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Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import analysis.inner_product_space.projection
import analysis.special_functions.trigonometric.inverse
import algebra.quadratic_discriminant
import linear_algebra.affine_space.finite_dimensional
import analysis.calculus.conformal.normed_space
/-!
# Euclidean spaces
This file makes some definitions and proves very basic geometrical
results about real inner product spaces and Euclidean affine spaces.
Results about real inner product spaces that involve the norm and
inner product but not angles generally go in
`analysis.normed_space.inner_product`. Results with longer
proofs or more geometrical content generally go in separate files.
## Main definitions
* `inner_product_geometry.angle` is the undirected angle between two
vectors.
* `euclidean_geometry.angle`, with notation `∠`, is the undirected
angle determined by three points.
* `euclidean_geometry.orthogonal_projection` is the orthogonal
projection of a point onto an affine subspace.
* `euclidean_geometry.reflection` is the reflection of a point in an
affine subspace.
## Implementation notes
To declare `P` as the type of points in a Euclidean affine space with
`V` as the type of vectors, use `[inner_product_space ℝ V] [metric_space P]
[normed_add_torsor V P]`. This works better with `out_param` to make
`V` implicit in most cases than having a separate type alias for
Euclidean affine spaces.
Rather than requiring Euclidean affine spaces to be finite-dimensional
(as in the definition on Wikipedia), this is specified only for those
theorems that need it.
## References
* https://en.wikipedia.org/wiki/Euclidean_space
-/
noncomputable theory
open_locale big_operators
open_locale classical
open_locale real
open_locale real_inner_product_space
namespace inner_product_geometry
/-!
### Geometrical results on real inner product spaces
This section develops some geometrical definitions and results on real
inner product spaces, where those definitions and results can most
conveniently be developed in terms of vectors and then used to deduce
corresponding results for Euclidean affine spaces.
-/
variables {V : Type*} [inner_product_space ℝ V]
/-- The undirected angle between two vectors. If either vector is 0,
this is π/2. -/
def angle (x y : V) : ℝ := real.arccos (inner x y / (∥x∥ * ∥y∥))
lemma is_conformal_map.preserves_angle {E F : Type*}
[inner_product_space ℝ E] [inner_product_space ℝ F]
{f' : E →L[ℝ] F} (h : is_conformal_map f') (u v : E) :
angle (f' u) (f' v) = angle u v :=
begin
obtain ⟨c, hc, li, hcf⟩ := h,
suffices : c * (c * inner u v) / (∥c∥ * ∥u∥ * (∥c∥ * ∥v∥)) = inner u v / (∥u∥ * ∥v∥),
{ simp [this, angle, hcf, norm_smul, inner_smul_left, inner_smul_right] },
by_cases hu : ∥u∥ = 0,
{ simp [norm_eq_zero.mp hu] },
by_cases hv : ∥v∥ = 0,
{ simp [norm_eq_zero.mp hv] },
have hc : ∥c∥ ≠ 0 := λ w, hc (norm_eq_zero.mp w),
field_simp,
have : c * c = ∥c∥ * ∥c∥ := by simp [real.norm_eq_abs, abs_mul_abs_self],
convert congr_arg (λ x, x * ⟪u, v⟫ * ∥u∥ * ∥v∥) this using 1; ring,
end
/-- If a real differentiable map `f` is conformal at a point `x`,
then it preserves the angles at that point. -/
lemma conformal_at.preserves_angle {E F : Type*}
[inner_product_space ℝ E] [inner_product_space ℝ F]
{f : E → F} {x : E} {f' : E →L[ℝ] F}
(h : has_fderiv_at f f' x) (H : conformal_at f x) (u v : E) :
angle (f' u) (f' v) = angle u v :=
let ⟨f₁, h₁, c⟩ := H in h₁.unique h ▸ is_conformal_map.preserves_angle c u v
/-- The cosine of the angle between two vectors. -/
lemma cos_angle (x y : V) : real.cos (angle x y) = inner x y / (∥x∥ * ∥y∥) :=
real.cos_arccos (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1
(abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2
/-- The angle between two vectors does not depend on their order. -/
lemma angle_comm (x y : V) : angle x y = angle y x :=
begin
unfold angle,
rw [real_inner_comm, mul_comm]
end
/-- The angle between the negation of two vectors. -/
@[simp] lemma angle_neg_neg (x y : V) : angle (-x) (-y) = angle x y :=
begin
unfold angle,
rw [inner_neg_neg, norm_neg, norm_neg]
end
/-- The angle between two vectors is nonnegative. -/
lemma angle_nonneg (x y : V) : 0 ≤ angle x y :=
real.arccos_nonneg _
/-- The angle between two vectors is at most π. -/
lemma angle_le_pi (x y : V) : angle x y ≤ π :=
real.arccos_le_pi _
/-- The angle between a vector and the negation of another vector. -/
lemma angle_neg_right (x y : V) : angle x (-y) = π - angle x y :=
begin
unfold angle,
rw [←real.arccos_neg, norm_neg, inner_neg_right, neg_div]
end
/-- The angle between the negation of a vector and another vector. -/
lemma angle_neg_left (x y : V) : angle (-x) y = π - angle x y :=
by rw [←angle_neg_neg, neg_neg, angle_neg_right]
/-- The angle between the zero vector and a vector. -/
@[simp] lemma angle_zero_left (x : V) : angle 0 x = π / 2 :=
begin
unfold angle,
rw [inner_zero_left, zero_div, real.arccos_zero]
end
/-- The angle between a vector and the zero vector. -/
@[simp] lemma angle_zero_right (x : V) : angle x 0 = π / 2 :=
begin
unfold angle,
rw [inner_zero_right, zero_div, real.arccos_zero]
end
/-- The angle between a nonzero vector and itself. -/
@[simp] lemma angle_self {x : V} (hx : x ≠ 0) : angle x x = 0 :=
begin
unfold angle,
rw [←real_inner_self_eq_norm_mul_norm, div_self (λ h, hx (inner_self_eq_zero.1 h)),
real.arccos_one]
end
/-- The angle between a nonzero vector and its negation. -/
@[simp] lemma angle_self_neg_of_nonzero {x : V} (hx : x ≠ 0) : angle x (-x) = π :=
by rw [angle_neg_right, angle_self hx, sub_zero]
/-- The angle between the negation of a nonzero vector and that
vector. -/
@[simp] lemma angle_neg_self_of_nonzero {x : V} (hx : x ≠ 0) : angle (-x) x = π :=
by rw [angle_comm, angle_self_neg_of_nonzero hx]
/-- The angle between a vector and a positive multiple of a vector. -/
@[simp] lemma angle_smul_right_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
angle x (r • y) = angle x y :=
begin
unfold angle,
rw [inner_smul_right, norm_smul, real.norm_eq_abs, abs_of_nonneg (le_of_lt hr), ←mul_assoc,
mul_comm _ r, mul_assoc, mul_div_mul_left _ _ (ne_of_gt hr)]
end
/-- The angle between a positive multiple of a vector and a vector. -/
@[simp] lemma angle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
angle (r • x) y = angle x y :=
by rw [angle_comm, angle_smul_right_of_pos y x hr, angle_comm]
/-- The angle between a vector and a negative multiple of a vector. -/
@[simp] lemma angle_smul_right_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
angle x (r • y) = angle x (-y) :=
by rw [←neg_neg r, neg_smul, angle_neg_right, angle_smul_right_of_pos x y (neg_pos_of_neg hr),
angle_neg_right]
/-- The angle between a negative multiple of a vector and a vector. -/
@[simp] lemma angle_smul_left_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
angle (r • x) y = angle (-x) y :=
by rw [angle_comm, angle_smul_right_of_neg y x hr, angle_comm]
/-- The cosine of the angle between two vectors, multiplied by the
product of their norms. -/
lemma cos_angle_mul_norm_mul_norm (x y : V) : real.cos (angle x y) * (∥x∥ * ∥y∥) = inner x y :=
begin
rw [cos_angle, div_mul_cancel_of_imp],
simp [or_imp_distrib] { contextual := tt },
end
/-- The sine of the angle between two vectors, multiplied by the
product of their norms. -/
lemma sin_angle_mul_norm_mul_norm (x y : V) : real.sin (angle x y) * (∥x∥ * ∥y∥) =
real.sqrt (inner x x * inner y y - inner x y * inner x y) :=
begin
unfold angle,
rw [real.sin_arccos (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1
(abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2,
←real.sqrt_mul_self (mul_nonneg (norm_nonneg x) (norm_nonneg y)),
←real.sqrt_mul' _ (mul_self_nonneg _), sq,
real.sqrt_mul_self (mul_nonneg (norm_nonneg x) (norm_nonneg y)),
real_inner_self_eq_norm_mul_norm,
real_inner_self_eq_norm_mul_norm],
by_cases h : (∥x∥ * ∥y∥) = 0,
{ rw [(show ∥x∥ * ∥x∥ * (∥y∥ * ∥y∥) = (∥x∥ * ∥y∥) * (∥x∥ * ∥y∥), by ring), h, mul_zero, mul_zero,
zero_sub],
cases eq_zero_or_eq_zero_of_mul_eq_zero h with hx hy,
{ rw norm_eq_zero at hx,
rw [hx, inner_zero_left, zero_mul, neg_zero] },
{ rw norm_eq_zero at hy,
rw [hy, inner_zero_right, zero_mul, neg_zero] } },
{ field_simp [h], ring_nf }
end
/-- The angle between two vectors is zero if and only if they are
nonzero and one is a positive multiple of the other. -/
lemma angle_eq_zero_iff {x y : V} : angle x y = 0 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), 0 < r ∧ y = r • x) :=
begin
rw [angle, ← real_inner_div_norm_mul_norm_eq_one_iff, real.arccos_eq_zero, has_le.le.le_iff_eq,
eq_comm],
exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2
end
/-- The angle between two vectors is π if and only if they are nonzero
and one is a negative multiple of the other. -/
lemma angle_eq_pi_iff {x y : V} : angle x y = π ↔ (x ≠ 0 ∧ ∃ (r : ℝ), r < 0 ∧ y = r • x) :=
begin
rw [angle, ← real_inner_div_norm_mul_norm_eq_neg_one_iff, real.arccos_eq_pi, has_le.le.le_iff_eq],
exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1
end
/-- If the angle between two vectors is π, the angles between those
vectors and a third vector add to π. -/
lemma angle_add_angle_eq_pi_of_angle_eq_pi {x y : V} (z : V) (h : angle x y = π) :
angle x z + angle y z = π :=
begin
rcases angle_eq_pi_iff.1 h with ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩,
rw [angle_smul_left_of_neg x z hr, angle_neg_left, add_sub_cancel'_right]
end
/-- Two vectors have inner product 0 if and only if the angle between
them is π/2. -/
lemma inner_eq_zero_iff_angle_eq_pi_div_two (x y : V) : ⟪x, y⟫ = 0 ↔ angle x y = π / 2 :=
iff.symm $ by simp [angle, or_imp_distrib] { contextual := tt }
/-- If the angle between two vectors is π, the inner product equals the negative product
of the norms. -/
lemma inner_eq_neg_mul_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) : ⟪x, y⟫ = - (∥x∥ * ∥y∥) :=
by simp [← cos_angle_mul_norm_mul_norm, h]
/-- If the angle between two vectors is 0, the inner product equals the product of the norms. -/
lemma inner_eq_mul_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ⟪x, y⟫ = ∥x∥ * ∥y∥ :=
by simp [← cos_angle_mul_norm_mul_norm, h]
/-- The inner product of two non-zero vectors equals the negative product of their norms
if and only if the angle between the two vectors is π. -/
lemma inner_eq_neg_mul_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
⟪x, y⟫ = - (∥x∥ * ∥y∥) ↔ angle x y = π :=
begin
refine ⟨λ h, _, inner_eq_neg_mul_norm_of_angle_eq_pi⟩,
have h₁ : (∥x∥ * ∥y∥) ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne',
rw [angle, h, neg_div, div_self h₁, real.arccos_neg_one],
end
/-- The inner product of two non-zero vectors equals the product of their norms
if and only if the angle between the two vectors is 0. -/
lemma inner_eq_mul_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
⟪x, y⟫ = ∥x∥ * ∥y∥ ↔ angle x y = 0 :=
begin
refine ⟨λ h, _, inner_eq_mul_norm_of_angle_eq_zero⟩,
have h₁ : (∥x∥ * ∥y∥) ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne',
rw [angle, h, div_self h₁, real.arccos_one],
end
/-- If the angle between two vectors is π, the norm of their difference equals
the sum of their norms. -/
lemma norm_sub_eq_add_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) : ∥x - y∥ = ∥x∥ + ∥y∥ :=
begin
rw ← sq_eq_sq (norm_nonneg (x - y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)),
rw [norm_sub_pow_two_real, inner_eq_neg_mul_norm_of_angle_eq_pi h],
ring,
end
/-- If the angle between two vectors is 0, the norm of their sum equals
the sum of their norms. -/
lemma norm_add_eq_add_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ∥x + y∥ = ∥x∥ + ∥y∥ :=
begin
rw ← sq_eq_sq (norm_nonneg (x + y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)),
rw [norm_add_pow_two_real, inner_eq_mul_norm_of_angle_eq_zero h],
ring,
end
/-- If the angle between two vectors is 0, the norm of their difference equals
the absolute value of the difference of their norms. -/
lemma norm_sub_eq_abs_sub_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) :
∥x - y∥ = |∥x∥ - ∥y∥| :=
begin
rw [← sq_eq_sq (norm_nonneg (x - y)) (abs_nonneg (∥x∥ - ∥y∥)),
norm_sub_pow_two_real, inner_eq_mul_norm_of_angle_eq_zero h, sq_abs (∥x∥ - ∥y∥)],
ring,
end
/-- The norm of the difference of two non-zero vectors equals the sum of their norms
if and only the angle between the two vectors is π. -/
lemma norm_sub_eq_add_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
∥x - y∥ = ∥x∥ + ∥y∥ ↔ angle x y = π :=
begin
refine ⟨λ h, _, norm_sub_eq_add_norm_of_angle_eq_pi⟩,
rw ← inner_eq_neg_mul_norm_iff_angle_eq_pi hx hy,
obtain ⟨hxy₁, hxy₂⟩ := ⟨norm_nonneg (x - y), add_nonneg (norm_nonneg x) (norm_nonneg y)⟩,
rw [← sq_eq_sq hxy₁ hxy₂, norm_sub_pow_two_real] at h,
calc inner x y = (∥x∥ ^ 2 + ∥y∥ ^ 2 - (∥x∥ + ∥y∥) ^ 2) / 2 : by linarith
... = -(∥x∥ * ∥y∥) : by ring,
end
/-- The norm of the sum of two non-zero vectors equals the sum of their norms
if and only the angle between the two vectors is 0. -/
lemma norm_add_eq_add_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
∥x + y∥ = ∥x∥ + ∥y∥ ↔ angle x y = 0 :=
begin
refine ⟨λ h, _, norm_add_eq_add_norm_of_angle_eq_zero⟩,
rw ← inner_eq_mul_norm_iff_angle_eq_zero hx hy,
obtain ⟨hxy₁, hxy₂⟩ := ⟨norm_nonneg (x + y), add_nonneg (norm_nonneg x) (norm_nonneg y)⟩,
rw [← sq_eq_sq hxy₁ hxy₂, norm_add_pow_two_real] at h,
calc inner x y = ((∥x∥ + ∥y∥) ^ 2 - ∥x∥ ^ 2 - ∥y∥ ^ 2)/ 2 : by linarith
... = ∥x∥ * ∥y∥ : by ring,
end
/-- The norm of the difference of two non-zero vectors equals the absolute value
of the difference of their norms if and only the angle between the two vectors is 0. -/
lemma norm_sub_eq_abs_sub_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
∥x - y∥ = |∥x∥ - ∥y∥| ↔ angle x y = 0 :=
begin
refine ⟨λ h, _, norm_sub_eq_abs_sub_norm_of_angle_eq_zero⟩,
rw ← inner_eq_mul_norm_iff_angle_eq_zero hx hy,
have h1 : ∥x - y∥ ^ 2 = (∥x∥ - ∥y∥) ^ 2, { rw h, exact sq_abs (∥x∥ - ∥y∥) },
rw norm_sub_pow_two_real at h1,
calc inner x y = ((∥x∥ + ∥y∥) ^ 2 - ∥x∥ ^ 2 - ∥y∥ ^ 2)/ 2 : by linarith
... = ∥x∥ * ∥y∥ : by ring,
end
/-- The norm of the sum of two vectors equals the norm of their difference if and only if
the angle between them is π/2. -/
lemma norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (x y : V) :
∥x + y∥ = ∥x - y∥ ↔ angle x y = π / 2 :=
begin
rw [← sq_eq_sq (norm_nonneg (x + y)) (norm_nonneg (x - y)),
← inner_eq_zero_iff_angle_eq_pi_div_two x y, norm_add_pow_two_real, norm_sub_pow_two_real],
split; intro h; linarith,
end
end inner_product_geometry
namespace euclidean_geometry
/-!
### Geometrical results on Euclidean affine spaces
This section develops some geometrical definitions and results on
Euclidean affine spaces.
-/
open inner_product_geometry
variables {V : Type*} {P : Type*} [inner_product_space ℝ V] [metric_space P]
[normed_add_torsor V P]
local notation `⟪`x`, `y`⟫` := @inner ℝ V _ x y
include V
/-- The undirected angle at `p2` between the line segments to `p1` and
`p3`. If either of those points equals `p2`, this is π/2. Use
`open_locale euclidean_geometry` to access the `∠ p1 p2 p3`
notation. -/
def angle (p1 p2 p3 : P) : ℝ := angle (p1 -ᵥ p2 : V) (p3 -ᵥ p2)
localized "notation `∠` := euclidean_geometry.angle" in euclidean_geometry
/-- The angle at a point does not depend on the order of the other two
points. -/
lemma angle_comm (p1 p2 p3 : P) : ∠ p1 p2 p3 = ∠ p3 p2 p1 :=
angle_comm _ _
/-- The angle at a point is nonnegative. -/
lemma angle_nonneg (p1 p2 p3 : P) : 0 ≤ ∠ p1 p2 p3 :=
angle_nonneg _ _
/-- The angle at a point is at most π. -/
lemma angle_le_pi (p1 p2 p3 : P) : ∠ p1 p2 p3 ≤ π :=
angle_le_pi _ _
/-- The angle ∠AAB at a point. -/
lemma angle_eq_left (p1 p2 : P) : ∠ p1 p1 p2 = π / 2 :=
begin
unfold angle,
rw vsub_self,
exact angle_zero_left _
end
/-- The angle ∠ABB at a point. -/
lemma angle_eq_right (p1 p2 : P) : ∠ p1 p2 p2 = π / 2 :=
by rw [angle_comm, angle_eq_left]
/-- The angle ∠ABA at a point. -/
lemma angle_eq_of_ne {p1 p2 : P} (h : p1 ≠ p2) : ∠ p1 p2 p1 = 0 :=
angle_self (λ he, h (vsub_eq_zero_iff_eq.1 he))
/-- If the angle ∠ABC at a point is π, the angle ∠BAC is 0. -/
lemma angle_eq_zero_of_angle_eq_pi_left {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) :
∠ p2 p1 p3 = 0 :=
begin
unfold angle at h,
rw angle_eq_pi_iff at h,
rcases h with ⟨hp1p2, ⟨r, ⟨hr, hpr⟩⟩⟩,
unfold angle,
rw angle_eq_zero_iff,
rw [←neg_vsub_eq_vsub_rev, neg_ne_zero] at hp1p2,
use [hp1p2, -r + 1, add_pos (neg_pos_of_neg hr) zero_lt_one],
rw [add_smul, ←neg_vsub_eq_vsub_rev p1 p2, smul_neg],
simp [←hpr]
end
/-- If the angle ∠ABC at a point is π, the angle ∠BCA is 0. -/
lemma angle_eq_zero_of_angle_eq_pi_right {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) :
∠ p2 p3 p1 = 0 :=
begin
rw angle_comm at h,
exact angle_eq_zero_of_angle_eq_pi_left h
end
/-- If ∠BCD = π, then ∠ABC = ∠ABD. -/
lemma angle_eq_angle_of_angle_eq_pi (p1 : P) {p2 p3 p4 : P} (h : ∠ p2 p3 p4 = π) :
∠ p1 p2 p3 = ∠ p1 p2 p4 :=
begin
unfold angle at *,
rcases angle_eq_pi_iff.1 h with ⟨hp2p3, ⟨r, ⟨hr, hpr⟩⟩⟩,
rw [eq_comm],
convert angle_smul_right_of_pos (p1 -ᵥ p2) (p3 -ᵥ p2) (add_pos (neg_pos_of_neg hr) zero_lt_one),
rw [add_smul, ← neg_vsub_eq_vsub_rev p2 p3, smul_neg, neg_smul, ← hpr],
simp
end
/-- If ∠BCD = π, then ∠ACB + ∠ACD = π. -/
lemma angle_add_angle_eq_pi_of_angle_eq_pi (p1 : P) {p2 p3 p4 : P} (h : ∠ p2 p3 p4 = π) :
∠ p1 p3 p2 + ∠ p1 p3 p4 = π :=
begin
unfold angle at h,
rw [angle_comm p1 p3 p2, angle_comm p1 p3 p4],
unfold angle,
exact angle_add_angle_eq_pi_of_angle_eq_pi _ h
end
/-- Vertical Angles Theorem: angles opposite each other, formed by two intersecting straight
lines, are equal. -/
lemma angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi {p1 p2 p3 p4 p5 : P}
(hapc : ∠ p1 p5 p3 = π) (hbpd : ∠ p2 p5 p4 = π) : ∠ p1 p5 p2 = ∠ p3 p5 p4 :=
by linarith [angle_add_angle_eq_pi_of_angle_eq_pi p1 hbpd, angle_comm p4 p5 p1,
angle_add_angle_eq_pi_of_angle_eq_pi p4 hapc, angle_comm p4 p5 p3]
/-- If ∠ABC = π then dist A B ≠ 0. -/
lemma left_dist_ne_zero_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) : dist p1 p2 ≠ 0 :=
begin
by_contra heq,
rw [dist_eq_zero] at heq,
rw [heq, angle_eq_left] at h,
exact real.pi_ne_zero (by linarith),
end
/-- If ∠ABC = π then dist C B ≠ 0. -/
lemma right_dist_ne_zero_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) : dist p3 p2 ≠ 0 :=
left_dist_ne_zero_of_angle_eq_pi $ (angle_comm _ _ _).trans h
/-- If ∠ABC = π, then (dist A C) = (dist A B) + (dist B C). -/
lemma dist_eq_add_dist_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) :
dist p1 p3 = dist p1 p2 + dist p3 p2 :=
begin
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
exact norm_sub_eq_add_norm_of_angle_eq_pi h,
end
/-- If A ≠ B and C ≠ B then ∠ABC = π if and only if (dist A C) = (dist A B) + (dist B C). -/
lemma dist_eq_add_dist_iff_angle_eq_pi {p1 p2 p3 : P} (hp1p2 : p1 ≠ p2) (hp3p2 : p3 ≠ p2) :
dist p1 p3 = dist p1 p2 + dist p3 p2 ↔ ∠ p1 p2 p3 = π :=
begin
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
exact norm_sub_eq_add_norm_iff_angle_eq_pi
((λ he, hp1p2 (vsub_eq_zero_iff_eq.1 he))) (λ he, hp3p2 (vsub_eq_zero_iff_eq.1 he)),
end
/-- If ∠ABC = 0, then (dist A C) = abs ((dist A B) - (dist B C)). -/
lemma dist_eq_abs_sub_dist_of_angle_eq_zero {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = 0) :
(dist p1 p3) = |(dist p1 p2) - (dist p3 p2)| :=
begin
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
exact norm_sub_eq_abs_sub_norm_of_angle_eq_zero h,
end
/-- If A ≠ B and C ≠ B then ∠ABC = 0 if and only if (dist A C) = abs ((dist A B) - (dist B C)). -/
lemma dist_eq_abs_sub_dist_iff_angle_eq_zero {p1 p2 p3 : P} (hp1p2 : p1 ≠ p2) (hp3p2 : p3 ≠ p2) :
(dist p1 p3) = |(dist p1 p2) - (dist p3 p2)| ↔ ∠ p1 p2 p3 = 0 :=
begin
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
exact norm_sub_eq_abs_sub_norm_iff_angle_eq_zero
((λ he, hp1p2 (vsub_eq_zero_iff_eq.1 he))) (λ he, hp3p2 (vsub_eq_zero_iff_eq.1 he)),
end
/-- The midpoint of the segment AB is the same distance from A as it is from B. -/
lemma dist_left_midpoint_eq_dist_right_midpoint (p1 p2 : P) :
dist p1 (midpoint ℝ p1 p2) = dist p2 (midpoint ℝ p1 p2) :=
by rw [dist_left_midpoint p1 p2, dist_right_midpoint p1 p2]
/-- If M is the midpoint of the segment AB, then ∠AMB = π. -/
lemma angle_midpoint_eq_pi (p1 p2 : P) (hp1p2 : p1 ≠ p2) : ∠ p1 (midpoint ℝ p1 p2) p2 = π :=
have p2 -ᵥ midpoint ℝ p1 p2 = -(p1 -ᵥ midpoint ℝ p1 p2), by { rw neg_vsub_eq_vsub_rev, simp },
by simp [angle, this, hp1p2, -zero_lt_one]
/-- If M is the midpoint of the segment AB and C is the same distance from A as it is from B
then ∠CMA = π / 2. -/
lemma angle_left_midpoint_eq_pi_div_two_of_dist_eq {p1 p2 p3 : P} (h : dist p3 p1 = dist p3 p2) :
∠ p3 (midpoint ℝ p1 p2) p1 = π / 2 :=
begin
let m : P := midpoint ℝ p1 p2,
have h1 : p3 -ᵥ p1 = (p3 -ᵥ m) - (p1 -ᵥ m) := (vsub_sub_vsub_cancel_right p3 p1 m).symm,
have h2 : p3 -ᵥ p2 = (p3 -ᵥ m) + (p1 -ᵥ m),
{ rw [left_vsub_midpoint, ← midpoint_vsub_right, vsub_add_vsub_cancel] },
rw [dist_eq_norm_vsub V p3 p1, dist_eq_norm_vsub V p3 p2, h1, h2] at h,
exact (norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (p3 -ᵥ m) (p1 -ᵥ m)).mp h.symm,
end
/-- If M is the midpoint of the segment AB and C is the same distance from A as it is from B
then ∠CMB = π / 2. -/
lemma angle_right_midpoint_eq_pi_div_two_of_dist_eq {p1 p2 p3 : P} (h : dist p3 p1 = dist p3 p2) :
∠ p3 (midpoint ℝ p1 p2) p2 = π / 2 :=
by rw [midpoint_comm p1 p2, angle_left_midpoint_eq_pi_div_two_of_dist_eq h.symm]
/-- The inner product of two vectors given with `weighted_vsub`, in
terms of the pairwise distances. -/
lemma inner_weighted_vsub {ι₁ : Type*} {s₁ : finset ι₁} {w₁ : ι₁ → ℝ} (p₁ : ι₁ → P)
(h₁ : ∑ i in s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : finset ι₂} {w₂ : ι₂ → ℝ} (p₂ : ι₂ → P)
(h₂ : ∑ i in s₂, w₂ i = 0) :
inner (s₁.weighted_vsub p₁ w₁) (s₂.weighted_vsub p₂ w₂) =
(-∑ i₁ in s₁, ∑ i₂ in s₂,
w₁ i₁ * w₂ i₂ * (dist (p₁ i₁) (p₂ i₂) * dist (p₁ i₁) (p₂ i₂))) / 2 :=
begin
rw [finset.weighted_vsub_apply, finset.weighted_vsub_apply,
inner_sum_smul_sum_smul_of_sum_eq_zero _ h₁ _ h₂],
simp_rw [vsub_sub_vsub_cancel_right],
rcongr i₁ i₂; rw dist_eq_norm_vsub V (p₁ i₁) (p₂ i₂)
end
/-- The distance between two points given with `affine_combination`,
in terms of the pairwise distances between the points in that
combination. -/
lemma dist_affine_combination {ι : Type*} {s : finset ι} {w₁ w₂ : ι → ℝ} (p : ι → P)
(h₁ : ∑ i in s, w₁ i = 1) (h₂ : ∑ i in s, w₂ i = 1) :
dist (s.affine_combination p w₁) (s.affine_combination p w₂) *
dist (s.affine_combination p w₁) (s.affine_combination p w₂) =
(-∑ i₁ in s, ∑ i₂ in s,
(w₁ - w₂) i₁ * (w₁ - w₂) i₂ * (dist (p i₁) (p i₂) * dist (p i₁) (p i₂))) / 2 :=
begin
rw [dist_eq_norm_vsub V (s.affine_combination p w₁) (s.affine_combination p w₂),
←inner_self_eq_norm_mul_norm, finset.affine_combination_vsub],
have h : ∑ i in s, (w₁ - w₂) i = 0,
{ simp_rw [pi.sub_apply, finset.sum_sub_distrib, h₁, h₂, sub_self] },
exact inner_weighted_vsub p h p h
end
/-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same
applies to `c₂`. Then the vector between `c₁` and `c₂` is orthogonal
to that between `p₁` and `p₂`. (In two dimensions, this says that the
diagonals of a kite are orthogonal.) -/
lemma inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : dist p₁ c₁ = dist p₂ c₁)
(hc₂ : dist p₁ c₂ = dist p₂ c₂) : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
begin
have h : ⟪(c₂ -ᵥ c₁) + (c₂ -ᵥ c₁), p₂ -ᵥ p₁⟫ = 0,
{ conv_lhs { congr, congr, rw ←vsub_sub_vsub_cancel_right c₂ c₁ p₁,
skip, rw ←vsub_sub_vsub_cancel_right c₂ c₁ p₂ },
rw [←add_sub_comm, inner_sub_left],
conv_lhs { congr, rw ←vsub_sub_vsub_cancel_right p₂ p₁ c₂,
skip, rw ←vsub_sub_vsub_cancel_right p₂ p₁ c₁ },
rw [dist_comm p₁, dist_comm p₂, dist_eq_norm_vsub V _ p₁,
dist_eq_norm_vsub V _ p₂, ←real_inner_add_sub_eq_zero_iff] at hc₁ hc₂,
simp_rw [←neg_vsub_eq_vsub_rev c₁, ←neg_vsub_eq_vsub_rev c₂, sub_neg_eq_add,
neg_add_eq_sub, hc₁, hc₂, sub_zero] },
simpa [inner_add_left, ←mul_two, (by norm_num : (2 : ℝ) ≠ 0)] using h
end
/-- The squared distance between points on a line (expressed as a
multiple of a fixed vector added to a point) and another point,
expressed as a quadratic. -/
lemma dist_smul_vadd_sq (r : ℝ) (v : V) (p₁ p₂ : P) :
dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ =
⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫ :=
begin
rw [dist_eq_norm_vsub V _ p₂, ←real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,
real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right],
ring
end
/-- The condition for two points on a line to be equidistant from
another point. -/
lemma dist_smul_vadd_eq_dist {v : V} (p₁ p₂ : P) (hv : v ≠ 0) (r : ℝ) :
dist (r • v +ᵥ p₁) p₂ = dist p₁ p₂ ↔ (r = 0 ∨ r = -2 * ⟪v, p₁ -ᵥ p₂⟫ / ⟪v, v⟫) :=
begin
conv_lhs { rw [←mul_self_inj_of_nonneg dist_nonneg dist_nonneg, dist_smul_vadd_sq,
←sub_eq_zero, add_sub_assoc, dist_eq_norm_vsub V p₁ p₂,
←real_inner_self_eq_norm_mul_norm, sub_self] },
have hvi : ⟪v, v⟫ ≠ 0, by simpa using hv,
have hd : discrim ⟪v, v⟫ (2 * ⟪v, p₁ -ᵥ p₂⟫) 0 =
(2 * inner v (p₁ -ᵥ p₂)) * (2 * inner v (p₁ -ᵥ p₂)),
{ rw discrim, ring },
rw [quadratic_eq_zero_iff hvi hd, add_left_neg, zero_div, neg_mul_eq_neg_mul,
←mul_sub_right_distrib, sub_eq_add_neg, ←mul_two, mul_assoc, mul_div_assoc,
mul_div_mul_left, mul_div_assoc],
norm_num
end
open affine_subspace finite_dimensional
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
lemma eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : affine_subspace ℝ P}
[finite_dimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
(hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
(hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
(hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
(hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ :=
begin
have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm),
have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm),
let b : fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁],
have hb : linear_independent ℝ b,
{ refine linear_independent_of_ne_zero_of_inner_eq_zero _ _,
{ intro i,
fin_cases i; simp [b, hc.symm, hp.symm], },
{ intros i j hij,
fin_cases i; fin_cases j; try { exact false.elim (hij rfl) },
{ exact ho },
{ rw real_inner_comm, exact ho } } },
have hbs : submodule.span ℝ (set.range b) = s.direction,
{ refine eq_of_le_of_finrank_eq _ _,
{ rw [submodule.span_le, set.range_subset_iff],
intro i,
fin_cases i,
{ exact vsub_mem_direction hc₂s hc₁s },
{ exact vsub_mem_direction hp₂s hp₁s } },
{ rw [finrank_span_eq_card hb, fintype.card_fin, hd] } },
have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁),
{ intros v hv,
have hr : set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁},
{ have hu : (finset.univ : finset (fin 2)) = {0, 1}, by dec_trivial,
rw [←fintype.coe_image_univ, hu],
simp,
refl },
rw [←hbs, hr, submodule.mem_span_insert] at hv,
rcases hv with ⟨t₁, v', hv', hv⟩,
rw submodule.mem_span_singleton at hv',
rcases hv' with ⟨t₂, rfl⟩,
exact ⟨t₁, t₂, hv⟩ },
rcases hv (p -ᵥ p₁) (vsub_mem_direction hps hp₁s) with ⟨t₁, t₂, hpt⟩,
simp only [hpt, inner_add_right, inner_smul_right, ho, mul_zero, add_zero, mul_eq_zero,
inner_self_eq_zero, vsub_eq_zero_iff_eq, hc.symm, or_false] at hop,
rw [hop, zero_smul, zero_add, ←eq_vadd_iff_vsub_eq] at hpt,
subst hpt,
have hp' : (p₂ -ᵥ p₁ : V) ≠ 0, { simp [hp.symm] },
have hp₂ : dist ((1 : ℝ) • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁, { simp [hp₂c₁] },
rw [←hp₁c₁, dist_smul_vadd_eq_dist _ _ hp'] at hpc₁ hp₂,
simp only [one_ne_zero, false_or] at hp₂,
rw hp₂.symm at hpc₁,
cases hpc₁; simp [hpc₁]
end
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in two-dimensional space (two circles intersect in at
most two points). -/
lemma eq_of_dist_eq_of_dist_eq_of_finrank_eq_two [finite_dimensional ℝ V] (hd : finrank ℝ V = 2)
{c₁ c₂ p₁ p₂ p : P} {r₁ r₂ : ℝ} (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁)
(hp₂c₁ : dist p₂ c₁ = r₁) (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂)
(hp₂c₂ : dist p₂ c₂ = r₂) (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ :=
begin
have hd' : finrank ℝ (⊤ : affine_subspace ℝ P).direction = 2,
{ rw [direction_top, finrank_top],
exact hd },
exact eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two hd'
(mem_top ℝ V _) (mem_top ℝ V _) (mem_top ℝ V _) (mem_top ℝ V _) (mem_top ℝ V _)
hc hp hp₁c₁ hp₂c₁ hpc₁ hp₁c₂ hp₂c₂ hpc₂
end
variables {V}
/-- The orthogonal projection of a point onto a nonempty affine
subspace, whose direction is complete, as an unbundled function. This
definition is only intended for use in setting up the bundled version
`orthogonal_projection` and should not be used once that is
defined. -/
def orthogonal_projection_fn (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction]
(p : P) : P :=
classical.some $ inter_eq_singleton_of_nonempty_of_is_compl
(nonempty_subtype.mp ‹_›)
(mk'_nonempty p s.directionᗮ)
begin
rw direction_mk' p s.directionᗮ,
exact submodule.is_compl_orthogonal_of_complete_space,
end
/-- The intersection of the subspace and the orthogonal subspace
through the given point is the `orthogonal_projection_fn` of that
point onto the subspace. This lemma is only intended for use in
setting up the bundled version and should not be used once that is
defined. -/
lemma inter_eq_singleton_orthogonal_projection_fn {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : P) :
(s : set P) ∩ (mk' p s.directionᗮ) = {orthogonal_projection_fn s p} :=
classical.some_spec $ inter_eq_singleton_of_nonempty_of_is_compl
(nonempty_subtype.mp ‹_›)
(mk'_nonempty p s.directionᗮ)
begin
rw direction_mk' p s.directionᗮ,
exact submodule.is_compl_orthogonal_of_complete_space
end
/-- The `orthogonal_projection_fn` lies in the given subspace. This
lemma is only intended for use in setting up the bundled version and
should not be used once that is defined. -/
lemma orthogonal_projection_fn_mem {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : P) : orthogonal_projection_fn s p ∈ s :=
begin
rw [←mem_coe, ←set.singleton_subset_iff, ←inter_eq_singleton_orthogonal_projection_fn],
exact set.inter_subset_left _ _
end
/-- The `orthogonal_projection_fn` lies in the orthogonal
subspace. This lemma is only intended for use in setting up the
bundled version and should not be used once that is defined. -/
lemma orthogonal_projection_fn_mem_orthogonal {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : P) :
orthogonal_projection_fn s p ∈ mk' p s.directionᗮ :=
begin
rw [←mem_coe, ←set.singleton_subset_iff, ←inter_eq_singleton_orthogonal_projection_fn],
exact set.inter_subset_right _ _
end
/-- Subtracting `p` from its `orthogonal_projection_fn` produces a
result in the orthogonal direction. This lemma is only intended for
use in setting up the bundled version and should not be used once that
is defined. -/
lemma orthogonal_projection_fn_vsub_mem_direction_orthogonal {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : P) :
orthogonal_projection_fn s p -ᵥ p ∈ s.directionᗮ :=
direction_mk' p s.directionᗮ ▸
vsub_mem_direction (orthogonal_projection_fn_mem_orthogonal p) (self_mem_mk' _ _)
/-- The orthogonal projection of a point onto a nonempty affine
subspace, whose direction is complete. The corresponding linear map
(mapping a vector to the difference between the projections of two
points whose difference is that vector) is the `orthogonal_projection`
for real inner product spaces, onto the direction of the affine
subspace being projected onto. -/
def orthogonal_projection (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction] :
P →ᵃ[ℝ] s :=
{ to_fun := λ p, ⟨orthogonal_projection_fn s p, orthogonal_projection_fn_mem p⟩,
linear := orthogonal_projection s.direction,
map_vadd' := λ p v, begin
have hs : ((orthogonal_projection s.direction) v : V) +ᵥ orthogonal_projection_fn s p ∈ s :=
vadd_mem_of_mem_direction (orthogonal_projection s.direction v).2
(orthogonal_projection_fn_mem p),
have ho : ((orthogonal_projection s.direction) v : V) +ᵥ orthogonal_projection_fn s p ∈
mk' (v +ᵥ p) s.directionᗮ,
{ rw [←vsub_right_mem_direction_iff_mem (self_mem_mk' _ _) _, direction_mk',
vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_comm, add_sub_assoc],
refine submodule.add_mem _ (orthogonal_projection_fn_vsub_mem_direction_orthogonal p) _,
rw submodule.mem_orthogonal',
intros w hw,
rw [←neg_sub, inner_neg_left, orthogonal_projection_inner_eq_zero _ w hw, neg_zero], },
have hm : ((orthogonal_projection s.direction) v : V) +ᵥ orthogonal_projection_fn s p ∈
({orthogonal_projection_fn s (v +ᵥ p)} : set P),
{ rw ←inter_eq_singleton_orthogonal_projection_fn (v +ᵥ p),
exact set.mem_inter hs ho },
rw set.mem_singleton_iff at hm,
ext,
exact hm.symm
end }
@[simp] lemma orthogonal_projection_fn_eq {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : P) :
orthogonal_projection_fn s p = orthogonal_projection s p :=
rfl
/-- The linear map corresponding to `orthogonal_projection`. -/
@[simp] lemma orthogonal_projection_linear {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] :
(orthogonal_projection s).linear = _root_.orthogonal_projection s.direction :=
rfl
/-- The intersection of the subspace and the orthogonal subspace
through the given point is the `orthogonal_projection` of that point
onto the subspace. -/
lemma inter_eq_singleton_orthogonal_projection {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : P) :
(s : set P) ∩ (mk' p s.directionᗮ) = {orthogonal_projection s p} :=
begin
rw ←orthogonal_projection_fn_eq,
exact inter_eq_singleton_orthogonal_projection_fn p
end
/-- The `orthogonal_projection` lies in the given subspace. -/
lemma orthogonal_projection_mem {s : affine_subspace ℝ P} [nonempty s] [complete_space s.direction]
(p : P) : ↑(orthogonal_projection s p) ∈ s :=
(orthogonal_projection s p).2
/-- The `orthogonal_projection` lies in the orthogonal subspace. -/
lemma orthogonal_projection_mem_orthogonal (s : affine_subspace ℝ P) [nonempty s]
[complete_space s.direction] (p : P) :
↑(orthogonal_projection s p) ∈ mk' p s.directionᗮ :=
orthogonal_projection_fn_mem_orthogonal p
/-- Subtracting a point in the given subspace from the
`orthogonal_projection` produces a result in the direction of the
given subspace. -/
lemma orthogonal_projection_vsub_mem_direction {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p1 : P} (p2 : P) (hp1 : p1 ∈ s) :
↑(orthogonal_projection s p2 -ᵥ ⟨p1, hp1⟩ : s.direction) ∈ s.direction :=
(orthogonal_projection s p2 -ᵥ ⟨p1, hp1⟩ : s.direction).2
/-- Subtracting the `orthogonal_projection` from a point in the given
subspace produces a result in the direction of the given subspace. -/
lemma vsub_orthogonal_projection_mem_direction {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p1 : P} (p2 : P) (hp1 : p1 ∈ s) :
↑((⟨p1, hp1⟩ : s) -ᵥ orthogonal_projection s p2 : s.direction) ∈ s.direction :=
((⟨p1, hp1⟩ : s) -ᵥ orthogonal_projection s p2 : s.direction).2
/-- A point equals its orthogonal projection if and only if it lies in
the subspace. -/
lemma orthogonal_projection_eq_self_iff {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p : P} :
↑(orthogonal_projection s p) = p ↔ p ∈ s :=
begin
split,
{ exact λ h, h ▸ orthogonal_projection_mem p },
{ intro h,
have hp : p ∈ ((s : set P) ∩ mk' p s.directionᗮ) := ⟨h, self_mem_mk' p _⟩,
rw [inter_eq_singleton_orthogonal_projection p] at hp,
symmetry,
exact hp }
end
@[simp] lemma orthogonal_projection_mem_subspace_eq_self {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] (p : s) :
orthogonal_projection s p = p :=
begin
ext,
rw orthogonal_projection_eq_self_iff,
exact p.2
end
/-- Orthogonal projection is idempotent. -/
@[simp] lemma orthogonal_projection_orthogonal_projection (s : affine_subspace ℝ P) [nonempty s]
[complete_space s.direction] (p : P) :
orthogonal_projection s (orthogonal_projection s p) = orthogonal_projection s p :=
begin
ext,
rw orthogonal_projection_eq_self_iff,
exact orthogonal_projection_mem p,
end
lemma eq_orthogonal_projection_of_eq_subspace {s s' : affine_subspace ℝ P} [nonempty s]
[nonempty s'] [complete_space s.direction] [complete_space s'.direction] (h : s = s') (p : P) :
(orthogonal_projection s p : P) = (orthogonal_projection s' p : P) :=
begin
change orthogonal_projection_fn s p = orthogonal_projection_fn s' p,
congr,
exact h
end
/-- The distance to a point's orthogonal projection is 0 iff it lies in the subspace. -/
lemma dist_orthogonal_projection_eq_zero_iff {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p : P} :
dist p (orthogonal_projection s p) = 0 ↔ p ∈ s :=
by rw [dist_comm, dist_eq_zero, orthogonal_projection_eq_self_iff]
/-- The distance between a point and its orthogonal projection is
nonzero if it does not lie in the subspace. -/
lemma dist_orthogonal_projection_ne_zero_of_not_mem {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p : P} (hp : p ∉ s) :
dist p (orthogonal_projection s p) ≠ 0 :=
mt dist_orthogonal_projection_eq_zero_iff.mp hp
/-- Subtracting `p` from its `orthogonal_projection` produces a result
in the orthogonal direction. -/
lemma orthogonal_projection_vsub_mem_direction_orthogonal (s : affine_subspace ℝ P) [nonempty s]
[complete_space s.direction] (p : P) :
(orthogonal_projection s p : P) -ᵥ p ∈ s.directionᗮ :=
orthogonal_projection_fn_vsub_mem_direction_orthogonal p
/-- Subtracting the `orthogonal_projection` from `p` produces a result
in the orthogonal direction. -/
lemma vsub_orthogonal_projection_mem_direction_orthogonal (s : affine_subspace ℝ P) [nonempty s]
[complete_space s.direction] (p : P) :
p -ᵥ orthogonal_projection s p ∈ s.directionᗮ :=
direction_mk' p s.directionᗮ ▸
vsub_mem_direction (self_mem_mk' _ _) (orthogonal_projection_mem_orthogonal s p)
/-- Subtracting the `orthogonal_projection` from `p` produces a result in the kernel of the linear
part of the orthogonal projection. -/
lemma orthogonal_projection_vsub_orthogonal_projection (s : affine_subspace ℝ P) [nonempty s]
[complete_space s.direction] (p : P) :
_root_.orthogonal_projection s.direction (p -ᵥ orthogonal_projection s p) = 0 :=
begin
apply orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero,
intros c hc,
rw [← neg_vsub_eq_vsub_rev, inner_neg_right,
(orthogonal_projection_vsub_mem_direction_orthogonal s p c hc), neg_zero]
end
/-- Adding a vector to a point in the given subspace, then taking the
orthogonal projection, produces the original point if the vector was
in the orthogonal direction. -/
lemma orthogonal_projection_vadd_eq_self {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p : P} (hp : p ∈ s) {v : V}
(hv : v ∈ s.directionᗮ) :
orthogonal_projection s (v +ᵥ p) = ⟨p, hp⟩ :=
begin
have h := vsub_orthogonal_projection_mem_direction_orthogonal s (v +ᵥ p),
rw [vadd_vsub_assoc, submodule.add_mem_iff_right _ hv] at h,
refine (eq_of_vsub_eq_zero _).symm,
ext,
refine submodule.disjoint_def.1 s.direction.orthogonal_disjoint _ _ h,
exact (_ : s.direction).2
end
/-- Adding a vector to a point in the given subspace, then taking the
orthogonal projection, produces the original point if the vector is a
multiple of the result of subtracting a point's orthogonal projection
from that point. -/
lemma orthogonal_projection_vadd_smul_vsub_orthogonal_projection {s : affine_subspace ℝ P}
[nonempty s] [complete_space s.direction] {p1 : P} (p2 : P) (r : ℝ) (hp : p1 ∈ s) :
orthogonal_projection s (r • (p2 -ᵥ orthogonal_projection s p2 : V) +ᵥ p1) = ⟨p1, hp⟩ :=
orthogonal_projection_vadd_eq_self hp
(submodule.smul_mem _ _ (vsub_orthogonal_projection_mem_direction_orthogonal s _))
/-- The square of the distance from a point in `s` to `p2` equals the
sum of the squares of the distances of the two points to the
`orthogonal_projection`. -/
lemma dist_sq_eq_dist_orthogonal_projection_sq_add_dist_orthogonal_projection_sq
{s : affine_subspace ℝ P} [nonempty s] [complete_space s.direction] {p1 : P}
(p2 : P) (hp1 : p1 ∈ s) :
dist p1 p2 * dist p1 p2 =
dist p1 (orthogonal_projection s p2) * dist p1 (orthogonal_projection s p2) +
dist p2 (orthogonal_projection s p2) * dist p2 (orthogonal_projection s p2) :=
begin
rw [pseudo_metric_space.dist_comm p2 _, dist_eq_norm_vsub V p1 _, dist_eq_norm_vsub V p1 _,
dist_eq_norm_vsub V _ p2, ← vsub_add_vsub_cancel p1 (orthogonal_projection s p2) p2,
norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero],
exact submodule.inner_right_of_mem_orthogonal
(vsub_orthogonal_projection_mem_direction p2 hp1)
(orthogonal_projection_vsub_mem_direction_orthogonal s p2),
end
/-- The square of the distance between two points constructed by
adding multiples of the same orthogonal vector to points in the same
subspace. -/
lemma dist_sq_smul_orthogonal_vadd_smul_orthogonal_vadd {s : affine_subspace ℝ P}
{p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) (r1 r2 : ℝ) {v : V}
(hv : v ∈ s.directionᗮ) :
dist (r1 • v +ᵥ p1) (r2 • v +ᵥ p2) * dist (r1 • v +ᵥ p1) (r2 • v +ᵥ p2) =
dist p1 p2 * dist p1 p2 + (r1 - r2) * (r1 - r2) * (∥v∥ * ∥v∥) :=
calc dist (r1 • v +ᵥ p1) (r2 • v +ᵥ p2) * dist (r1 • v +ᵥ p1) (r2 • v +ᵥ p2)
= ∥(p1 -ᵥ p2) + (r1 - r2) • v∥ * ∥(p1 -ᵥ p2) + (r1 - r2) • v∥
: by rw [dist_eq_norm_vsub V (r1 • v +ᵥ p1), vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, sub_smul,
add_comm, add_sub_assoc]
... = ∥p1 -ᵥ p2∥ * ∥p1 -ᵥ p2∥ + ∥(r1 - r2) • v∥ * ∥(r1 - r2) • v∥
: norm_add_sq_eq_norm_sq_add_norm_sq_real
(submodule.inner_right_of_mem_orthogonal (vsub_mem_direction hp1 hp2)
(submodule.smul_mem _ _ hv))
... = ∥(p1 -ᵥ p2 : V)∥ * ∥(p1 -ᵥ p2 : V)∥ + |r1 - r2| * |r1 - r2| * ∥v∥ * ∥v∥
: by { rw [norm_smul, real.norm_eq_abs], ring }
... = dist p1 p2 * dist p1 p2 + (r1 - r2) * (r1 - r2) * (∥v∥ * ∥v∥)
: by { rw [dist_eq_norm_vsub V p1, abs_mul_abs_self, mul_assoc] }
/-- Reflection in an affine subspace, which is expected to be nonempty
and complete. The word "reflection" is sometimes understood to mean
specifically reflection in a codimension-one subspace, and sometimes
more generally to cover operations such as reflection in a point. The
definition here, of reflection in an affine subspace, is a more
general sense of the word that includes both those common cases. -/
def reflection (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction] :
P ≃ᵃⁱ[ℝ] P :=
affine_isometry_equiv.mk'
(λ p, (↑(orthogonal_projection s p) -ᵥ p) +ᵥ orthogonal_projection s p)
(_root_.reflection s.direction)
↑(classical.arbitrary s)
begin
intros p,
let v := p -ᵥ ↑(classical.arbitrary s),
let a : V := _root_.orthogonal_projection s.direction v,
let b : P := ↑(classical.arbitrary s),
have key : a +ᵥ b -ᵥ (v +ᵥ b) +ᵥ (a +ᵥ b) = a + a - v +ᵥ (b -ᵥ b +ᵥ b),
{ rw [← add_vadd, vsub_vadd_eq_vsub_sub, vsub_vadd, vadd_vsub],
congr' 1,
abel },
have : p = v +ᵥ ↑(classical.arbitrary s) := (vsub_vadd p ↑(classical.arbitrary s)).symm,
simpa only [coe_vadd, reflection_apply, affine_map.map_vadd, orthogonal_projection_linear,
orthogonal_projection_mem_subspace_eq_self, vadd_vsub, continuous_linear_map.coe_coe,
continuous_linear_equiv.coe_coe, this] using key,
end
/-- The result of reflecting. -/
lemma reflection_apply (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction] (p : P) :
reflection s p = (↑(orthogonal_projection s p) -ᵥ p) +ᵥ orthogonal_projection s p :=
rfl
lemma eq_reflection_of_eq_subspace {s s' : affine_subspace ℝ P} [nonempty s]
[nonempty s'] [complete_space s.direction] [complete_space s'.direction] (h : s = s') (p : P) :
(reflection s p : P) = (reflection s' p : P) :=
by unfreezingI { subst h }
/-- Reflecting twice in the same subspace. -/
@[simp] lemma reflection_reflection (s : affine_subspace ℝ P) [nonempty s]
[complete_space s.direction] (p : P) :
reflection s (reflection s p) = p :=
begin
have : ∀ a : s, ∀ b : V, (_root_.orthogonal_projection s.direction) b = 0
→ reflection s (reflection s (b +ᵥ a)) = b +ᵥ a,
{ intros a b h,
have : (a:P) -ᵥ (b +ᵥ a) = - b,
{ rw [vsub_vadd_eq_vsub_sub, vsub_self, zero_sub] },
simp [reflection, h, this] },
rw ← vsub_vadd p (orthogonal_projection s p),
exact this (orthogonal_projection s p) _ (orthogonal_projection_vsub_orthogonal_projection s p),
end
/-- Reflection is its own inverse. -/
@[simp] lemma reflection_symm (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction] :
(reflection s).symm = reflection s :=
by { ext, rw ← (reflection s).injective.eq_iff, simp }
/-- Reflection is involutive. -/
lemma reflection_involutive (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction] :
function.involutive (reflection s) :=
reflection_reflection s
/-- A point is its own reflection if and only if it is in the
subspace. -/
lemma reflection_eq_self_iff {s : affine_subspace ℝ P} [nonempty s] [complete_space s.direction]
(p : P) : reflection s p = p ↔ p ∈ s :=
begin
rw [←orthogonal_projection_eq_self_iff, reflection_apply],
split,
{ intro h,
rw [←@vsub_eq_zero_iff_eq V, vadd_vsub_assoc,
←two_smul ℝ (↑(orthogonal_projection s p) -ᵥ p), smul_eq_zero] at h,
norm_num at h,
exact h },
{ intro h,
simp [h] }
end
/-- Reflecting a point in two subspaces produces the same result if
and only if the point has the same orthogonal projection in each of
those subspaces. -/
lemma reflection_eq_iff_orthogonal_projection_eq (s₁ s₂ : affine_subspace ℝ P)
[nonempty s₁] [nonempty s₂] [complete_space s₁.direction] [complete_space s₂.direction] (p : P) :
reflection s₁ p = reflection s₂ p ↔
(orthogonal_projection s₁ p : P) = orthogonal_projection s₂ p :=
begin
rw [reflection_apply, reflection_apply],
split,
{ intro h,
rw [←@vsub_eq_zero_iff_eq V, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_comm,
add_sub_assoc, vsub_sub_vsub_cancel_right,
←two_smul ℝ ((orthogonal_projection s₁ p : P) -ᵥ orthogonal_projection s₂ p),
smul_eq_zero] at h,
norm_num at h,
exact h },
{ intro h,
rw h }
end
/-- The distance between `p₁` and the reflection of `p₂` equals that
between the reflection of `p₁` and `p₂`. -/
lemma dist_reflection (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction]
(p₁ p₂ : P) :
dist p₁ (reflection s p₂) = dist (reflection s p₁) p₂ :=
begin
conv_lhs { rw ←reflection_reflection s p₁ },
exact (reflection s).dist_map _ _
end
/-- A point in the subspace is equidistant from another point and its
reflection. -/
lemma dist_reflection_eq_of_mem (s : affine_subspace ℝ P) [nonempty s] [complete_space s.direction]
{p₁ : P} (hp₁ : p₁ ∈ s) (p₂ : P) :
dist p₁ (reflection s p₂) = dist p₁ p₂ :=
begin
rw ←reflection_eq_self_iff p₁ at hp₁,
convert (reflection s).dist_map p₁ p₂,
rw hp₁
end
/-- The reflection of a point in a subspace is contained in any larger
subspace containing both the point and the subspace reflected in. -/
lemma reflection_mem_of_le_of_mem {s₁ s₂ : affine_subspace ℝ P} [nonempty s₁]
[complete_space s₁.direction] (hle : s₁ ≤ s₂) {p : P}
(hp : p ∈ s₂) : reflection s₁ p ∈ s₂ :=
begin
rw [reflection_apply],
have ho : ↑(orthogonal_projection s₁ p) ∈ s₂ := hle (orthogonal_projection_mem p),
exact vadd_mem_of_mem_direction (vsub_mem_direction ho hp) ho
end
/-- Reflecting an orthogonal vector plus a point in the subspace
produces the negation of that vector plus the point. -/
lemma reflection_orthogonal_vadd {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p : P} (hp : p ∈ s) {v : V}
(hv : v ∈ s.directionᗮ) : reflection s (v +ᵥ p) = -v +ᵥ p :=
begin
rw [reflection_apply, orthogonal_projection_vadd_eq_self hp hv, vsub_vadd_eq_vsub_sub],
simp
end
/-- Reflecting a vector plus a point in the subspace produces the
negation of that vector plus the point if the vector is a multiple of
the result of subtracting a point's orthogonal projection from that
point. -/
lemma reflection_vadd_smul_vsub_orthogonal_projection {s : affine_subspace ℝ P} [nonempty s]
[complete_space s.direction] {p₁ : P} (p₂ : P) (r : ℝ) (hp₁ : p₁ ∈ s) :
reflection s (r • (p₂ -ᵥ orthogonal_projection s p₂) +ᵥ p₁) =
-(r • (p₂ -ᵥ orthogonal_projection s p₂)) +ᵥ p₁ :=
reflection_orthogonal_vadd hp₁
(submodule.smul_mem _ _ (vsub_orthogonal_projection_mem_direction_orthogonal s _))
omit V
/-- A set of points is cospherical if they are equidistant from some
point. In two dimensions, this is the same thing as being
concyclic. -/
def cospherical (ps : set P) : Prop :=
∃ (center : P) (radius : ℝ), ∀ p ∈ ps, dist p center = radius
/-- The definition of `cospherical`. -/
lemma cospherical_def (ps : set P) :
cospherical ps ↔ ∃ (center : P) (radius : ℝ), ∀ p ∈ ps, dist p center = radius :=
iff.rfl
/-- A subset of a cospherical set is cospherical. -/
lemma cospherical_subset {ps₁ ps₂ : set P} (hs : ps₁ ⊆ ps₂) (hc : cospherical ps₂) :
cospherical ps₁ :=
begin
rcases hc with ⟨c, r, hcr⟩,
exact ⟨c, r, λ p hp, hcr p (hs hp)⟩
end
include V
/-- The empty set is cospherical. -/
lemma cospherical_empty : cospherical (∅ : set P) :=
begin
use add_torsor.nonempty.some,
simp,
end
omit V
/-- A single point is cospherical. -/
lemma cospherical_singleton (p : P) : cospherical ({p} : set P) :=
begin
use p,
simp
end
include V
/-- Two points are cospherical. -/
lemma cospherical_insert_singleton (p₁ p₂ : P) : cospherical ({p₁, p₂} : set P) :=
begin
use [(2⁻¹ : ℝ) • (p₂ -ᵥ p₁) +ᵥ p₁, (2⁻¹ : ℝ) * (dist p₂ p₁)],
intro p,
rw [set.mem_insert_iff, set.mem_singleton_iff],
rintro ⟨_|_⟩,
{ rw [dist_eq_norm_vsub V p₁, vsub_vadd_eq_vsub_sub, vsub_self, zero_sub, norm_neg, norm_smul,
dist_eq_norm_vsub V p₂],
simp },
{ rw [H, dist_eq_norm_vsub V p₂, vsub_vadd_eq_vsub_sub, dist_eq_norm_vsub V p₂],
conv_lhs { congr, congr, rw ←one_smul ℝ (p₂ -ᵥ p₁ : V) },
rw [←sub_smul, norm_smul],
norm_num }
end
/-- Any three points in a cospherical set are affinely independent. -/
lemma cospherical.affine_independent {s : set P} (hs : cospherical s) {p : fin 3 → P}
(hps : set.range p ⊆ s) (hpi : function.injective p) :
affine_independent ℝ p :=
begin
rw affine_independent_iff_not_collinear,
intro hc,
rw collinear_iff_of_mem ℝ (set.mem_range_self (0 : fin 3)) at hc,
rcases hc with ⟨v, hv⟩,
rw set.forall_range_iff at hv,
have hv0 : v ≠ 0,
{ intro h,
have he : p 1 = p 0, by simpa [h] using hv 1,
exact (dec_trivial : (1 : fin 3) ≠ 0) (hpi he) },
rcases hs with ⟨c, r, hs⟩,
have hs' := λ i, hs (p i) (set.mem_of_mem_of_subset (set.mem_range_self _) hps),
choose f hf using hv,
have hsd : ∀ i, dist ((f i • v) +ᵥ p 0) c = r,
{ intro i,
rw ←hf,
exact hs' i },
have hf0 : f 0 = 0,
{ have hf0' := hf 0,
rw [eq_comm, ←@vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0',
simpa [hv0] using hf0' },
have hfi : function.injective f,
{ intros i j h,
have hi := hf i,
rw [h, ←hf j] at hi,
exact hpi hi },
simp_rw [←hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd,
have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := λ i, (hfi.ne_iff' hf0).2,
have hfn0' : ∀ i, i ≠ 0 → f i = (-2) * ⟪v, (p 0 -ᵥ c)⟫ / ⟪v, v⟫,
{ intros i hi,
have hsdi := hsd i,
simpa [hfn0, hi] using hsdi },
have hf12 : f 1 = f 2, { rw [hfn0' 1 dec_trivial, hfn0' 2 dec_trivial] },
exact (dec_trivial : (1 : fin 3) ≠ 2) (hfi hf12)
end
end euclidean_geometry
|
24f64c38b06cbbbe3711d30c9678a9e160aa7d05 | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/data/set/basic.lean | 9b437b04c483c9f4c370c7f9784efbf2599eb695 | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 81,242 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import logic.unique
import order.boolean_algebra
/-!
# Basic properties of sets
Sets in Lean are homogeneous; all their elements have the same type. Sets whose elements
have type `X` are thus defined as `set X := X → Prop`. Note that this function need not
be decidable. The definition is in the core library.
This file provides some basic definitions related to sets and functions not present in the core
library, as well as extra lemmas for functions in the core library (empty set, univ, union,
intersection, insert, singleton, set-theoretic difference, complement, and powerset).
Note that a set is a term, not a type. There is a coersion from `set α` to `Type*` sending
`s` to the corresponding subtype `↥s`.
See also the file `set_theory/zfc.lean`, which contains an encoding of ZFC set theory in Lean.
## Main definitions
Notation used here:
- `f : α → β` is a function,
- `s : set α` and `s₁ s₂ : set α` are subsets of `α`
- `t : set β` is a subset of `β`.
Definitions in the file:
* `strict_subset s₁ s₂ : Prop` : the predicate `s₁ ⊆ s₂` but `s₁ ≠ s₂`.
* `nonempty s : Prop` : the predicate `s ≠ ∅`. Note that this is the preferred way to express the
fact that `s` has an element (see the Implementation Notes).
* `preimage f t : set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `subsingleton s : Prop` : the predicate saying that `s` has at most one element.
* `range f : set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
* `prod s t : set (α × β)` : the subset `s × t`.
* `inclusion s₁ s₂ : ↥s₁ → ↥s₂` : the map `↥s₁ → ↥s₂` induced by an inclusion `s₁ ⊆ s₂`.
## Notation
* `f ⁻¹' t` for `preimage f t`
* `f '' s` for `image f s`
* `sᶜ` for the complement of `s`
## Implementation notes
* `s.nonempty` is to be preferred to `s ≠ ∅` or `∃ x, x ∈ s`. It has the advantage that
the `s.nonempty` dot notation can be used.
* For `s : set α`, do not use `subtype s`. Instead use `↥s` or `(s : Type*)` or `s`.
## Tags
set, sets, subset, subsets, image, preimage, pre-image, range, union, intersection, insert,
singleton, complement, powerset
-/
/-! ### Set coercion to a type -/
open function
universe variables u v w x
run_cmd do e ← tactic.get_env,
tactic.set_env $ e.mk_protected `set.compl
namespace set
variable {α : Type*}
instance : has_le (set α) := ⟨(⊆)⟩
instance : has_lt (set α) := ⟨λ s t, s ≤ t ∧ ¬t ≤ s⟩
instance {α : Type*} : boolean_algebra (set α) :=
{ sup := (∪),
le := (≤),
lt := (<),
inf := (∩),
bot := ∅,
compl := set.compl,
top := univ,
sdiff := (\),
.. (infer_instance : boolean_algebra (α → Prop)) }
@[simp] lemma bot_eq_empty : (⊥ : set α) = ∅ := rfl
@[simp] lemma sup_eq_union (s t : set α) : s ⊔ t = s ∪ t := rfl
@[simp] lemma inf_eq_inter (s t : set α) : s ⊓ t = s ∩ t := rfl
@[simp] lemma le_eq_subset (s t : set α) : s ≤ t = (s ⊆ t) := rfl
/-- Coercion from a set to the corresponding subtype. -/
instance {α : Type*} : has_coe_to_sort (set α) := ⟨_, λ s, {x // x ∈ s}⟩
end set
section set_coe
variables {α : Type u}
theorem set.set_coe_eq_subtype (s : set α) :
coe_sort.{(u+1) (u+2)} s = {x // x ∈ s} := rfl
@[simp] theorem set_coe.forall {s : set α} {p : s → Prop} :
(∀ x : s, p x) ↔ (∀ x (h : x ∈ s), p ⟨x, h⟩) :=
subtype.forall
@[simp] theorem set_coe.exists {s : set α} {p : s → Prop} :
(∃ x : s, p x) ↔ (∃ x (h : x ∈ s), p ⟨x, h⟩) :=
subtype.exists
theorem set_coe.exists' {s : set α} {p : Π x, x ∈ s → Prop} :
(∃ x (h : x ∈ s), p x h) ↔ (∃ x : s, p x.1 x.2) :=
(@set_coe.exists _ _ $ λ x, p x.1 x.2).symm
@[simp] theorem set_coe_cast : ∀ {s t : set α} (H' : s = t) (H : @eq (Type u) s t) (x : s),
cast H x = ⟨x.1, H' ▸ x.2⟩
| s _ rfl _ ⟨x, h⟩ := rfl
theorem set_coe.ext {s : set α} {a b : s} : (↑a : α) = ↑b → a = b :=
subtype.eq
theorem set_coe.ext_iff {s : set α} {a b : s} : (↑a : α) = ↑b ↔ a = b :=
iff.intro set_coe.ext (assume h, h ▸ rfl)
end set_coe
/-- See also `subtype.prop` -/
lemma subtype.mem {α : Type*} {s : set α} (p : s) : (p : α) ∈ s := p.prop
namespace set
variables {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x} {a : α} {s t : set α}
instance : inhabited (set α) := ⟨∅⟩
@[ext]
theorem ext {a b : set α} (h : ∀ x, x ∈ a ↔ x ∈ b) : a = b :=
funext (assume x, propext (h x))
theorem ext_iff {s t : set α} : s = t ↔ ∀ x, x ∈ s ↔ x ∈ t :=
⟨λ h x, by rw h, ext⟩
@[trans] theorem mem_of_mem_of_subset {x : α} {s t : set α} (hx : x ∈ s) (h : s ⊆ t) : x ∈ t := h hx
/-! ### Lemmas about `mem` and `set_of` -/
@[simp] theorem mem_set_of_eq {a : α} {p : α → Prop} : a ∈ {a | p a} = p a := rfl
theorem nmem_set_of_eq {a : α} {P : α → Prop} : a ∉ {a : α | P a} = ¬ P a := rfl
@[simp] theorem set_of_mem_eq {s : set α} : {x | x ∈ s} = s := rfl
theorem set_of_set {s : set α} : set_of s = s := rfl
lemma set_of_app_iff {p : α → Prop} {x : α} : { x | p x } x ↔ p x := iff.rfl
theorem mem_def {a : α} {s : set α} : a ∈ s ↔ s a := iff.rfl
instance decidable_mem (s : set α) [H : decidable_pred s] : ∀ a, decidable (a ∈ s) := H
instance decidable_set_of (p : α → Prop) [H : decidable_pred p] : decidable_pred {a | p a} := H
@[simp] theorem set_of_subset_set_of {p q : α → Prop} :
{a | p a} ⊆ {a | q a} ↔ (∀a, p a → q a) := iff.rfl
@[simp] lemma sep_set_of {α} {p q : α → Prop} : {a ∈ {a | p a } | q a} = {a | p a ∧ q a} := rfl
/-! ### Lemmas about subsets -/
-- TODO(Jeremy): write a tactic to unfold specific instances of generic notation?
theorem subset_def {s t : set α} : (s ⊆ t) = ∀ x, x ∈ s → x ∈ t := rfl
@[refl] theorem subset.refl (a : set α) : a ⊆ a := assume x, id
theorem subset.rfl {s : set α} : s ⊆ s := subset.refl s
@[trans] theorem subset.trans {a b c : set α} (ab : a ⊆ b) (bc : b ⊆ c) : a ⊆ c :=
assume x h, bc (ab h)
@[trans] theorem mem_of_eq_of_mem {x y : α} {s : set α} (hx : x = y) (h : y ∈ s) : x ∈ s :=
hx.symm ▸ h
theorem subset.antisymm {a b : set α} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
ext (λ x, iff.intro (λ ina, h₁ ina) (λ inb, h₂ inb))
theorem subset.antisymm_iff {a b : set α} : a = b ↔ a ⊆ b ∧ b ⊆ a :=
⟨λ e, e ▸ ⟨subset.refl _, subset.refl _⟩,
λ ⟨h₁, h₂⟩, subset.antisymm h₁ h₂⟩
-- an alternative name
theorem eq_of_subset_of_subset {a b : set α} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
subset.antisymm h₁ h₂
theorem mem_of_subset_of_mem {s₁ s₂ : set α} {a : α} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
assume h₁ h₂, h₁ h₂
theorem not_subset : (¬ s ⊆ t) ↔ ∃a ∈ s, a ∉ t :=
by simp [subset_def, classical.not_forall]
/-! ### Definition of strict subsets `s ⊂ t` and basic properties. -/
instance : has_ssubset (set α) := ⟨(<)⟩
@[simp] lemma lt_eq_ssubset (s t : set α) : s < t = (s ⊂ t) := rfl
theorem ssubset_def : (s ⊂ t) = (s ⊆ t ∧ ¬ (t ⊆ s)) := rfl
theorem eq_or_ssubset_of_subset (h : s ⊆ t) : s = t ∨ s ⊂ t :=
classical.by_cases
(λ H : t ⊆ s, or.inl $ subset.antisymm h H)
(λ H, or.inr ⟨h, H⟩)
lemma exists_of_ssubset {s t : set α} (h : s ⊂ t) : (∃x∈t, x ∉ s) :=
not_subset.1 h.2
lemma ssubset_iff_subset_ne {s t : set α} : s ⊂ t ↔ s ⊆ t ∧ s ≠ t :=
by split; simp [set.ssubset_def, ne.def, set.subset.antisymm_iff] {contextual := tt}
lemma ssubset_iff_of_subset {s t : set α} (h : s ⊆ t) : s ⊂ t ↔ ∃ x ∈ t, x ∉ s :=
⟨exists_of_ssubset, λ ⟨x, hxt, hxs⟩, ⟨h, λ h, hxs $ h hxt⟩⟩
theorem not_mem_empty (x : α) : ¬ (x ∈ (∅ : set α)) :=
assume h : x ∈ ∅, h
@[simp] theorem not_not_mem : ¬ (a ∉ s) ↔ a ∈ s :=
by { classical, exact not_not }
/-! ### Non-empty sets -/
/-- The property `s.nonempty` expresses the fact that the set `s` is not empty. It should be used
in theorem assumptions instead of `∃ x, x ∈ s` or `s ≠ ∅` as it gives access to a nice API thanks
to the dot notation. -/
protected def nonempty (s : set α) : Prop := ∃ x, x ∈ s
lemma nonempty_def : s.nonempty ↔ ∃ x, x ∈ s := iff.rfl
lemma nonempty_of_mem {x} (h : x ∈ s) : s.nonempty := ⟨x, h⟩
theorem nonempty.not_subset_empty : s.nonempty → ¬(s ⊆ ∅)
| ⟨x, hx⟩ hs := hs hx
theorem nonempty.ne_empty : s.nonempty → s ≠ ∅
| ⟨x, hx⟩ hs := by { rw hs at hx, exact hx }
/-- Extract a witness from `s.nonempty`. This function might be used instead of case analysis
on the argument. Note that it makes a proof depend on the `classical.choice` axiom. -/
protected noncomputable def nonempty.some (h : s.nonempty) : α := classical.some h
protected lemma nonempty.some_mem (h : s.nonempty) : h.some ∈ s := classical.some_spec h
lemma nonempty.mono (ht : s ⊆ t) (hs : s.nonempty) : t.nonempty := hs.imp ht
lemma nonempty_of_not_subset (h : ¬s ⊆ t) : (s \ t).nonempty :=
let ⟨x, xs, xt⟩ := not_subset.1 h in ⟨x, xs, xt⟩
lemma nonempty_of_ssubset (ht : s ⊂ t) : (t \ s).nonempty :=
nonempty_of_not_subset ht.2
lemma nonempty.of_diff (h : (s \ t).nonempty) : s.nonempty := h.imp $ λ _, and.left
lemma nonempty_of_ssubset' (ht : s ⊂ t) : t.nonempty := (nonempty_of_ssubset ht).of_diff
lemma nonempty.inl (hs : s.nonempty) : (s ∪ t).nonempty := hs.imp $ λ _, or.inl
lemma nonempty.inr (ht : t.nonempty) : (s ∪ t).nonempty := ht.imp $ λ _, or.inr
@[simp] lemma union_nonempty : (s ∪ t).nonempty ↔ s.nonempty ∨ t.nonempty := exists_or_distrib
lemma nonempty.left (h : (s ∩ t).nonempty) : s.nonempty := h.imp $ λ _, and.left
lemma nonempty.right (h : (s ∩ t).nonempty) : t.nonempty := h.imp $ λ _, and.right
lemma nonempty_inter_iff_exists_right : (s ∩ t).nonempty ↔ ∃ x : t, ↑x ∈ s :=
⟨λ ⟨x, xs, xt⟩, ⟨⟨x, xt⟩, xs⟩, λ ⟨⟨x, xt⟩, xs⟩, ⟨x, xs, xt⟩⟩
lemma nonempty_inter_iff_exists_left : (s ∩ t).nonempty ↔ ∃ x : s, ↑x ∈ t :=
⟨λ ⟨x, xs, xt⟩, ⟨⟨x, xs⟩, xt⟩, λ ⟨⟨x, xt⟩, xs⟩, ⟨x, xt, xs⟩⟩
lemma nonempty_iff_univ_nonempty : nonempty α ↔ (univ : set α).nonempty :=
⟨λ ⟨x⟩, ⟨x, trivial⟩, λ ⟨x, _⟩, ⟨x⟩⟩
@[simp] lemma univ_nonempty : ∀ [h : nonempty α], (univ : set α).nonempty
| ⟨x⟩ := ⟨x, trivial⟩
lemma nonempty.to_subtype (h : s.nonempty) : nonempty s :=
nonempty_subtype.2 h
/-! ### Lemmas about the empty set -/
theorem empty_def : (∅ : set α) = {x | false} := rfl
@[simp] theorem mem_empty_eq (x : α) : x ∈ (∅ : set α) = false := rfl
@[simp] theorem set_of_false : {a : α | false} = ∅ := rfl
theorem eq_empty_iff_forall_not_mem {s : set α} : s = ∅ ↔ ∀ x, x ∉ s :=
by simp [ext_iff]
@[simp] theorem empty_subset (s : set α) : ∅ ⊆ s :=
assume x, assume h, false.elim h
theorem subset_empty_iff {s : set α} : s ⊆ ∅ ↔ s = ∅ :=
by simp [subset.antisymm_iff]
theorem eq_empty_of_subset_empty {s : set α} : s ⊆ ∅ → s = ∅ :=
subset_empty_iff.1
theorem eq_empty_of_not_nonempty (h : ¬nonempty α) (s : set α) : s = ∅ :=
eq_empty_of_subset_empty $ λ x hx, h ⟨x⟩
lemma not_nonempty_iff_eq_empty {s : set α} : ¬s.nonempty ↔ s = ∅ :=
by simp only [set.nonempty, eq_empty_iff_forall_not_mem, not_exists]
lemma empty_not_nonempty : ¬(∅ : set α).nonempty :=
not_nonempty_iff_eq_empty.2 rfl
lemma eq_empty_or_nonempty (s : set α) : s = ∅ ∨ s.nonempty :=
classical.by_cases or.inr (λ h, or.inl $ not_nonempty_iff_eq_empty.1 h)
theorem ne_empty_iff_nonempty : s ≠ ∅ ↔ s.nonempty :=
(not_congr not_nonempty_iff_eq_empty.symm).trans classical.not_not
theorem subset_eq_empty {s t : set α} (h : t ⊆ s) (e : s = ∅) : t = ∅ :=
subset_empty_iff.1 $ e ▸ h
theorem ball_empty_iff {p : α → Prop} :
(∀ x ∈ (∅ : set α), p x) ↔ true :=
by simp [iff_def]
/-!
### Universal set.
In Lean `@univ α` (or `univ : set α`) is the set that contains all elements of type `α`.
Mathematically it is the same as `α` but it has a different type.
