -1) : y > 0 ∨ y < -1 := - Or.inr h - - example : x < abs y → x < y ∨ x < -y := by - cases' le_or_gt 0 y with h h - · rw [abs_of_nonneg h] - intro h - left - exact h - rw [abs_of_neg h] - intro h; right; exact h - - namespace MyAbs - - theorem le_abs_self (x : ℝ) : x ≤ abs x := by - sorry - - theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by - sorry - - theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by - sorry - - theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by - sorry - - theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by - sorry - - end MyAbs - - end - - example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by - rcases lt_trichotomy x 0 with (xlt | xeq | xgt) - · left - exact xlt - · contradiction - right; exact xgt - - example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by - rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) - · rw [mul_assoc] - apply dvd_mul_right - rw [mul_comm, mul_assoc] - apply dvd_mul_right - - example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by - sorry - - example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by - sorry - - example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by - sorry - - section - - variable {R : Type _} [CommRing R] [IsDomain R] - - variable (x y : R) - - example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by - sorry - - example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by - sorry - - end - - example (P : Prop) : ¬¬P → P := by - intro h - cases em P - · assumption - contradiction - - example (P : Prop) : ¬¬P → P := by - intro h - by_cases h' : P - · assumption - contradiction - - example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by - sorry -
src/C_Logic/S06_Sequences_and_Convergence.lean (deleted)
- @@ -1,96 +0,0 @@
 - import Mathlib.Data.Real.Basic - - def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := - ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε - - example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by - ext - ring - - example (a b : ℝ) : abs a = abs (a - b + b) := by - congr - ring - - example {a : ℝ} (h : 1 < a) : a < a * a := by - convert(mul_lt_mul_right _).2 h - · rw [one_mul] - exact lt_trans zero_lt_one h - - theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by - intro ε εpos - use 0 - intro n nge; dsimp - rw [sub_self, abs_zero] - apply εpos - - theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : - ConvergesTo (fun n => s n + t n) (a + b) := by - intro ε εpos - dsimp - have ε2pos : 0 < ε / 2 := by linarith - cases' cs (ε / 2) ε2pos with Ns hs - cases' ct (ε / 2) ε2pos with Nt ht - use max Ns Nt - sorry - - theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : - ConvergesTo (fun n => c * s n) (c * a) := by - by_cases h : c = 0 - · convert convergesTo_const 0 - · rw [h, MulZeroClass.zero_mul] - rw [h, MulZeroClass.zero_mul] - have acpos : 0 < abs c := abs_pos.mpr h - sorry - - theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : - ∃ N b, ∀ n, N ≤ n → abs (s n) s n * t n) 0 := by - intro ε εpos - dsimp - rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ - have Bpos : 0 0 := div_pos εpos Bpos - cases' ct _ pos₀ with N₁ h₁ - sorry - - theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : - ConvergesTo (fun n => s n * t n) (a * b) := by - have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by - apply aux cs - convert convergesTo_add ct (convergesTo_const (-b)) - ring - have := convergesTo_add h₁ (convergesTo_mul_const b cs) - convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 - · ext; ring - ring - - theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : - a = b := by - by_contra abne - have : abs (a - b) > 0 := by sorry - let ε := abs (a - b) / 2 - have εpos : ε > 0 := by - change abs (a - b) / 2 > 0 - linarith - cases' sa ε εpos with Na hNa - cases' sb ε εpos with Nb hNb - let N := max Na Nb - have absa : abs (s N - a) < ε := by sorry - have absb : abs (s N - b) < ε := by sorry - have : abs (a - b) < abs (a - b) := by sorry - exact lt_irrefl _ this - - section - - variable {α : Type _} [LinearOrder α] - - def ConvergesTo' (s : α → ℝ) (a : ℝ) := - ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε - - end -
src/C_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean (deleted)
- @@ -1,131 +0,0 @@
  - import Mathlib.Data.Real.Basic - - def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, f x ≤ a - - def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, a ≤ f x - - section - variable (f g : ℝ → ℝ) (a b : ℝ) - - example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := by - intro x - apply add_le_add - apply hfa - apply hgb - - example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := by - intro x - apply mul_nonneg - apply nnf - apply nng - - example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : - FnUb (fun x => f x * g x) (a * b) := by - intro x - apply mul_le_mul - apply hfa - apply hfb - apply nng - apply nna - - end - - section - variable (f g : ℝ → ℝ) - - example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := by - intro a b aleb - apply mul_le_mul_of_nonneg_left _ nnc - apply mf aleb - - example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := - fun a b aleb => mul_le_mul_of_nonneg_left (mf aleb) nnc - - example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := by - intro a b aleb - apply mf - apply mg - apply aleb - - example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := - fun a b aleb => mf (mg aleb) - - def FnEven (f : ℝ → ℝ) : Prop := - ∀ x, f x = f (-x) - - def FnOdd (f : ℝ → ℝ) : Prop := - ∀ x, f x = -f (-x) - - example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by - intro x - calc - (fun x => f x * g x) x = f x * g x := rfl - _ = f (-x) * g (-x) := by rw [of, og, neg_mul_neg] - - - example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by - intro x - dsimp - rw [ef, og, neg_mul_eq_mul_neg] - - example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by - intro x - dsimp - rw [og, ← ef] - - end - - section - - variable {α : Type _} (r s t : Set α) - - example : r ⊆ s → s ⊆ t → r ⊆ t := by - intro rsubs ssubt x xr - apply ssubt - apply rsubs - apply xr - - theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := - fun rsubs ssubt x xr => ssubt (rsubs xr) - - end - - section - - variable {α : Type _} [PartialOrder α] - - variable (s : Set α) (a b : α) - - def SetUb (s : Set α) (a : α) := - ∀ x, x ∈ s → x ≤ a - - example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := by - intro x xs - apply le_trans (h x xs) h' - - example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := - fun x xs => le_trans (h x xs) h' - - end - - section - - open Function - - example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by - intro x₁ x₂ h' - apply (mul_right_inj' h).mp h' - - variable {α : Type _} {β : Type _} {γ : Type _} - - variable {g : β → γ} {f : α → β} - - example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by - intro x₁ x₂ h - apply injf - apply injg - apply h - - end
src/C_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean (deleted)
- @@ -1,85 +0,0 @@
  - import Mathlib.Data.Real.Basic - - def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, f x ≤ a - - def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, a ≤ f x - - def FnHasUb (f : ℝ → ℝ) := - ∃ a, FnUb f a - - def FnHasLb (f : ℝ → ℝ) := - ∃ a, FnLb f a - - theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : - FnUb (fun x => f x + g x) (a + b) := - fun x => add_le_add (hfa x) (hgb x) - - section - - variable {f g : ℝ → ℝ} - - example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by - cases' lbf with a lbfa - cases' lbg with b lbgb - use a + b - intro x - exact add_le_add (lbfa x) (lbgb x) - - example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by - cases' ubf with a lbfa - use c * a - intro x - exact mul_le_mul_of_nonneg_left (lbfa x) h - - end - - section - - variable {a b c : ℕ} - - example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by - rcases divab with ⟨d, rfl⟩ - rcases divbc with ⟨e, rfl⟩ - use d * e; ring - - example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by - rcases divab with ⟨d, rfl⟩ - rcases divac with ⟨e, rfl⟩ - use d + e; ring - - end - - section - - 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] - - example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by - intro x - use x / c - field_simp [h] ; ring - - end - - section - - open Function - - variable {α : Type _} {β : Type _} {γ : Type _} - - variable {g : β → γ} {f : α → β} - - example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by - intro z - rcases surjg z with ⟨y, rfl⟩ - rcases surjf y with ⟨x, rfl⟩ - use x - - end -
src/C_Logic/solutions/Solutions_S03_Negation.lean (deleted)
- @@ -1,113 +0,0 @@
 - import Mathlib.Data.Real.Basic - - section - - variable (a b : ℝ) - - def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, f x ≤ a - - def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := - ∀ x, a ≤ f x - - def FnHasUb (f : ℝ → ℝ) := - ∃ a, FnUb f a - - def FnHasLb (f : ℝ → ℝ) := - ∃ a, FnLb f a - - variable (f : ℝ → ℝ) - - example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by - rintro ⟨a, ha⟩ - rcases h a with ⟨x, hx⟩ - have := ha x - linarith - - example : ¬FnHasUb fun x => x := by - rintro ⟨a, ha⟩ - have : a + 1 ≤ a := ha (a + 1) - linarith - - example (h : Monotone f) (h' : f a < f b) : a < b := by - apply lt_of_not_ge - intro h'' - apply absurd h' - apply not_lt_of_ge (h h'') - - example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by - intro h'' - apply absurd h' - apply not_lt_of_ge - apply h'' h - - example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by - intro h - let f := fun x : ℝ => (0 : ℝ) - have monof : Monotone f := by - intro a b leab - rfl - have h' : f 1 ≤ f 0 := le_refl _ - have : (1 : ℝ) ≤ 0 := h monof h' - linarith - - example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by - apply le_of_not_gt - intro h' - linarith [h _ h'] - - end - - section - - variable {α : Type _} (P : α → Prop) (Q : Prop) - - example (h : ¬∃ x, P x) : ∀ x, ¬P x := by - intro x Px - apply h - use x - exact Px - - example (h : ∀ x, ¬P x) : ¬∃ x, P x := by - rintro ⟨x, Px⟩ - exact h x Px - - example (h : ∃ x, ¬P x) : ¬∀ x, P x := by - intro h' - rcases h with ⟨x, nPx⟩ - apply nPx - apply h' - - example (h : ¬¬Q) : Q := by - by_contra h' - exact h h' - - example (h : Q) : ¬¬Q := by - intro h' - exact h' h - - end - - section - - variable (f : ℝ → ℝ) - - example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by - intro a - by_contra h' - apply h - use a - intro x - apply le_of_not_gt - intro h'' - apply h' - use x - exact h'' - - example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by - rw [Monotone] at h - push_neg at h - exact h - - end -
src/C_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean (deleted)
- @@ -1,97 +0,0 @@
 - import Mathlib.Data.Real.Basic - import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum - - example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by - cases' h with h0 h1 - constructor - · exact h0 - intro h2 - apply h1 - apply Nat.dvd_antisymm h0 h2 - - example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := by - constructor - · rintro ⟨h0, h1⟩ - constructor - · exact h0 - intro h2 - apply h1 - rw [h2] - rintro ⟨h0, h1⟩ - constructor - · exact h0 - intro h2 - apply h1 - apply le_antisymm h0 h2 - - theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := - have h' : x ^ 2 = 0 := by linarith [pow_two_nonneg x, pow_two_nonneg y] - pow_eq_zero h' - - example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - constructor - · intro h - constructor - · exact aux h - rw [add_comm] at h - exact aux h - rintro ⟨rfl, rfl⟩ - norm_num - - theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by - rw [Monotone] - push_neg - rfl - - example : ¬Monotone fun x : ℝ => -x := by - rw [not_monotone_iff] - use 0, 1 - norm_num - - section - - variable {α : Type _} [PartialOrder α] - - variable (a b : α) - - example : a < b ↔ a ≤ b ∧ a ≠ b := by - rw [lt_iff_le_not_le] - constructor - · rintro ⟨h0, h1⟩ - constructor - · exact h0 - intro h2 - apply h1 - rw [h2] - rintro ⟨h0, h1⟩ - constructor - · exact h0 - intro h2 - apply h1 - apply le_antisymm h0 h2 - - end - - section - - variable {α : Type _} [Preorder α] - - variable (a b c : α) - - example : ¬a < a := by - rw [lt_iff_le_not_le] - rintro ⟨h0, h1⟩ - exact h1 h0 - - example : a < b → b < c → a < c := by - simp only [lt_iff_le_not_le] - rintro ⟨h0, h1⟩ ⟨h2, h3⟩ - constructor - · apply le_trans h0 h2 - intro h4 - apply h1 - apply le_trans h2 h4 - - end -
src/C_Logic/solutions/Solutions_S05_Disjunction.lean (deleted)
- @@ -1,142 +0,0 @@