-/
@[simp] theorem set_of_true : {x : α | true} = univ := rfl
@[simp] theorem mem_univ (x : α) : x ∈ @univ α := trivial
theorem empty_ne_univ [h : nonempty α] : (∅ : set α) ≠ univ :=
by simp [ext_iff]
@[simp] theorem subset_univ (s : set α) : s ⊆ univ := λ x H, trivial
theorem univ_subset_iff {s : set α} : univ ⊆ s ↔ s = univ :=
by simp [subset.antisymm_iff]
theorem eq_univ_of_univ_subset {s : set α} : univ ⊆ s → s = univ :=
univ_subset_iff.1
theorem eq_univ_iff_forall {s : set α} : s = univ ↔ ∀ x, x ∈ s :=
by simp [ext_iff]
theorem eq_univ_of_forall {s : set α} : (∀ x, x ∈ s) → s = univ := eq_univ_iff_forall.2
lemma eq_univ_of_subset {s t : set α} (h : s ⊆ t) (hs : s = univ) : t = univ :=
eq_univ_of_univ_subset $ hs ▸ h
@[simp] lemma univ_eq_empty_iff : (univ : set α) = ∅ ↔ ¬ nonempty α :=
eq_empty_iff_forall_not_mem.trans ⟨λ H ⟨x⟩, H x trivial, λ H x _, H ⟨x⟩⟩
lemma exists_mem_of_nonempty (α) : ∀ [nonempty α], ∃x:α, x ∈ (univ : set α)
| ⟨x⟩ := ⟨x, trivial⟩
instance univ_decidable : decidable_pred (@set.univ α) :=
λ x, is_true trivial
/-- `diagonal α` is the subset of `α × α` consisting of all pairs of the form `(a, a)`. -/
def diagonal (α : Type*) : set (α × α) := {p | p.1 = p.2}
@[simp]
lemma mem_diagonal {α : Type*} (x : α) : (x, x) ∈ diagonal α :=
by simp [diagonal]
/-! ### Lemmas about union -/
theorem union_def {s₁ s₂ : set α} : s₁ ∪ s₂ = {a | a ∈ s₁ ∨ a ∈ s₂} := rfl
theorem mem_union_left {x : α} {a : set α} (b : set α) : x ∈ a → x ∈ a ∪ b := or.inl
theorem mem_union_right {x : α} {b : set α} (a : set α) : x ∈ b → x ∈ a ∪ b := or.inr
theorem mem_or_mem_of_mem_union {x : α} {a b : set α} (H : x ∈ a ∪ b) : x ∈ a ∨ x ∈ b := H
theorem mem_union.elim {x : α} {a b : set α} {P : Prop}
(H₁ : x ∈ a ∪ b) (H₂ : x ∈ a → P) (H₃ : x ∈ b → P) : P :=
or.elim H₁ H₂ H₃
theorem mem_union (x : α) (a b : set α) : x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b := iff.rfl
@[simp] theorem mem_union_eq (x : α) (a b : set α) : x ∈ a ∪ b = (x ∈ a ∨ x ∈ b) := rfl
@[simp] theorem union_self (a : set α) : a ∪ a = a :=
ext (assume x, or_self _)
@[simp] theorem union_empty (a : set α) : a ∪ ∅ = a :=
ext (assume x, or_false _)
@[simp] theorem empty_union (a : set α) : ∅ ∪ a = a :=
ext (assume x, false_or _)
theorem union_comm (a b : set α) : a ∪ b = b ∪ a :=
ext (assume x, or.comm)
theorem union_assoc (a b c : set α) : (a ∪ b) ∪ c = a ∪ (b ∪ c) :=
ext (assume x, or.assoc)
instance union_is_assoc : is_associative (set α) (∪) :=
⟨union_assoc⟩
instance union_is_comm : is_commutative (set α) (∪) :=
⟨union_comm⟩
theorem union_left_comm (s₁ s₂ s₃ : set α) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) :=
by finish
theorem union_right_comm (s₁ s₂ s₃ : set α) : (s₁ ∪ s₂) ∪ s₃ = (s₁ ∪ s₃) ∪ s₂ :=
by finish
theorem union_eq_self_of_subset_left {s t : set α} (h : s ⊆ t) : s ∪ t = t :=
by finish [subset_def, ext_iff, iff_def]
theorem union_eq_self_of_subset_right {s t : set α} (h : t ⊆ s) : s ∪ t = s :=
by finish [subset_def, ext_iff, iff_def]
@[simp] theorem subset_union_left (s t : set α) : s ⊆ s ∪ t := λ x, or.inl
@[simp] theorem subset_union_right (s t : set α) : t ⊆ s ∪ t := λ x, or.inr
theorem union_subset {s t r : set α} (sr : s ⊆ r) (tr : t ⊆ r) : s ∪ t ⊆ r :=
by finish [subset_def, union_def]
@[simp] theorem union_subset_iff {s t u : set α} : s ∪ t ⊆ u ↔ s ⊆ u ∧ t ⊆ u :=
by finish [iff_def, subset_def]
theorem union_subset_union {s₁ s₂ t₁ t₂ : set α} (h₁ : s₁ ⊆ s₂) (h₂ : t₁ ⊆ t₂) : s₁ ∪ t₁ ⊆ s₂ ∪ t₂ :=
by finish [subset_def]
theorem union_subset_union_left {s₁ s₂ : set α} (t) (h : s₁ ⊆ s₂) : s₁ ∪ t ⊆ s₂ ∪ t :=
union_subset_union h (by refl)
theorem union_subset_union_right (s) {t₁ t₂ : set α} (h : t₁ ⊆ t₂) : s ∪ t₁ ⊆ s ∪ t₂ :=
union_subset_union (by refl) h
lemma subset_union_of_subset_left {s t : set α} (h : s ⊆ t) (u : set α) : s ⊆ t ∪ u :=
subset.trans h (subset_union_left t u)
lemma subset_union_of_subset_right {s u : set α} (h : s ⊆ u) (t : set α) : s ⊆ t ∪ u :=
subset.trans h (subset_union_right t u)
@[simp] theorem union_empty_iff {s t : set α} : s ∪ t = ∅ ↔ s = ∅ ∧ t = ∅ :=
⟨by finish [ext_iff], by finish [ext_iff]⟩
/-! ### Lemmas about intersection -/
theorem inter_def {s₁ s₂ : set α} : s₁ ∩ s₂ = {a | a ∈ s₁ ∧ a ∈ s₂} := rfl
theorem mem_inter_iff (x : α) (a b : set α) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b := iff.rfl
@[simp] theorem mem_inter_eq (x : α) (a b : set α) : x ∈ a ∩ b = (x ∈ a ∧ x ∈ b) := rfl
theorem mem_inter {x : α} {a b : set α} (ha : x ∈ a) (hb : x ∈ b) : x ∈ a ∩ b :=
⟨ha, hb⟩
theorem mem_of_mem_inter_left {x : α} {a b : set α} (h : x ∈ a ∩ b) : x ∈ a :=
h.left
theorem mem_of_mem_inter_right {x : α} {a b : set α} (h : x ∈ a ∩ b) : x ∈ b :=
h.right
@[simp] theorem inter_self (a : set α) : a ∩ a = a :=
ext (assume x, and_self _)
@[simp] theorem inter_empty (a : set α) : a ∩ ∅ = ∅ :=
ext (assume x, and_false _)
@[simp] theorem empty_inter (a : set α) : ∅ ∩ a = ∅ :=
ext (assume x, false_and _)
theorem inter_comm (a b : set α) : a ∩ b = b ∩ a :=
ext (assume x, and.comm)
theorem inter_assoc (a b c : set α) : (a ∩ b) ∩ c = a ∩ (b ∩ c) :=
ext (assume x, and.assoc)
instance inter_is_assoc : is_associative (set α) (∩) :=
⟨inter_assoc⟩
instance inter_is_comm : is_commutative (set α) (∩) :=
⟨inter_comm⟩
theorem inter_left_comm (s₁ s₂ s₃ : set α) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) :=
by finish
theorem inter_right_comm (s₁ s₂ s₃ : set α) : (s₁ ∩ s₂) ∩ s₃ = (s₁ ∩ s₃) ∩ s₂ :=
by finish
@[simp] theorem inter_subset_left (s t : set α) : s ∩ t ⊆ s := λ x H, and.left H
@[simp] theorem inter_subset_right (s t : set α) : s ∩ t ⊆ t := λ x H, and.right H
theorem subset_inter {s t r : set α} (rs : r ⊆ s) (rt : r ⊆ t) : r ⊆ s ∩ t :=
by finish [subset_def, inter_def]
@[simp] theorem subset_inter_iff {s t r : set α} : r ⊆ s ∩ t ↔ r ⊆ s ∧ r ⊆ t :=
⟨λ h, ⟨subset.trans h (inter_subset_left _ _), subset.trans h (inter_subset_right _ _)⟩,
λ ⟨h₁, h₂⟩, subset_inter h₁ h₂⟩
@[simp] theorem inter_univ (a : set α) : a ∩ univ = a :=
ext (assume x, and_true _)
@[simp] theorem univ_inter (a : set α) : univ ∩ a = a :=
ext (assume x, true_and _)
theorem inter_subset_inter_left {s t : set α} (u : set α) (H : s ⊆ t) : s ∩ u ⊆ t ∩ u :=
by finish [subset_def]
theorem inter_subset_inter_right {s t : set α} (u : set α) (H : s ⊆ t) : u ∩ s ⊆ u ∩ t :=
by finish [subset_def]
theorem inter_subset_inter {s₁ s₂ t₁ t₂ : set α} (h₁ : s₁ ⊆ t₁) (h₂ : s₂ ⊆ t₂) : s₁ ∩ s₂ ⊆ t₁ ∩ t₂ :=
by finish [subset_def]
theorem inter_eq_self_of_subset_left {s t : set α} (h : s ⊆ t) : s ∩ t = s :=
by finish [subset_def, ext_iff, iff_def]
theorem inter_eq_self_of_subset_right {s t : set α} (h : t ⊆ s) : s ∩ t = t :=
by finish [subset_def, ext_iff, iff_def]
lemma inter_compl_nonempty_iff {s t : set α} : (s ∩ tᶜ).nonempty ↔ ¬ s ⊆ t :=
begin
split,
{ rintros ⟨x ,xs, xt⟩ sub,
exact xt (sub xs) },
{ intros h,
rcases not_subset.mp h with ⟨x, xs, xt⟩,
exact ⟨x, xs, xt⟩ }
end
theorem union_inter_cancel_left {s t : set α} : (s ∪ t) ∩ s = s :=
by finish [ext_iff, iff_def]
theorem union_inter_cancel_right {s t : set α} : (s ∪ t) ∩ t = t :=
by finish [ext_iff, iff_def]
/-! ### Distributivity laws -/
theorem inter_distrib_left (s t u : set α) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
ext (assume x, and_or_distrib_left)
theorem inter_distrib_right (s t u : set α) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
ext (assume x, or_and_distrib_right)
theorem union_distrib_left (s t u : set α) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
ext (assume x, or_and_distrib_left)
theorem union_distrib_right (s t u : set α) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
ext (assume x, and_or_distrib_right)
/-!
### Lemmas about `insert`
`insert α s` is the set `{α} ∪ s`.
-/
theorem insert_def (x : α) (s : set α) : insert x s = { y | y = x ∨ y ∈ s } := rfl
@[simp] theorem subset_insert (x : α) (s : set α) : s ⊆ insert x s :=
assume y ys, or.inr ys
theorem mem_insert (x : α) (s : set α) : x ∈ insert x s :=
or.inl rfl
theorem mem_insert_of_mem {x : α} {s : set α} (y : α) : x ∈ s → x ∈ insert y s := or.inr
theorem eq_or_mem_of_mem_insert {x a : α} {s : set α} : x ∈ insert a s → x = a ∨ x ∈ s := id
theorem mem_of_mem_insert_of_ne {x a : α} {s : set α} (xin : x ∈ insert a s) : x ≠ a → x ∈ s :=
by finish [insert_def]
@[simp] theorem mem_insert_iff {x a : α} {s : set α} : x ∈ insert a s ↔ (x = a ∨ x ∈ s) := iff.rfl
@[simp] theorem insert_eq_of_mem {a : α} {s : set α} (h : a ∈ s) : insert a s = s :=
by finish [ext_iff, iff_def]
lemma ne_insert_of_not_mem {s : set α} (t : set α) {a : α} (h : a ∉ s) :
s ≠ insert a t :=
by { contrapose! h, simp [h] }
theorem insert_subset : insert a s ⊆ t ↔ (a ∈ t ∧ s ⊆ t) :=
by simp [subset_def, or_imp_distrib, forall_and_distrib]
theorem insert_subset_insert (h : s ⊆ t) : insert a s ⊆ insert a t :=
assume a', or.imp_right (@h a')
theorem ssubset_iff_insert {s t : set α} : s ⊂ t ↔ ∃ a ∉ s, insert a s ⊆ t :=
begin
simp only [insert_subset, exists_and_distrib_right, ssubset_def, not_subset],
simp only [exists_prop, and_comm]
end
theorem ssubset_insert {s : set α} {a : α} (h : a ∉ s) : s ⊂ insert a s :=
ssubset_iff_insert.2 ⟨a, h, subset.refl _⟩
theorem insert_comm (a b : α) (s : set α) : insert a (insert b s) = insert b (insert a s) :=
ext $ by simp [or.left_comm]
theorem insert_union : insert a s ∪ t = insert a (s ∪ t) :=
ext $ assume a, by simp [or.comm, or.left_comm]
@[simp] theorem union_insert : s ∪ insert a t = insert a (s ∪ t) :=
ext $ assume a, by simp [or.comm, or.left_comm]
theorem insert_nonempty (a : α) (s : set α) : (insert a s).nonempty :=
⟨a, mem_insert a s⟩
lemma insert_inter (x : α) (s t : set α) : insert x (s ∩ t) = insert x s ∩ insert x t :=
by { ext y, simp, tauto! }
-- useful in proofs by induction
theorem forall_of_forall_insert {P : α → Prop} {a : α} {s : set α} (h : ∀ x, x ∈ insert a s → P x) :
∀ x, x ∈ s → P x :=
by finish
theorem forall_insert_of_forall {P : α → Prop} {a : α} {s : set α} (h : ∀ x, x ∈ s → P x) (ha : P a) :
∀ x, x ∈ insert a s → P x :=
by finish
theorem bex_insert_iff {P : α → Prop} {a : α} {s : set α} :
(∃ x ∈ insert a s, P x) ↔ (∃ x ∈ s, P x) ∨ P a :=
by finish [iff_def]
theorem ball_insert_iff {P : α → Prop} {a : α} {s : set α} :
(∀ x ∈ insert a s, P x) ↔ P a ∧ (∀x ∈ s, P x) :=
by finish [iff_def]
/-! ### Lemmas about singletons -/
theorem singleton_def (a : α) : ({a} : set α) = insert a ∅ :=
(insert_emptyc_eq _).symm
@[simp] theorem mem_singleton_iff {a b : α} : a ∈ ({b} : set α) ↔ a = b :=
iff.rfl
@[simp]
lemma set_of_eq_eq_singleton {a : α} : {n | n = a} = {a} := set.ext $ λ n, (set.mem_singleton_iff).symm
-- TODO: again, annotation needed
@[simp] theorem mem_singleton (a : α) : a ∈ ({a} : set α) := by finish
theorem eq_of_mem_singleton {x y : α} (h : x ∈ ({y} : set α)) : x = y :=
by finish
@[simp] theorem singleton_eq_singleton_iff {x y : α} : {x} = ({y} : set α) ↔ x = y :=
by finish [ext_iff, iff_def]
theorem mem_singleton_of_eq {x y : α} (H : x = y) : x ∈ ({y} : set α) :=
by finish
theorem insert_eq (x : α) (s : set α) : insert x s = ({x} : set α) ∪ s :=
by finish [ext_iff, or_comm]
@[simp] theorem pair_eq_singleton (a : α) : ({a, a} : set α) = {a} :=
by finish
@[simp] theorem singleton_nonempty (a : α) : ({a} : set α).nonempty :=
⟨a, rfl⟩
@[simp] theorem singleton_subset_iff {a : α} {s : set α} : {a} ⊆ s ↔ a ∈ s :=
⟨λh, h (by simp), λh b e, by simp at e; simp [*]⟩
theorem set_compr_eq_eq_singleton {a : α} : {b | b = a} = {a} :=
ext $ by simp
@[simp] theorem singleton_union : {a} ∪ s = insert a s :=
rfl
@[simp] theorem union_singleton : s ∪ {a} = insert a s :=
by rw [union_comm, singleton_union]
theorem singleton_inter_eq_empty : {a} ∩ s = ∅ ↔ a ∉ s :=
by simp [eq_empty_iff_forall_not_mem]
theorem inter_singleton_eq_empty : s ∩ {a} = ∅ ↔ a ∉ s :=
by rw [inter_comm, singleton_inter_eq_empty]
lemma nmem_singleton_empty {s : set α} : s ∉ ({∅} : set (set α)) ↔ s.nonempty :=
by rw [mem_singleton_iff, ← ne.def, ne_empty_iff_nonempty]
instance unique_singleton (a : α) : unique ↥({a} : set α) :=
{ default := ⟨a, mem_singleton a⟩,
uniq :=
begin
intros x,
apply subtype.ext,
apply eq_of_mem_singleton (subtype.mem x),
end}
/-! ### Lemmas about sets defined as `{x ∈ s | p x}`. -/
theorem mem_sep {s : set α} {p : α → Prop} {x : α} (xs : x ∈ s) (px : p x) : x ∈ {x ∈ s | p x} :=
⟨xs, px⟩
@[simp] theorem sep_mem_eq {s t : set α} : {x ∈ s | x ∈ t} = s ∩ t := rfl
@[simp] theorem mem_sep_eq {s : set α} {p : α → Prop} {x : α} : x ∈ {x ∈ s | p x} = (x ∈ s ∧ p x) := rfl
theorem mem_sep_iff {s : set α} {p : α → Prop} {x : α} : x ∈ {x ∈ s | p x} ↔ x ∈ s ∧ p x :=
iff.rfl
theorem eq_sep_of_subset {s t : set α} (ssubt : s ⊆ t) : s = {x ∈ t | x ∈ s} :=
by finish [ext_iff, iff_def, subset_def]
theorem sep_subset (s : set α) (p : α → Prop) : {x ∈ s | p x} ⊆ s :=
assume x, and.left
theorem forall_not_of_sep_empty {s : set α} {p : α → Prop} (h : {x ∈ s | p x} = ∅) :
∀ x ∈ s, ¬ p x :=
by finish [ext_iff]
@[simp] lemma sep_univ {α} {p : α → Prop} : {a ∈ (univ : set α) | p a} = {a | p a} :=
set.ext $ by simp
/-! ### Lemmas about complement -/
theorem mem_compl {s : set α} {x : α} (h : x ∉ s) : x ∈ sᶜ := h
lemma compl_set_of {α} (p : α → Prop) : {a | p a}ᶜ = { a | ¬ p a } := rfl
theorem not_mem_of_mem_compl {s : set α} {x : α} (h : x ∈ sᶜ) : x ∉ s := h
@[simp] theorem mem_compl_eq (s : set α) (x : α) : x ∈ sᶜ = (x ∉ s) := rfl
theorem mem_compl_iff (s : set α) (x : α) : x ∈ sᶜ ↔ x ∉ s := iff.rfl
@[simp] theorem inter_compl_self (s : set α) : s ∩ sᶜ = ∅ :=
by finish [ext_iff]
@[simp] theorem compl_inter_self (s : set α) : sᶜ ∩ s = ∅ :=
by finish [ext_iff]
@[simp] theorem compl_empty : (∅ : set α)ᶜ = univ :=
by finish [ext_iff]
@[simp] theorem compl_union (s t : set α) : (s ∪ t)ᶜ = sᶜ ∩ tᶜ :=
by finish [ext_iff]
local attribute [simp] -- Will be generalized to lattices in `compl_compl'`
theorem compl_compl (s : set α) : sᶜᶜ = s :=
by finish [ext_iff]
-- ditto
theorem compl_inter (s t : set α) : (s ∩ t)ᶜ = sᶜ ∪ tᶜ :=
by finish [ext_iff]
@[simp] theorem compl_univ : (univ : set α)ᶜ = ∅ :=
by finish [ext_iff]
lemma compl_empty_iff {s : set α} : sᶜ = ∅ ↔ s = univ :=
by { split, intro h, rw [←compl_compl s, h, compl_empty], intro h, rw [h, compl_univ] }
lemma compl_univ_iff {s : set α} : sᶜ = univ ↔ s = ∅ :=
by rw [←compl_empty_iff, compl_compl]
lemma nonempty_compl {s : set α} : sᶜ.nonempty ↔ s ≠ univ :=
ne_empty_iff_nonempty.symm.trans $ not_congr $ compl_empty_iff
lemma mem_compl_singleton_iff {a x : α} : x ∈ ({a} : set α)ᶜ ↔ x ≠ a :=
not_iff_not_of_iff mem_singleton_iff
lemma compl_singleton_eq (a : α) : ({a} : set α)ᶜ = {x | x ≠ a} :=
ext $ λ x, mem_compl_singleton_iff
theorem union_eq_compl_compl_inter_compl (s t : set α) : s ∪ t = (sᶜ ∩ tᶜ)ᶜ :=
by simp [compl_inter, compl_compl]
theorem inter_eq_compl_compl_union_compl (s t : set α) : s ∩ t = (sᶜ ∪ tᶜ)ᶜ :=
by simp [compl_compl]
@[simp] theorem union_compl_self (s : set α) : s ∪ sᶜ = univ :=
by finish [ext_iff]
@[simp] theorem compl_union_self (s : set α) : sᶜ ∪ s = univ :=
by finish [ext_iff]
theorem compl_comp_compl : compl ∘ compl = @id (set α) :=
funext compl_compl
theorem compl_subset_comm {s t : set α} : sᶜ ⊆ t ↔ tᶜ ⊆ s :=
by haveI := classical.prop_decidable; exact
forall_congr (λ a, not_imp_comm)
lemma compl_subset_compl {s t : set α} : sᶜ ⊆ tᶜ ↔ t ⊆ s :=
by rw [compl_subset_comm, compl_compl]
theorem compl_subset_iff_union {s t : set α} : sᶜ ⊆ t ↔ s ∪ t = univ :=
iff.symm $ eq_univ_iff_forall.trans $ forall_congr $ λ a,
by haveI := classical.prop_decidable; exact or_iff_not_imp_left
theorem subset_compl_comm {s t : set α} : s ⊆ tᶜ ↔ t ⊆ sᶜ :=
forall_congr $ λ a, imp_not_comm
theorem subset_compl_iff_disjoint {s t : set α} : s ⊆ tᶜ ↔ s ∩ t = ∅ :=
iff.trans (forall_congr $ λ a, and_imp.symm) subset_empty_iff
lemma subset_compl_singleton_iff {a : α} {s : set α} : s ⊆ {a}ᶜ ↔ a ∉ s :=
by { rw subset_compl_comm, simp }
theorem inter_subset (a b c : set α) : a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c :=
begin
classical,
split,
{ intros h x xa, by_cases h' : x ∈ b, simp [h ⟨xa, h'⟩], simp [h'] },
intros h x, rintro ⟨xa, xb⟩, cases h xa, contradiction, assumption
end
/-! ### Lemmas about set difference -/
theorem diff_eq (s t : set α) : s \ t = s ∩ tᶜ := rfl
@[simp] theorem mem_diff {s t : set α} (x : α) : x ∈ s \ t ↔ x ∈ s ∧ x ∉ t := iff.rfl
theorem mem_diff_of_mem {s t : set α} {x : α} (h1 : x ∈ s) (h2 : x ∉ t) : x ∈ s \ t :=
⟨h1, h2⟩
theorem mem_of_mem_diff {s t : set α} {x : α} (h : x ∈ s \ t) : x ∈ s :=
h.left
theorem not_mem_of_mem_diff {s t : set α} {x : α} (h : x ∈ s \ t) : x ∉ t :=
h.right
theorem nonempty_diff {s t : set α} : (s \ t).nonempty ↔ ¬ (s ⊆ t) :=
⟨λ ⟨x, xs, xt⟩, not_subset.2 ⟨x, xs, xt⟩,
λ h, let ⟨x, xs, xt⟩ := not_subset.1 h in ⟨x, xs, xt⟩⟩
theorem union_diff_cancel {s t : set α} (h : s ⊆ t) : s ∪ (t \ s) = t :=
by finish [ext_iff, iff_def, subset_def]
theorem union_diff_cancel_left {s t : set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ s = t :=
by finish [ext_iff, iff_def, subset_def]
theorem union_diff_cancel_right {s t : set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ t = s :=
by finish [ext_iff, iff_def, subset_def]
theorem union_diff_left {s t : set α} : (s ∪ t) \ s = t \ s :=
by finish [ext_iff, iff_def]
theorem union_diff_right {s t : set α} : (s ∪ t) \ t = s \ t :=
by finish [ext_iff, iff_def]
theorem union_diff_distrib {s t u : set α} : (s ∪ t) \ u = s \ u ∪ t \ u :=
inter_distrib_right _ _ _
theorem inter_union_distrib_left {s t u : set α} : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
set.ext $ λ _, and_or_distrib_left
theorem inter_union_distrib_right {s t u : set α} : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
set.ext $ λ _, and_or_distrib_right
theorem union_inter_distrib_left {s t u : set α} : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
set.ext $ λ _, or_and_distrib_left
theorem union_inter_distrib_right {s t u : set α} : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
set.ext $ λ _, or_and_distrib_right
theorem inter_diff_assoc (a b c : set α) : (a ∩ b) \ c = a ∩ (b \ c) :=
inter_assoc _ _ _
theorem inter_diff_self (a b : set α) : a ∩ (b \ a) = ∅ :=
by finish [ext_iff]
theorem inter_union_diff (s t : set α) : (s ∩ t) ∪ (s \ t) = s :=
by finish [ext_iff, iff_def]
theorem diff_subset (s t : set α) : s \ t ⊆ s :=
by finish [subset_def]
theorem diff_subset_diff {s₁ s₂ t₁ t₂ : set α} : s₁ ⊆ s₂ → t₂ ⊆ t₁ → s₁ \ t₁ ⊆ s₂ \ t₂ :=
by finish [subset_def]
theorem diff_subset_diff_left {s₁ s₂ t : set α} (h : s₁ ⊆ s₂) : s₁ \ t ⊆ s₂ \ t :=
diff_subset_diff h (by refl)
theorem diff_subset_diff_right {s t u : set α} (h : t ⊆ u) : s \ u ⊆ s \ t :=
diff_subset_diff (subset.refl s) h
theorem compl_eq_univ_diff (s : set α) : sᶜ = univ \ s :=
by finish [ext_iff]
@[simp] lemma empty_diff (s : set α) : (∅ \ s : set α) = ∅ :=
eq_empty_of_subset_empty $ assume x ⟨hx, _⟩, hx
theorem diff_eq_empty {s t : set α} : s \ t = ∅ ↔ s ⊆ t :=
⟨assume h x hx, classical.by_contradiction $ assume : x ∉ t, show x ∈ (∅ : set α), from h ▸ ⟨hx, this⟩,
assume h, eq_empty_of_subset_empty $ assume x ⟨hx, hnx⟩, hnx $ h hx⟩
@[simp] theorem diff_empty {s : set α} : s \ ∅ = s :=
ext $ assume x, ⟨assume ⟨hx, _⟩, hx, assume h, ⟨h, not_false⟩⟩
theorem diff_diff {u : set α} : s \ t \ u = s \ (t ∪ u) :=
ext $ by simp [not_or_distrib, and.comm, and.left_comm]
lemma diff_subset_iff {s t u : set α} : s \ t ⊆ u ↔ s ⊆ t ∪ u :=
⟨assume h x xs, classical.by_cases or.inl (assume nxt, or.inr (h ⟨xs, nxt⟩)),
assume h x ⟨xs, nxt⟩, or.resolve_left (h xs) nxt⟩
lemma subset_diff_union (s t : set α) : s ⊆ (s \ t) ∪ t :=
by rw [union_comm, ←diff_subset_iff]
@[simp] lemma diff_singleton_subset_iff {x : α} {s t : set α} : s \ {x} ⊆ t ↔ s ⊆ insert x t :=
by { rw [←union_singleton, union_comm], apply diff_subset_iff }
lemma subset_diff_singleton {x : α} {s t : set α} (h : s ⊆ t) (hx : x ∉ s) : s ⊆ t \ {x} :=
subset_inter h $ subset_compl_comm.1 $ singleton_subset_iff.2 hx
lemma subset_insert_diff_singleton (x : α) (s : set α) : s ⊆ insert x (s \ {x}) :=
by rw [←diff_singleton_subset_iff]
lemma diff_subset_comm {s t u : set α} : s \ t ⊆ u ↔ s \ u ⊆ t :=
by rw [diff_subset_iff, diff_subset_iff, union_comm]
lemma diff_inter {s t u : set α} : s \ (t ∩ u) = (s \ t) ∪ (s \ u) :=
ext $ λ x, by simp [classical.not_and_distrib, and_or_distrib_left]
lemma diff_inter_diff {s t u : set α} : s \ t ∩ (s \ u) = s \ (t ∪ u) :=
by { ext x, simp only [mem_inter_eq, mem_union_eq, mem_diff, not_or_distrib],
exact ⟨λ ⟨⟨h1, h2⟩, _, h3⟩, ⟨h1, h2, h3⟩, λ ⟨h1, h2, h3⟩, ⟨⟨h1, h2⟩, h1, h3⟩⟩ }
lemma diff_compl : s \ tᶜ = s ∩ t := by rw [diff_eq, compl_compl]
lemma diff_diff_right {s t u : set α} : s \ (t \ u) = (s \ t) ∪ (s ∩ u) :=
by rw [diff_eq t u, diff_inter, diff_compl]
@[simp] theorem insert_diff_of_mem (s) (h : a ∈ t) : insert a s \ t = s \ t :=
ext $ by intro; constructor; simp [or_imp_distrib, h] {contextual := tt}
theorem insert_diff_of_not_mem (s) (h : a ∉ t) : insert a s \ t = insert a (s \ t) :=
begin
classical,
ext x,
by_cases h' : x ∈ t,
{ have : x ≠ a,
{ assume H,
rw H at h',
exact h h' },
simp [h, h', this] },
{ simp [h, h'] }
end
lemma insert_diff_self_of_not_mem {a : α} {s : set α} (h : a ∉ s) :
insert a s \ {a} = s :=
ext $ λ x, by simp [and_iff_left_of_imp (λ hx : x ∈ s, show x ≠ a, from λ hxa, h $ hxa ▸ hx)]
theorem union_diff_self {s t : set α} : s ∪ (t \ s) = s ∪ t :=
by finish [ext_iff, iff_def]
theorem diff_union_self {s t : set α} : (s \ t) ∪ t = s ∪ t :=
by rw [union_comm, union_diff_self, union_comm]
theorem diff_inter_self {a b : set α} : (b \ a) ∩ a = ∅ :=
ext $ by simp [iff_def] {contextual:=tt}
theorem diff_eq_self {s t : set α} : s \ t = s ↔ t ∩ s ⊆ ∅ :=
by finish [ext_iff, iff_def, subset_def]
@[simp] theorem diff_singleton_eq_self {a : α} {s : set α} (h : a ∉ s) : s \ {a} = s :=
diff_eq_self.2 $ by simp [singleton_inter_eq_empty.2 h]
@[simp] theorem insert_diff_singleton {a : α} {s : set α} :
insert a (s \ {a}) = insert a s :=
by simp [insert_eq, union_diff_self, -union_singleton, -singleton_union]
@[simp] lemma diff_self {s : set α} : s \ s = ∅ := ext $ by simp
lemma diff_diff_cancel_left {s t : set α} (h : s ⊆ t) : t \ (t \ s) = s :=
by simp only [diff_diff_right, diff_self, inter_eq_self_of_subset_right h, empty_union]
lemma mem_diff_singleton {x y : α} {s : set α} : x ∈ s \ {y} ↔ (x ∈ s ∧ x ≠ y) :=
iff.rfl
lemma mem_diff_singleton_empty {s : set α} {t : set (set α)} :
s ∈ t \ {∅} ↔ (s ∈ t ∧ s.nonempty) :=
mem_diff_singleton.trans $ and_congr iff.rfl ne_empty_iff_nonempty
/-! ### Powerset -/
theorem mem_powerset {x s : set α} (h : x ⊆ s) : x ∈ powerset s := h
theorem subset_of_mem_powerset {x s : set α} (h : x ∈ powerset s) : x ⊆ s := h
theorem mem_powerset_iff (x s : set α) : x ∈ powerset s ↔ x ⊆ s := iff.rfl
/-! ### Inverse image -/
/-- The preimage of `s : set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : set β) : set α := {x | f x ∈ s}
infix ` ⁻¹' `:80 := preimage
section preimage
variables {f : α → β} {g : β → γ}
@[simp] theorem preimage_empty : f ⁻¹' ∅ = ∅ := rfl
@[simp] theorem mem_preimage {s : set β} {a : α} : (a ∈ f ⁻¹' s) ↔ (f a ∈ s) := iff.rfl
theorem preimage_mono {s t : set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t :=
assume x hx, h hx
@[simp] theorem preimage_univ : f ⁻¹' univ = univ := rfl
theorem subset_preimage_univ {s : set α} : s ⊆ f ⁻¹' univ := subset_univ _
@[simp] theorem preimage_inter {s t : set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t := rfl
@[simp] theorem preimage_union {s t : set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t := rfl
@[simp] theorem preimage_compl {s : set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ := rfl
@[simp] theorem preimage_diff (f : α → β) (s t : set β) :
f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t := rfl
@[simp] theorem preimage_set_of_eq {p : α → Prop} {f : β → α} : f ⁻¹' {a | p a} = {a | p (f a)} :=
rfl
@[simp] theorem preimage_id {s : set α} : id ⁻¹' s = s := rfl
@[simp] theorem preimage_id' {s : set α} : (λ x, x) ⁻¹' s = s := rfl
theorem preimage_const_of_mem {b : β} {s : set β} (h : b ∈ s) :
(λ (x : α), b) ⁻¹' s = univ :=
eq_univ_of_forall $ λ x, h
theorem preimage_const_of_not_mem {b : β} {s : set β} (h : b ∉ s) :
(λ (x : α), b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty $ λ x hx, h hx
theorem preimage_const (b : β) (s : set β) [decidable (b ∈ s)] :
(λ (x : α), b) ⁻¹' s = if b ∈ s then univ else ∅ :=
by { split_ifs with hb hb, exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb] }
theorem preimage_comp {s : set γ} : (g ∘ f) ⁻¹' s = f ⁻¹' (g ⁻¹' s) := rfl
lemma preimage_preimage {g : β → γ} {f : α → β} {s : set γ} :
f ⁻¹' (g ⁻¹' s) = (λ x, g (f x)) ⁻¹' s :=
preimage_comp.symm
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : set (subtype p)} {t : set α} :
s = subtype.val ⁻¹' t ↔ (∀x (h : p x), (⟨x, h⟩ : subtype p) ∈ s ↔ x ∈ t) :=
⟨assume s_eq x h, by rw [s_eq]; simp,
assume h, ext $ assume ⟨x, hx⟩, by simp [h]⟩
lemma preimage_coe_coe_diagonal {α : Type*} (s : set α) :
(prod.map coe coe) ⁻¹' (diagonal α) = diagonal s :=
begin
ext ⟨⟨x, x_in⟩, ⟨y, y_in⟩⟩,
simp [set.diagonal],
end
end preimage
/-! ### Image of a set under a function -/
section image
infix ` '' `:80 := image
theorem mem_image_iff_bex {f : α → β} {s : set α} {y : β} :
y ∈ f '' s ↔ ∃ x (_ : x ∈ s), f x = y := bex_def.symm
theorem mem_image_eq (f : α → β) (s : set α) (y: β) : y ∈ f '' s = ∃ x, x ∈ s ∧ f x = y := rfl
@[simp] theorem mem_image (f : α → β) (s : set α) (y : β) :
y ∈ f '' s ↔ ∃ x, x ∈ s ∧ f x = y := iff.rfl
lemma image_eta (f : α → β) : f '' s = (λ x, f x) '' s := rfl
theorem mem_image_of_mem (f : α → β) {x : α} {a : set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
theorem mem_image_of_injective {f : α → β} {a : α} {s : set α} (hf : injective f) :
f a ∈ f '' s ↔ a ∈ s :=
iff.intro
(assume ⟨b, hb, eq⟩, (hf eq) ▸ hb)
(assume h, mem_image_of_mem _ h)
theorem ball_image_of_ball {f : α → β} {s : set α} {p : β → Prop}
(h : ∀ x ∈ s, p (f x)) : ∀ y ∈ f '' s, p y :=
by finish [mem_image_eq]
theorem ball_image_iff {f : α → β} {s : set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ (∀ x ∈ s, p (f x)) :=
iff.intro
(assume h a ha, h _ $ mem_image_of_mem _ ha)
(assume h b ⟨a, ha, eq⟩, eq ▸ h a ha)
theorem bex_image_iff {f : α → β} {s : set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ (∃ x ∈ s, p (f x)) :=
by simp
theorem mem_image_elim {f : α → β} {s : set α} {C : β → Prop} (h : ∀ (x : α), x ∈ s → C (f x)) :
∀{y : β}, y ∈ f '' s → C y
| ._ ⟨a, a_in, rfl⟩ := h a a_in
theorem mem_image_elim_on {f : α → β} {s : set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ (x : α), x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
@[congr] lemma image_congr {f g : α → β} {s : set α}
(h : ∀a∈s, f a = g a) : f '' s = g '' s :=
by safe [ext_iff, iff_def]
/-- A common special case of `image_congr` -/
lemma image_congr' {f g : α → β} {s : set α} (h : ∀ (x : α), f x = g x) : f '' s = g '' s :=
image_congr (λx _, h x)
theorem image_comp (f : β → γ) (g : α → β) (a : set α) : (f ∘ g) '' a = f '' (g '' a) :=
subset.antisymm
(ball_image_of_ball $ assume a ha, mem_image_of_mem _ $ mem_image_of_mem _ ha)
(ball_image_of_ball $ ball_image_of_ball $ assume a ha, mem_image_of_mem _ ha)
/-- A variant of `image_comp`, useful for rewriting -/
lemma image_image (g : β → γ) (f : α → β) (s : set α) : g '' (f '' s) = (λ x, g (f x)) '' s :=
(image_comp g f s).symm
/-- Image is monotone with respect to `⊆`. See `set.monotone_image` for the statement in
terms of `≤`. -/
theorem image_subset {a b : set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b :=
by finish [subset_def, mem_image_eq]
theorem image_union (f : α → β) (s t : set α) :
f '' (s ∪ t) = f '' s ∪ f '' t :=
by finish [ext_iff, iff_def, mem_image_eq]
@[simp] theorem image_empty (f : α → β) : f '' ∅ = ∅ := ext $ by simp
lemma image_inter_subset (f : α → β) (s t : set α) :
f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ $ inter_subset_left _ _) (image_subset _ $ inter_subset_right _ _)
theorem image_inter_on {f : α → β} {s t : set α} (h : ∀x∈t, ∀y∈s, f x = f y → x = y) :
f '' s ∩ f '' t = f '' (s ∩ t) :=
subset.antisymm
(assume b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩,
have a₂ = a₁, from h _ ha₂ _ ha₁ (by simp *),
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩)
(image_inter_subset _ _ _)
theorem image_inter {f : α → β} {s t : set α} (H : injective f) :
f '' s ∩ f '' t = f '' (s ∩ t) :=
image_inter_on (assume x _ y _ h, H h)
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : surjective f) : f '' univ = univ :=
eq_univ_of_forall $ by simp [image]; exact H
@[simp] theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} :=
ext $ λ x, by simp [image]; rw eq_comm
theorem nonempty.image_const {s : set α} (hs : s.nonempty) (a : β) : (λ _, a) '' s = {a} :=
ext $ λ x, ⟨λ ⟨y, _, h⟩, h ▸ mem_singleton _,
λ h, (eq_of_mem_singleton h).symm ▸ hs.imp (λ y hy, ⟨hy, rfl⟩)⟩
@[simp] lemma image_eq_empty {α β} {f : α → β} {s : set α} : f '' s = ∅ ↔ s = ∅ :=
by simp only [eq_empty_iff_forall_not_mem]; exact
⟨λ H a ha, H _ ⟨_, ha, rfl⟩, λ H b ⟨_, ha, _⟩, H _ ha⟩
lemma inter_singleton_nonempty {s : set α} {a : α} : (s ∩ {a}).nonempty ↔ a ∈ s :=
by finish [set.nonempty]
-- TODO(Jeremy): there is an issue with - t unfolding to compl t
theorem mem_compl_image (t : set α) (S : set (set α)) :
t ∈ compl '' S ↔ tᶜ ∈ S :=
begin
suffices : ∀ x, xᶜ = t ↔ tᶜ = x, {simp [this]},
intro x, split; { intro e, subst e, simp }
end
/-- A variant of `image_id` -/
@[simp] lemma image_id' (s : set α) : (λx, x) '' s = s := ext $ by simp
theorem image_id (s : set α) : id '' s = s := by simp
theorem compl_compl_image (S : set (set α)) :
compl '' (compl '' S) = S :=
by rw [← image_comp, compl_comp_compl, image_id]
theorem image_insert_eq {f : α → β} {a : α} {s : set α} :
f '' (insert a s) = insert (f a) (f '' s) :=
ext $ by simp [and_or_distrib_left, exists_or_distrib, eq_comm, or_comm, and_comm]
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} :=
by simp only [image_insert_eq, image_singleton]
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α}
(I : left_inverse g f) (s : set α) : f '' s ⊆ g ⁻¹' s :=
λ b ⟨a, h, e⟩, e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α}
(I : left_inverse g f) (s : set β) : f ⁻¹' s ⊆ g '' s :=
λ b h, ⟨f b, h, I b⟩
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α}
(h₁ : left_inverse g f) (h₂ : right_inverse g f) :
image f = preimage g :=
funext $ λ s, subset.antisymm
(image_subset_preimage_of_inverse h₁ s)
(preimage_subset_image_of_inverse h₂ s)
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : set α}
(h₁ : left_inverse g f) (h₂ : right_inverse g f) :
b ∈ f '' s ↔ g b ∈ s :=
by rw image_eq_preimage_of_inverse h₁ h₂; refl
theorem image_compl_subset {f : α → β} {s : set α} (H : injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
subset_compl_iff_disjoint.2 $ by simp [image_inter H]
theorem subset_image_compl {f : α → β} {s : set α} (H : surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 $
by rw ← image_union; simp [image_univ_of_surjective H]
theorem image_compl_eq {f : α → β} {s : set α} (H : bijective f) : f '' sᶜ = (f '' s)ᶜ :=
subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
theorem subset_image_diff (f : α → β) (s t : set α) :
f '' s \ f '' t ⊆ f '' (s \ t) :=
begin
rw [diff_subset_iff, ← image_union, union_diff_self],
exact image_subset f (subset_union_right t s)
end
theorem image_diff {f : α → β} (hf : injective f) (s t : set α) :
f '' (s \ t) = f '' s \ f '' t :=
subset.antisymm
(subset.trans (image_inter_subset _ _ _) $ inter_subset_inter_right _ $ image_compl_subset hf)
(subset_image_diff f s t)
lemma nonempty.image (f : α → β) {s : set α} : s.nonempty → (f '' s).nonempty
| ⟨x, hx⟩ := ⟨f x, mem_image_of_mem f hx⟩
lemma nonempty.of_image {f : α → β} {s : set α} : (f '' s).nonempty → s.nonempty
| ⟨y, x, hx, _⟩ := ⟨x, hx⟩
@[simp] lemma nonempty_image_iff {f : α → β} {s : set α} :
(f '' s).nonempty ↔ s.nonempty :=
⟨nonempty.of_image, λ h, h.image f⟩
/-- image and preimage are a Galois connection -/
@[simp] theorem image_subset_iff {s : set α} {t : set β} {f : α → β} :
f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
theorem image_preimage_subset (f : α → β) (s : set β) :
f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 (subset.refl _)
theorem subset_preimage_image (f : α → β) (s : set α) :
s ⊆ f ⁻¹' (f '' s) :=
λ x, mem_image_of_mem f
theorem preimage_image_eq {f : α → β} (s : set α) (h : injective f) : f ⁻¹' (f '' s) = s :=
subset.antisymm
(λ x ⟨y, hy, e⟩, h e ▸ hy)
(subset_preimage_image f s)
theorem image_preimage_eq {f : α → β} {s : set β} (h : surjective f) : f '' (f ⁻¹' s) = s :=
subset.antisymm
(image_preimage_subset f s)
(λ x hx, let ⟨y, e⟩ := h x in ⟨y, (e.symm ▸ hx : f y ∈ s), e⟩)
lemma preimage_eq_preimage {f : β → α} (hf : surjective f) : f ⁻¹' s = preimage f t ↔ s = t :=
iff.intro
(assume eq, by rw [← @image_preimage_eq β α f s hf, ← @image_preimage_eq β α f t hf, eq])
(assume eq, eq ▸ rfl)
protected lemma push_pull (f : α → β) (s : set α) (t : set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t :=
begin
apply subset.antisymm,
{ calc f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ (f '' (f⁻¹' t)) : image_inter_subset _ _ _
... ⊆ f '' s ∩ t : inter_subset_inter_right _ (image_preimage_subset f t) },
{ rintros _ ⟨⟨x, h', rfl⟩, h⟩,
exact ⟨x, ⟨h', h⟩, rfl⟩ }
end
protected lemma push_pull' (f : α → β) (s : set α) (t : set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s :=
by simp only [inter_comm, set.push_pull]
theorem compl_image : image (compl : set α → set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
theorem compl_image_set_of {p : set α → Prop} :
compl '' {s | p s} = {s | p sᶜ} :=
congr_fun compl_image p
theorem inter_preimage_subset (s : set α) (t : set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) :=
λ x h, ⟨mem_image_of_mem _ h.left, h.right⟩
theorem union_preimage_subset (s : set α) (t : set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) :=
λ x h, or.elim h (λ l, or.inl $ mem_image_of_mem _ l) (λ r, or.inr r)
theorem subset_image_union (f : α → β) (s : set α) (t : set β) :
f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
lemma preimage_subset_iff {A : set α} {B : set β} {f : α → β} :
f⁻¹' B ⊆ A ↔ (∀ a : α, f a ∈ B → a ∈ A) := iff.rfl
lemma image_eq_image {f : α → β} (hf : injective f) : f '' s = f '' t ↔ s = t :=
iff.symm $ iff.intro (assume eq, eq ▸ rfl) $ assume eq,
by rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
lemma image_subset_image_iff {f : α → β} (hf : injective f) : f '' s ⊆ f '' t ↔ s ⊆ t :=
begin
refine (iff.symm $ iff.intro (image_subset f) $ assume h, _),
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf],
exact preimage_mono h
end
lemma prod_quotient_preimage_eq_image [s : setoid α] (g : quotient s → β) {h : α → β}
(Hh : h = g ∘ quotient.mk) (r : set (β × β)) :
{x : quotient s × quotient s | (g x.1, g x.2) ∈ r} =
(λ a : α × α, (⟦a.1⟧, ⟦a.2⟧)) '' ((λ a : α × α, (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸ set.ext (λ ⟨a₁, a₂⟩, ⟨quotient.induction_on₂ a₁ a₂
(λ a₁ a₂ h, ⟨(a₁, a₂), h, rfl⟩),
λ ⟨⟨b₁, b₂⟩, h₁, h₂⟩, show (g a₁, g a₂) ∈ r, from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := prod.ext_iff.1 h₂,
h₃.1 ▸ h₃.2 ▸ h₁⟩)
/-- Restriction of `f` to `s` factors through `s.image_factorization f : s → f '' s`. -/
def image_factorization (f : α → β) (s : set α) : s → f '' s :=
λ p, ⟨f p.1, mem_image_of_mem f p.2⟩
lemma image_factorization_eq {f : α → β} {s : set α} :
subtype.val ∘ image_factorization f s = f ∘ subtype.val :=
funext $ λ p, rfl
lemma surjective_onto_image {f : α → β} {s : set α} :
surjective (image_factorization f s) :=
λ ⟨_, ⟨a, ha, rfl⟩⟩, ⟨⟨a, ha⟩, rfl⟩
end image
/-! ### Subsingleton -/
/-- A set `s` is a `subsingleton`, if it has at most one element. -/
protected def subsingleton (s : set α) : Prop :=
∀ ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s), x = y
lemma subsingleton.mono (ht : t.subsingleton) (hst : s ⊆ t) : s.subsingleton :=
λ x hx y hy, ht (hst hx) (hst hy)
lemma subsingleton.image (hs : s.subsingleton) (f : α → β) : (f '' s).subsingleton :=
λ _ ⟨x, hx, Hx⟩ _ ⟨y, hy, Hy⟩, Hx ▸ Hy ▸ congr_arg f (hs hx hy)
lemma subsingleton.eq_singleton_of_mem (hs : s.subsingleton) {x:α} (hx : x ∈ s) :
s = {x} :=
ext $ λ y, ⟨λ hy, (hs hx hy) ▸ mem_singleton _, λ hy, (eq_of_mem_singleton hy).symm ▸ hx⟩
lemma subsingleton_empty : (∅ : set α).subsingleton := λ x, false.elim
lemma subsingleton_singleton {a} : ({a} : set α).subsingleton :=
λ x hx y hy, (eq_of_mem_singleton hx).symm ▸ (eq_of_mem_singleton hy).symm ▸ rfl
lemma subsingleton.eq_empty_or_singleton (hs : s.subsingleton) :
s = ∅ ∨ ∃ x, s = {x} :=
s.eq_empty_or_nonempty.elim or.inl (λ ⟨x, hx⟩, or.inr ⟨x, hs.eq_singleton_of_mem hx⟩)
lemma subsingleton_univ [subsingleton α] : (univ : set α).subsingleton :=
λ x hx y hy, subsingleton.elim x y
theorem univ_eq_true_false : univ = ({true, false} : set Prop) :=
eq.symm $ eq_univ_of_forall $ classical.cases (by simp) (by simp)
/-! ### Lemmas about range of a function. -/
section range
variables {f : ι → α}
open function
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : set α := {x | ∃y, f y = x}
@[simp] theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x := iff.rfl
@[simp] theorem mem_range_self (i : ι) : f i ∈ range f := ⟨i, rfl⟩
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ (∀ i, p (f i)) :=
⟨assume h i, h (f i) (mem_range_self _), assume h a ⟨i, (hi : f i = a)⟩, hi ▸ h i⟩
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ (∃ i, p (f i)) :=
by simp
lemma exists_range_iff' {p : α → Prop} :
(∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) :=
by simpa only [exists_prop] using exists_range_iff
theorem range_iff_surjective : range f = univ ↔ surjective f :=
eq_univ_iff_forall
@[simp] theorem range_id : range (@id α) = univ := range_iff_surjective.2 surjective_id
theorem range_inl_union_range_inr : range (@sum.inl α β) ∪ range sum.inr = univ :=
ext $ λ x, by cases x; simp
@[simp] theorem range_quot_mk (r : α → α → Prop) : range (quot.mk r) = univ :=
range_iff_surjective.2 quot.exists_rep
@[simp] theorem image_univ {ι : Type*} {f : ι → β} : f '' univ = range f :=
ext $ by simp [image, range]
theorem image_subset_range {ι : Type*} (f : ι → β) (s : set ι) : f '' s ⊆ range f :=
by rw ← image_univ; exact image_subset _ (subset_univ _)
theorem range_comp {g : α → β} : range (g ∘ f) = g '' range f :=
subset.antisymm
(forall_range_iff.mpr $ assume i, mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr $ forall_range_iff.mpr mem_range_self)
theorem range_subset_iff {s : set α} : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
lemma range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g :=
by rw range_comp; apply image_subset_range
lemma range_nonempty_iff_nonempty : (range f).nonempty ↔ nonempty ι :=
⟨λ ⟨y, x, hxy⟩, ⟨x⟩, λ ⟨x⟩, ⟨f x, mem_range_self x⟩⟩
lemma range_nonempty [h : nonempty ι] (f : ι → α) : (range f).nonempty :=
range_nonempty_iff_nonempty.2 h
@[simp] lemma range_eq_empty {f : ι → α} : range f = ∅ ↔ ¬ nonempty ι :=
not_nonempty_iff_eq_empty.symm.trans $ not_congr range_nonempty_iff_nonempty
theorem image_preimage_eq_inter_range {f : α → β} {t : set β} :
f '' (f ⁻¹' t) = t ∩ range f :=
ext $ assume x, ⟨assume ⟨x, hx, heq⟩, heq ▸ ⟨hx, mem_range_self _⟩,
assume ⟨hx, ⟨y, h_eq⟩⟩, h_eq ▸ mem_image_of_mem f $
show y ∈ f ⁻¹' t, by simp [preimage, h_eq, hx]⟩
lemma image_preimage_eq_of_subset {f : α → β} {s : set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s :=
by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
lemma preimage_subset_preimage_iff {s t : set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t :=
begin
split,
{ intros h x hx, rcases hs hx with ⟨y, rfl⟩, exact h hx },
intros h x, apply h
end
lemma preimage_eq_preimage' {s t : set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t :=
begin
split,
{ intro h, apply subset.antisymm, rw [←preimage_subset_preimage_iff hs, h],
rw [←preimage_subset_preimage_iff ht, h] },
rintro rfl, refl
end
@[simp] theorem preimage_inter_range {f : α → β} {s : set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
set.ext $ λ x, and_iff_left ⟨x, rfl⟩
@[simp] theorem preimage_range_inter {f : α → β} {s : set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s :=
by rw [inter_comm, preimage_inter_range]
theorem preimage_image_preimage {f : α → β} {s : set β} :
f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s :=
by rw [image_preimage_eq_inter_range, preimage_inter_range]
@[simp] theorem quot_mk_range_eq [setoid α] : range (λx : α, ⟦x⟧) = univ :=
range_iff_surjective.2 quot.exists_rep
lemma range_const_subset {c : α} : range (λx:ι, c) ⊆ {c} :=
range_subset_iff.2 $ λ x, rfl
@[simp] lemma range_const : ∀ [nonempty ι] {c : α}, range (λx:ι, c) = {c}
| ⟨x⟩ c := subset.antisymm range_const_subset $
assume y hy, (mem_singleton_iff.1 hy).symm ▸ mem_range_self x
lemma diagonal_eq_range {α : Type*} : diagonal α = range (λ x, (x, x)) :=
by { ext ⟨x, y⟩, simp [diagonal, eq_comm] }
theorem preimage_singleton_nonempty {f : α → β} {y : β} :
(f ⁻¹' {y}).nonempty ↔ y ∈ range f :=
iff.rfl
theorem preimage_singleton_eq_empty {f : α → β} {y : β} :
f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans $ not_congr preimage_singleton_nonempty
/-- Any map `f : ι → β` factors through a map `range_factorization f : ι → range f`. -/
def range_factorization (f : ι → β) : ι → range f :=
λ i, ⟨f i, mem_range_self i⟩
lemma range_factorization_eq {f : ι → β} :
subtype.val ∘ range_factorization f = f :=
funext $ λ i, rfl
lemma surjective_onto_range : surjective (range_factorization f) :=
λ ⟨_, ⟨i, rfl⟩⟩, ⟨i, rfl⟩
lemma image_eq_range (f : α → β) (s : set α) : f '' s = range (λ(x : s), f x) :=
by { ext, split, rintro ⟨x, h1, h2⟩, exact ⟨⟨x, h1⟩, h2⟩, rintro ⟨⟨x, h1⟩, h2⟩, exact ⟨x, h1, h2⟩ }
@[simp] lemma sum.elim_range {α β γ : Type*} (f : α → γ) (g : β → γ) :
range (sum.elim f g) = range f ∪ range g :=
by simp [set.ext_iff, mem_range]
lemma range_ite_subset' {p : Prop} [decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g :=
begin
by_cases h : p, {rw if_pos h, exact subset_union_left _ _},
{rw if_neg h, exact subset_union_right _ _}
end
lemma range_ite_subset {p : α → Prop} [decidable_pred p] {f g : α → β} :
range (λ x, if p x then f x else g x) ⊆ range f ∪ range g :=
begin
rw range_subset_iff, intro x, by_cases h : p x,
simp [if_pos h, mem_union, mem_range_self],
simp [if_neg h, mem_union, mem_range_self]
end
@[simp] lemma preimage_range (f : α → β) : f ⁻¹' (range f) = univ :=
eq_univ_of_forall mem_range_self
end range
/-- The set `s` is pairwise `r` if `r x y` for all *distinct* `x y ∈ s`. -/
def pairwise_on (s : set α) (r : α → α → Prop) := ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r x y
theorem pairwise_on.mono {s t : set α} {r}
(h : t ⊆ s) (hp : pairwise_on s r) : pairwise_on t r :=
λ x xt y yt, hp x (h xt) y (h yt)
theorem pairwise_on.mono' {s : set α} {r r' : α → α → Prop}
(H : ∀ a b, r a b → r' a b) (hp : pairwise_on s r) : pairwise_on s r' :=
λ x xs y ys h, H _ _ (hp x xs y ys h)
end set
open set
namespace function
variables {ι : Sort*} {α : Type*} {β : Type*} {f : α → β}
lemma surjective.preimage_injective (hf : surjective f) : injective (preimage f) :=
assume s t, (preimage_eq_preimage hf).1
lemma injective.preimage_surjective (hf : injective f) : surjective (preimage f) :=
by { intro s, use f '' s, rw preimage_image_eq _ hf }
lemma surjective.image_surjective (hf : surjective f) : surjective (image f) :=
by { intro s, use f ⁻¹' s, rw image_preimage_eq hf }
lemma injective.image_injective (hf : injective f) : injective (image f) :=
by { intros s t h, rw [←preimage_image_eq s hf, ←preimage_image_eq t hf, h] }
lemma surjective.range_eq {f : ι → α} (hf : surjective f) : range f = univ :=
range_iff_surjective.2 hf
lemma surjective.range_comp {ι' : Sort*} {f : ι → ι'} (hf : surjective f) (g : ι' → α) :
range (g ∘ f) = range g :=
ext $ λ y, (@surjective.exists _ _ _ hf (λ x, g x = y)).symm
lemma injective.nonempty_apply_iff {f : set α → set β} (hf : injective f)
(h2 : f ∅ = ∅) {s : set α} : (f s).nonempty ↔ s.nonempty :=
by rw [← ne_empty_iff_nonempty, ← h2, ← ne_empty_iff_nonempty, hf.ne_iff]
end function
open function
/-! ### Image and preimage on subtypes -/
namespace subtype
variable {α : Type*}
lemma coe_image {p : α → Prop} {s : set (subtype p)} :
coe '' s = {x | ∃h : p x, (⟨x, h⟩ : subtype p) ∈ s} :=
set.ext $ assume a,
⟨assume ⟨⟨a', ha'⟩, in_s, h_eq⟩, h_eq ▸ ⟨ha', in_s⟩,
assume ⟨ha, in_s⟩, ⟨⟨a, ha⟩, in_s, rfl⟩⟩
lemma range_coe {s : set α} :
range (coe : s → α) = s :=
by { rw ← set.image_univ, simp [-set.image_univ, coe_image] }
/-- A variant of `range_coe`. Try to use `range_coe` if possible.
This version is useful when defining a new type that is defined as the subtype of something.
In that case, the coercion doesn't fire anymore. -/
lemma range_val {s : set α} :
range (subtype.val : s → α) = s :=
range_coe
/-- We make this the simp lemma instead of `range_coe`. The reason is that if we write
for `s : set α` the function `coe : s → α`, then the inferred implicit arguments of `coe` are
`coe α (λ x, x ∈ s)`. -/
@[simp] lemma range_coe_subtype {p : α → Prop} :
range (coe : subtype p → α) = {x | p x} :=
range_coe
lemma range_val_subtype {p : α → Prop} :
range (subtype.val : subtype p → α) = {x | p x} :=
range_coe
theorem coe_image_subset (s : set α) (t : set s) : t.image coe ⊆ s :=
λ x ⟨y, yt, yvaleq⟩, by rw ←yvaleq; exact y.property
theorem coe_image_univ (s : set α) : (coe : s → α) '' set.univ = s :=
image_univ.trans range_coe
@[simp] theorem image_preimage_coe (s t : set α) :
(coe : s → α) '' (coe ⁻¹' t) = t ∩ s :=
image_preimage_eq_inter_range.trans $ congr_arg _ range_coe
theorem image_preimage_val (s t : set α) :
(subtype.val : s → α) '' (subtype.val ⁻¹' t) = t ∩ s :=
image_preimage_coe s t
theorem preimage_coe_eq_preimage_coe_iff {s t u : set α} :
((coe : s → α) ⁻¹' t = coe ⁻¹' u) ↔ t ∩ s = u ∩ s :=
begin
rw [←image_preimage_coe, ←image_preimage_coe],
split, { intro h, rw h },
intro h, exact coe_injective.image_injective h
end
theorem preimage_val_eq_preimage_val_iff (s t u : set α) :
((subtype.val : s → α) ⁻¹' t = subtype.val ⁻¹' u) ↔ (t ∩ s = u ∩ s) :=
preimage_coe_eq_preimage_coe_iff
lemma exists_set_subtype {t : set α} (p : set α → Prop) :
(∃(s : set t), p (coe '' s)) ↔ ∃(s : set α), s ⊆ t ∧ p s :=
begin
split,
{ rintro ⟨s, hs⟩, refine ⟨coe '' s, _, hs⟩,
convert image_subset_range _ _, rw [range_coe] },
rintro ⟨s, hs₁, hs₂⟩, refine ⟨coe ⁻¹' s, _⟩,
rw [image_preimage_eq_of_subset], exact hs₂, rw [range_coe], exact hs₁
end
end subtype
namespace set
/-! ### Lemmas about cartesian product of sets -/
section prod
variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variables {s s₁ s₂ : set α} {t t₁ t₂ : set β}
/-- The cartesian product `prod s t` is the set of `(a, b)`
such that `a ∈ s` and `b ∈ t`. -/
protected def prod (s : set α) (t : set β) : set (α × β) :=
{p | p.1 ∈ s ∧ p.2 ∈ t}
lemma prod_eq (s : set α) (t : set β) : set.prod s t = prod.fst ⁻¹' s ∩ prod.snd ⁻¹' t := rfl
theorem mem_prod_eq {p : α × β} : p ∈ set.prod s t = (p.1 ∈ s ∧ p.2 ∈ t) := rfl
@[simp] theorem mem_prod {p : α × β} : p ∈ set.prod s t ↔ p.1 ∈ s ∧ p.2 ∈ t := iff.rfl
lemma mk_mem_prod {a : α} {b : β} (a_in : a ∈ s) (b_in : b ∈ t) : (a, b) ∈ set.prod s t := ⟨a_in, b_in⟩
lemma prod_subset_iff {P : set (α × β)} :
(set.prod s t ⊆ P) ↔ ∀ (x ∈ s) (y ∈ t), (x, y) ∈ P :=
⟨λ h _ xin _ yin, h (mk_mem_prod xin yin),
λ h _ pin, by { cases mem_prod.1 pin with hs ht, simpa using h _ hs _ ht }⟩
@[simp] theorem prod_empty : set.prod s ∅ = (∅ : set (α × β)) :=
ext $ by simp [set.prod]
@[simp] theorem empty_prod : set.prod ∅ t = (∅ : set (α × β)) :=
ext $ by simp [set.prod]
theorem insert_prod {a : α} {s : set α} {t : set β} :
set.prod (insert a s) t = (prod.mk a '' t) ∪ set.prod s t :=
ext begin simp [set.prod, image, iff_def, or_imp_distrib] {contextual := tt}; cc end
theorem prod_insert {b : β} {s : set α} {t : set β} :
set.prod s (insert b t) = ((λa, (a, b)) '' s) ∪ set.prod s t :=
ext begin simp [set.prod, image, iff_def, or_imp_distrib] {contextual := tt}; cc end
theorem prod_preimage_eq {f : γ → α} {g : δ → β} :
set.prod (preimage f s) (preimage g t) = preimage (λp, (f p.1, g p.2)) (set.prod s t) := rfl
theorem prod_mono {s₁ s₂ : set α} {t₁ t₂ : set β} (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) :
set.prod s₁ t₁ ⊆ set.prod s₂ t₂ :=
assume x ⟨h₁, h₂⟩, ⟨hs h₁, ht h₂⟩
theorem prod_inter_prod : set.prod s₁ t₁ ∩ set.prod s₂ t₂ = set.prod (s₁ ∩ s₂) (t₁ ∩ t₂) :=
subset.antisymm
(assume ⟨a, b⟩ ⟨⟨ha₁, hb₁⟩, ⟨ha₂, hb₂⟩⟩, ⟨⟨ha₁, ha₂⟩, ⟨hb₁, hb₂⟩⟩)
(subset_inter
(prod_mono (inter_subset_left _ _) (inter_subset_left _ _))
(prod_mono (inter_subset_right _ _) (inter_subset_right _ _)))
theorem image_swap_prod : (λp:β×α, (p.2, p.1)) '' set.prod t s = set.prod s t :=
ext $ assume ⟨a, b⟩, by simp [mem_image_eq, set.prod, and_comm]; exact
⟨ assume ⟨b', a', ⟨h_a, h_b⟩, h⟩, by subst a'; subst b'; assumption,
assume h, ⟨b, a, ⟨rfl, rfl⟩, h⟩⟩
theorem image_swap_eq_preimage_swap : image (@prod.swap α β) = preimage prod.swap :=
image_eq_preimage_of_inverse prod.swap_left_inverse prod.swap_right_inverse
theorem prod_image_image_eq {m₁ : α → γ} {m₂ : β → δ} :
set.prod (image m₁ s) (image m₂ t) = image (λp:α×β, (m₁ p.1, m₂ p.2)) (set.prod s t) :=
ext $ by simp [-exists_and_distrib_right, exists_and_distrib_right.symm, and.left_comm, and.assoc, and.comm]
theorem prod_range_range_eq {α β γ δ} {m₁ : α → γ} {m₂ : β → δ} :
set.prod (range m₁) (range m₂) = range (λp:α×β, (m₁ p.1, m₂ p.2)) :=
ext $ by simp [range]
theorem prod_range_univ_eq {α β γ} {m₁ : α → γ} :
set.prod (range m₁) (univ : set β) = range (λp:α×β, (m₁ p.1, p.2)) :=
ext $ by simp [range]
theorem prod_univ_range_eq {α β δ} {m₂ : β → δ} :
set.prod (univ : set α) (range m₂) = range (λp:α×β, (p.1, m₂ p.2)) :=
ext $ by simp [range]
@[simp] theorem prod_singleton_singleton {a : α} {b : β} :
set.prod {a} {b} = ({(a, b)} : set (α×β)) :=
ext $ by simp [set.prod]
theorem nonempty.prod : s.nonempty → t.nonempty → (s.prod t).nonempty
| ⟨x, hx⟩ ⟨y, hy⟩ := ⟨(x, y), ⟨hx, hy⟩⟩
theorem nonempty.fst : (s.prod t).nonempty → s.nonempty
| ⟨p, hp⟩ := ⟨p.1, hp.1⟩
theorem nonempty.snd : (s.prod t).nonempty → t.nonempty
| ⟨p, hp⟩ := ⟨p.2, hp.2⟩
theorem prod_nonempty_iff : (s.prod t).nonempty ↔ s.nonempty ∧ t.nonempty :=
⟨λ h, ⟨h.fst, h.snd⟩, λ h, nonempty.prod h.1 h.2⟩
theorem prod_eq_empty_iff {s : set α} {t : set β} :
set.prod s t = ∅ ↔ (s = ∅ ∨ t = ∅) :=
by simp only [not_nonempty_iff_eq_empty.symm, prod_nonempty_iff, classical.not_and_distrib]
@[simp] theorem prod_mk_mem_set_prod_eq {a : α} {b : β} {s : set α} {t : set β} :
(a, b) ∈ set.prod s t = (a ∈ s ∧ b ∈ t) := rfl
@[simp] theorem univ_prod_univ : set.prod (@univ α) (@univ β) = univ :=
ext $ assume ⟨a, b⟩, by simp
lemma prod_sub_preimage_iff {W : set γ} {f : α × β → γ} :
set.prod s t ⊆ f ⁻¹' W ↔ ∀ a b, a ∈ s → b ∈ t → f (a, b) ∈ W :=
by simp [subset_def]
lemma fst_image_prod_subset (s : set α) (t : set β) :
prod.fst '' (set.prod s t) ⊆ s :=
λ _ h, let ⟨_, ⟨h₂, _⟩, h₁⟩ := (set.mem_image _ _ _).1 h in h₁ ▸ h₂
lemma prod_subset_preimage_fst (s : set α) (t : set β) :
set.prod s t ⊆ prod.fst ⁻¹' s :=
image_subset_iff.1 (fst_image_prod_subset s t)
lemma fst_image_prod (s : set β) {t : set α} (ht : t.nonempty) :
prod.fst '' (set.prod s t) = s :=
set.subset.antisymm (fst_image_prod_subset _ _)
$ λ y y_in, let ⟨x, x_in⟩ := ht in
⟨(y, x), ⟨y_in, x_in⟩, rfl⟩
lemma snd_image_prod_subset (s : set α) (t : set β) :
prod.snd '' (set.prod s t) ⊆ t :=
λ _ h, let ⟨_, ⟨_, h₂⟩, h₁⟩ := (set.mem_image _ _ _).1 h in h₁ ▸ h₂
lemma prod_subset_preimage_snd (s : set α) (t : set β) :
set.prod s t ⊆ prod.snd ⁻¹' t :=
image_subset_iff.1 (snd_image_prod_subset s t)
lemma snd_image_prod {s : set α} (hs : s.nonempty) (t : set β) :
prod.snd '' (set.prod s t) = t :=
set.subset.antisymm (snd_image_prod_subset _ _)
$ λ y y_in, let ⟨x, x_in⟩ := hs in
⟨(x, y), ⟨x_in, y_in⟩, rfl⟩
/-- A product set is included in a product set if and only factors are included, or a factor of the
first set is empty. -/
lemma prod_subset_prod_iff :
(set.prod s t ⊆ set.prod s₁ t₁) ↔ (s ⊆ s₁ ∧ t ⊆ t₁) ∨ (s = ∅) ∨ (t = ∅) :=
begin
classical,
cases (set.prod s t).eq_empty_or_nonempty with h h,
{ simp [h, prod_eq_empty_iff.1 h] },
{ have st : s.nonempty ∧ t.nonempty, by rwa [prod_nonempty_iff] at h,
split,
{ assume H : set.prod s t ⊆ set.prod s₁ t₁,
have h' : s₁.nonempty ∧ t₁.nonempty := prod_nonempty_iff.1 (h.mono H),
refine or.inl ⟨_, _⟩,
show s ⊆ s₁,
{ have := image_subset (prod.fst : α × β → α) H,
rwa [fst_image_prod _ st.2, fst_image_prod _ h'.2] at this },
show t ⊆ t₁,
{ have := image_subset (prod.snd : α × β → β) H,
rwa [snd_image_prod st.1, snd_image_prod h'.1] at this } },
{ assume H,
simp only [st.1.ne_empty, st.2.ne_empty, or_false] at H,
exact prod_mono H.1 H.2 } }
end
end prod
/-! ### Lemmas about set-indexed products of sets -/
section pi
variables {α : Type*} {π : α → Type*}
/-- Given an index set `i` and a family of sets `s : Πa, set (π a)`, `pi i s`
is the set of dependent functions `f : Πa, π a` such that `f a` belongs to `π a`
whenever `a ∈ i`. -/
def pi (i : set α) (s : Πa, set (π a)) : set (Πa, π a) := { f | ∀a∈i, f a ∈ s a }
@[simp] lemma pi_empty_index (s : Πa, set (π a)) : pi ∅ s = univ := by ext; simp [pi]
@[simp] lemma pi_insert_index (a : α) (i : set α) (s : Πa, set (π a)) :
pi (insert a i) s = ((λf, f a) ⁻¹' s a) ∩ pi i s :=
by ext; simp [pi, or_imp_distrib, forall_and_distrib]
@[simp] lemma pi_singleton_index (a : α) (s : Πa, set (π a)) :
pi {a} s = ((λf:(Πa, π a), f a) ⁻¹' s a) :=
by ext; simp [pi]
lemma pi_if {p : α → Prop} [h : decidable_pred p] (i : set α) (s t : Πa, set (π a)) :
pi i (λa, if p a then s a else t a) = pi {a ∈ i | p a} s ∩ pi {a ∈ i | ¬ p a} t :=
begin
ext f,
split,
{ assume h, split; { rintros a ⟨hai, hpa⟩, simpa [*] using h a } },
{ rintros ⟨hs, ht⟩ a hai,
by_cases p a; simp [*, pi] at * }
end
end pi
/-! ### Lemmas about `inclusion`, the injection of subtypes induced by `⊆` -/
section inclusion
variable {α : Type*}
/-- `inclusion` is the "identity" function between two subsets `s` and `t`, where `s ⊆ t` -/
def inclusion {s t : set α} (h : s ⊆ t) : s → t :=
λ x : s, (⟨x, h x.2⟩ : t)
@[simp] lemma inclusion_self {s : set α} (x : s) :
inclusion (set.subset.refl _) x = x := by cases x; refl
@[simp] lemma inclusion_inclusion {s t u : set α} (hst : s ⊆ t) (htu : t ⊆ u)
(x : s) : inclusion htu (inclusion hst x) = inclusion (set.subset.trans hst htu) x :=
by cases x; refl
@[simp] lemma coe_inclusion {s t : set α} (h : s ⊆ t) (x : s) :
(inclusion h x : α) = (x : α) := rfl
lemma inclusion_injective {s t : set α} (h : s ⊆ t) :
function.injective (inclusion h)
| ⟨_, _⟩ ⟨_, _⟩ := subtype.ext_iff_val.2 ∘ subtype.ext_iff_val.1
lemma range_inclusion {s t : set α} (h : s ⊆ t) : range (inclusion h) = {x : t | (x:α) ∈ s} :=
ext $ λ ⟨x, hx⟩ , by simp [inclusion]
end inclusion
/-! ### Injectivity and surjectivity lemmas for image and preimage -/
section image_preimage
variables {α : Type u} {β : Type v} {f : α → β}
@[simp]
lemma preimage_injective : injective (preimage f) ↔ surjective f :=
begin
refine ⟨λ h y, _, surjective.preimage_injective⟩,
obtain ⟨x, hx⟩ : (f ⁻¹' {y}).nonempty,
{ rw [h.nonempty_apply_iff preimage_empty], apply singleton_nonempty },
exact ⟨x, hx⟩
end
@[simp]
lemma preimage_surjective : surjective (preimage f) ↔ injective f :=
begin
refine ⟨λ h x x' hx, _, injective.preimage_surjective⟩,
cases h {x} with s hs, have := mem_singleton x,
rwa [← hs, mem_preimage, hx, ← mem_preimage, hs, mem_singleton_iff, eq_comm] at this
end
@[simp] lemma image_surjective : surjective (image f) ↔ surjective f :=
begin
refine ⟨λ h y, _, surjective.image_surjective⟩,
cases h {y} with s hs,
have := mem_singleton y, rw [← hs] at this, rcases this with ⟨x, h1x, h2x⟩,
exact ⟨x, h2x⟩
end
@[simp] lemma image_injective : injective (image f) ↔ injective f :=
begin
refine ⟨λ h x x' hx, _, injective.image_injective⟩,
rw [← singleton_eq_singleton_iff], apply h,
rw [image_singleton, image_singleton, hx]
end
end image_preimage
/-! ### Lemmas about images of binary and ternary functions -/
section n_ary_image
variables {α β γ δ ε : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variables {s s' : set α} {t t' : set β} {u u' : set γ} {a a' : α} {b b' : β} {c c' : γ} {d d' : δ}
/-- The image of a binary function `f : α → β → γ` as a function `set α → set β → set γ`.
Mathematically this should be thought of as the image of the corresponding function `α × β → γ`.
-/
def image2 (f : α → β → γ) (s : set α) (t : set β) : set γ :=
{c | ∃ a b, a ∈ s ∧ b ∈ t ∧ f a b = c }
lemma mem_image2_eq : c ∈ image2 f s t = ∃ a b, a ∈ s ∧ b ∈ t ∧ f a b = c := rfl
@[simp] lemma mem_image2 : c ∈ image2 f s t ↔ ∃ a b, a ∈ s ∧ b ∈ t ∧ f a b = c := iff.rfl
lemma mem_image2_of_mem (h1 : a ∈ s) (h2 : b ∈ t) : f a b ∈ image2 f s t :=
⟨a, b, h1, h2, rfl⟩
lemma mem_image2_iff (hf : injective2 f) : f a b ∈ image2 f s t ↔ a ∈ s ∧ b ∈ t :=
⟨ by { rintro ⟨a', b', ha', hb', h⟩, rcases hf h with ⟨rfl, rfl⟩, exact ⟨ha', hb'⟩ },
λ ⟨ha, hb⟩, mem_image2_of_mem ha hb⟩
/-- image2 is monotone with respect to `⊆`. -/
lemma image2_subset (hs : s ⊆ s') (ht : t ⊆ t') : image2 f s t ⊆ image2 f s' t' :=
by { rintro _ ⟨a, b, ha, hb, rfl⟩, exact mem_image2_of_mem (hs ha) (ht hb) }
lemma image2_union_left : image2 f (s ∪ s') t = image2 f s t ∪ image2 f s' t :=
begin
ext c, split,
{ rintros ⟨a, b, h1a|h2a, hb, rfl⟩;[left, right]; exact ⟨_, _, ‹_›, ‹_›, rfl⟩ },
{ rintro (⟨_, _, _, _, rfl⟩|⟨_, _, _, _, rfl⟩); refine ⟨_, _, _, ‹_›, rfl⟩; simp [mem_union, *] }
end
lemma image2_union_right : image2 f s (t ∪ t') = image2 f s t ∪ image2 f s t' :=
begin
ext c, split,
{ rintros ⟨a, b, ha, h1b|h2b, rfl⟩;[left, right]; exact ⟨_, _, ‹_›, ‹_›, rfl⟩ },
{ rintro (⟨_, _, _, _, rfl⟩|⟨_, _, _, _, rfl⟩); refine ⟨_, _, ‹_›, _, rfl⟩; simp [mem_union, *] }
end
@[simp] lemma image2_empty_left : image2 f ∅ t = ∅ := ext $ by simp
@[simp] lemma image2_empty_right : image2 f s ∅ = ∅ := ext $ by simp
lemma image2_inter_subset_left : image2 f (s ∩ s') t ⊆ image2 f s t ∩ image2 f s' t :=
by { rintro _ ⟨a, b, ⟨h1a, h2a⟩, hb, rfl⟩, split; exact ⟨_, _, ‹_›, ‹_›, rfl⟩ }
lemma image2_inter_subset_right : image2 f s (t ∩ t') ⊆ image2 f s t ∩ image2 f s t' :=
by { rintro _ ⟨a, b, ha, ⟨h1b, h2b⟩, rfl⟩, split; exact ⟨_, _, ‹_›, ‹_›, rfl⟩ }
@[simp] lemma image2_singleton : image2 f {a} {b} = {f a b} :=
ext $ λ x, by simp [eq_comm]
lemma image2_singleton_left : image2 f {a} t = f a '' t :=
ext $ λ x, by simp
lemma image2_singleton_right : image2 f s {b} = (λ a, f a b) '' s :=
ext $ λ x, by simp
@[congr] lemma image2_congr (h : ∀ (a ∈ s) (b ∈ t), f a b = f' a b) :
image2 f s t = image2 f' s t :=
by { ext, split; rintro ⟨a, b, ha, hb, rfl⟩; refine ⟨a, b, ha, hb, by rw h a ha b hb⟩ }
/-- A common special case of `image2_congr` -/
lemma image2_congr' (h : ∀ a b, f a b = f' a b) : image2 f s t = image2 f' s t :=
image2_congr (λ a _ b _, h a b)
/-- The image of a ternary function `f : α → β → γ → δ` as a function
`set α → set β → set γ → set δ`. Mathematically this should be thought of as the image of the
corresponding function `α × β × γ → δ`.
-/
def image3 (g : α → β → γ → δ) (s : set α) (t : set β) (u : set γ) : set δ :=
{d | ∃ a b c, a ∈ s ∧ b ∈ t ∧ c ∈ u ∧ g a b c = d }
@[simp] lemma mem_image3 : d ∈ image3 g s t u ↔ ∃ a b c, a ∈ s ∧ b ∈ t ∧ c ∈ u ∧ g a b c = d :=
iff.rfl
@[congr] lemma image3_congr (h : ∀ (a ∈ s) (b ∈ t) (c ∈ u), g a b c = g' a b c) :
image3 g s t u = image3 g' s t u :=
by { ext x,
split; rintro ⟨a, b, c, ha, hb, hc, rfl⟩; refine ⟨a, b, c, ha, hb, hc, by rw h a ha b hb c hc⟩ }
/-- A common special case of `image3_congr` -/
lemma image3_congr' (h : ∀ a b c, g a b c = g' a b c) : image3 g s t u = image3 g' s t u :=
image3_congr (λ a _ b _ c _, h a b c)
lemma image2_image2_left (f : δ → γ → ε) (g : α → β → δ) :
image2 f (image2 g s t) u = image3 (λ a b c, f (g a b) c) s t u :=
begin
ext, split,
{ rintro ⟨_, c, ⟨a, b, ha, hb, rfl⟩, hc, rfl⟩, refine ⟨a, b, c, ha, hb, hc, rfl⟩ },
{ rintro ⟨a, b, c, ha, hb, hc, rfl⟩, refine ⟨_, c, ⟨a, b, ha, hb, rfl⟩, hc, rfl⟩ }
end
lemma image2_image2_right (f : α → δ → ε) (g : β → γ → δ) :
image2 f s (image2 g t u) = image3 (λ a b c, f a (g b c)) s t u :=
begin
ext, split,
{ rintro ⟨a, _, ha, ⟨b, c, hb, hc, rfl⟩, rfl⟩, refine ⟨a, b, c, ha, hb, hc, rfl⟩ },
{ rintro ⟨a, b, c, ha, hb, hc, rfl⟩, refine ⟨a, _, ha, ⟨b, c, hb, hc, rfl⟩, rfl⟩ }
end
lemma image_image2 (f : α → β → γ) (g : γ → δ) :
g '' image2 f s t = image2 (λ a b, g (f a b)) s t :=
begin
ext, split,
{ rintro ⟨_, ⟨a, b, ha, hb, rfl⟩, rfl⟩, refine ⟨a, b, ha, hb, rfl⟩ },
{ rintro ⟨a, b, ha, hb, rfl⟩, refine ⟨_, ⟨a, b, ha, hb, rfl⟩, rfl⟩ }
end
lemma image2_image_left (f : γ → β → δ) (g : α → γ) :
image2 f (g '' s) t = image2 (λ a b, f (g a) b) s t :=
begin
ext, split,
{ rintro ⟨_, b, ⟨a, ha, rfl⟩, hb, rfl⟩, refine ⟨a, b, ha, hb, rfl⟩ },
{ rintro ⟨a, b, ha, hb, rfl⟩, refine ⟨_, b, ⟨a, ha, rfl⟩, hb, rfl⟩ }
end
lemma image2_image_right (f : α → γ → δ) (g : β → γ) :
image2 f s (g '' t) = image2 (λ a b, f a (g b)) s t :=
begin
ext, split,
{ rintro ⟨a, _, ha, ⟨b, hb, rfl⟩, rfl⟩, refine ⟨a, b, ha, hb, rfl⟩ },
{ rintro ⟨a, b, ha, hb, rfl⟩, refine ⟨a, _, ha, ⟨b, hb, rfl⟩, rfl⟩ }
end
lemma image2_swap (f : α → β → γ) (s : set α) (t : set β) :
image2 f s t = image2 (λ a b, f b a) t s :=
by { ext, split; rintro ⟨a, b, ha, hb, rfl⟩; refine ⟨b, a, hb, ha, rfl⟩ }
@[simp] lemma image2_left (h : t.nonempty) : image2 (λ x y, x) s t = s :=
by simp [nonempty_def.mp h, ext_iff]
@[simp] lemma image2_right (h : s.nonempty) : image2 (λ x y, y) s t = t :=
by simp [nonempty_def.mp h, ext_iff]
@[simp] lemma image_prod (f : α → β → γ) : (λ x : α × β, f x.1 x.2) '' s.prod t = image2 f s t :=
set.ext $ λ a,
⟨ by { rintros ⟨_, _, rfl⟩, exact ⟨_, _, (mem_prod.mp ‹_›).1, (mem_prod.mp ‹_›).2, rfl⟩ },
by { rintros ⟨_, _, _, _, rfl⟩, exact ⟨(_, _), mem_prod.mpr ⟨‹_›, ‹_›⟩, rfl⟩ }⟩
lemma nonempty.image2 (hs : s.nonempty) (ht : t.nonempty) : (image2 f s t).nonempty :=
by { cases hs with a ha, cases ht with b hb, exact ⟨f a b, ⟨a, b, ha, hb, rfl⟩⟩ }
end n_ary_image
end set
namespace subsingleton
variables {α : Type*} [subsingleton α]
lemma eq_univ_of_nonempty {s : set α} : s.nonempty → s = univ :=
λ ⟨x, hx⟩, eq_univ_of_forall $ λ y, subsingleton.elim x y ▸ hx
@[elab_as_eliminator]
lemma set_cases {p : set α → Prop} (h0 : p ∅) (h1 : p univ) (s) : p s :=
s.eq_empty_or_nonempty.elim (λ h, h.symm ▸ h0) $ λ h, (eq_univ_of_nonempty h).symm ▸ h1
end subsingleton
|
b96e175a48bb966408e02cc7db851f61a0e505cc | 6fca17f8d5025f89be1b2d9d15c9e0c4b4900cbf | /src/game/world2/level3.lean | 09c1383141373f8bf3bd9dc0579c061219c477cd | [
"Apache-2.0"
] | permissive | arolihas/natural_number_game | 4f0c93feefec93b8824b2b96adff8b702b8b43ce | 8e4f7b4b42888a3b77429f90cce16292bd288138 | refs/heads/master | 1,621,872,426,808 | 1,586,270,467,000 | 1,586,270,467,000 | 253,648,466 | 0 | 0 | null | 1,586,219,694,000 | 1,586,219,694,000 | null | UTF-8 | Lean | false | false | 1,290 | lean | import mynat.definition -- hide
import mynat.add -- hide
import game.world2.level2 -- hide
namespace mynat -- hide
/-
# Addition World
## Level 3: `succ_add`
Oh no! On the way to `add_comm`, a wild `succ_add` appears. `succ_add`
is the proof that `succ(a) + b = succ(a + b)` for `a` and `b` in your
natural number type. We need to prove this now, because we will need
to use this result in our proof that `a + b = b + a` in the next level.
NB: think about why computer scientists called this result `succ_add` .
There is a logic to all the names.
Note that if you want to be more precise about exactly where you want
to rewrite something like `add_succ` (the proof you already have),
you can do things like `rw add_succ (succ a)` or
`rw add_succ (succ a) d`, telling Lean explicitly what to use for
the input variables for the function `add_succ`. Indeed, `add_succ`
is a function -- it takes as input two variables `a` and `b` and outputs a proof
that `a + succ(b) = succ(a + b)`. The tactic `rw add_succ` just says to Lean "guess
what the variables are".
-/
/- Lemma
For all natural numbers $a, b$, we have
$$ \operatorname{succ}(a) + b = \operatorname{succ}(a + b). $$
-/
lemma succ_add (a b : mynat) : succ a + b = succ (a + b) :=
begin [nat_num_game]
end
end mynat -- hide
|
86f615407a37d3a8223de746697c998eb84d53a0 | ce6917c5bacabee346655160b74a307b4a5ab620 | /src/ch6/ex0308.lean | 95a86e4cd174e505f5d0cb8d2479ab90ada583b3 | [] | no_license | Ailrun/Theorem_Proving_in_Lean | ae6a23f3c54d62d401314d6a771e8ff8b4132db2 | 2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68 | refs/heads/master | 1,609,838,270,467 | 1,586,846,743,000 | 1,586,846,743,000 | 240,967,761 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 26 | lean | export nat (succ add sub)
|
fce0ac415cdd463b8f117e503d23326a3cea4fa1 | 35677d2df3f081738fa6b08138e03ee36bc33cad | /src/tactic/basic.lean | d58af9230685e9ec29c3a008e7d0950d2a1ac67a | [
"Apache-2.0"
] | permissive | gebner/mathlib | eab0150cc4f79ec45d2016a8c21750244a2e7ff0 | cc6a6edc397c55118df62831e23bfbd6e6c6b4ab | refs/heads/master | 1,625,574,853,976 | 1,586,712,827,000 | 1,586,712,827,000 | 99,101,412 | 1 | 0 | Apache-2.0 | 1,586,716,389,000 | 1,501,667,958,000 | Lean | UTF-8 | Lean | false | false | 613 | lean | import
tactic.alias
tactic.cache
tactic.clear
tactic.converter.interactive
tactic.converter.apply_congr
tactic.core
tactic.ext
tactic.elide
tactic.explode
tactic.show_term
tactic.find
tactic.generalize_proofs
tactic.interactive
tactic.suggest
tactic.lift
tactic.localized
tactic.mk_iff_of_inductive_prop
tactic.push_neg
tactic.rcases
tactic.rename
tactic.replacer
tactic.rename_var
tactic.restate_axiom
tactic.rewrite
tactic.lint
tactic.simp_rw
tactic.simpa
tactic.simps
tactic.split_ifs
tactic.squeeze
tactic.trunc_cases
tactic.where
tactic.hint
|
59f3b89d25f9dd760361944fdfdefe8b7b7b362c | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/algebraic_geometry/elliptic_curve/weierstrass.lean | ea856a6ac27ac54aeca61960909f21a62f2eb854 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 33,547 | lean | /-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import algebra.cubic_discriminant
import ring_theory.norm
import tactic.linear_combination
/-!
# Weierstrass equations of elliptic curves
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file defines the structure of an elliptic curve as a nonsingular Weierstrass curve given by a
Weierstrass equation, which is mathematically accurate in many cases but also good for computation.
## Mathematical background
Let `S` be a scheme. The actual category of elliptic curves over `S` is a large category, whose
objects are schemes `E` equipped with a map `E → S`, a section `S → E`, and some axioms (the map
is smooth and proper and the fibres are geometrically-connected one-dimensional group varieties). In
the special case where `S` is the spectrum of some commutative ring `R` whose Picard group is zero
(this includes all fields, all PIDs, and many other commutative rings) it can be shown (using a lot
of algebro-geometric machinery) that every elliptic curve `E` is a projective plane cubic isomorphic
to a Weierstrass curve given by the equation $Y^2 + a_1XY + a_3Y = X^3 + a_2X^2 + a_4X + a_6$ for
some $a_i$ in `R`, and such that a certain quantity called the discriminant of `E` is a unit in `R`.
If `R` is a field, this quantity divides the discriminant of a cubic polynomial whose roots over a
splitting field of `R` are precisely the $X$-coordinates of the non-zero 2-torsion points of `E`.
## Main definitions
* `weierstrass_curve`: a Weierstrass curve over a commutative ring.
* `weierstrass_curve.Δ`: the discriminant of a Weierstrass curve.
* `weierstrass_curve.variable_change`: the Weierstrass curve induced by a change of variables.
* `weierstrass_curve.base_change`: the Weierstrass curve base changed over an algebra.
* `weierstrass_curve.two_torsion_polynomial`: the 2-torsion polynomial of a Weierstrass curve.
* `weierstrass_curve.polynomial`: the polynomial associated to a Weierstrass curve.
* `weierstrass_curve.equation`: the Weirstrass equation of a Weierstrass curve.
* `weierstrass_curve.nonsingular`: the nonsingular condition at a point on a Weierstrass curve.
* `weierstrass_curve.coordinate_ring`: the coordinate ring of a Weierstrass curve.
* `weierstrass_curve.function_field`: the function field of a Weierstrass curve.
* `weierstrass_curve.coordinate_ring.basis`: the power basis of the coordinate ring over `R[X]`.
* `elliptic_curve`: an elliptic curve over a commutative ring.
* `elliptic_curve.j`: the j-invariant of an elliptic curve.
## Main statements
* `weierstrass_curve.two_torsion_polynomial_disc`: the discriminant of a Weierstrass curve is a
constant factor of the cubic discriminant of its 2-torsion polynomial.
* `weierstrass_curve.nonsingular_of_Δ_ne_zero`: a Weierstrass curve is nonsingular at every point
if its discriminant is non-zero.
* `weierstrass_curve.coordinate_ring.is_domain`: the coordinate ring of a Weierstrass curve is
an integral domain.
* `weierstrass_curve.coordinate_ring.degree_norm_smul_basis`: the degree of the norm of an element
in the coordinate ring in terms of the power basis.
* `elliptic_curve.nonsingular`: an elliptic curve is nonsingular at every point.
* `elliptic_curve.variable_change_j`: the j-invariant of an elliptic curve is invariant under an
admissible linear change of variables.
## Implementation notes
The definition of elliptic curves in this file makes sense for all commutative rings `R`, but it
only gives a type which can be beefed up to a category which is equivalent to the category of
elliptic curves over the spectrum $\mathrm{Spec}(R)$ of `R` in the case that `R` has trivial Picard
group $\mathrm{Pic}(R)$ or, slightly more generally, when its 12-torsion is trivial. The issue is
that for a general ring `R`, there might be elliptic curves over $\mathrm{Spec}(R)$ in the sense of
algebraic geometry which are not globally defined by a cubic equation valid over the entire base.
## References
* [N Katz and B Mazur, *Arithmetic Moduli of Elliptic Curves*][katz_mazur]
* [P Deligne, *Courbes Elliptiques: Formulaire (d'après J. Tate)*][deligne_formulaire]
* [J Silverman, *The Arithmetic of Elliptic Curves*][silverman2009]
## Tags
elliptic curve, weierstrass equation, j invariant
-/
private meta def map_simp : tactic unit :=
`[simp only [map_one, map_bit0, map_bit1, map_neg, map_add, map_sub, map_mul, map_pow]]
private meta def eval_simp : tactic unit :=
`[simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, eval_pow]]
private meta def C_simp : tactic unit := `[simp only [C_0, C_1, C_neg, C_add, C_sub, C_mul, C_pow]]
universes u v w
variable {R : Type u}
/-! ## Weierstrass curves -/
/-- A Weierstrass curve $Y^2 + a_1XY + a_3Y = X^3 + a_2X^2 + a_4X + a_6$ with parameters $a_i$. -/
@[ext] structure weierstrass_curve (R : Type u) := (a₁ a₂ a₃ a₄ a₆ : R)
instance [inhabited R] : inhabited $ weierstrass_curve R :=
⟨⟨default, default, default, default, default⟩⟩
namespace weierstrass_curve
variables [comm_ring R] (W : weierstrass_curve R)
section quantity
/-! ### Standard quantities -/
/-- The `b₂` coefficient of a Weierstrass curve. -/
@[simp] def b₂ : R := W.a₁ ^ 2 + 4 * W.a₂
/-- The `b₄` coefficient of a Weierstrass curve. -/
@[simp] def b₄ : R := 2 * W.a₄ + W.a₁ * W.a₃
/-- The `b₆` coefficient of a Weierstrass curve. -/
@[simp] def b₆ : R := W.a₃ ^ 2 + 4 * W.a₆
/-- The `b₈` coefficient of a Weierstrass curve. -/
@[simp] def b₈ : R :=
W.a₁ ^ 2 * W.a₆ + 4 * W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2
lemma b_relation : 4 * W.b₈ = W.b₂ * W.b₆ - W.b₄ ^ 2 := by { simp only [b₂, b₄, b₆, b₈], ring1 }
/-- The `c₄` coefficient of a Weierstrass curve. -/
@[simp] def c₄ : R := W.b₂ ^ 2 - 24 * W.b₄
/-- The `c₆` coefficient of a Weierstrass curve. -/
@[simp] def c₆ : R := -W.b₂ ^ 3 + 36 * W.b₂ * W.b₄ - 216 * W.b₆
/-- The discriminant `Δ` of a Weierstrass curve. If `R` is a field, then this polynomial vanishes
if and only if the cubic curve cut out by this equation is singular. Sometimes only defined up to
sign in the literature; we choose the sign used by the LMFDB. For more discussion, see
[the LMFDB page on discriminants](https://www.lmfdb.org/knowledge/show/ec.discriminant). -/
@[simp] def Δ : R := -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 - 27 * W.b₆ ^ 2 + 9 * W.b₂ * W.b₄ * W.b₆
lemma c_relation : 1728 * W.Δ = W.c₄ ^ 3 - W.c₆ ^ 2 :=
by { simp only [b₂, b₄, b₆, b₈, c₄, c₆, Δ], ring1 }
end quantity
section variable_change
/-! ### Variable changes -/
variables (u : Rˣ) (r s t : R)
/-- The Weierstrass curve over `R` induced by an admissible linear change of variables
$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$. -/
@[simps] def variable_change : weierstrass_curve R :=
{ a₁ := ↑u⁻¹ * (W.a₁ + 2 * s),
a₂ := ↑u⁻¹ ^ 2 * (W.a₂ - s * W.a₁ + 3 * r - s ^ 2),
a₃ := ↑u⁻¹ ^ 3 * (W.a₃ + r * W.a₁ + 2 * t),
a₄ := ↑u⁻¹ ^ 4 * (W.a₄ - s * W.a₃ + 2 * r * W.a₂ - (t + r * s) * W.a₁ + 3 * r ^ 2 - 2 * s * t),
a₆ := ↑u⁻¹ ^ 6 * (W.a₆ + r * W.a₄ + r ^ 2 * W.a₂ + r ^ 3 - t * W.a₃ - t ^ 2 - r * t * W.a₁) }
@[simp] lemma variable_change_b₂ : (W.variable_change u r s t).b₂ = ↑u⁻¹ ^ 2 * (W.b₂ + 12 * r) :=
by { simp only [b₂, variable_change_a₁, variable_change_a₂], ring1 }
@[simp] lemma variable_change_b₄ :
(W.variable_change u r s t).b₄ = ↑u⁻¹ ^ 4 * (W.b₄ + r * W.b₂ + 6 * r ^ 2) :=
by { simp only [b₂, b₄, variable_change_a₁, variable_change_a₃, variable_change_a₄], ring1 }
@[simp] lemma variable_change_b₆ :
(W.variable_change u r s t).b₆ = ↑u⁻¹ ^ 6 * (W.b₆ + 2 * r * W.b₄ + r ^ 2 * W.b₂ + 4 * r ^ 3) :=
by { simp only [b₂, b₄, b₆, variable_change_a₃, variable_change_a₆], ring1 }
@[simp] lemma variable_change_b₈ :
(W.variable_change u r s t).b₈
= ↑u⁻¹ ^ 8 * (W.b₈ + 3 * r * W.b₆ + 3 * r ^ 2 * W.b₄ + r ^ 3 * W.b₂ + 3 * r ^ 4) :=
by { simp only [b₂, b₄, b₆, b₈, variable_change_a₁, variable_change_a₂, variable_change_a₃,
variable_change_a₄, variable_change_a₆], ring1 }
@[simp] lemma variable_change_c₄ : (W.variable_change u r s t).c₄ = ↑u⁻¹ ^ 4 * W.c₄ :=
by { simp only [c₄, variable_change_b₂, variable_change_b₄], ring1 }
@[simp] lemma variable_change_c₆ : (W.variable_change u r s t).c₆ = ↑u⁻¹ ^ 6 * W.c₆ :=
by { simp only [c₆, variable_change_b₂, variable_change_b₄, variable_change_b₆], ring1 }
@[simp] lemma variable_change_Δ : (W.variable_change u r s t).Δ = ↑u⁻¹ ^ 12 * W.Δ :=
by { dsimp, ring1 }
end variable_change
variables (A : Type v) [comm_ring A] [algebra R A] (B : Type w) [comm_ring B] [algebra R B]
[algebra A B] [is_scalar_tower R A B]
section base_change
/-! ### Base changes -/
/-- The Weierstrass curve over `R` base changed to `A`. -/
@[simps] def base_change : weierstrass_curve A :=
⟨algebra_map R A W.a₁, algebra_map R A W.a₂, algebra_map R A W.a₃, algebra_map R A W.a₄,
algebra_map R A W.a₆⟩
@[simp] lemma base_change_b₂ : (W.base_change A).b₂ = algebra_map R A W.b₂ :=
by { simp only [b₂, base_change_a₁, base_change_a₂], map_simp }
@[simp] lemma base_change_b₄ : (W.base_change A).b₄ = algebra_map R A W.b₄ :=
by { simp only [b₄, base_change_a₁, base_change_a₃, base_change_a₄], map_simp }
@[simp] lemma base_change_b₆ : (W.base_change A).b₆ = algebra_map R A W.b₆ :=
by { simp only [b₆, base_change_a₃, base_change_a₆], map_simp }
@[simp] lemma base_change_b₈ : (W.base_change A).b₈ = algebra_map R A W.b₈ :=
by { simp only [b₈, base_change_a₁, base_change_a₂, base_change_a₃, base_change_a₄, base_change_a₆],
map_simp }
@[simp] lemma base_change_c₄ : (W.base_change A).c₄ = algebra_map R A W.c₄ :=
by { simp only [c₄, base_change_b₂, base_change_b₄], map_simp }
@[simp] lemma base_change_c₆ : (W.base_change A).c₆ = algebra_map R A W.c₆ :=
by { simp only [c₆, base_change_b₂, base_change_b₄, base_change_b₆], map_simp }
@[simp, nolint simp_nf] lemma base_change_Δ : (W.base_change A).Δ = algebra_map R A W.Δ :=
by { simp only [Δ, base_change_b₂, base_change_b₄, base_change_b₆, base_change_b₈], map_simp }
lemma base_change_self : W.base_change R = W := by ext; refl
lemma base_change_base_change : (W.base_change A).base_change B = W.base_change B :=
by ext; exact (is_scalar_tower.algebra_map_apply R A B _).symm
end base_change
section torsion_polynomial
/-! ### 2-torsion polynomials -/
/-- A cubic polynomial whose discriminant is a multiple of the Weierstrass curve discriminant. If
`W` is an elliptic curve over a field `R` of characteristic different from 2, then its roots over a
splitting field of `R` are precisely the $X$-coordinates of the non-zero 2-torsion points of `W`. -/
def two_torsion_polynomial : cubic R := ⟨4, W.b₂, 2 * W.b₄, W.b₆⟩
lemma two_torsion_polynomial_disc : W.two_torsion_polynomial.disc = 16 * W.Δ :=
by { dsimp [two_torsion_polynomial, cubic.disc], ring1 }
lemma two_torsion_polynomial_disc_is_unit [invertible (2 : R)] :
is_unit W.two_torsion_polynomial.disc ↔ is_unit W.Δ :=
begin
rw [two_torsion_polynomial_disc, is_unit.mul_iff, show (16 : R) = 2 ^ 4, by norm_num1],
exact and_iff_right (is_unit_of_invertible $ 2 ^ 4)
end
lemma two_torsion_polynomial_disc_ne_zero [nontrivial R] [invertible (2 : R)] (hΔ : is_unit W.Δ) :
W.two_torsion_polynomial.disc ≠ 0 :=
(W.two_torsion_polynomial_disc_is_unit.mpr hΔ).ne_zero
end torsion_polynomial
localized "notation (name := outer_variable) `Y` := polynomial.X" in polynomial_polynomial
localized "notation (name := polynomial_polynomial) R`[X][Y]` := polynomial (polynomial R)"
in polynomial_polynomial
section polynomial
/-! ### Weierstrass equations -/
open polynomial
open_locale polynomial polynomial_polynomial
/-- The polynomial $W(X, Y) := Y^2 + a_1XY + a_3Y - (X^3 + a_2X^2 + a_4X + a_6)$ associated to a
Weierstrass curve `W` over `R`. For ease of polynomial manipulation, this is represented as a term
of type `R[X][X]`, where the inner variable represents $X$ and the outer variable represents $Y$.
For clarity, the alternative notations `Y` and `R[X][Y]` are provided in the `polynomial_polynomial`
locale to represent the outer variable and the bivariate polynomial ring `R[X][X]` respectively. -/
protected noncomputable def polynomial : R[X][Y] :=
Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)
lemma polynomial_eq : W.polynomial = cubic.to_poly
⟨0, 1, cubic.to_poly ⟨0, 0, W.a₁, W.a₃⟩, cubic.to_poly ⟨-1, -W.a₂, -W.a₄, -W.a₆⟩⟩ :=
by { simp only [weierstrass_curve.polynomial, cubic.to_poly], C_simp, ring1 }
lemma polynomial_ne_zero [nontrivial R] : W.polynomial ≠ 0 :=
by { rw [polynomial_eq], exact cubic.ne_zero_of_b_ne_zero one_ne_zero }
@[simp] lemma degree_polynomial [nontrivial R] : W.polynomial.degree = 2 :=
by { rw [polynomial_eq], exact cubic.degree_of_b_ne_zero' one_ne_zero }
@[simp] lemma nat_degree_polynomial [nontrivial R] : W.polynomial.nat_degree = 2 :=
by { rw [polynomial_eq], exact cubic.nat_degree_of_b_ne_zero' one_ne_zero }
lemma monic_polynomial : W.polynomial.monic :=
by { nontriviality R, simpa only [polynomial_eq] using cubic.monic_of_b_eq_one' }
lemma irreducible_polynomial [is_domain R] : irreducible W.polynomial :=
begin
by_contra h,
rcases (W.monic_polynomial.not_irreducible_iff_exists_add_mul_eq_coeff W.nat_degree_polynomial).mp
h with ⟨f, g, h0, h1⟩,
simp only [polynomial_eq, cubic.coeff_eq_c, cubic.coeff_eq_d] at h0 h1,
apply_fun degree at h0 h1,
rw [cubic.degree_of_a_ne_zero' $ neg_ne_zero.mpr $ one_ne_zero' R, degree_mul] at h0,
apply (h1.symm.le.trans cubic.degree_of_b_eq_zero').not_lt,
rcases nat.with_bot.add_eq_three_iff.mp h0.symm with h | h | h | h,
any_goals { rw [degree_add_eq_left_of_degree_lt]; simp only [h]; dec_trivial },
any_goals { rw [degree_add_eq_right_of_degree_lt]; simp only [h]; dec_trivial }
end
@[simp] lemma eval_polynomial (x y : R) :
(W.polynomial.eval $ C y).eval x
= y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) :=
by { simp only [weierstrass_curve.polynomial], eval_simp, rw [add_mul, ← add_assoc] }
@[simp] lemma eval_polynomial_zero : (W.polynomial.eval 0).eval 0 = -W.a₆ :=
by simp only [← C_0, eval_polynomial, zero_add, zero_sub, mul_zero, zero_pow (nat.zero_lt_succ _)]
/-- The proposition that an affine point $(x, y)$ lies in `W`. In other words, $W(x, y) = 0$. -/
def equation (x y : R) : Prop := (W.polynomial.eval $ C y).eval x = 0
lemma equation_iff' (x y : R) :
W.equation x y ↔ y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) = 0 :=
by rw [equation, eval_polynomial]
@[simp] lemma equation_iff (x y : R) :
W.equation x y ↔ y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆ :=
by rw [equation_iff', sub_eq_zero]
@[simp] lemma equation_zero : W.equation 0 0 ↔ W.a₆ = 0 :=
by rw [equation, C_0, eval_polynomial_zero, neg_eq_zero]
lemma equation_iff_variable_change (x y : R) :
W.equation x y ↔ (W.variable_change 1 x 0 y).equation 0 0 :=
begin
rw [equation_iff', ← neg_eq_zero, equation_zero, variable_change_a₆, inv_one, units.coe_one],
congr' 2,
ring1
end
lemma equation_iff_base_change [nontrivial A] [no_zero_smul_divisors R A] (x y : R) :
W.equation x y ↔ (W.base_change A).equation (algebra_map R A x) (algebra_map R A y) :=
begin
simp only [equation_iff],
refine ⟨λ h, _, λ h, _⟩,
{ convert congr_arg (algebra_map R A) h; { map_simp, refl } },
{ apply no_zero_smul_divisors.algebra_map_injective R A, map_simp, exact h }
end
lemma equation_iff_base_change_of_base_change [nontrivial B] [no_zero_smul_divisors A B] (x y : A) :
(W.base_change A).equation x y
↔ (W.base_change B).equation (algebra_map A B x) (algebra_map A B y) :=
by rw [equation_iff_base_change (W.base_change A) B, base_change_base_change]
/-! ### Nonsingularity of Weierstrass curves -/
/-- The partial derivative $W_X(X, Y)$ of $W(X, Y)$ with respect to $X$.
TODO: define this in terms of `polynomial.derivative`. -/
noncomputable def polynomial_X : R[X][Y] :=
C (C W.a₁) * Y - C (C 3 * X ^ 2 + C (2 * W.a₂) * X + C W.a₄)
@[simp] lemma eval_polynomial_X (x y : R) :
(W.polynomial_X.eval $ C y).eval x = W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) :=
by { simp only [polynomial_X], eval_simp }
@[simp] lemma eval_polynomial_X_zero : (W.polynomial_X.eval 0).eval 0 = -W.a₄ :=
by simp only [← C_0, eval_polynomial_X, zero_add, zero_sub, mul_zero, zero_pow zero_lt_two]
/-- The partial derivative $W_Y(X, Y)$ of $W(X, Y)$ with respect to $Y$.
TODO: define this in terms of `polynomial.derivative`. -/
noncomputable def polynomial_Y : R[X][Y] := C (C 2) * Y + C (C W.a₁ * X + C W.a₃)
@[simp] lemma eval_polynomial_Y (x y : R) :
(W.polynomial_Y.eval $ C y).eval x = 2 * y + W.a₁ * x + W.a₃ :=
by { simp only [polynomial_Y], eval_simp, rw [← add_assoc] }
@[simp] lemma eval_polynomial_Y_zero : (W.polynomial_Y.eval 0).eval 0 = W.a₃ :=
by simp only [← C_0, eval_polynomial_Y, zero_add, mul_zero]
/-- The proposition that an affine point $(x, y)$ on `W` is nonsingular.
In other words, either $W_X(x, y) \ne 0$ or $W_Y(x, y) \ne 0$. -/
def nonsingular (x y : R) : Prop :=
W.equation x y ∧ ((W.polynomial_X.eval $ C y).eval x ≠ 0 ∨ (W.polynomial_Y.eval $ C y).eval x ≠ 0)
lemma nonsingular_iff' (x y : R) :
W.nonsingular x y
↔ W.equation x y
∧ (W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) ≠ 0 ∨ 2 * y + W.a₁ * x + W.a₃ ≠ 0) :=
by rw [nonsingular, equation_iff', eval_polynomial_X, eval_polynomial_Y]
@[simp] lemma nonsingular_iff (x y : R) :
W.nonsingular x y
↔ W.equation x y ∧ (W.a₁ * y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ -y - W.a₁ * x - W.a₃) :=
by { rw [nonsingular_iff', sub_ne_zero, ← @sub_ne_zero _ _ y], congr' 4; ring1 }
@[simp] lemma nonsingular_zero : W.nonsingular 0 0 ↔ W.a₆ = 0 ∧ (W.a₃ ≠ 0 ∨ W.a₄ ≠ 0) :=
by rw [nonsingular, equation_zero, C_0, eval_polynomial_X_zero, neg_ne_zero, eval_polynomial_Y_zero,
or_comm]
lemma nonsingular_iff_variable_change (x y : R) :
W.nonsingular x y ↔ (W.variable_change 1 x 0 y).nonsingular 0 0 :=
begin
rw [nonsingular_iff', equation_iff_variable_change, equation_zero, ← neg_ne_zero, or_comm,
nonsingular_zero, variable_change_a₃, variable_change_a₄, inv_one, units.coe_one],
congr' 4; ring1
end
lemma nonsingular_iff_base_change [nontrivial A] [no_zero_smul_divisors R A] (x y : R) :
W.nonsingular x y ↔ (W.base_change A).nonsingular (algebra_map R A x) (algebra_map R A y) :=
begin
rw [nonsingular_iff, nonsingular_iff, and_congr $ W.equation_iff_base_change A x y],
refine ⟨or.imp (not_imp_not.mpr $ λ h, _) (not_imp_not.mpr $ λ h, _),
or.imp (not_imp_not.mpr $ λ h, _) (not_imp_not.mpr $ λ h, _)⟩,
any_goals { apply no_zero_smul_divisors.algebra_map_injective R A, map_simp, exact h },
any_goals { convert congr_arg (algebra_map R A) h; { map_simp, refl } }
end
lemma nonsingular_iff_base_change_of_base_change [nontrivial B] [no_zero_smul_divisors A B]
(x y : A) : (W.base_change A).nonsingular x y
↔ (W.base_change B).nonsingular (algebra_map A B x) (algebra_map A B y) :=
by rw [nonsingular_iff_base_change (W.base_change A) B, base_change_base_change]
lemma nonsingular_zero_of_Δ_ne_zero (h : W.equation 0 0) (hΔ : W.Δ ≠ 0) : W.nonsingular 0 0 :=
by { simp only [equation_zero, nonsingular_zero] at *, contrapose! hΔ, simp [h, hΔ] }
/-- A Weierstrass curve is nonsingular at every point if its discriminant is non-zero. -/
lemma nonsingular_of_Δ_ne_zero {x y : R} (h : W.equation x y) (hΔ : W.Δ ≠ 0) : W.nonsingular x y :=
(W.nonsingular_iff_variable_change x y).mpr $
nonsingular_zero_of_Δ_ne_zero _ ((W.equation_iff_variable_change x y).mp h) $
by rwa [variable_change_Δ, inv_one, units.coe_one, one_pow, one_mul]
/-! ### Ideals in the coordinate ring -/
/-- The coordinate ring $R[W] := R[X, Y] / \langle W(X, Y) \rangle$ of `W`.
Note that `derive comm_ring` generates a reducible instance of `comm_ring` for `coordinate_ring`.
In certain circumstances this might be extremely slow, because all instances in its definition are
unified exponentially many times. In this case, one solution is to manually add the local attribute
`local attribute [irreducible] coordinate_ring.comm_ring` to block this type-level unification.
TODO Lean 4: verify if the new def-eq cache (lean4#1102) fixed this issue.
See Zulip thread:
https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/.E2.9C.94.20class_group.2Emk -/
@[derive [inhabited, comm_ring]] def coordinate_ring : Type u := adjoin_root W.polynomial
/-- The function field $R(W) := \mathrm{Frac}(R[W])$ of `W`. -/
abbreviation function_field : Type u := fraction_ring W.coordinate_ring
namespace coordinate_ring
open ideal
instance [is_domain R] [normalized_gcd_monoid R] : is_domain W.coordinate_ring :=
(quotient.is_domain_iff_prime _).mpr $
by simpa only [span_singleton_prime W.polynomial_ne_zero, ← gcd_monoid.irreducible_iff_prime]
using W.irreducible_polynomial
instance is_domain_of_field {F : Type u} [field F] (W : weierstrass_curve F) :
is_domain W.coordinate_ring :=
by { classical, apply_instance }
variables (x : R) (y : R[X])
/-- The class of the element $X - x$ in $R[W]$ for some $x \in R$. -/
@[simp] noncomputable def X_class : W.coordinate_ring := adjoin_root.mk W.polynomial $ C $ X - C x
lemma X_class_ne_zero [nontrivial R] : X_class W x ≠ 0 :=
adjoin_root.mk_ne_zero_of_nat_degree_lt W.monic_polynomial (C_ne_zero.mpr $ X_sub_C_ne_zero x) $
by { rw [nat_degree_polynomial, nat_degree_C], norm_num1 }
/-- The class of the element $Y - y(X)$ in $R[W]$ for some $y(X) \in R[X]$. -/
@[simp] noncomputable def Y_class : W.coordinate_ring := adjoin_root.mk W.polynomial $ Y - C y
lemma Y_class_ne_zero [nontrivial R] : Y_class W y ≠ 0 :=
adjoin_root.mk_ne_zero_of_nat_degree_lt W.monic_polynomial (X_sub_C_ne_zero y) $
by { rw [nat_degree_polynomial, nat_degree_X_sub_C], norm_num1 }
/-- The ideal $\langle X - x \rangle$ of $R[W]$ for some $x \in R$. -/
@[simp] noncomputable def X_ideal : ideal W.coordinate_ring := span {X_class W x}
/-- The ideal $\langle Y - y(X) \rangle$ of $R[W]$ for some $y(X) \in R[X]$. -/
@[simp] noncomputable def Y_ideal : ideal W.coordinate_ring := span {Y_class W y}
/-- The ideal $\langle X - x, Y - y(X) \rangle$ of $R[W]$ for some $x \in R$ and $y(X) \in R[X]$. -/
@[simp] noncomputable def XY_ideal (x : R) (y : R[X]) : ideal W.coordinate_ring :=
span {X_class W x, Y_class W y}
/-! ### The coordinate ring as an `R[X]`-algebra -/
noncomputable instance : algebra R[X] W.coordinate_ring := quotient.algebra R[X]
noncomputable instance algebra' : algebra R W.coordinate_ring := quotient.algebra R
instance : is_scalar_tower R R[X] W.coordinate_ring := quotient.is_scalar_tower R R[X] _
instance [subsingleton R] : subsingleton W.coordinate_ring := module.subsingleton R[X] _
/-- The $R$-algebra isomorphism from $R[W] / \langle X - x, Y - y(X) \rangle$ to $R$ obtained by
evaluation at $y(X)$ and at $x$ provided that $W(x, y(x)) = 0$. -/
noncomputable def quotient_XY_ideal_equiv {x : R} {y : R[X]}
(h : (W.polynomial.eval y).eval x = 0) : (W.coordinate_ring ⧸ XY_ideal W x y) ≃ₐ[R] R :=
(quotient_equiv_alg_of_eq R $
by simpa only [XY_ideal, X_class, Y_class, ← set.image_pair, ← map_span]).trans $
(double_quot.quot_quot_equiv_quot_of_leₐ R $ (span_singleton_le_iff_mem _).mpr $
mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero.mpr h).trans $
((quotient_span_C_X_sub_C_alg_equiv (X - C x) y).restrict_scalars R).trans $
quotient_span_X_sub_C_alg_equiv x
/-- The basis $\{1, Y\}$ for the coordinate ring $R[W]$ over the polynomial ring $R[X]$.
Given a Weierstrass curve `W`, write `W^.coordinate_ring.basis` for this basis. -/
protected noncomputable def basis : basis (fin 2) R[X] W.coordinate_ring :=
(subsingleton_or_nontrivial R).by_cases (λ _, by exactI default) $ λ _, by exactI
((adjoin_root.power_basis' W.monic_polynomial).basis.reindex $
fin_congr W.nat_degree_polynomial)
lemma basis_apply (n : fin 2) :
W^.coordinate_ring.basis n = (adjoin_root.power_basis' W.monic_polynomial).gen ^ (n : ℕ) :=
begin
classical,
nontriviality R,
simpa only [coordinate_ring.basis, or.by_cases, dif_neg (not_subsingleton R),
basis.reindex_apply, power_basis.basis_eq_pow]
end
lemma basis_zero : W^.coordinate_ring.basis 0 = 1 := by simpa only [basis_apply] using pow_zero _
lemma basis_one : W^.coordinate_ring.basis 1 = adjoin_root.mk W.polynomial Y :=
by simpa only [basis_apply] using pow_one _
@[simp] lemma coe_basis :
(W^.coordinate_ring.basis : fin 2 → W.coordinate_ring) = ![1, adjoin_root.mk W.polynomial Y] :=
by { ext n, fin_cases n, exacts [basis_zero W, basis_one W] }
variable {W}
lemma smul (x : R[X]) (y : W.coordinate_ring) : x • y = adjoin_root.mk W.polynomial (C x) * y :=
(algebra_map_smul W.coordinate_ring x y).symm
lemma smul_basis_eq_zero {p q : R[X]}
(hpq : p • 1 + q • adjoin_root.mk W.polynomial Y = 0) : p = 0 ∧ q = 0 :=
begin
have h := fintype.linear_independent_iff.mp (coordinate_ring.basis W).linear_independent ![p, q],
erw [fin.sum_univ_succ, basis_zero, fin.sum_univ_one, basis_one] at h,
exact ⟨h hpq 0, h hpq 1⟩
end
lemma exists_smul_basis_eq (x : W.coordinate_ring) :
∃ p q : R[X], p • 1 + q • adjoin_root.mk W.polynomial Y = x :=
begin
have h := (coordinate_ring.basis W).sum_equiv_fun x,
erw [fin.sum_univ_succ, fin.sum_univ_one, basis_zero, basis_one] at h,
exact ⟨_, _, h⟩
end
variable (W)
lemma smul_basis_mul_C (p q : R[X]) :
(p • 1 + q • adjoin_root.mk W.polynomial Y) * adjoin_root.mk W.polynomial (C y)
= ((p * y) • 1 + (q * y) • adjoin_root.mk W.polynomial Y) :=
by { simp only [smul, _root_.map_mul], ring1 }
lemma smul_basis_mul_Y (p q : R[X]) :
(p • 1 + q • adjoin_root.mk W.polynomial Y) * adjoin_root.mk W.polynomial Y
= (q * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)) • 1
+ (p - q * (C W.a₁ * X + C W.a₃)) • adjoin_root.mk W.polynomial Y :=
begin
have Y_sq : adjoin_root.mk W.polynomial Y ^ 2 = adjoin_root.mk W.polynomial
(C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆) - C (C W.a₁ * X + C W.a₃) * Y) :=
adjoin_root.mk_eq_mk.mpr ⟨1, by { simp only [weierstrass_curve.polynomial], ring1 }⟩,
simp only [smul, add_mul, mul_assoc, ← sq, Y_sq, map_sub, _root_.map_mul],
ring1
end
/-! ### Norms on the coordinate ring -/
lemma norm_smul_basis (p q : R[X]) :
algebra.norm R[X] (p • 1 + q • adjoin_root.mk W.polynomial Y)
= p ^ 2 - p * q * (C W.a₁ * X + C W.a₃)
- q ^ 2 * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆) :=
begin
simp_rw [algebra.norm_eq_matrix_det W^.coordinate_ring.basis, matrix.det_fin_two,
algebra.left_mul_matrix_eq_repr_mul, basis_zero, mul_one, basis_one, smul_basis_mul_Y,
map_add, finsupp.add_apply, map_smul, finsupp.smul_apply, ← basis_zero, ← basis_one,
basis.repr_self_apply, if_pos, if_neg one_ne_zero, if_neg zero_ne_one, smul_eq_mul],
ring1
end
lemma coe_norm_smul_basis (p q : R[X]) :
↑(algebra.norm R[X] $ p • 1 + q • adjoin_root.mk W.polynomial Y)
= adjoin_root.mk W.polynomial
((C p + C q * X) * (C p + C q * (-Y - C (C W.a₁ * X + C W.a₃)))) :=
adjoin_root.mk_eq_mk.mpr
⟨C q ^ 2, by { rw [norm_smul_basis, weierstrass_curve.polynomial], C_simp, ring1 }⟩
lemma degree_norm_smul_basis [is_domain R] (p q : R[X]) :
(algebra.norm R[X] $ p • 1 + q • adjoin_root.mk W.polynomial Y).degree
= max (2 • p.degree) (2 • q.degree + 3) :=
begin
have hdp : (p ^ 2).degree = 2 • p.degree := degree_pow p 2,
have hdpq : (p * q * (C W.a₁ * X + C W.a₃)).degree ≤ p.degree + q.degree + 1,
{ simpa only [degree_mul] using add_le_add_left degree_linear_le (p.degree + q.degree) },
have hdq : (q ^ 2 * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)).degree = 2 • q.degree + 3,
{ rw [degree_mul, degree_pow, ← one_mul $ X ^ 3, ← C_1, degree_cubic $ one_ne_zero' R] },
rw [norm_smul_basis],
by_cases hp : p = 0, { simpa only [hp, hdq, neg_zero, zero_sub, zero_mul, zero_pow zero_lt_two,
degree_neg] using (max_bot_left _).symm },
by_cases hq : q = 0, { simpa only [hq, hdp, sub_zero, zero_mul, mul_zero, zero_pow zero_lt_two]
using (max_bot_right _).symm },
rw [← not_iff_not_of_iff degree_eq_bot] at hp hq,
cases p.degree with dp, { exact (hp rfl).elim },
cases q.degree with dq, { exact (hq rfl).elim },
cases le_or_lt dp (dq + 1) with hpq hpq,
{ convert (degree_sub_eq_right_of_degree_lt $ (degree_sub_le _ _).trans_lt $
max_lt_iff.mpr ⟨hdp.trans_lt _, hdpq.trans_lt _⟩).trans (max_eq_right_of_lt _).symm; rw [hdq];
exact with_bot.coe_lt_coe.mpr (by linarith only [hpq]) },
{ rw [sub_sub],
convert (degree_sub_eq_left_of_degree_lt $ (degree_add_le _ _).trans_lt $
max_lt_iff.mpr ⟨hdpq.trans_lt _, hdq.trans_lt _⟩).trans (max_eq_left_of_lt _).symm; rw [hdp];
exact with_bot.coe_lt_coe.mpr (by linarith only [hpq]) }
end
variable {W}
lemma degree_norm_ne_one [is_domain R] (x : W.coordinate_ring) : (algebra.norm R[X] x).degree ≠ 1 :=
begin
rcases exists_smul_basis_eq x with ⟨p, q, rfl⟩,
rw [degree_norm_smul_basis],
rcases p.degree with (_ | _ | _ | _); cases q.degree,
any_goals { rintro (_ | _) },
exact (lt_max_of_lt_right dec_trivial).ne'
end
lemma nat_degree_norm_ne_one [is_domain R] (x : W.coordinate_ring) :
(algebra.norm R[X] x).nat_degree ≠ 1 :=
mt (degree_eq_iff_nat_degree_eq_of_pos zero_lt_one).mpr $ degree_norm_ne_one x
end coordinate_ring
end polynomial
end weierstrass_curve
/-! ## Elliptic curves -/
/-- An elliptic curve over a commutative ring. Note that this definition is only mathematically
accurate for certain rings whose Picard group has trivial 12-torsion, such as a field or a PID. -/
@[ext] structure elliptic_curve (R : Type u) [comm_ring R] extends weierstrass_curve R :=
(Δ' : Rˣ) (coe_Δ' : ↑Δ' = to_weierstrass_curve.Δ)
instance : inhabited $ elliptic_curve ℚ :=
⟨⟨⟨0, 0, 1, -1, 0⟩, ⟨37, 37⁻¹, by norm_num1, by norm_num1⟩, by { dsimp, ring1 }⟩⟩
namespace elliptic_curve
variables [comm_ring R] (E : elliptic_curve R)
/-- The j-invariant `j` of an elliptic curve, which is invariant under isomorphisms over `R`. -/
@[simp] def j : R := ↑E.Δ'⁻¹ * E.c₄ ^ 3
lemma two_torsion_polynomial_disc_ne_zero [nontrivial R] [invertible (2 : R)] :
E.two_torsion_polynomial.disc ≠ 0 :=
E.two_torsion_polynomial_disc_ne_zero $ E.coe_Δ' ▸ E.Δ'.is_unit
lemma nonsingular [nontrivial R] {x y : R} (h : E.equation x y) : E.nonsingular x y :=
E.nonsingular_of_Δ_ne_zero h $ E.coe_Δ' ▸ E.Δ'.ne_zero
section variable_change
/-! ### Variable changes -/
variables (u : Rˣ) (r s t : R)
/-- The elliptic curve over `R` induced by an admissible linear change of variables
$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$.
When `R` is a field, any two Weierstrass equations isomorphic to `E` are related by this. -/
@[simps] def variable_change : elliptic_curve R :=
⟨E.variable_change u r s t, u⁻¹ ^ 12 * E.Δ',
by rw [units.coe_mul, units.coe_pow, coe_Δ', E.variable_change_Δ]⟩
lemma coe_variable_change_Δ' : (↑(E.variable_change u r s t).Δ' : R) = ↑u⁻¹ ^ 12 * E.Δ' :=
by rw [variable_change_Δ', units.coe_mul, units.coe_pow]
lemma coe_inv_variable_change_Δ' : (↑(E.variable_change u r s t).Δ'⁻¹ : R) = u ^ 12 * ↑E.Δ'⁻¹ :=
by rw [variable_change_Δ', mul_inv, inv_pow, inv_inv, units.coe_mul, units.coe_pow]
@[simp] lemma variable_change_j : (E.variable_change u r s t).j = E.j :=
begin
rw [j, coe_inv_variable_change_Δ'],
have hu : (u * ↑u⁻¹ : R) ^ 12 = 1 := by rw [u.mul_inv, one_pow],
linear_combination E.j * hu with { normalization_tactic := `[dsimp, ring1] }
end
end variable_change
section base_change
/-! ### Base changes -/
variables (A : Type v) [comm_ring A] [algebra R A]
/-- The elliptic curve over `R` base changed to `A`. -/
@[simps] def base_change : elliptic_curve A :=
⟨E.base_change A, units.map ↑(algebra_map R A) E.Δ',
by rw [units.coe_map, ring_hom.coe_monoid_hom, coe_Δ', E.base_change_Δ]⟩
lemma coe_base_change_Δ' : ↑(E.base_change A).Δ' = algebra_map R A E.Δ' := rfl
lemma coe_inv_base_change_Δ' : ↑(E.base_change A).Δ'⁻¹ = algebra_map R A ↑E.Δ'⁻¹ := rfl
@[simp] lemma base_change_j : (E.base_change A).j = algebra_map R A E.j :=
by { simp only [j, coe_inv_base_change_Δ', base_change_to_weierstrass_curve, E.base_change_c₄],
map_simp }
end base_change
end elliptic_curve
|
4d6a15872cdbf3b6213a05b58d85374a6dbe5754 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/algebraic_topology/cech_nerve.lean | fd2452124d89424eceacb2fbd909f2799017283e | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 16,350 | lean | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import algebraic_topology.simplicial_object
import category_theory.limits.shapes.wide_pullbacks
import category_theory.limits.shapes.finite_products
import category_theory.arrow
/-!
# The Čech Nerve
This file provides a definition of the Čech nerve associated to an arrow, provided
the base category has the correct wide pullbacks.
Several variants are provided, given `f : arrow C`:
1. `f.cech_nerve` is the Čech nerve, considered as a simplicial object in `C`.
2. `f.augmented_cech_nerve` is the augmented Čech nerve, considered as an
augmented simplicial object in `C`.
3. `simplicial_object.cech_nerve` and `simplicial_object.augmented_cech_nerve` are
functorial versions of 1 resp. 2.
We end the file with a description of the Čech nerve of an arrow `X ⟶ ⊤_ C` to a terminal
object, when `C` has finite products. We call this `cech_nerve_terminal_from`. When `C` is
`G`-Set this gives us `EG` (the universal cover of the classifying space of `G`) as a simplicial
`G`-set, which is useful for group cohomology.
-/
open category_theory
open category_theory.limits
noncomputable theory
universes v u w
variables {C : Type u} [category.{v} C]
namespace category_theory.arrow
variables (f : arrow C)
variables [∀ n : ℕ, has_wide_pullback.{0} f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
/-- The Čech nerve associated to an arrow. -/
@[simps]
def cech_nerve : simplicial_object C :=
{ obj := λ n, wide_pullback.{0} f.right
(λ i : fin (n.unop.len + 1), f.left) (λ i, f.hom),
map := λ m n g, wide_pullback.lift (wide_pullback.base _)
(λ i, wide_pullback.π (λ i, f.hom) $ g.unop.to_order_hom i) $ λ j, by simp,
map_id' := λ x, by { ext ⟨⟩, { simpa }, { simp } },
map_comp' := λ x y z f g, by { ext ⟨⟩, { simpa }, { simp } } }
/-- The morphism between Čech nerves associated to a morphism of arrows. -/
@[simps]
def map_cech_nerve {f g : arrow C}
[∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pullback g.right (λ i : fin (n+1), g.left) (λ i, g.hom)]
(F : f ⟶ g) : f.cech_nerve ⟶ g.cech_nerve :=
{ app := λ n, wide_pullback.lift (wide_pullback.base _ ≫ F.right)
(λ i, wide_pullback.π _ i ≫ F.left) $ λ j, by simp,
naturality' := λ x y f, by { ext ⟨⟩, { simp }, { simp } } }
/-- The augmented Čech nerve associated to an arrow. -/
@[simps]
def augmented_cech_nerve : simplicial_object.augmented C :=
{ left := f.cech_nerve,
right := f.right,
hom :=
{ app := λ i, wide_pullback.base _,
naturality' := λ x y f, by { dsimp, simp } } }
/-- The morphism between augmented Čech nerve associated to a morphism of arrows. -/
@[simps]
def map_augmented_cech_nerve {f g : arrow C}
[∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pullback g.right (λ i : fin (n+1), g.left) (λ i, g.hom)]
(F : f ⟶ g) : f.augmented_cech_nerve ⟶ g.augmented_cech_nerve :=
{ left := map_cech_nerve F,
right := F.right,
w' := by { ext, simp } }
end category_theory.arrow
namespace category_theory
namespace simplicial_object
variables [∀ (n : ℕ) (f : arrow C),
has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
/-- The Čech nerve construction, as a functor from `arrow C`. -/
@[simps]
def cech_nerve : arrow C ⥤ simplicial_object C :=
{ obj := λ f, f.cech_nerve,
map := λ f g F, arrow.map_cech_nerve F,
map_id' := λ i, by { ext, { simp }, { simp } },
map_comp' := λ x y z f g, by { ext, { simp }, { simp } } }
/-- The augmented Čech nerve construction, as a functor from `arrow C`. -/
@[simps]
def augmented_cech_nerve : arrow C ⥤ simplicial_object.augmented C :=
{ obj := λ f, f.augmented_cech_nerve,
map := λ f g F, arrow.map_augmented_cech_nerve F,
map_id' := λ x, by { ext, { simp }, { simp }, { refl } },
map_comp' := λ x y z f g, by { ext, { simp }, { simp }, { refl } } }
/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def equivalence_right_to_left (X : simplicial_object.augmented C) (F : arrow C)
(G : X ⟶ F.augmented_cech_nerve) : augmented.to_arrow.obj X ⟶ F :=
{ left := G.left.app _ ≫ wide_pullback.π (λ i, F.hom) 0,
right := G.right,
w' := begin
have := G.w,
apply_fun (λ e, e.app (opposite.op $ simplex_category.mk 0)) at this,
simpa using this,
end }
/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def equivalence_left_to_right (X : simplicial_object.augmented C) (F : arrow C)
(G : augmented.to_arrow.obj X ⟶ F) : X ⟶ F.augmented_cech_nerve :=
{ left :=
{ app := λ x, limits.wide_pullback.lift (X.hom.app _ ≫ G.right)
(λ i, X.left.map (simplex_category.const x.unop i).op ≫ G.left)
(λ i, by { dsimp, erw [category.assoc, arrow.w,
augmented.to_arrow_obj_hom, nat_trans.naturality_assoc,
functor.const_obj_map, category.id_comp] } ),
naturality' := begin
intros x y f,
ext,
{ dsimp,
simp only [wide_pullback.lift_π, category.assoc],
rw [← category.assoc, ← X.left.map_comp],
refl },
{ dsimp,
simp only [functor.const_obj_map, nat_trans.naturality_assoc,
wide_pullback.lift_base, category.assoc],
erw category.id_comp }
end },
right := G.right,
w' := by { ext, dsimp, simp } }
/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def cech_nerve_equiv (X : simplicial_object.augmented C) (F : arrow C) :
(augmented.to_arrow.obj X ⟶ F) ≃ (X ⟶ F.augmented_cech_nerve) :=
{ to_fun := equivalence_left_to_right _ _,
inv_fun := equivalence_right_to_left _ _,
left_inv := begin
intro A,
dsimp,
ext,
{ dsimp,
erw wide_pullback.lift_π,
nth_rewrite 1 ← category.id_comp A.left,
congr' 1,
convert X.left.map_id _,
rw ← op_id,
congr' 1,
ext ⟨a,ha⟩,
change a < 1 at ha,
change 0 = a,
linarith },
{ refl, }
end,
right_inv := begin
intro A,
ext _ ⟨j⟩,
{ dsimp,
simp only [arrow.cech_nerve_map, wide_pullback.lift_π, nat_trans.naturality_assoc],
erw wide_pullback.lift_π,
refl },
{ erw wide_pullback.lift_base,
have := A.w,
apply_fun (λ e, e.app x) at this,
rw nat_trans.comp_app at this,
erw this,
refl },
{ refl }
end }
/-- The augmented Čech nerve construction is right adjoint to the `to_arrow` functor. -/
abbreviation cech_nerve_adjunction :
(augmented.to_arrow : _ ⥤ arrow C) ⊣ augmented_cech_nerve :=
adjunction.mk_of_hom_equiv
{ hom_equiv := cech_nerve_equiv,
hom_equiv_naturality_left_symm' := λ x y f g h, by { ext, { simp }, { simp } },
hom_equiv_naturality_right' := λ x y f g h, by { ext, { simp }, { simp }, { refl } } }
end simplicial_object
end category_theory
namespace category_theory.arrow
variables (f : arrow C)
variables [∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
/-- The Čech conerve associated to an arrow. -/
@[simps]
def cech_conerve : cosimplicial_object C :=
{ obj := λ n, wide_pushout f.left
(λ i : fin (n.len + 1), f.right) (λ i, f.hom),
map := λ m n g, wide_pushout.desc (wide_pushout.head _)
(λ i, wide_pushout.ι (λ i, f.hom) $ g.to_order_hom i) $
λ i, by { rw [wide_pushout.arrow_ι (λ i, f.hom)] },
map_id' := λ x, by { ext ⟨⟩, { simpa }, { simp } },
map_comp' := λ x y z f g, by { ext ⟨⟩, { simpa }, { simp } } }
/-- The morphism between Čech conerves associated to a morphism of arrows. -/
@[simps]
def map_cech_conerve {f g : arrow C}
[∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pushout g.left (λ i : fin (n+1), g.right) (λ i, g.hom)]
(F : f ⟶ g) : f.cech_conerve ⟶ g.cech_conerve :=
{ app := λ n, wide_pushout.desc (F.left ≫ wide_pushout.head _)
(λ i, F.right ≫ wide_pushout.ι _ i) $
λ i, by { rw [← arrow.w_assoc F, wide_pushout.arrow_ι (λ i, g.hom)] },
naturality' := λ x y f, by { ext, { simp }, { simp } } }
/-- The augmented Čech conerve associated to an arrow. -/
@[simps]
def augmented_cech_conerve : cosimplicial_object.augmented C :=
{ left := f.left,
right := f.cech_conerve,
hom :=
{ app := λ i, wide_pushout.head _,
naturality' := λ x y f, by { dsimp, simp } } }
/-- The morphism between augmented Čech conerves associated to a morphism of arrows. -/
@[simps]
def map_augmented_cech_conerve {f g : arrow C}
[∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pushout g.left (λ i : fin (n+1), g.right) (λ i, g.hom)]
(F : f ⟶ g) : f.augmented_cech_conerve ⟶ g.augmented_cech_conerve :=
{ left := F.left,
right := map_cech_conerve F,
w' := by { ext, simp } }
end category_theory.arrow
namespace category_theory
namespace cosimplicial_object
variables [∀ (n : ℕ) (f : arrow C),
has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
/-- The Čech conerve construction, as a functor from `arrow C`. -/
@[simps]
def cech_conerve : arrow C ⥤ cosimplicial_object C :=
{ obj := λ f, f.cech_conerve,
map := λ f g F, arrow.map_cech_conerve F,
map_id' := λ i, by { ext, { dsimp, simp }, { dsimp, simp } },
map_comp' := λ f g h F G, by { ext, { simp }, { simp } } }
/-- The augmented Čech conerve construction, as a functor from `arrow C`. -/
@[simps]
def augmented_cech_conerve : arrow C ⥤ cosimplicial_object.augmented C :=
{ obj := λ f, f.augmented_cech_conerve,
map := λ f g F, arrow.map_augmented_cech_conerve F,
map_id' := λ f, by { ext, { refl }, { dsimp, simp }, { dsimp, simp } },
map_comp' := λ f g h F G, by { ext, { refl }, { simp }, { simp } } }
/-- A helper function used in defining the Čech conerve adjunction. -/
@[simps]
def equivalence_left_to_right (F : arrow C) (X : cosimplicial_object.augmented C)
(G : F.augmented_cech_conerve ⟶ X) : F ⟶ augmented.to_arrow.obj X :=
{ left := G.left,
right :=
(wide_pushout.ι (λ i, F.hom) 0 ≫ G.right.app (simplex_category.mk 0) : _),
w' := begin
have := G.w,
apply_fun (λ e, e.app (simplex_category.mk 0)) at this,
simpa only [category_theory.functor.id_map, augmented.to_arrow_obj_hom,
wide_pushout.arrow_ι_assoc (λ i, F.hom)],
end }
/-- A helper function used in defining the Čech conerve adjunction. -/
@[simps]
def equivalence_right_to_left (F : arrow C) (X : cosimplicial_object.augmented C)
(G : F ⟶ augmented.to_arrow.obj X) : F.augmented_cech_conerve ⟶ X :=
{ left := G.left,
right := { app := λ x, limits.wide_pushout.desc (G.left ≫ X.hom.app _)
(λ i, G.right ≫ X.right.map (simplex_category.const x i))
begin
rintros j,
rw ← arrow.w_assoc G,
have t := X.hom.naturality (x.const j),
dsimp at t ⊢,
simp only [category.id_comp] at t,
rw ← t,
end,
naturality' := begin
intros x y f,
ext,
{ dsimp,
simp only [wide_pushout.ι_desc_assoc, wide_pushout.ι_desc],
rw [category.assoc, ←X.right.map_comp],
refl },
{ dsimp,
simp only [functor.const_obj_map, ←nat_trans.naturality,
wide_pushout.head_desc_assoc, wide_pushout.head_desc, category.assoc],
erw category.id_comp }
end },
w' := by { ext, simp } }
/-- A helper function used in defining the Čech conerve adjunction. -/
@[simps]
def cech_conerve_equiv (F : arrow C) (X : cosimplicial_object.augmented C) :
(F.augmented_cech_conerve ⟶ X) ≃ (F ⟶ augmented.to_arrow.obj X) :=
{ to_fun := equivalence_left_to_right _ _,
inv_fun := equivalence_right_to_left _ _,
left_inv := begin
intro A,
dsimp,
ext _, { refl }, ext _ ⟨⟩, -- A bug in the `ext` tactic?
{ dsimp,
simp only [arrow.cech_conerve_map, wide_pushout.ι_desc, category.assoc,
← nat_trans.naturality, wide_pushout.ι_desc_assoc],
refl },
{ erw wide_pushout.head_desc,
have := A.w,
apply_fun (λ e, e.app x) at this,
rw nat_trans.comp_app at this,
erw this,
refl },
end,
right_inv := begin
intro A,
ext,
{ refl, },
{ dsimp,
erw wide_pushout.ι_desc,
nth_rewrite 1 ← category.comp_id A.right,
congr' 1,
convert X.right.map_id _,
ext ⟨a,ha⟩,
change a < 1 at ha,
change 0 = a,
linarith },
end }
/-- The augmented Čech conerve construction is left adjoint to the `to_arrow` functor. -/
abbreviation cech_conerve_adjunction :
augmented_cech_conerve ⊣ (augmented.to_arrow : _ ⥤ arrow C) :=
adjunction.mk_of_hom_equiv
{ hom_equiv := cech_conerve_equiv,
hom_equiv_naturality_left_symm' := λ x y f g h, by { ext, { refl }, { simp }, { simp } },
hom_equiv_naturality_right' := λ x y f g h, by { ext, { simp }, { simp } } }
end cosimplicial_object
/-- Given an object `X : C`, the natural simplicial object sending `[n]` to `Xⁿ⁺¹`. -/
def cech_nerve_terminal_from {C : Type u} [category.{v} C] [has_finite_products C] (X : C) :
simplicial_object C :=
{ obj := λ n, ∏ (λ i : fin (n.unop.len + 1), X),
map := λ m n f, limits.pi.lift (λ i, limits.pi.π _ (f.unop.to_order_hom i)),
map_id' := λ f, limit.hom_ext $ λ j, by discrete_cases;
simpa only [limit.lift_π, category.id_comp],
map_comp' := λ m n o f g, limit.hom_ext $ λ j, by discrete_cases;
simpa only [category.assoc, limit.lift_π, fan.mk_π_app] }
namespace cech_nerve_terminal_from
variables [has_terminal C] (ι : Type w)
/-- The diagram `option ι ⥤ C` sending `none` to the terminal object and `some j` to `X`. -/
def wide_cospan (X : C) : wide_pullback_shape ι ⥤ C :=
wide_pullback_shape.wide_cospan (terminal C) (λ i : ι, X) (λ i, terminal.from X)
instance unique_to_wide_cospan_none (X Y : C) : unique (Y ⟶ (wide_cospan ι X).obj none) :=
by unfold wide_cospan; dsimp; apply_instance
variables [has_finite_products C]
/-- The product `Xᶥ` is the vertex of a limit cone on `wide_cospan ι X`. -/
def wide_cospan.limit_cone [fintype ι] (X : C) : limit_cone (wide_cospan ι X) :=
{ cone :=
{ X := ∏ (λ i : ι, X),
π :=
{ app := λ X, option.cases_on X (terminal.from _) (λ i, limit.π _ ⟨i⟩),
naturality' := λ i j f,
begin
cases f,
{ cases i,
all_goals { dsimp, simp }},
{ dsimp,
simp only [terminal.comp_from],
exact subsingleton.elim _ _ }
end } },
is_limit :=
{ lift := λ s, limits.pi.lift (λ j, s.π.app (some j)),
fac' := λ s j, option.cases_on j (subsingleton.elim _ _) (λ j, limit.lift_π _ _),
uniq' := λ s f h,
begin
ext j,
dunfold limits.pi.lift,
rw limit.lift_π,
dsimp,
rw ←h (some j.as),
congr,
ext,
refl,
end } }
instance has_wide_pullback [finite ι] (X : C) :
has_wide_pullback (arrow.mk (terminal.from X)).right
(λ i : ι, (arrow.mk (terminal.from X)).left) (λ i, (arrow.mk (terminal.from X)).hom) :=
begin
casesI nonempty_fintype ι,
exact ⟨⟨wide_cospan.limit_cone ι X⟩⟩,
end
/-- Given an object `X : C`, the Čech nerve of the hom to the terminal object `X ⟶ ⊤_ C` is
naturally isomorphic to a simplicial object sending `[n]` to `Xⁿ⁺¹` (when `C` is `G-Set`, this is
`EG`, the universal cover of the classifying space of `G`. -/
def iso (X : C) :
(arrow.mk (terminal.from X)).cech_nerve ≅ cech_nerve_terminal_from X :=
iso.symm (nat_iso.of_components (λ m, ((limit.is_limit _).cone_point_unique_up_to_iso
(wide_cospan.limit_cone (fin (m.unop.len + 1)) X).2).symm) $ λ m n f, wide_pullback.hom_ext _ _ _
(begin
intro j,
simp only [category.assoc],
dunfold cech_nerve_terminal_from wide_pullback.π pi.lift,
erw [wide_pullback.lift_π, limit.cone_point_unique_up_to_iso_inv_comp
(wide_cospan.limit_cone _ _).2, (limit.is_limit _).cone_point_unique_up_to_iso_inv_comp
(wide_cospan.limit_cone _ _).2, limit.lift_π],
refl,
end)
(@subsingleton.elim _ (@unique.subsingleton _ (limits.unique_to_terminal _)) _ _))
end cech_nerve_terminal_from
end category_theory
|
bbd417d7dec3fb31fcba17c0fa1fc1b37684b31a | acc85b4be2c618b11fc7cb3005521ae6858a8d07 | /data/sigma/basic.lean | ef374a733dc0cc46cd318b0a79e0033588849543 | [
"Apache-2.0"
] | permissive | linpingchuan/mathlib | d49990b236574df2a45d9919ba43c923f693d341 | 5ad8020f67eb13896a41cc7691d072c9331b1f76 | refs/heads/master | 1,626,019,377,808 | 1,508,048,784,000 | 1,508,048,784,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,449 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Johannes Hölzl
-/
variables {α : Type*} {β : α → Type*}
instance [inhabited α] [inhabited (β (default α))] : inhabited (sigma β) :=
⟨⟨default α, default (β (default α))⟩⟩
instance [h₁ : decidable_eq α] [h₂ : ∀a, decidable_eq (β a)] : decidable_eq (sigma β)
| ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ := match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, is_true (eq.refl a) :=
match b₁, b₂, h₂ a b₁ b₂ with
| _, _, is_true (eq.refl b) := is_true rfl
| b₁, b₂, is_false n := is_false (assume h, sigma.no_confusion h (λe₁ e₂, n $ eq_of_heq e₂))
end
| a₁, _, a₂, _, is_false n := is_false (assume h, sigma.no_confusion h (λe₁ e₂, n e₁))
end
namespace sigma
lemma mk_eq_mk_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :
@sigma.mk α β a₁ b₁ = @sigma.mk α β a₂ b₂ ↔ (a₁ = a₂ ∧ b₁ == b₂) :=
iff.intro sigma.mk.inj $
assume ⟨h₁, h₂⟩, match a₁, a₂, b₁, b₂, h₁, h₂ with _, _, _, _, eq.refl a, heq.refl b := rfl end
variables {α₁ : Type*} {α₂ : Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
def map (f₁ : α₁ → α₂) (f₂ : Πa, β₁ a → β₂ (f₁ a)) : sigma β₁ → sigma β₂
| ⟨a, b⟩ := ⟨f₁ a, f₂ a b⟩
end sigma |
a4b734071706a8a34a84e46e04a8868a345a7d51 | 2c096fdfecf64e46ea7bc6ce5521f142b5926864 | /src/Lean/Compiler/IR/LLVMBindings.lean | dec031c79843665e4ed1b3a8bda5892688d7bfac | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | Kha/lean4 | 1005785d2c8797ae266a303968848e5f6ce2fe87 | b99e11346948023cd6c29d248cd8f3e3fb3474cf | refs/heads/master | 1,693,355,498,027 | 1,669,080,461,000 | 1,669,113,138,000 | 184,748,176 | 0 | 0 | Apache-2.0 | 1,665,995,520,000 | 1,556,884,930,000 | Lean | UTF-8 | Lean | false | false | 14,358 | lean | /-
Copyright (c) 2022 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddharth Bhat
-/
namespace LLVM
/-!
This file defines bindings for LLVM. The main mechanisms
are to:
- Mirror the values of C enumerations.
- Create opaque values for LLVM's pointer based API
- Write bindings that are `IO` based, which mutate the underlying in-memory
data structures to build IR.
-/
-- https://github.com/llvm/llvm-project/blob/c3e073bcbdc523b0f758d44a89a6333e38bff863/llvm/include/llvm-c/TargetMachine.h#L64
structure CodegenFileType where
private mk :: val : UInt64
def CodegenFileType.AssemblyFile : CodegenFileType := { val := 0 }
def CodegenFileType.ObjectFile : CodegenFileType := { val := 1 }
-- https://github.com/llvm/llvm-project/blob/c3e073bcbdc523b0f758d44a89a6333e38bff863/llvm/include/llvm-c/Core.h#L290
structure IntPredicate where
private mk :: val : UInt64
def IntPredicate.EQ : IntPredicate := { val := 32 }
def IntPredicate.NE : IntPredicate := { val := IntPredicate.EQ.val + 1 }
def IntPredicate.UGT : IntPredicate := { val := IntPredicate.NE.val + 1 }
structure BasicBlock (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (BasicBlock ctx) := ⟨{ ptr := default }⟩
structure Builder (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (Builder ctx) := ⟨{ ptr := default }⟩
structure Context where
private mk :: ptr : USize
instance : Nonempty Context := ⟨{ ptr := default }⟩
structure LLVMType (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (LLVMType ctx) := ⟨{ ptr := default }⟩
structure MemoryBuffer (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (MemoryBuffer ctx) := ⟨{ ptr := default }⟩
structure Module (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (Module ctx) := ⟨{ ptr := default }⟩
structure PassManager (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (PassManager ctx) := ⟨{ ptr := default }⟩
structure PassManagerBuilder (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (PassManagerBuilder ctx) := ⟨{ ptr := default }⟩
structure Target (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (Target ctx) := ⟨{ ptr := default }⟩
structure TargetMachine (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (TargetMachine ctx) := ⟨{ ptr := default }⟩
structure Value (ctx : Context) where
private mk :: ptr : USize
instance : Nonempty (Value ctx) := ⟨{ ptr := default }⟩
@[extern "lean_llvm_create_context"]
opaque createContext : BaseIO (Context)
@[extern "lean_llvm_create_module"]
opaque createModule (ctx : Context) (name : @&String) : BaseIO (Module ctx)
@[extern "lean_llvm_module_to_string"]
opaque moduleToString (m : Module ctx) : BaseIO String
@[extern "lean_llvm_write_bitcode_to_file"]
opaque writeBitcodeToFile (m : Module ctx) (path : @&String) : BaseIO Unit
@[extern "lean_llvm_add_function"]
opaque addFunction (m : Module ctx) (name : @&String) (type : LLVMType ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_get_named_function"]
opaque getNamedFunction (m : Module ctx) (name : @&String) : BaseIO (Option (Value ctx))
@[extern "lean_llvm_add_global"]
opaque addGlobal (m : Module ctx) (name : @&String) (type : LLVMType ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_get_named_global"]
opaque getNamedGlobal (m : Module ctx) (name : @&String) : BaseIO (Option (Value ctx))
@[extern "lean_llvm_build_global_string"]
opaque buildGlobalString (builder : Builder ctx) (value : @&String) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_set_initializer"]
opaque setInitializer (glbl : Value ctx) (val : Value ctx) : BaseIO Unit
@[extern "lean_llvm_function_type"]
opaque functionType (retty : LLVMType ctx) (args : @&Array (LLVMType ctx)) (isVarArg : Bool := false) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_void_type_in_context"]
opaque voidType (ctx : Context) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_int_type_in_context"]
opaque intTypeInContext (ctx : Context) (width : UInt64) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_float_type_in_context"]
opaque floatTypeInContext (ctx : Context) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_double_type_in_context"]
opaque doubleTypeInContext (ctx : Context) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_pointer_type"]
opaque pointerType (elemty : LLVMType ctx) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_array_type"]
opaque arrayType (elemty : LLVMType ctx) (nelem : UInt64) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_const_array"]
opaque constArray (elemty : LLVMType ctx) (vals : @&Array (Value ctx)) : BaseIO (LLVMType ctx)
-- `constString` provides a `String` as a constant array of element type `i8`
@[extern "lean_llvm_const_string"]
opaque constString (ctx : Context) (str : @&String) : BaseIO (Value ctx)
@[extern "lean_llvm_const_pointer_null"]
opaque constPointerNull (elemty : LLVMType ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_get_undef"]
opaque getUndef (elemty : LLVMType ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_create_builder_in_context"]
opaque createBuilderInContext (ctx : Context) : BaseIO (Builder ctx)
@[extern "lean_llvm_append_basic_block_in_context"]
opaque appendBasicBlockInContext (ctx : Context) (fn : Value ctx) (name : @&String) : BaseIO (BasicBlock ctx)
@[extern "lean_llvm_position_builder_at_end"]
opaque positionBuilderAtEnd (builder : Builder ctx) (bb : BasicBlock ctx) : BaseIO Unit
@[extern "lean_llvm_build_call"]
opaque buildCall (builder : Builder ctx) (fn : Value ctx) (args : @&Array (Value ctx)) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_set_tail_call"]
opaque setTailCall (fn : Value ctx) (istail : Bool) : BaseIO Unit
@[extern "lean_llvm_build_cond_br"]
opaque buildCondBr (builder : Builder ctx) (if_ : Value ctx) (thenbb : BasicBlock ctx) (elsebb : BasicBlock ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_build_br"]
opaque buildBr (builder : Builder ctx) (bb : BasicBlock ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_build_alloca"]
opaque buildAlloca (builder : Builder ctx) (ty : LLVMType ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_load"]
opaque buildLoad (builder : Builder ctx) (val : Value ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_store"]
opaque buildStore (builder : Builder ctx) (val : Value ctx) (store_loc_ptr : Value ctx) : BaseIO Unit
@[extern "lean_llvm_build_ret"]
opaque buildRet (builder : Builder ctx) (val : Value ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_build_unreachable"]
opaque buildUnreachable (builder : Builder ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_build_gep"]
opaque buildGEP (builder : Builder ctx) (base : Value ctx) (ixs : @&Array (Value ctx)) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_inbounds_gep"]
opaque buildInBoundsGEP (builder : Builder ctx) (base : Value ctx) (ixs : @&Array (Value ctx)) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_pointer_cast"]
opaque buildPointerCast (builder : Builder ctx) (val : Value ctx) (destTy : LLVMType ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_sext"]
opaque buildSext (builder : Builder ctx) (val : Value ctx) (destTy : LLVMType ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_zext"]
opaque buildZext (builder : Builder ctx) (val : Value ctx) (destTy : LLVMType ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_sext_or_trunc"]
opaque buildSextOrTrunc (builder : Builder ctx) (val : Value ctx) (destTy : LLVMType ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_switch"]
opaque buildSwitch (builder : Builder ctx) (val : Value ctx) (elseBB : BasicBlock ctx) (numCasesHint : UInt64) : BaseIO (Value ctx)
@[extern "lean_llvm_build_ptr_to_int"]
opaque buildPtrToInt (builder : Builder ctx) (ptr : Value ctx) (destTy : LLVMType ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_mul"]
opaque buildMul (builder : Builder ctx) (x y : Value ctx) (name : @&String) : BaseIO (Value ctx)
@[extern "lean_llvm_build_add"]
opaque buildAdd (builder : Builder ctx) (x y : Value ctx) (name : @&String) : BaseIO (Value ctx)
@[extern "lean_llvm_build_sub"]
opaque buildSub (builder : Builder ctx) (x y : Value ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_not"]
opaque buildNot (builder : Builder ctx) (x : Value ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_build_icmp"]
opaque buildICmp (builder : Builder ctx) (predicate : IntPredicate) (x y : Value ctx) (name : @&String := "") : BaseIO (Value ctx)
@[extern "lean_llvm_add_case"]
opaque addCase (switch onVal : Value ctx) (destBB : BasicBlock ctx) : BaseIO Unit
@[extern "lean_llvm_get_insert_block"]
opaque getInsertBlock (builder : Builder ctx) : BaseIO (BasicBlock ctx)
@[extern "lean_llvm_clear_insertion_position"]
opaque clearInsertionPosition (builder : Builder ctx) : BaseIO Unit
@[extern "lean_llvm_get_basic_block_parent"]
opaque getBasicBlockParent (bb : BasicBlock ctx) : BaseIO (Value ctx)
@[extern "lean_llvm_type_of"]
opaque typeOf (val : Value ctx) : BaseIO (LLVMType ctx)
@[extern "lean_llvm_const_int"]
opaque constInt (intty : LLVMType ctx) (value : UInt64) (signExtend : @Bool := false) : BaseIO (Value ctx)
@[extern "lean_llvm_print_module_to_string"]
opaque printModuletoString (mod : Module ctx) : BaseIO (String)
@[extern "lean_llvm_print_module_to_file"]
opaque printModuletoFile (mod : Module ctx) (file : @&String) : BaseIO Unit
@[extern "llvm_count_params"]
opaque countParams (fn : Value ctx) : UInt64
@[extern "llvm_get_param"]
opaque getParam (fn : Value ctx) (ix : UInt64) : BaseIO (Value ctx)
@[extern "lean_llvm_create_memory_buffer_with_contents_of_file"]
opaque createMemoryBufferWithContentsOfFile (path : @&String) : BaseIO (MemoryBuffer ctx)
@[extern "lean_llvm_parse_bitcode"]
opaque parseBitcode (ctx : Context) (membuf : MemoryBuffer ctx) : BaseIO (Module ctx)
@[extern "lean_llvm_link_modules"]
opaque linkModules (dest : Module ctx) (src : Module ctx) : BaseIO Unit
@[extern "lean_llvm_get_default_target_triple"]
opaque getDefaultTargetTriple : BaseIO String
@[extern "lean_llvm_get_target_from_triple"]
opaque getTargetFromTriple (triple : @&String) : BaseIO (Target ctx)
@[extern "lean_llvm_create_target_machine"]
opaque createTargetMachine (target : Target ctx) (tripleStr : @&String) (cpu : @&String) (features : @&String) : BaseIO (TargetMachine ctx)
@[extern "lean_llvm_target_machine_emit_to_file"]
opaque targetMachineEmitToFile (targetMachine : TargetMachine ctx) (module : Module ctx) (filepath : @&String) (codegenType : LLVM.CodegenFileType) : BaseIO Unit
@[extern "lean_llvm_create_pass_manager"]
opaque createPassManager : BaseIO (PassManager ctx)
@[extern "lean_llvm_dispose_pass_manager"]
opaque disposePassManager (pm : PassManager ctx) : BaseIO Unit
@[extern "lean_llvm_run_pass_manager"]
opaque runPassManager (pm : PassManager ctx) (mod : Module ctx): BaseIO Unit
@[extern "lean_llvm_create_pass_manager_builder"]
opaque createPassManagerBuilder : BaseIO (PassManagerBuilder ctx)
@[extern "lean_llvm_dispose_pass_manager_builder"]
opaque disposePassManagerBuilder (pmb : PassManagerBuilder ctx) : BaseIO Unit
@[extern "lean_llvm_pass_manager_builder_set_opt_level"]
opaque PassManagerBuilder.setOptLevel (pmb : PassManagerBuilder ctx) (optLevel : unsigned) : BaseIO Unit
@[extern "lean_llvm_pass_manager_builder_populate_module_pass_manager"]
opaque PassManagerBuilder.populateModulePassManager (pmb : PassManagerBuilder ctx) (pm : PassManager ctx): BaseIO Unit
@[extern "lean_llvm_dispose_target_machine"]
opaque disposeTargetMachine (tm : TargetMachine ctx) : BaseIO Unit
@[extern "lean_llvm_dispose_module"]
opaque disposeModule (m : Module ctx) : BaseIO Unit
-- Helper to add a function if it does not exist, and to return the function handle if it does.
def getOrAddFunction(m : Module ctx) (name : String) (type : LLVMType ctx) : BaseIO (Value ctx) := do
match (← getNamedFunction m name) with
| .some fn => return fn
| .none => addFunction m name type
def getOrAddGlobal(m : Module ctx) (name : String) (type : LLVMType ctx) : BaseIO (Value ctx) := do
match (← getNamedGlobal m name) with
| .some fn => return fn
| .none => addGlobal m name type
def i1Type (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
LLVM.intTypeInContext ctx 1
def i8Type (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
LLVM.intTypeInContext ctx 8
def i16Type (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
LLVM.intTypeInContext ctx 16
def i32Type (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
LLVM.intTypeInContext ctx 32
def i64Type (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
LLVM.intTypeInContext ctx 64
def voidPtrType (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
do LLVM.pointerType (← LLVM.intTypeInContext ctx 8)
def i8PtrType (ctx : LLVM.Context) : BaseIO (LLVM.LLVMType ctx) :=
voidPtrType ctx
def constTrue (ctx : Context) : BaseIO (Value ctx) :=
do constInt (← i1Type ctx) 1 (signExtend := false)
def constFalse (ctx : Context) : BaseIO (Value ctx) :=
do constInt (← i1Type ctx) 0 (signExtend := false)
def constInt' (ctx : Context) (width : UInt64) (value : UInt64) (signExtend : Bool := false) : BaseIO (Value ctx) :=
do constInt (← LLVM.intTypeInContext ctx width) value signExtend
def constInt1 (ctx : Context) (value : UInt64) (signExtend : Bool := false) : BaseIO (Value ctx) :=
constInt' ctx 1 value signExtend
def constInt8 (ctx : Context) (value : UInt64) (signExtend : Bool := false) : BaseIO (Value ctx) :=
constInt' ctx 8 value signExtend
def constInt32 (ctx : Context) (value : UInt64) (signExtend : Bool := false) : BaseIO (Value ctx) :=
constInt' ctx 32 value signExtend
def constInt64 (ctx : Context) (value : UInt64) (signExtend : Bool := false) : BaseIO (Value ctx) :=
constInt' ctx 64 value signExtend
def constIntUnsigned (ctx : Context) (value : UInt64) (signExtend : Bool := false) : BaseIO (Value ctx) :=
constInt' ctx 64 value signExtend
end LLVM
|
b18ba8705b13425431aab6845b1237d14efd34cf | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/topology/metric_space/basic.lean | 91fb9271b1a34a5b4ec6f271cf8fba923151fe26 | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 101,122 | lean | /-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import topology.metric_space.emetric_space
import topology.algebra.ordered.basic
import data.fintype.intervals
/-!