 - import Mathlib.Data.Real.Basic - - section - - variable {x y : ℝ} - - namespace MyAbs - - theorem le_abs_self (x : ℝ) : x ≤ abs x := by - cases' le_or_gt 0 x with h h - · rw [abs_of_nonneg h] - rw [abs_of_neg h] - linarith - - theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by - cases' le_or_gt 0 x with h h - · rw [abs_of_nonneg h] - linarith - rw [abs_of_neg h] - - theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by - cases' le_or_gt 0 (x + y) with h h - · rw [abs_of_nonneg h] - linarith [le_abs_self x, le_abs_self y] - rw [abs_of_neg h] - linarith [neg_le_abs_self x, neg_le_abs_self y] - - theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by - cases' le_or_gt 0 y with h h - · rw [abs_of_nonneg h] - constructor - · intro h' - left - exact h' - intro h' - cases' h' with h' h' - · exact h' - linarith - rw [abs_of_neg h] - constructor - · intro h' - right - exact h' - intro h' - cases' h' with h' h' - · linarith - exact h' - - theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by - cases' le_or_gt 0 x with h h - · rw [abs_of_nonneg h] - constructor - · intro h' - constructor - · linarith - exact h' - intro h' - cases' h' with h1 h2 - exact h2 - rw [abs_of_neg h] - constructor - · intro h' - constructor - · linarith - linarith - intro h' - linarith - - end MyAbs - - end - - example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by - rcases h with ⟨x, y, rfl | rfl⟩ <;> linarith [sq_nonneg x, sq_nonneg y] - - example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by - have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] - have h'' : (x + 1) * (x - 1) = 0 := by - rw [← h'] - ring - cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 - · right - exact eq_neg_iff_add_eq_zero.mpr h1 - left - exact eq_of_sub_eq_zero h1 - - example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by - have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] - have h'' : (x + y) * (x - y) = 0 := by - rw [← h'] - ring - cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 - · right - exact eq_neg_iff_add_eq_zero.mpr h1 - left - exact eq_of_sub_eq_zero h1 - - section - - variable {R : Type _} [CommRing R] [IsDomain R] - - variable (x y : R) - - example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by - have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] - have h'' : (x + 1) * (x - 1) = 0 := by - rw [← h'] - ring - cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 - · right - exact eq_neg_iff_add_eq_zero.mpr h1 - left - exact eq_of_sub_eq_zero h1 - - example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by - have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] - have h'' : (x + y) * (x - y) = 0 := by - rw [← h'] - ring - cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 - · right - exact eq_neg_iff_add_eq_zero.mpr h1 - left - exact eq_of_sub_eq_zero h1 - - end - - example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by - constructor - · intro h - by_cases h' : P - · right - exact h h' - left - exact h' - rintro (h | h) - · intro h' - exact absurd h' h - intro - exact h - -
src/C_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean (deleted)
- @@ -1,126 +0,0 @@
 - import Mathlib.Data.Real.Basic - - def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := - ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε - - theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by - intro ε εpos - use 0 - intro n nge; dsimp - rw [sub_self, abs_zero] - apply εpos - - theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : - ConvergesTo (fun n => s n + t n) (a + b) := by - intro ε εpos - dsimp - have ε2pos : 0 < ε / 2 := by linarith - cases' cs (ε / 2) ε2pos with Ns hs - cases' ct (ε / 2) ε2pos with Nt ht - use max Ns Nt - intro n hn - have ngeNs : n ≥ Ns := le_of_max_le_left hn - have ngeNt : n ≥ Nt := le_of_max_le_right hn - calc - |s n + t n - (a + b)| = |s n - a + (t n - b)| := by - congr - ring - _ ≤ |s n - a| + |t n - b| := (abs_add _ _) - _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt)) - _ = ε := by norm_num - - - theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : - ConvergesTo (fun n => c * s n) (c * a) := by - by_cases h : c = 0 - · convert convergesTo_const 0 - · rw [h, MulZeroClass.zero_mul] - rw [h, MulZeroClass.zero_mul] - have acpos : 0 < abs c := abs_pos.mpr h - intro ε εpos - dsimp - have εcpos : 0 < ε / abs c := by apply div_pos εpos acpos - cases' cs (ε / abs c) εcpos with Ns hs - use Ns - intro n ngt - 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 - - - theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : - ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by - cases' cs 1 zero_lt_one with N h - use N, abs a + 1 - intro n ngt - calc - |s n| = |s n - a + a| := by - congr - abel - _ ≤ |s n - a| + |a| := (abs_add _ _) - _ < |a| + 1 := by linarith [h n ngt] - - - theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : - ConvergesTo (fun n => s n * t n) 0 := by - intro ε εpos - dsimp - rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ - have Bpos : 0 0 := div_pos εpos Bpos - cases' ct _ pos₀ with N₁ h₁ - use max N₀ N₁ - intro n ngt - have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt - have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt - 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 - - theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : - ConvergesTo (fun n => s n * t n) (a * b) := by - have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by - apply aux cs - convert convergesTo_add ct (convergesTo_const (-b)) - ring - have := convergesTo_add h₁ (convergesTo_mul_const b cs) - convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 - · ext; ring - ring - - theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : - a = b := by - by_contra abne - have : abs (a - b) > 0 := by - apply lt_of_le_of_ne - · apply abs_nonneg - intro h'' - apply abne - apply eq_of_abs_sub_eq_zero h''.symm - let ε := abs (a - b) / 2 - have εpos : ε > 0 := by - change abs (a - b) / 2 > 0 - linarith - cases' sa ε εpos with Na hNa - cases' sb ε εpos with Nb hNb - let N := max Na Nb - have absa : abs (s N - a) < ε := by - apply hNa - apply le_max_left - have absb : abs (s N - b) < ε := by - apply hNb - apply le_max_right - have : abs (a - b) < abs (a - b) - calc - abs (a - b) = abs (-(s N - a) + (s N - b)) := by - congr - ring - _ ≤ abs (-(s N - a)) + abs (s N - b) := (abs_add _ _) - _ = abs (s N - a) + abs (s N - b) := by rw [abs_neg] - _ < ε + ε := (add_lt_add absa absb) - _ = abs (a - b) := by norm_num - - exact lt_irrefl _ this -
src/C_Number_Theory/S01_Irrational_Roots.lean (deleted)
- @@ -1,118 +0,0 @@
 - import Mathlib.Data.Nat.Factorization.Basic - import Mathlib.Data.Nat.Prime - import Mathlib.Tactic.NormNum.GCD - import Mathlib.Tactic.NormNum.Prime - - #print Nat.coprime - - example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := - h - - example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := by - rw [Nat.coprime] at h - exact h - - example : Nat.coprime 12 7 := by norm_num - - example : Nat.gcd 12 8 = 4 := by norm_num - - #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 - - #check Nat.Prime.eq_one_or_self_of_dvd - - example (p : ℕ) (prime_p : Nat.Prime p) : ∀ m : ℕ, m ∣ p → m = 1 ∨ m = p := - prime_p.eq_one_or_self_of_dvd - - example : Nat.Prime 17 := by norm_num - - -- commonly used - example : Nat.Prime 2 := - Nat.prime_two - - example : Nat.Prime 3 := - Nat.prime_three - - #check @Nat.Prime.dvd_mul - - #check Nat.Prime.dvd_mul Nat.prime_two - - #check Nat.prime_two.dvd_mul - - theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by - rw [pow_two, Nat.prime_two.dvd_mul] at h - cases h <;> assumption - - example {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := - Nat.Prime.dvd_of_dvd_pow Nat.prime_two h - - example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := - -- library_search suggests the following: - (mul_right_inj' h').mp h - - example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by - intro sqr_eq - have : 2 ∣ m := by - sorry, - obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this - have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by - rw [← sqr_eq, meq] - ring - have : 2 * k ^ 2 = n ^ 2 := - sorry, - have : 2 ∣ n := by - sorry, - have : 2 ∣ m.gcd n := by - sorry, - have : 2 ∣ 1 := by - sorry, - norm_num at this - - example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by - sorry - #check Nat.factors - #check Nat.prime_of_mem_factors - #check Nat.prod_factors - #check Nat.factors_unique - - theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : - (m * n).factorization p = m.factorization p + n.factorization p := by - rw [Nat.factorization_mul mnez nnez] - rfl - - theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by - rw [Nat.factorization_pow] - rfl - - theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by - rw [prime_p.factorization] - simp - - example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by - intro sqr_eq - have nsqr_nez : n ^ 2 ≠ 0 := by simpa - have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by - sorry, - have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by - sorry, - have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] - rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this - norm_num at this - - example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : - k ∣ r.factorization p := by - cases' r with r - · simp - have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) - have eq1 : (m ^ k).factorization p = k * m.factorization p := by - sorry, - have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by - sorry, - have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by - rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] - rw [this] - sorry - #check multiplicity -
src/C_Number_Theory/S02_Induction_and_Recursion.lean (deleted)
- @@ -1,148 +0,0 @@
 - import Mathlib.Data.Nat.GCD.Basic - import Mathlib.Algebra.BigOperators.Basic - import Mathlib.Tactic - - example (n : Nat) : n.succ ≠ Nat.zero := - Nat.succ_ne_zero n - - example (m n : Nat) (h : m.succ = n.succ) : m = n := - Nat.succ.inj h - - def fac : ℕ → ℕ - | 0 => 1 - | n + 1 => (n + 1) * fac n - - example : fac 0 = 1 := - rfl - - example : fac 0 = 1 := by - rw [fac] - - example : fac 0 = 1 := by - simp [fac] - - example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := - rfl - - example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by - rw [fac] - - example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by - simp [fac] - - theorem fac_pos (n : ℕ) : 0 < fac n := by - induction' n with n ih - · rw [fac] - exact zero_lt_one - rw [fac] - exact mul_pos n.succ_pos ih - - theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := by - induction' n with n ih - · exact absurd ipos (not_lt_of_ge ile) - rw [fac] - cases' Nat.of_le_succ ile with h h - · apply dvd_mul_of_dvd_right (ih h) - rw [h] - apply dvd_mul_right - - theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by - cases' n with n - · simp [fac] - sorry - section - - variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) - - #check Finset.sum s f - #check Finset.prod s f - - open BigOperators - open Finset - - example : s.sum f = ∑ x in s, f x := - rfl - - example : s.prod f = ∏ x in s, f x := - rfl - - example : (range n).sum f = ∑ x in range n, f x := - rfl - - example : (range n).prod f = ∏ x in range n, f x := - rfl - - example (f : ℕ → ℕ) : (∑ x in range 0, f x) = 0 := - Finset.sum_range_zero f - - example (f : ℕ → ℕ) (n : ℕ) : (∑ x in range n.succ, f x) = (∑ x in range n, f x) + f n := - Finset.sum_range_succ f n - - example (f : ℕ → ℕ) : (∏ x in range 0, f x) = 1 := - Finset.prod_range_zero f - - example (f : ℕ → ℕ) (n : ℕ) : (∏ x in range n.succ, f x) = (∏ x in range n, f x) * f n := - Finset.prod_range_succ f n - - example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by - induction' n with n ih - · rw [fac, prod_range_zero] - rw [fac, ih, prod_range_succ, mul_comm] - - example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by - simp [mul_assoc, mul_comm, mul_left_comm] - - 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] - ring - - theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by - sorry - end - - inductive MyNat - | zero : MyNat - | succ : MyNat → MyNat - - namespace MyNat - - def add : MyNat → MyNat → MyNat - | x, zero => x - | x, succ y => succ (add x y) - - def mul : MyNat → MyNat → MyNat - | x, zero => zero - | x, succ y => add (mul x y) x - - theorem zero_add (n : MyNat) : add zero n = n := by - induction' n with n ih - · rfl - rw [add, ih] - - theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by - induction' n with n ih - · rfl - rw [add, ih] - rfl - - theorem add_comm (m n : MyNat) : add m n = add n m := by - induction' n with n ih - · rw [zero_add] - rfl - rw [add, succ_add, ih] - - theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by - sorry - theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by - sorry - theorem zero_mul (n : MyNat) : mul zero n = zero := by - sorry - theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by - sorry - theorem mul_comm (m n : MyNat) : mul m n = mul n m := by - sorry - end MyNat -
src/C_Number_Theory/S03_Infinitely_Many_Primes.lean (deleted)
- @@ -1,231 +0,0 @@
 - import Mathlib.Data.Nat.Prime - import Mathlib.Algebra.BigOperators.Order - import Mathlib.Tactic - import Mathlib.Tactic.IntervalCases - - open BigOperators - - theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by - cases m; contradiction - case succ m => - cases m; contradiction - repeat' apply Nat.succ_le_succ - apply zero_le - - example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by - by_contra h - push_neg at h - interval_cases m <;> contradiction - - example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by - by_contra h - push_neg at h - revert h0 h1 - revert h m - decide - - theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by - by_cases np : n.Prime - · use n, np - induction' n using Nat.strong_induction_on with n ih - dsimp at ih - rw [Nat.prime_def_lt] at np - push_neg at np - rcases np h with ⟨m, mltn, mdvdn, mne1⟩ - have : m ≠ 0 := by - intro mz - rw [mz, zero_dvd_iff] at mdvdn - linarith - have mgt2 : 2 ≤ m := two_le this mne1 - by_cases mp : m.Prime - · use m, mp - exact mdvdn - . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ - use p, pp - apply pdvd.trans mdvdn - - theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by - intro n - have : 2 ≤ Nat.factorial (n + 1) + 1 := by - sorry, - rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ - refine' ⟨p, _, pp⟩ - show p > n - by_contra ple - push_neg at ple - have : p ∣ Nat.factorial (n + 1) := by - sorry, - have : p ∣ 1 := by - sorry, - show False - sorry - open Finset - - section - - variable {α : Type _} [DecidableEq α] (r s t : Finset α) - - example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by - rw [subset_iff] - intro x - rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter] - tauto - - example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by - simp [subset_iff] - intro x - tauto - - example : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by - simp [subset_iff] - intro x - tauto - - example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by - ext x - simp - tauto - - end - - section - - variable {α : Type _} [DecidableEq α] (r s t : Finset α) - - example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by - sorry - example : (r \ s) \ t = r \ (s ∪ t) := by - sorry - end - - example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := - Finset.dvd_prod_of_mem _ h - - theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) - (h : p ∣ q) : p = q := by - sorry - theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : - (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by - intro h₀ h₁ - induction' s using Finset.induction_on with a s ans ih - · simp at h₁ - linarith [prime_p.two_le] - simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ - rw [mem_insert] - sorry - example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.Prime := - mem_filter - - theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by - intro s - by_contra h - push_neg at h - set s' := s.filter Nat.Prime with s'_def - have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by - intro n - simp [s'_def] - apply h - have : 2 ≤ (∏ i in s', i) + 1 := by - sorry, - rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ - have : p ∣ ∏ i in s', i := by - sorry, - have : p ∣ 1 := by - convert Nat.dvd_sub' pdvd this - simp - show False - sorry - theorem bounded_of_ex_finset (Q : ℕ → Prop) : - (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by - rintro ⟨s, hs⟩ - use s.sup id + 1 - intro k Qk - apply Nat.lt_succ_of_le - show id k ≤ s.sup id - apply le_sup (hs k Qk) - - theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : - (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by - rintro ⟨n, hn⟩ - use (range (n + 1)).filter Q - intro k - simp [Nat.lt_succ_iff] - exact hn k - - 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] - have : n % 4 < 4 := Nat.mod_lt n (by norm_num) - interval_cases hn : n % 4 <;> simp [hn] - - theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by - apply two_le <;> - · intro neq - rw [neq] at h - norm_num at h - - theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by - sorry - theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : - ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by - by_cases np : n.Prime - · use n - exact ⟨np, dvd_rfl, h⟩ - induction' n using Nat.strong_induction_on with n ih - dsimp at ih - rw [Nat.prime_def_lt] at np - push_neg at np - rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ - have mge2 : 2 ≤ m := by - apply two_le _ mne1 - intro mz - rw [mz, zero_dvd_iff] at mdvdn - linarith - have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn - have : m % 4 = 3 ∨ n / m % 4 = 3 := by - apply mod_4_eq_3_or_mod_4_eq_3 - rw [neq, h] - cases' this with h1 h1 - . sorry - sorry - example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by - rwa [mem_erase] at h - - example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by - simp at h - assumption - - theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by - by_contra h - push_neg at h - cases' h with n hn - have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by - apply ex_finset_of_bounded - use n - contrapose! hn - rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ - exact ⟨p, pltn, pp, p4⟩ - cases' this with s hs - have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by - sorry, - rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ - have ps : p ∈ s := by - sorry, - have pne3 : p ≠ 3 := by - sorry, - have : p ∣ 4 * ∏ i in erase s 3, i := by - sorry, - have : p ∣ 3 := by - sorry, - have : p = 3 := by - sorry, - contradiction -
src/C_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean (deleted)
- @@ -1,101 +0,0 @@
 - 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 - cases h <;> assumption - - example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by - intro sqr_eq - have : 2 ∣ m := by - apply even_of_even_sqr - rw [sqr_eq] - apply dvd_mul_right - obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this - have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by - rw [← sqr_eq, meq] - ring - have : 2 * k ^ 2 = n ^ 2 := - (mul_right_inj' (by norm_num)).mp this - have : 2 ∣ n := by - apply even_of_even_sqr - rw [← this] - apply dvd_mul_right - have : 2 ∣ m.gcd n := by - apply Nat.dvd_gcd <;> - assumption - have : 2 ∣ 1 := by - convert this - symm - exact coprime_mn - norm_num at this - - example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by - intro sqr_eq - have : p ∣ m := by - apply prime_p.dvd_of_dvd_pow - rw [sqr_eq] - apply dvd_mul_right - obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this - have : p * (p * k ^ 2) = p * n ^ 2 := by - rw [← sqr_eq, meq] - ring - have : p * k ^ 2 = n ^ 2 := by - apply (mul_right_inj' _).mp this - exact prime_p.ne_zero - have : p ∣ n := by - apply prime_p.dvd_of_dvd_pow - rw [← this] - apply dvd_mul_right - have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption - have : p ∣ 1 := by - convert this - symm - exact coprime_mn - have : 2 ≤ 1 := by - apply prime_p.two_le.trans - exact Nat.le_of_dvd zero_lt_one this - norm_num at this - - theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : - (m * n).factorization p = m.factorization p + n.factorization p := by - rw [Nat.factorization_mul mnez nnez] - rfl - - theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by - rw [Nat.factorization_pow] - rfl - - theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by - rw [prime_p.factorization] - simp - - example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by - intro sqr_eq - have nsqr_nez : n ^ 2 ≠ 0 := by simpa - have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by - rw [factorization_pow'] - have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by - rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow', - add_comm] - have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] - rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this - norm_num at this - - example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : - k ∣ r.factorization p := by - cases' r with r - · simp - have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) - have eq1 : (m ^ k).factorization p = k * m.factorization p := by - rw [factorization_pow'] - have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by - rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm] - have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by - rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] - rw [this] - apply Nat.dvd_sub' <;> - apply Nat.dvd_mul_right -
src/C_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean (deleted)
- @@ -1,98 +0,0 @@
 - import Mathlib.Data.Nat.GCD.Basic - import Mathlib.Algebra.BigOperators.Basic - import Mathlib.Tactic - - def fac : ℕ → ℕ - | 0 => 1 - | n + 1 => (n + 1) * fac n - - theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by - cases' n with n - · simp [fac] - induction' n with n ih - · simp [fac] - simp at * - rw [pow_succ, fac] - apply Nat.mul_le_mul _ ih - repeat' apply Nat.succ_le_succ - apply zero_le - - section - - variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) - - open BigOperators - open Finset - - theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by - symm; - 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] - ring - - end - - inductive MyNat - | zero : MyNat - | succ : MyNat → MyNat - - namespace MyNat - - def add : MyNat → MyNat → MyNat - | x, zero => x - | x, succ y => succ (add x y) - - def mul : MyNat → MyNat → MyNat - | x, zero => zero - | x, succ y => add (mul x y) x - - theorem zero_add (n : MyNat) : add zero n = n := by - induction' n with n ih - · rfl - rw [add, ih] - - theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by - induction' n with n ih - · rfl - rw [add, ih] - rfl - - theorem add_comm (m n : MyNat) : add m n = add n m := by - induction' n with n ih - · rw [zero_add] - rfl - rw [add, succ_add, ih] - - theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by - induction' k with k ih - · rfl - rw [add, ih] - rfl - - theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by - induction' k with k ih - · rfl - rw [add, mul, mul, ih, add_assoc] - - theorem zero_mul (n : MyNat) : mul zero n = zero := by - induction' n with n ih - · rfl - rw [mul, ih] - rfl - - theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by - induction' n with n ih - · rfl - rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] - rfl - - theorem mul_comm (m n : MyNat) : mul m n = mul n m := by - induction' n with n ih - · rw [zero_mul] - rfl - rw [mul, ih, succ_mul] - - end MyNat -
src/C_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean (deleted)
- @@ -1,241 +0,0 @@
 - import Mathlib.Data.Nat.Prime - import Mathlib.Algebra.BigOperators.Order - import Mathlib.Tactic - import Mathlib.Tactic.IntervalCases - - open BigOperators - - theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by - cases m; contradiction - case succ m => - cases m; contradiction - repeat' apply Nat.succ_le_succ - apply zero_le - - theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by - by_cases np : n.Prime - · use n, np - induction' n using Nat.strong_induction_on with n ih - dsimp at ih - rw [Nat.prime_def_lt] at np - push_neg at np - rcases np h with ⟨m, mltn, mdvdn, mne1⟩ - have : m ≠ 0 := by - intro mz - rw [mz, zero_dvd_iff] at mdvdn - linarith - have mgt2 : 2 ≤ m := two_le this mne1 - by_cases mp : m.Prime - · use m, mp - exact mdvdn - . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ - use p, pp - apply pdvd.trans mdvdn - - theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by - intro n - have : 2 ≤ Nat.factorial (n + 1) + 1 := by - apply Nat.succ_le_succ - exact Nat.succ_le_of_lt (Nat.factorial_pos _) - rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ - refine' ⟨p, _, pp⟩ - show p > n - by_contra ple - push_neg at ple - have : p ∣ Nat.factorial (n + 1) := by - apply Nat.dvd_factorial - apply pp.pos - linarith - have : p ∣ 1 := by - convert Nat.dvd_sub' pdvd this - simp - show False - have := Nat.le_of_dvd zero_lt_one this - linarith [pp.two_le] - - open Finset - - section - - variable {α : Type _} [DecidableEq α] (r s t : Finset α) - - example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by - ext x - rw [mem_inter, mem_union, mem_union, mem_union, mem_inter] - tauto - - example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by - ext x - simp - tauto - - example : (r \ s) \ t = r \ (s ∪ t) := by - ext x - rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union] - tauto - - example : (r \ s) \ t = r \ (s ∪ t) := by - ext x - simp - tauto - - end - - theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) - (h : p ∣ q) : p = q := by - cases prime_q.eq_one_or_self_of_dvd _ h - · linarith [prime_p.two_le] - assumption - - theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : - (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by - intro h₀ h₁ - induction' s using Finset.induction_on with a s ans ih - · simp at h₁ - linarith [prime_p.two_le] - simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ - rw [mem_insert] - cases' h₁ with h₁ h₁ - · left - exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ - right - exact ih h₀.2 h₁ - - theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by - intro s - by_contra h - push_neg at h - set s' := s.filter Nat.Prime with s'_def - have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by - intro n - simp [s'_def] - apply h - have : 2 ≤ (∏ i in s', i) + 1 := by - apply Nat.succ_le_succ - apply Nat.succ_le_of_lt - apply Finset.prod_pos - intro n ns' - apply (mem_s'.mp ns').pos - rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ - have : p ∣ ∏ i in s', i := by - apply dvd_prod_of_mem - rw [mem_s'] - apply pp - have : p ∣ 1 := by - convert Nat.dvd_sub' pdvd this - simp - show False - have := Nat.le_of_dvd zero_lt_one this - linarith [pp.two_le] - - theorem bounded_of_ex_finset (Q : ℕ → Prop) : - (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by - rintro ⟨s, hs⟩ - use s.sup id + 1 - intro k Qk - apply Nat.lt_succ_of_le - show id k ≤ s.sup id - apply le_sup (hs k Qk) - - theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : - (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by - rintro ⟨n, hn⟩ - use (range (n + 1)).filter Q - intro k - simp [Nat.lt_succ_iff] - exact hn k - - 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] - have : n % 4 < 4 := Nat.mod_lt n (by norm_num) - interval_cases hn : n % 4 <;> simp [hn] - - theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by - apply two_le <;> - · intro neq - rw [neq] at h - norm_num at h - - theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by - constructor - · exact Nat.div_dvd_of_dvd h₀ - exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ - - theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : - ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by - by_cases np : n.Prime - · use n - exact ⟨np, dvd_rfl, h⟩ - induction' n using Nat.strong_induction_on with n ih - dsimp at ih - rw [Nat.prime_def_lt] at np - push_neg at np - rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ - have mge2 : 2 ≤ m := by - apply two_le _ mne1 - intro mz - rw [mz, zero_dvd_iff] at mdvdn - linarith - have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn - have : m % 4 = 3 ∨ n / m % 4 = 3 := by - apply mod_4_eq_3_or_mod_4_eq_3 - rw [neq, h] - cases' this with h1 h1 - · by_cases mp : m.Prime - · use m - exact ⟨mp, mdvdn, h1⟩ - rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩ - use p - exact ⟨pp, pdvd.trans mdvdn, p4eq⟩ - obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn - by_cases nmp : (n / m).Prime - · use n / m - exact ⟨nmp, nmdvdn, h1⟩ - rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩ - use p - exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩ - - theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by - by_contra h - push_neg at h - cases' h with n hn - have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by - apply ex_finset_of_bounded - use n - contrapose! hn - rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ - exact ⟨p, pltn, pp, p4⟩ - cases' 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] - exact ⟨pp, p4eq⟩ - have pne3 : p ≠ 3 := by - intro peq - rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd - rw [Nat.prime_three.dvd_mul] at pdvd - norm_num at pdvd - have : 3 ∈ s.erase 3 := by - apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd - intro n - simp [← hs n] - tauto - simp at this - have : p ∣ 4 * ∏ i in erase s 3, i := by - apply dvd_trans _ (dvd_mul_left _ _) - apply dvd_prod_of_mem - simp - constructor <;> assumption - have : p ∣ 3 := by - convert Nat.dvd_sub' pdvd this - simp - have : p = 3 := by - apply pp.eq_of_dvd_of_prime Nat.prime_three this - contradiction -