# Metric spaces
This file defines metric spaces. Many definitions and theorems expected
on metric spaces are already introduced on uniform spaces and topological spaces.
For example: open and closed sets, compactness, completeness, continuity and uniform continuity
## Main definitions
* `has_dist α`: Endows a space `α` with a function `dist a b`.
* `pseudo_metric_space α`: A space endowed with a distance function, which can
be zero even if the two elements are non-equal.
* `metric.ball x ε`: The set of all points `y` with `dist y x < ε`.
* `metric.bounded s`: Whether a subset of a `pseudo_metric_space` is bounded.
* `metric_space α`: A `pseudo_metric_space` with the guarantee `dist x y = 0 → x = y`.
Additional useful definitions:
* `nndist a b`: `dist` as a function to the non-negative reals.
* `metric.closed_ball x ε`: The set of all points `y` with `dist y x ≤ ε`.
* `metric.sphere x ε`: The set of all points `y` with `dist y x = ε`.
* `proper_space α`: A `pseudo_metric_space` where all closed balls are compact.
* `metric.diam s` : The `supr` of the distances of members of `s`.
Defined in terms of `emetric.diam`, for better handling of the case when it should be infinite.
TODO (anyone): Add "Main results" section.
## Implementation notes
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory of `pseudo_metric_space`, where we don't require `dist x y = 0 → x = y` and we specialize
to `metric_space` at the end.
## Tags
metric, pseudo_metric, dist
-/
open set filter topological_space
noncomputable theory
open_locale uniformity topological_space big_operators filter nnreal ennreal
universes u v w
variables {α : Type u} {β : Type v}
/-- Construct a uniform structure core from a distance function and metric space axioms.
This is a technical construction that can be immediately used to construct a uniform structure
from a distance function and metric space axioms but is also useful when discussing
metrizable topologies, see `pseudo_metric_space.of_metrizable`. -/
def uniform_space.core_of_dist {α : Type*} (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : uniform_space.core α :=
{ uniformity := (⨅ ε>0, 𝓟 {p:α×α | dist p.1 p.2 < ε}),
refl := le_infi $ assume ε, le_infi $
by simp [set.subset_def, id_rel, dist_self, (>)] {contextual := tt},
comp := le_infi $ assume ε, le_infi $ assume h, lift'_le
(mem_infi_of_mem (ε / 2) $ mem_infi_of_mem (div_pos h zero_lt_two) (subset.refl _)) $
have ∀ (a b c : α), dist a c < ε / 2 → dist c b < ε / 2 → dist a b < ε,
from assume a b c hac hcb,
calc dist a b ≤ dist a c + dist c b : dist_triangle _ _ _
... < ε / 2 + ε / 2 : add_lt_add hac hcb
... = ε : by rw [div_add_div_same, add_self_div_two],
by simpa [comp_rel],
symm := tendsto_infi.2 $ assume ε, tendsto_infi.2 $ assume h,
tendsto_infi' ε $ tendsto_infi' h $ tendsto_principal_principal.2 $ by simp [dist_comm] }
/-- Construct a uniform structure from a distance function and metric space axioms -/
def uniform_space_of_dist
(dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : uniform_space α :=
uniform_space.of_core (uniform_space.core_of_dist dist dist_self dist_comm dist_triangle)
/-- The distance function (given an ambient metric space on `α`), which returns
a nonnegative real number `dist x y` given `x y : α`. -/
class has_dist (α : Type*) := (dist : α → α → ℝ)
export has_dist (dist)
-- the uniform structure and the emetric space structure are embedded in the metric space structure
-- to avoid instance diamond issues. See Note [forgetful inheritance].
/-- Metric space
Each metric space induces a canonical `uniform_space` and hence a canonical `topological_space`.
This is enforced in the type class definition, by extending the `uniform_space` structure. When
instantiating a `metric_space` structure, the uniformity fields are not necessary, they will be
filled in by default. In the same way, each metric space induces an emetric space structure.
It is included in the structure, but filled in by default.
-/
class pseudo_metric_space (α : Type u) extends has_dist α : Type u :=
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(edist : α → α → ℝ≥0∞ := λx y, ennreal.of_real (dist x y))
(edist_dist : ∀ x y : α, edist x y = ennreal.of_real (dist x y) . control_laws_tac)
(to_uniform_space : uniform_space α := uniform_space_of_dist dist dist_self dist_comm dist_triangle)
(uniformity_dist : 𝓤 α = ⨅ ε>0, 𝓟 {p:α×α | dist p.1 p.2 < ε} . control_laws_tac)
variables [pseudo_metric_space α]
@[priority 100] -- see Note [lower instance priority]
instance metric_space.to_uniform_space' : uniform_space α :=
pseudo_metric_space.to_uniform_space
@[priority 200] -- see Note [lower instance priority]
instance pseudo_metric_space.to_has_edist : has_edist α := ⟨pseudo_metric_space.edist⟩
/-- Construct a pseudo-metric space structure whose underlying topological space structure
(definitionally) agrees which a pre-existing topology which is compatible with a given distance
function. -/
def pseudo_metric_space.of_metrizable {α : Type*} [topological_space α] (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(H : ∀ s : set α, is_open s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) :
pseudo_metric_space α :=
{ dist := dist,
dist_self := dist_self,
dist_comm := dist_comm,
dist_triangle := dist_triangle,
to_uniform_space := { is_open_uniformity := begin
dsimp only [uniform_space.core_of_dist],
intros s,
change is_open s ↔ _,
rw H s,
apply forall_congr, intro x,
apply forall_congr, intro x_in,
erw (has_basis_binfi_principal _ nonempty_Ioi).mem_iff,
{ apply exists_congr, intros ε,
apply exists_congr, intros ε_pos,
simp only [prod.forall, set_of_subset_set_of],
split,
{ rintros h _ y H rfl,
exact h y H },
{ intros h y hxy,
exact h _ _ hxy rfl } },
{ exact λ r (hr : 0 < r) p (hp : 0 < p), ⟨min r p, lt_min hr hp,
λ x (hx : dist _ _ < _), lt_of_lt_of_le hx (min_le_left r p),
λ x (hx : dist _ _ < _), lt_of_lt_of_le hx (min_le_right r p)⟩ },
{ apply_instance }
end,
..uniform_space.core_of_dist dist dist_self dist_comm dist_triangle },
uniformity_dist := rfl }
@[simp] theorem dist_self (x : α) : dist x x = 0 := pseudo_metric_space.dist_self x
theorem dist_comm (x y : α) : dist x y = dist y x := pseudo_metric_space.dist_comm x y
theorem edist_dist (x y : α) : edist x y = ennreal.of_real (dist x y) :=
pseudo_metric_space.edist_dist x y
theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
pseudo_metric_space.dist_triangle x y z
theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y :=
by rw dist_comm z; apply dist_triangle
theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z :=
by rw dist_comm y; apply dist_triangle
lemma dist_triangle4 (x y z w : α) :
dist x w ≤ dist x y + dist y z + dist z w :=
calc dist x w ≤ dist x z + dist z w : dist_triangle x z w
... ≤ (dist x y + dist y z) + dist z w : add_le_add_right (dist_triangle x y z) _
lemma dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) :=
by { rw [add_left_comm, dist_comm x₁, ← add_assoc], apply dist_triangle4 }
lemma dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ :=
by { rw [add_right_comm, dist_comm y₁], apply dist_triangle4 }
/-- The triangle (polygon) inequality for sequences of points; `finset.Ico` version. -/
lemma dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
dist (f m) (f n) ≤ ∑ i in finset.Ico m n, dist (f i) (f (i + 1)) :=
begin
revert n,
apply nat.le_induction,
{ simp only [finset.sum_empty, finset.Ico.self_eq_empty, dist_self] },
{ assume n hn hrec,
calc dist (f m) (f (n+1)) ≤ dist (f m) (f n) + dist _ _ : dist_triangle _ _ _
... ≤ ∑ i in finset.Ico m n, _ + _ : add_le_add hrec (le_refl _)
... = ∑ i in finset.Ico m (n+1), _ :
by rw [finset.Ico.succ_top hn, finset.sum_insert, add_comm]; simp }
end
/-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/
lemma dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
dist (f 0) (f n) ≤ ∑ i in finset.range n, dist (f i) (f (i + 1)) :=
finset.Ico.zero_bot n ▸ dist_le_Ico_sum_dist f (nat.zero_le n)
/-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
with an upper estimate. -/
lemma dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n)
{d : ℕ → ℝ} (hd : ∀ {k}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f m) (f n) ≤ ∑ i in finset.Ico m n, d i :=
le_trans (dist_le_Ico_sum_dist f hmn) $
finset.sum_le_sum $ λ k hk, hd (finset.Ico.mem.1 hk).1 (finset.Ico.mem.1 hk).2
/-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced
with an upper estimate. -/
lemma dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ)
{d : ℕ → ℝ} (hd : ∀ {k}, k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f 0) (f n) ≤ ∑ i in finset.range n, d i :=
finset.Ico.zero_bot n ▸ dist_le_Ico_sum_of_dist_le (zero_le n) (λ _ _, hd)
theorem swap_dist : function.swap (@dist α _) = dist :=
by funext x y; exact dist_comm _ _
theorem abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
abs_sub_le_iff.2
⟨sub_le_iff_le_add.2 (dist_triangle _ _ _),
sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
have 2 * dist x y ≥ 0,
from calc 2 * dist x y = dist x y + dist y x : by rw [dist_comm x y, two_mul]
... ≥ 0 : by rw ← dist_self x; apply dist_triangle,
nonneg_of_mul_nonneg_left this zero_lt_two
@[simp] theorem abs_dist {a b : α} : |dist a b| = dist a b :=
abs_of_nonneg dist_nonneg
/-- A version of `has_dist` that takes value in `ℝ≥0`. -/
class has_nndist (α : Type*) := (nndist : α → α → ℝ≥0)
export has_nndist (nndist)
/-- Distance as a nonnegative real number. -/
@[priority 100] -- see Note [lower instance priority]
instance pseudo_metric_space.to_has_nndist : has_nndist α := ⟨λ a b, ⟨dist a b, dist_nonneg⟩⟩
/--Express `nndist` in terms of `edist`-/
lemma nndist_edist (x y : α) : nndist x y = (edist x y).to_nnreal :=
by simp [nndist, edist_dist, real.to_nnreal, max_eq_left dist_nonneg, ennreal.of_real]
/--Express `edist` in terms of `nndist`-/
lemma edist_nndist (x y : α) : edist x y = ↑(nndist x y) :=
by { simpa only [edist_dist, ennreal.of_real_eq_coe_nnreal dist_nonneg] }
@[simp, norm_cast] lemma coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
(edist_nndist x y).symm
@[simp, norm_cast] lemma edist_lt_coe {x y : α} {c : ℝ≥0} :
edist x y < c ↔ nndist x y < c :=
by rw [edist_nndist, ennreal.coe_lt_coe]
@[simp, norm_cast] lemma edist_le_coe {x y : α} {c : ℝ≥0} :
edist x y ≤ c ↔ nndist x y ≤ c :=
by rw [edist_nndist, ennreal.coe_le_coe]
/--In a pseudometric space, the extended distance is always finite-/
lemma edist_lt_top {α : Type*} [pseudo_metric_space α] (x y : α) : edist x y < ⊤ :=
(edist_dist x y).symm ▸ ennreal.of_real_lt_top
/--In a pseudometric space, the extended distance is always finite-/
lemma edist_ne_top (x y : α) : edist x y ≠ ⊤ := (edist_lt_top x y).ne
/--`nndist x x` vanishes-/
@[simp] lemma nndist_self (a : α) : nndist a a = 0 := (nnreal.coe_eq_zero _).1 (dist_self a)
/--Express `dist` in terms of `nndist`-/
lemma dist_nndist (x y : α) : dist x y = ↑(nndist x y) := rfl
@[simp, norm_cast] lemma coe_nndist (x y : α) : ↑(nndist x y) = dist x y :=
(dist_nndist x y).symm
@[simp, norm_cast] lemma dist_lt_coe {x y : α} {c : ℝ≥0} :
dist x y < c ↔ nndist x y < c :=
iff.rfl
@[simp, norm_cast] lemma dist_le_coe {x y : α} {c : ℝ≥0} :
dist x y ≤ c ↔ nndist x y ≤ c :=
iff.rfl
/--Express `nndist` in terms of `dist`-/
lemma nndist_dist (x y : α) : nndist x y = real.to_nnreal (dist x y) :=
by rw [dist_nndist, real.to_nnreal_coe]
theorem nndist_comm (x y : α) : nndist x y = nndist y x :=
by simpa only [dist_nndist, nnreal.coe_eq] using dist_comm x y
/--Triangle inequality for the nonnegative distance-/
theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
dist_triangle _ _ _
theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
dist_triangle_left _ _ _
theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
dist_triangle_right _ _ _
/--Express `dist` in terms of `edist`-/
lemma dist_edist (x y : α) : dist x y = (edist x y).to_real :=
by rw [edist_dist, ennreal.to_real_of_real (dist_nonneg)]
namespace metric
/- instantiate pseudometric space as a topology -/
variables {x y z : α} {ε ε₁ ε₂ : ℝ} {s : set α}
/-- `ball x ε` is the set of all points `y` with `dist y x < ε` -/
def ball (x : α) (ε : ℝ) : set α := {y | dist y x < ε}
@[simp] theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε := iff.rfl
theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw dist_comm; refl
theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
dist_nonneg.trans_lt hy
theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε :=
show dist x x < ε, by rw dist_self; assumption
@[simp] lemma nonempty_ball : (ball x ε).nonempty ↔ 0 < ε :=
⟨λ ⟨x, hx⟩, pos_of_mem_ball hx, λ h, ⟨x, mem_ball_self h⟩⟩
@[simp] lemma ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 :=
by rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt]
@[simp] lemma ball_zero : ball x 0 = ∅ :=
by rw [ball_eq_empty]
lemma ball_eq_ball (ε : ℝ) (x : α) :
uniform_space.ball x {p | dist p.2 p.1 < ε} = metric.ball x ε := rfl
lemma ball_eq_ball' (ε : ℝ) (x : α) :
uniform_space.ball x {p | dist p.1 p.2 < ε} = metric.ball x ε :=
by { ext, simp [dist_comm, uniform_space.ball] }
@[simp] lemma Union_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
Union_eq_univ_iff.2 $ λ y, exists_nat_gt (dist y x)
@[simp] lemma Union_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
Union_eq_univ_iff.2 $ λ y, (exists_nat_gt (dist y x)).imp $ λ n hn,
hn.trans (lt_add_one _)
/-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
def closed_ball (x : α) (ε : ℝ) := {y | dist y x ≤ ε}
@[simp] theorem mem_closed_ball : y ∈ closed_ball x ε ↔ dist y x ≤ ε := iff.rfl
/-- `sphere x ε` is the set of all points `y` with `dist y x = ε` -/
def sphere (x : α) (ε : ℝ) := {y | dist y x = ε}
@[simp] theorem mem_sphere : y ∈ sphere x ε ↔ dist y x = ε := iff.rfl
theorem mem_closed_ball' : y ∈ closed_ball x ε ↔ dist x y ≤ ε :=
by { rw dist_comm, refl }
theorem mem_closed_ball_self (h : 0 ≤ ε) : x ∈ closed_ball x ε :=
show dist x x ≤ ε, by rw dist_self; assumption
@[simp] lemma nonempty_closed_ball : (closed_ball x ε).nonempty ↔ 0 ≤ ε :=
⟨λ ⟨x, hx⟩, dist_nonneg.trans hx, λ h, ⟨x, mem_closed_ball_self h⟩⟩
@[simp] lemma closed_ball_eq_empty : closed_ball x ε = ∅ ↔ ε < 0 :=
by rw [← not_nonempty_iff_eq_empty, nonempty_closed_ball, not_le]
theorem ball_subset_closed_ball : ball x ε ⊆ closed_ball x ε :=
assume y (hy : _ < _), le_of_lt hy
theorem sphere_subset_closed_ball : sphere x ε ⊆ closed_ball x ε :=
λ y, le_of_eq
lemma ball_disjoint_ball (x y : α) (rx ry : ℝ) (h : rx + ry ≤ dist x y) :
disjoint (ball x rx) (ball y ry) :=
begin
rw disjoint_left,
assume a ax ay,
apply lt_irrefl (dist x y),
calc dist x y ≤ dist x a + dist a y : dist_triangle _ _ _
... < rx + ry : add_lt_add (mem_ball'.1 ax) (mem_ball.1 ay)
... ≤ dist x y : h
end
theorem sphere_disjoint_ball : disjoint (sphere x ε) (ball x ε) :=
λ y ⟨hy₁, hy₂⟩, absurd hy₁ $ ne_of_lt hy₂
@[simp] theorem ball_union_sphere : ball x ε ∪ sphere x ε = closed_ball x ε :=
set.ext $ λ y, (@le_iff_lt_or_eq ℝ _ _ _).symm
@[simp] theorem sphere_union_ball : sphere x ε ∪ ball x ε = closed_ball x ε :=
by rw [union_comm, ball_union_sphere]
@[simp] theorem closed_ball_diff_sphere : closed_ball x ε \ sphere x ε = ball x ε :=
by rw [← ball_union_sphere, set.union_diff_cancel_right sphere_disjoint_ball.symm]
@[simp] theorem closed_ball_diff_ball : closed_ball x ε \ ball x ε = sphere x ε :=
by rw [← ball_union_sphere, set.union_diff_cancel_left sphere_disjoint_ball.symm]
theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε :=
by simp [dist_comm]
theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ :=
λ y (yx : _ < ε₁), lt_of_lt_of_le yx h
lemma ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ :=
λ z hz, calc
dist z y ≤ dist z x + dist x y : dist_triangle _ _ _
... < ε₁ + dist x y : add_lt_add_right hz _
... ≤ ε₂ : h
theorem closed_ball_subset_closed_ball (h : ε₁ ≤ ε₂) :
closed_ball x ε₁ ⊆ closed_ball x ε₂ :=
λ y (yx : _ ≤ ε₁), le_trans yx h
lemma closed_ball_subset_closed_ball' (h : ε₁ + dist x y ≤ ε₂) :
closed_ball x ε₁ ⊆ closed_ball y ε₂ :=
λ z hz, calc
dist z y ≤ dist z x + dist x y : dist_triangle _ _ _
... ≤ ε₁ + dist x y : add_le_add_right hz _
... ≤ ε₂ : h
theorem closed_ball_subset_ball (h : ε₁ < ε₂) :
closed_ball x ε₁ ⊆ ball x ε₂ :=
λ y (yh : dist y x ≤ ε₁), lt_of_le_of_lt yh h
lemma dist_le_add_of_nonempty_closed_ball_inter_closed_ball
(h : (closed_ball x ε₁ ∩ closed_ball y ε₂).nonempty) :
dist x y ≤ ε₁ + ε₂ :=
let ⟨z, hz⟩ := h in calc
dist x y ≤ dist z x + dist z y : dist_triangle_left _ _ _
... ≤ ε₁ + ε₂ : add_le_add hz.1 hz.2
lemma dist_lt_add_of_nonempty_closed_ball_inter_ball (h : (closed_ball x ε₁ ∩ ball y ε₂).nonempty) :
dist x y < ε₁ + ε₂ :=
let ⟨z, hz⟩ := h in calc
dist x y ≤ dist z x + dist z y : dist_triangle_left _ _ _
... < ε₁ + ε₂ : add_lt_add_of_le_of_lt hz.1 hz.2
lemma dist_lt_add_of_nonempty_ball_inter_closed_ball (h : (ball x ε₁ ∩ closed_ball y ε₂).nonempty) :
dist x y < ε₁ + ε₂ :=
begin
rw inter_comm at h,
rw [add_comm, dist_comm],
exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
end
lemma dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).nonempty) :
dist x y < ε₁ + ε₂ :=
dist_lt_add_of_nonempty_closed_ball_inter_ball $
h.mono (inter_subset_inter ball_subset_closed_ball subset.rfl)
@[simp] lemma Union_closed_ball_nat (x : α) : (⋃ n : ℕ, closed_ball x n) = univ :=
Union_eq_univ_iff.2 $ λ y, exists_nat_ge (dist y x)
theorem ball_disjoint (h : ε₁ + ε₂ ≤ dist x y) : ball x ε₁ ∩ ball y ε₂ = ∅ :=
eq_empty_iff_forall_not_mem.2 $ λ z ⟨h₁, h₂⟩,
not_lt_of_le (dist_triangle_left x y z)
(lt_of_lt_of_le (add_lt_add h₁ h₂) h)
theorem ball_disjoint_same (h : ε ≤ dist x y / 2) : ball x ε ∩ ball y ε = ∅ :=
ball_disjoint $ by rwa [← two_mul, ← le_div_iff' (@zero_lt_two ℝ _ _)]
theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ :=
λ z zx, by rw ← add_sub_cancel'_right ε₁ ε₂; exact
lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
ball_subset $ by rw sub_self_div_two; exact le_of_lt h
theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
⟨_, sub_pos.2 h, ball_subset $ by rw sub_sub_self⟩
theorem uniformity_basis_dist :
(𝓤 α).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {p:α×α | dist p.1 p.2 < ε}) :=
begin
rw ← pseudo_metric_space.uniformity_dist.symm,
refine has_basis_binfi_principal _ nonempty_Ioi,
exact λ r (hr : 0 < r) p (hp : 0 < p), ⟨min r p, lt_min hr hp,
λ x (hx : dist _ _ < _), lt_of_lt_of_le hx (min_le_left r p),
λ x (hx : dist _ _ < _), lt_of_lt_of_le hx (min_le_right r p)⟩
end
/-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,
and `uniformity_basis_dist_inv_nat_pos`. -/
protected theorem mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i (hi : p i), f i ≤ ε) :
(𝓤 α).has_basis p (λ i, {p:α×α | dist p.1 p.2 < f i}) :=
begin
refine ⟨λ s, uniformity_basis_dist.mem_iff.trans _⟩,
split,
{ rintros ⟨ε, ε₀, hε⟩,
obtain ⟨i, hi, H⟩ : ∃ i (hi : p i), f i ≤ ε, from hf ε₀,
exact ⟨i, hi, λ x (hx : _ < _), hε $ lt_of_lt_of_le hx H⟩ },
{ exact λ ⟨i, hi, H⟩, ⟨f i, hf₀ i hi, H⟩ }
end
theorem uniformity_basis_dist_inv_nat_succ :
(𝓤 α).has_basis (λ _, true) (λ n:ℕ, {p:α×α | dist p.1 p.2 < 1 / (↑n+1) }) :=
metric.mk_uniformity_basis (λ n _, div_pos zero_lt_one $ nat.cast_add_one_pos n)
(λ ε ε0, (exists_nat_one_div_lt ε0).imp $ λ n hn, ⟨trivial, le_of_lt hn⟩)
theorem uniformity_basis_dist_inv_nat_pos :
(𝓤 α).has_basis (λ n:ℕ, 0<n) (λ n:ℕ, {p:α×α | dist p.1 p.2 < 1 / ↑n }) :=
metric.mk_uniformity_basis (λ n hn, div_pos zero_lt_one $ nat.cast_pos.2 hn)
(λ ε ε0, let ⟨n, hn⟩ := exists_nat_one_div_lt ε0 in ⟨n+1, nat.succ_pos n, hn.le⟩)
theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).has_basis (λ n:ℕ, true) (λ n:ℕ, {p:α×α | dist p.1 p.2 < r ^ n }) :=
metric.mk_uniformity_basis (λ n hn, pow_pos h0 _)
(λ ε ε0, let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1 in ⟨n, trivial, hn.le⟩)
theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
(𝓤 α).has_basis (λ r : ℝ, 0 < r ∧ r < R) (λ r, {p : α × α | dist p.1 p.2 < r}) :=
metric.mk_uniformity_basis (λ r, and.left) $ λ r hr,
⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 $ or.inr (half_lt_self hR)⟩,
min_le_left _ _⟩
/-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
form a basis of `𝓤 α`.
Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
More can be easily added if needed in the future. -/
protected theorem mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x (hx : p x), f x ≤ ε) :
(𝓤 α).has_basis p (λ x, {p:α×α | dist p.1 p.2 ≤ f x}) :=
begin
refine ⟨λ s, uniformity_basis_dist.mem_iff.trans _⟩,
split,
{ rintros ⟨ε, ε₀, hε⟩,
rcases exists_between ε₀ with ⟨ε', hε'⟩,
rcases hf ε' hε'.1 with ⟨i, hi, H⟩,
exact ⟨i, hi, λ x (hx : _ ≤ _), hε $ lt_of_le_of_lt (le_trans hx H) hε'.2⟩ },
{ exact λ ⟨i, hi, H⟩, ⟨f i, hf₀ i hi, λ x (hx : _ < _), H (le_of_lt hx)⟩ }
end
/-- Contant size closed neighborhoods of the diagonal form a basis
of the uniformity filter. -/
theorem uniformity_basis_dist_le :
(𝓤 α).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {p:α×α | dist p.1 p.2 ≤ ε}) :=
metric.mk_uniformity_basis_le (λ _, id) (λ ε ε₀, ⟨ε, ε₀, le_refl ε⟩)
theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).has_basis (λ n:ℕ, true) (λ n:ℕ, {p:α×α | dist p.1 p.2 ≤ r ^ n }) :=
metric.mk_uniformity_basis_le (λ n hn, pow_pos h0 _)
(λ ε ε0, let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1 in ⟨n, trivial, hn.le⟩)
theorem mem_uniformity_dist {s : set (α×α)} :
s ∈ 𝓤 α ↔ (∃ε>0, ∀{a b:α}, dist a b < ε → (a, b) ∈ s) :=
uniformity_basis_dist.mem_uniformity_iff
/-- A constant size neighborhood of the diagonal is an entourage. -/
theorem dist_mem_uniformity {ε:ℝ} (ε0 : 0 < ε) :
{p:α×α | dist p.1 p.2 < ε} ∈ 𝓤 α :=
mem_uniformity_dist.2 ⟨ε, ε0, λ a b, id⟩
theorem uniform_continuous_iff [pseudo_metric_space β] {f : α → β} :
uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0,
∀{a b:α}, dist a b < δ → dist (f a) (f b) < ε :=
uniformity_basis_dist.uniform_continuous_iff uniformity_basis_dist
lemma uniform_continuous_on_iff [pseudo_metric_space β] {f : α → β} {s : set α} :
uniform_continuous_on f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x y ∈ s, dist x y < δ → dist (f x) (f y) < ε :=
metric.uniformity_basis_dist.uniform_continuous_on_iff metric.uniformity_basis_dist
lemma uniform_continuous_on_iff_le [pseudo_metric_space β] {f : α → β} {s : set α} :
uniform_continuous_on f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε :=
metric.uniformity_basis_dist_le.uniform_continuous_on_iff metric.uniformity_basis_dist_le
theorem uniform_embedding_iff [pseudo_metric_space β] {f : α → β} :
uniform_embedding f ↔ function.injective f ∧ uniform_continuous f ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
uniform_embedding_def'.trans $ and_congr iff.rfl $ and_congr iff.rfl
⟨λ H δ δ0, let ⟨t, tu, ht⟩ := H _ (dist_mem_uniformity δ0),
⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 tu in
⟨ε, ε0, λ a b h, ht _ _ (hε h)⟩,
λ H s su, let ⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 su, ⟨ε, ε0, hε⟩ := H _ δ0 in
⟨_, dist_mem_uniformity ε0, λ a b h, hδ (hε h)⟩⟩
/-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y`. -/
theorem controlled_of_uniform_embedding [pseudo_metric_space β] {f : α → β} :
uniform_embedding f →
(∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
(∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ) :=
begin
assume h,
exact ⟨uniform_continuous_iff.1 (uniform_embedding_iff.1 h).2.1, (uniform_embedding_iff.1 h).2.2⟩
end
theorem totally_bounded_iff {s : set α} :
totally_bounded s ↔ ∀ ε > 0, ∃t : set α, finite t ∧ s ⊆ ⋃y∈t, ball y ε :=
⟨λ H ε ε0, H _ (dist_mem_uniformity ε0),
λ H r ru, let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 ru,
⟨t, ft, h⟩ := H ε ε0 in
⟨t, ft, subset.trans h $ Union_subset_Union $ λ y, Union_subset_Union $ λ yt z, hε⟩⟩
/-- A pseudometric space space is totally bounded if one can reconstruct up to any ε>0 any element
of the space from finitely many data. -/
lemma totally_bounded_of_finite_discretization {s : set α}
(H : ∀ε > (0 : ℝ), ∃ (β : Type u) (_ : fintype β) (F : s → β),
∀x y, F x = F y → dist (x:α) y < ε) :
totally_bounded s :=
begin
cases s.eq_empty_or_nonempty with hs hs,
{ rw hs, exact totally_bounded_empty },
rcases hs with ⟨x0, hx0⟩,
haveI : inhabited s := ⟨⟨x0, hx0⟩⟩,
refine totally_bounded_iff.2 (λ ε ε0, _),
rcases H ε ε0 with ⟨β, fβ, F, hF⟩,
resetI,
let Finv := function.inv_fun F,
refine ⟨range (subtype.val ∘ Finv), finite_range _, λ x xs, _⟩,
let x' := Finv (F ⟨x, xs⟩),
have : F x' = F ⟨x, xs⟩ := function.inv_fun_eq ⟨⟨x, xs⟩, rfl⟩,
simp only [set.mem_Union, set.mem_range],
exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
end
theorem finite_approx_of_totally_bounded {s : set α} (hs : totally_bounded s) :
∀ ε > 0, ∃ t ⊆ s, finite t ∧ s ⊆ ⋃y∈t, ball y ε :=
begin
intros ε ε_pos,
rw totally_bounded_iff_subset at hs,
exact hs _ (dist_mem_uniformity ε_pos),
end
/-- Expressing locally uniform convergence on a set using `dist`. -/
lemma tendsto_locally_uniformly_on_iff {ι : Type*} [topological_space β]
{F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :
tendsto_locally_uniformly_on F f p s ↔
∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε :=
begin
refine ⟨λ H ε hε, H _ (dist_mem_uniformity hε), λ H u hu x hx, _⟩,
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩,
rcases H ε εpos x hx with ⟨t, ht, Ht⟩,
exact ⟨t, ht, Ht.mono (λ n hs x hx, hε (hs x hx))⟩
end
/-- Expressing uniform convergence on a set using `dist`. -/
lemma tendsto_uniformly_on_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :
tendsto_uniformly_on F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε :=
begin
refine ⟨λ H ε hε, H _ (dist_mem_uniformity hε), λ H u hu, _⟩,
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩,
exact (H ε εpos).mono (λ n hs x hx, hε (hs x hx))
end
/-- Expressing locally uniform convergence using `dist`. -/
lemma tendsto_locally_uniformly_iff {ι : Type*} [topological_space β]
{F : ι → β → α} {f : β → α} {p : filter ι} :
tendsto_locally_uniformly F f p ↔
∀ ε > 0, ∀ (x : β), ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε :=
by simp only [← tendsto_locally_uniformly_on_univ, tendsto_locally_uniformly_on_iff,
nhds_within_univ, mem_univ, forall_const, exists_prop]
/-- Expressing uniform convergence using `dist`. -/
lemma tendsto_uniformly_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} :
tendsto_uniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε :=
by { rw [← tendsto_uniformly_on_univ, tendsto_uniformly_on_iff], simp }
protected lemma cauchy_iff {f : filter α} :
cauchy f ↔ ne_bot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x y ∈ t, dist x y < ε :=
uniformity_basis_dist.cauchy_iff
theorem nhds_basis_ball : (𝓝 x).has_basis (λ ε:ℝ, 0 < ε) (ball x) :=
nhds_basis_uniformity uniformity_basis_dist
theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ε>0, ball x ε ⊆ s :=
nhds_basis_ball.mem_iff
theorem eventually_nhds_iff {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ε>0, ∀ ⦃y⦄, dist y x < ε → p y :=
mem_nhds_iff
lemma eventually_nhds_iff_ball {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε>0, ∀ y ∈ ball x ε, p y :=
mem_nhds_iff
theorem nhds_basis_closed_ball : (𝓝 x).has_basis (λ ε:ℝ, 0 < ε) (closed_ball x) :=
nhds_basis_uniformity uniformity_basis_dist_le
theorem nhds_basis_ball_inv_nat_succ :
(𝓝 x).has_basis (λ _, true) (λ n:ℕ, ball x (1 / (↑n+1))) :=
nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ
theorem nhds_basis_ball_inv_nat_pos :
(𝓝 x).has_basis (λ n, 0<n) (λ n:ℕ, ball x (1 / ↑n)) :=
nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos
theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).has_basis (λ n, true) (λ n:ℕ, ball x (r ^ n)) :=
nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1)
theorem nhds_basis_closed_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).has_basis (λ n, true) (λ n:ℕ, closed_ball x (r ^ n)) :=
nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1)
theorem is_open_iff : is_open s ↔ ∀x∈s, ∃ε>0, ball x ε ⊆ s :=
by simp only [is_open_iff_mem_nhds, mem_nhds_iff]
theorem is_open_ball : is_open (ball x ε) :=
is_open_iff.2 $ λ y, exists_ball_subset_ball
theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
is_open.mem_nhds is_open_ball (mem_ball_self ε0)
theorem closed_ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closed_ball x ε ∈ 𝓝 x :=
mem_of_superset (ball_mem_nhds x ε0) ball_subset_closed_ball
theorem nhds_within_basis_ball {s : set α} :
(𝓝[s] x).has_basis (λ ε:ℝ, 0 < ε) (λ ε, ball x ε ∩ s) :=
nhds_within_has_basis nhds_basis_ball s
theorem mem_nhds_within_iff {t : set α} : s ∈ 𝓝[t] x ↔ ∃ε>0, ball x ε ∩ t ⊆ s :=
nhds_within_basis_ball.mem_iff
theorem tendsto_nhds_within_nhds_within [pseudo_metric_space β] {t : set β} {f : α → β} {a b} :
tendsto f (𝓝[s] a) (𝓝[t] b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε :=
(nhds_within_basis_ball.tendsto_iff nhds_within_basis_ball).trans $
by simp only [inter_comm, mem_inter_iff, and_imp, mem_ball]
theorem tendsto_nhds_within_nhds [pseudo_metric_space β] {f : α → β} {a b} :
tendsto f (𝓝[s] a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → dist x a < δ → dist (f x) b < ε :=
by { rw [← nhds_within_univ b, tendsto_nhds_within_nhds_within],
simp only [mem_univ, true_and] }
theorem tendsto_nhds_nhds [pseudo_metric_space β] {f : α → β} {a b} :
tendsto f (𝓝 a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, dist x a < δ → dist (f x) b < ε :=
nhds_basis_ball.tendsto_iff nhds_basis_ball
theorem continuous_at_iff [pseudo_metric_space β] {f : α → β} {a : α} :
continuous_at f a ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, dist x a < δ → dist (f x) (f a) < ε :=
by rw [continuous_at, tendsto_nhds_nhds]
theorem continuous_within_at_iff [pseudo_metric_space β] {f : α → β} {a : α} {s : set α} :
continuous_within_at f s a ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε :=
by rw [continuous_within_at, tendsto_nhds_within_nhds]
theorem continuous_on_iff [pseudo_metric_space β] {f : α → β} {s : set α} :
continuous_on f s ↔
∀ (b ∈ s) (ε > 0), ∃ δ > 0, ∀a ∈ s, dist a b < δ → dist (f a) (f b) < ε :=
by simp [continuous_on, continuous_within_at_iff]
theorem continuous_iff [pseudo_metric_space β] {f : α → β} :
continuous f ↔
∀b (ε > 0), ∃ δ > 0, ∀a, dist a b < δ → dist (f a) (f b) < ε :=
continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds_nhds
theorem tendsto_nhds {f : filter β} {u : β → α} {a : α} :
tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε :=
nhds_basis_ball.tendsto_right_iff
theorem continuous_at_iff' [topological_space β] {f : β → α} {b : β} :
continuous_at f b ↔
∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε :=
by rw [continuous_at, tendsto_nhds]
theorem continuous_within_at_iff' [topological_space β] {f : β → α} {b : β} {s : set β} :
continuous_within_at f s b ↔
∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε :=
by rw [continuous_within_at, tendsto_nhds]
theorem continuous_on_iff' [topological_space β] {f : β → α} {s : set β} :
continuous_on f s ↔
∀ (b ∈ s) (ε > 0), ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε :=
by simp [continuous_on, continuous_within_at_iff']
theorem continuous_iff' [topological_space β] {f : β → α} :
continuous f ↔ ∀a (ε > 0), ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε :=
continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds
theorem tendsto_at_top [nonempty β] [semilattice_sup β] {u : β → α} {a : α} :
tendsto u at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n≥N, dist (u n) a < ε :=
(at_top_basis.tendsto_iff nhds_basis_ball).trans $
by { simp only [exists_prop, true_and], refl }
/--
A variant of `tendsto_at_top` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
-/
theorem tendsto_at_top' [nonempty β] [semilattice_sup β] [no_top_order β] {u : β → α} {a : α} :
tendsto u at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n>N, dist (u n) a < ε :=
(at_top_basis_Ioi.tendsto_iff nhds_basis_ball).trans $
by { simp only [exists_prop, true_and], refl }
lemma is_open_singleton_iff {α : Type*} [pseudo_metric_space α] {x : α} :
is_open ({x} : set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x :=
by simp [is_open_iff, subset_singleton_iff, mem_ball]
/-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
centered at `x` and intersecting `s` only at `x`. -/
lemma exists_ball_inter_eq_singleton_of_mem_discrete [discrete_topology s] {x : α} (hx : x ∈ s) :
∃ ε > 0, metric.ball x ε ∩ s = {x} :=
nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx
/-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
of positive radius centered at `x` and intersecting `s` only at `x`. -/
lemma exists_closed_ball_inter_eq_singleton_of_discrete [discrete_topology s] {x : α} (hx : x ∈ s) :
∃ ε > 0, metric.closed_ball x ε ∩ s = {x} :=
nhds_basis_closed_ball.exists_inter_eq_singleton_of_mem_discrete hx
end metric
open metric
/-Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance,
we need to show that the uniform structure coming from the edistance and the
distance coincide. -/
/-- Expressing the uniformity in terms of `edist` -/
protected lemma pseudo_metric.uniformity_basis_edist :
(𝓤 α).has_basis (λ ε:ℝ≥0∞, 0 < ε) (λ ε, {p | edist p.1 p.2 < ε}) :=
⟨begin
intro t,
refine mem_uniformity_dist.trans ⟨_, _⟩; rintro ⟨ε, ε0, Hε⟩,
{ use [ennreal.of_real ε, ennreal.of_real_pos.2 ε0],
rintros ⟨a, b⟩,
simp only [edist_dist, ennreal.of_real_lt_of_real_iff ε0],
exact Hε },
{ rcases ennreal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩,
rw [ennreal.of_real_pos] at ε0',
refine ⟨ε', ε0', λ a b h, Hε (lt_trans _ hε)⟩,
rwa [edist_dist, ennreal.of_real_lt_of_real_iff ε0'] }
end⟩
theorem metric.uniformity_edist : 𝓤 α = (⨅ ε>0, 𝓟 {p:α×α | edist p.1 p.2 < ε}) :=
pseudo_metric.uniformity_basis_edist.eq_binfi
/-- A pseudometric space induces a pseudoemetric space -/
@[priority 100] -- see Note [lower instance priority]
instance pseudo_metric_space.to_pseudo_emetric_space : pseudo_emetric_space α :=
{ edist := edist,
edist_self := by simp [edist_dist],
edist_comm := by simp only [edist_dist, dist_comm]; simp,
edist_triangle := assume x y z, begin
simp only [edist_dist, ← ennreal.of_real_add, dist_nonneg],
rw ennreal.of_real_le_of_real_iff _,
{ exact dist_triangle _ _ _ },
{ simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg }
end,
uniformity_edist := metric.uniformity_edist,
..‹pseudo_metric_space α› }
/-- In a pseudometric space, an open ball of infinite radius is the whole space -/
lemma metric.eball_top_eq_univ (x : α) :
emetric.ball x ∞ = set.univ :=
set.eq_univ_iff_forall.mpr (λ y, edist_lt_top y x)
/-- Balls defined using the distance or the edistance coincide -/
@[simp] lemma metric.emetric_ball {x : α} {ε : ℝ} : emetric.ball x (ennreal.of_real ε) = ball x ε :=
begin
ext y,
simp only [emetric.mem_ball, mem_ball, edist_dist],
exact ennreal.of_real_lt_of_real_iff_of_nonneg dist_nonneg
end
/-- Balls defined using the distance or the edistance coincide -/
@[simp] lemma metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : emetric.ball x ε = ball x ε :=
by { convert metric.emetric_ball, simp }
/-- Closed balls defined using the distance or the edistance coincide -/
lemma metric.emetric_closed_ball {x : α} {ε : ℝ} (h : 0 ≤ ε) :
emetric.closed_ball x (ennreal.of_real ε) = closed_ball x ε :=
by ext y; simp [edist_dist]; rw ennreal.of_real_le_of_real_iff h
/-- Closed balls defined using the distance or the edistance coincide -/
@[simp] lemma metric.emetric_closed_ball_nnreal {x : α} {ε : ℝ≥0} :
emetric.closed_ball x ε = closed_ball x ε :=
by { convert metric.emetric_closed_ball ε.2, simp }
@[simp] lemma metric.emetric_ball_top (x : α) : emetric.ball x ⊤ = univ :=
eq_univ_of_forall $ λ y, edist_lt_top _ _
/-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
(but typically non-definitionaly) equal to some given uniform structure.
See Note [forgetful inheritance].
-/
def pseudo_metric_space.replace_uniformity {α} [U : uniform_space α] (m : pseudo_metric_space α)
(H : @uniformity _ U = @uniformity _ pseudo_emetric_space.to_uniform_space') :
pseudo_metric_space α :=
{ dist := @dist _ m.to_has_dist,
dist_self := dist_self,
dist_comm := dist_comm,
dist_triangle := dist_triangle,
edist := edist,
edist_dist := edist_dist,
to_uniform_space := U,
uniformity_dist := H.trans pseudo_metric_space.uniformity_dist }
/-- One gets a pseudometric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
distance is given separately, to be able to prescribe some expression which is not defeq to the
push-forward of the edistance to reals. -/
def pseudo_emetric_space.to_pseudo_metric_space_of_dist {α : Type u} [e : pseudo_emetric_space α]
(dist : α → α → ℝ)
(edist_ne_top : ∀x y: α, edist x y ≠ ⊤)
(h : ∀x y, dist x y = ennreal.to_real (edist x y)) :
pseudo_metric_space α :=
let m : pseudo_metric_space α :=
{ dist := dist,
dist_self := λx, by simp [h],
dist_comm := λx y, by simp [h, pseudo_emetric_space.edist_comm],
dist_triangle := λx y z, begin
simp only [h],
rw [← ennreal.to_real_add (edist_ne_top _ _) (edist_ne_top _ _),
ennreal.to_real_le_to_real (edist_ne_top _ _)],
{ exact edist_triangle _ _ _ },
{ simp [ennreal.add_eq_top, edist_ne_top] }
end,
edist := λx y, edist x y,
edist_dist := λx y, by simp [h, ennreal.of_real_to_real, edist_ne_top] } in
m.replace_uniformity $ by { rw [uniformity_pseudoedist, metric.uniformity_edist], refl }
/-- One gets a pseudometric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the pseudometric space and the emetric space. -/
def pseudo_emetric_space.to_pseudo_metric_space {α : Type u} [e : emetric_space α]
(h : ∀x y: α, edist x y ≠ ⊤) : pseudo_metric_space α :=
pseudo_emetric_space.to_pseudo_metric_space_of_dist
(λx y, ennreal.to_real (edist x y)) h (λx y, rfl)
/-- A very useful criterion to show that a space is complete is to show that all sequences
which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
`0`, which makes it possible to use arguments of converging series, while this is impossible
to do in general for arbitrary Cauchy sequences. -/
theorem metric.complete_of_convergent_controlled_sequences (B : ℕ → real) (hB : ∀n, 0 < B n)
(H : ∀u : ℕ → α, (∀N n m : ℕ, N ≤ n → N ≤ m → dist (u n) (u m) < B N) →
∃x, tendsto u at_top (𝓝 x)) :
complete_space α :=
begin
-- this follows from the same criterion in emetric spaces. We just need to translate
-- the convergence assumption from `dist` to `edist`
apply emetric.complete_of_convergent_controlled_sequences (λn, ennreal.of_real (B n)),
{ simp [hB] },
{ assume u Hu,
apply H,
assume N n m hn hm,
rw [← ennreal.of_real_lt_of_real_iff (hB N), ← edist_dist],
exact Hu N n m hn hm }
end
theorem metric.complete_of_cauchy_seq_tendsto :
(∀ u : ℕ → α, cauchy_seq u → ∃a, tendsto u at_top (𝓝 a)) → complete_space α :=
emetric.complete_of_cauchy_seq_tendsto
section real
/-- Instantiate the reals as a pseudometric space. -/
instance real.pseudo_metric_space : pseudo_metric_space ℝ :=
{ dist := λx y, |x - y|,
dist_self := by simp [abs_zero],
dist_comm := assume x y, abs_sub_comm _ _,
dist_triangle := assume x y z, abs_sub_le _ _ _ }
theorem real.dist_eq (x y : ℝ) : dist x y = |x - y| := rfl
theorem real.nndist_eq (x y : ℝ) : nndist x y = real.nnabs (x - y) := rfl
theorem real.nndist_eq' (x y : ℝ) : nndist x y = real.nnabs (y - x) := nndist_comm _ _
theorem real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| :=
by simp [real.dist_eq]
theorem real.dist_left_le_of_mem_interval {x y z : ℝ} (h : y ∈ interval x z) :
dist x y ≤ dist x z :=
by simpa only [dist_comm x] using abs_sub_left_of_mem_interval h
theorem real.dist_right_le_of_mem_interval {x y z : ℝ} (h : y ∈ interval x z) :
dist y z ≤ dist x z :=
by simpa only [dist_comm _ z] using abs_sub_right_of_mem_interval h
theorem real.dist_le_of_mem_interval {x y x' y' : ℝ} (hx : x ∈ interval x' y')
(hy : y ∈ interval x' y') : dist x y ≤ dist x' y' :=
abs_sub_le_of_subinterval $ interval_subset_interval (by rwa interval_swap) (by rwa interval_swap)
theorem real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
dist x y ≤ y' - x' :=
by simpa only [real.dist_eq, abs_of_nonpos (sub_nonpos.2 $ hx.1.trans hx.2), neg_sub]
using real.dist_le_of_mem_interval (Icc_subset_interval hx) (Icc_subset_interval hy)
theorem real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0:ℝ) 1) (hy : y ∈ Icc (0:ℝ) 1) :
dist x y ≤ 1 :=
by simpa only [sub_zero] using real.dist_le_of_mem_Icc hx hy
instance : order_topology ℝ :=
order_topology_of_nhds_abs $ λ x,
by simp only [nhds_basis_ball.eq_binfi, ball, real.dist_eq, abs_sub_comm]
lemma real.ball_eq (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
set.ext $ λ y, by rw [mem_ball, dist_comm, real.dist_eq,
abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add', sub_lt]
lemma real.closed_ball_eq {x r : ℝ} : closed_ball x r = Icc (x - r) (x + r) :=
by ext y; rw [mem_closed_ball, dist_comm, real.dist_eq,
abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add', sub_le]
section metric_ordered
variables [conditionally_complete_linear_order α] [order_topology α]
lemma totally_bounded_Icc (a b : α) : totally_bounded (Icc a b) :=
is_compact_Icc.totally_bounded
lemma totally_bounded_Ico (a b : α) : totally_bounded (Ico a b) :=
totally_bounded_subset Ico_subset_Icc_self (totally_bounded_Icc a b)
lemma totally_bounded_Ioc (a b : α) : totally_bounded (Ioc a b) :=
totally_bounded_subset Ioc_subset_Icc_self (totally_bounded_Icc a b)
lemma totally_bounded_Ioo (a b : α) : totally_bounded (Ioo a b) :=
totally_bounded_subset Ioo_subset_Icc_self (totally_bounded_Icc a b)
end metric_ordered
/-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
general case. -/
lemma squeeze_zero' {α} {f g : α → ℝ} {t₀ : filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
(hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : tendsto g t₀ (nhds 0)) : tendsto f t₀ (𝓝 0) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft
/-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
lemma squeeze_zero {α} {f g : α → ℝ} {t₀ : filter α} (hf : ∀t, 0 ≤ f t) (hft : ∀t, f t ≤ g t)
(g0 : tendsto g t₀ (𝓝 0)) : tendsto f t₀ (𝓝 0) :=
squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0
theorem metric.uniformity_eq_comap_nhds_zero :
𝓤 α = comap (λp:α×α, dist p.1 p.2) (𝓝 (0 : ℝ)) :=
by { ext s,
simp [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff, subset_def, real.dist_0_eq_abs] }
lemma cauchy_seq_iff_tendsto_dist_at_top_0 [nonempty β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ tendsto (λ (n : β × β), dist (u n.1) (u n.2)) at_top (𝓝 0) :=
by rw [cauchy_seq_iff_tendsto, metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff,
prod.map_def]
lemma tendsto_uniformity_iff_dist_tendsto_zero {ι : Type*} {f : ι → α × α} {p : filter ι} :
tendsto f p (𝓤 α) ↔ tendsto (λ x, dist (f x).1 (f x).2) p (𝓝 0) :=
by rw [metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]
lemma filter.tendsto.congr_dist {ι : Type*} {f₁ f₂ : ι → α} {p : filter ι} {a : α}
(h₁ : tendsto f₁ p (𝓝 a)) (h : tendsto (λ x, dist (f₁ x) (f₂ x)) p (𝓝 0)) :
tendsto f₂ p (𝓝 a) :=
h₁.congr_uniformity $ tendsto_uniformity_iff_dist_tendsto_zero.2 h
alias filter.tendsto.congr_dist ← tendsto_of_tendsto_of_dist
lemma tendsto_iff_of_dist {ι : Type*} {f₁ f₂ : ι → α} {p : filter ι} {a : α}
(h : tendsto (λ x, dist (f₁ x) (f₂ x)) p (𝓝 0)) :
tendsto f₁ p (𝓝 a) ↔ tendsto f₂ p (𝓝 a) :=
uniform.tendsto_congr $ tendsto_uniformity_iff_dist_tendsto_zero.2 h
end real
section cauchy_seq
variables [nonempty β] [semilattice_sup β]
/-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
the distance between its elements is arbitrarily small -/
@[nolint ge_or_gt] -- see Note [nolint_ge]
theorem metric.cauchy_seq_iff {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀m n≥N, dist (u m) (u n) < ε :=
uniformity_basis_dist.cauchy_seq_iff
/-- A variation around the pseudometric characterization of Cauchy sequences -/
theorem metric.cauchy_seq_iff' {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀n≥N, dist (u n) (u N) < ε :=
uniformity_basis_dist.cauchy_seq_iff'
/-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
and `b` converges to zero, then `s` is a Cauchy sequence. -/
lemma cauchy_seq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
(h : ∀ n m N : β, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : tendsto b at_top (nhds 0)) :
cauchy_seq s :=
metric.cauchy_seq_iff.2 $ λ ε ε0,
(metric.tendsto_at_top.1 h₀ ε ε0).imp $ λ N hN m n hm hn,
calc dist (s m) (s n) ≤ b N : h m n N hm hn
... ≤ |b N| : le_abs_self _
... = dist (b N) 0 : by rw real.dist_0_eq_abs; refl
... < ε : (hN _ (le_refl N))
/-- A Cauchy sequence on the natural numbers is bounded. -/
theorem cauchy_seq_bdd {u : ℕ → α} (hu : cauchy_seq u) :
∃ R > 0, ∀ m n, dist (u m) (u n) < R :=
begin
rcases metric.cauchy_seq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩,
suffices : ∃ R > 0, ∀ n, dist (u n) (u N) < R,
{ rcases this with ⟨R, R0, H⟩,
exact ⟨_, add_pos R0 R0, λ m n,
lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩ },
let R := finset.sup (finset.range N) (λ n, nndist (u n) (u N)),
refine ⟨↑R + 1, add_pos_of_nonneg_of_pos R.2 zero_lt_one, λ n, _⟩,
cases le_or_lt N n,
{ exact lt_of_lt_of_le (hN _ h) (le_add_of_nonneg_left R.2) },
{ have : _ ≤ R := finset.le_sup (finset.mem_range.2 h),
exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one) }
end
/-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. -/
lemma cauchy_seq_iff_le_tendsto_0 {s : ℕ → α} : cauchy_seq s ↔ ∃ b : ℕ → ℝ,
(∀ n, 0 ≤ b n) ∧
(∀ n m N : ℕ, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧
tendsto b at_top (𝓝 0) :=
⟨λ hs, begin
/- `s` is a Cauchy sequence. The sequence `b` will be constructed by taking
the supremum of the distances between `s n` and `s m` for `n m ≥ N`.
First, we prove that all these distances are bounded, as otherwise the Sup
would not make sense. -/
let S := λ N, (λ(p : ℕ × ℕ), dist (s p.1) (s p.2)) '' {p | p.1 ≥ N ∧ p.2 ≥ N},
have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x,
{ rcases cauchy_seq_bdd hs with ⟨R, R0, hR⟩,
refine λ N, ⟨R, _⟩, rintro _ ⟨⟨m, n⟩, _, rfl⟩,
exact le_of_lt (hR m n) },
have bdd : bdd_above (range (λ(p : ℕ × ℕ), dist (s p.1) (s p.2))),
{ rcases cauchy_seq_bdd hs with ⟨R, R0, hR⟩,
use R, rintro _ ⟨⟨m, n⟩, rfl⟩, exact le_of_lt (hR m n) },
-- Prove that it bounds the distances of points in the Cauchy sequence
have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ Sup (S N) :=
λ m n N hm hn, le_cSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩,
have S0m : ∀ n, (0:ℝ) ∈ S n := λ n, ⟨⟨n, n⟩, ⟨le_refl _, le_refl _⟩, dist_self _⟩,
have S0 := λ n, le_cSup (hS n) (S0m n),
-- Prove that it tends to `0`, by using the Cauchy property of `s`
refine ⟨λ N, Sup (S N), S0, ub, metric.tendsto_at_top.2 (λ ε ε0, _)⟩,
refine (metric.cauchy_seq_iff.1 hs (ε/2) (half_pos ε0)).imp (λ N hN n hn, _),
rw [real.dist_0_eq_abs, abs_of_nonneg (S0 n)],
refine lt_of_le_of_lt (cSup_le ⟨_, S0m _⟩ _) (half_lt_self ε0),
rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩,
exact le_of_lt (hN _ _ (le_trans hn hm') (le_trans hn hn'))
end,
λ ⟨b, _, b_bound, b_lim⟩, cauchy_seq_of_le_tendsto_0 b b_bound b_lim⟩
end cauchy_seq
/-- Pseudometric space structure pulled back by a function. -/
def pseudo_metric_space.induced {α β} (f : α → β)
(m : pseudo_metric_space β) : pseudo_metric_space α :=
{ dist := λ x y, dist (f x) (f y),
dist_self := λ x, dist_self _,
dist_comm := λ x y, dist_comm _ _,
dist_triangle := λ x y z, dist_triangle _ _ _,
edist := λ x y, edist (f x) (f y),
edist_dist := λ x y, edist_dist _ _,
to_uniform_space := uniform_space.comap f m.to_uniform_space,
uniformity_dist := begin
apply @uniformity_dist_of_mem_uniformity _ _ _ _ _ (λ x y, dist (f x) (f y)),
refine λ s, mem_comap.trans _,
split; intro H,
{ rcases H with ⟨r, ru, rs⟩,
rcases mem_uniformity_dist.1 ru with ⟨ε, ε0, hε⟩,
refine ⟨ε, ε0, λ a b h, rs (hε _)⟩, exact h },
{ rcases H with ⟨ε, ε0, hε⟩,
exact ⟨_, dist_mem_uniformity ε0, λ ⟨a, b⟩, hε⟩ }
end }
/-- Pull back a pseudometric space structure by a uniform inducing map. This is a version of
`pseudo_metric_space.induced` useful in case if the domain already has a `uniform_space`
structure. -/
def uniform_inducing.comap_pseudo_metric_space {α β} [uniform_space α] [pseudo_metric_space β]
(f : α → β) (h : uniform_inducing f) : pseudo_metric_space α :=
(pseudo_metric_space.induced f ‹_›).replace_uniformity h.comap_uniformity.symm
instance subtype.psudo_metric_space {α : Type*} {p : α → Prop} [t : pseudo_metric_space α] :
pseudo_metric_space (subtype p) :=
pseudo_metric_space.induced coe t
theorem subtype.pseudo_dist_eq {p : α → Prop} (x y : subtype p) : dist x y = dist (x : α) y := rfl
section nnreal
instance : pseudo_metric_space ℝ≥0 := by unfold nnreal; apply_instance
lemma nnreal.dist_eq (a b : ℝ≥0) : dist a b = |(a:ℝ) - b| := rfl
lemma nnreal.nndist_eq (a b : ℝ≥0) :
nndist a b = max (a - b) (b - a) :=
begin
wlog h : a ≤ b,
{ apply nnreal.coe_eq.1,
rw [sub_eq_zero_iff_le.2 h, max_eq_right (zero_le $ b - a), ← dist_nndist, nnreal.dist_eq,
nnreal.coe_sub h, abs_eq_max_neg, neg_sub],
apply max_eq_right,
linarith [nnreal.coe_le_coe.2 h] },
rwa [nndist_comm, max_comm]
end
end nnreal
section prod
instance prod.pseudo_metric_space_max [pseudo_metric_space β] : pseudo_metric_space (α × β) :=
{ dist := λ x y, max (dist x.1 y.1) (dist x.2 y.2),
dist_self := λ x, by simp,
dist_comm := λ x y, by simp [dist_comm],
dist_triangle := λ x y z, max_le
(le_trans (dist_triangle _ _ _) (add_le_add (le_max_left _ _) (le_max_left _ _)))
(le_trans (dist_triangle _ _ _) (add_le_add (le_max_right _ _) (le_max_right _ _))),
edist := λ x y, max (edist x.1 y.1) (edist x.2 y.2),
edist_dist := assume x y, begin
have : monotone ennreal.of_real := assume x y h, ennreal.of_real_le_of_real h,
rw [edist_dist, edist_dist, ← this.map_max]
end,
uniformity_dist := begin
refine uniformity_prod.trans _,
simp only [uniformity_basis_dist.eq_binfi, comap_infi],
rw ← infi_inf_eq, congr, funext,
rw ← infi_inf_eq, congr, funext,
simp [inf_principal, ext_iff, max_lt_iff]
end,
to_uniform_space := prod.uniform_space }
lemma prod.dist_eq [pseudo_metric_space β] {x y : α × β} :
dist x y = max (dist x.1 y.1) (dist x.2 y.2) := rfl
theorem ball_prod_same [pseudo_metric_space β] (x : α) (y : β) (r : ℝ) :
(ball x r).prod (ball y r) = ball (x, y) r :=
ext $ λ z, by simp [prod.dist_eq]
theorem closed_ball_prod_same [pseudo_metric_space β] (x : α) (y : β) (r : ℝ) :
(closed_ball x r).prod (closed_ball y r) = closed_ball (x, y) r :=
ext $ λ z, by simp [prod.dist_eq]
end prod
theorem uniform_continuous_dist : uniform_continuous (λp:α×α, dist p.1 p.2) :=
metric.uniform_continuous_iff.2 (λ ε ε0, ⟨ε/2, half_pos ε0,
begin
suffices,
{ intros p q h, cases p with p₁ p₂, cases q with q₁ q₂,
cases max_lt_iff.1 h with h₁ h₂, clear h,
dsimp at h₁ h₂ ⊢,
rw real.dist_eq,
refine abs_sub_lt_iff.2 ⟨_, _⟩,
{ revert p₁ p₂ q₁ q₂ h₁ h₂, exact this },
{ apply this; rwa dist_comm } },
intros p₁ p₂ q₁ q₂ h₁ h₂,
have := add_lt_add
(abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₁ q₁ p₂) h₁)).1
(abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₂ q₂ q₁) h₂)).1,
rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this
end⟩)
theorem uniform_continuous.dist [uniform_space β] {f g : β → α}
(hf : uniform_continuous f) (hg : uniform_continuous g) :
uniform_continuous (λb, dist (f b) (g b)) :=
uniform_continuous_dist.comp (hf.prod_mk hg)
@[continuity]
theorem continuous_dist : continuous (λp:α×α, dist p.1 p.2) :=
uniform_continuous_dist.continuous
@[continuity]
theorem continuous.dist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, dist (f b) (g b)) :=
continuous_dist.comp (hf.prod_mk hg : _)
theorem filter.tendsto.dist {f g : β → α} {x : filter β} {a b : α}
(hf : tendsto f x (𝓝 a)) (hg : tendsto g x (𝓝 b)) :
tendsto (λx, dist (f x) (g x)) x (𝓝 (dist a b)) :=
(continuous_dist.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
lemma nhds_comap_dist (a : α) : (𝓝 (0 : ℝ)).comap (λa', dist a' a) = 𝓝 a :=
by simp only [@nhds_eq_comap_uniformity α, metric.uniformity_eq_comap_nhds_zero,
comap_comap, (∘), dist_comm]
lemma tendsto_iff_dist_tendsto_zero {f : β → α} {x : filter β} {a : α} :
(tendsto f x (𝓝 a)) ↔ (tendsto (λb, dist (f b) a) x (𝓝 0)) :=
by rw [← nhds_comap_dist a, tendsto_comap_iff]
lemma uniform_continuous_nndist : uniform_continuous (λp:α×α, nndist p.1 p.2) :=
uniform_continuous_subtype_mk uniform_continuous_dist _
lemma uniform_continuous.nndist [uniform_space β] {f g : β → α} (hf : uniform_continuous f)
(hg : uniform_continuous g) :
uniform_continuous (λ b, nndist (f b) (g b)) :=
uniform_continuous_nndist.comp (hf.prod_mk hg)
lemma continuous_nndist : continuous (λp:α×α, nndist p.1 p.2) :=
uniform_continuous_nndist.continuous
lemma continuous.nndist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, nndist (f b) (g b)) :=
continuous_nndist.comp (hf.prod_mk hg : _)
theorem filter.tendsto.nndist {f g : β → α} {x : filter β} {a b : α}
(hf : tendsto f x (𝓝 a)) (hg : tendsto g x (𝓝 b)) :
tendsto (λx, nndist (f x) (g x)) x (𝓝 (nndist a b)) :=
(continuous_nndist.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
namespace metric
variables {x y z : α} {ε ε₁ ε₂ : ℝ} {s : set α}
theorem is_closed_ball : is_closed (closed_ball x ε) :=
is_closed_le (continuous_id.dist continuous_const) continuous_const
lemma is_closed_sphere : is_closed (sphere x ε) :=
is_closed_eq (continuous_id.dist continuous_const) continuous_const
@[simp] theorem closure_closed_ball : closure (closed_ball x ε) = closed_ball x ε :=
is_closed_ball.closure_eq
theorem closure_ball_subset_closed_ball : closure (ball x ε) ⊆ closed_ball x ε :=
closure_minimal ball_subset_closed_ball is_closed_ball
theorem frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε :=
frontier_lt_subset_eq (continuous_id.dist continuous_const) continuous_const
theorem frontier_closed_ball_subset_sphere : frontier (closed_ball x ε) ⊆ sphere x ε :=
frontier_le_subset_eq (continuous_id.dist continuous_const) continuous_const
theorem ball_subset_interior_closed_ball : ball x ε ⊆ interior (closed_ball x ε) :=
interior_maximal ball_subset_closed_ball is_open_ball
/-- ε-characterization of the closure in pseudometric spaces-/
theorem mem_closure_iff {α : Type u} [pseudo_metric_space α] {s : set α} {a : α} :
a ∈ closure s ↔ ∀ε>0, ∃b ∈ s, dist a b < ε :=
(mem_closure_iff_nhds_basis nhds_basis_ball).trans $
by simp only [mem_ball, dist_comm]
lemma mem_closure_range_iff {α : Type u} [pseudo_metric_space α] {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ε>0, ∃ k : β, dist a (e k) < ε :=
by simp only [mem_closure_iff, exists_range_iff]
lemma mem_closure_range_iff_nat {α : Type u} [pseudo_metric_space α] {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
(mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans $
by simp only [mem_ball, dist_comm, exists_range_iff, forall_const]
theorem mem_of_closed' {α : Type u} [pseudo_metric_space α] {s : set α} (hs : is_closed s)
{a : α} : a ∈ s ↔ ∀ε>0, ∃b ∈ s, dist a b < ε :=
by simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a
end metric
section pi
open finset
variables {π : β → Type*} [fintype β] [∀b, pseudo_metric_space (π b)]
/-- A finite product of pseudometric spaces is a pseudometric space, with the sup distance. -/
instance pseudo_metric_space_pi : pseudo_metric_space (Πb, π b) :=
begin
/- we construct the instance from the pseudoemetric space instance to avoid checking again that
the uniformity is the same as the product uniformity, but we register nevertheless a nice formula
for the distance -/
refine pseudo_emetric_space.to_pseudo_metric_space_of_dist
(λf g, ((sup univ (λb, nndist (f b) (g b)) : ℝ≥0) : ℝ)) _ _,
show ∀ (x y : Π (b : β), π b), edist x y ≠ ⊤,
{ assume x y,
rw ← lt_top_iff_ne_top,
have : (⊥ : ℝ≥0∞) < ⊤ := ennreal.coe_lt_top,
simp [edist_pi_def, finset.sup_lt_iff this, edist_lt_top] },
show ∀ (x y : Π (b : β), π b), ↑(sup univ (λ (b : β), nndist (x b) (y b))) =
ennreal.to_real (sup univ (λ (b : β), edist (x b) (y b))),
{ assume x y,
simp only [edist_nndist],
norm_cast }
end
lemma nndist_pi_def (f g : Πb, π b) : nndist f g = sup univ (λb, nndist (f b) (g b)) :=
subtype.eta _ _
lemma dist_pi_def (f g : Πb, π b) :
dist f g = (sup univ (λb, nndist (f b) (g b)) : ℝ≥0) := rfl
@[simp] lemma dist_pi_const [nonempty β] (a b : α) : dist (λ x : β, a) (λ _, b) = dist a b :=
by simpa only [dist_edist] using congr_arg ennreal.to_real (edist_pi_const a b)
@[simp] lemma nndist_pi_const [nonempty β] (a b : α) :
nndist (λ x : β, a) (λ _, b) = nndist a b := nnreal.eq $ dist_pi_const a b
lemma dist_pi_lt_iff {f g : Πb, π b} {r : ℝ} (hr : 0 < r) :
dist f g < r ↔ ∀b, dist (f b) (g b) < r :=
begin
lift r to ℝ≥0 using hr.le,
simp [dist_pi_def, finset.sup_lt_iff (show ⊥ < r, from hr)],
end
lemma dist_pi_le_iff {f g : Πb, π b} {r : ℝ} (hr : 0 ≤ r) :
dist f g ≤ r ↔ ∀b, dist (f b) (g b) ≤ r :=
begin
lift r to ℝ≥0 using hr,
simp [nndist_pi_def]
end
lemma nndist_le_pi_nndist (f g : Πb, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g :=
by { rw [nndist_pi_def], exact finset.le_sup (finset.mem_univ b) }
lemma dist_le_pi_dist (f g : Πb, π b) (b : β) : dist (f b) (g b) ≤ dist f g :=
by simp only [dist_nndist, nnreal.coe_le_coe, nndist_le_pi_nndist f g b]
/-- An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
for a version assuming `nonempty β` instead of `0 < r`. -/
lemma ball_pi (x : Πb, π b) {r : ℝ} (hr : 0 < r) :
ball x r = set.pi univ (λ b, ball (x b) r) :=
by { ext p, simp [dist_pi_lt_iff hr] }
/-- An open ball in a product space is a product of open balls. See also `metric.ball_pi`
for a version assuming `0 < r` instead of `nonempty β`. -/
lemma ball_pi' [nonempty β] (x : Π b, π b) (r : ℝ) :
ball x r = set.pi univ (λ b, ball (x b) r) :=
(lt_or_le 0 r).elim (ball_pi x) $ λ hr, by simp [ball_eq_empty.2 hr]
/-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
for a version assuming `nonempty β` instead of `0 ≤ r`. -/
lemma closed_ball_pi (x : Πb, π b) {r : ℝ} (hr : 0 ≤ r) :
closed_ball x r = set.pi univ (λ b, closed_ball (x b) r) :=
by { ext p, simp [dist_pi_le_iff hr] }
/-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi`
for a version assuming `0 ≤ r` instead of `nonempty β`. -/
lemma closed_ball_pi' [nonempty β] (x : Π b, π b) (r : ℝ) :
closed_ball x r = set.pi univ (λ b, closed_ball (x b) r) :=
(le_or_lt 0 r).elim (closed_ball_pi x) $ λ hr, by simp [closed_ball_eq_empty.2 hr]
end pi
section compact
/-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
positive radius -/
lemma finite_cover_balls_of_compact {α : Type u} [pseudo_metric_space α] {s : set α}
(hs : is_compact s) {e : ℝ} (he : 0 < e) :
∃t ⊆ s, finite t ∧ s ⊆ ⋃x∈t, ball x e :=
begin
apply hs.elim_finite_subcover_image,
{ simp [is_open_ball] },
{ intros x xs,
simp,
exact ⟨x, ⟨xs, by simpa⟩⟩ }
end
alias finite_cover_balls_of_compact ← is_compact.finite_cover_balls
end compact
section proper_space
open metric
/-- A pseudometric space is proper if all closed balls are compact. -/
class proper_space (α : Type u) [pseudo_metric_space α] : Prop :=
(is_compact_closed_ball : ∀x:α, ∀r, is_compact (closed_ball x r))
/-- In a proper pseudometric space, all spheres are compact. -/
lemma is_compact_sphere {α : Type*} [pseudo_metric_space α] [proper_space α] (x : α) (r : ℝ) :
is_compact (sphere x r) :=
compact_of_is_closed_subset (proper_space.is_compact_closed_ball x r) is_closed_sphere
sphere_subset_closed_ball
/-- In a proper pseudometric space, any sphere is a `compact_space` when considered as a subtype. -/
instance {α : Type*} [pseudo_metric_space α] [proper_space α] (x : α) (r : ℝ) :
compact_space (sphere x r) :=
is_compact_iff_compact_space.mp (is_compact_sphere _ _)
/-- A proper pseudo metric space is sigma compact, and therefore second countable. -/
@[priority 100] -- see Note [lower instance priority]
instance second_countable_of_proper [proper_space α] :
second_countable_topology α :=
begin
-- We already have `sigma_compact_space_of_locally_compact_second_countable`, so we don't
-- add an instance for `sigma_compact_space`.