src/C_Sets_and_Functions/S01_Sets.lean (deleted)
- @@ -1,254 +0,0 @@
 - import Mathlib.Data.Set.Lattice - import Mathlib.Data.Nat.Prime - import Mathlib.Data.Nat.Parity - import Mathlib.Tactic - - section - - variable {α : Type _} - - variable (s t u : Set α) - - open Set - - example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by - rw [subset_def, inter_def, inter_def] - rw [subset_def] at h - dsimp - rintro x ⟨xs, xu⟩ - exact ⟨h _ xs, xu⟩ - - example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by - simp only [subset_def, mem_inter_iff] at * - rintro x ⟨xs, xu⟩ - exact ⟨h _ xs, xu⟩ - - example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by - intro x xsu - exact ⟨h xsu.1, xsu.2⟩ - - theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := - fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ - - example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := - fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ - - example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by - intro x hx - have xs : x ∈ s := hx.1 - have xtu : x ∈ t ∪ u := hx.2 - cases' xtu with xt xu - · left - show x ∈ s ∩ t - exact ⟨xs, xt⟩ - right - show x ∈ s ∩ u - exact ⟨xs, xu⟩ - - example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by - rintro x ⟨xs, xt | xu⟩ - · left - exact ⟨xs, xt⟩ - right; exact ⟨xs, xu⟩ - - example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by - sorry - example : (s \ t) \ u ⊆ s \ (t ∪ u) := by - intro x xstu - have xs : x ∈ s := xstu.1.1 - have xnt : x ∉ t := xstu.1.2 - have xnu : x ∉ u := xstu.2 - constructor - · exact xs - intro xtu - -- x ∈ t ∨ x ∈ u - cases' xtu with xt xu - · show False - exact xnt xt - show False; exact xnu xu - - example : (s \ t) \ u ⊆ s \ (t ∪ u) := by - rintro x ⟨⟨xs, xnt⟩, xnu⟩ - use xs - rintro (xt | xu) <;> contradiction - - example : s \ (t ∪ u) ⊆ (s \ t) \ u := by - sorry - example : s ∩ t = t ∩ s := by - ext x - simp only [mem_inter_iff] - constructor - · rintro ⟨xs, xt⟩ - exact ⟨xt, xs⟩ - rintro ⟨xt, xs⟩; exact ⟨xs, xt⟩ - - example : s ∩ t = t ∩ s := - Set.ext fun x => ⟨fun ⟨xs, xt⟩ => ⟨xt, xs⟩, fun ⟨xt, xs⟩ => ⟨xs, xt⟩⟩ - - example : s ∩ t = t ∩ s := by ext x; simp [and_comm] - - example : s ∩ t = t ∩ s := by - apply Subset.antisymm - · rintro x ⟨xs, xt⟩ - exact ⟨xt, xs⟩ - rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ - - example : s ∩ t = t ∩ s := - Subset.antisymm sorry sorry - example : s ∩ (s ∪ t) = s := by - sorry - - example : s ∪ s ∩ t = s := by - sorry - - example : s \ t ∪ t = s ∪ t := by - sorry - - example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by - sorry - - def evens : Set ℕ := - { n | Even n } - - def odds : Set ℕ := - { n | ¬Even n } - - example : evens ∪ odds = univ := by - rw [evens, odds] - ext n - simp - apply Classical.em - - example (x : ℕ) (h : x ∈ (∅ : Set ℕ)) : False := - h - - example (x : ℕ) : x ∈ (univ : Set ℕ) := - trivial - - example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by - sorry - - #print Prime - - #print Nat.Prime - - example (n : ℕ) : Prime n ↔ Nat.Prime n := - Nat.prime_iff.symm - - example (n : ℕ) (h : Prime n) : Nat.Prime n := by - rw [Nat.prime_iff] - exact h - - example (n : ℕ) (h : Prime n) : Nat.Prime n := by - rwa [Nat.prime_iff] - - end - - section - - variable (s t : Set ℕ) - - example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by - intro x xs - constructor - · apply h₀ x xs - apply h₁ x xs - - example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by - rcases h with ⟨x, xs, _, prime_x⟩ - use x, xs - exact prime_x - - section - - variable (ssubt : s ⊆ t) - - example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by - sorry - - example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by - sorry - - end - - end - - section - - variable {α I : Type _} - - variable (A B : I → Set α) - - variable (s : Set α) - - open Set - - example : (s ∩ ⋃ i, A i) = ⋃ i, A i ∩ s := by - ext x - simp only [mem_inter_iff, mem_iUnion] - constructor - · rintro ⟨xs, ⟨i, xAi⟩⟩ - exact ⟨i, xAi, xs⟩ - rintro ⟨i, xAi, xs⟩ - exact ⟨xs, ⟨i, xAi⟩⟩ - - example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ ⋂ i, B i := by - ext x - simp only [mem_inter_iff, mem_iInter] - constructor - · intro h - constructor - · intro i - exact (h i).1 - intro i - exact (h i).2 - rintro ⟨h1, h2⟩ i - constructor - · exact h1 i - exact h2 i - - - example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by - sorry - - def primes : Set ℕ := - { x | Nat.Prime x } - - example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } :=by - ext - rw [mem_iUnion₂] - simp - - example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by - ext - simp - - example : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by - intro x - contrapose! - simp - apply Nat.exists_prime_and_dvd - - example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by - sorry - - end - - section - - open Set - - variable {α : Type _} (s : Set (Set α)) - - example : ⋃₀ s = ⋃ t ∈ s, t := by - ext x - rw [mem_iUnion₂] - simp - - example : ⋂₀ s = ⋂ t ∈ s, t := by - ext x - rw [mem_iInter₂] - rfl - - end -
src/C_Sets_and_Functions/S02_Functions.lean (deleted)
- @@ -1,217 +0,0 @@
  - import Mathlib.Data.Set.Lattice - import Mathlib.Data.Set.Function - import Mathlib.Analysis.SpecialFunctions.Log.Basic - - section - - variable {α β : Type _} - variable (f : α → β) - variable (s t : Set α) - variable (u v : Set β) - - open Function - open Set - - example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by - ext - rfl - - example : f '' (s ∪ t) = f '' s ∪ f '' t := by - ext y; constructor - · rintro ⟨x, xs | xt, rfl⟩ - · left - use x, xs - right - use x, xt - rintro (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩) - · use x, Or.inl xs - use x, Or.inr xt - - example : s ⊆ f ⁻¹' (f '' s) := by - intro x xs - show f x ∈ f '' s - use x, xs - - example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by - sorry - - example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by - sorry - - example : f '' (f ⁻¹' u) ⊆ u := by - sorry - - example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by - sorry - - example (h : s ⊆ t) : f '' s ⊆ f '' t := by - sorry - - example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by - sorry - - example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by - sorry - - example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by - sorry - - example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by - sorry - - example : f '' s \ f '' t ⊆ f '' (s \ t) := by - sorry - - example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := by - sorry - - example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by - sorry - - example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := by - sorry - - example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by - sorry - - example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by - sorry - - variable {I : Type _} (A : I → Set α) (B : I → Set β) - - example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by - ext y; simp - constructor - · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ - use i, x - exact ⟨xAi, fxeq⟩ - rintro ⟨i, x, xAi, fxeq⟩ - exact ⟨x, ⟨i, xAi⟩, fxeq⟩ - - example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by - intro y; simp - intro x h fxeq i - use x - exact ⟨h i, fxeq⟩ - - example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by - intro y; simp - intro h - rcases h i with ⟨x, xAi, fxeq⟩ - use x; constructor - · intro i' - rcases h i' with ⟨x', x'Ai, fx'eq⟩ - have : f x = f x' := by rw [fxeq, fx'eq] - have : x = x' := injf this - rw [this] - exact x'Ai - exact fxeq - - example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by - ext x - simp - - example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by - ext x - simp - - example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := - Iff.refl _ - - end - - section - - open Set Real - - example : InjOn log { x | x > 0 } := by - intro x xpos y ypos - intro e - -- log x = log y - calc - x = exp (log x) := by rw [exp_log xpos] - _ = exp (log y) := by rw [e] - _ = y := by rw [exp_log ypos] - - - example : range exp = { y | y > 0 } := by - ext y; constructor - · rintro ⟨x, rfl⟩ - apply exp_pos - intro ypos - use log y - rw [exp_log ypos] - - example : InjOn sqrt { x | x ≥ 0 } := by - sorry - - example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by - sorry - - example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by - sorry - - example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by - sorry - - end - - section - - variable {α β : Type _} [Inhabited α] - - #check (default : α) - - variable (P : α → Prop) (h : ∃ x, P x) - - #check Classical.choose h - - example : P (Classical.choose h) := - Classical.choose_spec h - - noncomputable section - - open Classical - - def inverse (f : α → β) : β → α := fun y : β => - if h : ∃ x, f x = y then Classical.choose h else default - - theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by - rw [inverse]; dsimp; rw [dif_pos h] - exact Classical.choose_spec h - - variable (f : α → β) - - open Function - - example : Injective f ↔ LeftInverse (inverse f) f := - sorry - - example : Surjective f ↔ RightInverse (inverse f) f := - sorry - - end - - section - - variable {α : Type _} - - open Function - - theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by - intro f surjf - let S := { i | i ∉ f i } - rcases surjf S with ⟨j, h⟩ - have h₁ : j ∉ f j := by - intro h' - have : j ∉ f j := by rwa [h] at h' - contradiction - have h₂ : j ∈ S - sorry - have h₃ : j ∉ S - sorry - contradiction - - -- COMMENTS: TODO: improve this - end -
src/C_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean (deleted)
- @@ -1,99 +0,0 @@
  - import Mathlib.Data.Set.Lattice - import Mathlib.Data.Set.Function - import Mathlib.Tactic - - open Set - - open Function - - noncomputable section - - open Classical - - variable {α β : Type _} [Nonempty β] - - section - - variable (f : α → β) (g : β → α) - - def sbAux : ℕ → Set α - | 0 => univ \ g '' univ - | n + 1 => g '' (f '' sbAux n) - - def sbSet := - ⋃ n, sbAux f g n - - def sbFun (x : α) : β := - if x ∈ sbSet f g then f x else invFun g x - - theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by - have : x ∈ g '' univ := by - contrapose! hx - rw [sbSet, mem_iUnion] - use 0 - rw [sbAux, mem_diff] - sorry }, - have : ∃ y, g y = x := by - { sorry }, - sorry - theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by - set A := sbSet f g with A_def - set h := sbFun f g with h_def - intro x₁ x₂ - intro (hxeq : h x₁ = h x₂) - show x₁ = x₂ - simp only [h_def, sbFun, ← A_def] at hxeq - by_cases xA : x₁ ∈ A ∨ x₂ ∈ A - · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA - · symm - apply this hxeq.symm xA.symm (xA.resolve_left x₁A) - have x₂A : x₂ ∈ A := by - apply 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 - have x₂eq : x₂ = g (f x₁) := by - . sorry - rcases x₁A with ⟨n, hn⟩ - rw [A_def, sbSet, mem_iUnion] - use n + 1 - simp [sbAux] - exact ⟨x₁, hn, x₂eq.symm⟩ - . sorry, - push_neg at xA - sorry - theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by - set A := sbSet f g with A_def - set h := sbFun f g with h_def - intro y - by_cases gyA : g y ∈ A - · rw [A_def, sbSet, mem_iUnion] at gyA - rcases gyA with ⟨n, hn⟩ - cases' n with n - · simp [sbAux] at hn - simp [sbAux] at hn - rcases hn with ⟨x, xmem, hx⟩ - use x - have : x ∈ A := by - rw [A_def, sbSet, mem_iUnion] - exact ⟨n, xmem⟩ - simp only [h_def, sbFun, if_pos this] - exact hg hx - sorry - end - - theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Injective f) (hg : Injective g) : - ∃ h : α → β, Bijective h := - ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ - - -- Auxiliary information - section - variable (g : β → α) (x : α) - - #check (invFun g : α → β) - #check (leftInverse_invFun : Injective g → LeftInverse (invFun g) g) - #check (leftInverse_invFun : Injective g → ∀ y, invFun g (g y) = y) - #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) - - end -
src/C_Sets_and_Functions/solutions/Solutions_S01_Sets.lean (deleted)
- @@ -1,165 +0,0 @@
 - import Mathlib.Data.Set.Lattice - import Mathlib.Data.Nat.Prime - import Mathlib.Data.Nat.Parity - import Mathlib.Tactic - - section - - variable {α : Type _} - - variable (s t u : Set α) - - open Set - - example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by - rintro x (⟨xs, xt⟩ | ⟨xs, xu⟩) - · use xs - left - exact xt - use xs; right; exact xu - - example : s \ (t ∪ u) ⊆ (s \ t) \ u := by - rintro x ⟨xs, xntu⟩ - constructor - use xs - · intro xt - exact xntu (Or.inl xt) - intro xu - apply xntu (Or.inr xu) - - example : s ∩ t = t ∩ s := - Subset.antisymm - (fun x ⟨xs, xt⟩ => ⟨xt, xs⟩) fun x ⟨xt, xs⟩ => ⟨xs, xt⟩ - - example : s ∩ (s ∪ t) = s := by - ext x; constructor - · rintro ⟨xs, _⟩ - exact xs - intro xs - use xs; left; exact xs - - example : s ∪ s ∩ t = s := by - ext x; constructor - · rintro (xs | ⟨xs, xt⟩) <;> exact xs - intro xs; left; exact xs - - example : s \ t ∪ t = s ∪ t := by - ext x; constructor - · rintro (⟨xs, nxt⟩ | xt) - · left - exact xs - right - exact xt - by_cases h : x ∈ t - · intro - right - exact h - rintro (xs | xt) - · left - use xs - exact h - right; exact xt - - example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by - ext x; constructor - · rintro (⟨xs, xnt⟩ | ⟨xt, xns⟩) - · constructor - left - exact xs - rintro ⟨_, xt⟩ - contradiction - constructor - right - exact xt - rintro ⟨xs, _⟩ - contradiction - rintro ⟨xs | xt, nxst⟩ - · left - use xs - intro xt - apply nxst - constructor <;> assumption - right; use xt; intro xs - apply nxst - constructor <;> assumption - - example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by - intro n - simp - intro nprime - cases' Nat.Prime.eq_two_or_odd nprime with h h - · rw [h] - intro - linarith - rw [Nat.even_iff, h] - norm_num - - end - - section - - variable (s t : Set ℕ) - - section - - variable (ssubt : s ⊆ t) - - example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by - intro x xs - constructor - · apply h₀ x (ssubt xs) - apply h₁ x (ssubt xs) - - example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by - rcases h with ⟨x, xs, _, px⟩ - use x, ssubt xs - exact px - - end - - end - - section - - variable {α I : Type _} - - variable (A B : I → Set α) - - variable (s : Set α) - - open Set - - example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by - ext x - simp only [mem_union, mem_iInter] - constructor - · rintro (xs | xI) - · intro i - right - exact xs - intro i - left - exact xI i - intro h - by_cases xs : x ∈ s - · left - exact xs - right - intro i - cases h i - · assumption - contradiction - - def primes : Set ℕ := - { x | Nat.Prime x } - - example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by - apply eq_univ_of_forall - intro x - simp - rcases Nat.exists_infinite_primes x with ⟨p, primep, pge⟩ - use p, pge - exact primep - - end -
src/C_Sets_and_Functions/solutions/Solutions_S02_Functions.lean (deleted)
- @@ -1,256 +0,0 @@
  - import Mathlib.Data.Set.Lattice - import Mathlib.Data.Set.Function - import Mathlib.Analysis.SpecialFunctions.Log.Basic - - section - - variable {α β : Type _} - variable (f : α → β) - variable (s t : Set α) - variable (u v : Set β) - - open Function - open Set - - example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by - constructor - · intro h x xs - have : f x ∈ f '' s := mem_image_of_mem _ xs - exact h this - intro h y ymem - rcases ymem with ⟨x, xs, fxeq⟩ - rw [← fxeq] - apply h xs - - example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by - rintro x ⟨y, ys, fxeq⟩ - rw [← h fxeq] - exact ys - - example : f '' (f ⁻¹' u) ⊆ u := by - rintro y ⟨x, xmem, rfl⟩ - exact xmem - - example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by - intro y yu - rcases h y with ⟨x, fxeq⟩ - use x - constructor - · show f x ∈ u - rw [fxeq] - exact yu - exact fxeq - - example (h : s ⊆ t) : f '' s ⊆ f '' t := by - rintro y ⟨x, xs, fxeq⟩ - use x, h xs - exact fxeq - - example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by - intro x; apply h - - example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by - ext x; rfl - - example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by - rintro y ⟨x, ⟨xs, xt⟩, rfl⟩ - constructor - . use x, xs - . use x, xt - - example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by - rintro y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩ - use x₁ - constructor - . use x₁s - rw [← h fx₂eq] - exact x₂t - . rfl - - example : f '' s \ f '' t ⊆ f '' (s \ t) := by - rintro y ⟨⟨x₁, x₁s, rfl⟩, h⟩ - use x₁ - constructor - . constructor - . exact x₁s - . intro h' - apply h - use x₁, h' - . rfl - - example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := - fun x => id - - example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by - ext y; constructor - · rintro ⟨⟨x, xs, rfl⟩, fxv⟩ - use x, ⟨xs, fxv⟩ - rintro ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩ - exact ⟨⟨x, xs, rfl⟩, fxv⟩ - - example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := by - rintro y ⟨x, ⟨xs, fxu⟩, rfl⟩ - exact ⟨⟨x, xs, rfl⟩, fxu⟩ - - example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by - rintro x ⟨xs, fxu⟩ - exact ⟨⟨x, xs, rfl⟩, fxu⟩ - - example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by - rintro x (xs | fxu) - · left - exact ⟨x, xs, rfl⟩ - right; exact fxu - - variable {I : Type _} (A : I → Set α) (B : I → Set β) - - example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by - ext y; simp - constructor - · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ - use i, x - exact ⟨xAi, fxeq⟩ - rintro ⟨i, x, xAi, fxeq⟩ - exact ⟨x, ⟨i, xAi⟩, fxeq⟩ - - example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by - intro y; simp - intro x h fxeq i - use x - exact ⟨h i, fxeq⟩ - - example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by - intro y; simp - intro h - rcases h i with ⟨x, xAi, fxeq⟩ - use x; constructor - · intro i' - rcases h i' with ⟨x', x'Ai, fx'eq⟩ - have : f x = f x' := by rw [fxeq, fx'eq] - have : x = x' := injf this - rw [this] - exact x'Ai - exact fxeq - - example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by - ext x - simp - - example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by - ext x - simp - - end - - section - - open Set Real - - example : InjOn sqrt { x | x ≥ 0 } := by - intro x xnonneg y ynonneg - intro e - calc - x = sqrt x ^ 2 := by rw [sq_sqrt xnonneg] - _ = sqrt y ^ 2 := by rw [e] - _ = y := by rw [sq_sqrt ynonneg] - - - example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by - intro x xnonneg y ynonneg - intro e - dsimp at * - calc - x = sqrt (x ^ 2) := by rw [sqrt_sq xnonneg] - _ = sqrt (y ^ 2) := by rw [e] - _ = y := by rw [sqrt_sq ynonneg] - - - example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by - ext y; constructor - · rintro ⟨x, ⟨xnonneg, rfl⟩⟩ - apply sqrt_nonneg - intro ynonneg - use y ^ 2 - dsimp at * - constructor - apply pow_nonneg ynonneg - apply sqrt_sq - assumption - - example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by - ext y - constructor - · rintro ⟨x, rfl⟩ - dsimp at * - apply pow_two_nonneg - intro ynonneg - use sqrt y - exact sq_sqrt ynonneg - - end - - section - - variable {α β : Type _} [Inhabited α] - - noncomputable section - - open Classical - - def inverse (f : α → β) : β → α := fun y : β => - if h : ∃ x, f x = y then Classical.choose h else default - - theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by - rw [inverse]; dsimp; rw [dif_pos h] - exact Classical.choose_spec h - - variable (f : α → β) - - open Function - - example : Injective f ↔ LeftInverse (inverse f) f := by - constructor - · intro h y - apply h - apply inverse_spec - use y - intro h x1 x2 e - rw [← h x1, ← h x2, e] - - example : Injective f ↔ LeftInverse (inverse f) f := - ⟨fun h y => h (inverse_spec _ ⟨y, rfl⟩), fun h x1 x2 e => by rw [← h x1, ← h x2, e]⟩ - - example : Surjective f ↔ RightInverse (inverse f) f := by - constructor - · intro h y - apply inverse_spec - apply h - intro h y - use inverse f y - apply h - - example : Surjective f ↔ RightInverse (inverse f) f := - ⟨fun h y => inverse_spec _ (h _), fun h y => ⟨inverse f y, h _⟩⟩ - - end - - section - - variable {α : Type _} - - open Function - - theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by - intro f surjf - let S := { i | i ∉ f i } - rcases surjf S with ⟨j, h⟩ - have h₁ : j ∉ f j := by - intro h' - have : j ∉ f j := by rwa [h] at h' - contradiction - have h₂ : j ∈ S := h₁ - have h₃ : j ∉ S := by rwa [h] at h₁ - contradiction - - end -
src/C_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean (deleted)
- @@ -1,92 +0,0 @@
  - import Mathlib.Data.Set.Lattice - import Mathlib.Data.Set.Function - import Mathlib.Tactic - - open Set - - open Function - - noncomputable section - - open Classical - - variable {α β : Type _} [Nonempty β] - - section - - variable (f : α → β) (g : β → α) - - def sbAux : ℕ → Set α - | 0 => univ \ g '' univ - | n + 1 => g '' (f '' sbAux n) - - def sbSet := - ⋃ n, sbAux f g n - - def sbFun (x : α) : β := - if x ∈ sbSet f g then f x else invFun g x - - theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by - have : x ∈ g '' univ := by - contrapose! hx - rw [sbSet, mem_iUnion] - use 0 - rw [sbAux, mem_diff] - exact ⟨mem_univ _, hx⟩ - have : ∃ y, g y = x := by - simp at this - assumption - exact invFun_eq this - - theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by - set A := sbSet f g with A_def - set h := sbFun f g with h_def - intro x₁ x₂ - intro (hxeq : h x₁ = h x₂) - show x₁ = x₂ - simp only [h_def, sbFun, ← A_def] at hxeq - by_cases xA : x₁ ∈ A ∨ x₂ ∈ A - · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA - · symm - apply this hxeq.symm xA.symm (xA.resolve_left x₁A) - have x₂A : x₂ ∈ A := by - apply 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 - have x₂eq : x₂ = g (f x₁) := by - rw [hxeq, sb_right_inv f g x₂nA] - rcases x₁A with ⟨n, hn⟩ - rw [A_def, sbSet, mem_iUnion] - use n + 1 - simp [sbAux] - exact ⟨x₁, hn, x₂eq.symm⟩ - rw [if_pos x₁A, if_pos x₂A] at hxeq - exact hf hxeq - push_neg at xA - rw [if_neg xA.1, if_neg xA.2] at hxeq - rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] - - theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by - set A := sbSet f g with A_def - set h := sbFun f g with h_def - intro y - by_cases gyA : g y ∈ A - · rw [A_def, sbSet, mem_iUnion] at gyA - rcases gyA with ⟨n, hn⟩ - cases' n with n - · simp [sbAux] at hn - simp [sbAux] at hn - rcases hn with ⟨x, xmem, hx⟩ - use x - have : x ∈ A := by - rw [A_def, sbSet, mem_iUnion] - exact ⟨n, xmem⟩ - simp only [h_def, sbFun, if_pos this] - exact hg hx - use g y - simp only [h_def, sbFun, if_neg gyA] - apply leftInverse_invFun hg - - end -
src/C_Structures/S01_Structures.lean (deleted)
- @@ -1,227 +0,0 @@
 - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - noncomputable section - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - #check Point.ext - - example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by - ext - repeat' assumption - - def myPoint1 : Point where - x := 2 - y := -1 - z := 4 - - def myPoint2 : Point := - ⟨2, -1, 4⟩ - - def myPoint3 := - Point.mk 2 (-1) 4 - - structure Point' where build :: - x : ℝ - y : ℝ - z : ℝ - - #check Point'.build 2 (-1) 4 - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def add' (a b : Point) : Point where - x := a.x + b.x - y := a.y + b.y - z := a.z + b.z - - #check add myPoint1 myPoint2 - - #check myPoint1.add myPoint2 - - end Point - - #check Point.add myPoint1 myPoint2 - - #check myPoint1.add myPoint2 - - namespace Point - - protected theorem add_comm (a b : Point) : add a b = add b a := by - rw [add, add] - ext <;> dsimp - repeat' apply add_comm - - example (a b : Point) : add a b = add b a := by simp [add, add_comm] - - theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := - rfl - - def addAlt : Point → Point → Point - | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ - - def addAlt' : Point → Point → Point - | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ - - theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by - cases a - cases b - rfl - - theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by - rcases a with ⟨xa, ya, za⟩ - rcases b with ⟨xb, yb, zb⟩ - rw [addAlt, addAlt] - ext <;> dsimp - apply add_comm - repeat' apply add_comm - - example (a b : Point) : addAlt a b = addAlt b a := by - rcases a with ⟨xa, ya, za⟩ - rcases b with ⟨xb, yb, zb⟩ - simp [addAlt, add_comm] - - example : ∀ a b : Point, addAlt a b = addAlt b a := by - rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ - simp [addAlt, add_comm] - - example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by - simp [add, add_comm] - - protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by - sorry - def smul (r : ℝ) (a : Point) : Point := - sorry - theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by - sorry - end Point - - structure StandardTwoSimplex where - x : ℝ - y : ℝ - z : ℝ - x_nonneg : 0 ≤ x - y_nonneg : 0 ≤ y - z_nonneg : 0 ≤ z - sum_eq : x + y + z = 1 - - namespace StandardTwoSimplex - - def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex - where - x := a.y - y := a.x - z := a.z - x_nonneg := a.y_nonneg - y_nonneg := a.x_nonneg - z_nonneg := a.z_nonneg - sum_eq := by rw [add_comm a.y a.x, a.sum_eq] - - noncomputable section - - def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex - where - x := (a.x + b.x) / 2 - y := (a.y + b.y) / 2 - z := (a.z + b.z) / 2 - x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) - y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) - z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) - sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] - - def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where - sorry - end - - end StandardTwoSimplex - - open BigOperators - - structure StandardSimplex (n : ℕ) where - V : Fin n → ℝ - NonNeg : ∀ i : Fin n, 0 ≤ V i - sum_eq_one : (∑ i, V i) = 1 - - namespace StandardSimplex - - def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n - where - V i := (a.V i + b.V i) / 2 - NonNeg := by - intro i - apply div_nonneg - · linarith [a.NonNeg i, b.NonNeg i] - norm_num - sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] - field_simp - - end StandardSimplex - - structure IsLinear (f : ℝ → ℝ) where - is_additive : ∀ x y, f (x + y) = f x + f y - preserves_mul : ∀ x c, f (c * x) = c * f x - - section - - variable (f : ℝ → ℝ) (linf : IsLinear f) - - #check linf.is_additive - - #check linf.preserves_mul - - end - - def Point'' := - ℝ × ℝ × ℝ - - def IsLinear' (f : ℝ → ℝ) := - (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x - - def PReal := - { y : ℝ // 0 < y } - - section - - variable (x : PReal) - - #check x.val - - #check x.property - - #check x.1 - - #check x.2 - - end - - def StandardTwoSimplex' := - { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } - - def StandardSimplex' (n : ℕ) := - { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } - - def StdSimplex := Σ n : ℕ, StandardSimplex n - - section - - variable (s : StdSimplex) - - #check s.fst - #check s.snd - - #check s.1 - #check s.2 - - end -