suffices : sigma_compact_space α, by exactI emetric.second_countable_of_sigma_compact α,
rcases em (nonempty α) with ⟨⟨x⟩⟩|hn,
{ exact ⟨⟨λ n, closed_ball x n, λ n, proper_space.is_compact_closed_ball _ _,
Union_closed_ball_nat _⟩⟩ },
{ exact ⟨⟨λ n, ∅, λ n, is_compact_empty, Union_eq_univ_iff.2 $ λ x, (hn ⟨x⟩).elim⟩⟩ }
end
lemma tendsto_dist_right_cocompact_at_top [proper_space α] (x : α) :
tendsto (λ y, dist y x) (cocompact α) at_top :=
(has_basis_cocompact.tendsto_iff at_top_basis).2 $ λ r hr,
⟨closed_ball x r, proper_space.is_compact_closed_ball x r,
λ y hy, (not_le.1 $ mt mem_closed_ball.2 hy).le⟩
lemma tendsto_dist_left_cocompact_at_top [proper_space α] (x : α) :
tendsto (dist x) (cocompact α) at_top :=
by simpa only [dist_comm] using tendsto_dist_right_cocompact_at_top x
/-- If all closed balls of large enough radius are compact, then the space is proper. Especially
useful when the lower bound for the radius is 0. -/
lemma proper_space_of_compact_closed_ball_of_le
(R : ℝ) (h : ∀x:α, ∀r, R ≤ r → is_compact (closed_ball x r)) :
proper_space α :=
⟨begin
assume x r,
by_cases hr : R ≤ r,
{ exact h x r hr },
{ have : closed_ball x r = closed_ball x R ∩ closed_ball x r,
{ symmetry,
apply inter_eq_self_of_subset_right,
exact closed_ball_subset_closed_ball (le_of_lt (not_le.1 hr)) },
rw this,
exact (h x R (le_refl _)).inter_right is_closed_ball }
end⟩
/- A compact pseudometric space is proper -/
@[priority 100] -- see Note [lower instance priority]
instance proper_of_compact [compact_space α] : proper_space α :=
⟨assume x r, is_closed_ball.is_compact⟩
/-- A proper space is locally compact -/
@[priority 100] -- see Note [lower instance priority]
instance locally_compact_of_proper [proper_space α] :
locally_compact_space α :=
locally_compact_space_of_has_basis (λ x, nhds_basis_closed_ball) $
λ x ε ε0, proper_space.is_compact_closed_ball _ _
/-- A proper space is complete -/
@[priority 100] -- see Note [lower instance priority]
instance complete_of_proper [proper_space α] : complete_space α :=
⟨begin
intros f hf,
/- We want to show that the Cauchy filter `f` is converging. It suffices to find a closed
ball (therefore compact by properness) where it is nontrivial. -/
obtain ⟨t, t_fset, ht⟩ : ∃ t ∈ f, ∀ x y ∈ t, dist x y < 1 :=
(metric.cauchy_iff.1 hf).2 1 zero_lt_one,
rcases hf.1.nonempty_of_mem t_fset with ⟨x, xt⟩,
have : closed_ball x 1 ∈ f := mem_of_superset t_fset (λ y yt, (ht y x yt xt).le),
rcases (compact_iff_totally_bounded_complete.1 (proper_space.is_compact_closed_ball x 1)).2 f hf
(le_principal_iff.2 this) with ⟨y, -, hy⟩,
exact ⟨y, hy⟩
end⟩
/-- A finite product of proper spaces is proper. -/
instance pi_proper_space {π : β → Type*} [fintype β] [∀b, pseudo_metric_space (π b)]
[h : ∀b, proper_space (π b)] : proper_space (Πb, π b) :=
begin
refine proper_space_of_compact_closed_ball_of_le 0 (λx r hr, _),
rw closed_ball_pi _ hr,
apply is_compact_univ_pi (λb, _),
apply (h b).is_compact_closed_ball
end
variables [proper_space α] {x : α} {r : ℝ} {s : set α}
/-- If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
ball with the same center and a strictly smaller radius that includes `s`. -/
lemma exists_pos_lt_subset_ball (hr : 0 < r) (hs : is_closed s) (h : s ⊆ ball x r) :
∃ r' ∈ Ioo 0 r, s ⊆ ball x r' :=
begin
unfreezingI { rcases eq_empty_or_nonempty s with rfl|hne },
{ exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩ },
have : is_compact s,
from compact_of_is_closed_subset (proper_space.is_compact_closed_ball x r) hs
(subset.trans h ball_subset_closed_ball),
obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closed_ball x (dist y x),
from this.exists_forall_ge hne (continuous_id.dist continuous_const).continuous_on,
have hyr : dist y x < r, from h hys,
rcases exists_between hyr with ⟨r', hyr', hrr'⟩,
exact ⟨r', ⟨dist_nonneg.trans_lt hyr', hrr'⟩, subset.trans hy $ closed_ball_subset_ball hyr'⟩
end
/-- If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
center and a strictly smaller radius that includes `s`. -/
lemma exists_lt_subset_ball (hs : is_closed s) (h : s ⊆ ball x r) :
∃ r' < r, s ⊆ ball x r' :=
begin
cases le_or_lt r 0 with hr hr,
{ rw [ball_eq_empty.2 hr, subset_empty_iff] at h, unfreezingI { subst s },
exact (no_bot r).imp (λ r' hr', ⟨hr', empty_subset _⟩) },
{ exact (exists_pos_lt_subset_ball hr hs h).imp (λ r' hr', ⟨hr'.fst.2, hr'.snd⟩) }
end
end proper_space
namespace metric
section second_countable
open topological_space
/-- A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
is `ε`-dense. -/
lemma second_countable_of_almost_dense_set
(H : ∀ε > (0 : ℝ), ∃ s : set α, countable s ∧ (∀x, ∃y ∈ s, dist x y ≤ ε)) :
second_countable_topology α :=
begin
refine emetric.second_countable_of_almost_dense_set (λ ε ε0, _),
rcases ennreal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩,
choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0),
refine ⟨s, hsc, bUnion_eq_univ_iff.2 (λ x, ⟨y x, hys _, le_trans _ ε'ε.le⟩)⟩,
exact_mod_cast hyx x
end
end second_countable
end metric
lemma lebesgue_number_lemma_of_metric
{s : set α} {ι} {c : ι → set α} (hs : is_compact s)
(hc₁ : ∀ i, is_open (c i)) (hc₂ : s ⊆ ⋃ i, c i) :
∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
let ⟨n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂,
⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 en in
⟨δ, δ0, assume x hx, let ⟨i, hi⟩ := hn x hx in
⟨i, assume y hy, hi (hδ (mem_ball'.mp hy))⟩⟩
lemma lebesgue_number_lemma_of_metric_sUnion
{s : set α} {c : set (set α)} (hs : is_compact s)
(hc₁ : ∀ t ∈ c, is_open t) (hc₂ : s ⊆ ⋃₀ c) :
∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t :=
by rw sUnion_eq_Union at hc₂;
simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
namespace metric
/-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
even in the empty space. -/
def bounded (s : set α) : Prop :=
∃C, ∀x y ∈ s, dist x y ≤ C
section bounded
variables {x : α} {s t : set α} {r : ℝ}
@[simp] lemma bounded_empty : bounded (∅ : set α) :=
⟨0, by simp⟩
lemma bounded_iff_mem_bounded : bounded s ↔ ∀ x ∈ s, bounded s :=
⟨λ h _ _, h, λ H,
s.eq_empty_or_nonempty.elim
(λ hs, hs.symm ▸ bounded_empty)
(λ ⟨x, hx⟩, H x hx)⟩
/-- Subsets of a bounded set are also bounded -/
lemma bounded.subset (incl : s ⊆ t) : bounded t → bounded s :=
Exists.imp $ λ C hC x y hx hy, hC x y (incl hx) (incl hy)
/-- Closed balls are bounded -/
lemma bounded_closed_ball : bounded (closed_ball x r) :=
⟨r + r, λ y z hy hz, begin
simp only [mem_closed_ball] at *,
calc dist y z ≤ dist y x + dist z x : dist_triangle_right _ _ _
... ≤ r + r : add_le_add hy hz
end⟩
/-- Open balls are bounded -/
lemma bounded_ball : bounded (ball x r) :=
bounded_closed_ball.subset ball_subset_closed_ball
/-- Given a point, a bounded subset is included in some ball around this point -/
lemma bounded_iff_subset_ball (c : α) : bounded s ↔ ∃r, s ⊆ closed_ball c r :=
begin
split; rintro ⟨C, hC⟩,
{ cases s.eq_empty_or_nonempty with h h,
{ subst s, exact ⟨0, by simp⟩ },
{ rcases h with ⟨x, hx⟩,
exact ⟨C + dist x c, λ y hy, calc
dist y c ≤ dist y x + dist x c : dist_triangle _ _ _
... ≤ C + dist x c : add_le_add_right (hC y x hy hx) _⟩ } },
{ exact bounded_closed_ball.subset hC }
end
lemma bounded.subset_ball (h : bounded s) (c : α) : ∃ r, s ⊆ closed_ball c r :=
(bounded_iff_subset_ball c).1 h
lemma bounded_closure_of_bounded (h : bounded s) : bounded (closure s) :=
let ⟨C, h⟩ := h in
⟨C, λ a b ha hb, (is_closed_le' C).closure_subset $ map_mem_closure2 continuous_dist ha hb h⟩
alias bounded_closure_of_bounded ← metric.bounded.closure
@[simp] lemma bounded_closure_iff : bounded (closure s) ↔ bounded s :=
⟨λ h, h.subset subset_closure, λ h, h.closure⟩
/-- The union of two bounded sets is bounded iff each of the sets is bounded -/
@[simp] lemma bounded_union :
bounded (s ∪ t) ↔ bounded s ∧ bounded t :=
⟨λh, ⟨h.subset (by simp), h.subset (by simp)⟩,
begin
rintro ⟨hs, ht⟩,
refine bounded_iff_mem_bounded.2 (λ x _, _),
rw bounded_iff_subset_ball x at hs ht ⊢,
rcases hs with ⟨Cs, hCs⟩, rcases ht with ⟨Ct, hCt⟩,
exact ⟨max Cs Ct, union_subset
(subset.trans hCs $ closed_ball_subset_closed_ball $ le_max_left _ _)
(subset.trans hCt $ closed_ball_subset_closed_ball $ le_max_right _ _)⟩,
end⟩
/-- A finite union of bounded sets is bounded -/
lemma bounded_bUnion {I : set β} {s : β → set α} (H : finite I) :
bounded (⋃i∈I, s i) ↔ ∀i ∈ I, bounded (s i) :=
finite.induction_on H (by simp) $ λ x I _ _ IH,
by simp [or_imp_distrib, forall_and_distrib, IH]
/-- A totally bounded set is bounded -/
lemma _root_.totally_bounded.bounded {s : set α} (h : totally_bounded s) : bounded s :=
-- We cover the totally bounded set by finitely many balls of radius 1,
-- and then argue that a finite union of bounded sets is bounded
let ⟨t, fint, subs⟩ := (totally_bounded_iff.mp h) 1 zero_lt_one in
bounded.subset subs $ (bounded_bUnion fint).2 $ λ i hi, bounded_ball
/-- A compact set is bounded -/
lemma _root_.is_compact.bounded {s : set α} (h : is_compact s) : bounded s :=
-- A compact set is totally bounded, thus bounded
h.totally_bounded.bounded
/-- A finite set is bounded -/
lemma bounded_of_finite {s : set α} (h : finite s) : bounded s :=
h.is_compact.bounded
alias bounded_of_finite ← set.finite.bounded
/-- A singleton is bounded -/
lemma bounded_singleton {x : α} : bounded ({x} : set α) :=
bounded_of_finite $ finite_singleton _
/-- Characterization of the boundedness of the range of a function -/
lemma bounded_range_iff {f : β → α} : bounded (range f) ↔ ∃C, ∀x y, dist (f x) (f y) ≤ C :=
exists_congr $ λ C, ⟨
λ H x y, H _ _ ⟨x, rfl⟩ ⟨y, rfl⟩,
by rintro H _ _ ⟨x, rfl⟩ ⟨y, rfl⟩; exact H x y⟩
/-- In a compact space, all sets are bounded -/
lemma bounded_of_compact_space [compact_space α] : bounded s :=
compact_univ.bounded.subset (subset_univ _)
lemma is_compact_of_is_closed_bounded [proper_space α] (hc : is_closed s) (hb : bounded s) :
is_compact s :=
begin
unfreezingI { rcases eq_empty_or_nonempty s with (rfl|⟨x, hx⟩) },
{ exact is_compact_empty },
{ rcases hb.subset_ball x with ⟨r, hr⟩,
exact compact_of_is_closed_subset (proper_space.is_compact_closed_ball x r) hc hr }
end
/-- The Heine–Borel theorem:
In a proper space, a set is compact if and only if it is closed and bounded -/
lemma compact_iff_closed_bounded [t2_space α] [proper_space α] :
is_compact s ↔ is_closed s ∧ bounded s :=
⟨λ h, ⟨h.is_closed, h.bounded⟩, λ h, is_compact_of_is_closed_bounded h.1 h.2⟩
lemma compact_space_iff_bounded_univ [proper_space α] : compact_space α ↔ bounded (univ : set α) :=
⟨@bounded_of_compact_space α _ _, λ hb, ⟨is_compact_of_is_closed_bounded is_closed_univ hb⟩⟩
section conditionally_complete_linear_order
variables [conditionally_complete_linear_order α] [order_topology α]
lemma bounded_Icc (a b : α) : bounded (Icc a b) :=
(totally_bounded_Icc a b).bounded
lemma bounded_Ico (a b : α) : bounded (Ico a b) :=
(totally_bounded_Ico a b).bounded
lemma bounded_Ioc (a b : α) : bounded (Ioc a b) :=
(totally_bounded_Ioc a b).bounded
lemma bounded_Ioo (a b : α) : bounded (Ioo a b) :=
(totally_bounded_Ioo a b).bounded
/-- In a pseudo metric space with a conditionally complete linear order such that the order and the
metric structure give the same topology, any order-bounded set is metric-bounded. -/
lemma bounded_of_bdd_above_of_bdd_below {s : set α} (h₁ : bdd_above s) (h₂ : bdd_below s) :
bounded s :=
let ⟨u, hu⟩ := h₁, ⟨l, hl⟩ := h₂ in
bounded.subset (λ x hx, mem_Icc.mpr ⟨hl hx, hu hx⟩) (bounded_Icc l u)
end conditionally_complete_linear_order
end bounded
section diam
variables {s : set α} {x y z : α}
/-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
should be infinite, we express it in terms of the emetric.diameter -/
def diam (s : set α) : ℝ := ennreal.to_real (emetric.diam s)
/-- The diameter of a set is always nonnegative -/
lemma diam_nonneg : 0 ≤ diam s := ennreal.to_real_nonneg
lemma diam_subsingleton (hs : s.subsingleton) : diam s = 0 :=
by simp only [diam, emetric.diam_subsingleton hs, ennreal.zero_to_real]
/-- The empty set has zero diameter -/
@[simp] lemma diam_empty : diam (∅ : set α) = 0 :=
diam_subsingleton subsingleton_empty
/-- A singleton has zero diameter -/
@[simp] lemma diam_singleton : diam ({x} : set α) = 0 :=
diam_subsingleton subsingleton_singleton
-- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
lemma diam_pair : diam ({x, y} : set α) = dist x y :=
by simp only [diam, emetric.diam_pair, dist_edist]
-- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
lemma diam_triple :
metric.diam ({x, y, z} : set α) = max (max (dist x y) (dist x z)) (dist y z) :=
begin
simp only [metric.diam, emetric.diam_triple, dist_edist],
rw [ennreal.to_real_max, ennreal.to_real_max];
apply_rules [ne_of_lt, edist_lt_top, max_lt]
end
/-- If the distance between any two points in a set is bounded by some constant `C`,
then `ennreal.of_real C` bounds the emetric diameter of this set. -/
lemma ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ (x ∈ s) (y ∈ s), dist x y ≤ C) :
emetric.diam s ≤ ennreal.of_real C :=
emetric.diam_le $
λ x hx y hy, (edist_dist x y).symm ▸ ennreal.of_real_le_of_real (h x hx y hy)
/-- If the distance between any two points in a set is bounded by some non-negative constant,
this constant bounds the diameter. -/
lemma diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ (x ∈ s) (y ∈ s), dist x y ≤ C) :
diam s ≤ C :=
ennreal.to_real_le_of_le_of_real h₀ (ediam_le_of_forall_dist_le h)
/-- If the distance between any two points in a nonempty set is bounded by some constant,
this constant bounds the diameter. -/
lemma diam_le_of_forall_dist_le_of_nonempty (hs : s.nonempty) {C : ℝ}
(h : ∀ (x ∈ s) (y ∈ s), dist x y ≤ C) : diam s ≤ C :=
have h₀ : 0 ≤ C, from let ⟨x, hx⟩ := hs in le_trans dist_nonneg (h x hx x hx),
diam_le_of_forall_dist_le h₀ h
/-- The distance between two points in a set is controlled by the diameter of the set. -/
lemma dist_le_diam_of_mem' (h : emetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
dist x y ≤ diam s :=
begin
rw [diam, dist_edist],
rw ennreal.to_real_le_to_real (edist_ne_top _ _) h,
exact emetric.edist_le_diam_of_mem hx hy
end
/-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
lemma bounded_iff_ediam_ne_top : bounded s ↔ emetric.diam s ≠ ⊤ :=
iff.intro
(λ ⟨C, hC⟩, ne_top_of_le_ne_top ennreal.of_real_ne_top
(ediam_le_of_forall_dist_le $ λ x hx y hy, hC x y hx hy))
(λ h, ⟨diam s, λ x y hx hy, dist_le_diam_of_mem' h hx hy⟩)
lemma bounded.ediam_ne_top (h : bounded s) : emetric.diam s ≠ ⊤ :=
bounded_iff_ediam_ne_top.1 h
/-- The distance between two points in a set is controlled by the diameter of the set. -/
lemma dist_le_diam_of_mem (h : bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
dist_le_diam_of_mem' h.ediam_ne_top hx hy
lemma ediam_of_unbounded (h : ¬(bounded s)) : emetric.diam s = ∞ :=
by rwa [bounded_iff_ediam_ne_top, not_not] at h
/-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
This lemma makes it possible to avoid side conditions in some situations -/
lemma diam_eq_zero_of_unbounded (h : ¬(bounded s)) : diam s = 0 :=
by rw [diam, ediam_of_unbounded h, ennreal.top_to_real]
/-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
lemma diam_mono {s t : set α} (h : s ⊆ t) (ht : bounded t) : diam s ≤ diam t :=
begin
unfold diam,
rw ennreal.to_real_le_to_real (bounded.subset h ht).ediam_ne_top ht.ediam_ne_top,
exact emetric.diam_mono h
end
/-- The diameter of a union is controlled by the sum of the diameters, and the distance between
any two points in each of the sets. This lemma is true without any side condition, since it is
obviously true if `s ∪ t` is unbounded. -/
lemma diam_union {t : set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + dist x y + diam t :=
begin
by_cases H : bounded (s ∪ t),
{ have hs : bounded s, from H.subset (subset_union_left _ _),
have ht : bounded t, from H.subset (subset_union_right _ _),
rw [bounded_iff_ediam_ne_top] at H hs ht,
rw [dist_edist, diam, diam, diam, ← ennreal.to_real_add, ← ennreal.to_real_add,
ennreal.to_real_le_to_real];
repeat { apply ennreal.add_ne_top.2; split }; try { assumption };
try { apply edist_ne_top },
exact emetric.diam_union xs yt },
{ rw [diam_eq_zero_of_unbounded H],
apply_rules [add_nonneg, diam_nonneg, dist_nonneg] }
end
/-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
lemma diam_union' {t : set α} (h : (s ∩ t).nonempty) : diam (s ∪ t) ≤ diam s + diam t :=
begin
rcases h with ⟨x, ⟨xs, xt⟩⟩,
simpa using diam_union xs xt
end
/-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
lemma diam_closed_ball {r : ℝ} (h : 0 ≤ r) : diam (closed_ball x r) ≤ 2 * r :=
diam_le_of_forall_dist_le (mul_nonneg (le_of_lt zero_lt_two) h) $ λa ha b hb, calc
dist a b ≤ dist a x + dist b x : dist_triangle_right _ _ _
... ≤ r + r : add_le_add ha hb
... = 2 * r : by simp [mul_two, mul_comm]
/-- The diameter of a ball of radius `r` is at most `2 r`. -/
lemma diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
le_trans (diam_mono ball_subset_closed_ball bounded_closed_ball) (diam_closed_ball h)
end diam
end metric
lemma comap_dist_right_at_top_le_cocompact (x : α) : comap (λ y, dist y x) at_top ≤ cocompact α :=
begin
refine filter.has_basis_cocompact.ge_iff.2 (λ s hs, mem_comap.2 _),
rcases hs.bounded.subset_ball x with ⟨r, hr⟩,
exact ⟨Ioi r, Ioi_mem_at_top r, λ y hy hys, (mem_closed_ball.1 $ hr hys).not_lt hy⟩
end
lemma comap_dist_left_at_top_le_cocompact (x : α) : comap (dist x) at_top ≤ cocompact α :=
by simpa only [dist_comm _ x] using comap_dist_right_at_top_le_cocompact x
lemma comap_dist_right_at_top_eq_cocompact [proper_space α] (x : α) :
comap (λ y, dist y x) at_top = cocompact α :=
(comap_dist_right_at_top_le_cocompact x).antisymm $ (tendsto_dist_right_cocompact_at_top x).le_comap
lemma comap_dist_left_at_top_eq_cocompact [proper_space α] (x : α) :
comap (dist x) at_top = cocompact α :=
(comap_dist_left_at_top_le_cocompact x).antisymm $ (tendsto_dist_left_cocompact_at_top x).le_comap
lemma tendsto_cocompact_of_tendsto_dist_comp_at_top {f : β → α} {l : filter β} (x : α)
(h : tendsto (λ y, dist (f y) x) l at_top) : tendsto f l (cocompact α) :=
by { refine tendsto.mono_right _ (comap_dist_right_at_top_le_cocompact x), rwa tendsto_comap_iff }
namespace int
open metric
/-- Under the coercion from `ℤ` to `ℝ`, inverse images of compact sets are finite. -/
lemma tendsto_coe_cofinite : tendsto (coe : ℤ → ℝ) cofinite (cocompact ℝ) :=
begin
refine tendsto_cocompact_of_tendsto_dist_comp_at_top (0 : ℝ) _,
simp only [filter.tendsto_at_top, eventually_cofinite, not_le, ← mem_ball],
change ∀ r : ℝ, finite (coe ⁻¹' (ball (0 : ℝ) r)),
simp [real.ball_eq, Ioo_ℤ_finite]
end
end int
/-- We now define `metric_space`, extending `pseudo_metric_space`. -/
class metric_space (α : Type u) extends pseudo_metric_space α : Type u :=
(eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y)
/-- Construct a metric space structure whose underlying topological space structure
(definitionally) agrees which a pre-existing topology which is compatible with a given distance
function. -/
def metric_space.of_metrizable {α : Type*} [topological_space α] (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(H : ∀ s : set α, is_open s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s)
(eq_of_dist_eq_zero : ∀ x y : α, dist x y = 0 → x = y) : metric_space α :=
{ eq_of_dist_eq_zero := eq_of_dist_eq_zero,
..pseudo_metric_space.of_metrizable dist dist_self dist_comm dist_triangle H }
variables {γ : Type w} [metric_space γ]
theorem eq_of_dist_eq_zero {x y : γ} : dist x y = 0 → x = y :=
metric_space.eq_of_dist_eq_zero
@[simp] theorem dist_eq_zero {x y : γ} : dist x y = 0 ↔ x = y :=
iff.intro eq_of_dist_eq_zero (assume : x = y, this ▸ dist_self _)
@[simp] theorem zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y :=
by rw [eq_comm, dist_eq_zero]
theorem dist_ne_zero {x y : γ} : dist x y ≠ 0 ↔ x ≠ y :=
by simpa only [not_iff_not] using dist_eq_zero
@[simp] theorem dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y :=
by simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y
@[simp] theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y :=
by simpa only [not_le] using not_congr dist_le_zero
theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y :=
eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
/--Deduce the equality of points with the vanishing of the nonnegative distance-/
theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y :=
by simp only [← nnreal.eq_iff, ← dist_nndist, imp_self, nnreal.coe_zero, dist_eq_zero]
/--Characterize the equality of points with the vanishing of the nonnegative distance-/
@[simp] theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y :=
by simp only [← nnreal.eq_iff, ← dist_nndist, imp_self, nnreal.coe_zero, dist_eq_zero]
@[simp] theorem zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y :=
by simp only [← nnreal.eq_iff, ← dist_nndist, imp_self, nnreal.coe_zero, zero_eq_dist]
namespace metric
variables {x : γ} {s : set γ}
@[simp] lemma closed_ball_zero : closed_ball x 0 = {x} :=
set.ext $ λ y, dist_le_zero
/-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
theorem uniform_embedding_iff' [metric_space β] {f : γ → β} :
uniform_embedding f ↔
(∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
(∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ) :=
begin
split,
{ assume h,
exact ⟨uniform_continuous_iff.1 (uniform_embedding_iff.1 h).2.1,
(uniform_embedding_iff.1 h).2.2⟩ },
{ rintros ⟨h₁, h₂⟩,
refine uniform_embedding_iff.2 ⟨_, uniform_continuous_iff.2 h₁, h₂⟩,
assume x y hxy,
have : dist x y ≤ 0,
{ refine le_of_forall_lt' (λδ δpos, _),
rcases h₂ δ δpos with ⟨ε, εpos, hε⟩,
have : dist (f x) (f y) < ε, by simpa [hxy],
exact hε this },
simpa using this }
end
@[priority 100] -- see Note [lower instance priority]
instance metric_space.to_separated : separated_space γ :=
separated_def.2 $ λ x y h, eq_of_forall_dist_le $
λ ε ε0, le_of_lt (h _ (dist_mem_uniformity ε0))
/-- If a `pseudo_metric_space` is separated, then it is a `metric_space`. -/
def of_t2_pseudo_metric_space {α : Type*} [pseudo_metric_space α]
(h : separated_space α) : metric_space α :=
{ eq_of_dist_eq_zero := λ x y hdist,
begin
refine separated_def.1 h x y (λ s hs, _),
obtain ⟨ε, hε, H⟩ := mem_uniformity_dist.1 hs,
exact H (show dist x y < ε, by rwa [hdist])
end
..‹pseudo_metric_space α› }
/-- A metric space induces an emetric space -/
@[priority 100] -- see Note [lower instance priority]
instance metric_space.to_emetric_space : emetric_space γ :=
{ eq_of_edist_eq_zero := assume x y h, by simpa [edist_dist] using h,
..pseudo_metric_space.to_pseudo_emetric_space, }
end metric
/-- Build a new metric space from an old one where the bundled uniform structure is provably
(but typically non-definitionaly) equal to some given uniform structure.
See Note [forgetful inheritance].
-/
def metric_space.replace_uniformity {γ} [U : uniform_space γ] (m : metric_space γ)
(H : @uniformity _ U = @uniformity _ emetric_space.to_uniform_space') :
metric_space γ :=
{ eq_of_dist_eq_zero := @eq_of_dist_eq_zero _ _,
..pseudo_metric_space.replace_uniformity m.to_pseudo_metric_space H, }
/-- One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. In this definition, the distance
is given separately, to be able to prescribe some expression which is not defeq to the push-forward
of the edistance to reals. -/
def emetric_space.to_metric_space_of_dist {α : Type u} [e : emetric_space α]
(dist : α → α → ℝ)
(edist_ne_top : ∀x y: α, edist x y ≠ ⊤)
(h : ∀x y, dist x y = ennreal.to_real (edist x y)) :
metric_space α :=
{ dist := dist,
eq_of_dist_eq_zero := λx y hxy,
by simpa [h, ennreal.to_real_eq_zero_iff, edist_ne_top x y] using hxy,
..pseudo_emetric_space.to_pseudo_metric_space_of_dist dist edist_ne_top h, }
/-- One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. -/
def emetric_space.to_metric_space {α : Type u} [e : emetric_space α] (h : ∀x y: α, edist x y ≠ ⊤) :
metric_space α :=
emetric_space.to_metric_space_of_dist (λx y, ennreal.to_real (edist x y)) h (λx y, rfl)
/-- Metric space structure pulled back by an injective function. Injectivity is necessary to
ensure that `dist x y = 0` only if `x = y`. -/
def metric_space.induced {γ β} (f : γ → β) (hf : function.injective f)
(m : metric_space β) : metric_space γ :=
{ eq_of_dist_eq_zero := λ x y h, hf (dist_eq_zero.1 h),
..pseudo_metric_space.induced f m.to_pseudo_metric_space }
/-- Pull back a metric space structure by a uniform embedding. This is a version of
`metric_space.induced` useful in case if the domain already has a `uniform_space` structure. -/
def uniform_embedding.comap_metric_space {α β} [uniform_space α] [metric_space β] (f : α → β)
(h : uniform_embedding f) : metric_space α :=
(metric_space.induced f h.inj ‹_›).replace_uniformity h.comap_uniformity.symm
instance subtype.metric_space {α : Type*} {p : α → Prop} [t : metric_space α] :
metric_space (subtype p) :=
metric_space.induced coe (λ x y, subtype.ext) t
theorem subtype.dist_eq {p : α → Prop} (x y : subtype p) : dist x y = dist (x : α) y := rfl
instance : metric_space empty :=
{ dist := λ _ _, 0,
dist_self := λ _, rfl,
dist_comm := λ _ _, rfl,
eq_of_dist_eq_zero := λ _ _ _, subsingleton.elim _ _,
dist_triangle := λ _ _ _, show (0:ℝ) ≤ 0 + 0, by rw add_zero, }
instance : metric_space punit :=
{ dist := λ _ _, 0,
dist_self := λ _, rfl,
dist_comm := λ _ _, rfl,
eq_of_dist_eq_zero := λ _ _ _, subsingleton.elim _ _,
dist_triangle := λ _ _ _, show (0:ℝ) ≤ 0 + 0, by rw add_zero, }
section real
/-- Instantiate the reals as a metric space. -/
instance real.metric_space : metric_space ℝ :=
{ eq_of_dist_eq_zero := λ x y h, by simpa [dist, sub_eq_zero] using h,
..real.pseudo_metric_space }
end real
section nnreal
instance : metric_space ℝ≥0 := subtype.metric_space
end nnreal
section prod
instance prod.metric_space_max [metric_space β] : metric_space (γ × β) :=
{ eq_of_dist_eq_zero := λ x y h, begin
cases max_le_iff.1 (le_of_eq h) with h₁ h₂,
exact prod.ext_iff.2 ⟨dist_le_zero.1 h₁, dist_le_zero.1 h₂⟩
end,
..prod.pseudo_metric_space_max, }
end prod
section pi
open finset
variables {π : β → Type*} [fintype β] [∀b, metric_space (π b)]
/-- A finite product of metric spaces is a metric space, with the sup distance. -/
instance metric_space_pi : metric_space (Πb, π b) :=
/- we construct the instance from the emetric space instance to avoid checking again that the
uniformity is the same as the product uniformity, but we register nevertheless a nice formula
for the distance -/
{ eq_of_dist_eq_zero := assume f g eq0,
begin
have eq1 : edist f g = 0 := by simp only [edist_dist, eq0, ennreal.of_real_zero],
have eq2 : sup univ (λ (b : β), edist (f b) (g b)) ≤ 0 := le_of_eq eq1,
simp only [finset.sup_le_iff] at eq2,
exact (funext $ assume b, edist_le_zero.1 $ eq2 b $ mem_univ b)
end,
..pseudo_metric_space_pi }
end pi
namespace metric
section second_countable
open topological_space
/-- A metric space space is second countable if one can reconstruct up to any `ε>0` any element of
the space from countably many data. -/
lemma second_countable_of_countable_discretization {α : Type u} [metric_space α]
(H : ∀ε > (0 : ℝ), ∃ (β : Type*) (_ : encodable β) (F : α → β), ∀x y, F x = F y → dist x y ≤ ε) :
second_countable_topology α :=
begin
cases (univ : set α).eq_empty_or_nonempty with hs hs,
{ haveI : compact_space α := ⟨by rw hs; exact is_compact_empty⟩, by apply_instance },
rcases hs with ⟨x0, hx0⟩,
letI : inhabited α := ⟨x0⟩,
refine second_countable_of_almost_dense_set (λε ε0, _),
rcases H ε ε0 with ⟨β, fβ, F, hF⟩,
resetI,
let Finv := function.inv_fun F,
refine ⟨range Finv, ⟨countable_range _, λx, _⟩⟩,
let x' := Finv (F x),
have : F x' = F x := function.inv_fun_eq ⟨x, rfl⟩,
exact ⟨x', mem_range_self _, hF _ _ this.symm⟩
end
end second_countable
end metric
section eq_rel
/-- The canonical equivalence relation on a pseudometric space. -/
def pseudo_metric.dist_setoid (α : Type u) [pseudo_metric_space α] : setoid α :=
setoid.mk (λx y, dist x y = 0)
begin
unfold equivalence,
repeat { split },
{ exact pseudo_metric_space.dist_self },
{ assume x y h, rwa pseudo_metric_space.dist_comm },
{ assume x y z hxy hyz,
refine le_antisymm _ dist_nonneg,
calc dist x z ≤ dist x y + dist y z : pseudo_metric_space.dist_triangle _ _ _
... = 0 + 0 : by rw [hxy, hyz]
... = 0 : by simp }
end
local attribute [instance] pseudo_metric.dist_setoid
/-- The canonical quotient of a pseudometric space, identifying points at distance `0`. -/
@[reducible] definition pseudo_metric_quot (α : Type u) [pseudo_metric_space α] : Type* :=
quotient (pseudo_metric.dist_setoid α)
instance has_dist_metric_quot {α : Type u} [pseudo_metric_space α] :
has_dist (pseudo_metric_quot α) :=
{ dist := quotient.lift₂ (λp q : α, dist p q)
begin
assume x y x' y' hxx' hyy',
have Hxx' : dist x x' = 0 := hxx',
have Hyy' : dist y y' = 0 := hyy',
have A : dist x y ≤ dist x' y' := calc
dist x y ≤ dist x x' + dist x' y : pseudo_metric_space.dist_triangle _ _ _
... = dist x' y : by simp [Hxx']
... ≤ dist x' y' + dist y' y : pseudo_metric_space.dist_triangle _ _ _
... = dist x' y' : by simp [pseudo_metric_space.dist_comm, Hyy'],
have B : dist x' y' ≤ dist x y := calc
dist x' y' ≤ dist x' x + dist x y' : pseudo_metric_space.dist_triangle _ _ _
... = dist x y' : by simp [pseudo_metric_space.dist_comm, Hxx']
... ≤ dist x y + dist y y' : pseudo_metric_space.dist_triangle _ _ _
... = dist x y : by simp [Hyy'],
exact le_antisymm A B
end }
lemma pseudo_metric_quot_dist_eq {α : Type u} [pseudo_metric_space α] (p q : α) :
dist ⟦p⟧ ⟦q⟧ = dist p q := rfl
instance metric_space_quot {α : Type u} [pseudo_metric_space α] :
metric_space (pseudo_metric_quot α) :=
{ dist_self := begin
refine quotient.ind (λy, _),
exact pseudo_metric_space.dist_self _
end,
eq_of_dist_eq_zero := λxc yc, by exact quotient.induction_on₂ xc yc (λx y H, quotient.sound H),
dist_comm :=
λxc yc, quotient.induction_on₂ xc yc (λx y, pseudo_metric_space.dist_comm _ _),
dist_triangle :=
λxc yc zc, quotient.induction_on₃ xc yc zc (λx y z, pseudo_metric_space.dist_triangle _ _ _) }
end eq_rel
|
06cb46383c44d6cda6362961fc896a75d8426a07 | f7c63613a4933f66368ef2802a50c417a46cddfc | /library/init/meta/derive.lean | 776a8e1b34eafa73a074a63ba8c1e48240537c0a | [
"Apache-2.0"
] | permissive | avigad/lean | 28cef71cd8c4eae53f342c87f81ca78d7cc75874 | 5410178203ab5ae854b5804586ace074ffd63aae | refs/heads/master | 1,608,801,379,340 | 1,504,894,459,000 | 1,505,174,163,000 | 22,361,408 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,773 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Ullrich
Attribute that can automatically derive typeclass instances.
-/
prelude
import init.meta.attribute
import init.meta.interactive_base
import init.meta.mk_has_reflect_instance
import init.meta.mk_has_sizeof_instance
import init.meta.mk_inhabited_instance
open lean
open interactive.types
open tactic
/-- A handler that may or may not be able to implement the typeclass `cls` for `decl`.
It should return `tt` if it was able to derive `cls` and `ff` if it does not know
how to derive `cls`, in which case lower-priority handlers will be tried next. -/
meta def derive_handler := Π (cls : pexpr) (decl : name), tactic bool
@[user_attribute]
meta def derive_handler_attr : user_attribute :=
{ name := `derive_handler, descr := "register a definition of type `derive_handler` for use in the [derive] attribute" }
private meta def try_handlers (p : pexpr) (n : name) : list derive_handler → tactic unit
| [] := fail format!"failed to find a derive handler for '{p}'"
| (h::hs) :=
do success ← h p n,
when (¬success) $
try_handlers hs
@[user_attribute] meta def derive_attr : user_attribute unit (list pexpr) :=
{ name := `derive, descr := "automatically derive typeclass instances",
parser := pexpr_list_or_texpr,
after_set := some (λ n _ _,
do ps ← derive_attr.get_param n,
handlers ← attribute.get_instances `derive_handler,
handlers ← handlers.mmap (λ n, eval_expr derive_handler (expr.const n [])),
ps.mmap' (λ p, try_handlers p n handlers)) }
meta def instance_derive_handler (cls : name) (tac : tactic unit) (univ_poly := tt)
(modify_target : name → list expr → expr → tactic expr := λ _ _, pure) : derive_handler :=
λ p n,
if p.is_constant_of cls then
do decl ← get_decl n,
cls_decl ← get_decl cls,
env ← get_env,
guard (env.is_inductive n) <|> fail format!"failed to derive '{cls}', '{n}' is not an inductive type",
let ls := decl.univ_params.map $ λ n, if univ_poly then level.param n else level.zero,
let tgt : expr := expr.const n ls,
⟨params, _⟩ ← mk_local_pis (decl.type.instantiate_univ_params (decl.univ_params.zip ls)),
let params := params.take (env.inductive_num_params n),
let tgt := tgt.mk_app params,
tgt ← mk_app cls [tgt],
tgt ← modify_target n params tgt,
tgt ← params.mfoldr (λ param tgt,
do param_cls ← mk_app cls [param],
-- TODO(sullrich): omit some typeclass parameters based on usage of `param`?
let tgt := expr.pi `a binder_info.inst_implicit param_cls tgt,
pure $ tgt.bind_pi param
) tgt,
(_, val) ← tactic.solve_aux tgt (intros >> tac),
val ← instantiate_mvars val,
let trusted := decl.is_trusted ∧ cls_decl.is_trusted,
add_decl (declaration.defn (n ++ cls)
(if univ_poly then decl.univ_params else [])
tgt val reducibility_hints.abbrev trusted),
set_basic_attribute `instance (n ++ cls) tt,
pure true
else pure false
@[derive_handler] meta def has_reflect_derive_handler :=
instance_derive_handler ``has_reflect mk_has_reflect_instance ff (λ n params tgt,
-- add additional `reflected` assumption for each parameter
params.mfoldr (λ param tgt,
do param_cls ← mk_app `reflected [param],
let tgt := expr.pi `a binder_info.inst_implicit param_cls tgt,
pure $ tgt
) tgt)
@[derive_handler] meta def has_sizeof_derive_handler :=
instance_derive_handler ``has_sizeof mk_has_sizeof_instance
@[derive_handler] meta def inhabited_derive_handler :=
instance_derive_handler ``inhabited mk_inhabited_instance
attribute [derive has_reflect] bool prod sum option interactive.loc pos
|
ba2ec7761a38e38484bfe8272f796d7ea8cedc85 | 08bd4ba4ca87dba1f09d2c96a26f5d65da81f4b4 | /src/Init/WF.lean | 1ba5015043cec366eb4879d3dfa645326cddca2a | [
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"Apache-2.0",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | gebner/lean4 | d51c4922640a52a6f7426536ea669ef18a1d9af5 | 8cd9ce06843c9d42d6d6dc43d3e81e3b49dfc20f | refs/heads/master | 1,685,732,780,391 | 1,672,962,627,000 | 1,673,459,398,000 | 373,307,283 | 0 | 0 | Apache-2.0 | 1,691,316,730,000 | 1,622,669,271,000 | Lean | UTF-8 | Lean | false | false | 12,142 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import Init.SizeOf
import Init.Data.Nat.Basic
universe u v
inductive Acc {α : Sort u} (r : α → α → Prop) : α → Prop where
| intro (x : α) (h : (y : α) → r y x → Acc r y) : Acc r x
noncomputable abbrev Acc.ndrec.{u1, u2} {α : Sort u2} {r : α → α → Prop} {C : α → Sort u1}
(m : (x : α) → ((y : α) → r y x → Acc r y) → ((y : α) → (a : r y x) → C y) → C x)
{a : α} (n : Acc r a) : C a :=
n.rec m
noncomputable abbrev Acc.ndrecOn.{u1, u2} {α : Sort u2} {r : α → α → Prop} {C : α → Sort u1}
{a : α} (n : Acc r a)
(m : (x : α) → ((y : α) → r y x → Acc r y) → ((y : α) → (a : r y x) → C y) → C x)
: C a :=
n.rec m
namespace Acc
variable {α : Sort u} {r : α → α → Prop}
def inv {x y : α} (h₁ : Acc r x) (h₂ : r y x) : Acc r y :=
h₁.recOn (fun _ ac₁ _ h₂ => ac₁ y h₂) h₂
end Acc
inductive WellFounded {α : Sort u} (r : α → α → Prop) : Prop where
| intro (h : ∀ a, Acc r a) : WellFounded r
class WellFoundedRelation (α : Sort u) where
rel : α → α → Prop
wf : WellFounded rel
namespace WellFounded
def apply {α : Sort u} {r : α → α → Prop} (wf : WellFounded r) (a : α) : Acc r a :=
wf.rec (fun p => p) a
section
variable {α : Sort u} {r : α → α → Prop} (hwf : WellFounded r)
theorem recursion {C : α → Sort v} (a : α) (h : ∀ x, (∀ y, r y x → C y) → C x) : C a := by
induction (apply hwf a) with
| intro x₁ _ ih => exact h x₁ ih
theorem induction {C : α → Prop} (a : α) (h : ∀ x, (∀ y, r y x → C y) → C x) : C a :=
recursion hwf a h
variable {C : α → Sort v}
variable (F : ∀ x, (∀ y, r y x → C y) → C x)
noncomputable def fixF (x : α) (a : Acc r x) : C x := by
induction a with
| intro x₁ _ ih => exact F x₁ ih
def fixFEq (x : α) (acx : Acc r x) : fixF F x acx = F x (fun (y : α) (p : r y x) => fixF F y (Acc.inv acx p)) := by
induction acx with
| intro x r _ => exact rfl
end
variable {α : Sort u} {C : α → Sort v} {r : α → α → Prop}
-- Well-founded fixpoint
noncomputable def fix (hwf : WellFounded r) (F : ∀ x, (∀ y, r y x → C y) → C x) (x : α) : C x :=
fixF F x (apply hwf x)
-- Well-founded fixpoint satisfies fixpoint equation
theorem fix_eq (hwf : WellFounded r) (F : ∀ x, (∀ y, r y x → C y) → C x) (x : α) :
fix hwf F x = F x (fun y _ => fix hwf F y) :=
fixFEq F x (apply hwf x)
end WellFounded
open WellFounded
-- Empty relation is well-founded
def emptyWf {α : Sort u} : WellFoundedRelation α where
rel := emptyRelation
wf := by
apply WellFounded.intro
intro a
apply Acc.intro a
intro b h
cases h
-- Subrelation of a well-founded relation is well-founded
namespace Subrelation
variable {α : Sort u} {r q : α → α → Prop}
def accessible {a : α} (h₁ : Subrelation q r) (ac : Acc r a) : Acc q a := by
induction ac with
| intro x _ ih =>
apply Acc.intro
intro y h
exact ih y (h₁ h)
def wf (h₁ : Subrelation q r) (h₂ : WellFounded r) : WellFounded q :=
⟨fun a => accessible @h₁ (apply h₂ a)⟩
end Subrelation
-- The inverse image of a well-founded relation is well-founded
namespace InvImage
variable {α : Sort u} {β : Sort v} {r : β → β → Prop}
private def accAux (f : α → β) {b : β} (ac : Acc r b) : (x : α) → f x = b → Acc (InvImage r f) x := by
induction ac with
| intro x acx ih =>
intro z e
apply Acc.intro
intro y lt
subst x
apply ih (f y) lt y rfl
def accessible {a : α} (f : α → β) (ac : Acc r (f a)) : Acc (InvImage r f) a :=
accAux f ac a rfl
def wf (f : α → β) (h : WellFounded r) : WellFounded (InvImage r f) :=
⟨fun a => accessible f (apply h (f a))⟩
end InvImage
@[reducible] def invImage (f : α → β) (h : WellFoundedRelation β) : WellFoundedRelation α where
rel := InvImage h.rel f
wf := InvImage.wf f h.wf
-- The transitive closure of a well-founded relation is well-founded
namespace TC
variable {α : Sort u} {r : α → α → Prop}
def accessible {z : α} (ac : Acc r z) : Acc (TC r) z := by
induction ac with
| intro x acx ih =>
apply Acc.intro x
intro y rel
induction rel with
| base a b rab => exact ih a rab
| trans a b c rab _ _ ih₂ => apply Acc.inv (ih₂ acx ih) rab
def wf (h : WellFounded r) : WellFounded (TC r) :=
⟨fun a => accessible (apply h a)⟩
end TC
namespace Nat
-- less-than is well-founded
def lt_wfRel : WellFoundedRelation Nat where
rel := Nat.lt
wf := by
apply WellFounded.intro
intro n
induction n with
| zero =>
apply Acc.intro 0
intro _ h
apply absurd h (Nat.not_lt_zero _)
| succ n ih =>
apply Acc.intro (Nat.succ n)
intro m h
have : m = n ∨ m < n := Nat.eq_or_lt_of_le (Nat.le_of_succ_le_succ h)
match this with
| Or.inl e => subst e; assumption
| Or.inr e => exact Acc.inv ih e
protected theorem strongInductionOn
{motive : Nat → Sort u}
(n : Nat)
(ind : ∀ n, (∀ m, m < n → motive m) → motive n) : motive n :=
Nat.lt_wfRel.wf.fix ind n
protected theorem caseStrongInductionOn
{motive : Nat → Sort u}
(a : Nat)
(zero : motive 0)
(ind : ∀ n, (∀ m, m ≤ n → motive m) → motive (succ n)) : motive a :=
Nat.strongInductionOn a fun n =>
match n with
| 0 => fun _ => zero
| n+1 => fun h₁ => ind n (λ _ h₂ => h₁ _ (lt_succ_of_le h₂))
end Nat
def Measure {α : Sort u} : (α → Nat) → α → α → Prop :=
InvImage (fun a b => a < b)
abbrev measure {α : Sort u} (f : α → Nat) : WellFoundedRelation α :=
invImage f Nat.lt_wfRel
def SizeOfRef (α : Sort u) [SizeOf α] : α → α → Prop :=
Measure sizeOf
abbrev sizeOfWFRel {α : Sort u} [SizeOf α] : WellFoundedRelation α :=
measure sizeOf
instance (priority := low) [SizeOf α] : WellFoundedRelation α :=
sizeOfWFRel
namespace Prod
open WellFounded
section
variable {α : Type u} {β : Type v}
variable (ra : α → α → Prop)
variable (rb : β → β → Prop)
-- Lexicographical order based on ra and rb
protected inductive Lex : α × β → α × β → Prop where
| left {a₁} (b₁) {a₂} (b₂) (h : ra a₁ a₂) : Prod.Lex (a₁, b₁) (a₂, b₂)
| right (a) {b₁ b₂} (h : rb b₁ b₂) : Prod.Lex (a, b₁) (a, b₂)
-- TODO: generalize
def Lex.right' {a₁ : Nat} {b₁ : β} (h₁ : a₁ ≤ a₂) (h₂ : rb b₁ b₂) : Prod.Lex Nat.lt rb (a₁, b₁) (a₂, b₂) :=
match Nat.eq_or_lt_of_le h₁ with
| Or.inl h => h ▸ Prod.Lex.right a₁ h₂
| Or.inr h => Prod.Lex.left b₁ _ h
-- relational product based on ra and rb
inductive RProd : α × β → α × β → Prop where
| intro {a₁ b₁ a₂ b₂} (h₁ : ra a₁ a₂) (h₂ : rb b₁ b₂) : RProd (a₁, b₁) (a₂, b₂)
end
section
variable {α : Type u} {β : Type v}
variable {ra : α → α → Prop} {rb : β → β → Prop}
def lexAccessible (aca : (a : α) → Acc ra a) (acb : (b : β) → Acc rb b) (a : α) (b : β) : Acc (Prod.Lex ra rb) (a, b) := by
induction (aca a) generalizing b with
| intro xa _ iha =>
induction (acb b) with
| intro xb _ ihb =>
apply Acc.intro (xa, xb)
intro p lt
cases lt with
| left _ _ h => apply iha _ h
| right _ h => apply ihb _ h
-- The lexicographical order of well founded relations is well-founded
@[reducible] def lex (ha : WellFoundedRelation α) (hb : WellFoundedRelation β) : WellFoundedRelation (α × β) where
rel := Prod.Lex ha.rel hb.rel
wf := ⟨fun (a, b) => lexAccessible (WellFounded.apply ha.wf) (WellFounded.apply hb.wf) a b⟩
instance [ha : WellFoundedRelation α] [hb : WellFoundedRelation β] : WellFoundedRelation (α × β) :=
lex ha hb
-- relational product is a Subrelation of the Lex
def RProdSubLex (a : α × β) (b : α × β) (h : RProd ra rb a b) : Prod.Lex ra rb a b := by
cases h with
| intro h₁ h₂ => exact Prod.Lex.left _ _ h₁
-- The relational product of well founded relations is well-founded
def rprod (ha : WellFoundedRelation α) (hb : WellFoundedRelation β) : WellFoundedRelation (α × β) where
rel := RProd ha.rel hb.rel
wf := by
apply Subrelation.wf (r := Prod.Lex ha.rel hb.rel) (h₂ := (lex ha hb).wf)
intro a b h
exact RProdSubLex a b h
end
end Prod
namespace PSigma
section
variable {α : Sort u} {β : α → Sort v}
variable (r : α → α → Prop)
variable (s : ∀ a, β a → β a → Prop)
-- Lexicographical order based on r and s
inductive Lex : PSigma β → PSigma β → Prop where
| left : ∀ {a₁ : α} (b₁ : β a₁) {a₂ : α} (b₂ : β a₂), r a₁ a₂ → Lex ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
| right : ∀ (a : α) {b₁ b₂ : β a}, s a b₁ b₂ → Lex ⟨a, b₁⟩ ⟨a, b₂⟩
end
section
variable {α : Sort u} {β : α → Sort v}
variable {r : α → α → Prop} {s : ∀ (a : α), β a → β a → Prop}
def lexAccessible {a} (aca : Acc r a) (acb : (a : α) → WellFounded (s a)) (b : β a) : Acc (Lex r s) ⟨a, b⟩ := by
induction aca with
| intro xa _ iha =>
induction (WellFounded.apply (acb xa) b) with
| intro xb _ ihb =>
apply Acc.intro
intro p lt
cases lt with
| left => apply iha; assumption
| right => apply ihb; assumption
-- The lexicographical order of well founded relations is well-founded
@[reducible] def lex (ha : WellFoundedRelation α) (hb : (a : α) → WellFoundedRelation (β a)) : WellFoundedRelation (PSigma β) where
rel := Lex ha.rel (fun a => hb a |>.rel)
wf := WellFounded.intro fun ⟨a, b⟩ => lexAccessible (WellFounded.apply ha.wf a) (fun a => hb a |>.wf) b
instance [ha : WellFoundedRelation α] [hb : (a : α) → WellFoundedRelation (β a)] : WellFoundedRelation (PSigma β) :=
lex ha hb
end
section
variable {α : Sort u} {β : Sort v}
def lexNdep (r : α → α → Prop) (s : β → β → Prop) :=
Lex r (fun _ => s)
def lexNdepWf {r : α → α → Prop} {s : β → β → Prop} (ha : WellFounded r) (hb : WellFounded s) : WellFounded (lexNdep r s) :=
WellFounded.intro fun ⟨a, b⟩ => lexAccessible (WellFounded.apply ha a) (fun _ => hb) b
end
section
variable {α : Sort u} {β : Sort v}
-- Reverse lexicographical order based on r and s
inductive RevLex (r : α → α → Prop) (s : β → β → Prop) : @PSigma α (fun _ => β) → @PSigma α (fun _ => β) → Prop where
| left : {a₁ a₂ : α} → (b : β) → r a₁ a₂ → RevLex r s ⟨a₁, b⟩ ⟨a₂, b⟩
| right : (a₁ : α) → {b₁ : β} → (a₂ : α) → {b₂ : β} → s b₁ b₂ → RevLex r s ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
end
section
open WellFounded
variable {α : Sort u} {β : Sort v}
variable {r : α → α → Prop} {s : β → β → Prop}
def revLexAccessible {b} (acb : Acc s b) (aca : (a : α) → Acc r a): (a : α) → Acc (RevLex r s) ⟨a, b⟩ := by
induction acb with
| intro xb _ ihb =>
intro a
induction (aca a) with
| intro xa _ iha =>
apply Acc.intro
intro p lt
cases lt with
| left => apply iha; assumption
| right => apply ihb; assumption
def revLex (ha : WellFounded r) (hb : WellFounded s) : WellFounded (RevLex r s) :=
WellFounded.intro fun ⟨a, b⟩ => revLexAccessible (apply hb b) (WellFounded.apply ha) a
end
section
def SkipLeft (α : Type u) {β : Type v} (s : β → β → Prop) : @PSigma α (fun _ => β) → @PSigma α (fun _ => β) → Prop :=
RevLex emptyRelation s
def skipLeft (α : Type u) {β : Type v} (hb : WellFoundedRelation β) : WellFoundedRelation (PSigma fun _ : α => β) where
rel := SkipLeft α hb.rel
wf := revLex emptyWf.wf hb.wf
def mkSkipLeft {α : Type u} {β : Type v} {b₁ b₂ : β} {s : β → β → Prop} (a₁ a₂ : α) (h : s b₁ b₂) : SkipLeft α s ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ :=
RevLex.right _ _ h
end
end PSigma
|
a19d26f2f97740b1491bf007c39d05f2ff32b881 | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /tests/lean/run/PPTopDownAnalyze.lean | 668e7b39f9be2c6614f6fa676e06bfec24d2fe16 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 13,869 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Selsam
-/
import Lean
import Std
open Lean Lean.Meta Lean.Elab Lean.Elab.Term Lean.Elab.Command
open Lean.PrettyPrinter
def checkDelab (e : Expr) (tgt? : Option Term) (name? : Option Name := none) : TermElabM Unit := do
let pfix := "[checkDelab" ++ (match name? with | some n => ("." ++ toString n) | none => "") ++ "]"
if e.hasMVar then throwError "{pfix} original term has mvars, {e}"
let stx ← delab e
match tgt? with
| some tgt =>
if toString (← PrettyPrinter.ppTerm stx) != toString (← PrettyPrinter.ppTerm tgt?.get!) then
throwError "{pfix} missed target\n{← PrettyPrinter.ppTerm stx}\n!=\n{← PrettyPrinter.ppTerm tgt}\n\nExpr: {e}\nType: {← inferType e}"
| _ => pure ()
let e' ←
try
let e' ← elabTerm stx (some (← inferType e))
synthesizeSyntheticMVarsNoPostponing
let e' ← instantiateMVars e'
-- let ⟨e', _⟩ ← levelMVarToParam e'
throwErrorIfErrors
pure e'
catch ex => throwError "{pfix} failed to re-elaborate,\n{stx}\n{← ex.toMessageData.toString}"
withTheReader Core.Context (fun ctx => { ctx with options := ctx.options.setBool `pp.all true }) do
if not (← isDefEq e e') then
println! "{pfix} {← inferType e} {← inferType e'}"
throwError "{pfix} roundtrip not structurally equal\n\nOriginal: {e}\n\nSyntax: {stx}\n\nNew: {e'}"
let e' ← instantiateMVars e'
if e'.hasMVar then throwError "{pfix} elaborated term still has mvars\n\nSyntax: {stx}\n\nExpression: {e'}"
syntax (name := testDelabTD) "#testDelab " term " expecting " term : command
@[commandElab testDelabTD] def elabTestDelabTD : CommandElab
| `(#testDelab $stx:term expecting $tgt:term) => liftTermElabM do withDeclName `delabTD do
let e ← elabTerm stx none
let ⟨e, _⟩ ← levelMVarToParam e
let e ← instantiateMVars e
checkDelab e (some tgt)
| _ => throwUnsupportedSyntax
syntax (name := testDelabTDN) "#testDelabN " ident : command
@[commandElab testDelabTDN] def elabTestDelabTDN : CommandElab
| `(#testDelabN $name:ident) => liftTermElabM do withDeclName `delabTD do
let name := name.getId
let [name] ← resolveGlobalConst (mkIdent name) | throwError "cannot resolve name"
let some cInfo := (← getEnv).find? name | throwError "no decl for name"
let some value ← pure cInfo.value? | throwError "decl has no value"
modify fun s => { s with levelNames := cInfo.levelParams }
withTheReader Core.Context (fun ctx => { ctx with currNamespace := name.getPrefix, openDecls := [] }) do
checkDelab value none (some name)
| _ => throwUnsupportedSyntax
-------------------------------------------------
-------------------------------------------------
-------------------------------------------------
universe u v
set_option pp.analyze true
set_option pp.analyze.checkInstances true
set_option pp.analyze.explicitHoles true
set_option pp.proofs true
#testDelab @Nat.brecOn (fun x => Nat) 0 (fun x ih => x)
expecting Nat.brecOn (motive := fun x => Nat) 0 fun x ih => x
#testDelab @Nat.brecOn (fun x => Nat → Nat) 0 (fun x ih => fun y => y + x)
expecting Nat.brecOn (motive := fun x => Nat → Nat) 0 fun x ih y => y + x
#testDelab @Nat.brecOn (fun x => Nat → Nat) 0 (fun x ih => fun y => y + x) 0
expecting Nat.brecOn (motive := fun x => Nat → Nat) 0 (fun x ih y => y + x) 0
#testDelab let xs := #[]; xs.push (5 : Nat)
expecting let xs : Array Nat := #[]; Array.push xs 5
#testDelab let x := Nat.zero; x + Nat.zero
expecting let x := Nat.zero; x + Nat.zero
def fHole (α : Type) (x : α) : α := x
#testDelab fHole Nat Nat.zero
expecting fHole _ Nat.zero
def fPoly {α : Type u} (x : α) : α := x
#testDelab fPoly Nat.zero
expecting fPoly Nat.zero
#testDelab fPoly (id Nat.zero)
expecting fPoly (id Nat.zero)
def fPoly2 {α : Type u} {β : Type v} (x : α) : α := x
#testDelab @fPoly2 _ (Type 3) Nat.zero
expecting fPoly2 (β := Type 3) Nat.zero
def fPolyInst {α : Type u} [Add α] : α → α → α := Add.add
#testDelab @fPolyInst Nat ⟨Nat.add⟩
expecting fPolyInst
def fPolyNotInst {α : Type u} (inst : Add α) : α → α → α := Add.add
#testDelab @fPolyNotInst Nat ⟨Nat.add⟩
expecting fPolyNotInst { add := Nat.add }
#testDelab (fun (x : Nat) => x) Nat.zero
expecting (fun (x : Nat) => x) Nat.zero
#testDelab (fun (α : Type) (x : α) => x) Nat Nat.zero
expecting (fun (α : Type) (x : α) => x) _ Nat.zero
#testDelab (fun {α : Type} (x : α) => x) Nat.zero
expecting (fun {α : Type} (x : α) => x) Nat.zero
#testDelab ((@Add.mk Nat Nat.add).1 : Nat → Nat → Nat)
expecting Add.add
class Foo (α : Type v) where foo : α
instance : Foo Bool := ⟨true⟩
#testDelab @Foo.foo Bool ⟨true⟩
expecting Foo.foo
#testDelab @Foo.foo Bool ⟨false⟩
expecting Foo.foo (self := { foo := false })
axiom wild {α : Type u} {f : α → Type v} {x : α} [_inst_1 : Foo (f x)] : Nat
abbrev nat2bool : Nat → Type := fun _ => Bool
#testDelab @wild Nat nat2bool Nat.zero ⟨false⟩
expecting wild (f := nat2bool) (x := Nat.zero) (_inst_1 := { foo := false })
#testDelab @wild Nat (fun (n : Nat) => Bool) Nat.zero ⟨false⟩
expecting wild (f := fun n => Bool) (x := Nat.zero) (_inst_1 := { foo := false })
def takesFooUnnamed {Impl : Type} (Expl : Type) [Foo Nat] (x : Impl) (y : Expl) : Impl × Expl := (x, y)
#testDelab @takesFooUnnamed _ Nat (Foo.mk 7) false 5
expecting @takesFooUnnamed _ _ { foo := 7 } false 5
#testDelab (fun {α : Type u} (x : α) => x : ∀ {α : Type u}, α → α)
expecting fun {α} x => x
#testDelab (fun {α : Type} (x : α) => x) Nat.zero
expecting (fun {α : Type} (x : α) => x) Nat.zero
#testDelab (fun {α : Type} [Add α] (x : α) => x + x) (0 : Nat)
expecting (fun {α : Type} [Add α] (x : α) => x + x) 0
#testDelab (fun {α : Type} [Add α] (x : α) => x + x) Nat.zero
expecting (fun {α : Type} [Add α] (x : α) => x + x) Nat.zero
#testDelab id id id id Nat.zero
expecting id id id id Nat.zero
def zzz : Unit := ()
def Z1.zzz : Unit := ()
def Z1.Z2.zzz : Unit := ()
namespace Z1.Z2
#testDelab _root_.zzz
expecting _root_.zzz
#testDelab Z1.zzz
expecting Z1.zzz
#testDelab zzz
expecting zzz
end Z1.Z2
#testDelab fun {σ : Type u} {m : Type u → Type v} [Monad m] {α : Type u} (f : σ → α × σ) (s : σ) => pure (f := m) (f s)
expecting fun {σ} {m} [Monad m] {α} f s => pure (f s)
set_option pp.analyze.trustSubst false in
#testDelab (fun (x y z : Nat) (hxy : x = y) (hyz : x = z) => hxy ▸ hyz : ∀ (x y z : Nat), x = y → x = z → y = z)
expecting fun x y z hxy hyz => Eq.rec (motive := fun x_1 h => x_1 = z) hyz hxy
set_option pp.analyze.trustSubst true in
#testDelab (fun (x y z : Nat) (hxy : x = y) (hyz : x = z) => hxy ▸ hyz : ∀ (x y z : Nat), x = y → x = z → y = z)
expecting fun x y z hxy hyz => hxy ▸ hyz
set_option pp.analyze.trustId true in
#testDelab Sigma.mk (β := fun α => α) Bool true
expecting { fst := _, snd := true }
set_option pp.analyze.trustId false in
#testDelab Sigma.mk (β := fun α => α) Bool true
expecting Sigma.mk (β := fun α => α) _ true
#testDelab let xs := #[true]; xs
expecting let xs := #[true]; xs
def fooReadGetModify : ReaderT Unit (StateT Unit IO) Unit := do
let _ ← read
let _ ← get
modify fun s => s
#testDelab
(do discard read
pure () : ReaderT Bool IO Unit)
expecting
do discard read
pure ()
#testDelab
((do let ctx ← read
let s ← get
modify fun s => s : ReaderT Bool (StateT Bool IO) Unit))
expecting
do let _ ← read
let _ ← get
modify fun s => s
set_option pp.analyze.typeAscriptions true in
#testDelab (fun (x : Unit) => @id (ReaderT Bool IO Bool) (do read : ReaderT Bool IO Bool)) ()
expecting (fun (x : Unit) => (id read : ReaderT Bool IO Bool)) ()
set_option pp.analyze.typeAscriptions false in
#testDelab (fun (x : Unit) => @id (ReaderT Bool IO Bool) (do read : ReaderT Bool IO Bool)) ()
expecting (fun (x : Unit) => id read) ()
instance : CoeFun Bool (fun b => Bool → Bool) := { coe := fun b x => b && x }
#testDelab fun (xs : List Nat) => xs ≠ xs
expecting fun xs => xs ≠ xs
structure S1 where x : Unit
structure S2 where x : Unit
#testDelab { x := () : S1 }
expecting { x := () }
#testDelab (fun (u : Unit) => { x := () : S2 }) ()
expecting (fun (u : Unit) => { x := () : S2 }) ()
#testDelab Eq.refl True
expecting Eq.refl _
#testDelab (fun (u : Unit) => Eq.refl True) ()
expecting (fun (u : Unit) => Eq.refl True) ()
inductive NeedsAnalysis {α : Type} : Prop
| mk : NeedsAnalysis
set_option pp.proofs false in
#testDelab @NeedsAnalysis.mk Unit
expecting (_ : NeedsAnalysis (α := Unit))
set_option pp.proofs false in
set_option pp.proofs.withType false in
#testDelab @NeedsAnalysis.mk Unit
expecting _
#testDelab ∀ (α : Type u) (vals vals_1 : List α), { data := vals : Array α } = { data := vals_1 : Array α }
expecting ∀ (α : Type u) (vals vals_1 : List α), { data := vals : Array α } = { data := vals_1 }
#testDelab (do let ctxCore ← readThe Core.Context; pure ctxCore.currNamespace : MetaM Name)
expecting do
let ctxCore ← readThe Core.Context
pure ctxCore.currNamespace
structure SubtypeLike1 {α : Sort u} (p : α → Prop) where
#testDelab SubtypeLike1 fun (x : Nat) => x < 10
expecting SubtypeLike1 fun (x : Nat) => x < 10
#eval "prevent comment from parsing as part of previous expression"
--Note: currently we do not try "bottom-upping" inside lambdas
--(so we will always annotate the binder type)
#testDelab SubtypeLike1 fun (x : Nat) => Nat.succ x = x
expecting SubtypeLike1 fun (x : Nat) => Nat.succ x = x
structure SubtypeLike3 {α β γ : Sort u} (p : α → β → γ → Prop) where
#testDelab SubtypeLike3 fun (x y z : Nat) => x + y < z
expecting SubtypeLike3 fun (x y z : Nat) => x + y < z
structure SubtypeLike3Double {α β γ : Sort u} (p₁ : α → β → Prop) (p₂ : β → γ → Prop) where
#testDelab SubtypeLike3Double (fun (x y : Nat) => x = y) (fun (y z : Nat) => y = z)
expecting SubtypeLike3Double (fun (x y : Nat) => x = y) fun y (z : Nat) => y = z
def takesStricts ⦃α : Type⦄ {β : Type} ⦃γ : Type⦄ : Unit := ()
#testDelab takesStricts expecting takesStricts
#testDelab @takesStricts expecting takesStricts
#testDelab @takesStricts Unit Unit expecting takesStricts (α := Unit) (β := Unit)
#testDelab @takesStricts Unit Unit Unit expecting takesStricts (α := Unit) (β := Unit) (γ := Unit)
def takesStrictMotive ⦃motive : Nat → Type⦄ {n : Nat} (x : motive n) : motive n := x
#testDelab takesStrictMotive expecting takesStrictMotive
#testDelab @takesStrictMotive (fun x => Unit) 0 expecting takesStrictMotive (motive := fun x => Unit) (n := 0)
#testDelab @takesStrictMotive (fun x => Unit) 0 () expecting takesStrictMotive (motive := fun x => Unit) (n := 0) ()
def stackMkInjEqSnippet :=
fun {α : Type} (xs : Array α) => Eq.ndrec (motive := fun _ => (Std.Stack.mk xs = Std.Stack.mk xs)) (Eq.refl (Std.Stack.mk xs)) (rfl : xs = xs)
#testDelabN stackMkInjEqSnippet
def typeAs (α : Type u) (a : α) := ()
set_option pp.analyze.explicitHoles false in
#testDelab ∀ {α : Sort u} {β : α → Sort v} {f₁ f₂ : (x : α) → β x}, (∀ (x : α), f₁ x = f₂ _) → f₁ = f₂
expecting ∀ {α : Sort u} {β : α → Sort v} {f₁ f₂ : (x : α) → β x}, (∀ (x : α), f₁ x = f₂ x) → f₁ = f₂
set_option pp.analyze.trustSubtypeMk true in
#testDelab fun (n : Nat) (val : List Nat) (property : List.length val = n) => List.length { val := val, property := property : { x : List Nat // List.length x = n } }.val = n
expecting fun n val property => List.length { val := val, property := property : { x : List Nat // List.length x = n } }.val = n
#testDelabN Nat.brecOn
#testDelabN Nat.below
#testDelabN Nat.mod_lt
#testDelabN Array.qsort
#testDelabN List.partition
#testDelabN List.partitionAux
#testDelabN StateT.modifyGet
#testDelabN Nat.gcd_one_left
#testDelabN List.hasDecidableLt
#testDelabN Lean.Xml.parse
#testDelabN Add.noConfusionType
#testDelabN List.filterMapM.loop
#testDelabN instMonadReaderOf
#testDelabN instInhabitedPUnit
#testDelabN Lean.Syntax.getOptionalIdent?
#testDelabN Lean.Meta.ppExpr
#testDelabN MonadLift.noConfusion
#testDelabN MonadLift.noConfusionType
#testDelabN MonadExcept.noConfusion
#testDelabN MonadFinally.noConfusion
#testDelabN Lean.Elab.InfoTree.goalsAt?.match_1
#testDelabN Std.ShareCommon.ObjectMap.find?
#testDelabN Std.ShareCommon.ObjectMap.insert
#testDelabN Std.Stack.mk.injEq
#testDelabN Lean.PrefixTree.empty
#testDelabN Std.PersistentHashMap.getCollisionNodeSize.match_1
#testDelabN Std.HashMap.size.match_1
-- TODO: for some reason this *only* works when trusting subst
set_option pp.analyze.trustSubst true in
#testDelabN and_false
-- TODO: this one prints out a structure instance with keyword field `end`
set_option pp.structureInstances false in
#testDelabN Lean.Server.FileWorker.handlePlainTermGoal
-- TODO: this one desugars to a `doLet` in an assignment
set_option pp.notation false in
#testDelabN Lean.Server.FileWorker.handlePlainGoal
-- TODO: this error occurs because we use a term's type to determine `blockImplicit` (@),
-- whereas we should actually use the expected type based on the function being applied.
-- #testDelabN HEq.subst
-- TODO: this error occurs because it cannot solve the universe constraints
-- (unclear if it is too few or too many annotations)
-- #testDelabN ExceptT.seqRight_eq
-- TODO: this error occurs because a function has explicit binders while its type has
-- implicit binders. This may be an issue in the elaborator.
-- #testDelabN Char.eqOfVeq
|
3137af4669eb074e78acbfc5a8c802fd2bbcb5bb | d1a52c3f208fa42c41df8278c3d280f075eb020c | /tests/lean/run/trace.lean | 694b771188d3142b2d950bf93c715e3229c54446 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | cipher1024/lean4 | 6e1f98bb58e7a92b28f5364eb38a14c8d0aae393 | 69114d3b50806264ef35b57394391c3e738a9822 | refs/heads/master | 1,642,227,983,603 | 1,642,011,696,000 | 1,642,011,696,000 | 228,607,691 | 0 | 0 | Apache-2.0 | 1,576,584,269,000 | 1,576,584,268,000 | null | UTF-8 | Lean | false | false | 1,489 | lean | import Lean.CoreM
open Lean
structure MyState :=
(trace_state : TraceState := {})
(s : Nat := 0)
abbrev M := CoreM
def tst1 : M Unit :=
do trace[module] (m!"hello" ++ MessageData.nest 9 (m!"\n" ++ "world"));
trace[module.aux] "another message";
pure ()
def tst2 (b : Bool) : M Unit :=
traceCtx `module $ do
tst1;
trace[bughunt] "at test2";
if b then throwError "error";
tst1;
pure ()
partial def ack : Nat → Nat → Nat
| 0, n => n+1
| m+1, 0 => ack m 1
| m+1, n+1 => ack m (ack (m+1) n)
def slow (b : Bool) : Nat :=
ack 4 (cond b 0 1)
def tst3 (b : Bool) : M Unit :=
do traceCtx `module $ do {
tst2 b;
tst1
};
trace[bughunt] "at end of tst3";
-- Messages are computed lazily. The following message will only be computed
-- if `trace.slow is active.
trace[slow] (m!"slow message: " ++ toString (slow b))
-- This is true even if it is a monad computation:
trace[slow] (m!"slow message: " ++ (← toString (slow b)))
def run (x : M Unit) : M Unit :=
withReader
(fun ctx =>
-- Try commeting/uncommeting the following `setBool`s
let opts := ctx.options;
let opts := opts.setBool `trace.module true;
-- let opts := opts.setBool `trace.module.aux false;
let opts := opts.setBool `trace.bughunt true;
-- let opts := opts.setBool `trace.slow true;
{ ctx with options := opts })
(tryCatch (tryFinally x printTraces) (fun _ => IO.println "ERROR"))
#eval run (tst3 true)
#eval run (tst3 false)
|
b8037f08428a038869ef448187260492d18911df | 94e33a31faa76775069b071adea97e86e218a8ee | /src/algebraic_geometry/AffineScheme.lean | 654dc6e6d7b16609cb41cc639ace59f6c6a5012e | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 23,128 | lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import algebraic_geometry.Gamma_Spec_adjunction
import algebraic_geometry.open_immersion
import category_theory.limits.opposites
import ring_theory.localization.inv_submonoid
/-!
# Affine schemes
We define the category of `AffineScheme`s as the essential image of `Spec`.
We also define predicates about affine schemes and affine open sets.
## Main definitions
* `algebraic_geometry.AffineScheme`: The category of affine schemes.
* `algebraic_geometry.is_affine`: A scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an
isomorphism.
* `algebraic_geometry.Scheme.iso_Spec`: The canonical isomorphism `X ≅ Spec Γ(X)` for an affine
scheme.
* `algebraic_geometry.AffineScheme.equiv_CommRing`: The equivalence of categories
`AffineScheme ≌ CommRingᵒᵖ` given by `AffineScheme.Spec : CommRingᵒᵖ ⥤ AffineScheme` and
`AffineScheme.Γ : AffineSchemeᵒᵖ ⥤ CommRing`.
* `algebraic_geometry.is_affine_open`: An open subset of a scheme is affine if the open subscheme is
affine.
* `algebraic_geometry.is_affine_open.from_Spec`: The immersion `Spec 𝒪ₓ(U) ⟶ X` for an affine `U`.
-/
noncomputable theory
open category_theory category_theory.limits opposite topological_space
universe u
namespace algebraic_geometry
open Spec (structure_sheaf)
/-- The category of affine schemes -/
def AffineScheme := Scheme.Spec.ess_image
/-- A Scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism. -/
class is_affine (X : Scheme) : Prop :=
(affine : is_iso (Γ_Spec.adjunction.unit.app X))
attribute [instance] is_affine.affine
/-- The canonical isomorphism `X ≅ Spec Γ(X)` for an affine scheme. -/
def Scheme.iso_Spec (X : Scheme) [is_affine X] :
X ≅ Scheme.Spec.obj (op $ Scheme.Γ.obj $ op X) :=
as_iso (Γ_Spec.adjunction.unit.app X)
lemma mem_AffineScheme (X : Scheme) : X ∈ AffineScheme ↔ is_affine X :=
⟨λ h, ⟨functor.ess_image.unit_is_iso h⟩, λ h, @@mem_ess_image_of_unit_is_iso _ _ _ X h.1⟩
instance is_affine_AffineScheme (X : AffineScheme.{u}) : is_affine (X : Scheme.{u}) :=
(mem_AffineScheme _).mp X.prop
instance Spec_is_affine (R : CommRingᵒᵖ) : is_affine (Scheme.Spec.obj R) :=
(mem_AffineScheme _).mp (Scheme.Spec.obj_mem_ess_image R)
lemma is_affine_of_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] [h : is_affine Y] :
is_affine X :=
by { rw [← mem_AffineScheme] at h ⊢, exact functor.ess_image.of_iso (as_iso f).symm h }
namespace AffineScheme
/-- The `Spec` functor into the category of affine schemes. -/
@[derive [full, faithful, ess_surj]]
def Spec : CommRingᵒᵖ ⥤ AffineScheme := Scheme.Spec.to_ess_image
/-- The forgetful functor `AffineScheme ⥤ Scheme`. -/
@[derive [full, faithful], simps]
def forget_to_Scheme : AffineScheme ⥤ Scheme := Scheme.Spec.ess_image_inclusion
/-- The global section functor of an affine scheme. -/
def Γ : AffineSchemeᵒᵖ ⥤ CommRing := forget_to_Scheme.op ⋙ Scheme.Γ
/-- The category of affine schemes is equivalent to the category of commutative rings. -/
def equiv_CommRing : AffineScheme ≌ CommRingᵒᵖ :=
equiv_ess_image_of_reflective.symm
instance Γ_is_equiv : is_equivalence Γ.{u} :=
begin
haveI : is_equivalence Γ.{u}.right_op.op := is_equivalence.of_equivalence equiv_CommRing.op,
exact (functor.is_equivalence_trans Γ.{u}.right_op.op (op_op_equivalence _).functor : _),
end
instance : has_colimits AffineScheme.{u} :=
begin
haveI := adjunction.has_limits_of_equivalence.{u} Γ.{u},
haveI : has_colimits AffineScheme.{u} ᵒᵖᵒᵖ := has_colimits_op_of_has_limits,
exactI adjunction.has_colimits_of_equivalence.{u} (op_op_equivalence AffineScheme.{u}).inverse
end
instance : has_limits AffineScheme.{u} :=
begin
haveI := adjunction.has_colimits_of_equivalence Γ.{u},
haveI : has_limits AffineScheme.{u} ᵒᵖᵒᵖ := limits.has_limits_op_of_has_colimits,
exactI adjunction.has_limits_of_equivalence (op_op_equivalence AffineScheme.{u}).inverse
end
end AffineScheme
/-- An open subset of a scheme is affine if the open subscheme is affine. -/
def is_affine_open {X : Scheme} (U : opens X.carrier) : Prop :=
is_affine (X.restrict U.open_embedding)
lemma range_is_affine_open_of_open_immersion {X Y : Scheme} [is_affine X] (f : X ⟶ Y)
[H : is_open_immersion f] : is_affine_open ⟨set.range f.1.base, H.base_open.open_range⟩ :=
begin
refine is_affine_of_iso (is_open_immersion.iso_of_range_eq f (Y.of_restrict _) _).inv,
exact subtype.range_coe.symm,
apply_instance
end
lemma top_is_affine_open (X : Scheme) [is_affine X] : is_affine_open (⊤ : opens X.carrier) :=
begin
convert range_is_affine_open_of_open_immersion (𝟙 X),
ext1,
exact set.range_id.symm
end
instance Scheme.affine_basis_cover_is_affine (X : Scheme) (i : X.affine_basis_cover.J) :
is_affine (X.affine_basis_cover.obj i) :=
algebraic_geometry.Spec_is_affine _
lemma is_basis_affine_open (X : Scheme) :
opens.is_basis { U : opens X.carrier | is_affine_open U } :=
begin
rw opens.is_basis_iff_nbhd,
rintros U x (hU : x ∈ (U : set X.carrier)),
obtain ⟨S, hS, hxS, hSU⟩ := X.affine_basis_cover_is_basis.exists_subset_of_mem_open hU U.prop,
refine ⟨⟨S, X.affine_basis_cover_is_basis.is_open hS⟩, _, hxS, hSU⟩,
rcases hS with ⟨i, rfl⟩,
exact range_is_affine_open_of_open_immersion _,
end
/-- The open immersion `Spec 𝒪ₓ(U) ⟶ X` for an affine `U`. -/
def is_affine_open.from_Spec {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
Scheme.Spec.obj (op $ X.presheaf.obj $ op U) ⟶ X :=
begin
haveI : is_affine (X.restrict U.open_embedding) := hU,
have : U.open_embedding.is_open_map.functor.obj ⊤ = U,
{ ext1, exact set.image_univ.trans subtype.range_coe },
exact Scheme.Spec.map (X.presheaf.map (eq_to_hom this.symm).op).op ≫
(X.restrict U.open_embedding).iso_Spec.inv ≫ X.of_restrict _
end
instance is_affine_open.is_open_immersion_from_Spec {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
is_open_immersion hU.from_Spec :=
by { delta is_affine_open.from_Spec, apply_instance }
lemma is_affine_open.from_Spec_range {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
set.range hU.from_Spec.1.base = (U : set X.carrier) :=
begin
delta is_affine_open.from_Spec,
erw [← category.assoc, Scheme.comp_val_base],
rw [coe_comp, set.range_comp, set.range_iff_surjective.mpr, set.image_univ],
exact subtype.range_coe,
rw ← Top.epi_iff_surjective,
apply_instance
end
lemma is_affine_open.from_Spec_image_top {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
hU.is_open_immersion_from_Spec.base_open.is_open_map.functor.obj ⊤ = U :=
by { ext1, exact set.image_univ.trans hU.from_Spec_range }
lemma is_affine_open.is_compact {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
is_compact (U : set X.carrier) :=
begin
convert @is_compact.image _ _ _ _ set.univ hU.from_Spec.1.base
prime_spectrum.compact_space.1 (by continuity),
convert hU.from_Spec_range.symm,
exact set.image_univ
end
instance Scheme.quasi_compact_of_affine (X : Scheme) [is_affine X] : compact_space X.carrier :=
⟨(top_is_affine_open X).is_compact⟩
lemma is_affine_open.from_Spec_base_preimage
{X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
(opens.map hU.from_Spec.val.base).obj U = ⊤ :=
begin
ext1,
change hU.from_Spec.1.base ⁻¹' (U : set X.carrier) = set.univ,
rw [← hU.from_Spec_range, ← set.image_univ],
exact set.preimage_image_eq _ PresheafedSpace.is_open_immersion.base_open.inj
end
lemma Scheme.Spec_map_presheaf_map_eq_to_hom {X : Scheme} {U V : opens X.carrier} (h : U = V) (W) :
(Scheme.Spec.map (X.presheaf.map (eq_to_hom h).op).op).val.c.app W =
eq_to_hom (by { cases h, dsimp, induction W using opposite.rec, congr, ext1, simpa }) :=
begin
have : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 _,
{ rw [X.presheaf.map_id, op_id, Scheme.Spec.map_id] },
cases h,
refine (Scheme.congr_app this _).trans _,
erw category.id_comp,
simpa [eq_to_hom_map],
end
lemma is_affine_open.Spec_Γ_identity_hom_app_from_Spec {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
(Spec_Γ_identity.hom.app (X.presheaf.obj $ op U)) ≫ hU.from_Spec.1.c.app (op U) =
(Scheme.Spec.obj _).presheaf.map (eq_to_hom hU.from_Spec_base_preimage).op :=
begin
haveI : is_affine _ := hU,
have e₁ :=
Spec_Γ_identity.hom.naturality (X.presheaf.map (eq_to_hom U.open_embedding_obj_top).op),
rw ← is_iso.comp_inv_eq at e₁,
have e₂ := Γ_Spec.adjunction_unit_app_app_top (X.restrict U.open_embedding),
erw ← e₂ at e₁,
simp only [functor.id_map, quiver.hom.unop_op, functor.comp_map, ← functor.map_inv, ← op_inv,
LocallyRingedSpace.Γ_map, category.assoc, functor.right_op_map, inv_eq_to_hom] at e₁,
delta is_affine_open.from_Spec Scheme.iso_Spec,
rw [Scheme.comp_val_c_app, Scheme.comp_val_c_app, ← e₁],
simp_rw category.assoc,
erw ← X.presheaf.map_comp_assoc,
rw ← op_comp,
have e₃ : U.open_embedding.is_open_map.adjunction.counit.app U ≫
eq_to_hom U.open_embedding_obj_top.symm =
U.open_embedding.is_open_map.functor.map (eq_to_hom U.inclusion_map_eq_top) :=
subsingleton.elim _ _,
have e₄ : X.presheaf.map _ ≫ _ = _ :=
(as_iso (Γ_Spec.adjunction.unit.app (X.restrict U.open_embedding)))
.inv.1.c.naturality_assoc (eq_to_hom U.inclusion_map_eq_top).op _,
erw [e₃, e₄, ← Scheme.comp_val_c_app_assoc, iso.inv_hom_id],
simp only [eq_to_hom_map, eq_to_hom_op, Scheme.Spec_map_presheaf_map_eq_to_hom],
erw [Scheme.Spec_map_presheaf_map_eq_to_hom, category.id_comp],
simpa only [eq_to_hom_trans]
end
@[elementwise]
lemma is_affine_open.from_Spec_app_eq {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
hU.from_Spec.1.c.app (op U) = Spec_Γ_identity.inv.app (X.presheaf.obj $ op U) ≫
(Scheme.Spec.obj _).presheaf.map (eq_to_hom hU.from_Spec_base_preimage).op :=
by rw [← hU.Spec_Γ_identity_hom_app_from_Spec, iso.inv_hom_id_app_assoc]
lemma is_affine_open.basic_open_is_affine {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) : is_affine_open (X.basic_open f) :=
begin
convert range_is_affine_open_of_open_immersion (Scheme.Spec.map (CommRing.of_hom
(algebra_map (X.presheaf.obj (op U)) (localization.away f))).op ≫ hU.from_Spec),
ext1,
have : hU.from_Spec.val.base '' (hU.from_Spec.val.base ⁻¹' (X.basic_open f : set X.carrier)) =
(X.basic_open f : set X.carrier),
{ rw [set.image_preimage_eq_inter_range, set.inter_eq_left_iff_subset, hU.from_Spec_range],
exact Scheme.basic_open_subset _ _ },
rw [subtype.coe_mk, Scheme.comp_val_base, ← this, coe_comp, set.range_comp],
congr' 1,
refine (congr_arg coe $ Scheme.preimage_basic_open hU.from_Spec f).trans _,
refine eq.trans _ (prime_spectrum.localization_away_comap_range (localization.away f) f).symm,
congr' 1,
have : (opens.map hU.from_Spec.val.base).obj U = ⊤,
{ ext1,
change hU.from_Spec.1.base ⁻¹' (U : set X.carrier) = set.univ,
rw [← hU.from_Spec_range, ← set.image_univ],
exact set.preimage_image_eq _ PresheafedSpace.is_open_immersion.base_open.inj },
refine eq.trans _ (basic_open_eq_of_affine f),
have lm : ∀ s, (opens.map hU.from_Spec.val.base).obj U ⊓ s = s := λ s, this.symm ▸ top_inf_eq,
refine eq.trans _ (lm _),
refine eq.trans _
((Scheme.Spec.obj $ op $ X.presheaf.obj $ op U).basic_open_res _ (eq_to_hom this).op),
rw ← comp_apply,
congr' 2,
rw iso.eq_inv_comp,
erw hU.Spec_Γ_identity_hom_app_from_Spec,
end
lemma Scheme.map_prime_spectrum_basic_open_of_affine (X : Scheme) [is_affine X]
(f : Scheme.Γ.obj (op X)) :
(opens.map X.iso_Spec.hom.1.base).obj (prime_spectrum.basic_open f) = X.basic_open f :=
begin
rw ← basic_open_eq_of_affine,
transitivity (opens.map X.iso_Spec.hom.1.base).obj ((Scheme.Spec.obj
(op (Scheme.Γ.obj (op X)))).basic_open ((inv (X.iso_Spec.hom.1.c.app
(op ((opens.map (inv X.iso_Spec.hom).val.base).obj ⊤)))) ((X.presheaf.map (eq_to_hom _)) f))),
congr,
{ rw [← is_iso.inv_eq_inv, is_iso.inv_inv, is_iso.iso.inv_inv, nat_iso.app_hom],
erw ← Γ_Spec.adjunction_unit_app_app_top,
refl },
{ rw eq_to_hom_map, refl },
{ dsimp, congr },
{ refine (Scheme.preimage_basic_open _ _).trans _,
rw [is_iso.inv_hom_id_apply, Scheme.basic_open_res_eq] }
end
lemma is_basis_basic_open (X : Scheme) [is_affine X] :
opens.is_basis (set.range (X.basic_open : X.presheaf.obj (op ⊤) → opens X.carrier)) :=
begin
delta opens.is_basis,
convert prime_spectrum.is_basis_basic_opens.inducing
(Top.homeo_of_iso (Scheme.forget_to_Top.map_iso X.iso_Spec)).inducing using 1,
ext,
simp only [set.mem_image, exists_exists_eq_and],
split,
{ rintro ⟨_, ⟨x, rfl⟩, rfl⟩,
refine ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, _⟩,
exact congr_arg subtype.val (X.map_prime_spectrum_basic_open_of_affine x) },
{ rintro ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, rfl⟩,
refine ⟨_, ⟨x, rfl⟩, _⟩,
exact congr_arg subtype.val (X.map_prime_spectrum_basic_open_of_affine x).symm }
end
lemma is_affine_open.exists_basic_open_subset {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) {V : opens X.carrier} (x : V) (h : ↑x ∈ U) :
∃ f : X.presheaf.obj (op U), X.basic_open f ⊆ V ∧ ↑x ∈ X.basic_open f :=
begin
haveI : is_affine _ := hU,
obtain ⟨_, ⟨_, ⟨r, rfl⟩, rfl⟩, h₁, h₂⟩ := (is_basis_basic_open (X.restrict U.open_embedding))
.exists_subset_of_mem_open _ ((opens.map U.inclusion).obj V).prop,
swap, exact ⟨x, h⟩,
have : U.open_embedding.is_open_map.functor.obj ((X.restrict U.open_embedding).basic_open r)
= X.basic_open (X.presheaf.map (eq_to_hom U.open_embedding_obj_top.symm).op r),
{ refine (is_open_immersion.image_basic_open (X.of_restrict U.open_embedding) r).trans _,
erw ← Scheme.basic_open_res_eq _ _ (eq_to_hom U.open_embedding_obj_top).op,
rw [← comp_apply, ← category_theory.functor.map_comp, ← op_comp, eq_to_hom_trans,
eq_to_hom_refl, op_id, category_theory.functor.map_id],
erw PresheafedSpace.is_open_immersion.of_restrict_inv_app,
congr },
use X.presheaf.map (eq_to_hom U.open_embedding_obj_top.symm).op r,
rw ← this,
exact ⟨set.image_subset_iff.mpr h₂, set.mem_image_of_mem _ h₁⟩,
exact x.prop,
end
instance {X : Scheme} {U : opens X.carrier} (f : X.presheaf.obj (op U)) :
algebra (X.presheaf.obj (op U)) (X.presheaf.obj (op $ X.basic_open f)) :=
(X.presheaf.map (hom_of_le $ RingedSpace.basic_open_subset _ f : _ ⟶ U).op).to_algebra
lemma is_affine_open.opens_map_from_Spec_basic_open {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) :
(opens.map hU.from_Spec.val.base).obj (X.basic_open f) =
RingedSpace.basic_open _ (Spec_Γ_identity.inv.app (X.presheaf.obj $ op U) f) :=
begin
erw LocallyRingedSpace.preimage_basic_open,
refine eq.trans _ (RingedSpace.basic_open_res_eq (Scheme.Spec.obj $ op $ X.presheaf.obj (op U))
.to_LocallyRingedSpace.to_RingedSpace (eq_to_hom hU.from_Spec_base_preimage).op _),
congr,
rw ← comp_apply,
congr,
erw ← hU.Spec_Γ_identity_hom_app_from_Spec,
rw iso.inv_hom_id_app_assoc,
end
/-- The canonical map `Γ(𝒪ₓ, D(f)) ⟶ Γ(Spec 𝒪ₓ(U), D(Spec_Γ_identity.inv f))`
This is an isomorphism, as witnessed by an `is_iso` instance. -/
def basic_open_sections_to_affine {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U)
(f : X.presheaf.obj (op U)) : X.presheaf.obj (op $ X.basic_open f) ⟶
(Scheme.Spec.obj $ op $ X.presheaf.obj (op U)).presheaf.obj
(op $ Scheme.basic_open _ $ Spec_Γ_identity.inv.app (X.presheaf.obj (op U)) f) :=
hU.from_Spec.1.c.app (op $ X.basic_open f) ≫ (Scheme.Spec.obj $ op $ X.presheaf.obj (op U))
.presheaf.map (eq_to_hom $ (hU.opens_map_from_Spec_basic_open f).symm).op
instance {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U)
(f : X.presheaf.obj (op U)) : is_iso (basic_open_sections_to_affine hU f) :=
begin
delta basic_open_sections_to_affine,
apply_with is_iso.comp_is_iso { instances := ff },
{ apply PresheafedSpace.is_open_immersion.is_iso_of_subset,
rw hU.from_Spec_range,
exact RingedSpace.basic_open_subset _ _ },
apply_instance
end
.