src/C_Structures/S02_Algebraic_Structures.lean (deleted)
- @@ -1,172 +0,0 @@
  - import Mathlib.Data.Real.Basic - - structure Group₁ (α : Type _) where - mul : α → α → α - one : α - inv : α → α - mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) - mul_one : ∀ x : α, mul x one = x - one_mul : ∀ x : α, mul one x = x - mul_left_inv : ∀ x : α, mul (inv x) x = one - - structure Group₁Cat where - α : Type _ - str : Group₁ α - - section - - variable (α β γ : Type _) - variable (f : α ≃ β) (g : β ≃ γ) - - #check Equiv α β - #check (f.toFun : α → β) - #check (f.invFun : β → α) - #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) - #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) - #check (Equiv.refl α : α ≃ α) - #check (f.symm : β ≃ α) - #check (f.trans g : α ≃ γ) - - example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := - rfl - - example (x : α) : (f.trans g) x = g (f x) := - rfl - - example : (f.trans g : α → γ) = g ∘ f := - rfl - - end - - example (α : Type _) : Equiv.Perm α = (α ≃ α) := - rfl - - def permGroup {α : Type _} : Group₁ (Equiv.Perm α) - where - mul f g := Equiv.trans g f - one := Equiv.refl α - inv := Equiv.symm - mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm - one_mul := Equiv.trans_refl - mul_one := Equiv.refl_trans - mul_left_inv := Equiv.self_trans_symm - - structure AddGroup₁ (α : Type _) where - (add : α → α → α) - -- fill in the rest - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def neg (a : point) : point := sorry - - def zero : point := sorry - - def add_group_point : add_group₁ point := sorry - end Point - - section - - variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) - - #check f * g - #check mul_assoc f g g⁻¹ - - -- group power, defined for any group - #check g ^ n - - example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] - - example : f * g * g⁻¹ = f := - mul_inv_cancel_right f g - - example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := - mul_inv_cancel_right f g - - end - - class Group₂ (α : Type _) where - mul : α → α → α - one : α - inv : α → α - mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) - mul_one : ∀ x : α, mul x one = x - one_mul : ∀ x : α, mul one x = x - mul_left_inv : ∀ x : α, mul (inv x) x = one - - instance {α : Type _} : Group₂ (Equiv.Perm α) where - mul f g := Equiv.trans g f - one := Equiv.refl α - inv := Equiv.symm - mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm - one_mul := Equiv.trans_refl - mul_one := Equiv.refl_trans - mul_left_inv := Equiv.self_trans_symm - - #check @Group₂.mul - - def mySquare {α : Type _} [Group₂ α] (x : α) := - Group₂.mul x x - - #check @mySquare - - section - variable {β : Type _} (f g : Equiv.Perm β) - - example : Group₂.mul f g = g.trans f := - rfl - - example : mySquare f = f.trans f := - rfl - - end - - instance : Inhabited Point where default := ⟨0, 0, 0⟩ - - #check (default : Point) - - example : ([] : List Point).headI = default := - rfl - - instance : Add Point where add := Point.add - - section - variable (x y : Point) - - #check x + y - - example : x + y = Point.add x y := - rfl - - end - - instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := - ⟨Group₂.mul⟩ - - instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := - ⟨Group₂.one⟩ - - instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := - ⟨Group₂.inv⟩ - - section - - variable {α : Type _} (f g : Equiv.Perm α) - - #check f * 1 * g⁻¹ - - def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := - rfl - - end - - class AddGroup₂ (α : Type _) where - add : α → α → α - -- fill in the rest
src/C_Structures/S03_Building_the_Gaussian_Integers.lean (deleted)
- @@ -1,271 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intros - ext <;> simp - add_zero := by - intros - ext <;> simp - add_left_neg := by - intros - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intros - ext <;> simp - mul_one := by - intros - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - example (a b : ℤ) : a = b * (a / b) + a % b := - Eq.symm <| Int.ediv_add_emod a b - - example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := - Int.emod_nonneg a - - example (a b : ℤ) : b ≠ 0 → a % b < abs b := - Int.emod_lt a - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - sorry - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - sorry - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - sorry - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - sorry - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - sorry - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C_Structures/solutions/Solutions_S01_Structures.lean (deleted)
- @@ -1,96 +0,0 @@
  - import Mathlib.Algebra.BigOperators.Ring - import Mathlib.Data.Real.Basic - - noncomputable section - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by - simp [add, add_assoc] - - def smul (r : ℝ) (a : Point) : Point := - ⟨r * a.x, r * a.y, r * a.z⟩ - - theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by - simp [add, smul, mul_add] - - end Point - - structure StandardTwoSimplex where - x : ℝ - y : ℝ - z : ℝ - x_nonneg : 0 ≤ x - y_nonneg : 0 ≤ y - z_nonneg : 0 ≤ z - sum_eq : x + y + z = 1 - - namespace StandardTwoSimplex - - noncomputable section - - def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where - x := lambda * a.x + (1 - lambda) * b.x - y := lambda * a.y + (1 - lambda) * b.y - z := lambda * a.z + (1 - lambda) * b.z - x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) - y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) - z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) - sum_eq := by - trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) - · ring - simp [a.sum_eq, b.sum_eq] - - end - - end StandardTwoSimplex - - open BigOperators - - structure StandardSimplex (n : ℕ) where - V : Fin n → ℝ - NonNeg : ∀ i : Fin n, 0 ≤ V i - sum_eq_one : (∑ i, V i) = 1 - - namespace StandardSimplex - - def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n - where - V i := (a.V i + b.V i) / 2 - NonNeg := by - intro i - apply div_nonneg - · linarith [a.NonNeg i, b.NonNeg i] - norm_num - sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] - field_simp - - end StandardSimplex - - namespace StandardSimplex - - def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardSimplex n) : StandardSimplex n - where - V i := lambda * a.V i + (1 - lambda) * b.V i - NonNeg i := - add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) - sum_eq_one := by - trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i - · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] - simp [a.sum_eq_one, b.sum_eq_one] - - end StandardSimplex -
src/C_Structures/solutions/Solutions_S02_Algebraic_Structures.lean (deleted)
- @@ -1,73 +0,0 @@
  - import Mathlib.Data.Real.Basic - - structure AddGroup₁ (α : Type _) where - add : α → α → α - zero : α - neg : α → α - add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) - add_zero : ∀ x : α, add x zero = x - zero_add : ∀ x : α, add x zero = x - add_left_neg : ∀ x : α, add (neg x) x = zero - - @[ext] - structure Point where - x : ℝ - y : ℝ - z : ℝ - - namespace Point - - def add (a b : Point) : Point := - ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - - def neg (a : Point) : Point := - ⟨-a.x, -a.y, -a.z⟩ - - def zero : Point := - ⟨0, 0, 0⟩ - - def addGroupPoint : AddGroup₁ Point where - add := Point.add - zero := Point.zero - neg := Point.neg - add_assoc := by simp [Point.add, add_assoc] - add_zero := by simp [Point.add, Point.zero] - zero_add := by simp [Point.add, Point.zero] - add_left_neg := by simp [Point.add, Point.neg, Point.zero] - - end Point - - class AddGroup₂ (α : Type _) where - add : α → α → α - zero : α - neg : α → α - add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) - add_zero : ∀ x : α, add x zero = x - zero_add : ∀ x : α, add x zero = x - add_left_neg : ∀ x : α, add (neg x) x = zero - - instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := - ⟨AddGroup₂.add⟩ - - instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := - ⟨AddGroup₂.zero⟩ - - instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := - ⟨AddGroup₂.neg⟩ - - instance : AddGroup₂ Point where - add := Point.add - zero := Point.zero - neg := Point.neg - add_assoc := by simp [Point.add, add_assoc] - add_zero := by simp [Point.add, Point.zero] - zero_add := by simp [Point.add, Point.zero] - add_left_neg := by simp [Point.add, Point.neg, Point.zero] - - section - variable (x y : Point) - - #check x + -y + 0 - - end -
src/C_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (deleted)
- @@ -1,285 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intros - ext <;> simp - add_zero := by - intros - ext <;> simp - add_left_neg := by - intros - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intros - ext <;> simp - mul_one := by - intros - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := - haveI h' : x ^ 2 = 0 := by - apply le_antisymm _ (sq_nonneg x) - rw [← h] - apply le_add_of_nonneg_right (sq_nonneg y) - pow_eq_zero h' - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - constructor - · intro h - constructor - · exact aux h - rw [add_comm] at h - exact aux h - rintro ⟨rfl, rfl⟩ - norm_num - - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - apply add_nonneg <;> - apply sq_nonneg - - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] - rfl - - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] - simp [norm_nonneg] - - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - simp [norm] - ring - - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C_Topology/S01_Filters.lean (deleted)
- @@ -1,110 +0,0 @@
 - import Mathlib.Topology.Instances.Real - - open Set Filter Topology - - def principal {α : Type _} (s : Set α) : Filter α - where - sets := { t | s ⊆ t } - univ_sets := sorry - sets_of_superset := sorry - inter_sets := sorry - - example : Filter ℕ := - { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } - univ_sets := sorry - sets_of_superset := sorry - inter_sets := sorry } - - def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := - ∀ V ∈ G, f ⁻¹' V ∈ F - - def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := - map f F ≤ G - - example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : - Tendsto₂ f F G ↔ Tendsto₁ f F G := - Iff.rfl - - #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) - - #check - (@Filter.map_map : - ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) - - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} - (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := - sorry - - variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) - - #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) - - #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) - - section - - variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} - - #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) - - end - - example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := - nhds_prod_eq - - #check le_inf_iff - - example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ - Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := - sorry - - example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := - nhds_basis_Ioo_pos x₀ - - example (u : ℕ → ℝ) (x₀ : ℝ) : - Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by - have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis - rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] - simp - - example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : - ∀ᶠ n in atTop, P n ∧ Q n := - hP.and hQ - - example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : - Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := - tendsto_congr' h - - 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 - - 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 - apply (hP.and (hQ.and hR)).mono - rintro n ⟨h, h', h''⟩ - exact h'' ⟨h, h'⟩ - - 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 - filter_upwards [hP, hQ, hR] - intro n h h' h'' - exact h'' ⟨h, h'⟩ - - #check mem_closure_iff_clusterPt - - #check le_principal_iff - - #check neBot_of_le - - example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) - (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := - sorry -
src/C_Topology/S02_Metric_Spaces.lean (deleted)
- @@ -1,200 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter - - open Topology Filter - - variable {X : Type _} [MetricSpace X] (a b c : X) - - #check (dist a b : ℝ) - - #check (dist_nonneg : 0 ≤ dist a b) - - #check (dist_eq_zero : dist a b = 0 ↔ a = b) - - #check (dist_comm a b : dist a b = dist b a) - - #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) - - -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. - #check EMetricSpace - - #check PseudoMetricSpace - - #check PseudoEMetricSpace - - example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := - Metric.tendsto_atTop - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : - Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := - Metric.continuous_iff - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by - apply Continuous.dist - exact hf.comp continuous_fst - exact hf.comp continuous_snd - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - (hf.comp continuous_fst).dist (hf.comp continuous_snd) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - hf.fst'.dist hf.snd' - - example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := - sorry - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : - ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := - Metric.continuousAt_iff - - variable (r : ℝ) - - example : Metric.ball a r = { b | dist b a < r } := - rfl - - example : Metric.closedBall a r = { b | dist b a ≤ r } := - rfl - - example (hr : 0 < r) : a ∈ Metric.ball a r := - Metric.mem_ball_self hr - - example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := - Metric.mem_closedBall_self hr - - example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.isOpen_iff - - example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := - isOpen_compl_iff.symm - - example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) - (hus : ∀ n, u n ∈ s) : a ∈ s := - hs.mem_of_tendsto hu (eventually_of_forall hus) - - example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := - Metric.mem_closure_iff - - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := - sorry - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.nhds_basis_ball.mem_iff - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := - Metric.nhds_basis_closedBall.mem_iff - - example : IsCompact (Set.Icc 0 1 : Set ℝ) := - isCompact_Icc - - example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : - ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := - hs.exists_forall_le hs' hfs - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := - hs.exists_forall_ge hs' hfs - - example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed - - example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ - - #check IsCompact.isClosed - - example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : - UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := - Metric.uniformContinuous_iff - - example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} - (hf : Continuous f) : UniformContinuous f := - sorry - - example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := - Metric.cauchySeq_iff - - example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := - Metric.cauchySeq_iff' - - example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := - cauchySeq_tendsto_of_complete hu - - open BigOperators - - open Finset - - theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} - (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by - rw [Metric.cauchySeq_iff'] - intro ε ε_pos - obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry - use N - intro n hn - obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn - calc - dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry - _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry - _ ≤ 1 / 2 ^ N * 2 := sorry - _ < ε := sorry - - - open Metric - - example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : - Dense (⋂ n, f n) := by - let B : ℕ → ℝ := fun n => (1 / 2) ^ n - have Bpos : ∀ n, 0 < B n - sorry - /- Translate the density assumption into two functions `center` and `radius` associating - to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. - We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ - have : - ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := - by sorry - choose! center radius Hpos HB Hball using this - intro x - rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] - intro ε εpos - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all - `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball - `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that - `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a - limit which belongs to all the `f n`. -/ - let F : ℕ → X × ℝ := fun n => - Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) - let c : ℕ → X := fun n => (F n).1 - let r : ℕ → ℝ := fun n => (F n).2 - have rpos : ∀ n, 0 < r n := by sorry - have rB : ∀ n, r n ≤ B n := by sorry - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry - have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry - have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist - -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. - rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ - -- this point `y` will be the desired point. We will check that it belongs to all - -- `f n` and to `ball x ε`. - use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry - sorry -
src/C_Topology/S03_Topological_Spaces.lean (deleted)
- @@ -1,151 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter - - open Topology Filter - - section - - variable {X : Type _} [TopologicalSpace X] - - example : IsOpen (univ : Set X) := - isOpen_univ - - example : IsOpen (∅ : Set X) := - isOpen_empty - - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := - isOpen_iUnion hs - - example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := - isOpen_iInter hs - - variable {Y : Type _} [TopologicalSpace Y] - - example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := - continuous_def - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := - Iff.rfl - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := - Iff.rfl - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := - mem_nhds_iff - - example (x : X) : pure x ≤ 𝓝 x := - pure_le_nhds x - - example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := - pure_le_nhds x h - - example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := - eventually_eventually_nhds.mpr h - - #check TopologicalSpace.mkOfNhds - - #check TopologicalSpace.nhds_mkOfNhds - - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) - (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : - ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := - sorry - - end - - -- BOTH. - variable {X Y : Type _} - - example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := - TopologicalSpace.coinduced f - - example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := - TopologicalSpace.induced f - - example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : - TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := - coinduced_le_iff_le_induced - - #check coinduced_compose - - #check induced_compose - - example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := - Iff.rfl - - example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : - Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := - continuous_iff_coinduced_le - - example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : - @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by - rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] - - example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : - (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = - ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := - rfl - - example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) - (hb : Tendsto u atTop (𝓝 b)) : a = b := - tendsto_nhds_unique ha hb - - example [TopologicalSpace X] [RegularSpace X] (a : X) : - (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := - closed_nhds_basis a - - example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := - nhds_basis_opens' x - - theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : - ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := - sorry - - example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace 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) : - ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := - sorry - - #check @HasBasis.tendsto_right_iff - - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : - a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := - mem_closure_iff_seq_limit - - variable [TopologicalSpace X] - - example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := - Iff.rfl - - example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := - Iff.rfl - - example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) - (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - variable [TopologicalSpace Y] - - example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} - (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := - ClusterPt.map H hfx hf - - example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : - IsCompact (f '' s) := by - intro F F_ne F_le - have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry - have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry - have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left - sorry - - example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) - (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := - hs.elim_finite_subcover U hUo hsU - - example [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ -
src/C_Topology/solutions/Solutions_S01_Filters.lean (deleted)
- @@ -1,71 +0,0 @@
 - import Mathlib.Topology.Instances.Real - - open Set Filter Topology - - -- In the next example we could use `tauto` in each proof instead of knowing the lemmas - example {α : Type _} (s : Set α) : Filter α := - { sets := { t | s ⊆ t } - univ_sets := subset_univ s - sets_of_superset := fun hU hUV => Subset.trans hU hUV - inter_sets := fun hU hV => subset_inter hU hV } - - example : Filter ℕ := - { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } - univ_sets := by - use 42 - simp - sets_of_superset := by - rintro U V ⟨N, hN⟩ hUV - use N - tauto - inter_sets := by - rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ - use max N N' - intro b hb - rw [max_le_iff] at hb - constructor <;> tauto } - - def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := - ∀ V ∈ G, f ⁻¹' V ∈ F - - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} - (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := - calc - map (g ∘ f) F = map g (map f F) := by rw [map_map] - _ ≤ map g G := (map_mono hf) - _ ≤ H := hg - - - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} - (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by - intro V hV - rw [preimage_comp] - apply hf - apply hg - exact hV - - example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ - Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := - calc - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl - _ ↔ map f atTop ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ := by rw [nhds_prod_eq] - _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl - _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff - _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by - rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] - _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by - rw [map_map, map_map] - - - -- an alternative solution - example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : - Tendsto f atTop (𝓝 (x₀, y₀)) ↔ - Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by - rw [nhds_prod_eq] - unfold Tendsto Filter.prod - rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] - - example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) - (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := - mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
src/C_Topology/solutions/Solutions_S02_Metric_Spaces.lean (deleted)
- @@ -1,365 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter - - open Topology Filter - - variable {X : Type _} [MetricSpace X] (a b c : X) - - #check (dist a b : ℝ) - - #check (dist_nonneg : 0 ≤ dist a b) - - #check (dist_eq_zero : dist a b = 0 ↔ a = b) - - #check (dist_comm a b : dist a b = dist b a) - - #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) - - -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. - #check EMetricSpace - - #check PseudoMetricSpace - - #check PseudoEMetricSpace - - example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := - Metric.tendsto_atTop - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : - Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := - Metric.continuous_iff - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := by - apply Continuous.dist - exact hf.comp continuous_fst - exact hf.comp continuous_snd - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - (hf.comp continuous_fst).dist (hf.comp continuous_snd) - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : - Continuous fun p : X × X => dist (f p.1) (f p.2) := - hf.fst'.dist hf.snd' - - example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := - sorry - - example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := - hf.comp <| (continuous_pow 2).add continuous_id - - example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : - ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := - Metric.continuousAt_iff - - variable (r : ℝ) - - example : Metric.ball a r = { b | dist b a < r } := - rfl - - example : Metric.closedBall a r = { b | dist b a ≤ r } := - rfl - - example (hr : 0 < r) : a ∈ Metric.ball a r := - Metric.mem_ball_self hr - - example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := - Metric.mem_closedBall_self hr - - example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.isOpen_iff - - example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := - isOpen_compl_iff.symm - - example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) - (hus : ∀ n, u n ∈ s) : a ∈ s := - hs.mem_of_tendsto hu (eventually_of_forall hus) - - example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := - Metric.mem_closure_iff - - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := - sorry - - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by - rw [Metric.tendsto_atTop] at hu - rw [Metric.mem_closure_iff] - intro ε ε_pos - rcases hu ε ε_pos with ⟨N, hN⟩ - refine' ⟨u N, hs _, _⟩ - rw [dist_comm] - exact hN N le_rfl - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := - Metric.nhds_basis_ball.mem_iff - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := - Metric.nhds_basis_closedBall.mem_iff - - example : IsCompact (Set.Icc 0 1 : Set ℝ) := - isCompact_Icc - - example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : - ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := - hs.exists_forall_le hs' hfs - - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : - ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := - hs.exists_forall_ge hs' hfs - - example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed - - example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ - - #check IsCompact.isClosed - - example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : - UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := - Metric.uniformContinuous_iff - - example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} - (hf : Continuous f) : UniformContinuous f := - sorry - - example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} - (hf : Continuous f) : UniformContinuous f := by - rw [Metric.uniformContinuous_iff] - intro ε ε_pos - let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) - have φ_cont : Continuous φ := hf.fst'.dist hf.snd' - let K := { p : X × X | ε ≤ φ p } - have K_closed : IsClosed K := isClosed_le continuous_const φ_cont - have K_cpct : IsCompact K := K_closed.isCompact - cases' eq_empty_or_nonempty K with hK hK - · use 1, by norm_num - intro x y _ - have : (x, y) ∉ K := by simp [hK] - simpa 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 - linarith - · intro x x' - contrapose! - intro hxx' - exact H (x, x') hxx' - - example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := - Metric.cauchySeq_iff - - example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := - Metric.cauchySeq_iff' - - example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := - cauchySeq_tendsto_of_complete hu - - open BigOperators - - open Finset - - theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} - (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by - rw [Metric.cauchySeq_iff'] - intro ε ε_pos - obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry - use N - intro n hn - obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn - calc - dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry - _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry - _ ≤ 1 / 2 ^ N * 2 := sorry - _ < ε := sorry - - - example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by - rw [Metric.cauchySeq_iff'] - intro ε ε_pos - obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by - have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by - rw [← MulZeroClass.zero_mul (2 : ℝ)] - apply Tendsto.mul - simp_rw [← one_div_pow (2 : ℝ)] - apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith - exact tendsto_const_nhds - rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ - exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ - use N - intro n hn - obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn - calc - dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] - _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := - (dist_le_range_sum_dist (fun i => u (N + i)) k) - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] - _ ≤ 1 / 2 ^ N * 2 := - (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) - (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) - _ < ε := hN - - - open Metric - - example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : - Dense (⋂ n, f n) := by - let B : ℕ → ℝ := fun n => (1 / 2) ^ n - have Bpos : ∀ n, 0 < B n - sorry - /- Translate the density assumption into two functions `center` and `radius` associating - to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. - We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ - have : - ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := - by sorry - choose! center radius Hpos HB Hball using this - intro x - rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] - intro ε εpos - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all - `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball - `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that - `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a - limit which belongs to all the `f n`. -/ - let F : ℕ → X × ℝ := fun n => - Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) - let c : ℕ → X := fun n => (F n).1 - let r : ℕ → ℝ := fun n => (F n).2 - have rpos : ∀ n, 0 < r n := by sorry - have rB : ∀ n, r n ≤ B n := by sorry - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry - have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry - have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist - -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. - rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ - -- this point `y` will be the desired point. We will check that it belongs to all - -- `f n` and to `ball x ε`. - use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry - sorry - - example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : - Dense (⋂ n, f n) := by - let B : ℕ → ℝ := fun n => (1 / 2) ^ n - have Bpos : ∀ n, 0 pow_pos sorry n - /- Translate the density assumption into two functions `center` and `radius` associating - to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. - We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ - have : - ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by - intro n x δ δpos - have : x ∈ closure (f n) := hd n x - rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ - rw [dist_comm] at xy - obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := - nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) - refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ - show 0 < min (min (δ / 2) r) (B (n + 1)) - exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) - show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) - exact min_le_right _ _ - show z ∈ closedBall x δ - exact - calc - dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ - _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) - _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) - _ = δ := add_halves δ - - show z ∈ f n - exact - hr - (calc - dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz - _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) - ) - choose! center radius Hpos HB Hball using this - refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all - `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball - `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that - `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a - limit which belongs to all the `f n`. -/ - let F : ℕ → X × ℝ := fun n => - Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) - let c : ℕ → X := fun n => (F n).1 - let r : ℕ → ℝ := fun n => (F n).2 - have rpos : ∀ n, 0 < r n := by - intro n - induction' n with n hn - exact lt_min εpos (Bpos 0) - exact Hpos n (c n) (r n) hn - have rB : ∀ n, r n ≤ B n := by - intro n - induction' n with n hn - exact min_le_right _ _ - exact HB n (c n) (r n) (rpos n) - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => - Hball n (c n) (r n) (rpos n) - have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by - intro n - rw [dist_comm] - have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := - mem_closedBall_self (rpos <| n + 1).le - have I := - calc - closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := - (incl n).trans (inter_subset_left _ _) - _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) - - exact I A - have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist - -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. - rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ - -- this point `y` will be the desired point. We will check that it belongs to all - -- `f n` and to `ball x ε`. - use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by - intro n - refine' Nat.le_induction _ fun m hnm h => _ - · exact Subset.rfl - · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by - intro n - refine' isClosed_ball.mem_of_tendsto ylim _ - refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ - exact I n m hm (mem_closedBall_self (rpos _).le) - constructor - · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] - intro n - have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := - Subset.trans (incl n) (inter_subset_right _ _) - exact this (yball (n + 1)) - calc - dist y x ≤ r 0 := yball 0 - _ ≤ ε := min_le_left _ _ - -
src/C_Topology/solutions/Solutions_S03_Topological_Spaces.lean (deleted)
- @@ -1,202 +0,0 @@
 - import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - - open Set Filter - - open Topology Filter - - section - - variable {X : Type _} [TopologicalSpace X] - - example : IsOpen (univ : Set X) := - isOpen_univ - - example : IsOpen (∅ : Set X) := - isOpen_empty - - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := - isOpen_iUnion hs - - example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := - isOpen_iInter hs - - variable {Y : Type _} [TopologicalSpace Y] - - example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := - continuous_def - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := - Iff.rfl - - example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := - Iff.rfl - - example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := - mem_nhds_iff - - example (x : X) : pure x ≤ 𝓝 x := - pure_le_nhds x - - example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := - pure_le_nhds x h - - example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := - eventually_eventually_nhds.mpr h - - #check TopologicalSpace.mkOfNhds - - #check TopologicalSpace.nhds_mkOfNhds - - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) - (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : - ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := - sorry - - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) - (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : - ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by - intro a s s_in - refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ - rintro y (hy : s ∈ n y) - exact H₀ y hy - - end - - -- BOTH. - variable {X Y : Type _} - - example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := - TopologicalSpace.coinduced f - - example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := - TopologicalSpace.induced f - - example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : - TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := - coinduced_le_iff_le_induced - - #check coinduced_compose - - #check induced_compose - - example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := - Iff.rfl - - example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : - Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := - continuous_iff_coinduced_le - - example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : - @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by - rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] - - example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : - (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = - ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := - rfl - - example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) - (hb : Tendsto u atTop (𝓝 b)) : a = b := - tendsto_nhds_unique ha hb - - example [TopologicalSpace X] [RegularSpace X] (a : X) : - (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := - closed_nhds_basis a - - example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := - nhds_basis_opens' x - - theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : - ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := - sorry - - example {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : - ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by - simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in - - example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace 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) : - ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := - sorry - - #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) - (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : - ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by - choose φ hφ using hf - use φ - constructor - · rw [continuous_iff_continuousAt] - intro x - suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by - simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] - intro V' V'_in V'_closed - obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in - suffices : ∀ y ∈ V, φ y ∈ V' - exact mem_of_superset V_in this - intro y y_in - have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in - haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y - apply V'_closed.mem_of_tendsto (hφ y) - exact mem_of_superset (preimage_mem_comap hVx) hV - · intro a - have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a - exact tendsto_nhds_unique lim f_cont.continuousAt - - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : - a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := - mem_closure_iff_seq_limit - - variable [TopologicalSpace X] - - example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := - Iff.rfl - - example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := - Iff.rfl - - example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) - (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := - hs.tendsto_subseq hu - - variable [TopologicalSpace Y] - - example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} - (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := - ClusterPt.map H hfx hf - - example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : - IsCompact (f '' s) := by - intro F F_ne F_le - have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry - have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry - have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left - sorry - - example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : - IsCompact (f '' s) := by - intro F F_ne F_le - have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] - have Hne : (𝓟 s ⊓ comap f F).NeBot := by - apply NeBot.of_map - rwa [map_eq, inf_of_le_right F_le] - have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left - rcases hs Hle with ⟨x, x_in, hx⟩ - refine' ⟨f x, mem_image_of_mem f x_in, _⟩ - apply hx.map hf.continuousAt - rw [Tendsto, map_eq] - exact inf_le_right - - example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) - (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := - hs.elim_finite_subcover U hUo hsU - - example [CompactSpace X] : IsCompact (univ : Set X) := - isCompact_univ -