lemma is_localization_basic_open {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U)
(f : X.presheaf.obj (op U)) :
is_localization.away f (X.presheaf.obj (op $ X.basic_open f)) :=
begin
apply (is_localization.is_localization_iff_of_ring_equiv (submonoid.powers f)
(as_iso $ basic_open_sections_to_affine hU f ≫ (Scheme.Spec.obj _).presheaf.map
(eq_to_hom (basic_open_eq_of_affine _).symm).op).CommRing_iso_to_ring_equiv).mpr,
convert structure_sheaf.is_localization.to_basic_open _ f,
change _ ≫ (basic_open_sections_to_affine hU f ≫ _) = _,
delta basic_open_sections_to_affine,
erw ring_hom.algebra_map_to_algebra,
simp only [Scheme.comp_val_c_app, category.assoc],
erw hU.from_Spec.val.c.naturality_assoc,
rw hU.from_Spec_app_eq,
dsimp,
simp only [category.assoc, ← functor.map_comp, ← op_comp],
apply structure_sheaf.to_open_res,
end
lemma basic_open_basic_open_is_basic_open {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) (g : X.presheaf.obj (op $ X.basic_open f)) :
∃ f' : X.presheaf.obj (op U), X.basic_open f' = X.basic_open g :=
begin
haveI := is_localization_basic_open hU f,
obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := is_localization.surj' (submonoid.powers f) g,
use f * x,
rw [algebra.smul_def, Scheme.basic_open_mul, Scheme.basic_open_mul],
erw Scheme.basic_open_res,
refine (inf_eq_left.mpr _).symm,
convert inf_le_left using 1,
apply Scheme.basic_open_of_is_unit,
apply submonoid.left_inv_le_is_unit _ (is_localization.to_inv_submonoid (submonoid.powers f)
(X.presheaf.obj (op $ X.basic_open f)) _).prop
end
lemma exists_basic_open_subset_affine_inter {X : Scheme} {U V : opens X.carrier}
(hU : is_affine_open U) (hV : is_affine_open V) (x : X.carrier) (hx : x ∈ U ∩ V) :
∃ (f : X.presheaf.obj $ op U) (g : X.presheaf.obj $ op V),
X.basic_open f = X.basic_open g ∧ x ∈ X.basic_open f :=
begin
obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basic_open_subset ⟨x, hx.2⟩ hx.1,
obtain ⟨g, hg₁, hg₂⟩ := hV.exists_basic_open_subset ⟨x, hf₂⟩ hx.2,
obtain ⟨f', hf'⟩ := basic_open_basic_open_is_basic_open hU f
(X.presheaf.map (hom_of_le hf₁ : _ ⟶ V).op g),
replace hf' := (hf'.trans (RingedSpace.basic_open_res _ _ _)).trans (inf_eq_right.mpr hg₁),
exact ⟨f', g, hf', hf'.symm ▸ hg₂⟩
end
/-- The prime ideal of `𝒪ₓ(U)` corresponding to a point `x : U`. -/
noncomputable
def is_affine_open.prime_ideal_of {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (x : U) :
prime_spectrum (X.presheaf.obj $ op U) :=
((Scheme.Spec.map (X.presheaf.map (eq_to_hom $
show U.open_embedding.is_open_map.functor.obj ⊤ = U, from
opens.ext (set.image_univ.trans subtype.range_coe)).op).op).1.base
((@@Scheme.iso_Spec (X.restrict U.open_embedding) hU).hom.1.base x))
lemma is_affine_open.from_Spec_prime_ideal_of {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (x : U) :
hU.from_Spec.val.base (hU.prime_ideal_of x) = x.1 :=
begin
dsimp only [is_affine_open.from_Spec, subtype.coe_mk],
erw [← Scheme.comp_val_base_apply, ← Scheme.comp_val_base_apply],
simpa only [← functor.map_comp_assoc, ← functor.map_comp, ← op_comp, eq_to_hom_trans, op_id,
eq_to_hom_refl, category_theory.functor.map_id, category.id_comp, iso.hom_inv_id_assoc]
end
lemma is_affine_open.is_localization_stalk_aux {X : Scheme} (U : opens X.carrier)
[is_affine (X.restrict U.open_embedding)] :
(inv (Γ_Spec.adjunction.unit.app (X.restrict U.open_embedding))).1.c.app
(op ((opens.map U.inclusion).obj U)) =
X.presheaf.map (eq_to_hom $ by rw opens.inclusion_map_eq_top :
U.open_embedding.is_open_map.functor.obj ⊤ ⟶
(U.open_embedding.is_open_map.functor.obj ((opens.map U.inclusion).obj U))).op ≫
to_Spec_Γ (X.presheaf.obj $ op (U.open_embedding.is_open_map.functor.obj ⊤)) ≫
(Scheme.Spec.obj $ op $ X.presheaf.obj $ _).presheaf.map
(eq_to_hom (by { rw [opens.inclusion_map_eq_top], refl }) : unop _ ⟶ ⊤).op :=
begin
have e : (opens.map (inv (Γ_Spec.adjunction.unit.app (X.restrict U.open_embedding))).1.base).obj
((opens.map U.inclusion).obj U) = ⊤,
by { rw [opens.inclusion_map_eq_top], refl },
rw [Scheme.inv_val_c_app, is_iso.comp_inv_eq, Scheme.app_eq _ e,
Γ_Spec.adjunction_unit_app_app_top],
simp only [category.assoc, eq_to_hom_op],
erw ← functor.map_comp_assoc,
rw [eq_to_hom_trans, eq_to_hom_refl, category_theory.functor.map_id,
category.id_comp],
erw Spec_Γ_identity.inv_hom_id_app_assoc,
simp only [eq_to_hom_map, eq_to_hom_trans],
end
lemma is_affine_open.is_localization_stalk {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (x : U) :
is_localization.at_prime (X.presheaf.stalk x) (hU.prime_ideal_of x).as_ideal :=
begin
haveI : is_affine _ := hU,
haveI : nonempty U := ⟨x⟩,
rcases x with ⟨x, hx⟩,
let y := hU.prime_ideal_of ⟨x, hx⟩,
have : hU.from_Spec.val.base y = x := hU.from_Spec_prime_ideal_of ⟨x, hx⟩,
change is_localization y.as_ideal.prime_compl _,
clear_value y,
subst this,
apply (is_localization.is_localization_iff_of_ring_equiv _
(as_iso $ PresheafedSpace.stalk_map hU.from_Spec.1 y).CommRing_iso_to_ring_equiv).mpr,
convert structure_sheaf.is_localization.to_stalk _ _ using 1,
delta structure_sheaf.stalk_algebra,
congr' 1,
rw ring_hom.algebra_map_to_algebra,
refine (PresheafedSpace.stalk_map_germ hU.from_Spec.1 _ ⟨_, _⟩).trans _,
delta is_affine_open.from_Spec Scheme.iso_Spec structure_sheaf.to_stalk,
simp only [Scheme.comp_val_c_app, category.assoc],
dsimp only [functor.op, as_iso_inv, unop_op],
erw is_affine_open.is_localization_stalk_aux,
simp only [category.assoc],
conv_lhs { rw ← category.assoc },
erw [← X.presheaf.map_comp, Spec_Γ_naturality_assoc],
congr' 1,
simp only [← category.assoc],
transitivity _ ≫ (structure_sheaf (X.presheaf.obj $ op U)).1.germ ⟨_, _⟩,
{ refl },
convert ((structure_sheaf (X.presheaf.obj $ op U)).1.germ_res (hom_of_le le_top) ⟨_, _⟩) using 2,
rw category.assoc,
erw nat_trans.naturality,
rw [← LocallyRingedSpace.Γ_map_op, ← LocallyRingedSpace.Γ.map_comp_assoc, ← op_comp],
erw ← Scheme.Spec.map_comp,
rw [← op_comp, ← X.presheaf.map_comp],
transitivity LocallyRingedSpace.Γ.map (quiver.hom.op $ Scheme.Spec.map
(X.presheaf.map (𝟙 (op U))).op) ≫ _,
{ congr },
simp only [category_theory.functor.map_id, op_id],
erw category_theory.functor.map_id,
rw category.id_comp,
refl
end
end algebraic_geometry
|
f99ff67ba21bd7674cd70eaae48a1b3a469c6081 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/algebra/big_operators/nat_antidiagonal.lean | a4ab8866ce7afe13c6b1420bd8da218f1c204a1b | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 2,739 | lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import data.finset.nat_antidiagonal
import algebra.big_operators.basic
/-!
# Big operators for `nat_antidiagonal`
This file contains theorems relevant to big operators over `finset.nat.antidiagonal`.
-/
open_locale big_operators
variables {M N : Type*} [comm_monoid M] [add_comm_monoid N]
namespace finset
namespace nat
lemma prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :
∏ p in antidiagonal (n + 1), f p = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) :=
begin
rw [antidiagonal_succ, prod_insert, prod_map], refl,
intro con, rcases mem_map.1 con with ⟨⟨a,b⟩, ⟨h1, h2⟩⟩,
simp only [prod.mk.inj_iff, function.embedding.coe_prod_map, prod.map_mk] at h2,
apply nat.succ_ne_zero a h2.1,
end
lemma sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :
∑ p in antidiagonal (n + 1), f p = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=
@prod_antidiagonal_succ (multiplicative N) _ _ _
@[to_additive]
lemma prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :
∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p :=
by { nth_rewrite 1 ← map_swap_antidiagonal, rw [prod_map], refl }
lemma prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} :
∏ p in antidiagonal (n + 1), f p = f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) :=
begin
rw [← prod_antidiagonal_swap, prod_antidiagonal_succ, ← prod_antidiagonal_swap],
refl
end
lemma sum_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → N} :
∑ p in antidiagonal (n + 1), f p = f (n + 1, 0) + ∑ p in antidiagonal n, f (p.1, p.2 + 1) :=
@prod_antidiagonal_succ' (multiplicative N) _ _ _
@[to_additive]
lemma prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :
∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) :=
prod_congr rfl $ λ p hp, by rw [nat.mem_antidiagonal.1 hp]
@[to_additive]
lemma prod_antidiagonal_eq_prod_range_succ_mk {M : Type*} [comm_monoid M] (f : ℕ × ℕ → M) (n : ℕ) :
∏ ij in finset.nat.antidiagonal n, f ij = ∏ k in range n.succ, f (k, n - k) :=
begin
convert prod_map _ ⟨λ i, (i, n - i), λ x y h, (prod.mk.inj h).1⟩ _,
refl,
end
/-- This lemma matches more generally than `finset.nat.prod_antidiagonal_eq_prod_range_succ_mk` when
using `rw ←`. -/
@[to_additive]
lemma prod_antidiagonal_eq_prod_range_succ {M : Type*} [comm_monoid M] (f : ℕ → ℕ → M) (n : ℕ) :
∏ ij in finset.nat.antidiagonal n, f ij.1 ij.2 = ∏ k in range n.succ, f k (n - k) :=
prod_antidiagonal_eq_prod_range_succ_mk _ _
end nat
end finset
|
fe0a002184d262629cb3d74341b73a0f53044485 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/algebra/linear_recurrence.lean | cd96d5ffd884daa9072906b5f12e4fdd9fdfd2de | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 7,816 | lean | /-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import data.polynomial.ring_division
import linear_algebra.dimension
import algebra.polynomial.big_operators
/-!
# Linear recurrence
Informally, a "linear recurrence" is an assertion of the form
`∀ n : ℕ, u (n + d) = a 0 * u n + a 1 * u (n+1) + ... + a (d-1) * u (n+d-1)`,
where `u` is a sequence, `d` is the *order* of the recurrence and the `a i`
are its *coefficients*.
In this file, we define the structure `linear_recurrence` so that
`linear_recurrence.mk d a` represents the above relation, and we call
a sequence `u` which verifies it a *solution* of the linear recurrence.
We prove a few basic lemmas about this concept, such as :
* the space of solutions is a submodule of `(ℕ → α)` (i.e a vector space if `α`
is a field)
* the function that maps a solution `u` to its first `d` terms builds a `linear_equiv`
between the solution space and `fin d → α`, aka `α ^ d`. As a consequence, two
solutions are equal if and only if their first `d` terms are equals.
* a geometric sequence `q ^ n` is solution iff `q` is a root of a particular polynomial,
which we call the *characteristic polynomial* of the recurrence
Of course, although we can inductively generate solutions (cf `mk_sol`), the
interesting part would be to determinate closed-forms for the solutions.
This is currently *not implemented*, as we are waiting for definition and
properties of eigenvalues and eigenvectors.
-/
noncomputable theory
open finset
open_locale big_operators
/-- A "linear recurrence relation" over a commutative semiring is given by its
order `n` and `n` coefficients. -/
structure linear_recurrence (α : Type*) [comm_semiring α] := (order : ℕ) (coeffs : fin order → α)
instance (α : Type*) [comm_semiring α] : inhabited (linear_recurrence α) :=
⟨⟨0, default _⟩⟩
namespace linear_recurrence
section comm_semiring
variables {α : Type*} [comm_semiring α] (E : linear_recurrence α)
/-- We say that a sequence `u` is solution of `linear_recurrence order coeffs` when we have
`u (n + order) = ∑ i : fin order, coeffs i * u (n + i)` for any `n`. -/
def is_solution (u : ℕ → α) :=
∀ n, u (n + E.order) = ∑ i, E.coeffs i * u (n + i)
/-- A solution of a `linear_recurrence` which satisfies certain initial conditions.
We will prove this is the only such solution. -/
def mk_sol (init : fin E.order → α) : ℕ → α
| n := if h : n < E.order then init ⟨n, h⟩ else
∑ k : fin E.order, have n - E.order + k < n, by have := k.is_lt; omega,
E.coeffs k * mk_sol (n - E.order + k)
/-- `E.mk_sol` indeed gives solutions to `E`. -/
lemma is_sol_mk_sol (init : fin E.order → α) : E.is_solution (E.mk_sol init) :=
λ n, by rw mk_sol; simp
/-- `E.mk_sol init`'s first `E.order` terms are `init`. -/
lemma mk_sol_eq_init (init : fin E.order → α) : ∀ n : fin E.order, E.mk_sol init n = init n :=
λ n, by { rw mk_sol, simp only [n.is_lt, dif_pos, fin.mk_coe] }
/-- If `u` is a solution to `E` and `init` designates its first `E.order` values,
then `∀ n, u n = E.mk_sol init n`. -/
lemma eq_mk_of_is_sol_of_eq_init {u : ℕ → α} {init : fin E.order → α}
(h : E.is_solution u) (heq : ∀ n : fin E.order, u n = init n) :
∀ n, u n = E.mk_sol init n
| n := if h' : n < E.order
then by rw mk_sol; simp only [h', dif_pos]; exact_mod_cast heq ⟨n, h'⟩
else begin
rw [mk_sol, ← nat.sub_add_cancel (le_of_not_lt h'), h (n-E.order)],
simp [h'],
congr' with k,
exact have wf : n - E.order + k < n, by have := k.is_lt; omega,
by rw eq_mk_of_is_sol_of_eq_init
end
/-- If `u` is a solution to `E` and `init` designates its first `E.order` values,
then `u = E.mk_sol init`. This proves that `E.mk_sol init` is the only solution
of `E` whose first `E.order` values are given by `init`. -/
lemma eq_mk_of_is_sol_of_eq_init' {u : ℕ → α} {init : fin E.order → α}
(h : E.is_solution u) (heq : ∀ n : fin E.order, u n = init n) : u = E.mk_sol init :=
funext (E.eq_mk_of_is_sol_of_eq_init h heq)
/-- The space of solutions of `E`, as a `submodule` over `α` of the semimodule `ℕ → α`. -/
def sol_space : submodule α (ℕ → α) :=
{ carrier := {u | E.is_solution u},
zero_mem' := λ n, by simp,
add_mem' := λ u v hu hv n, by simp [mul_add, sum_add_distrib, hu n, hv n],
smul_mem' := λ a u hu n, by simp [hu n, mul_sum]; congr'; ext; ac_refl }
/-- Defining property of the solution space : `u` is a solution
iff it belongs to the solution space. -/
lemma is_sol_iff_mem_sol_space (u : ℕ → α) : E.is_solution u ↔ u ∈ E.sol_space :=
iff.rfl
/-- The function that maps a solution `u` of `E` to its first
`E.order` terms as a `linear_equiv`. -/
def to_init :
E.sol_space ≃ₗ[α] (fin E.order → α) :=
{ to_fun := λ u x, (u : ℕ → α) x,
map_add' := λ u v, by { ext, simp },
map_smul' := λ a u, by { ext, simp },
inv_fun := λ u, ⟨E.mk_sol u, E.is_sol_mk_sol u⟩,
left_inv := λ u, by ext n; symmetry; apply E.eq_mk_of_is_sol_of_eq_init u.2; intros k; refl,
right_inv := λ u, function.funext_iff.mpr (λ n, E.mk_sol_eq_init u n) }
/-- Two solutions are equal iff they are equal on `range E.order`. -/
lemma sol_eq_of_eq_init (u v : ℕ → α) (hu : E.is_solution u) (hv : E.is_solution v) :
u = v ↔ set.eq_on u v ↑(range E.order) :=
begin
refine iff.intro (λ h x hx, h ▸ rfl) _,
intro h,
set u' : ↥(E.sol_space) := ⟨u, hu⟩,
set v' : ↥(E.sol_space) := ⟨v, hv⟩,
change u'.val = v'.val,
suffices h' : u' = v', from h' ▸ rfl,
rw [← E.to_init.to_equiv.apply_eq_iff_eq, linear_equiv.coe_to_equiv],
ext x,
exact_mod_cast h (mem_range.mpr x.2)
end
/-! `E.tuple_succ` maps `![s₀, s₁, ..., sₙ]` to `![s₁, ..., sₙ, ∑ (E.coeffs i) * sᵢ]`,
where `n := E.order`. This operation is quite useful for determining closed-form
solutions of `E`. -/
/-- `E.tuple_succ` maps `![s₀, s₁, ..., sₙ]` to `![s₁, ..., sₙ, ∑ (E.coeffs i) * sᵢ]`,
where `n := E.order`. -/
def tuple_succ : (fin E.order → α) →ₗ[α] (fin E.order → α) :=
{ to_fun := λ X i, if h : (i : ℕ) + 1 < E.order then X ⟨i+1, h⟩ else (∑ i, E.coeffs i * X i),
map_add' := λ x y,
begin
ext i,
split_ifs ; simp [h, mul_add, sum_add_distrib],
end,
map_smul' := λ x y,
begin
ext i,
split_ifs ; simp [h, mul_sum],
exact sum_congr rfl (λ x _, by ac_refl),
end }
end comm_semiring
section field
variables {α : Type*} [field α] (E : linear_recurrence α)
/-- The dimension of `E.sol_space` is `E.order`. -/
lemma sol_space_dim : vector_space.dim α E.sol_space = E.order :=
@dim_fin_fun α _ E.order ▸ E.to_init.dim_eq
end field
section comm_ring
variables {α : Type*} [comm_ring α] (E : linear_recurrence α)
/-- The characteristic polynomial of `E` is `X ^ E.order - ∑ i : fin E.order, (E.coeffs i) * X ^ i`. -/
def char_poly : polynomial α :=
polynomial.monomial E.order 1 - (∑ i : fin E.order, polynomial.monomial i (E.coeffs i))
/-- The geometric sequence `q^n` is a solution of `E` iff
`q` is a root of `E`'s characteristic polynomial. -/
lemma geom_sol_iff_root_char_poly (q : α) : E.is_solution (λ n, q^n) ↔ E.char_poly.is_root q :=
begin
rw [char_poly, polynomial.is_root.def, polynomial.eval],
simp only [polynomial.eval₂_finset_sum, one_mul,
ring_hom.id_apply, polynomial.eval₂_monomial, polynomial.eval₂_sub],
split,
{ intro h,
simpa [sub_eq_zero_iff_eq] using h 0 },
{ intros h n,
simp only [pow_add, sub_eq_zero_iff_eq.mp h, mul_sum],
exact sum_congr rfl (λ _ _, by ring) }
end
end comm_ring
end linear_recurrence
|
707cbecdf3adecd61c7d02a239e48173a636e2d4 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/category_theory/bicategory/functor_bicategory.lean | 0d1cd9f0e4618ed987435b97fc07f79b17c02f8b | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 2,769 | lean | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import category_theory.bicategory.natural_transformation
/-!
# The bicategory of oplax functors between two bicategories
Given bicategories `B` and `C`, we give a bicategory structure on `oplax_functor B C` whose
* objects are oplax functors,
* 1-morphisms are oplax natural transformations, and
* 2-morphisms are modifications.
-/
namespace category_theory
open category bicategory
open_locale bicategory
universes w₁ w₂ v₁ v₂ u₁ u₂
variables {B : Type u₁} [bicategory.{w₁ v₁} B] {C : Type u₂} [bicategory.{w₂ v₂} C]
variables {F G H I : oplax_functor B C}
namespace oplax_nat_trans
/-- Left whiskering of an oplax natural transformation and a modification. -/
@[simps]
def whisker_left (η : F ⟶ G) {θ ι : G ⟶ H} (Γ : θ ⟶ ι) : η ≫ θ ⟶ η ≫ ι :=
{ app := λ a, η.app a ◁ Γ.app a,
naturality' := λ a b f, by
{ dsimp, rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc], simp } }
/-- Right whiskering of an oplax natural transformation and a modification. -/
@[simps]
def whisker_right {η θ : F ⟶ G} (Γ : η ⟶ θ) (ι : G ⟶ H) : η ≫ ι ⟶ θ ≫ ι :=
{ app := λ a, Γ.app a ▷ ι.app a,
naturality' := λ a b f, by
{ dsimp, simp_rw [assoc, ←associator_inv_naturality_left, whisker_exchange_assoc], simp } }
/-- Associator for the vertical composition of oplax natural transformations. -/
@[simps]
def associator (η : F ⟶ G) (θ : G ⟶ H) (ι : H ⟶ I) : (η ≫ θ) ≫ ι ≅ η ≫ (θ ≫ ι) :=
modification_iso.of_components (λ a, α_ (η.app a) (θ.app a) (ι.app a)) (by tidy)
/-- Left unitor for the vertical composition of oplax natural transformations. -/
@[simps]
def left_unitor (η : F ⟶ G) : 𝟙 F ≫ η ≅ η :=
modification_iso.of_components (λ a, λ_ (η.app a)) (by tidy)
/-- Right unitor for the vertical composition of oplax natural transformations. -/
@[simps]
def right_unitor (η : F ⟶ G) : η ≫ 𝟙 G ≅ η :=
modification_iso.of_components (λ a, ρ_ (η.app a)) (by tidy)
end oplax_nat_trans
variables (B C)
/-- A bicategory structure on the oplax functors between bicategories. -/
@[simps]
instance oplax_functor.bicategory : bicategory (oplax_functor B C) :=
{ whisker_left := λ F G H η _ _ Γ, oplax_nat_trans.whisker_left η Γ,
whisker_right := λ F G H _ _ Γ η, oplax_nat_trans.whisker_right Γ η,
associator := λ F G H I, oplax_nat_trans.associator,
left_unitor := λ F G, oplax_nat_trans.left_unitor,
right_unitor := λ F G, oplax_nat_trans.right_unitor,
whisker_exchange' := by { intros, ext, apply whisker_exchange } }
end category_theory
|
52fdef2ce7066e7188b7be84cda8e2c92a4c5ac6 | 69d4931b605e11ca61881fc4f66db50a0a875e39 | /src/algebra/hierarchy_design.lean | 151bd8f90095e173c5399e0a1305d496307a5b42 | [
"Apache-2.0"
] | permissive | abentkamp/mathlib | d9a75d291ec09f4637b0f30cc3880ffb07549ee5 | 5360e476391508e092b5a1e5210bd0ed22dc0755 | refs/heads/master | 1,682,382,954,948 | 1,622,106,077,000 | 1,622,106,077,000 | 149,285,665 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 7,844 | lean | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Eric Weiser
-/
import tactic.doc_commands
/-!
# Documentation of the algebraic hierarchy
A library note giving advice on modifying the algebraic hierarchy.
(It is not intended as a "tour".)
TODO: Add sections about interactions with topological typeclasses, and order typeclasses.
-/
/--
# The algebraic hierarchy
In any theorem proving environment,
there are difficult decisions surrounding the design of the "algebraic hierarchy".
There is a danger of exponential explosion in the number of gadgets,
especially once interactions between algebraic and order/topological/etc structures are considered.
In mathlib, we try to avoid this by only introducing new algebraic typeclasses either
1. when there is "real mathematics" to be done with them, or
2. when there is a meaninful gain in simplicity by factoring out a common substructure.
(As examples, at this point we don't have `loop`, or `unital_magma`,
but we do have `lie_submodule` and `topological_field`!
We also have `group_with_zero`, as an exemplar of point 2.)
Generally in mathlib we use the extension mechanism (so `comm_ring` extends `ring`)
rather than mixins (e.g. with separate `ring` and `comm_mul` classes),
in part because of the potential blow-up in term sizes described at
https://www.ralfj.de/blog/2019/05/15/typeclasses-exponential-blowup.html
However there is tension here, as it results in considerable duplication in the API,
particularly in the interaction with order structures.
This library note is not intended as a design document
justifying and explaining the history of mathlib's algebraic hierarchy!
Instead it is intended as a developer's guide, for contributors wanting to extend
(either new leaves, or new intermediate classes) the algebraic hierarchy as it exists.
(Ideally we would have both a tour guide to the existing hierarchy,
and an account of the design choices.
See https://arxiv.org/abs/1910.09336 for an overview of mathlib as a whole,
with some attention to the algebraic hierarchy and
https://leanprover-community.github.io/mathlib-overview.html
for a summary of what is in mathlib today.)
## Instances
When adding a new typeclass `Z` to the algebraic hierarchy
one should attempt to add the following constructions and results,
when applicable:
* Instances transferred elementwise to products, like `prod.monoid`.
See `algebra.group.prod` for more examples.
```
instance prod.Z [Z M] [Z N] : Z (M × N) := ...
```
* Instances transferred elementwise to pi types, like `pi.monoid`.
See `algebra.group.pi` for more examples.
```
instance pi.Z [∀ i, Z $ f i] : Z (Π i : I, f i) := ...
```
* Instances transferred to `opposite M`, like `opposite.monoid`.
See `algebra.opposites` for more examples.
```
instance opposite.Z [Z M] : Z (opposite M) := ...
```
* Instances transferred to `ulift M`, like `ulift.monoid`.
See `algebra.group.ulift` for more examples.
```
instance ulift.Z [Z M] : Z (ulift M) := ...
```
* Definitions for transferring the proof fields of instances along
injective or surjective functions that agree on the data fields,
like `function.injective.monoid` and `function.surjective.monoid`.
See ``algebra.group.inj_surj` for more examples.
```
def function.injective.Z [Z M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) : Z M₁ := ...
def function.surjective.Z [Z M₁] (f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) : Z M₂ := ...
```
* Instances transferred elementwise to `finsupp`s, like `finsupp.semigroup`.
See `data.finsupp.pointwise` for more examples.
```
instance finsupp.Z [Z β] : Z (α →₀ β) := ...
```
* Instances transferred elementwise to `set`s, like `set.monoid`.
See `algebra.pointwise` for more examples.
```
instance set.Z [Z α] : Z (set α) := ...
```
* Definitions for transferring the entire structure across an equivalence, like `equiv.monoid`.
See `data.equiv.transfer_instance` for more examples. See also the `transport` tactic.
```
def equiv.Z (e : α ≃ β) [Z β] : Z α := ...
/- When there is a new notion of `Z`-equiv: -/
def equiv.Z_equiv (e : α ≃ β) [Z β] : by { letI := equiv.Z e, exact α ≃Z β } := ...
```
## Subobjects
When a new typeclass `Z` adds new data fields,
you should also create a new `sub_Z` `structure` with a `carrier` field.
This can be a lot of work; for now try to closely follow the existing examples
(e.g. `submonoid`, `subring`, `subalgebra`).
We would very much like to provide some automation here, but a prerequisite will be making
all the existing APIs more uniform.
If `Z` extends `Y`, then `sub_Z` should usually extend `sub_Y`.
When `Z` adds only new proof fields to an existing structure `Y`,
you should provide instances transferring
`Z α` to `Z (sub_Y α)`, like `submonoid.to_comm_monoid`.
Typically this is done using the `function.injective.Z` definition mentioned above.
```
instance sub_Y.to_Z [Z α] : Z (sub_Y α) :=
coe_injective.Z coe ...
```
## Morphisms and equivalences
## Category theory
For many algebraic structures, particularly ones used in representation theory, algebraic geometry,
etc., we also define "bundled" versions, which carry `category` instances.
These bundled versions are usually named in camel case,
so for example we have `AddCommGroup` as a bundled `add_comm_group`,
and `TopCommRing` (which bundles together `comm_ring`, `topological_space`, and `topological_ring`).
These bundled versions have many appealing features:
* a uniform notation for morphisms `X ⟶ Y`
* a uniform notation (and definition) for isomorphisms `X ≅ Y`
* a uniform API for subobjects, via the partial order `subobject X`
* interoperability with unbundled structures, via coercions to `Type`
(so if `G : AddCommGroup`, you can treat `G` as a type,
and it automatically has an `add_comm_group` instance)
and lifting maps `AddCommGroup.of G`, when `G` is a type with an `add_comm_group` instance.
If, for example you do the work of proving that a typeclass `Z` has a good notion of tensor product,
you are strongly encouraged to provide the corresponding `monoidal_category` instance
on a bundled version.
This ensures that the API for tensor products is complete, and enables use of general machinery.
Similarly if you prove universal properties, or adjunctions, you are encouraged to state these
using categorical language!
One disadvantage of the bundled approach is that we can only speak of morphisms between
objects living in the same type-theoretic universe.
In practice this is rarely a problem.
# Making a pull request
With so many moving parts, how do you actually go about changing the algebraic hierarchy?
We're still evolving how to handle this, but the current suggestion is:
* If you're adding a new "leaf" class, the requirements are lower,
and an initial PR can just add whatever is immediately needed.
* A new "intermediate" class, especially low down in the hierarchy,
needs to be careful about leaving gaps.
In a perfect world, there would be a group of simultaneous PRs that basically cover everything!
(Or at least an expectation that PRs may not be merged immediately while waiting on other
PRs that fill out the API.)
However "perfect is the enemy of good", and it would also be completely reasonable
to add a TODO list in the main module doc-string for the new class,
briefly listing the parts of the API which still need to be provided.
Hopefully this document makes it easy to assemble this list.
Another alternative to a TODO list in the doc-strings is adding github issues.
-/
library_note "the algebraic hierarchy"
|
caf4bb2ef5c024de1ecf8e2bb7af3cb8b4047171 | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/run/occurs_check_bug1.lean | d686ebc6f8da9b8131f92cbde996cb87523b6087 | [
"Apache-2.0"
] | permissive | soonhokong/lean-osx | 4a954262c780e404c1369d6c06516161d07fcb40 | 3670278342d2f4faa49d95b46d86642d7875b47c | refs/heads/master | 1,611,410,334,552 | 1,474,425,686,000 | 1,474,425,686,000 | 12,043,103 | 5 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 511 | lean | open nat prod
open decidable
constant modulo1 (x : ℕ) (y : ℕ) : ℕ
infixl `mod`:70 := modulo1
constant gcd_aux : ℕ × ℕ → ℕ
noncomputable definition gcd (x y : ℕ) : ℕ := gcd_aux (x, y)
theorem gcd_def (x y : ℕ) : gcd x y = @ite (y = 0) (nat.has_decidable_eq (pr2 (x, y)) 0) nat x (gcd y (x mod y)) :=
sorry
constant succ_ne_zero (a : nat) : succ a ≠ 0
theorem gcd_succ (m n : ℕ) : gcd m (succ n) = gcd (succ n) (m mod succ n) :=
eq.trans (gcd_def _ _) (if_neg (nat.succ_ne_zero _))
|
45426e2774e2f1716231c2eff8fd5dbda3120b08 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/order/filter/basic_auto.lean | b82fac908f2333af2d4b5ed0eaee6265efc26d39 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 105,471 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.order.zorn
import Mathlib.order.copy
import Mathlib.data.set.finite
import Mathlib.tactic.monotonicity.default
import Mathlib.PostPort
universes u_1 l u v x w u_2 u_3 u_4
namespace Mathlib
/-!
# Theory of filters on sets
## Main definitions
* `filter` : filters on a set;
* `at_top`, `at_bot`, `cofinite`, `principal` : specific filters;
* `map`, `comap`, `prod` : operations on filters;
* `tendsto` : limit with respect to filters;
* `eventually` : `f.eventually p` means `{x | p x} ∈ f`;
* `frequently` : `f.frequently p` means `{x | ¬p x} ∉ f`;
* `filter_upwards [h₁, ..., hₙ]` : takes a list of proofs `hᵢ : sᵢ ∈ f`, and replaces a goal `s ∈ f`
with `∀ x, x ∈ s₁ → ... → x ∈ sₙ → x ∈ s`;
* `ne_bot f` : an utility class stating that `f` is a non-trivial filter.
Filters on a type `X` are sets of sets of `X` satisfying three conditions. They are mostly used to
abstract two related kinds of ideas:
* *limits*, including finite or infinite limits of sequences, finite or infinite limits of functions
at a point or at infinity, etc...
* *things happening eventually*, including things happening for large enough `n : ℕ`, or near enough
a point `x`, or for close enough pairs of points, or things happening almost everywhere in the
sense of measure theory. Dually, filters can also express the idea of *things happening often*:
for arbitrarily large `n`, or at a point in any neighborhood of given a point etc...
In this file, we define the type `filter X` of filters on `X`, and endow it with a complete lattice
structure. This structure is lifted from the lattice structure on `set (set X)` using the Galois
insertion which maps a filter to its elements in one direction, and an arbitrary set of sets to
the smallest filter containing it in the other direction.
We also prove `filter` is a monadic functor, with a push-forward operation
`filter.map` and a pull-back operation `filter.comap` that form a Galois connections for the
order on filters.
Finally we describe a product operation `filter X → filter Y → filter (X × Y)`.
The examples of filters appearing in the description of the two motivating ideas are:
* `(at_top : filter ℕ)` : made of sets of `ℕ` containing `{n | n ≥ N}` for some `N`
* `𝓝 x` : made of neighborhoods of `x` in a topological space (defined in topology.basic)
* `𝓤 X` : made of entourages of a uniform space (those space are generalizations of metric spaces
defined in topology.uniform_space.basic)
* `μ.ae` : made of sets whose complement has zero measure with respect to `μ` (defined in
`measure_theory.measure_space`)
The general notion of limit of a map with respect to filters on the source and target types
is `filter.tendsto`. It is defined in terms of the order and the push-forward operation.
The predicate "happening eventually" is `filter.eventually`, and "happening often" is
`filter.frequently`, whose definitions are immediate after `filter` is defined (but they come
rather late in this file in order to immediately relate them to the lattice structure).
For instance, anticipating on topology.basic, the statement: "if a sequence `u` converges to
some `x` and `u n` belongs to a set `M` for `n` large enough then `x` is in the closure of
`M`" is formalized as: `tendsto u at_top (𝓝 x) → (∀ᶠ n in at_top, u n ∈ M) → x ∈ closure M`,
which is a special case of `mem_closure_of_tendsto` from topology.basic.
## Notations
* `∀ᶠ x in f, p x` : `f.eventually p`;
* `∃ᶠ x in f, p x` : `f.frequently p`;
* `f =ᶠ[l] g` : `∀ᶠ x in l, f x = g x`;
* `f ≤ᶠ[l] g` : `∀ᶠ x in l, f x ≤ g x`;
* `f ×ᶠ g` : `filter.prod f g`, localized in `filter`;
* `𝓟 s` : `principal s`, localized in `filter`.
## References
* [N. Bourbaki, *General Topology*][bourbaki1966]
Important note: Bourbaki requires that a filter on `X` cannot contain all sets of `X`, which
we do *not* require. This gives `filter X` better formal properties, in particular a bottom element
`⊥` for its lattice structure, at the cost of including the assumption
`[ne_bot f]` in a number of lemmas and definitions.
-/
/-- A filter `F` on a type `α` is a collection of sets of `α` which contains the whole `α`,
is upwards-closed, and is stable under intersection. We do not forbid this collection to be
all sets of `α`. -/
structure filter (α : Type u_1) where
sets : set (set α)
univ_sets : set.univ ∈ sets
sets_of_superset : ∀ {x y : set α}, x ∈ sets → x ⊆ y → y ∈ sets
inter_sets : ∀ {x y : set α}, x ∈ sets → y ∈ sets → x ∩ y ∈ sets
/-- If `F` is a filter on `α`, and `U` a subset of `α` then we can write `U ∈ F` as on paper. -/
protected instance filter.has_mem {α : Type u_1} : has_mem (set α) (filter α) :=
has_mem.mk fun (U : set α) (F : filter α) => U ∈ filter.sets F
namespace filter
@[simp] protected theorem mem_mk {α : Type u} {s : set α} {t : set (set α)} {h₁ : set.univ ∈ t}
{h₂ : ∀ {x y : set α}, x ∈ t → x ⊆ y → y ∈ t}
{h₃ : ∀ {x y : set α}, x ∈ t → y ∈ t → x ∩ y ∈ t} : s ∈ mk t h₁ h₂ h₃ ↔ s ∈ t :=
iff.rfl
@[simp] protected theorem mem_sets {α : Type u} {f : filter α} {s : set α} : s ∈ sets f ↔ s ∈ f :=
iff.rfl
protected instance inhabited_mem {α : Type u} {f : filter α} :
Inhabited (Subtype fun (s : set α) => s ∈ f) :=
{ default := { val := set.univ, property := univ_sets f } }
theorem filter_eq {α : Type u} {f : filter α} {g : filter α} : sets f = sets g → f = g := sorry
theorem filter_eq_iff {α : Type u} {f : filter α} {g : filter α} : f = g ↔ sets f = sets g :=
{ mp := congr_arg fun {f : filter α} => sets f, mpr := filter_eq }
protected theorem ext_iff {α : Type u} {f : filter α} {g : filter α} :
f = g ↔ ∀ (s : set α), s ∈ f ↔ s ∈ g :=
sorry
protected theorem ext {α : Type u} {f : filter α} {g : filter α} :
(∀ (s : set α), s ∈ f ↔ s ∈ g) → f = g :=
iff.mpr filter.ext_iff
@[simp] theorem univ_mem_sets {α : Type u} {f : filter α} : set.univ ∈ f := univ_sets f
theorem mem_sets_of_superset {α : Type u} {f : filter α} {x : set α} {y : set α} :
x ∈ f → x ⊆ y → y ∈ f :=
sets_of_superset f
theorem inter_mem_sets {α : Type u} {f : filter α} {s : set α} {t : set α} :
s ∈ f → t ∈ f → s ∩ t ∈ f :=
inter_sets f
@[simp] theorem inter_mem_sets_iff {α : Type u} {f : filter α} {s : set α} {t : set α} :
s ∩ t ∈ f ↔ s ∈ f ∧ t ∈ f :=
sorry
theorem univ_mem_sets' {α : Type u} {f : filter α} {s : set α} (h : ∀ (a : α), a ∈ s) : s ∈ f :=
mem_sets_of_superset univ_mem_sets fun (x : α) (_x : x ∈ set.univ) => h x
theorem mp_sets {α : Type u} {f : filter α} {s : set α} {t : set α} (hs : s ∈ f)
(h : (set_of fun (x : α) => x ∈ s → x ∈ t) ∈ f) : t ∈ f :=
sorry
theorem congr_sets {α : Type u} {f : filter α} {s : set α} {t : set α}
(h : (set_of fun (x : α) => x ∈ s ↔ x ∈ t) ∈ f) : s ∈ f ↔ t ∈ f :=
{ mp := fun (hs : s ∈ f) => mp_sets hs (mem_sets_of_superset h fun (x : α) => iff.mp),
mpr := fun (hs : t ∈ f) => mp_sets hs (mem_sets_of_superset h fun (x : α) => iff.mpr) }
@[simp] theorem bInter_mem_sets {α : Type u} {f : filter α} {β : Type v} {s : β → set α}
{is : set β} (hf : set.finite is) :
(set.Inter fun (i : β) => set.Inter fun (H : i ∈ is) => s i) ∈ f ↔
∀ (i : β), i ∈ is → s i ∈ f :=
sorry
@[simp] theorem bInter_finset_mem_sets {α : Type u} {f : filter α} {β : Type v} {s : β → set α}
(is : finset β) :
(set.Inter fun (i : β) => set.Inter fun (H : i ∈ is) => s i) ∈ f ↔
∀ (i : β), i ∈ is → s i ∈ f :=
bInter_mem_sets (finset.finite_to_set is)
protected theorem Mathlib.finset.Inter_mem_sets {α : Type u} {f : filter α} {β : Type v}
{s : β → set α} (is : finset β) :
(set.Inter fun (i : β) => set.Inter fun (H : i ∈ is) => s i) ∈ f ↔
∀ (i : β), i ∈ is → s i ∈ f :=
bInter_finset_mem_sets
@[simp] theorem sInter_mem_sets {α : Type u} {f : filter α} {s : set (set α)}
(hfin : set.finite s) : ⋂₀s ∈ f ↔ ∀ (U : set α), U ∈ s → U ∈ f :=
sorry
@[simp] theorem Inter_mem_sets {α : Type u} {f : filter α} {β : Type v} {s : β → set α}
[fintype β] : (set.Inter fun (i : β) => s i) ∈ f ↔ ∀ (i : β), s i ∈ f :=
sorry
theorem exists_sets_subset_iff {α : Type u} {f : filter α} {s : set α} :
(∃ (t : set α), ∃ (H : t ∈ f), t ⊆ s) ↔ s ∈ f :=
sorry
theorem monotone_mem_sets {α : Type u} {f : filter α} : monotone fun (s : set α) => s ∈ f :=
fun (s t : set α) (hst : s ≤ t) (h : s ∈ f) => mem_sets_of_superset h hst
end filter
namespace tactic.interactive
/-- `filter_upwards [h1, ⋯, hn]` replaces a goal of the form `s ∈ f`
and terms `h1 : t1 ∈ f, ⋯, hn : tn ∈ f` with `∀x, x ∈ t1 → ⋯ → x ∈ tn → x ∈ s`.
`filter_upwards [h1, ⋯, hn] e` is a short form for `{ filter_upwards [h1, ⋯, hn], exact e }`.
-/
end tactic.interactive
namespace filter
/-- The principal filter of `s` is the collection of all supersets of `s`. -/
def principal {α : Type u} (s : set α) : filter α :=
mk (set_of fun (t : set α) => s ⊆ t) (set.subset_univ s) sorry sorry
protected instance inhabited {α : Type u} : Inhabited (filter α) := { default := principal ∅ }
@[simp] theorem mem_principal_sets {α : Type u} {s : set α} {t : set α} : s ∈ principal t ↔ t ⊆ s :=
iff.rfl
theorem mem_principal_self {α : Type u} (s : set α) : s ∈ principal s := set.subset.refl s
/-- The join of a filter of filters is defined by the relation `s ∈ join f ↔ {t | s ∈ t} ∈ f`. -/
def join {α : Type u} (f : filter (filter α)) : filter α :=
mk (set_of fun (s : set α) => (set_of fun (t : filter α) => s ∈ t) ∈ f) sorry sorry sorry
@[simp] theorem mem_join_sets {α : Type u} {s : set α} {f : filter (filter α)} :
s ∈ join f ↔ (set_of fun (t : filter α) => s ∈ t) ∈ f :=
iff.rfl
protected instance partial_order {α : Type u} : partial_order (filter α) :=
partial_order.mk (fun (f g : filter α) => ∀ {U : set α}, U ∈ g → U ∈ f)
(preorder.lt._default fun (f g : filter α) => ∀ {U : set α}, U ∈ g → U ∈ f) sorry sorry sorry
theorem le_def {α : Type u} {f : filter α} {g : filter α} : f ≤ g ↔ ∀ (x : set α), x ∈ g → x ∈ f :=
iff.rfl
/-- `generate_sets g s`: `s` is in the filter closure of `g`. -/
inductive generate_sets {α : Type u} (g : set (set α)) : set α → Prop where
| basic : ∀ {s : set α}, s ∈ g → generate_sets g s
| univ : generate_sets g set.univ
| superset : ∀ {s t : set α}, generate_sets g s → s ⊆ t → generate_sets g t
| inter : ∀ {s t : set α}, generate_sets g s → generate_sets g t → generate_sets g (s ∩ t)
/-- `generate g` is the smallest filter containing the sets `g`. -/
def generate {α : Type u} (g : set (set α)) : filter α :=
mk (generate_sets g) generate_sets.univ sorry sorry
theorem sets_iff_generate {α : Type u} {s : set (set α)} {f : filter α} :
f ≤ generate s ↔ s ⊆ sets f :=
sorry
theorem mem_generate_iff {α : Type u} {s : set (set α)} {U : set α} :
U ∈ generate s ↔ ∃ (t : set (set α)), ∃ (H : t ⊆ s), set.finite t ∧ ⋂₀t ⊆ U :=
sorry
/-- `mk_of_closure s hs` constructs a filter on `α` whose elements set is exactly
`s : set (set α)`, provided one gives the assumption `hs : (generate s).sets = s`. -/
protected def mk_of_closure {α : Type u} (s : set (set α)) (hs : sets (generate s) = s) :
filter α :=
mk s sorry sorry sorry
theorem mk_of_closure_sets {α : Type u} {s : set (set α)} {hs : sets (generate s) = s} :
filter.mk_of_closure s hs = generate s :=
filter.ext
fun (u : set α) =>
(fun (this : u ∈ sets (filter.mk_of_closure s hs) ↔ u ∈ sets (generate s)) => this)
(Eq.symm hs ▸ iff.rfl)
/-- Galois insertion from sets of sets into filters. -/
def gi_generate (α : Type u_1) : galois_insertion generate sets :=
galois_insertion.mk
(fun (s : set (set α)) (hs : sets (generate s) ≤ s) => filter.mk_of_closure s sorry) sorry sorry
sorry
/-- The infimum of filters is the filter generated by intersections
of elements of the two filters. -/
protected instance has_inf {α : Type u} : has_inf (filter α) :=
has_inf.mk
fun (f g : filter α) =>
mk
(set_of
fun (s : set α) => ∃ (a : set α), ∃ (H : a ∈ f), ∃ (b : set α), ∃ (H : b ∈ g), a ∩ b ⊆ s)
sorry sorry sorry
@[simp] theorem mem_inf_sets {α : Type u} {f : filter α} {g : filter α} {s : set α} :
s ∈ f ⊓ g ↔ ∃ (t₁ : set α), ∃ (H : t₁ ∈ f), ∃ (t₂ : set α), ∃ (H : t₂ ∈ g), t₁ ∩ t₂ ⊆ s :=
iff.rfl
theorem mem_inf_sets_of_left {α : Type u} {f : filter α} {g : filter α} {s : set α} (h : s ∈ f) :
s ∈ f ⊓ g :=
Exists.intro s
(Exists.intro h
(Exists.intro set.univ (Exists.intro univ_mem_sets (set.inter_subset_left s set.univ))))
theorem mem_inf_sets_of_right {α : Type u} {f : filter α} {g : filter α} {s : set α} (h : s ∈ g) :
s ∈ f ⊓ g :=
Exists.intro set.univ
(Exists.intro univ_mem_sets
(Exists.intro s (Exists.intro h (set.inter_subset_right set.univ s))))
theorem inter_mem_inf_sets {α : Type u} {f : filter α} {g : filter α} {s : set α} {t : set α}
(hs : s ∈ f) (ht : t ∈ g) : s ∩ t ∈ f ⊓ g :=
inter_mem_sets (mem_inf_sets_of_left hs) (mem_inf_sets_of_right ht)
protected instance has_top {α : Type u} : has_top (filter α) :=
has_top.mk (mk (set_of fun (s : set α) => ∀ (x : α), x ∈ s) sorry sorry sorry)
theorem mem_top_sets_iff_forall {α : Type u} {s : set α} : s ∈ ⊤ ↔ ∀ (x : α), x ∈ s := iff.rfl
@[simp] theorem mem_top_sets {α : Type u} {s : set α} : s ∈ ⊤ ↔ s = set.univ :=
eq.mpr (id (Eq._oldrec (Eq.refl (s ∈ ⊤ ↔ s = set.univ)) (propext mem_top_sets_iff_forall)))
(eq.mpr
(id
(Eq._oldrec (Eq.refl ((∀ (x : α), x ∈ s) ↔ s = set.univ)) (propext set.eq_univ_iff_forall)))
(iff.refl (∀ (x : α), x ∈ s)))
/- We lift the complete lattice along the Galois connection `generate` / `sets`. Unfortunately,
we want to have different definitional equalities for the lattice operations. So we define them
upfront and change the lattice operations for the complete lattice instance. -/
protected instance complete_lattice {α : Type u} : complete_lattice (filter α) :=
complete_lattice.copy original_complete_lattice partial_order.le sorry ⊤ sorry
complete_lattice.bot sorry complete_lattice.sup sorry has_inf.inf sorry (join ∘ principal) sorry
complete_lattice.Inf sorry
/-- A filter is `ne_bot` if it is not equal to `⊥`, or equivalently the empty set
does not belong to the filter. Bourbaki include this assumption in the definition
of a filter but we prefer to have a `complete_lattice` structure on filter, so
we use a typeclass argument in lemmas instead. -/
def ne_bot {α : Type u} (f : filter α) := f ≠ ⊥
theorem ne_bot.ne {α : Type u} {f : filter α} (hf : ne_bot f) : f ≠ ⊥ := hf
@[simp] theorem not_ne_bot {α : Type u_1} {f : filter α} : ¬ne_bot f ↔ f = ⊥ := not_not
theorem ne_bot.mono {α : Type u} {f : filter α} {g : filter α} (hf : ne_bot f) (hg : f ≤ g) :
ne_bot g :=
ne_bot_of_le_ne_bot hf hg
theorem ne_bot_of_le {α : Type u} {f : filter α} {g : filter α} [hf : ne_bot f] (hg : f ≤ g) :
ne_bot g :=
ne_bot.mono hf hg
theorem bot_sets_eq {α : Type u} : sets ⊥ = set.univ := rfl
theorem sup_sets_eq {α : Type u} {f : filter α} {g : filter α} : sets (f ⊔ g) = sets f ∩ sets g :=
galois_connection.u_inf (galois_insertion.gc (gi_generate α))
theorem Sup_sets_eq {α : Type u} {s : set (filter α)} :
sets (Sup s) = set.Inter fun (f : filter α) => set.Inter fun (H : f ∈ s) => sets f :=
galois_connection.u_Inf (galois_insertion.gc (gi_generate α))
theorem supr_sets_eq {α : Type u} {ι : Sort x} {f : ι → filter α} :
sets (supr f) = set.Inter fun (i : ι) => sets (f i) :=
galois_connection.u_infi (galois_insertion.gc (gi_generate α))
theorem generate_empty {α : Type u} : generate ∅ = ⊤ :=
galois_connection.l_bot (galois_insertion.gc (gi_generate α))
theorem generate_univ {α : Type u} : generate set.univ = ⊥ := Eq.symm mk_of_closure_sets
theorem generate_union {α : Type u} {s : set (set α)} {t : set (set α)} :
generate (s ∪ t) = generate s ⊓ generate t :=
galois_connection.l_sup (galois_insertion.gc (gi_generate α))
theorem generate_Union {α : Type u} {ι : Sort x} {s : ι → set (set α)} :
generate (set.Union fun (i : ι) => s i) = infi fun (i : ι) => generate (s i) :=
galois_connection.l_supr (galois_insertion.gc (gi_generate α))
@[simp] theorem mem_bot_sets {α : Type u} {s : set α} : s ∈ ⊥ := trivial
@[simp] theorem mem_sup_sets {α : Type u} {f : filter α} {g : filter α} {s : set α} :
s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g :=
iff.rfl
theorem union_mem_sup {α : Type u} {f : filter α} {g : filter α} {s : set α} {t : set α}
(hs : s ∈ f) (ht : t ∈ g) : s ∪ t ∈ f ⊔ g :=
{ left := mem_sets_of_superset hs (set.subset_union_left s t),
right := mem_sets_of_superset ht (set.subset_union_right s t) }
@[simp] theorem mem_Sup_sets {α : Type u} {x : set α} {s : set (filter α)} :
x ∈ Sup s ↔ ∀ (f : filter α), f ∈ s → x ∈ f :=
iff.rfl
@[simp] theorem mem_supr_sets {α : Type u} {ι : Sort x} {x : set α} {f : ι → filter α} :
x ∈ supr f ↔ ∀ (i : ι), x ∈ f i :=
sorry
theorem infi_eq_generate {α : Type u} {ι : Sort x} (s : ι → filter α) :
infi s = generate (set.Union fun (i : ι) => sets (s i)) :=
sorry
theorem mem_infi_iff {α : Type u} {ι : Type u_1} {s : ι → filter α} {U : set α} :
(U ∈ infi fun (i : ι) => s i) ↔
∃ (I : set ι),
set.finite I ∧
∃ (V : ↥(set_of fun (i : ι) => i ∈ I) → set α),
(∀ (i : ↥(set_of fun (i : ι) => i ∈ I)), V i ∈ s ↑i) ∧
(set.Inter fun (i : ↥(set_of fun (i : ι) => i ∈ I)) => V i) ⊆ U :=
sorry
@[simp] theorem le_principal_iff {α : Type u} {s : set α} {f : filter α} :
f ≤ principal s ↔ s ∈ f :=
(fun (this : (∀ {t : set α}, s ⊆ t → t ∈ f) ↔ s ∈ f) => this)
{ mp := fun (h : ∀ {t : set α}, s ⊆ t → t ∈ f) => h (set.subset.refl s),
mpr := fun (hs : s ∈ f) (t : set α) (ht : s ⊆ t) => mem_sets_of_superset hs ht }
theorem principal_mono {α : Type u} {s : set α} {t : set α} : principal s ≤ principal t ↔ s ⊆ t :=
sorry
theorem monotone_principal {α : Type u} : monotone principal :=
fun (_x _x_1 : set α) => iff.mpr principal_mono
@[simp] theorem principal_eq_iff_eq {α : Type u} {s : set α} {t : set α} :
principal s = principal t ↔ s = t :=
sorry
@[simp] theorem join_principal_eq_Sup {α : Type u} {s : set (filter α)} :
join (principal s) = Sup s :=
rfl
@[simp] theorem principal_univ {α : Type u} : principal set.univ = ⊤ := sorry
@[simp] theorem principal_empty {α : Type u} : principal ∅ = ⊥ :=
bot_unique fun (s : set α) (_x : s ∈ ⊥) => set.empty_subset s
/-! ### Lattice equations -/
theorem empty_in_sets_eq_bot {α : Type u} {f : filter α} : ∅ ∈ f ↔ f = ⊥ :=
{ mp :=
fun (h : ∅ ∈ f) =>
bot_unique fun (s : set α) (_x : s ∈ ⊥) => mem_sets_of_superset h (set.empty_subset s),
mpr := fun (this : f = ⊥) => Eq.symm this ▸ mem_bot_sets }
theorem nonempty_of_mem_sets {α : Type u} {f : filter α} [hf : ne_bot f] {s : set α} (hs : s ∈ f) :
set.nonempty s :=
or.elim (set.eq_empty_or_nonempty s)
(fun (h : s = ∅) => absurd hs (Eq.symm h ▸ mt (iff.mp empty_in_sets_eq_bot) hf)) id
theorem ne_bot.nonempty_of_mem {α : Type u} {f : filter α} (hf : ne_bot f) {s : set α}
(hs : s ∈ f) : set.nonempty s :=
nonempty_of_mem_sets hs
@[simp] theorem empty_nmem_sets {α : Type u} (f : filter α) [ne_bot f] : ¬∅ ∈ f :=
fun (h : ∅ ∈ f) => set.nonempty.ne_empty (nonempty_of_mem_sets h) rfl
theorem nonempty_of_ne_bot {α : Type u} (f : filter α) [ne_bot f] : Nonempty α :=
nonempty_of_exists (nonempty_of_mem_sets univ_mem_sets)
theorem compl_not_mem_sets {α : Type u} {f : filter α} {s : set α} [ne_bot f] (h : s ∈ f) :
¬sᶜ ∈ f :=
fun (hsc : sᶜ ∈ f) =>
set.nonempty.ne_empty (nonempty_of_mem_sets (inter_mem_sets h hsc)) (set.inter_compl_self s)
theorem filter_eq_bot_of_not_nonempty {α : Type u} (f : filter α) (ne : ¬Nonempty α) : f = ⊥ :=
iff.mp empty_in_sets_eq_bot (univ_mem_sets' fun (x : α) => false.elim (ne (Nonempty.intro x)))
theorem forall_sets_nonempty_iff_ne_bot {α : Type u} {f : filter α} :
(∀ (s : set α), s ∈ f → set.nonempty s) ↔ ne_bot f :=
sorry
theorem nontrivial_iff_nonempty {α : Type u} : nontrivial (filter α) ↔ Nonempty α := sorry
theorem mem_sets_of_eq_bot {α : Type u} {f : filter α} {s : set α} (h : f ⊓ principal (sᶜ) = ⊥) :
s ∈ f :=
sorry
theorem eq_Inf_of_mem_sets_iff_exists_mem {α : Type u} {S : set (filter α)} {l : filter α}
(h : ∀ {s : set α}, s ∈ l ↔ ∃ (f : filter α), ∃ (H : f ∈ S), s ∈ f) : l = Inf S :=
sorry
theorem eq_infi_of_mem_sets_iff_exists_mem {α : Type u} {ι : Sort x} {f : ι → filter α}
{l : filter α} (h : ∀ {s : set α}, s ∈ l ↔ ∃ (i : ι), s ∈ f i) : l = infi f :=
eq_Inf_of_mem_sets_iff_exists_mem fun (s : set α) => iff.trans h (iff.symm set.exists_range_iff)
theorem eq_binfi_of_mem_sets_iff_exists_mem {α : Type u} {ι : Sort x} {f : ι → filter α}
{p : ι → Prop} {l : filter α} (h : ∀ {s : set α}, s ∈ l ↔ ∃ (i : ι), ∃ (_x : p i), s ∈ f i) :
l = infi fun (i : ι) => infi fun (_x : p i) => f i :=
sorry
theorem infi_sets_eq {α : Type u} {ι : Sort x} {f : ι → filter α} (h : directed ge f)
[ne : Nonempty ι] : sets (infi f) = set.Union fun (i : ι) => sets (f i) :=
sorry
theorem mem_infi {α : Type u} {ι : Sort x} {f : ι → filter α} (h : directed ge f) [Nonempty ι]
(s : set α) : s ∈ infi f ↔ ∃ (i : ι), s ∈ f i :=
sorry
theorem mem_binfi {α : Type u} {β : Type v} {f : β → filter α} {s : set β}
(h : directed_on (f ⁻¹'o ge) s) (ne : set.nonempty s) {t : set α} :
(t ∈ infi fun (i : β) => infi fun (H : i ∈ s) => f i) ↔ ∃ (i : β), ∃ (H : i ∈ s), t ∈ f i :=
sorry
theorem binfi_sets_eq {α : Type u} {β : Type v} {f : β → filter α} {s : set β}
(h : directed_on (f ⁻¹'o ge) s) (ne : set.nonempty s) :
sets (infi fun (i : β) => infi fun (H : i ∈ s) => f i) =
set.Union fun (i : β) => set.Union fun (H : i ∈ s) => sets (f i) :=
sorry
theorem infi_sets_eq_finite {α : Type u} {ι : Type u_1} (f : ι → filter α) :
sets (infi fun (i : ι) => f i) =
set.Union fun (t : finset ι) => sets (infi fun (i : ι) => infi fun (H : i ∈ t) => f i) :=
sorry
theorem infi_sets_eq_finite' {α : Type u} {ι : Sort x} (f : ι → filter α) :
sets (infi fun (i : ι) => f i) =
set.Union
fun (t : finset (plift ι)) =>
sets (infi fun (i : plift ι) => infi fun (H : i ∈ t) => f (plift.down i)) :=
sorry
theorem mem_infi_finite {α : Type u} {ι : Type u_1} {f : ι → filter α} (s : set α) :
s ∈ infi f ↔ ∃ (t : finset ι), s ∈ infi fun (i : ι) => infi fun (H : i ∈ t) => f i :=
iff.trans (iff.mp set.ext_iff (infi_sets_eq_finite f) s) set.mem_Union
theorem mem_infi_finite' {α : Type u} {ι : Sort x} {f : ι → filter α} (s : set α) :
s ∈ infi f ↔
∃ (t : finset (plift ι)),
s ∈ infi fun (i : plift ι) => infi fun (H : i ∈ t) => f (plift.down i) :=
iff.trans (iff.mp set.ext_iff (infi_sets_eq_finite' f) s) set.mem_Union
@[simp] theorem sup_join {α : Type u} {f₁ : filter (filter α)} {f₂ : filter (filter α)} :
join f₁ ⊔ join f₂ = join (f₁ ⊔ f₂) :=
sorry
@[simp] theorem supr_join {α : Type u} {ι : Sort w} {f : ι → filter (filter α)} :
(supr fun (x : ι) => join (f x)) = join (supr fun (x : ι) => f x) :=
sorry
protected instance bounded_distrib_lattice {α : Type u} : bounded_distrib_lattice (filter α) :=
bounded_distrib_lattice.mk complete_lattice.sup complete_lattice.le complete_lattice.lt sorry
sorry sorry sorry sorry sorry complete_lattice.inf sorry sorry sorry sorry complete_lattice.top
sorry complete_lattice.bot sorry
/- the complementary version with ⨆i, f ⊓ g i does not hold! -/
theorem infi_sup_left {α : Type u} {ι : Sort x} {f : filter α} {g : ι → filter α} :
(infi fun (x : ι) => f ⊔ g x) = f ⊔ infi g :=
sorry
theorem infi_sup_right {α : Type u} {ι : Sort x} {f : filter α} {g : ι → filter α} :
(infi fun (x : ι) => g x ⊔ f) = infi g ⊔ f :=
sorry
theorem binfi_sup_right {α : Type u} {ι : Sort x} (p : ι → Prop) (f : ι → filter α) (g : filter α) :
(infi fun (i : ι) => infi fun (h : p i) => f i ⊔ g) =
(infi fun (i : ι) => infi fun (h : p i) => f i) ⊔ g :=
sorry
theorem binfi_sup_left {α : Type u} {ι : Sort x} (p : ι → Prop) (f : ι → filter α) (g : filter α) :
(infi fun (i : ι) => infi fun (h : p i) => g ⊔ f i) =
g ⊔ infi fun (i : ι) => infi fun (h : p i) => f i :=
sorry
theorem mem_infi_sets_finset {α : Type u} {β : Type v} {s : finset α} {f : α → filter β}
(t : set β) :
(t ∈ infi fun (a : α) => infi fun (H : a ∈ s) => f a) ↔
∃ (p : α → set β),
(∀ (a : α), a ∈ s → p a ∈ f a) ∧
(set.Inter fun (a : α) => set.Inter fun (H : a ∈ s) => p a) ⊆ t :=
sorry
/-- If `f : ι → filter α` is directed, `ι` is not empty, and `∀ i, f i ≠ ⊥`, then `infi f ≠ ⊥`.
See also `infi_ne_bot_of_directed` for a version assuming `nonempty α` instead of `nonempty ι`. -/
theorem infi_ne_bot_of_directed' {α : Type u} {ι : Sort x} {f : ι → filter α} [Nonempty ι]
(hd : directed ge f) (hb : ∀ (i : ι), ne_bot (f i)) : ne_bot (infi f) :=
sorry
/-- If `f : ι → filter α` is directed, `α` is not empty, and `∀ i, f i ≠ ⊥`, then `infi f ≠ ⊥`.
See also `infi_ne_bot_of_directed'` for a version assuming `nonempty ι` instead of `nonempty α`. -/
theorem infi_ne_bot_of_directed {α : Type u} {ι : Sort x} {f : ι → filter α} [hn : Nonempty α]
(hd : directed ge f) (hb : ∀ (i : ι), ne_bot (f i)) : ne_bot (infi f) :=
sorry
theorem infi_ne_bot_iff_of_directed' {α : Type u} {ι : Sort x} {f : ι → filter α} [Nonempty ι]
(hd : directed ge f) : ne_bot (infi f) ↔ ∀ (i : ι), ne_bot (f i) :=
{ mp := fun (H : ne_bot (infi f)) (i : ι) => ne_bot.mono H (infi_le f i),
mpr := infi_ne_bot_of_directed' hd }
theorem infi_ne_bot_iff_of_directed {α : Type u} {ι : Sort x} {f : ι → filter α} [Nonempty α]
(hd : directed ge f) : ne_bot (infi f) ↔ ∀ (i : ι), ne_bot (f i) :=
{ mp := fun (H : ne_bot (infi f)) (i : ι) => ne_bot.mono H (infi_le f i),
mpr := infi_ne_bot_of_directed hd }
theorem mem_infi_sets {α : Type u} {ι : Sort x} {f : ι → filter α} (i : ι) {s : set α} :
s ∈ f i → s ∈ infi fun (i : ι) => f i :=
(fun (this : (infi fun (i : ι) => f i) ≤ f i) => this) (infi_le (fun (i : ι) => f i) i)
theorem infi_sets_induct {α : Type u} {ι : Sort x} {f : ι → filter α} {s : set α} (hs : s ∈ infi f)
{p : set α → Prop} (uni : p set.univ)
(ins : ∀ {i : ι} {s₁ s₂ : set α}, s₁ ∈ f i → p s₂ → p (s₁ ∩ s₂))
(upw : ∀ {s₁ s₂ : set α}, s₁ ⊆ s₂ → p s₁ → p s₂) : p s :=
sorry
/- principal equations -/
@[simp] theorem inf_principal {α : Type u} {s : set α} {t : set α} :
principal s ⊓ principal t = principal (s ∩ t) :=
sorry
@[simp] theorem sup_principal {α : Type u} {s : set α} {t : set α} :
principal s ⊔ principal t = principal (s ∪ t) :=
sorry
@[simp] theorem supr_principal {α : Type u} {ι : Sort w} {s : ι → set α} :
(supr fun (x : ι) => principal (s x)) = principal (set.Union fun (i : ι) => s i) :=
sorry
@[simp] theorem principal_eq_bot_iff {α : Type u} {s : set α} : principal s = ⊥ ↔ s = ∅ :=
iff.trans (iff.symm empty_in_sets_eq_bot) (iff.trans mem_principal_sets set.subset_empty_iff)
@[simp] theorem principal_ne_bot_iff {α : Type u} {s : set α} :
ne_bot (principal s) ↔ set.nonempty s :=
iff.trans (not_congr principal_eq_bot_iff) set.ne_empty_iff_nonempty
theorem is_compl_principal {α : Type u} (s : set α) : is_compl (principal s) (principal (sᶜ)) :=
sorry
theorem inf_principal_eq_bot {α : Type u} {f : filter α} {s : set α} (hs : sᶜ ∈ f) :
f ⊓ principal s = ⊥ :=
sorry
theorem mem_inf_principal {α : Type u} {f : filter α} {s : set α} {t : set α} :
s ∈ f ⊓ principal t ↔ (set_of fun (x : α) => x ∈ t → x ∈ s) ∈ f :=
sorry
theorem diff_mem_inf_principal_compl {α : Type u} {f : filter α} {s : set α} (hs : s ∈ f)
(t : set α) : s \ t ∈ f ⊓ principal (tᶜ) :=
eq.mpr (id (Eq._oldrec (Eq.refl (s \ t ∈ f ⊓ principal (tᶜ))) (propext mem_inf_principal)))
(mp_sets hs
(univ_mem_sets'
(id fun (a : α) (has : a ∈ s) (hat : a ∈ (tᶜ)) => { left := has, right := hat })))
theorem principal_le_iff {α : Type u} {s : set α} {f : filter α} :
principal s ≤ f ↔ ∀ (V : set α), V ∈ f → s ⊆ V :=
sorry
@[simp] theorem infi_principal_finset {α : Type u} {ι : Type w} (s : finset ι) (f : ι → set α) :
(infi fun (i : ι) => infi fun (H : i ∈ s) => principal (f i)) =
principal (set.Inter fun (i : ι) => set.Inter fun (H : i ∈ s) => f i) :=
sorry
@[simp] theorem infi_principal_fintype {α : Type u} {ι : Type w} [fintype ι] (f : ι → set α) :
(infi fun (i : ι) => principal (f i)) = principal (set.Inter fun (i : ι) => f i) :=
sorry
theorem join_mono {α : Type u} {f₁ : filter (filter α)} {f₂ : filter (filter α)} (h : f₁ ≤ f₂) :
join f₁ ≤ join f₂ :=
fun (s : set α) (hs : s ∈ join f₂) => h hs
/-! ### Eventually -/
/-- `f.eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in at_top, p x`
means that `p` holds true for sufficiently large `x`. -/
protected def eventually {α : Type u} (p : α → Prop) (f : filter α) :=
(set_of fun (x : α) => p x) ∈ f
theorem eventually_iff {α : Type u} {f : filter α} {P : α → Prop} :
filter.eventually (fun (x : α) => P x) f ↔ (set_of fun (x : α) => P x) ∈ f :=
iff.rfl
protected theorem ext' {α : Type u} {f₁ : filter α} {f₂ : filter α}
(h :
∀ (p : α → Prop),
filter.eventually (fun (x : α) => p x) f₁ ↔ filter.eventually (fun (x : α) => p x) f₂) :
f₁ = f₂ :=
filter.ext h
theorem eventually.filter_mono {α : Type u} {f₁ : filter α} {f₂ : filter α} (h : f₁ ≤ f₂)
{p : α → Prop} (hp : filter.eventually (fun (x : α) => p x) f₂) :
filter.eventually (fun (x : α) => p x) f₁ :=
h hp
theorem eventually_of_mem {α : Type u} {f : filter α} {P : α → Prop} {U : set α} (hU : U ∈ f)
(h : ∀ (x : α), x ∈ U → P x) : filter.eventually (fun (x : α) => P x) f :=
mem_sets_of_superset hU h
protected theorem eventually.and {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α} :
filter.eventually p f →
filter.eventually q f → filter.eventually (fun (x : α) => p x ∧ q x) f :=
inter_mem_sets
@[simp] theorem eventually_true {α : Type u} (f : filter α) :
filter.eventually (fun (x : α) => True) f :=
univ_mem_sets
theorem eventually_of_forall {α : Type u} {p : α → Prop} {f : filter α} (hp : ∀ (x : α), p x) :
filter.eventually (fun (x : α) => p x) f :=
univ_mem_sets' hp
@[simp] theorem eventually_false_iff_eq_bot {α : Type u} {f : filter α} :
filter.eventually (fun (x : α) => False) f ↔ f = ⊥ :=
empty_in_sets_eq_bot
@[simp] theorem eventually_const {α : Type u} {f : filter α} [ne_bot f] {p : Prop} :
filter.eventually (fun (x : α) => p) f ↔ p :=
sorry
theorem eventually_iff_exists_mem {α : Type u} {p : α → Prop} {f : filter α} :
filter.eventually (fun (x : α) => p x) f ↔
∃ (v : set α), ∃ (H : v ∈ f), ∀ (y : α), y ∈ v → p y :=
iff.symm exists_sets_subset_iff
theorem eventually.exists_mem {α : Type u} {p : α → Prop} {f : filter α}
(hp : filter.eventually (fun (x : α) => p x) f) :
∃ (v : set α), ∃ (H : v ∈ f), ∀ (y : α), y ∈ v → p y :=
iff.mp eventually_iff_exists_mem hp
theorem eventually.mp {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α}
(hp : filter.eventually (fun (x : α) => p x) f)
(hq : filter.eventually (fun (x : α) => p x → q x) f) :
filter.eventually (fun (x : α) => q x) f :=
mp_sets hp hq
theorem eventually.mono {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α}
(hp : filter.eventually (fun (x : α) => p x) f) (hq : ∀ (x : α), p x → q x) :
filter.eventually (fun (x : α) => q x) f :=
eventually.mp hp (eventually_of_forall hq)
@[simp] theorem eventually_and {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α} :
filter.eventually (fun (x : α) => p x ∧ q x) f ↔
filter.eventually (fun (x : α) => p x) f ∧ filter.eventually (fun (x : α) => q x) f :=
inter_mem_sets_iff
theorem eventually.congr {α : Type u} {f : filter α} {p : α → Prop} {q : α → Prop}
(h' : filter.eventually (fun (x : α) => p x) f)
(h : filter.eventually (fun (x : α) => p x ↔ q x) f) :
filter.eventually (fun (x : α) => q x) f :=
eventually.mp h' (eventually.mono h fun (x : α) (hx : p x ↔ q x) => iff.mp hx)
theorem eventually_congr {α : Type u} {f : filter α} {p : α → Prop} {q : α → Prop}
(h : filter.eventually (fun (x : α) => p x ↔ q x) f) :
filter.eventually (fun (x : α) => p x) f ↔ filter.eventually (fun (x : α) => q x) f :=
sorry
@[simp] theorem eventually_all {α : Type u} {ι : Type u_1} [fintype ι] {l : filter α}
{p : ι → α → Prop} :
filter.eventually (fun (x : α) => ∀ (i : ι), p i x) l ↔
∀ (i : ι), filter.eventually (fun (x : α) => p i x) l :=
sorry
@[simp] theorem eventually_all_finite {α : Type u} {ι : Type u_1} {I : set ι} (hI : set.finite I)
{l : filter α} {p : ι → α → Prop} :
filter.eventually (fun (x : α) => ∀ (i : ι), i ∈ I → p i x) l ↔
∀ (i : ι), i ∈ I → filter.eventually (fun (x : α) => p i x) l :=
sorry
protected theorem Mathlib.set.finite.eventually_all {α : Type u} {ι : Type u_1} {I : set ι}
(hI : set.finite I) {l : filter α} {p : ι → α → Prop} :
filter.eventually (fun (x : α) => ∀ (i : ι), i ∈ I → p i x) l ↔
∀ (i : ι), i ∈ I → filter.eventually (fun (x : α) => p i x) l :=
eventually_all_finite
@[simp] theorem eventually_all_finset {α : Type u} {ι : Type u_1} (I : finset ι) {l : filter α}
{p : ι → α → Prop} :
filter.eventually (fun (x : α) => ∀ (i : ι), i ∈ I → p i x) l ↔
∀ (i : ι), i ∈ I → filter.eventually (fun (x : α) => p i x) l :=
set.finite.eventually_all (finset.finite_to_set I)
protected theorem Mathlib.finset.eventually_all {α : Type u} {ι : Type u_1} (I : finset ι)
{l : filter α} {p : ι → α → Prop} :
filter.eventually (fun (x : α) => ∀ (i : ι), i ∈ I → p i x) l ↔
∀ (i : ι), i ∈ I → filter.eventually (fun (x : α) => p i x) l :=
eventually_all_finset
@[simp] theorem eventually_or_distrib_left {α : Type u} {f : filter α} {p : Prop} {q : α → Prop} :
filter.eventually (fun (x : α) => p ∨ q x) f ↔ p ∨ filter.eventually (fun (x : α) => q x) f :=
sorry
@[simp] theorem eventually_or_distrib_right {α : Type u} {f : filter α} {p : α → Prop} {q : Prop} :
filter.eventually (fun (x : α) => p x ∨ q) f ↔ filter.eventually (fun (x : α) => p x) f ∨ q :=
sorry
@[simp] theorem eventually_imp_distrib_left {α : Type u} {f : filter α} {p : Prop} {q : α → Prop} :
filter.eventually (fun (x : α) => p → q x) f ↔ p → filter.eventually (fun (x : α) => q x) f :=
sorry
@[simp] theorem eventually_bot {α : Type u} {p : α → Prop} :
filter.eventually (fun (x : α) => p x) ⊥ :=
True.intro
@[simp] theorem eventually_top {α : Type u} {p : α → Prop} :
filter.eventually (fun (x : α) => p x) ⊤ ↔ ∀ (x : α), p x :=
iff.rfl
@[simp] theorem eventually_sup {α : Type u} {p : α → Prop} {f : filter α} {g : filter α} :
filter.eventually (fun (x : α) => p x) (f ⊔ g) ↔
filter.eventually (fun (x : α) => p x) f ∧ filter.eventually (fun (x : α) => p x) g :=
iff.rfl
@[simp] theorem eventually_Sup {α : Type u} {p : α → Prop} {fs : set (filter α)} :
filter.eventually (fun (x : α) => p x) (Sup fs) ↔
∀ (f : filter α), f ∈ fs → filter.eventually (fun (x : α) => p x) f :=
iff.rfl
@[simp] theorem eventually_supr {α : Type u} {β : Type v} {p : α → Prop} {fs : β → filter α} :
filter.eventually (fun (x : α) => p x) (supr fun (b : β) => fs b) ↔
∀ (b : β), filter.eventually (fun (x : α) => p x) (fs b) :=
mem_supr_sets
@[simp] theorem eventually_principal {α : Type u} {a : set α} {p : α → Prop} :
filter.eventually (fun (x : α) => p x) (principal a) ↔ ∀ (x : α), x ∈ a → p x :=
iff.rfl
theorem eventually_inf_principal {α : Type u} {f : filter α} {p : α → Prop} {s : set α} :
filter.eventually (fun (x : α) => p x) (f ⊓ principal s) ↔
filter.eventually (fun (x : α) => x ∈ s → p x) f :=
mem_inf_principal
/-! ### Frequently -/
/-- `f.frequently p` or `∃ᶠ x in f, p x` mean that `{x | ¬p x} ∉ f`. E.g., `∃ᶠ x in at_top, p x`
means that there exist arbitrarily large `x` for which `p` holds true. -/
protected def frequently {α : Type u} (p : α → Prop) (f : filter α) :=
¬filter.eventually (fun (x : α) => ¬p x) f
theorem eventually.frequently {α : Type u} {f : filter α} [ne_bot f] {p : α → Prop}
(h : filter.eventually (fun (x : α) => p x) f) : filter.frequently (fun (x : α) => p x) f :=
compl_not_mem_sets h
theorem frequently_of_forall {α : Type u} {f : filter α} [ne_bot f] {p : α → Prop}
(h : ∀ (x : α), p x) : filter.frequently (fun (x : α) => p x) f :=
eventually.frequently (eventually_of_forall h)
theorem frequently.mp {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α}
(h : filter.frequently (fun (x : α) => p x) f)
(hpq : filter.eventually (fun (x : α) => p x → q x) f) :
filter.frequently (fun (x : α) => q x) f :=
sorry
theorem frequently.mono {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α}
(h : filter.frequently (fun (x : α) => p x) f) (hpq : ∀ (x : α), p x → q x) :
filter.frequently (fun (x : α) => q x) f :=
frequently.mp h (eventually_of_forall hpq)
theorem frequently.and_eventually {α : Type u} {p : α → Prop} {q : α → Prop} {f : filter α}
(hp : filter.frequently (fun (x : α) => p x) f)
(hq : filter.eventually (fun (x : α) => q x) f) :
filter.frequently (fun (x : α) => p x ∧ q x) f :=
sorry
theorem frequently.exists {α : Type u} {p : α → Prop} {f : filter α}
(hp : filter.frequently (fun (x : α) => p x) f) : ∃ (x : α), p x :=
decidable.by_contradiction
fun (H : ¬∃ (x : α), p x) => hp (eventually_of_forall (iff.mp not_exists H))
theorem eventually.exists {α : Type u} {p : α → Prop} {f : filter α} [ne_bot f]
(hp : filter.eventually (fun (x : α) => p x) f) : ∃ (x : α), p x :=
frequently.exists (eventually.frequently hp)
theorem frequently_iff_forall_eventually_exists_and {α : Type u} {p : α → Prop} {f : filter α} :
filter.frequently (fun (x : α) => p x) f ↔
∀ {q : α → Prop}, filter.eventually (fun (x : α) => q x) f → ∃ (x : α), p x ∧ q x :=
sorry
theorem frequently_iff {α : Type u} {f : filter α} {P : α → Prop} :
filter.frequently (fun (x : α) => P x) f ↔
∀ {U : set α}, U ∈ f → ∃ (x : α), ∃ (H : x ∈ U), P x :=
sorry
@[simp] theorem not_eventually {α : Type u} {p : α → Prop} {f : filter α} :
¬filter.eventually (fun (x : α) => p x) f ↔ filter.frequently (fun (x : α) => ¬p x) f :=
sorry
@[simp] theorem not_frequently {α : Type u} {p : α → Prop} {f : filter α} :
¬filter.frequently (fun (x : α) => p x) f ↔ filter.eventually (fun (x : α) => ¬p x) f :=
sorry
@[simp] theorem frequently_true_iff_ne_bot {α : Type u} (f : filter α) :
filter.frequently (fun (x : α) => True) f ↔ ne_bot f :=
sorry
@[simp] theorem frequently_false {α : Type u} (f : filter α) :
¬filter.frequently (fun (x : α) => False) f :=
sorry
@[simp] theorem frequently_const {α : Type u} {f : filter α} [ne_bot f] {p : Prop} :
filter.frequently (fun (x : α) => p) f ↔ p :=
sorry
@[simp] theorem frequently_or_distrib {α : Type u} {f : filter α} {p : α → Prop} {q : α → Prop} :
filter.frequently (fun (x : α) => p x ∨ q x) f ↔
filter.frequently (fun (x : α) => p x) f ∨ filter.frequently (fun (x : α) => q x) f :=
sorry
theorem frequently_or_distrib_left {α : Type u} {f : filter α} [ne_bot f] {p : Prop}
{q : α → Prop} :
filter.frequently (fun (x : α) => p ∨ q x) f ↔ p ∨ filter.frequently (fun (x : α) => q x) f :=
sorry
theorem frequently_or_distrib_right {α : Type u} {f : filter α} [ne_bot f] {p : α → Prop}
{q : Prop} :
filter.frequently (fun (x : α) => p x ∨ q) f ↔ filter.frequently (fun (x : α) => p x) f ∨ q :=
sorry
@[simp] theorem frequently_imp_distrib {α : Type u} {f : filter α} {p : α → Prop} {q : α → Prop} :
filter.frequently (fun (x : α) => p x → q x) f ↔
filter.eventually (fun (x : α) => p x) f → filter.frequently (fun (x : α) => q x) f :=
sorry
theorem frequently_imp_distrib_left {α : Type u} {f : filter α} [ne_bot f] {p : Prop}
{q : α → Prop} :
filter.frequently (fun (x : α) => p → q x) f ↔ p → filter.frequently (fun (x : α) => q x) f :=
sorry
theorem frequently_imp_distrib_right {α : Type u} {f : filter α} [ne_bot f] {p : α → Prop}
{q : Prop} :
filter.frequently (fun (x : α) => p x → q) f ↔ filter.eventually (fun (x : α) => p x) f → q :=
sorry
@[simp] theorem eventually_imp_distrib_right {α : Type u} {f : filter α} {p : α → Prop} {q : Prop} :
filter.eventually (fun (x : α) => p x → q) f ↔ filter.frequently (fun (x : α) => p x) f → q :=
sorry
@[simp] theorem frequently_bot {α : Type u} {p : α → Prop} :
¬filter.frequently (fun (x : α) => p x) ⊥ :=
eq.mpr (id (Eq.trans (propext not_frequently) (propext (iff_true_intro eventually_bot)))) trivial
@[simp] theorem frequently_top {α : Type u} {p : α → Prop} :
filter.frequently (fun (x : α) => p x) ⊤ ↔ ∃ (x : α), p x :=
sorry
@[simp] theorem frequently_principal {α : Type u} {a : set α} {p : α → Prop} :
filter.frequently (fun (x : α) => p x) (principal a) ↔ ∃ (x : α), ∃ (H : x ∈ a), p x :=
sorry
theorem frequently_sup {α : Type u} {p : α → Prop} {f : filter α} {g : filter α} :
filter.frequently (fun (x : α) => p x) (f ⊔ g) ↔
filter.frequently (fun (x : α) => p x) f ∨ filter.frequently (fun (x : α) => p x) g :=
sorry
@[simp] theorem frequently_Sup {α : Type u} {p : α → Prop} {fs : set (filter α)} :
filter.frequently (fun (x : α) => p x) (Sup fs) ↔
∃ (f : filter α), ∃ (H : f ∈ fs), filter.frequently (fun (x : α) => p x) f :=
sorry
@[simp] theorem frequently_supr {α : Type u} {β : Type v} {p : α → Prop} {fs : β → filter α} :
filter.frequently (fun (x : α) => p x) (supr fun (b : β) => fs b) ↔
∃ (b : β), filter.frequently (fun (x : α) => p x) (fs b) :=
sorry
/-!
### Relation “eventually equal”
-/
/-- Two functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that
`f x = g x` belongs to `l`. -/
def eventually_eq {α : Type u} {β : Type v} (l : filter α) (f : α → β) (g : α → β) :=
filter.eventually (fun (x : α) => f x = g x) l
theorem eventually_eq.eventually {α : Type u} {β : Type v} {l : filter α} {f : α → β} {g : α → β}
(h : eventually_eq l f g) : filter.eventually (fun (x : α) => f x = g x) l :=
h
theorem eventually_eq.rw {α : Type u} {β : Type v} {l : filter α} {f : α → β} {g : α → β}
(h : eventually_eq l f g) (p : α → β → Prop)
(hf : filter.eventually (fun (x : α) => p x (f x)) l) :
filter.eventually (fun (x : α) => p x (g x)) l :=
eventually.congr hf (eventually.mono h fun (x : α) (hx : f x = g x) => hx ▸ iff.rfl)
theorem eventually_eq_set {α : Type u} {s : set α} {t : set α} {l : filter α} :
eventually_eq l s t ↔ filter.eventually (fun (x : α) => x ∈ s ↔ x ∈ t) l :=
eventually_congr (eventually_of_forall fun (x : α) => { mp := eq.to_iff, mpr := iff.to_eq })
theorem eventually.set_eq {α : Type u} {s : set α} {t : set α} {l : filter α} :
filter.eventually (fun (x : α) => x ∈ s ↔ x ∈ t) l → eventually_eq l s t :=
iff.mpr eventually_eq_set
theorem eventually_eq.exists_mem {α : Type u} {β : Type v} {l : filter α} {f : α → β} {g : α → β}
(h : eventually_eq l f g) : ∃ (s : set α), ∃ (H : s ∈ l), set.eq_on f g s :=
eventually.exists_mem h
theorem eventually_eq_of_mem {α : Type u} {β : Type v} {l : filter α} {f : α → β} {g : α → β}
{s : set α} (hs : s ∈ l) (h : set.eq_on f g s) : eventually_eq l f g :=
eventually_of_mem hs h
theorem eventually_eq_iff_exists_mem {α : Type u} {β : Type v} {l : filter α} {f : α → β}
{g : α → β} : eventually_eq l f g ↔ ∃ (s : set α), ∃ (H : s ∈ l), set.eq_on f g s :=
eventually_iff_exists_mem
theorem eventually_eq.filter_mono {α : Type u} {β : Type v} {l : filter α} {l' : filter α}
{f : α → β} {g : α → β} (h₁ : eventually_eq l f g) (h₂ : l' ≤ l) : eventually_eq l' f g :=
h₂ h₁
theorem eventually_eq.refl {α : Type u} {β : Type v} (l : filter α) (f : α → β) :
eventually_eq l f f :=
eventually_of_forall fun (x : α) => rfl
theorem eventually_eq.rfl {α : Type u} {β : Type v} {l : filter α} {f : α → β} :
eventually_eq l f f :=
eventually_eq.refl l f
theorem eventually_eq.symm {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : filter α}
(H : eventually_eq l f g) : eventually_eq l g f :=
eventually.mono H fun (_x : α) => Eq.symm
theorem eventually_eq.trans {α : Type u} {β : Type v} {f : α → β} {g : α → β} {h : α → β}
{l : filter α} (H₁ : eventually_eq l f g) (H₂ : eventually_eq l g h) : eventually_eq l f h :=
eventually_eq.rw H₂ (fun (x : α) (y : β) => f x = y) H₁
theorem eventually_eq.prod_mk {α : Type u} {β : Type v} {γ : Type w} {l : filter α} {f : α → β}
{f' : α → β} (hf : eventually_eq l f f') {g : α → γ} {g' : α → γ} (hg : eventually_eq l g g') :
eventually_eq l (fun (x : α) => (f x, g x)) fun (x : α) => (f' x, g' x) :=
sorry
theorem eventually_eq.fun_comp {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : α → β}
{l : filter α} (H : eventually_eq l f g) (h : β → γ) : eventually_eq l (h ∘ f) (h ∘ g) :=
eventually.mono H fun (x : α) (hx : f x = g x) => congr_arg h hx
theorem eventually_eq.comp₂ {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} {f : α → β}
{f' : α → β} {g : α → γ} {g' : α → γ} {l : filter α} (Hf : eventually_eq l f f') (h : β → γ → δ)
(Hg : eventually_eq l g g') :
eventually_eq l (fun (x : α) => h (f x) (g x)) fun (x : α) => h (f' x) (g' x) :=
eventually_eq.fun_comp (eventually_eq.prod_mk Hf Hg) (function.uncurry h)
theorem eventually_eq.add {α : Type u} {β : Type v} [Add β] {f : α → β} {f' : α → β} {g : α → β}
{g' : α → β} {l : filter α} (h : eventually_eq l f g) (h' : eventually_eq l f' g') :
eventually_eq l (fun (x : α) => f x + f' x) fun (x : α) => g x + g' x :=
eventually_eq.comp₂ h Add.add h'
theorem eventually_eq.neg {α : Type u} {β : Type v} [Neg β] {f : α → β} {g : α → β} {l : filter α}
(h : eventually_eq l f g) : eventually_eq l (fun (x : α) => -f x) fun (x : α) => -g x :=
eventually_eq.fun_comp h Neg.neg
theorem eventually_eq.div {α : Type u} {β : Type v} [group_with_zero β] {f : α → β} {f' : α → β}
{g : α → β} {g' : α → β} {l : filter α} (h : eventually_eq l f g) (h' : eventually_eq l f' g') :
eventually_eq l (fun (x : α) => f x / f' x) fun (x : α) => g x / g' x :=
sorry
theorem eventually_eq.sub {α : Type u} {β : Type v} [add_group β] {f : α → β} {f' : α → β}
{g : α → β} {g' : α → β} {l : filter α} (h : eventually_eq l f g) (h' : eventually_eq l f' g') :
eventually_eq l (fun (x : α) => f x - f' x) fun (x : α) => g x - g' x :=
sorry
theorem eventually_eq.inter {α : Type u} {s : set α} {t : set α} {s' : set α} {t' : set α}
{l : filter α} (h : eventually_eq l s t) (h' : eventually_eq l s' t') :
eventually_eq l (s ∩ s') (t ∩ t') :=
eventually_eq.comp₂ h And h'
theorem eventually_eq.union {α : Type u} {s : set α} {t : set α} {s' : set α} {t' : set α}
{l : filter α} (h : eventually_eq l s t) (h' : eventually_eq l s' t') :
eventually_eq l (s ∪ s') (t ∪ t') :=
eventually_eq.comp₂ h Or h'
theorem eventually_eq.compl {α : Type u} {s : set α} {t : set α} {l : filter α}
(h : eventually_eq l s t) : eventually_eq l (sᶜ) (tᶜ) :=
eventually_eq.fun_comp h Not
theorem eventually_eq.diff {α : Type u} {s : set α} {t : set α} {s' : set α} {t' : set α}
{l : filter α} (h : eventually_eq l s t) (h' : eventually_eq l s' t') :
eventually_eq l (s \ s') (t \ t') :=
eventually_eq.inter h (eventually_eq.compl h')
theorem eventually_eq_empty {α : Type u} {s : set α} {l : filter α} :
eventually_eq l s ∅ ↔ filter.eventually (fun (x : α) => ¬x ∈ s) l :=
sorry
@[simp] theorem eventually_eq_principal {α : Type u} {β : Type v} {s : set α} {f : α → β}
{g : α → β} : eventually_eq (principal s) f g ↔ set.eq_on f g s :=
iff.rfl
theorem eventually_eq_inf_principal_iff {α : Type u} {β : Type v} {F : filter α} {s : set α}
{f : α → β} {g : α → β} :
eventually_eq (F ⊓ principal s) f g ↔ filter.eventually (fun (x : α) => x ∈ s → f x = g x) F :=
eventually_inf_principal
/-- A function `f` is eventually less than or equal to a function `g` at a filter `l`. -/
def eventually_le {α : Type u} {β : Type v} [HasLessEq β] (l : filter α) (f : α → β) (g : α → β) :=
filter.eventually (fun (x : α) => f x ≤ g x) l
theorem eventually_le.congr {α : Type u} {β : Type v} [HasLessEq β] {l : filter α} {f : α → β}
{f' : α → β} {g : α → β} {g' : α → β} (H : eventually_le l f g) (hf : eventually_eq l f f')
(hg : eventually_eq l g g') : eventually_le l f' g' :=
sorry
theorem eventually_le_congr {α : Type u} {β : Type v} [HasLessEq β] {l : filter α} {f : α → β}
{f' : α → β} {g : α → β} {g' : α → β} (hf : eventually_eq l f f') (hg : eventually_eq l g g') :
eventually_le l f g ↔ eventually_le l f' g' :=
{ mp := fun (H : eventually_le l f g) => eventually_le.congr H hf hg,
mpr :=
fun (H : eventually_le l f' g') =>
eventually_le.congr H (eventually_eq.symm hf) (eventually_eq.symm hg) }
theorem eventually_eq.le {α : Type u} {β : Type v} [preorder β] {l : filter α} {f : α → β}
{g : α → β} (h : eventually_eq l f g) : eventually_le l f g :=
eventually.mono h fun (x : α) => le_of_eq
theorem eventually_le.refl {α : Type u} {β : Type v} [preorder β] (l : filter α) (f : α → β) :
eventually_le l f f :=
eventually_eq.le eventually_eq.rfl
theorem eventually_le.rfl {α : Type u} {β : Type v} [preorder β] {l : filter α} {f : α → β} :
eventually_le l f f :=
eventually_le.refl l f
theorem eventually_le.trans {α : Type u} {β : Type v} [preorder β] {l : filter α} {f : α → β}
{g : α → β} {h : α → β} (H₁ : eventually_le l f g) (H₂ : eventually_le l g h) :
eventually_le l f h :=
eventually.mp H₂ (eventually.mono H₁ fun (x : α) => le_trans)
theorem eventually_eq.trans_le {α : Type u} {β : Type v} [preorder β] {l : filter α} {f : α → β}
{g : α → β} {h : α → β} (H₁ : eventually_eq l f g) (H₂ : eventually_le l g h) :
eventually_le l f h :=
eventually_le.trans (eventually_eq.le H₁) H₂
theorem eventually_le.trans_eq {α : Type u} {β : Type v} [preorder β] {l : filter α} {f : α → β}
{g : α → β} {h : α → β} (H₁ : eventually_le l f g) (H₂ : eventually_eq l g h) :
eventually_le l f h :=
eventually_le.trans H₁ (eventually_eq.le H₂)
theorem eventually_le.antisymm {α : Type u} {β : Type v} [partial_order β] {l : filter α}
{f : α → β} {g : α → β} (h₁ : eventually_le l f g) (h₂ : eventually_le l g f) :
eventually_eq l f g :=
eventually.mp h₂ (eventually.mono h₁ fun (x : α) => le_antisymm)
theorem eventually_le_antisymm_iff {α : Type u} {β : Type v} [partial_order β] {l : filter α}
{f : α → β} {g : α → β} : eventually_eq l f g ↔ eventually_le l f g ∧ eventually_le l g f :=
sorry
theorem eventually_le.le_iff_eq {α : Type u} {β : Type v} [partial_order β] {l : filter α}
{f : α → β} {g : α → β} (h : eventually_le l f g) : eventually_le l g f ↔ eventually_eq l g f :=
{ mp := fun (h' : eventually_le l g f) => eventually_le.antisymm h' h, mpr := eventually_eq.le }
theorem eventually_le.inter {α : Type u} {s : set α} {t : set α} {s' : set α} {t' : set α}
{l : filter α} (h : eventually_le l s t) (h' : eventually_le l s' t') :
eventually_le l (s ∩ s') (t ∩ t') :=
eventually.mp h' (eventually.mono h fun (x : α) => and.imp)
theorem eventually_le.union {α : Type u} {s : set α} {t : set α} {s' : set α} {t' : set α}
{l : filter α} (h : eventually_le l s t) (h' : eventually_le l s' t') :
eventually_le l (s ∪ s') (t ∪ t') :=
eventually.mp h' (eventually.mono h fun (x : α) => or.imp)
theorem eventually_le.compl {α : Type u} {s : set α} {t : set α} {l : filter α}
(h : eventually_le l s t) : eventually_le l (tᶜ) (sᶜ) :=
eventually.mono h fun (x : α) => mt
theorem eventually_le.diff {α : Type u} {s : set α} {t : set α} {s' : set α} {t' : set α}
{l : filter α} (h : eventually_le l s t) (h' : eventually_le l t' s') :
eventually_le l (s \ s') (t \ t') :=
eventually_le.inter h (eventually_le.compl h')
theorem join_le {α : Type u} {f : filter (filter α)} {l : filter α}
(h : filter.eventually (fun (m : filter α) => m ≤ l) f) : join f ≤ l :=
fun (s : set α) (hs : s ∈ l) => eventually.mono h fun (m : filter α) (hm : m ≤ l) => hm hs
/-! ### Push-forwards, pull-backs, and the monad structure -/
/-- The forward map of a filter -/
def map {α : Type u} {β : Type v} (m : α → β) (f : filter α) : filter β :=
mk (set.preimage m ⁻¹' sets f) univ_mem_sets sorry sorry
@[simp] theorem map_principal {α : Type u} {β : Type v} {s : set α} {f : α → β} :
map f (principal s) = principal (f '' s) :=
filter_eq (set.ext fun (a : set β) => iff.symm set.image_subset_iff)
@[simp] theorem eventually_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} {P : β → Prop} :
filter.eventually (fun (b : β) => P b) (map m f) ↔
filter.eventually (fun (a : α) => P (m a)) f :=
iff.rfl
@[simp] theorem frequently_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} {P : β → Prop} :
filter.frequently (fun (b : β) => P b) (map m f) ↔
filter.frequently (fun (a : α) => P (m a)) f :=
iff.rfl
@[simp] theorem mem_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} {t : set β} :
t ∈ map m f ↔ (set_of fun (x : α) => m x ∈ t) ∈ f :=
iff.rfl
theorem image_mem_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} {s : set α}
(hs : s ∈ f) : m '' s ∈ map m f :=
sets_of_superset f hs (set.subset_preimage_image m s)
theorem image_mem_map_iff {α : Type u} {β : Type v} {f : filter α} {m : α → β} {s : set α}
(hf : function.injective m) : m '' s ∈ map m f ↔ s ∈ f :=
{ mp :=
fun (h : m '' s ∈ map m f) =>
eq.mpr (id (Eq._oldrec (Eq.refl (s ∈ f)) (Eq.symm (set.preimage_image_eq s hf)))) h,
mpr := image_mem_map }
theorem range_mem_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} :
set.range m ∈ map m f :=
eq.mpr (id (Eq._oldrec (Eq.refl (set.range m ∈ map m f)) (Eq.symm set.image_univ)))
(image_mem_map univ_mem_sets)
theorem mem_map_sets_iff {α : Type u} {β : Type v} {f : filter α} {m : α → β} {t : set β} :
t ∈ map m f ↔ ∃ (s : set α), ∃ (H : s ∈ f), m '' s ⊆ t :=
sorry
@[simp] theorem map_id {α : Type u} {f : filter α} : map id f = f := filter_eq rfl
@[simp] theorem map_id' {α : Type u} {f : filter α} : map (fun (x : α) => x) f = f := map_id
@[simp] theorem map_compose {α : Type u} {β : Type v} {γ : Type w} {m : α → β} {m' : β → γ} :
map m' ∘ map m = map (m' ∘ m) :=
funext fun (_x : filter α) => filter_eq rfl
@[simp] theorem map_map {α : Type u} {β : Type v} {γ : Type w} {f : filter α} {m : α → β}
{m' : β → γ} : map m' (map m f) = map (m' ∘ m) f :=
congr_fun map_compose f
/-- If functions `m₁` and `m₂` are eventually equal at a filter `f`, then
they map this filter to the same filter. -/
theorem map_congr {α : Type u} {β : Type v} {m₁ : α → β} {m₂ : α → β} {f : filter α}
(h : eventually_eq f m₁ m₂) : map m₁ f = map m₂ f :=
sorry
/-- The inverse map of a filter -/
def comap {α : Type u} {β : Type v} (m : α → β) (f : filter β) : filter α :=
mk (set_of fun (s : set α) => ∃ (t : set β), ∃ (H : t ∈ f), m ⁻¹' t ⊆ s) sorry sorry sorry
@[simp] theorem eventually_comap {α : Type u} {β : Type v} {f : filter β} {φ : α → β}
{P : α → Prop} :
filter.eventually (fun (a : α) => P a) (comap φ f) ↔
filter.eventually (fun (b : β) => ∀ (a : α), φ a = b → P a) f :=
sorry
@[simp] theorem frequently_comap {α : Type u} {β : Type v} {f : filter β} {φ : α → β}
{P : α → Prop} :
filter.frequently (fun (a : α) => P a) (comap φ f) ↔
filter.frequently (fun (b : β) => ∃ (a : α), φ a = b ∧ P a) f :=
sorry
/-- The monadic bind operation on filter is defined the usual way in terms of `map` and `join`.
Unfortunately, this `bind` does not result in the expected applicative. See `filter.seq` for the
applicative instance. -/
def bind {α : Type u} {β : Type v} (f : filter α) (m : α → filter β) : filter β := join (map m f)
/-- The applicative sequentiation operation. This is not induced by the bind operation. -/
def seq {α : Type u} {β : Type v} (f : filter (α → β)) (g : filter α) : filter β :=
mk
(set_of
fun (s : set β) =>
∃ (u : set (α → β)),
∃ (H : u ∈ f),
∃ (t : set α), ∃ (H : t ∈ g), ∀ (m : α → β), m ∈ u → ∀ (x : α), x ∈ t → m x ∈ s)
sorry sorry sorry
/-- `pure x` is the set of sets that contain `x`. It is equal to `𝓟 {x}` but
with this definition we have `s ∈ pure a` defeq `a ∈ s`. -/
protected instance has_pure : Pure filter :=
{ pure := fun (α : Type u) (x : α) => mk (set_of fun (s : set α) => x ∈ s) trivial sorry sorry }
protected instance has_bind : Bind filter := { bind := bind }
protected instance has_seq : Seq filter := { seq := seq }
protected instance functor : Functor filter :=
{ map := map, mapConst := fun (α β : Type u_1) => map ∘ function.const β }
theorem pure_sets {α : Type u} (a : α) : sets (pure a) = set_of fun (s : set α) => a ∈ s := rfl
@[simp] theorem mem_pure_sets {α : Type u} {a : α} {s : set α} : s ∈ pure a ↔ a ∈ s := iff.rfl
@[simp] theorem eventually_pure {α : Type u} {a : α} {p : α → Prop} :
filter.eventually (fun (x : α) => p x) (pure a) ↔ p a :=
iff.rfl
@[simp] theorem principal_singleton {α : Type u} (a : α) : principal (singleton a) = pure a := sorry
@[simp] theorem map_pure {α : Type u} {β : Type v} (f : α → β) (a : α) :
map f (pure a) = pure (f a) :=
rfl
@[simp] theorem join_pure {α : Type u} (f : filter α) : join (pure f) = f :=
filter.ext fun (s : set α) => iff.rfl
@[simp] theorem pure_bind {α : Type u} {β : Type v} (a : α) (m : α → filter β) :
bind (pure a) m = m a :=
sorry
-- this section needs to be before applicative, otherwise the wrong instance will be chosen
/-- The monad structure on filters. -/
protected def monad : Monad filter :=
{ toApplicative :=
{ toFunctor := { map := map, mapConst := fun (α β : Type u_1) => map ∘ function.const β },
toPure := filter.has_pure,
toSeq :=
{ seq :=
fun (α β : Type u_1) (f : filter (α → β)) (x : filter α) =>
do
let _x ← f
map _x x },
toSeqLeft :=
{ seqLeft :=
fun (α β : Type u_1) (a : filter α) (b : filter β) =>
(fun (α β : Type u_1) (f : filter (α → β)) (x : filter α) =>
do
let _x ← f
map _x x)
β α (map (function.const β) a) b },
toSeqRight :=
{ seqRight :=
fun (α β : Type u_1) (a : filter α) (b : filter β) =>
(fun (α β : Type u_1) (f : filter (α → β)) (x : filter α) =>
do
let _x ← f
map _x x)
β β (map (function.const α id) a) b } },
toBind := filter.has_bind }
protected theorem is_lawful_monad : is_lawful_monad filter :=
is_lawful_monad.mk (fun (α β : Type u_1) => pure_bind)
fun (α β γ : Type u_1) (f : filter α) (m₁ : α → filter β) (m₂ : β → filter γ) => filter_eq rfl
protected instance applicative : Applicative filter :=
{ toFunctor := { map := map, mapConst := fun (α β : Type u_1) => map ∘ function.const β },
toPure := filter.has_pure, toSeq := { seq := seq },
toSeqLeft :=
{ seqLeft :=
fun (α β : Type u_1) (a : filter α) (b : filter β) => seq (map (function.const β) a) b },
toSeqRight :=
{ seqRight :=
fun (α β : Type u_1) (a : filter α) (b : filter β) =>
seq (map (function.const α id) a) b } }
protected instance alternative : alternative filter := alternative.mk fun (α : Type u_1) => ⊥
@[simp] theorem map_def {α : Type u_1} {β : Type u_1} (m : α → β) (f : filter α) :
m <$> f = map m f :=
rfl
@[simp] theorem bind_def {α : Type u_1} {β : Type u_1} (f : filter α) (m : α → filter β) :
f >>= m = bind f m :=
rfl
/- map and comap equations -/
@[simp] theorem mem_comap_sets {α : Type u} {β : Type v} {g : filter β} {m : α → β} {s : set α} :
s ∈ comap m g ↔ ∃ (t : set β), ∃ (H : t ∈ g), m ⁻¹' t ⊆ s :=
iff.rfl
theorem preimage_mem_comap {α : Type u} {β : Type v} {g : filter β} {m : α → β} {t : set β}
(ht : t ∈ g) : m ⁻¹' t ∈ comap m g :=
Exists.intro t (Exists.intro ht (set.subset.refl (m ⁻¹' t)))
theorem comap_id {α : Type u} {f : filter α} : comap id f = f := sorry
theorem comap_const_of_not_mem {α : Type u} {x : α} {f : filter α} {V : set α} (hV : V ∈ f)
(hx : ¬x ∈ V) : comap (fun (y : α) => x) f = ⊥ :=
sorry
theorem comap_const_of_mem {α : Type u} {x : α} {f : filter α} (h : ∀ (V : set α), V ∈ f → x ∈ V) :
comap (fun (y : α) => x) f = ⊤ :=
sorry
theorem comap_comap {α : Type u} {β : Type v} {γ : Type w} {f : filter α} {m : γ → β} {n : β → α} :
comap m (comap n f) = comap (n ∘ m) f :=
sorry
@[simp] theorem comap_principal {α : Type u} {β : Type v} {m : α → β} {t : set β} :
comap m (principal t) = principal (m ⁻¹' t) :=
sorry
@[simp] theorem comap_pure {α : Type u} {β : Type v} {m : α → β} {b : β} :
comap m (pure b) = principal (m ⁻¹' singleton b) :=
sorry
theorem map_le_iff_le_comap {α : Type u} {β : Type v} {f : filter α} {g : filter β} {m : α → β} :
map m f ≤ g ↔ f ≤ comap m g :=
sorry
theorem gc_map_comap {α : Type u} {β : Type v} (m : α → β) : galois_connection (map m) (comap m) :=
fun (f : filter α) (g : filter β) => map_le_iff_le_comap
theorem map_mono {α : Type u} {β : Type v} {m : α → β} : monotone (map m) :=
galois_connection.monotone_l (gc_map_comap m)
theorem comap_mono {α : Type u} {β : Type v} {m : α → β} : monotone (comap m) :=
galois_connection.monotone_u (gc_map_comap m)
@[simp] theorem map_bot {α : Type u} {β : Type v} {m : α → β} : map m ⊥ = ⊥ :=
galois_connection.l_bot (gc_map_comap m)
@[simp] theorem map_sup {α : Type u} {β : Type v} {f₁ : filter α} {f₂ : filter α} {m : α → β} :
map m (f₁ ⊔ f₂) = map m f₁ ⊔ map m f₂ :=
galois_connection.l_sup (gc_map_comap m)
@[simp] theorem map_supr {α : Type u} {β : Type v} {ι : Sort x} {m : α → β} {f : ι → filter α} :
map m (supr fun (i : ι) => f i) = supr fun (i : ι) => map m (f i) :=
galois_connection.l_supr (gc_map_comap m)
@[simp] theorem comap_top {α : Type u} {β : Type v} {m : α → β} : comap m ⊤ = ⊤ :=
galois_connection.u_top (gc_map_comap m)
@[simp] theorem comap_inf {α : Type u} {β : Type v} {g₁ : filter β} {g₂ : filter β} {m : α → β} :
comap m (g₁ ⊓ g₂) = comap m g₁ ⊓ comap m g₂ :=
galois_connection.u_inf (gc_map_comap m)
@[simp] theorem comap_infi {α : Type u} {β : Type v} {ι : Sort x} {m : α → β} {f : ι → filter β} :
comap m (infi fun (i : ι) => f i) = infi fun (i : ι) => comap m (f i) :=
galois_connection.u_infi (gc_map_comap m)
theorem le_comap_top {α : Type u} {β : Type v} (f : α → β) (l : filter α) : l ≤ comap f ⊤ :=
eq.mpr (id (Eq._oldrec (Eq.refl (l ≤ comap f ⊤)) comap_top)) le_top
theorem map_comap_le {α : Type u} {β : Type v} {g : filter β} {m : α → β} : map m (comap m g) ≤ g :=
galois_connection.l_u_le (gc_map_comap m) g
theorem le_comap_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} : f ≤ comap m (map m f) :=
galois_connection.le_u_l (gc_map_comap m) f
@[simp] theorem comap_bot {α : Type u} {β : Type v} {m : α → β} : comap m ⊥ = ⊥ := sorry
theorem comap_supr {α : Type u} {β : Type v} {ι : Sort u_1} {f : ι → filter β} {m : α → β} :
comap m (supr f) = supr fun (i : ι) => comap m (f i) :=
sorry
theorem comap_Sup {α : Type u} {β : Type v} {s : set (filter β)} {m : α → β} :
comap m (Sup s) = supr fun (f : filter β) => supr fun (H : f ∈ s) => comap m f :=
sorry
theorem comap_sup {α : Type u} {β : Type v} {g₁ : filter β} {g₂ : filter β} {m : α → β} :
comap m (g₁ ⊔ g₂) = comap m g₁ ⊔ comap m g₂ :=
sorry
theorem map_comap {α : Type u} {β : Type v} {f : filter β} {m : α → β} (hf : set.range m ∈ f) :
map m (comap m f) = f :=
sorry
theorem image_mem_sets {α : Type u} {β : Type v} {f : filter α} {c : β → α} (h : set.range c ∈ f)
{W : set β} (W_in : W ∈ comap c f) : c '' W ∈ f :=
eq.mpr (id (Eq._oldrec (Eq.refl (c '' W ∈ f)) (Eq.symm (map_comap h)))) (image_mem_map W_in)
theorem image_coe_mem_sets {α : Type u} {f : filter α} {U : set α} (h : U ∈ f) {W : set ↥U}
(W_in : W ∈ comap coe f) : coe '' W ∈ f :=
sorry
theorem comap_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} (h : function.injective m) :
comap m (map m f) = f :=
sorry
theorem mem_comap_iff {α : Type u} {β : Type v} {f : filter β} {m : α → β}
(inj : function.injective m) (large : set.range m ∈ f) {S : set α} :
S ∈ comap m f ↔ m '' S ∈ f :=
eq.mpr
(id
(Eq._oldrec (Eq.refl (S ∈ comap m f ↔ m '' S ∈ f))
(Eq.symm (propext (image_mem_map_iff inj)))))
(eq.mpr (id (Eq._oldrec (Eq.refl (m '' S ∈ map m (comap m f) ↔ m '' S ∈ f)) (map_comap large)))
(iff.refl (m '' S ∈ f)))
theorem le_of_map_le_map_inj' {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β}
{s : set α} (hsf : s ∈ f) (hsg : s ∈ g)
(hm : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → m x = m y → x = y) (h : map m f ≤ map m g) :
f ≤ g :=
sorry
theorem le_of_map_le_map_inj_iff {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β}
{s : set α} (hsf : s ∈ f) (hsg : s ∈ g)
(hm : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → m x = m y → x = y) : map m f ≤ map m g ↔ f ≤ g :=
{ mp := le_of_map_le_map_inj' hsf hsg hm, mpr := fun (h : f ≤ g) => map_mono h }
theorem eq_of_map_eq_map_inj' {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β}
{s : set α} (hsf : s ∈ f) (hsg : s ∈ g)
(hm : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → m x = m y → x = y) (h : map m f = map m g) :
f = g :=
le_antisymm (le_of_map_le_map_inj' hsf hsg hm (le_of_eq h))
(le_of_map_le_map_inj' hsg hsf hm (le_of_eq (Eq.symm h)))
theorem map_inj {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β}
(hm : ∀ (x y : α), m x = m y → x = y) (h : map m f = map m g) : f = g :=
sorry
theorem le_map_comap_of_surjective' {α : Type u} {β : Type v} {f : α → β} {l : filter β} {u : set β}
(ul : u ∈ l) (hf : ∀ (y : β), y ∈ u → ∃ (x : α), f x = y) : l ≤ map f (comap f l) :=
sorry
theorem map_comap_of_surjective' {α : Type u} {β : Type v} {f : α → β} {l : filter β} {u : set β}
(ul : u ∈ l) (hf : ∀ (y : β), y ∈ u → ∃ (x : α), f x = y) : map f (comap f l) = l :=
le_antisymm map_comap_le (le_map_comap_of_surjective' ul hf)
theorem le_map_comap_of_surjective {α : Type u} {β : Type v} {f : α → β}
(hf : function.surjective f) (l : filter β) : l ≤ map f (comap f l) :=
le_map_comap_of_surjective' univ_mem_sets fun (y : β) (_x : y ∈ set.univ) => hf y
theorem map_comap_of_surjective {α : Type u} {β : Type v} {f : α → β} (hf : function.surjective f)
(l : filter β) : map f (comap f l) = l :=
le_antisymm map_comap_le (le_map_comap_of_surjective hf l)
theorem subtype_coe_map_comap {α : Type u} (s : set α) (f : filter α) :
map coe (comap coe f) = f ⊓ principal s :=
sorry
theorem subtype_coe_map_comap_prod {α : Type u} (s : set α) (f : filter (α × α)) :
map coe (comap coe f) = f ⊓ principal (set.prod s s) :=
sorry
theorem comap_ne_bot_iff {α : Type u} {β : Type v} {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ ∀ (t : set β), t ∈ f → ∃ (a : α), m a ∈ t :=
sorry
theorem comap_ne_bot {α : Type u} {β : Type v} {f : filter β} {m : α → β}
(hm : ∀ (t : set β), t ∈ f → ∃ (a : α), m a ∈ t) : ne_bot (comap m f) :=
iff.mpr comap_ne_bot_iff hm
theorem comap_ne_bot_iff_frequently {α : Type u} {β : Type v} {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ filter.frequently (fun (y : β) => y ∈ set.range m) f :=
sorry
theorem comap_ne_bot_iff_compl_range {α : Type u} {β : Type v} {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ ¬set.range mᶜ ∈ f :=
comap_ne_bot_iff_frequently
theorem ne_bot.comap_of_range_mem {α : Type u} {β : Type v} {f : filter β} {m : α → β}
(hf : ne_bot f) (hm : set.range m ∈ f) : ne_bot (comap m f) :=
iff.mpr comap_ne_bot_iff_frequently (eventually.frequently hm)
theorem comap_inf_principal_ne_bot_of_image_mem {α : Type u} {β : Type v} {f : filter β} {m : α → β}
(hf : ne_bot f) {s : set α} (hs : m '' s ∈ f) : ne_bot (comap m f ⊓ principal s) :=
sorry
theorem ne_bot.comap_of_surj {α : Type u} {β : Type v} {f : filter β} {m : α → β} (hf : ne_bot f)
(hm : function.surjective m) : ne_bot (comap m f) :=
ne_bot.comap_of_range_mem hf (univ_mem_sets' hm)
theorem ne_bot.comap_of_image_mem {α : Type u} {β : Type v} {f : filter β} {m : α → β}
(hf : ne_bot f) {s : set α} (hs : m '' s ∈ f) : ne_bot (comap m f) :=
ne_bot.comap_of_range_mem hf (mem_sets_of_superset hs (set.image_subset_range m s))
@[simp] theorem map_eq_bot_iff {α : Type u} {β : Type v} {f : filter α} {m : α → β} :
map m f = ⊥ ↔ f = ⊥ :=
sorry
theorem map_ne_bot_iff {α : Type u} {β : Type v} (f : α → β) {F : filter α} :
ne_bot (map f F) ↔ ne_bot F :=
not_congr map_eq_bot_iff
theorem ne_bot.map {α : Type u} {β : Type v} {f : filter α} (hf : ne_bot f) (m : α → β) :
ne_bot (map m f) :=
iff.mpr (map_ne_bot_iff m) hf
protected instance map_ne_bot {α : Type u} {β : Type v} {f : filter α} {m : α → β} [hf : ne_bot f] :
ne_bot (map m f) :=
ne_bot.map hf m
theorem sInter_comap_sets {α : Type u} {β : Type v} (f : α → β) (F : filter β) :
⋂₀sets (comap f F) = set.Inter fun (U : set β) => set.Inter fun (H : U ∈ F) => f ⁻¹' U :=
sorry
-- this is a generic rule for monotone functions:
theorem map_infi_le {α : Type u} {β : Type v} {ι : Sort x} {f : ι → filter α} {m : α → β} :
map m (infi f) ≤ infi fun (i : ι) => map m (f i) :=
le_infi fun (i : ι) => map_mono (infi_le f i)
theorem map_infi_eq {α : Type u} {β : Type v} {ι : Sort x} {f : ι → filter α} {m : α → β}
(hf : directed ge f) [Nonempty ι] : map m (infi f) = infi fun (i : ι) => map m (f i) :=
sorry
theorem map_binfi_eq {α : Type u} {β : Type v} {ι : Type w} {f : ι → filter α} {m : α → β}
{p : ι → Prop} (h : directed_on (f ⁻¹'o ge) (set_of fun (x : ι) => p x)) (ne : ∃ (x : ι), p x) :
map m (infi fun (i : ι) => infi fun (h : p i) => f i) =
infi fun (i : ι) => infi fun (h : p i) => map m (f i) :=
sorry
theorem map_inf_le {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β} :
map m (f ⊓ g) ≤ map m f ⊓ map m g :=
monotone.map_inf_le map_mono f g
theorem map_inf' {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β} {t : set α}
(htf : t ∈ f) (htg : t ∈ g) (h : ∀ (x : α), x ∈ t → ∀ (y : α), y ∈ t → m x = m y → x = y) :
map m (f ⊓ g) = map m f ⊓ map m g :=
sorry
theorem map_inf {α : Type u} {β : Type v} {f : filter α} {g : filter α} {m : α → β}
(h : function.injective m) : map m (f ⊓ g) = map m f ⊓ map m g :=
map_inf' univ_mem_sets univ_mem_sets
fun (x : α) (_x : x ∈ set.univ) (y : α) (_x : y ∈ set.univ) (hxy : m x = m y) => h hxy
theorem map_eq_comap_of_inverse {α : Type u} {β : Type v} {f : filter α} {m : α → β} {n : β → α}
(h₁ : m ∘ n = id) (h₂ : n ∘ m = id) : map m f = comap n f :=
sorry
theorem map_swap_eq_comap_swap {α : Type u} {β : Type v} {f : filter (α × β)} :
prod.swap <$> f = comap prod.swap f :=
map_eq_comap_of_inverse prod.swap_swap_eq prod.swap_swap_eq
theorem le_map {α : Type u} {β : Type v} {f : filter α} {m : α → β} {g : filter β}
(h : ∀ (s : set α), s ∈ f → m '' s ∈ g) : g ≤ map m f :=
fun (s : set β) (hs : s ∈ map m f) =>
mem_sets_of_superset (h (m ⁻¹' s) hs) (set.image_preimage_subset m s)
protected theorem push_pull {α : Type u} {β : Type v} (f : α → β) (F : filter α) (G : filter β) :
map f (F ⊓ comap f G) = map f F ⊓ G :=
sorry
protected theorem push_pull' {α : Type u} {β : Type v} (f : α → β) (F : filter α) (G : filter β) :
map f (comap f G ⊓ F) = G ⊓ map f F :=
sorry
theorem singleton_mem_pure_sets {α : Type u} {a : α} : singleton a ∈ pure a := set.mem_singleton a
theorem pure_injective {α : Type u} : function.injective pure :=
fun (a b : α) (hab : pure a = pure b) =>
iff.mp (iff.mp filter.ext_iff hab (set_of fun (x : α) => a = x)) rfl
protected instance pure_ne_bot {α : Type u} {a : α} : ne_bot (pure a) :=
mt (iff.mpr empty_in_sets_eq_bot) (set.not_mem_empty a)
@[simp] theorem le_pure_iff {α : Type u} {f : filter α} {a : α} : f ≤ pure a ↔ singleton a ∈ f :=
sorry
theorem mem_seq_sets_def {α : Type u} {β : Type v} {f : filter (α → β)} {g : filter α} {s : set β} :
s ∈ seq f g ↔
∃ (u : set (α → β)),
∃ (H : u ∈ f),
∃ (t : set α), ∃ (H : t ∈ g), ∀ (x : α → β), x ∈ u → ∀ (y : α), y ∈ t → x y ∈ s :=
iff.rfl
theorem mem_seq_sets_iff {α : Type u} {β : Type v} {f : filter (α → β)} {g : filter α} {s : set β} :
s ∈ seq f g ↔
∃ (u : set (α → β)), ∃ (H : u ∈ f), ∃ (t : set α), ∃ (H : t ∈ g), set.seq u t ⊆ s :=
sorry
theorem mem_map_seq_iff {α : Type u} {β : Type v} {γ : Type w} {f : filter α} {g : filter β}
{m : α → β → γ} {s : set γ} :
s ∈ seq (map m f) g ↔
∃ (t : set β),
∃ (u : set α), t ∈ g ∧ u ∈ f ∧ ∀ (x : α), x ∈ u → ∀ (y : β), y ∈ t → m x y ∈ s :=
sorry
theorem seq_mem_seq_sets {α : Type u} {β : Type v} {f : filter (α → β)} {g : filter α}
{s : set (α → β)} {t : set α} (hs : s ∈ f) (ht : t ∈ g) : set.seq s t ∈ seq f g :=
sorry
theorem le_seq {α : Type u} {β : Type v} {f : filter (α → β)} {g : filter α} {h : filter β}
(hh : ∀ (t : set (α → β)), t ∈ f → ∀ (u : set α), u ∈ g → set.seq t u ∈ h) : h ≤ seq f g :=
sorry
theorem seq_mono {α : Type u} {β : Type v} {f₁ : filter (α → β)} {f₂ : filter (α → β)}
{g₁ : filter α} {g₂ : filter α} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : seq f₁ g₁ ≤ seq f₂ g₂ :=
le_seq
fun (s : set (α → β)) (hs : s ∈ f₂) (t : set α) (ht : t ∈ g₂) =>
seq_mem_seq_sets (hf hs) (hg ht)
@[simp] theorem pure_seq_eq_map {α : Type u} {β : Type v} (g : α → β) (f : filter α) :
seq (pure g) f = map g f :=
sorry
@[simp] theorem seq_pure {α : Type u} {β : Type v} (f : filter (α → β)) (a : α) :
seq f (pure a) = map (fun (g : α → β) => g a) f :=
sorry
@[simp] theorem seq_assoc {α : Type u} {β : Type v} {γ : Type w} (x : filter α) (g : filter (α → β))
(h : filter (β → γ)) : seq h (seq g x) = seq (seq (map function.comp h) g) x :=
sorry
theorem prod_map_seq_comm {α : Type u} {β : Type v} (f : filter α) (g : filter β) :
seq (map Prod.mk f) g = seq (map (fun (b : β) (a : α) => (a, b)) g) f :=
sorry
protected instance is_lawful_functor : is_lawful_functor filter :=
is_lawful_functor.mk (fun (α : Type u) (f : filter α) => map_id)
fun (α β γ : Type u) (f : α → β) (g : β → γ) (a : filter α) => Eq.symm map_map
protected instance is_lawful_applicative : is_lawful_applicative filter :=
is_lawful_applicative.mk (fun (α β : Type u) => pure_seq_eq_map) (fun (α β : Type u) => map_pure)
(fun (α β : Type u) => seq_pure) fun (α β γ : Type u) => seq_assoc
protected instance is_comm_applicative : is_comm_applicative filter :=
is_comm_applicative.mk fun (α β : Type u) (f : filter α) (g : filter β) => prod_map_seq_comm f g
theorem seq_eq_filter_seq {α : Type l} {β : Type l} (f : filter (α → β)) (g : filter α) :
f <*> g = seq f g :=
rfl
/- bind equations -/
@[simp] theorem eventually_bind {α : Type u} {β : Type v} {f : filter α} {m : α → filter β}
{p : β → Prop} :
filter.eventually (fun (y : β) => p y) (bind f m) ↔
filter.eventually (fun (x : α) => filter.eventually (fun (y : β) => p y) (m x)) f :=
iff.rfl
@[simp] theorem eventually_eq_bind {α : Type u} {β : Type v} {γ : Type w} {f : filter α}
{m : α → filter β} {g₁ : β → γ} {g₂ : β → γ} :
eventually_eq (bind f m) g₁ g₂ ↔
filter.eventually (fun (x : α) => eventually_eq (m x) g₁ g₂) f :=
iff.rfl
@[simp] theorem eventually_le_bind {α : Type u} {β : Type v} {γ : Type w} [HasLessEq γ]
{f : filter α} {m : α → filter β} {g₁ : β → γ} {g₂ : β → γ} :
eventually_le (bind f m) g₁ g₂ ↔
filter.eventually (fun (x : α) => eventually_le (m x) g₁ g₂) f :=
iff.rfl
theorem mem_bind_sets' {α : Type u} {β : Type v} {s : set β} {f : filter α} {m : α → filter β} :
s ∈ bind f m ↔ (set_of fun (a : α) => s ∈ m a) ∈ f :=
iff.rfl
@[simp] theorem mem_bind_sets {α : Type u} {β : Type v} {s : set β} {f : filter α}
{m : α → filter β} : s ∈ bind f m ↔ ∃ (t : set α), ∃ (H : t ∈ f), ∀ (x : α), x ∈ t → s ∈ m x :=
iff.trans (iff.trans iff.rfl (iff.symm exists_sets_subset_iff)) iff.rfl
theorem bind_le {α : Type u} {β : Type v} {f : filter α} {g : α → filter β} {l : filter β}
(h : filter.eventually (fun (x : α) => g x ≤ l) f) : bind f g ≤ l :=
join_le (iff.mpr eventually_map h)
theorem bind_mono {α : Type u} {β : Type v} {f₁ : filter α} {f₂ : filter α} {g₁ : α → filter β}
{g₂ : α → filter β} (hf : f₁ ≤ f₂) (hg : eventually_le f₁ g₁ g₂) : bind f₁ g₁ ≤ bind f₂ g₂ :=
sorry
theorem bind_inf_principal {α : Type u} {β : Type v} {f : filter α} {g : α → filter β} {s : set β} :
(bind f fun (x : α) => g x ⊓ principal s) = bind f g ⊓ principal s :=
sorry
theorem sup_bind {α : Type u} {β : Type v} {f : filter α} {g : filter α} {h : α → filter β} :
bind (f ⊔ g) h = bind f h ⊔ bind g h :=
sorry
theorem principal_bind {α : Type u} {β : Type v} {s : set α} {f : α → filter β} :
bind (principal s) f = supr fun (x : α) => supr fun (H : x ∈ s) => f x :=
sorry
/- This is a separate section in order to open `list`, but mostly because of universe
equality requirements in `traverse` -/
theorem sequence_mono {α : Type u} (as : List (filter α)) (bs : List (filter α)) :
list.forall₂ LessEq as bs → sequence as ≤ sequence bs :=
sorry
theorem mem_traverse_sets {α' : Type u} {β' : Type u} {γ' : Type u} {f : β' → filter α'}
{s : γ' → set α'} (fs : List β') (us : List γ') :
list.forall₂ (fun (b : β') (c : γ') => s c ∈ f b) fs us → traverse s us ∈ traverse f fs :=
sorry
theorem mem_traverse_sets_iff {α' : Type u} {β' : Type u} {f : β' → filter α'} (fs : List β')
(t : set (List α')) :
t ∈ traverse f fs ↔
∃ (us : List (set α')),
list.forall₂ (fun (b : β') (s : set α') => s ∈ f b) fs us ∧ sequence us ⊆ t :=
sorry
/-! ### Limits -/
/-- `tendsto` is the generic "limit of a function" predicate.
`tendsto f l₁ l₂` asserts that for every `l₂` neighborhood `a`,
the `f`-preimage of `a` is an `l₁` neighborhood. -/
def tendsto {α : Type u} {β : Type v} (f : α → β) (l₁ : filter α) (l₂ : filter β) := map f l₁ ≤ l₂
theorem tendsto_def {α : Type u} {β : Type v} {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ ∀ (s : set β), s ∈ l₂ → f ⁻¹' s ∈ l₁ :=
iff.rfl
theorem tendsto_iff_eventually {α : Type u} {β : Type v} {f : α → β} {l₁ : filter α}
{l₂ : filter β} :
tendsto f l₁ l₂ ↔
∀ {p : β → Prop},
filter.eventually (fun (y : β) => p y) l₂ →
filter.eventually (fun (x : α) => p (f x)) l₁ :=
iff.rfl
theorem tendsto.eventually {α : Type u} {β : Type v} {f : α → β} {l₁ : filter α} {l₂ : filter β}
{p : β → Prop} (hf : tendsto f l₁ l₂) (h : filter.eventually (fun (y : β) => p y) l₂) :
filter.eventually (fun (x : α) => p (f x)) l₁ :=
hf h
theorem tendsto.frequently {α : Type u} {β : Type v} {f : α → β} {l₁ : filter α} {l₂ : filter β}
{p : β → Prop} (hf : tendsto f l₁ l₂) (h : filter.frequently (fun (x : α) => p (f x)) l₁) :
filter.frequently (fun (y : β) => p y) l₂ :=
mt (tendsto.eventually hf) h
@[simp] theorem tendsto_bot {α : Type u} {β : Type v} {f : α → β} {l : filter β} : tendsto f ⊥ l :=
sorry
@[simp] theorem tendsto_top {α : Type u} {β : Type v} {f : α → β} {l : filter α} : tendsto f l ⊤ :=
le_top
theorem le_map_of_right_inverse {α : Type u} {β : Type v} {mab : α → β} {mba : β → α} {f : filter α}
{g : filter β} (h₁ : eventually_eq g (mab ∘ mba) id) (h₂ : tendsto mba g f) : g ≤ map mab f :=
eq.mpr (id (Eq._oldrec (Eq.refl (g ≤ map mab f)) (Eq.symm map_id)))
(eq.mpr (id (Eq._oldrec (Eq.refl (map id g ≤ map mab f)) (Eq.symm (map_congr h₁))))
(eq.mpr (id (Eq._oldrec (Eq.refl (map (mab ∘ mba) g ≤ map mab f)) (Eq.symm map_map)))
(map_mono h₂)))
theorem tendsto_of_not_nonempty {α : Type u} {β : Type v} {f : α → β} {la : filter α}
{lb : filter β} (h : ¬Nonempty α) : tendsto f la lb :=
sorry
theorem eventually_eq_of_left_inv_of_right_inv {α : Type u} {β : Type v} {f : α → β} {g₁ : β → α}
{g₂ : β → α} {fa : filter α} {fb : filter β}
(hleft : filter.eventually (fun (x : α) => g₁ (f x) = x) fa)
(hright : filter.eventually (fun (y : β) => f (g₂ y) = y) fb) (htendsto : tendsto g₂ fb fa) :
eventually_eq fb g₁ g₂ :=
eventually.mp (tendsto.eventually htendsto hleft)
(eventually.mono hright
fun (y : β) (hr : f (g₂ y) = y) (hl : g₁ (f (g₂ y)) = g₂ y) =>
Eq.trans (congr_arg g₁ (Eq.symm hr)) hl)
theorem tendsto_iff_comap {α : Type u} {β : Type v} {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ l₁ ≤ comap f l₂ :=
map_le_iff_le_comap
theorem tendsto.le_comap {α : Type u} {β : Type v} {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ → l₁ ≤ comap f l₂ :=
iff.mp tendsto_iff_comap
theorem tendsto_congr' {α : Type u} {β : Type v} {f₁ : α → β} {f₂ : α → β} {l₁ : filter α}
{l₂ : filter β} (hl : eventually_eq l₁ f₁ f₂) : tendsto f₁ l₁ l₂ ↔ tendsto f₂ l₁ l₂ :=
eq.mpr
(id
(Eq._oldrec (Eq.refl (tendsto f₁ l₁ l₂ ↔ tendsto f₂ l₁ l₂))
(tendsto.equations._eqn_1 f₁ l₁ l₂)))
(eq.mpr
(id
(Eq._oldrec (Eq.refl (map f₁ l₁ ≤ l₂ ↔ tendsto f₂ l₁ l₂))
(tendsto.equations._eqn_1 f₂ l₁ l₂)))
(eq.mpr (id (Eq._oldrec (Eq.refl (map f₁ l₁ ≤ l₂ ↔ map f₂ l₁ ≤ l₂)) (map_congr hl)))
(iff.refl (map f₂ l₁ ≤ l₂))))
theorem tendsto.congr' {α : Type u} {β : Type v} {f₁ : α → β} {f₂ : α → β} {l₁ : filter α}
{l₂ : filter β} (hl : eventually_eq l₁ f₁ f₂) (h : tendsto f₁ l₁ l₂) : tendsto f₂ l₁ l₂ :=
iff.mp (tendsto_congr' hl) h
theorem tendsto_congr {α : Type u} {β : Type v} {f₁ : α → β} {f₂ : α → β} {l₁ : filter α}
{l₂ : filter β} (h : ∀ (x : α), f₁ x = f₂ x) : tendsto f₁ l₁ l₂ ↔ tendsto f₂ l₁ l₂ :=
tendsto_congr' (univ_mem_sets' h)
theorem tendsto.congr {α : Type u} {β : Type v} {f₁ : α → β} {f₂ : α → β} {l₁ : filter α}
{l₂ : filter β} (h : ∀ (x : α), f₁ x = f₂ x) : tendsto f₁ l₁ l₂ → tendsto f₂ l₁ l₂ :=
iff.mp (tendsto_congr h)
theorem tendsto_id' {α : Type u} {x : filter α} {y : filter α} : x ≤ y → tendsto id x y := sorry
theorem tendsto_id {α : Type u} {x : filter α} : tendsto id x x := tendsto_id' (le_refl x)
theorem tendsto.comp {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : β → γ} {x : filter α}
{y : filter β} {z : filter γ} (hg : tendsto g y z) (hf : tendsto f x y) : tendsto (g ∘ f) x z :=
sorry
theorem tendsto.mono_left {α : Type u} {β : Type v} {f : α → β} {x : filter α} {y : filter α}
{z : filter β} (hx : tendsto f x z) (h : y ≤ x) : tendsto f y z :=
le_trans (map_mono h) hx
theorem tendsto.mono_right {α : Type u} {β : Type v} {f : α → β} {x : filter α} {y : filter β}
{z : filter β} (hy : tendsto f x y) (hz : y ≤ z) : tendsto f x z :=
le_trans hy hz
theorem tendsto.ne_bot {α : Type u} {β : Type v} {f : α → β} {x : filter α} {y : filter β}
(h : tendsto f x y) [hx : ne_bot x] : ne_bot y :=
ne_bot.mono (ne_bot.map hx f) h
theorem tendsto_map {α : Type u} {β : Type v} {f : α → β} {x : filter α} : tendsto f x (map f x) :=
le_refl (map f x)
theorem tendsto_map' {α : Type u} {β : Type v} {γ : Type w} {f : β → γ} {g : α → β} {x : filter α}
{y : filter γ} (h : tendsto (f ∘ g) x y) : tendsto f (map g x) y :=
eq.mpr
(id (Eq._oldrec (Eq.refl (tendsto f (map g x) y)) (tendsto.equations._eqn_1 f (map g x) y)))
(eq.mpr (id (Eq._oldrec (Eq.refl (map f (map g x) ≤ y)) map_map)) h)
theorem tendsto_map'_iff {α : Type u} {β : Type v} {γ : Type w} {f : β → γ} {g : α → β}
{x : filter α} {y : filter γ} : tendsto f (map g x) y ↔ tendsto (f ∘ g) x y :=
sorry
theorem tendsto_comap {α : Type u} {β : Type v} {f : α → β} {x : filter β} :
tendsto f (comap f x) x :=
map_comap_le
theorem tendsto_comap_iff {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : β → γ}
{a : filter α} {c : filter γ} : tendsto f a (comap g c) ↔ tendsto (g ∘ f) a c :=
sorry
theorem tendsto_comap'_iff {α : Type u} {β : Type v} {γ : Type w} {m : α → β} {f : filter α}
{g : filter β} {i : γ → α} (h : set.range i ∈ f) :
tendsto (m ∘ i) (comap i f) g ↔ tendsto m f g :=
sorry
theorem comap_eq_of_inverse {α : Type u} {β : Type v} {f : filter α} {g : filter β} {φ : α → β}
(ψ : β → α) (eq : ψ ∘ φ = id) (hφ : tendsto φ f g) (hψ : tendsto ψ g f) : comap φ g = f :=
sorry
theorem map_eq_of_inverse {α : Type u} {β : Type v} {f : filter α} {g : filter β} {φ : α → β}
(ψ : β → α) (eq : φ ∘ ψ = id) (hφ : tendsto φ f g) (hψ : tendsto ψ g f) : map φ f = g :=
sorry
theorem tendsto_inf {α : Type u} {β : Type v} {f : α → β} {x : filter α} {y₁ : filter β}
{y₂ : filter β} : tendsto f x (y₁ ⊓ y₂) ↔ tendsto f x y₁ ∧ tendsto f x y₂ :=
sorry
theorem tendsto_inf_left {α : Type u} {β : Type v} {f : α → β} {x₁ : filter α} {x₂ : filter α}
{y : filter β} (h : tendsto f x₁ y) : tendsto f (x₁ ⊓ x₂) y :=
le_trans (map_mono inf_le_left) h
theorem tendsto_inf_right {α : Type u} {β : Type v} {f : α → β} {x₁ : filter α} {x₂ : filter α}
{y : filter β} (h : tendsto f x₂ y) : tendsto f (x₁ ⊓ x₂) y :=
le_trans (map_mono inf_le_right) h
theorem tendsto.inf {α : Type u} {β : Type v} {f : α → β} {x₁ : filter α} {x₂ : filter α}
{y₁ : filter β} {y₂ : filter β} (h₁ : tendsto f x₁ y₁) (h₂ : tendsto f x₂ y₂) :
tendsto f (x₁ ⊓ x₂) (y₁ ⊓ y₂) :=
iff.mpr tendsto_inf { left := tendsto_inf_left h₁, right := tendsto_inf_right h₂ }
@[simp] theorem tendsto_infi {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : filter α}
{y : ι → filter β} : tendsto f x (infi fun (i : ι) => y i) ↔ ∀ (i : ι), tendsto f x (y i) :=
sorry
theorem tendsto_infi' {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : ι → filter α}
{y : filter β} (i : ι) (hi : tendsto f (x i) y) : tendsto f (infi fun (i : ι) => x i) y :=
tendsto.mono_left hi (infi_le (fun (i : ι) => x i) i)
theorem tendsto_sup {α : Type u} {β : Type v} {f : α → β} {x₁ : filter α} {x₂ : filter α}
{y : filter β} : tendsto f (x₁ ⊔ x₂) y ↔ tendsto f x₁ y ∧ tendsto f x₂ y :=
sorry
theorem tendsto.sup {α : Type u} {β : Type v} {f : α → β} {x₁ : filter α} {x₂ : filter α}
{y : filter β} : tendsto f x₁ y → tendsto f x₂ y → tendsto f (x₁ ⊔ x₂) y :=
fun (h₁ : tendsto f x₁ y) (h₂ : tendsto f x₂ y) => iff.mpr tendsto_sup { left := h₁, right := h₂ }
@[simp] theorem tendsto_principal {α : Type u} {β : Type v} {f : α → β} {l : filter α} {s : set β} :
tendsto f l (principal s) ↔ filter.eventually (fun (a : α) => f a ∈ s) l :=
sorry
@[simp] theorem tendsto_principal_principal {α : Type u} {β : Type v} {f : α → β} {s : set α}
{t : set β} : tendsto f (principal s) (principal t) ↔ ∀ (a : α), a ∈ s → f a ∈ t :=
sorry
@[simp] theorem tendsto_pure {α : Type u} {β : Type v} {f : α → β} {a : filter α} {b : β} :
tendsto f a (pure b) ↔ filter.eventually (fun (x : α) => f x = b) a :=
sorry
theorem tendsto_pure_pure {α : Type u} {β : Type v} (f : α → β) (a : α) :
tendsto f (pure a) (pure (f a)) :=
iff.mpr tendsto_pure rfl
theorem tendsto_const_pure {α : Type u} {β : Type v} {a : filter α} {b : β} :
tendsto (fun (x : α) => b) a (pure b) :=
iff.mpr tendsto_pure (univ_mem_sets' fun (_x : α) => rfl)
theorem pure_le_iff {α : Type u} {a : α} {l : filter α} :
pure a ≤ l ↔ ∀ (s : set α), s ∈ l → a ∈ s :=
iff.rfl
theorem tendsto_pure_left {α : Type u} {β : Type v} {f : α → β} {a : α} {l : filter β} :
tendsto f (pure a) l ↔ ∀ (s : set β), s ∈ l → f a ∈ s :=
iff.rfl
/-- If two filters are disjoint, then a function cannot tend to both of them along a non-trivial
filter. -/
theorem tendsto.not_tendsto {α : Type u} {β : Type v} {f : α → β} {a : filter α} {b₁ : filter β}
{b₂ : filter β} (hf : tendsto f a b₁) [ne_bot a] (hb : disjoint b₁ b₂) : ¬tendsto f a b₂ :=
fun (hf' : tendsto f a b₂) =>
tendsto.ne_bot (iff.mpr tendsto_inf { left := hf, right := hf' }) (disjoint.eq_bot hb)
theorem tendsto_if {α : Type u} {β : Type v} {l₁ : filter α} {l₂ : filter β} {f : α → β} {g : α → β}
{p : α → Prop} [decidable_pred p] (h₀ : tendsto f (l₁ ⊓ principal p) l₂)
(h₁ : tendsto g (l₁ ⊓ principal (set_of fun (x : α) => ¬p x)) l₂) :
tendsto (fun (x : α) => ite (p x) (f x) (g x)) l₁ l₂ :=
sorry
/-! ### Products of filters -/
/- The product filter cannot be defined using the monad structure on filters. For example:
F := do {x ← seq, y ← top, return (x, y)}
hence:
s ∈ F ↔ ∃n, [n..∞] × univ ⊆ s
G := do {y ← top, x ← seq, return (x, y)}
hence:
s ∈ G ↔ ∀i:ℕ, ∃n, [n..∞] × {i} ⊆ s
Now ⋃i, [i..∞] × {i} is in G but not in F.
As product filter we want to have F as result.
-/
/-- Product of filters. This is the filter generated by cartesian products
of elements of the component filters. -/
protected def prod {α : Type u} {β : Type v} (f : filter α) (g : filter β) : filter (α × β) :=
comap prod.fst f ⊓ comap prod.snd g
theorem prod_mem_prod {α : Type u} {β : Type v} {s : set α} {t : set β} {f : filter α}
{g : filter β} (hs : s ∈ f) (ht : t ∈ g) : set.prod s t ∈ filter.prod f g :=
inter_mem_inf_sets (preimage_mem_comap hs) (preimage_mem_comap ht)
theorem mem_prod_iff {α : Type u} {β : Type v} {s : set (α × β)} {f : filter α} {g : filter β} :
s ∈ filter.prod f g ↔
∃ (t₁ : set α), ∃ (H : t₁ ∈ f), ∃ (t₂ : set β), ∃ (H : t₂ ∈ g), set.prod t₁ t₂ ⊆ s :=
sorry
theorem comap_prod {α : Type u} {β : Type v} {γ : Type w} (f : α → β × γ) (b : filter β)
(c : filter γ) : comap f (filter.prod b c) = comap (prod.fst ∘ f) b ⊓ comap (prod.snd ∘ f) c :=
sorry
theorem eventually_prod_iff {α : Type u} {β : Type v} {p : α × β → Prop} {f : filter α}
{g : filter β} :
filter.eventually (fun (x : α × β) => p x) (filter.prod f g) ↔
∃ (pa : α → Prop),
∃ (ha : filter.eventually (fun (x : α) => pa x) f),
∃ (pb : β → Prop),
∃ (hb : filter.eventually (fun (y : β) => pb y) g),
∀ {x : α}, pa x → ∀ {y : β}, pb y → p (x, y) :=
sorry
theorem tendsto_fst {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
tendsto prod.fst (filter.prod f g) f :=
tendsto_inf_left tendsto_comap
theorem tendsto_snd {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
tendsto prod.snd (filter.prod f g) g :=
tendsto_inf_right tendsto_comap
theorem tendsto.prod_mk {α : Type u} {β : Type v} {γ : Type w} {f : filter α} {g : filter β}
{h : filter γ} {m₁ : α → β} {m₂ : α → γ} (h₁ : tendsto m₁ f g) (h₂ : tendsto m₂ f h) :
tendsto (fun (x : α) => (m₁ x, m₂ x)) f (filter.prod g h) :=
iff.mpr tendsto_inf
{ left := iff.mpr tendsto_comap_iff h₁, right := iff.mpr tendsto_comap_iff h₂ }
theorem eventually.prod_inl {α : Type u} {β : Type v} {la : filter α} {p : α → Prop}
(h : filter.eventually (fun (x : α) => p x) la) (lb : filter β) :
filter.eventually (fun (x : α × β) => p (prod.fst x)) (filter.prod la lb) :=
tendsto.eventually tendsto_fst h
theorem eventually.prod_inr {α : Type u} {β : Type v} {lb : filter β} {p : β → Prop}
(h : filter.eventually (fun (x : β) => p x) lb) (la : filter α) :
filter.eventually (fun (x : α × β) => p (prod.snd x)) (filter.prod la lb) :=
tendsto.eventually tendsto_snd h
theorem eventually.prod_mk {α : Type u} {β : Type v} {la : filter α} {pa : α → Prop}
(ha : filter.eventually (fun (x : α) => pa x) la) {lb : filter β} {pb : β → Prop}
(hb : filter.eventually (fun (y : β) => pb y) lb) :
filter.eventually (fun (p : α × β) => pa (prod.fst p) ∧ pb (prod.snd p)) (filter.prod la lb) :=
eventually.and (eventually.prod_inl ha lb) (eventually.prod_inr hb la)
theorem eventually.curry {α : Type u} {β : Type v} {la : filter α} {lb : filter β}
{p : α × β → Prop} (h : filter.eventually (fun (x : α × β) => p x) (filter.prod la lb)) :
filter.eventually (fun (x : α) => filter.eventually (fun (y : β) => p (x, y)) lb) la :=
sorry
theorem prod_infi_left {α : Type u} {β : Type v} {ι : Sort x} [Nonempty ι] {f : ι → filter α}
{g : filter β} :
filter.prod (infi fun (i : ι) => f i) g = infi fun (i : ι) => filter.prod (f i) g :=
sorry
theorem prod_infi_right {α : Type u} {β : Type v} {ι : Sort x} [Nonempty ι] {f : filter α}
{g : ι → filter β} :
filter.prod f (infi fun (i : ι) => g i) = infi fun (i : ι) => filter.prod f (g i) :=
sorry
theorem prod_mono {α : Type u} {β : Type v} {f₁ : filter α} {f₂ : filter α} {g₁ : filter β}
{g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : filter.prod f₁ g₁ ≤ filter.prod f₂ g₂ :=
inf_le_inf (comap_mono hf) (comap_mono hg)
theorem prod_comap_comap_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} {f₁ : filter α₁}
{f₂ : filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} :
filter.prod (comap m₁ f₁) (comap m₂ f₂) =
comap (fun (p : β₁ × β₂) => (m₁ (prod.fst p), m₂ (prod.snd p))) (filter.prod f₁ f₂) :=
sorry
theorem prod_comm' {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
filter.prod f g = comap prod.swap (filter.prod g f) :=
sorry
theorem prod_comm {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
filter.prod f g = map (fun (p : β × α) => (prod.snd p, prod.fst p)) (filter.prod g f) :=
sorry
theorem prod_map_map_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} {f₁ : filter α₁}
{f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} :
filter.prod (map m₁ f₁) (map m₂ f₂) =
map (fun (p : α₁ × α₂) => (m₁ (prod.fst p), m₂ (prod.snd p))) (filter.prod f₁ f₂) :=
sorry
theorem prod_map_map_eq' {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4}
(f : α₁ → α₂) (g : β₁ → β₂) (F : filter α₁) (G : filter β₁) :
filter.prod (map f F) (map g G) = map (prod.map f g) (filter.prod F G) :=
eq.mpr
(id
(Eq._oldrec (Eq.refl (filter.prod (map f F) (map g G) = map (prod.map f g) (filter.prod F G)))
prod_map_map_eq))
(Eq.refl (map (fun (p : α₁ × β₁) => (f (prod.fst p), g (prod.snd p))) (filter.prod F G)))
theorem tendsto.prod_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} {f : α → γ}
{g : β → δ} {a : filter α} {b : filter β} {c : filter γ} {d : filter δ} (hf : tendsto f a c)
(hg : tendsto g b d) : tendsto (prod.map f g) (filter.prod a b) (filter.prod c d) :=
sorry
theorem map_prod {α : Type u} {β : Type v} {γ : Type w} (m : α × β → γ) (f : filter α)
(g : filter β) : map m (filter.prod f g) = seq (map (fun (a : α) (b : β) => m (a, b)) f) g :=
sorry
theorem prod_eq {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
filter.prod f g = seq (map Prod.mk f) g :=
(fun (h : map id (filter.prod f g) = seq (map (fun (a : α) (b : β) => id (a, b)) f) g) =>
eq.mp
(Eq._oldrec
(Eq.refl (map id (filter.prod f g) = seq (map (fun (a : α) (b : β) => id (a, b)) f) g))
map_id)
h)
(map_prod id f g)
theorem prod_inf_prod {α : Type u} {β : Type v} {f₁ : filter α} {f₂ : filter α} {g₁ : filter β}
{g₂ : filter β} : filter.prod f₁ g₁ ⊓ filter.prod f₂ g₂ = filter.prod (f₁ ⊓ f₂) (g₁ ⊓ g₂) :=
sorry
@[simp] theorem prod_bot {α : Type u} {β : Type v} {f : filter α} : filter.prod f ⊥ = ⊥ := sorry
@[simp] theorem bot_prod {α : Type u} {β : Type v} {g : filter β} : filter.prod ⊥ g = ⊥ := sorry
@[simp] theorem prod_principal_principal {α : Type u} {β : Type v} {s : set α} {t : set β} :
filter.prod (principal s) (principal t) = principal (set.prod s t) :=
sorry
@[simp] theorem pure_prod {α : Type u} {β : Type v} {a : α} {f : filter β} :
filter.prod (pure a) f = map (Prod.mk a) f :=
sorry
@[simp] theorem prod_pure {α : Type u} {β : Type v} {f : filter α} {b : β} :
filter.prod f (pure b) = map (fun (a : α) => (a, b)) f :=
sorry
theorem prod_pure_pure {α : Type u} {β : Type v} {a : α} {b : β} :
filter.prod (pure a) (pure b) = pure (a, b) :=
sorry
theorem prod_eq_bot {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
filter.prod f g = ⊥ ↔ f = ⊥ ∨ g = ⊥ :=
sorry
theorem prod_ne_bot {α : Type u} {β : Type v} {f : filter α} {g : filter β} :
ne_bot (filter.prod f g) ↔ ne_bot f ∧ ne_bot g :=
iff.trans (not_congr prod_eq_bot) not_or_distrib
theorem ne_bot.prod {α : Type u} {β : Type v} {f : filter α} {g : filter β} (hf : ne_bot f)
(hg : ne_bot g) : ne_bot (filter.prod f g) :=
iff.mpr prod_ne_bot { left := hf, right := hg }
protected instance prod_ne_bot' {α : Type u} {β : Type v} {f : filter α} {g : filter β}
[hf : ne_bot f] [hg : ne_bot g] : ne_bot (filter.prod f g) :=
ne_bot.prod hf hg
theorem tendsto_prod_iff {α : Type u} {β : Type v} {γ : Type w} {f : α × β → γ} {x : filter α}
{y : filter β} {z : filter γ} :
tendsto f (filter.prod x y) z ↔
∀ (W : set γ) (H : W ∈ z),
∃ (U : set α),
∃ (H : U ∈ x),
∃ (V : set β), ∃ (H : V ∈ y), ∀ (x : α) (y : β), x ∈ U → y ∈ V → f (x, y) ∈ W :=
sorry
end filter
theorem set.eq_on.eventually_eq {α : Type u_1} {β : Type u_2} {s : set α} {f : α → β} {g : α → β}
(h : set.eq_on f g s) : filter.eventually_eq (filter.principal s) f g :=
h
theorem set.eq_on.eventually_eq_of_mem {α : Type u_1} {β : Type u_2} {s : set α} {l : filter α}
{f : α → β} {g : α → β} (h : set.eq_on f g s) (hl : s ∈ l) : filter.eventually_eq l f g :=
filter.eventually_eq.filter_mono (set.eq_on.eventually_eq h) (iff.mpr filter.le_principal_iff hl)
theorem set.subset.eventually_le {α : Type u_1} {l : filter α} {s : set α} {t : set α} (h : s ⊆ t) :
filter.eventually_le l s t :=
filter.eventually_of_forall h
end Mathlib |
fc8d4dc45f69d9c76d76ea7abfba69c809b83ef2 | b7f22e51856f4989b970961f794f1c435f9b8f78 | /tests/lean/run/blast_cc2.lean | c9d018ffc53f8be4e191b8684f435f18061d2059 | [
"Apache-2.0"
] | permissive | soonhokong/lean | cb8aa01055ffe2af0fb99a16b4cda8463b882cd1 | 38607e3eb57f57f77c0ac114ad169e9e4262e24f | refs/heads/master | 1,611,187,284,081 | 1,450,766,737,000 | 1,476,122,547,000 | 11,513,992 | 2 | 0 | null | 1,401,763,102,000 | 1,374,182,235,000 | C++ | UTF-8 | Lean | false | false | 572 | lean | set_option blast.strategy "cc"
example (a b c d : nat) : a == b → b = c → c == d → a == d :=
by blast
example (a b c d : nat) : a = b → b = c → c == d → a == d :=
by blast
example (a b c d : nat) : a = b → b == c → c == d → a == d :=
by blast
example (a b c d : nat) : a == b → b == c → c = d → a == d :=
by blast
example (a b c d : nat) : a == b → b = c → c = d → a == d :=
by blast
example (a b c d : nat) : a = b → b = c → c = d → a == d :=
by blast
example (a b c d : nat) : a = b → b == c → c = d → a == d :=
by blast
|
018354c60a9d8faedd529caa8ed4f1bf1839992f | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/thunk.lean | 3a460e8253d7b103cb7840b5ad1d0d068644aa2c | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 525 | lean | #eval (Thunk.pure 1).get
#eval (Thunk.mk fun _ => 2).get
#eval
let t1 := Thunk.mk fun _ => dbg_trace 4; 5
let t2 := Thunk.mk fun _ => dbg_trace 3; 0
let v2 := t2.get
let v1 := t1.get
v1 + v2
#eval
let t1 := Thunk.pure 8 |>.map fun n => dbg_trace 7; n
let t2 := Thunk.mk fun _ => dbg_trace 6; 0
let v2 := t2.get
let v1 := t1.get
v1 + v2
#eval
let t1 := Thunk.pure 11 |>.bind fun n => dbg_trace 10; Thunk.pure n
let t2 := Thunk.mk fun _ => dbg_trace 9; 0
let v2 := t2.get
let v1 := t1.get
v1 + v2
|
ca92a5301f53957c57cda0df69ec3c20d991da5b | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/matchErrorLocation.lean | df15599fdc8e24dc63e3189e850ed3b9141ec5ef | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 133 | lean | --
def head {α} (xs : List α) (h : xs = [] → False) : α :=
match (generalizing := false) he:xs with
| [] => h he
| x::_ => x
|
d77e01476fa888b816535a909b77b711ae5853a8 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/category_theory/natural_transformation.lean | 7422dcd92a3e6d9f03c79f274d216ae859a09ad6 | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 2,969 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
Defines natural transformations between functors.
Introduces notations
`τ.app X` for the components of natural transformations,
`F ⟶ G` for the type of natural transformations between functors `F` and `G`,
`σ ≫ τ` for vertical compositions, and
`σ ◫ τ` for horizontal compositions.
-/
import category_theory.functor
namespace category_theory
universes v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄ -- declare the `v`'s first; see `category_theory.category` for an explanation
variables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]
/--
`nat_trans F G` represents a natural transformation between functors `F` and `G`.
The field `app` provides the components of the natural transformation.
Naturality is expressed by `α.naturality_lemma`.
-/
@[ext]
structure nat_trans (F G : C ⥤ D) : Type (max u₁ v₂) :=
(app : Π X : C, (F.obj X) ⟶ (G.obj X))
(naturality' : ∀ {{X Y : C}} (f : X ⟶ Y), (F.map f) ≫ (app Y) = (app X) ≫ (G.map f) . obviously)
restate_axiom nat_trans.naturality'
-- Rather arbitrarily, we say that the 'simpler' form is
-- components of natural transfomations moving earlier.
attribute [simp, reassoc] nat_trans.naturality
lemma congr_app {F G : C ⥤ D} {α β : nat_trans F G} (h : α = β) (X : C) : α.app X = β.app X :=
congr_fun (congr_arg nat_trans.app h) X
namespace nat_trans
/-- `nat_trans.id F` is the identity natural transformation on a functor `F`. -/
protected def id (F : C ⥤ D) : nat_trans F F :=
{ app := λ X, 𝟙 (F.obj X) }
@[simp] lemma id_app' (F : C ⥤ D) (X : C) : (nat_trans.id F).app X = 𝟙 (F.obj X) := rfl
instance (F : C ⥤ D) : inhabited (nat_trans F F) := ⟨nat_trans.id F⟩
open category
open category_theory.functor
section
variables {F G H I : C ⥤ D}
/-- `vcomp α β` is the vertical compositions of natural transformations. -/
def vcomp (α : nat_trans F G) (β : nat_trans G H) : nat_trans F H :=
{ app := λ X, (α.app X) ≫ (β.app X) }
-- functor_category will rewrite (vcomp α β) to (α ≫ β), so this is not a
-- suitable simp lemma. We will declare the variant vcomp_app' there.
lemma vcomp_app (α : nat_trans F G) (β : nat_trans G H) (X : C) :
(vcomp α β).app X = (α.app X) ≫ (β.app X) := rfl
end
/--
The diagram
F(f) F(g) F(h)
F X ----> F Y ----> F U ----> F U
| | | |
| α(X) | α(Y) | α(U) | α(V)
v v v v
G X ----> G Y ----> G U ----> G V
G(f) G(g) G(h)
commutes.
-/
example {F G : C ⥤ D} (α : nat_trans F G) {X Y U V : C} (f : X ⟶ Y) (g : Y ⟶ U) (h : U ⟶ V) :
α.app X ≫ G.map f ≫ G.map g ≫ G.map h =
F.map f ≫ F.map g ≫ F.map h ≫ α.app V :=
by simp
end nat_trans
end category_theory
|
ac72c594076a9b7bf88ad2ce74b1f34bdc78609e | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/analysis/calculus/darboux.lean | 39b859ab57c4a2341d2c1acebd4f7e0cef1bf2fd | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 4,805 | lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import analysis.calculus.mean_value
/-!
# Darboux's theorem
In this file we prove that the derivative of a differentiable function on an interval takes all
intermediate values. The proof is based on the
[Wikipedia](https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)) page about this theorem.
-/
open filter set
open_locale topological_space classical
variables {a b : ℝ} {f f' : ℝ → ℝ}
/-- Darboux's theorem: if `a ≤ b` and `f' a < m < f' b`, then `f' c = m` for some `c ∈ [a, b]`. -/
theorem exists_has_deriv_within_at_eq_of_gt_of_lt
(hab : a ≤ b) (hf : ∀ x ∈ (Icc a b), has_deriv_within_at f (f' x) (Icc a b) x)
{m : ℝ} (hma : f' a < m) (hmb : m < f' b) :
m ∈ f' '' (Icc a b) :=
begin
have hab' : a < b,
{ refine lt_of_le_of_ne hab (λ hab', _),
subst b,
exact lt_asymm hma hmb },
set g : ℝ → ℝ := λ x, f x - m * x,
have hg : ∀ x ∈ Icc a b, has_deriv_within_at g (f' x - m) (Icc a b) x,
{ intros x hx,
simpa using (hf x hx).sub ((has_deriv_within_at_id x _).const_mul m) },
obtain ⟨c, cmem, hc⟩ : ∃ c ∈ Icc a b, is_min_on g (Icc a b) c,
from is_compact_Icc.exists_forall_le (nonempty_Icc.2 $ hab)
(λ x hx, (hg x hx).continuous_within_at),
have cmem' : c ∈ Ioo a b,
{ cases eq_or_lt_of_le cmem.1 with hac hac,
-- Show that `c` can't be equal to `a`
{ subst c,
refine absurd (sub_nonneg.1 $ nonneg_of_mul_nonneg_left _ (sub_pos.2 hab'))
(not_le_of_lt hma),
have : b - a ∈ pos_tangent_cone_at (Icc a b) a,
from mem_pos_tangent_cone_at_of_segment_subset (segment_eq_Icc hab ▸ subset.refl _),
simpa [-sub_nonneg, -continuous_linear_map.map_sub]
using hc.localize.has_fderiv_within_at_nonneg (hg a (left_mem_Icc.2 hab)) this },
cases eq_or_lt_of_le cmem.2 with hbc hbc,
-- Show that `c` can't be equal to `b`
{ subst c,
refine absurd (sub_nonpos.1 $ nonpos_of_mul_nonneg_right _ (sub_lt_zero.2 hab'))
(not_le_of_lt hmb),
have : a - b ∈ pos_tangent_cone_at (Icc a b) b,
from mem_pos_tangent_cone_at_of_segment_subset (by rw [segment_symm, segment_eq_Icc hab]),
simpa [-sub_nonneg, -continuous_linear_map.map_sub]
using hc.localize.has_fderiv_within_at_nonneg (hg b (right_mem_Icc.2 hab)) this },
exact ⟨hac, hbc⟩ },
use [c, cmem],
rw [← sub_eq_zero],
have : Icc a b ∈ 𝓝 c, by rwa [← mem_interior_iff_mem_nhds, interior_Icc],
exact (hc.is_local_min this).has_deriv_at_eq_zero ((hg c cmem).has_deriv_at this)
end
/-- Darboux's theorem: if `a ≤ b` and `f' a > m > f' b`, then `f' c = m` for some `c ∈ [a, b]`. -/
theorem exists_has_deriv_within_at_eq_of_lt_of_gt
(hab : a ≤ b) (hf : ∀ x ∈ (Icc a b), has_deriv_within_at f (f' x) (Icc a b) x)
{m : ℝ} (hma : m < f' a) (hmb : f' b < m) :
m ∈ f' '' (Icc a b) :=
let ⟨c, cmem, hc⟩ := exists_has_deriv_within_at_eq_of_gt_of_lt hab (λ x hx, (hf x hx).neg)
(neg_lt_neg hma) (neg_lt_neg hmb)
in ⟨c, cmem, neg_injective hc⟩
/-- Darboux's theorem: the image of a convex set under `f'` is a convex set. -/
theorem convex_image_has_deriv_at {s : set ℝ} (hs : convex s)
(hf : ∀ x ∈ s, has_deriv_at f (f' x) x) :
convex (f' '' s) :=
begin
refine real.convex_iff_ord_connected.2 ⟨_⟩,
rintros _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ m ⟨hma, hmb⟩,
cases eq_or_lt_of_le hma with hma hma,
by exact hma ▸ mem_image_of_mem f' ha,
cases eq_or_lt_of_le hmb with hmb hmb,
by exact hmb.symm ▸ mem_image_of_mem f' hb,
cases le_total a b with hab hab,
{ have : Icc a b ⊆ s, from hs.ord_connected.out ha hb,
rcases exists_has_deriv_within_at_eq_of_gt_of_lt hab
(λ x hx, (hf x $ this hx).has_deriv_within_at) hma hmb
with ⟨c, cmem, hc⟩,
exact ⟨c, this cmem, hc⟩ },
{ have : Icc b a ⊆ s, from hs.ord_connected.out hb ha,
rcases exists_has_deriv_within_at_eq_of_lt_of_gt hab
(λ x hx, (hf x $ this hx).has_deriv_within_at) hmb hma
with ⟨c, cmem, hc⟩,
exact ⟨c, this cmem, hc⟩ }
end
/-- If the derivative of a function is never equal to `m`, then either
it is always greater than `m`, or it is always less than `m`. -/
theorem deriv_forall_lt_or_forall_gt_of_forall_ne {s : set ℝ} (hs : convex s)
(hf : ∀ x ∈ s, has_deriv_at f (f' x) x) {m : ℝ} (hf' : ∀ x ∈ s, f' x ≠ m) :
(∀ x ∈ s, f' x < m) ∨ (∀ x ∈ s, m < f' x) :=
begin
contrapose! hf',
rcases hf' with ⟨⟨b, hb, hmb⟩, ⟨a, ha, hma⟩⟩,
exact (convex_image_has_deriv_at hs hf).ord_connected.out (mem_image_of_mem f' ha)
(mem_image_of_mem f' hb) ⟨hma, hmb⟩
end
|
1e3de6af9fd6772eef2b5cd10930fd938b341343 | 22e97a5d648fc451e25a06c668dc03ac7ed7bc25 | /src/algebra/field.lean | 1c70c6da74753771a44804b7dccac85297a98d59 | [
"Apache-2.0"
] | permissive | keeferrowan/mathlib | f2818da875dbc7780830d09bd4c526b0764a4e50 | aad2dfc40e8e6a7e258287a7c1580318e865817e | refs/heads/master | 1,661,736,426,952 | 1,590,438,032,000 | 1,590,438,032,000 | 266,892,663 | 0 | 0 | Apache-2.0 | 1,590,445,835,000 | 1,590,445,835,000 | null | UTF-8 | Lean | false | false | 15,859 | lean | /-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import algebra.ring
import algebra.group_with_zero
open set
set_option default_priority 100 -- see Note [default priority]
set_option old_structure_cmd true
universe u
variables {α : Type u}
@[ancestor ring has_inv zero_ne_one_class]
class division_ring (α : Type u) extends ring α, has_inv α, zero_ne_one_class α :=
(mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1)
(inv_mul_cancel : ∀ {a : α}, a ≠ 0 → a⁻¹ * a = 1)
(inv_zero : (0 : α)⁻¹ = 0)
section division_ring
variables [division_ring α] {a b : α}
protected definition algebra.div (a b : α) : α :=
a * b⁻¹
instance division_ring_has_div : has_div α :=
⟨algebra.div⟩
lemma division_def (a b : α) : a / b = a * b⁻¹ :=
rfl
@[simp] lemma mul_inv_cancel (h : a ≠ 0) : a * a⁻¹ = 1 :=
division_ring.mul_inv_cancel h
@[simp] lemma inv_mul_cancel (h : a ≠ 0) : a⁻¹ * a = 1 :=
division_ring.inv_mul_cancel h
@[simp] lemma one_div_eq_inv (a : α) : 1 / a = a⁻¹ := one_mul a⁻¹
@[field_simps] lemma inv_eq_one_div (a : α) : a⁻¹ = 1 / a := by simp
/-- Every division ring is a `group_with_zero`. -/
@[priority 10] -- see Note [lower instance priority]
instance division_ring.to_group_with_zero :
group_with_zero α :=
{ mul_inv_cancel := λ _, mul_inv_cancel,
.. ‹division_ring α›,
.. (by apply_instance : semiring α) }
local attribute [simp]
division_def mul_comm mul_assoc
mul_left_comm mul_inv_cancel inv_mul_cancel
lemma div_eq_mul_one_div (a b : α) : a / b = a * (1 / b) := by simp
lemma mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 := by simp [h]
lemma one_div_mul_cancel (h : a ≠ 0) : (1 / a) * a = 1 := by simp [h]
@[simp] lemma div_self (h : a ≠ 0) : a / a = 1 := by simp [h]
lemma one_div_one : 1 / 1 = (1:α) :=
div_self (ne.symm zero_ne_one)
theorem inv_one : (1⁻¹ : α) = 1 := by rw [inv_eq_one_div, one_div_one]
lemma mul_div_assoc (a b c : α) : (a * b) / c = a * (b / c) := by simp
@[field_simps] lemma mul_div_assoc' (a b c : α) : a * (b / c) = (a * b) / c :=
by simp [mul_div_assoc]
lemma one_div_ne_zero (h : a ≠ 0) : 1 / a ≠ 0 :=
assume : 1 / a = 0,
have 0 = (1:α), from eq.symm (by rw [← mul_one_div_cancel h, this, mul_zero]),
absurd this zero_ne_one
lemma ne_zero_of_one_div_ne_zero (h : 1 / a ≠ 0) : a ≠ 0 :=
assume ha : a = 0, begin rw [ha, div_zero] at h, contradiction end
lemma inv_ne_zero (h : a ≠ 0) : a⁻¹ ≠ 0 :=
by rw inv_eq_one_div; exact one_div_ne_zero h
lemma eq_zero_of_one_div_eq_zero (h : 1 / a = 0) : a = 0 :=
classical.by_cases
(assume ha, ha)
(assume ha, false.elim ((one_div_ne_zero ha) h))
lemma one_inv_eq : 1⁻¹ = (1:α) :=
calc 1⁻¹ = 1 * 1⁻¹ : by rw [one_mul]
... = (1:α) : by simp
local attribute [simp] one_inv_eq
lemma div_one (a : α) : a / 1 = a := by simp
lemma zero_div (a : α) : 0 / a = 0 := by simp
-- note: integral domain has a "mul_ne_zero". a commutative division ring is an integral
-- domain, but let's not define that class for now.
lemma division_ring.mul_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 :=
assume : a * b = 0,
have a * 1 = 0, by rw [← mul_one_div_cancel hb, ← mul_assoc, this, zero_mul],
have a = 0, by rwa mul_one at this,
absurd this ha
lemma mul_ne_zero_comm (h : a * b ≠ 0) : b * a ≠ 0 :=
have h₁ : a ≠ 0, from ne_zero_of_mul_ne_zero_right h,
have h₂ : b ≠ 0, from ne_zero_of_mul_ne_zero_left h,
division_ring.mul_ne_zero h₂ h₁
lemma eq_one_div_of_mul_eq_one (h : a * b = 1) : b = 1 / a :=
have a ≠ 0, from
assume : a = 0,
have 0 = (1:α), by rwa [this, zero_mul] at h,
absurd this zero_ne_one,
have b = (1 / a) * a * b, by rw [one_div_mul_cancel this, one_mul],
show b = 1 / a, by rwa [mul_assoc, h, mul_one] at this
lemma eq_one_div_of_mul_eq_one_left (h : b * a = 1) : b = 1 / a :=
have a ≠ 0, from
assume : a = 0,
have 0 = (1:α), by rwa [this, mul_zero] at h,
absurd this zero_ne_one,
by rw [← h, mul_div_assoc, div_self this, mul_one]
lemma division_ring.one_div_mul_one_div : (1 / a) * (1 / b) = 1 / (b * a) :=
match classical.em (a = 0), classical.em (b = 0) with
| or.inr ha, or.inr hb :=
have (b * a) * ((1 / a) * (1 / b)) = 1,
by rw [mul_assoc, ← mul_assoc a, mul_one_div_cancel ha, one_mul, mul_one_div_cancel hb],
eq_one_div_of_mul_eq_one this
| or.inl ha, _ := by simp [ha]
| _ , or.inl hb := by simp [hb]
end
lemma one_div_neg_one_eq_neg_one : (1:α) / (-1) = -1 :=
have (-1) * (-1) = (1:α), by rw [neg_mul_neg, one_mul],
eq.symm (eq_one_div_of_mul_eq_one this)
lemma one_div_neg_eq_neg_one_div (a : α) : 1 / (- a) = - (1 / a) :=
calc
1 / (- a) = 1 / ((-1) * a) : by rw neg_eq_neg_one_mul
... = (1 / a) * (1 / (- 1)) : by rw division_ring.one_div_mul_one_div
... = (1 / a) * (-1) : by rw one_div_neg_one_eq_neg_one
... = - (1 / a) : by rw [mul_neg_eq_neg_mul_symm, mul_one]
lemma div_neg_eq_neg_div (a b : α) : b / (- a) = - (b / a) :=
calc
b / (- a) = b * (1 / (- a)) : by rw [← inv_eq_one_div, division_def]
... = b * -(1 / a) : by rw one_div_neg_eq_neg_one_div
... = -(b * (1 / a)) : by rw neg_mul_eq_mul_neg
... = - (b * a⁻¹) : by rw inv_eq_one_div
lemma neg_div (a b : α) : (-b) / a = - (b / a) :=
by rw [neg_eq_neg_one_mul, mul_div_assoc, ← neg_eq_neg_one_mul]
@[field_simps] lemma neg_div' {α : Type*} [division_ring α] (a b : α) : - (b / a) = (-b) / a :=
by simp [neg_div]
lemma neg_div_neg_eq (a b : α) : (-a) / (-b) = a / b :=
by rw [div_neg_eq_neg_div, neg_div, neg_neg]
lemma one_div_one_div (a : α) : 1 / (1 / a) = a :=
match classical.em (a = 0) with
| or.inl h := by simp [h]
| or.inr h := eq.symm (eq_one_div_of_mul_eq_one_left (mul_one_div_cancel h))
end
lemma inv_inv' (a : α) : a⁻¹⁻¹ = a :=
by rw [inv_eq_one_div, inv_eq_one_div, one_div_one_div]
lemma eq_of_one_div_eq_one_div (h : 1 / a = 1 / b) : a = b :=
by rw [← one_div_one_div a, h,one_div_one_div]
lemma mul_inv' (a b : α) : (b * a)⁻¹ = a⁻¹ * b⁻¹ :=
eq.symm $ calc
a⁻¹ * b⁻¹ = (1 / a) * (1 / b) : by simp
... = (1 / (b * a)) : division_ring.one_div_mul_one_div
... = (b * a)⁻¹ : by simp
lemma one_div_div (a b : α) : 1 / (a / b) = b / a :=
by rw [one_div_eq_inv, division_def, mul_inv',
inv_inv', division_def]
lemma div_helper (b : α) (h : a ≠ 0) : (1 / (a * b)) * a = 1 / b :=
by simp only [division_def, mul_inv', one_mul, mul_assoc, inv_mul_cancel h, mul_one]
lemma mul_div_cancel (a : α) {b : α} (hb : b ≠ 0) : a * b / b = a :=
by simp [hb]
lemma div_mul_cancel (a : α) {b : α} (hb : b ≠ 0) : a / b * b = a :=
by simp [hb]
@[field_simps] lemma div_div_eq_mul_div (a b c : α) : a / (b / c) = (a * c) / b :=
by rw [div_eq_mul_one_div, one_div_div, ← mul_div_assoc]
lemma div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
by simp only [division_def, mul_inv', ← mul_assoc, mul_inv_cancel hb]
lemma mul_div_mul_right (a : α) (b : α) {c : α} (hc : c ≠ 0) :
(a * c) / (b * c) = a / b :=
by rw [mul_div_assoc, div_mul_left hc, ← mul_div_assoc, mul_one]
@[field_simps] lemma div_add_div_same (a b c : α) : a / c + b / c = (a + b) / c :=
eq.symm $ right_distrib a b (c⁻¹)
lemma div_sub_div_same (a b c : α) : (a / c) - (b / c) = (a - b) / c :=
by rw [sub_eq_add_neg, ← neg_div, div_add_div_same, sub_eq_add_neg]
lemma neg_inv : - a⁻¹ = (- a)⁻¹ :=
by rw [inv_eq_one_div, inv_eq_one_div, div_neg_eq_neg_div]
lemma add_div (a b c : α) : (a + b) / c = a / c + b / c :=
(div_add_div_same _ _ _).symm
lemma sub_div (a b c : α) : (a - b) / c = a / c - b / c :=
(div_sub_div_same _ _ _).symm
lemma division_ring.inv_inj : a⁻¹ = b⁻¹ ↔ a = b :=
inv_inj'' _ _
lemma division_ring.inv_eq_iff : a⁻¹ = b ↔ b⁻¹ = a :=
inv_eq_iff
lemma div_neg (a : α) : a / -b = -(a / b) :=
by rw [← div_neg_eq_neg_div]
lemma inv_neg : (-a)⁻¹ = -(a⁻¹) :=
by rw neg_inv
lemma one_div_mul_add_mul_one_div_eq_one_div_add_one_div (ha : a ≠ 0) (hb : b ≠ 0) :
(1 / a) * (a + b) * (1 / b) = 1 / a + 1 / b :=
by rw [(left_distrib (1 / a)), (one_div_mul_cancel ha), right_distrib, one_mul,
mul_assoc, (mul_one_div_cancel hb), mul_one, add_comm]
lemma one_div_mul_sub_mul_one_div_eq_one_div_add_one_div (ha : a ≠ 0) (hb : b ≠ 0) :
(1 / a) * (b - a) * (1 / b) = 1 / a - 1 / b :=
by rw [(mul_sub_left_distrib (1 / a)), (one_div_mul_cancel ha), mul_sub_right_distrib,
one_mul, mul_assoc, (mul_one_div_cancel hb), mul_one]
lemma div_eq_one_iff_eq (a : α) {b : α} (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
iff.intro
(assume : a / b = 1, calc
a = a / b * b : by simp [hb]
... = 1 * b : by rw this
... = b : by simp)
(assume : a = b, by simp [this, hb])
lemma eq_of_div_eq_one (a : α) {b : α} (Hb : b ≠ 0) : a / b = 1 → a = b :=
iff.mp $ div_eq_one_iff_eq a Hb
lemma eq_div_iff_mul_eq (a b : α) {c : α} (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
iff.intro
(assume : a = b / c, by rw [this, (div_mul_cancel _ hc)])
(assume : a * c = b, by rw [← this, mul_div_cancel _ hc])
lemma eq_div_of_mul_eq (a b : α) {c : α} (hc : c ≠ 0) : a * c = b → a = b / c :=
iff.mpr $ eq_div_iff_mul_eq a b hc
lemma mul_eq_of_eq_div (a b: α) {c : α} (hc : c ≠ 0) : a = b / c → a * c = b :=
iff.mp $ eq_div_iff_mul_eq a b hc
lemma add_div_eq_mul_add_div (a b : α) {c : α} (hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
have (a + b / c) * c = a * c + b, by rw [right_distrib, (div_mul_cancel _ hc)],
(iff.mpr (eq_div_iff_mul_eq _ _ hc)) this
lemma mul_mul_div (a : α) {c : α} (hc : c ≠ 0) : a = a * c * (1 / c) :=
by simp [hc]
lemma eq_of_mul_eq_mul_of_nonzero_left {a b c : α} (h : a ≠ 0) (h₂ : a * b = a * c) : b = c :=
by rw [← one_mul b, ← inv_mul_cancel h, mul_assoc, h₂, ← mul_assoc, inv_mul_cancel h, one_mul]
lemma eq_of_mul_eq_mul_of_nonzero_right {a b c : α} (h : c ≠ 0) (h2 : a * c = b * c) : a = b :=
by rw [← mul_one a, ← mul_inv_cancel h, ← mul_assoc, h2, mul_assoc, mul_inv_cancel h, mul_one]
instance division_ring.to_domain : domain α :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := λ a b h,
classical.by_contradiction $ λ hn,
division_ring.mul_ne_zero (mt or.inl hn) (mt or.inr hn) h
..‹division_ring α›, ..(by apply_instance : semiring α) }
end division_ring
@[ancestor division_ring comm_ring]
class field (α : Type u) extends comm_ring α, has_inv α, zero_ne_one_class α :=
(mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1)
(inv_zero : (0 : α)⁻¹ = 0)
section field
variable [field α]
instance field.to_division_ring : division_ring α :=
{ inv_mul_cancel := λ _ h, by rw [mul_comm, field.mul_inv_cancel h]
..show field α, by apply_instance }
lemma one_div_mul_one_div (a b : α) : (1 / a) * (1 / b) = 1 / (a * b) :=
by rw [division_ring.one_div_mul_one_div, mul_comm b]
lemma div_mul_right {a : α} (b : α) (ha : a ≠ 0) : a / (a * b) = 1 / b :=
by rw [mul_comm, div_mul_left ha]
lemma mul_div_cancel_left {a : α} (b : α) (ha : a ≠ 0) : a * b / a = b :=
by rw [mul_comm a, (mul_div_cancel _ ha)]
lemma mul_div_cancel' (a : α) {b : α} (hb : b ≠ 0) : b * (a / b) = a :=
by rw [mul_comm, (div_mul_cancel _ hb)]
lemma one_div_add_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) :=
by rw [add_comm, ← div_mul_left ha, ← div_mul_right _ hb,
division_def, division_def, division_def, ← right_distrib, mul_comm a]
local attribute [simp] mul_assoc mul_comm mul_left_comm
lemma div_mul_div (a b c d : α) :
(a / b) * (c / d) = (a * c) / (b * d) :=
begin simp [division_def], rw [mul_inv', mul_comm d⁻¹] end
lemma mul_div_mul_left (a b : α) {c : α} (hc : c ≠ 0) :
(c * a) / (c * b) = a / b :=
by rw [← div_mul_div, div_self hc, one_mul]
@[field_simps] lemma div_mul_eq_mul_div (a b c : α) : (b / c) * a = (b * a) / c :=
by simp [division_def]
lemma div_mul_eq_mul_div_comm (a b c : α) :
(b / c) * a = b * (a / c) :=
by rw [div_mul_eq_mul_div, ← one_mul c, ← div_mul_div,
div_one, one_mul]
lemma div_add_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0) (hd : d ≠ 0) :
(a / b) + (c / d) = ((a * d) + (b * c)) / (b * d) :=
by rw [← mul_div_mul_right _ b hd, ← mul_div_mul_left c d hb, div_add_div_same]
@[field_simps] lemma div_sub_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0) (hd : d ≠ 0) :
(a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) :=
begin
simp [sub_eq_add_neg],
rw [neg_eq_neg_one_mul, ← mul_div_assoc, div_add_div _ _ hb hd,
← mul_assoc, mul_comm b, mul_assoc, ← neg_eq_neg_one_mul]
end
lemma mul_eq_mul_of_div_eq_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0)
(hd : d ≠ 0) (h : a / b = c / d) : a * d = c * b :=
by rw [← mul_one (a*d), mul_assoc, mul_comm d, ← mul_assoc, ← div_self hb,
← div_mul_eq_mul_div_comm, h, div_mul_eq_mul_div, div_mul_cancel _ hd]
@[field_simps] lemma div_div_eq_div_mul (a b c : α) : (a / b) / c = a / (b * c) :=
by rw [div_eq_mul_one_div, div_mul_div, mul_one]
lemma div_div_div_div_eq (a : α) {b c d : α} :
(a / b) / (c / d) = (a * d) / (b * c) :=
by rw [div_div_eq_mul_div, div_mul_eq_mul_div,
div_div_eq_div_mul]
lemma div_mul_eq_div_mul_one_div (a b c : α) :
a / (b * c) = (a / b) * (1 / c) :=
by rw [← div_div_eq_div_mul, ← div_eq_mul_one_div]
lemma inv_add_inv {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ + b⁻¹ = (a + b) / (a * b) :=
by rw [inv_eq_one_div, inv_eq_one_div, one_div_add_one_div ha hb]
lemma inv_sub_inv {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ - b⁻¹ = (b - a) / (a * b) :=
by rw [inv_eq_one_div, inv_eq_one_div, div_sub_div _ _ ha hb, one_mul, mul_one]
@[field_simps] lemma add_div' (a b c : α) (hc : c ≠ 0) : b + a / c = (b * c + a) / c :=
by simpa using div_add_div b a one_ne_zero hc
@[field_simps] lemma sub_div' (a b c : α) (hc : c ≠ 0) : b - a / c = (b * c - a) / c :=
by simpa using div_sub_div b a one_ne_zero hc
@[field_simps] lemma div_add' (a b c : α) (hc : c ≠ 0) : a / c + b = (a + b * c) / c :=
by rwa [add_comm, add_div', add_comm]
@[field_simps] lemma div_sub' (a b c : α) (hc : c ≠ 0) : a / c - b = (a - c * b) / c :=
by simpa using div_sub_div a b hc one_ne_zero
/-- Every field is a `comm_group_with_zero`. -/
instance field.to_comm_group_with_zero :
comm_group_with_zero α :=
{ .. (_ : group_with_zero α), .. ‹field α› }
@[priority 100] -- see Note [lower instance priority]
instance field.to_integral_domain : integral_domain α :=
{ ..‹field α›, ..division_ring.to_domain }
end field
namespace ring_hom
section
variables {β : Type*} [division_ring α] [division_ring β] (f : α →+* β) {x y : α}
lemma map_ne_zero : f x ≠ 0 ↔ x ≠ 0 :=
⟨mt $ λ h, h.symm ▸ f.map_zero,
λ x0 h, one_ne_zero $ by rw [← f.map_one, ← mul_inv_cancel x0, f.map_mul, h, zero_mul]⟩
lemma map_eq_zero : f x = 0 ↔ x = 0 :=
by haveI := classical.dec; exact not_iff_not.1 f.map_ne_zero
lemma map_inv : f x⁻¹ = (f x)⁻¹ :=
begin
classical, by_cases h : x = 0, by simp [h],
apply (domain.mul_right_inj (f.map_ne_zero.2 h)).1,
rw [mul_inv_cancel (f.map_ne_zero.2 h), ← f.map_mul, mul_inv_cancel h, f.map_one]
end
lemma map_div : f (x / y) = f x / f y :=
(f.map_mul _ _).trans $ congr_arg _ $ f.map_inv
lemma injective : function.injective f :=
f.injective_iff.2
(λ a ha, classical.by_contradiction $ λ ha0,
by simpa [ha, f.map_mul, f.map_one, zero_ne_one]
using congr_arg f (mul_inv_cancel ha0))
end
end ring_hom
|
6692b6b3e9d794ab6ad8898f39a0d84f18cc0719 | 5ae26df177f810c5006841e9c73dc56e01b978d7 | /src/data/set/disjointed.lean | 6b427e8a54f2ca7505ef67a28f079d4d07d82973 | [
"Apache-2.0"
] | permissive | ChrisHughes24/mathlib | 98322577c460bc6b1fe5c21f42ce33ad1c3e5558 | a2a867e827c2a6702beb9efc2b9282bd801d5f9a | refs/heads/master | 1,583,848,251,477 | 1,565,164,247,000 | 1,565,164,247,000 | 129,409,993 | 0 | 1 | Apache-2.0 | 1,565,164,817,000 | 1,523,628,059,000 | Lean | UTF-8 | Lean | false | false | 3,780 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Disjointed sets
-/
import data.set.lattice data.nat.basic tactic.wlog
open set classical lattice
local attribute [instance] prop_decidable
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
{s t u : set α}
/-- A relation `p` holds pairwise if `p i j` for all `i ≠ j`. -/
def pairwise {α : Type*} (p : α → α → Prop) := ∀i j, i ≠ j → p i j
theorem pairwise_on_bool {r} (hr : symmetric r) {a b : α} :
pairwise (r on (λ c, cond c a b)) ↔ r a b :=
by simp [pairwise, bool.forall_bool, function.on_fun];
exact and_iff_right_of_imp (@hr _ _)
theorem pairwise_disjoint_on_bool [semilattice_inf_bot α] {a b : α} :
pairwise (disjoint on (λ c, cond c a b)) ↔ disjoint a b :=
pairwise_on_bool $ λ _ _, disjoint.symm
namespace set
/-- If `f : ℕ → set α` is a sequence of sets, then `disjointed f` is
the sequence formed with each set subtracted from the later ones
in the sequence, to form a disjoint sequence. -/
def disjointed (f : ℕ → set α) (n : ℕ) : set α := f n ∩ (⋂i<n, - f i)
lemma disjoint_disjointed {f : ℕ → set α} : pairwise (disjoint on disjointed f) :=
λ i j h, begin
wlog h' : i ≤ j; [skip, {revert a, exact (this h.symm).symm}],
rintro a ⟨⟨h₁, _⟩, h₂, h₃⟩, simp at h₃,
exact h₃ _ (lt_of_le_of_ne h' h) h₁
end
lemma disjoint_disjointed' {f : ℕ → set α} :
∀ i j, i ≠ j → (disjointed f i) ∩ (disjointed f j) = ∅ :=
begin
assume i j hij, have := @disjoint_disjointed _ f i j hij,
rw [function.on_fun, disjoint_iff] at this, exact this
end
lemma disjointed_subset {f : ℕ → set α} {n : ℕ} : disjointed f n ⊆ f n := inter_subset_left _ _
lemma Union_lt_succ {f : ℕ → set α} {n} : (⋃i < nat.succ n, f i) = f n ∪ (⋃i < n, f i) :=
ext $ λ a, by simp [nat.lt_succ_iff_lt_or_eq, or_and_distrib_right, exists_or_distrib, or_comm]
lemma Inter_lt_succ {f : ℕ → set α} {n} : (⋂i < nat.succ n, f i) = f n ∩ (⋂i < n, f i) :=
ext $ λ a, by simp [nat.lt_succ_iff_lt_or_eq, or_imp_distrib, forall_and_distrib, and_comm]
lemma Union_disjointed {f : ℕ → set α} : (⋃n, disjointed f n) = (⋃n, f n) :=
subset.antisymm (Union_subset_Union $ assume i, inter_subset_left _ _) $
assume x h,
have ∃n, x ∈ f n, from mem_Union.mp h,
have hn : ∀ (i : ℕ), i < nat.find this → x ∉ f i,
from assume i, nat.find_min this,
mem_Union.mpr ⟨nat.find this, nat.find_spec this, by simp; assumption⟩
lemma disjointed_induct {f : ℕ → set α} {n : ℕ} {p : set α → Prop}
(h₁ : p (f n)) (h₂ : ∀t i, p t → p (t \ f i)) :
p (disjointed f n) :=
begin
rw disjointed,
generalize_hyp : f n = t at h₁ ⊢,
induction n,
case nat.zero { simp [nat.not_lt_zero, h₁] },
case nat.succ : n ih {
rw [Inter_lt_succ, inter_comm (-f n), ← inter_assoc],
exact h₂ _ n ih }
end
lemma disjointed_of_mono {f : ℕ → set α} {n : ℕ} (hf : monotone f) :
disjointed f (n + 1) = f (n + 1) \ f n :=
have (⋂i (h : i < n + 1), -f i) = - f n,
from le_antisymm
(infi_le_of_le n $ infi_le_of_le (nat.lt_succ_self _) $ subset.refl _)
(le_infi $ assume i, le_infi $ assume hi, neg_le_neg $ hf $ nat.le_of_succ_le_succ hi),
by simp [disjointed, this, diff_eq]
lemma Union_disjointed_of_mono {f : ℕ → set α} (hf : monotone f) :
∀ n : ℕ, (⋃i<n.succ, disjointed f i) = f n
| 0 := by rw [Union_lt_succ]; simp [disjointed, nat.not_lt_zero]
| (n+1) := by rw [Union_lt_succ,
disjointed_of_mono hf, Union_disjointed_of_mono n,
diff_union_self, union_eq_self_of_subset_right (hf (nat.le_succ _))]
end set
|
553ab8e5cd3c109d0e2b209baf0fc45392e8ad9a | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/attribute_bug1.lean | a85e401654b8efc35ce02252e874225e096d0594 | [
"Apache-2.0"
] | permissive | soonhokong/lean-osx | 4a954262c780e404c1369d6c06516161d07fcb40 | 3670278342d2f4faa49d95b46d86642d7875b47c | refs/heads/master | 1,611,410,334,552 | 1,474,425,686,000 | 1,474,425,686,000 | 12,043,103 | 5 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 476 | lean | open tactic
constant f : nat → nat
constant fdef : ∀ n, f n = n + 1
example (n : nat) : f n = n + 1 :=
by simp -- Failed as expected, since fdef is not marked as [simp]
local attribute [simp] fdef
example (n : nat) : f n = n + 1 :=
by simp -- Succeeded as expected
local attribute [-simp] fdef
print fdef -- we don't get the [simp] attribute when printing fdef
example (n : nat) : f n = n + 1 :=
by simp -- Failed as expected, since we removed [simp] attribute
|
d7a9f96b2c3ee131215068c0523e24970de63dda | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/Lean3Lib/init/data/array/basic_auto.lean | e5a7d663ec9df7f90ef825ff286a366cbe7e1ea4 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 10,633 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.data.nat.default
import Mathlib.Lean3Lib.init.data.bool.default
import Mathlib.Lean3Lib.init.ite_simp
universes u l u_1 w v
namespace Mathlib
/-- In the VM, d_array is implemented as a persistent array. -/
structure d_array (n : ℕ) (α : fin n → Type u) where
data : (i : fin n) → α i
namespace d_array
/-- The empty array. -/
def nil {α : fin 0 → Type u_1} : d_array 0 α := mk fun (_x : fin 0) => sorry
/-- `read a i` reads the `i`th member of `a`. Has builtin VM implementation. -/
def read {n : ℕ} {α : fin n → Type u} (a : d_array n α) (i : fin n) : α i := data a i
/-- `write a i v` sets the `i`th member of `a` to be `v`. Has builtin VM implementation. -/
def write {n : ℕ} {α : fin n → Type u} (a : d_array n α) (i : fin n) (v : α i) : d_array n α :=
mk fun (j : fin n) => dite (i = j) (fun (h : i = j) => eq.rec_on h v) fun (h : ¬i = j) => read a j
def iterate_aux {n : ℕ} {α : fin n → Type u} {β : Type w} (a : d_array n α)
(f : (i : fin n) → α i → β → β) (i : ℕ) : i ≤ n → β → β :=
sorry
/-- Fold over the elements of the given array in ascending order. Has builtin VM implementation. -/
def iterate {n : ℕ} {α : fin n → Type u} {β : Type w} (a : d_array n α) (b : β)
(f : (i : fin n) → α i → β → β) : β :=
iterate_aux a f n sorry b
/-- Map the array. Has builtin VM implementation. -/
def foreach {n : ℕ} {α : fin n → Type u} {α' : fin n → Type v} (a : d_array n α)
(f : (i : fin n) → α i → α' i) : d_array n α' :=
mk fun (i : fin n) => f i (read a i)
def map {n : ℕ} {α : fin n → Type u} {α' : fin n → Type v} (f : (i : fin n) → α i → α' i)
(a : d_array n α) : d_array n α' :=
foreach a f
def map₂ {n : ℕ} {α : fin n → Type u} {α' : fin n → Type v} {α'' : fin n → Type w}
(f : (i : fin n) → α i → α' i → α'' i) (a : d_array n α) (b : d_array n α') : d_array n α'' :=
foreach b fun (i : fin n) => f i (read a i)
def foldl {n : ℕ} {α : fin n → Type u} {β : Type w} (a : d_array n α) (b : β)
(f : (i : fin n) → α i → β → β) : β :=
iterate a b f
def rev_iterate_aux {n : ℕ} {α : fin n → Type u} {β : Type w} (a : d_array n α)
(f : (i : fin n) → α i → β → β) (i : ℕ) : i ≤ n → β → β :=
sorry
def rev_iterate {n : ℕ} {α : fin n → Type u} {β : Type w} (a : d_array n α) (b : β)
(f : (i : fin n) → α i → β → β) : β :=
rev_iterate_aux a f n sorry b
@[simp] theorem read_write {n : ℕ} {α : fin n → Type u} (a : d_array n α) (i : fin n) (v : α i) :
read (write a i v) i = v :=
sorry
@[simp] theorem read_write_of_ne {n : ℕ} {α : fin n → Type u} (a : d_array n α) {i : fin n}
{j : fin n} (v : α i) : i ≠ j → read (write a i v) j = read a j :=
sorry
protected theorem ext {n : ℕ} {α : fin n → Type u} {a : d_array n α} {b : d_array n α}
(h : ∀ (i : fin n), read a i = read b i) : a = b :=
sorry
protected theorem ext' {n : ℕ} {α : fin n → Type u} {a : d_array n α} {b : d_array n α}
(h :
∀ (i : ℕ) (h : i < n),
read a { val := i, property := h } = read b { val := i, property := h }) :
a = b :=
sorry
protected def beq_aux {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)]
(a : d_array n α) (b : d_array n α) (i : ℕ) : i ≤ n → Bool :=
sorry
/-- Boolean element-wise equality check. -/
protected def beq {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)] (a : d_array n α)
(b : d_array n α) : Bool :=
d_array.beq_aux a b n sorry
theorem of_beq_aux_eq_tt {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)]
{a : d_array n α} {b : d_array n α} (i : ℕ) (h : i ≤ n) :
d_array.beq_aux a b i h = tt →
∀ (j : ℕ) (h' : j < i),
read a { val := j, property := lt_of_lt_of_le h' h } =
read b { val := j, property := lt_of_lt_of_le h' h } :=
sorry
theorem of_beq_eq_tt {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)]
{a : d_array n α} {b : d_array n α} : d_array.beq a b = tt → a = b :=
sorry
theorem of_beq_aux_eq_ff {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)]
{a : d_array n α} {b : d_array n α} (i : ℕ) (h : i ≤ n) :
d_array.beq_aux a b i h = false →
∃ (j : ℕ),
∃ (h' : j < i),
read a { val := j, property := lt_of_lt_of_le h' h } ≠
read b { val := j, property := lt_of_lt_of_le h' h } :=
sorry
theorem of_beq_eq_ff {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)]
{a : d_array n α} {b : d_array n α} : d_array.beq a b = false → a ≠ b :=
sorry
protected instance decidable_eq {n : ℕ} {α : fin n → Type u} [(i : fin n) → DecidableEq (α i)] :
DecidableEq (d_array n α) :=
fun (a b : d_array n α) =>
dite (d_array.beq a b = tt) (fun (h : d_array.beq a b = tt) => is_true sorry)
fun (h : ¬d_array.beq a b = tt) => isFalse sorry
end d_array
/-- A non-dependent array (see `d_array`). Implemented in the VM as a persistent array. -/
def array (n : ℕ) (α : Type u) := d_array n fun (_x : fin n) => α
/-- `mk_array n v` creates a new array of length `n` where each element is `v`. Has builtin VM implementation. -/
def mk_array {α : Type u_1} (n : ℕ) (v : α) : array n α := d_array.mk fun (_x : fin n) => v
namespace array
def nil {α : Type u_1} : array 0 α := d_array.nil
def read {n : ℕ} {α : Type u} (a : array n α) (i : fin n) : α := d_array.read a i
def write {n : ℕ} {α : Type u} (a : array n α) (i : fin n) (v : α) : array n α :=
d_array.write a i v
/-- Fold array starting from 0, folder function includes an index argument. -/
def iterate {n : ℕ} {α : Type u} {β : Type v} (a : array n α) (b : β) (f : fin n → α → β → β) : β :=
d_array.iterate a b f
/-- Map each element of the given array with an index argument. -/
def foreach {n : ℕ} {α : Type u} {β : Type v} (a : array n α) (f : fin n → α → β) : array n β :=
d_array.foreach a f
def map₂ {n : ℕ} {α : Type u} (f : α → α → α) (a : array n α) (b : array n α) : array n α :=
foreach b fun (i : fin n) => f (read a i)
def foldl {n : ℕ} {α : Type u} {β : Type v} (a : array n α) (b : β) (f : α → β → β) : β :=
iterate a b fun (_x : fin n) => f
def rev_list {n : ℕ} {α : Type u} (a : array n α) : List α :=
foldl a [] fun (_x : α) (_y : List α) => _x :: _y
def rev_iterate {n : ℕ} {α : Type u} {β : Type v} (a : array n α) (b : β) (f : fin n → α → β → β) :
β :=
d_array.rev_iterate a b f
def rev_foldl {n : ℕ} {α : Type u} {β : Type v} (a : array n α) (b : β) (f : α → β → β) : β :=
rev_iterate a b fun (_x : fin n) => f
def to_list {n : ℕ} {α : Type u} (a : array n α) : List α :=
rev_foldl a [] fun (_x : α) (_y : List α) => _x :: _y
theorem push_back_idx {j : ℕ} {n : ℕ} (h₁ : j < n + 1) (h₂ : j ≠ n) : j < n :=
nat.lt_of_le_and_ne (nat.le_of_lt_succ h₁) h₂
/-- `push_back a v` pushes value `v` to the end of the array. Has builtin VM implementation. -/
def push_back {n : ℕ} {α : Type u} (a : array n α) (v : α) : array (n + 1) α :=
d_array.mk fun (_x : fin (n + 1)) => sorry
theorem pop_back_idx {j : ℕ} {n : ℕ} (h : j < n) : j < n + 1 := nat.lt.step h
/-- Discard _last_ element in the array. Has builtin VM implementation. -/
def pop_back {n : ℕ} {α : Type u} (a : array (n + 1) α) : array n α :=
d_array.mk fun (_x : fin n) => sorry
/-- Auxilliary function for monadically mapping a function over an array. -/
def mmap_core {n : ℕ} {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (a : array n α)
(f : α → m β) (i : ℕ) (H : i ≤ n) : m (array i β) :=
sorry
/-- Monadically map a function over the array. -/
def mmap {n : ℕ} {α : Type u} {β : Type v} {m : Type v → Type u_1} [Monad m] (a : array n α)
(f : α → m β) : m (array n β) :=
mmap_core a f n sorry
/-- Map a function over the array. -/
def map {n : ℕ} {α : Type u} {β : Type v} (a : array n α) (f : α → β) : array n β :=
d_array.map (fun (_x : fin n) => f) a
protected def mem {n : ℕ} {α : Type u} (v : α) (a : array n α) := ∃ (i : fin n), read a i = v
protected instance has_mem {n : ℕ} {α : Type u} : has_mem α (array n α) := has_mem.mk array.mem
theorem read_mem {n : ℕ} {α : Type u} (a : array n α) (i : fin n) : read a i ∈ a :=
exists.intro i rfl
protected instance has_repr {n : ℕ} {α : Type u} [has_repr α] : has_repr (array n α) :=
has_repr.mk (repr ∘ to_list)
@[simp] theorem read_write {n : ℕ} {α : Type u} (a : array n α) (i : fin n) (v : α) :
read (write a i v) i = v :=
d_array.read_write a i v
@[simp] theorem read_write_of_ne {n : ℕ} {α : Type u} (a : array n α) {i : fin n} {j : fin n}
(v : α) : i ≠ j → read (write a i v) j = read a j :=
d_array.read_write_of_ne a v
def read' {n : ℕ} {β : Type v} [Inhabited β] (a : array n β) (i : ℕ) : β :=
dite (i < n) (fun (h : i < n) => read a { val := i, property := h })
fun (h : ¬i < n) => Inhabited.default
def write' {n : ℕ} {α : Type u} (a : array n α) (i : ℕ) (v : α) : array n α :=
dite (i < n) (fun (h : i < n) => write a { val := i, property := h } v) fun (h : ¬i < n) => a
theorem read_eq_read' {n : ℕ} {α : Type u} [Inhabited α] (a : array n α) {i : ℕ} (h : i < n) :
read a { val := i, property := h } = read' a i :=
sorry
theorem write_eq_write' {n : ℕ} {α : Type u} (a : array n α) {i : ℕ} (h : i < n) (v : α) :
write a { val := i, property := h } v = write' a i v :=
sorry
protected theorem ext {n : ℕ} {α : Type u} {a : array n α} {b : array n α}
(h : ∀ (i : fin n), read a i = read b i) : a = b :=
d_array.ext h
protected theorem ext' {n : ℕ} {α : Type u} {a : array n α} {b : array n α}
(h :
∀ (i : ℕ) (h : i < n),
read a { val := i, property := h } = read b { val := i, property := h }) :
a = b :=
d_array.ext' h
protected instance decidable_eq {n : ℕ} {α : Type u} [DecidableEq α] : DecidableEq (array n α) :=
eq.mpr sorry fun (a b : d_array n fun (_x : fin n) => α) => d_array.decidable_eq a b
end Mathlib |
acfb515db4c41694c7097f17e9c37774e90f45a9 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/subst_tac1.lean | c0bad7e89782d36aeb8fa3282cfa9bc06524ba9a | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 372 | lean | constant p : nat → Prop
open tactic
set_option pp.all true
definition ex (a b c : nat) (H : p c) : a = b → p a → p b :=
by do intro `H1, intro `H2,
get_local `a >>= subst,
trace_state,
assumption
#print ex
example (a b c : nat) (H : p c) : a = b → p a → p b :=
by do intros,
get_local `b >>= subst,
trace_state,
assumption
|
2c5febe3071faa3046861b51bc774072196fb27c | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Lean/Util/FindLevelMVar.lean | f6ce8d040427dd07eddaaaf74db0a3e84703affb | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 1,566 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Selsam
-/
import Lean.Expr
namespace Lean
namespace FindLevelMVar
abbrev Visitor := Option LMVarId → Option LMVarId
mutual
partial def visit (p : LMVarId → Bool) (e : Expr) : Visitor := fun s =>
if s.isSome || !e.hasLevelMVar then s else main p e s
partial def main (p : LMVarId → Bool) : Expr → Visitor
| .sort l => visitLevel p l
| .const _ ls => ls.foldr (init := id) fun l acc => visitLevel p l ∘ acc
| .forallE _ d b _ => visit p b ∘ visit p d
| .lam _ d b _ => visit p b ∘ visit p d
| .letE _ t v b _ => visit p b ∘ visit p v ∘ visit p t
| .app f a => visit p a ∘ visit p f
| .mdata _ b => visit p b
| .proj _ _ e => visit p e
| _ => id
partial def visitLevel (p : LMVarId → Bool) (l : Level) : Visitor := fun s =>
if s.isSome || !l.hasMVar then s else mainLevel p l s
partial def mainLevel (p : LMVarId → Bool) : Level → Visitor
| .zero => id
| .succ l => visitLevel p l
| .max l₁ l₂ => visitLevel p l₁ ∘ visitLevel p l₂
| .imax l₁ l₂ => visitLevel p l₁ ∘ visitLevel p l₂
| .param _ => id
| .mvar mvarId => fun s => if p mvarId then some mvarId else s
end
end FindLevelMVar
@[inline] def Expr.findLevelMVar? (e : Expr) (p : LMVarId → Bool) : Option LMVarId :=
FindLevelMVar.main p e none
end Lean
|
8f9c8a238316daffa9ce722a0eeb4e538264219e | 4727251e0cd73359b15b664c3170e5d754078599 | /src/geometry/manifold/tangent_bundle.lean | db791abf647c96de735f4ac39ffb8f39ca7bd1cc | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 31,370 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import topology.vector_bundle
import geometry.manifold.smooth_manifold_with_corners
import data.set.prod
/-!
# Basic smooth bundles
In general, a smooth bundle is a bundle over a smooth manifold, whose fiber is a manifold, and
for which the coordinate changes are smooth. In this definition, there are charts involved at
several places: in the manifold structure of the base, in the manifold structure of the fibers, and
in the local trivializations. This makes it a complicated object in general. There is however a
specific situation where things are much simpler: when the fiber is a vector space (no need for
charts for the fibers), and when the local trivializations of the bundle and the charts of the base
coincide. Then everything is expressed in terms of the charts of the base, making for a much
simpler overall structure, which is easier to manipulate formally.
Most vector bundles that naturally occur in differential geometry are of this form:
the tangent bundle, the cotangent bundle, differential forms (used to define de Rham cohomology)
and the bundle of Riemannian metrics. Therefore, it is worth defining a specific constructor for
this kind of bundle, that we call basic smooth bundles.
A basic smooth bundle is thus a smooth bundle over a smooth manifold whose fiber is a vector space,
and which is trivial in the coordinate charts of the base. (We recall that in our notion of manifold
there is a distinguished atlas, which does not need to be maximal: we require the triviality above
this specific atlas). It can be constructed from a basic smooth bundled core, defined below,
specifying the changes in the fiber when one goes from one coordinate chart to another one.
## Main definitions
* `basic_smooth_vector_bundle_core I M F`: assuming that `M` is a smooth manifold over the model
with corners `I` on `(𝕜, E, H)`, and `F` is a normed vector space over `𝕜`, this structure
registers, for each pair of charts of `M`, a linear change of coordinates on `F` depending
smoothly on the base point. This is the core structure from which one will build a smooth vector
bundle with fiber `F` over `M`.
Let `Z` be a basic smooth bundle core over `M` with fiber `F`. We define
`Z.to_topological_vector_bundle_core`, the (topological) vector bundle core associated to `Z`. From
it, we get a space `Z.to_topological_vector_bundle_core.total_space` (which as a Type is just
`Σ (x : M), F`), with the fiber bundle topology. It inherits a manifold structure (where the
charts are in bijection with the charts of the basis). We show that this manifold is smooth.
Then we use this machinery to construct the tangent bundle of a smooth manifold.
* `tangent_bundle_core I M`: the basic smooth bundle core associated to a smooth manifold `M` over
a model with corners `I`.
* `tangent_bundle I M` : the total space of `tangent_bundle_core I M`. It is itself a
smooth manifold over the model with corners `I.tangent`, the product of `I` and the trivial model
with corners on `E`.
* `tangent_space I x` : the tangent space to `M` at `x`
* `tangent_bundle.proj I M`: the projection from the tangent bundle to the base manifold
## Implementation notes
We register the vector space structure on the fibers of the tangent bundle, but we do not register
the normed space structure coming from that of `F` (as it is not canonical, and we also want to
keep the possibility to add a Riemannian structure on the manifold later on without having two
competing normed space instances on the tangent spaces).
We require `F` to be a normed space, and not just a topological vector space, as we want to talk
about smooth functions on `F`. The notion of derivative requires a norm to be defined.
## TODO
construct the cotangent bundle, and the bundles of differential forms. They should follow
functorially from the description of the tangent bundle as a basic smooth bundle.
## Tags
Smooth fiber bundle, vector bundle, tangent space, tangent bundle
-/
noncomputable theory
universe u
open topological_space set
open_locale manifold topological_space
/-- Core structure used to create a smooth bundle above `M` (a manifold over the model with
corner `I`) with fiber the normed vector space `F` over `𝕜`, which is trivial in the chart domains
of `M`. This structure registers the changes in the fibers when one changes coordinate charts in the
base. We require the change of coordinates of the fibers to be linear, so that the resulting bundle
is a vector bundle. -/
structure basic_smooth_vector_bundle_core {𝕜 : Type*} [nondiscrete_normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E]
{H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H)
(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
(F : Type*) [normed_group F] [normed_space 𝕜 F] :=
(coord_change : atlas H M → atlas H M → H → (F →L[𝕜] F))
(coord_change_self : ∀ i : atlas H M, ∀ x ∈ i.1.target, ∀ v, coord_change i i x v = v)
(coord_change_comp : ∀ i j k : atlas H M,
∀ x ∈ ((i.1.symm.trans j.1).trans (j.1.symm.trans k.1)).source, ∀ v,
(coord_change j k ((i.1.symm.trans j.1) x)) (coord_change i j x v) = coord_change i k x v)
(coord_change_smooth_clm : ∀ i j : atlas H M,
cont_diff_on 𝕜 ∞ ((coord_change i j) ∘ I.symm) (I '' (i.1.symm.trans j.1).source))
/-- The trivial basic smooth bundle core, in which all the changes of coordinates are the
identity. -/
def trivial_basic_smooth_vector_bundle_core {𝕜 : Type*} [nondiscrete_normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E]
{H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H)
(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
(F : Type*) [normed_group F] [normed_space 𝕜 F] : basic_smooth_vector_bundle_core I M F :=
{ coord_change := λ i j x, continuous_linear_map.id 𝕜 F,
coord_change_self := λ i x hx v, rfl,
coord_change_comp := λ i j k x hx v, rfl,
coord_change_smooth_clm := λ i j, by { dsimp, exact cont_diff_on_const } }
namespace basic_smooth_vector_bundle_core
variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E]
{H : Type*} [topological_space H] {I : model_with_corners 𝕜 E H}
{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
{F : Type*} [normed_group F] [normed_space 𝕜 F]
(Z : basic_smooth_vector_bundle_core I M F)
instance : inhabited (basic_smooth_vector_bundle_core I M F) :=
⟨trivial_basic_smooth_vector_bundle_core I M F⟩
lemma coord_change_continuous (i j : atlas H M) :
continuous_on (Z.coord_change i j) (i.1.symm.trans j.1).source :=
begin
assume x hx,
apply (((Z.coord_change_smooth_clm i j).continuous_on.continuous_within_at
(mem_image_of_mem I hx)).comp I.continuous_within_at _).congr,
{ assume y hy,
simp only with mfld_simps },
{ simp only with mfld_simps },
{ exact maps_to_image I _ },
end
lemma coord_change_smooth (i j : atlas H M) :
cont_diff_on 𝕜 ∞ (λ p : E × F, Z.coord_change i j (I.symm p.1) p.2)
((I '' (i.1.symm.trans j.1).source) ×ˢ (univ : set F)) :=
begin
have A : cont_diff 𝕜 ∞ (λ p : (F →L[𝕜] F) × F, p.1 p.2),
{ apply is_bounded_bilinear_map.cont_diff,
exact is_bounded_bilinear_map_apply },
have B : cont_diff_on 𝕜 ∞ (λ (p : E × F), (Z.coord_change i j (I.symm p.1), p.snd))
((I '' (i.1.symm.trans j.1).source) ×ˢ (univ : set F)),
{ apply cont_diff_on.prod _ _,
{ exact (Z.coord_change_smooth_clm i j).comp cont_diff_fst.cont_diff_on
(prod_subset_preimage_fst _ _) },
{ exact is_bounded_linear_map.snd.cont_diff.cont_diff_on } },
exact A.comp_cont_diff_on B,
end
/-- Vector bundle core associated to a basic smooth bundle core -/
def to_topological_vector_bundle_core : topological_vector_bundle_core 𝕜 M F (atlas H M) :=
{ base_set := λ i, i.1.source,
is_open_base_set := λ i, i.1.open_source,
index_at := λ x, ⟨chart_at H x, chart_mem_atlas H x⟩,
mem_base_set_at := λ x, mem_chart_source H x,
coord_change := λ i j x, Z.coord_change i j (i.1 x),
coord_change_self := λ i x hx v, Z.coord_change_self i (i.1 x) (i.1.map_source hx) v,
coord_change_comp := λ i j k x ⟨⟨hx1, hx2⟩, hx3⟩ v, begin
have := Z.coord_change_comp i j k (i.1 x) _ v,
convert this using 2,
{ simp only [hx1] with mfld_simps },
{ simp only [hx1, hx2, hx3] with mfld_simps }
end,
coord_change_continuous := λ i j, begin
refine ((Z.coord_change_continuous i j).comp' i.1.continuous_on).mono _,
rintros p ⟨hp₁, hp₂⟩,
refine ⟨hp₁, i.1.maps_to hp₁, _⟩,
simp only [i.1.left_inv hp₁, hp₂] with mfld_simps
end }
@[simp, mfld_simps] lemma base_set (i : atlas H M) :
(Z.to_topological_vector_bundle_core.local_triv i).base_set = i.1.source := rfl
@[simp, mfld_simps] lemma target (i : atlas H M) :
(Z.to_topological_vector_bundle_core.local_triv i).target = i.1.source ×ˢ (univ : set F) := rfl
/-- Local chart for the total space of a basic smooth bundle -/
def chart {e : local_homeomorph M H} (he : e ∈ atlas H M) :
local_homeomorph (Z.to_topological_vector_bundle_core.total_space) (model_prod H F) :=
(Z.to_topological_vector_bundle_core.local_triv ⟨e, he⟩).to_local_homeomorph.trans
(local_homeomorph.prod e (local_homeomorph.refl F))
@[simp, mfld_simps] lemma chart_source (e : local_homeomorph M H) (he : e ∈ atlas H M) :
(Z.chart he).source = Z.to_topological_vector_bundle_core.proj ⁻¹' e.source :=
by { simp only [chart, mem_prod], mfld_set_tac }
@[simp, mfld_simps] lemma chart_target (e : local_homeomorph M H) (he : e ∈ atlas H M) :
(Z.chart he).target = e.target ×ˢ (univ : set F) :=
by { simp only [chart], mfld_set_tac }
/-- The total space of a basic smooth bundle is endowed with a charted space structure, where the
charts are in bijection with the charts of the basis. -/
instance to_charted_space :
charted_space (model_prod H F) Z.to_topological_vector_bundle_core.total_space :=
{ atlas := ⋃(e : local_homeomorph M H) (he : e ∈ atlas H M), {Z.chart he},
chart_at := λ p, Z.chart (chart_mem_atlas H p.1),
mem_chart_source := λ p, by simp [mem_chart_source],
chart_mem_atlas := λ p, begin
simp only [mem_Union, mem_singleton_iff, chart_mem_atlas],
exact ⟨chart_at H p.1, chart_mem_atlas H p.1, rfl⟩
end }
lemma mem_atlas_iff
(f : local_homeomorph Z.to_topological_vector_bundle_core.total_space (model_prod H F)) :
f ∈ atlas (model_prod H F) Z.to_topological_vector_bundle_core.total_space ↔
∃(e : local_homeomorph M H) (he : e ∈ atlas H M), f = Z.chart he :=
by simp only [atlas, mem_Union, mem_singleton_iff]
@[simp, mfld_simps] lemma mem_chart_source_iff
(p q : Z.to_topological_vector_bundle_core.total_space) :
p ∈ (chart_at (model_prod H F) q).source ↔ p.1 ∈ (chart_at H q.1).source :=
by simp only [chart_at] with mfld_simps
@[simp, mfld_simps] lemma mem_chart_target_iff
(p : H × F) (q : Z.to_topological_vector_bundle_core.total_space) :
p ∈ (chart_at (model_prod H F) q).target ↔ p.1 ∈ (chart_at H q.1).target :=
by simp only [chart_at] with mfld_simps
@[simp, mfld_simps] lemma coe_chart_at_fst (p q : Z.to_topological_vector_bundle_core.total_space) :
((chart_at (model_prod H F) q) p).1 = chart_at H q.1 p.1 := rfl
@[simp, mfld_simps] lemma coe_chart_at_symm_fst
(p : H × F) (q : Z.to_topological_vector_bundle_core.total_space) :
((chart_at (model_prod H F) q).symm p).1 = ((chart_at H q.1).symm : H → M) p.1 := rfl
/-- Smooth manifold structure on the total space of a basic smooth bundle -/
instance to_smooth_manifold :
smooth_manifold_with_corners (I.prod (𝓘(𝕜, F))) Z.to_topological_vector_bundle_core.total_space :=
begin
/- We have to check that the charts belong to the smooth groupoid, i.e., they are smooth on their
source, and their inverses are smooth on the target. Since both objects are of the same kind, it
suffices to prove the first statement in A below, and then glue back the pieces at the end. -/
let J := model_with_corners.to_local_equiv (I.prod (𝓘(𝕜, F))),
have A : ∀ (e e' : local_homeomorph M H) (he : e ∈ atlas H M) (he' : e' ∈ atlas H M),
cont_diff_on 𝕜 ∞
(J ∘ ((Z.chart he).symm.trans (Z.chart he')) ∘ J.symm)
(J.symm ⁻¹' ((Z.chart he).symm.trans (Z.chart he')).source ∩ range J),
{ assume e e' he he',
have : J.symm ⁻¹' ((chart Z he).symm.trans (chart Z he')).source ∩ range J =
(I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F),
by { simp only [J, chart, model_with_corners.prod], mfld_set_tac },
rw this,
-- check separately that the two components of the coordinate change are smooth
apply cont_diff_on.prod,
show cont_diff_on 𝕜 ∞ (λ (p : E × F), (I ∘ e' ∘ e.symm ∘ I.symm) p.1)
((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F)),
{ -- the coordinate change on the base is just a coordinate change for `M`, smooth since
-- `M` is smooth
have A : cont_diff_on 𝕜 ∞ (I ∘ (e.symm.trans e') ∘ I.symm)
(I.symm ⁻¹' (e.symm.trans e').source ∩ range I) :=
(has_groupoid.compatible (cont_diff_groupoid ∞ I) he he').1,
have B : cont_diff_on 𝕜 ∞ (λ p : E × F, p.1)
((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F)) :=
cont_diff_fst.cont_diff_on,
exact cont_diff_on.comp A B (prod_subset_preimage_fst _ _) },
show cont_diff_on 𝕜 ∞ (λ (p : E × F),
Z.coord_change ⟨chart_at H (e.symm (I.symm p.1)), _⟩ ⟨e', he'⟩
((chart_at H (e.symm (I.symm p.1)) : M → H) (e.symm (I.symm p.1)))
(Z.coord_change ⟨e, he⟩ ⟨chart_at H (e.symm (I.symm p.1)), _⟩
(e (e.symm (I.symm p.1))) p.2))
((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F)),
{ /- The coordinate change in the fiber is more complicated as its definition involves the
reference chart chosen at each point. However, it appears with its inverse, so using the
cocycle property one can get rid of it, and then conclude using the smoothness of the
cocycle as given in the definition of basic smooth bundles. -/
have := Z.coord_change_smooth ⟨e, he⟩ ⟨e', he'⟩,
rw I.image_eq at this,
apply cont_diff_on.congr this,
rintros ⟨x, v⟩ hx,
simp only with mfld_simps at hx,
let f := chart_at H (e.symm (I.symm x)),
have A : I.symm x ∈ ((e.symm.trans f).trans (f.symm.trans e')).source,
by simp only [hx.1.1, hx.1.2] with mfld_simps,
rw e.right_inv hx.1.1,
have := Z.coord_change_comp ⟨e, he⟩ ⟨f, chart_mem_atlas _ _⟩ ⟨e', he'⟩ (I.symm x) A v,
simpa only [] using this } },
refine @smooth_manifold_with_corners.mk _ _ _ _ _ _ _ _ _ _ _ ⟨_⟩,
assume e₀ e₀' he₀ he₀',
rcases (Z.mem_atlas_iff _).1 he₀ with ⟨e, he, rfl⟩,
rcases (Z.mem_atlas_iff _).1 he₀' with ⟨e', he', rfl⟩,
rw [cont_diff_groupoid, mem_groupoid_of_pregroupoid],
exact ⟨A e e' he he', A e' e he' he⟩
end
end basic_smooth_vector_bundle_core
section tangent_bundle
variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E]
{H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H)
(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
/-- Basic smooth bundle core version of the tangent bundle of a smooth manifold `M` modelled over a
model with corners `I` on `(E, H)`. The fibers are equal to `E`, and the coordinate change in the
fiber corresponds to the derivative of the coordinate change in `M`. -/
def tangent_bundle_core : basic_smooth_vector_bundle_core I M E :=
{ coord_change := λ i j x, (fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm) (range I) (I x)),
coord_change_smooth_clm := λ i j,
begin
rw I.image_eq,
have A : cont_diff_on 𝕜 ∞
(I ∘ (i.1.symm.trans j.1) ∘ I.symm)
(I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) :=
(has_groupoid.compatible (cont_diff_groupoid ∞ I) i.2 j.2).1,
have B : unique_diff_on 𝕜 (I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) :=
I.unique_diff_preimage_source,
have C : cont_diff_on 𝕜 ∞
(λ (p : E × E), (fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) p.1 : E → E) p.2)
((I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) ×ˢ (univ : set E)) :=
cont_diff_on_fderiv_within_apply A B le_top,
have D : ∀ x ∈ (I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I),
fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(range I) x =
fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) x,
{ assume x hx,
have N : I.symm ⁻¹' (i.1.symm.trans j.1).source ∈ nhds x :=
I.continuous_symm.continuous_at.preimage_mem_nhds
(is_open.mem_nhds (local_homeomorph.open_source _) hx.1),
symmetry,
rw inter_comm,
exact fderiv_within_inter N (I.unique_diff _ hx.2) },
apply (A.fderiv_within B le_top).congr,
assume x hx,
simp only with mfld_simps at hx,
simp only [hx, D] with mfld_simps,
end,
coord_change_self := λ i x hx v, begin
/- Locally, a self-change of coordinate is just the identity, thus its derivative is the
identity. One just needs to write this carefully, paying attention to the sets where the
functions are defined. -/
have A : I.symm ⁻¹' (i.1.symm.trans i.1).source ∩ range I ∈ 𝓝[range I] (I x),
{ rw inter_comm,
apply inter_mem_nhds_within,
apply I.continuous_symm.continuous_at.preimage_mem_nhds
(is_open.mem_nhds (local_homeomorph.open_source _) _),
simp only [hx, i.1.map_target] with mfld_simps },
have B : ∀ᶠ y in 𝓝[range I] (I x),
(I ∘ i.1 ∘ i.1.symm ∘ I.symm) y = (id : E → E) y,
{ filter_upwards [A] with _ hy,
rw ← I.image_eq at hy,
rcases hy with ⟨z, hz⟩,
simp only with mfld_simps at hz,
simp only [hz.2.symm, hz.1] with mfld_simps, },
have C : fderiv_within 𝕜 (I ∘ i.1 ∘ i.1.symm ∘ I.symm) (range I) (I x) =
fderiv_within 𝕜 (id : E → E) (range I) (I x) :=
filter.eventually_eq.fderiv_within_eq I.unique_diff_at_image B
(by simp only [hx] with mfld_simps),
rw fderiv_within_id I.unique_diff_at_image at C,
rw C,
refl
end,
coord_change_comp := λ i j u x hx, begin
/- The cocycle property is just the fact that the derivative of a composition is the product of
the derivatives. One needs however to check that all the functions one considers are smooth, and
to pay attention to the domains where these functions are defined, making this proof a little
bit cumbersome although there is nothing complicated here. -/
have M : I x ∈
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I) :=
⟨by simpa only [mem_preimage, model_with_corners.left_inv] using hx, mem_range_self _⟩,
have U : unique_diff_within_at 𝕜
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I) (I x) :=
I.unique_diff_preimage_source _ M,
have A : fderiv_within 𝕜 ((I ∘ u.1 ∘ j.1.symm ∘ I.symm) ∘ (I ∘ j.1 ∘ i.1.symm ∘ I.symm))
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x)
= (fderiv_within 𝕜 (I ∘ u.1 ∘ j.1.symm ∘ I.symm)
(I.symm ⁻¹' (j.1.symm.trans u.1).source ∩ range I)
((I ∘ j.1 ∘ i.1.symm ∘ I.symm) (I x))).comp
(fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x)),
{ apply fderiv_within.comp _ _ _ _ U,
show differentiable_within_at 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x),
{ have A : cont_diff_on 𝕜 ∞
(I ∘ (i.1.symm.trans j.1) ∘ I.symm)
(I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) :=
(has_groupoid.compatible (cont_diff_groupoid ∞ I) i.2 j.2).1,
have B : differentiable_on 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I),
{ apply (A.differentiable_on le_top).mono,
have : ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ⊆
(i.1.symm.trans j.1).source := inter_subset_left _ _,
exact inter_subset_inter (preimage_mono this) (subset.refl (range I)) },
apply B,
simpa only [] with mfld_simps using hx },
show differentiable_within_at 𝕜 (I ∘ u.1 ∘ j.1.symm ∘ I.symm)
(I.symm ⁻¹' (j.1.symm.trans u.1).source ∩ range I)
((I ∘ j.1 ∘ i.1.symm ∘ I.symm) (I x)),
{ have A : cont_diff_on 𝕜 ∞
(I ∘ (j.1.symm.trans u.1) ∘ I.symm)
(I.symm ⁻¹' (j.1.symm.trans u.1).source ∩ range I) :=
(has_groupoid.compatible (cont_diff_groupoid ∞ I) j.2 u.2).1,
apply A.differentiable_on le_top,
rw [local_homeomorph.trans_source] at hx,
simp only with mfld_simps,
exact hx.2 },
show (I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
⊆ (I ∘ j.1 ∘ i.1.symm ∘ I.symm) ⁻¹' (I.symm ⁻¹' (j.1.symm.trans u.1).source ∩ range I),
{ assume y hy,
simp only with mfld_simps at hy,
rw [local_homeomorph.left_inv] at hy,
{ simp only [hy] with mfld_simps },
{ exact hy.1.1.2 } } },
have B : fderiv_within 𝕜 ((I ∘ u.1 ∘ j.1.symm ∘ I.symm)
∘ (I ∘ j.1 ∘ i.1.symm ∘ I.symm))
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x)
= fderiv_within 𝕜 (I ∘ u.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x),
{ have E :
∀ y ∈ (I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I),
((I ∘ u.1 ∘ j.1.symm ∘ I.symm) ∘ (I ∘ j.1 ∘ i.1.symm ∘ I.symm)) y =
(I ∘ u.1 ∘ i.1.symm ∘ I.symm) y,
{ assume y hy,
simp only [function.comp_app, model_with_corners.left_inv],
rw [j.1.left_inv],
exact hy.1.1.2 },
exact fderiv_within_congr U E (E _ M) },
have C : fderiv_within 𝕜 (I ∘ u.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x) =
fderiv_within 𝕜 (I ∘ u.1 ∘ i.1.symm ∘ I.symm)
(range I) (I x),
{ rw inter_comm,
apply fderiv_within_inter _ I.unique_diff_at_image,
apply I.continuous_symm.continuous_at.preimage_mem_nhds
(is_open.mem_nhds (local_homeomorph.open_source _) _),
simpa only [model_with_corners.left_inv] using hx },
have D : fderiv_within 𝕜 (I ∘ u.1 ∘ j.1.symm ∘ I.symm)
(I.symm ⁻¹' (j.1.symm.trans u.1).source ∩ range I) ((I ∘ j.1 ∘ i.1.symm ∘ I.symm) (I x)) =
fderiv_within 𝕜 (I ∘ u.1 ∘ j.1.symm ∘ I.symm) (range I) ((I ∘ j.1 ∘ i.1.symm ∘ I.symm) (I x)),
{ rw inter_comm,
apply fderiv_within_inter _ I.unique_diff_at_image,
apply I.continuous_symm.continuous_at.preimage_mem_nhds
(is_open.mem_nhds (local_homeomorph.open_source _) _),
rw [local_homeomorph.trans_source] at hx,
simp only with mfld_simps,
exact hx.2 },
have E : fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm)
(I.symm ⁻¹' ((i.1.symm.trans j.1).trans (j.1.symm.trans u.1)).source ∩ range I)
(I x) =
fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm) (range I) (I x),
{ rw inter_comm,
apply fderiv_within_inter _ I.unique_diff_at_image,
apply I.continuous_symm.continuous_at.preimage_mem_nhds
(is_open.mem_nhds (local_homeomorph.open_source _) _),
simpa only [model_with_corners.left_inv] using hx },
rw [B, C, D, E] at A,
simp only [A, continuous_linear_map.coe_comp'] with mfld_simps,
end }
variable {M}
include I
/-- The tangent space at a point of the manifold `M`. It is just `E`. We could use instead
`(tangent_bundle_core I M).to_topological_vector_bundle_core.fiber x`, but we use `E` to help the
kernel.
-/
@[nolint unused_arguments]
def tangent_space (x : M) : Type* := E
omit I
variable (M)
/-- The tangent bundle to a smooth manifold, as a Sigma type. Defined in terms of
`bundle.total_space` to be able to put a suitable topology on it. -/
@[nolint has_inhabited_instance, reducible] -- is empty if the base manifold is empty
def tangent_bundle := bundle.total_space (tangent_space I : M → Type*)
local notation `TM` := tangent_bundle I M
/-- The projection from the tangent bundle of a smooth manifold to the manifold. As the tangent
bundle is represented internally as a sigma type, the notation `p.1` also works for the projection
of the point `p`. -/
def tangent_bundle.proj : TM → M :=
λ p, p.1
variable {M}
@[simp, mfld_simps] lemma tangent_bundle.proj_apply (x : M) (v : tangent_space I x) :
tangent_bundle.proj I M ⟨x, v⟩ = x :=
rfl
section tangent_bundle_instances
/- In general, the definition of tangent_bundle and tangent_space are not reducible, so that type
class inference does not pick wrong instances. In this section, we record the right instances for
them, noting in particular that the tangent bundle is a smooth manifold. -/
section
local attribute [reducible] tangent_space
variables {M} (x : M)
instance : topological_space (tangent_space I x) := by apply_instance
instance : add_comm_group (tangent_space I x) := by apply_instance
instance : topological_add_group (tangent_space I x) := by apply_instance
instance : module 𝕜 (tangent_space I x) := by apply_instance
instance : inhabited (tangent_space I x) := ⟨0⟩
end
variable (M)
instance : topological_space TM :=
(tangent_bundle_core I M).to_topological_vector_bundle_core.to_topological_space (atlas H M)
instance : charted_space (model_prod H E) TM :=
(tangent_bundle_core I M).to_charted_space
instance : smooth_manifold_with_corners I.tangent TM :=
(tangent_bundle_core I M).to_smooth_manifold
instance : topological_vector_bundle 𝕜 E (tangent_space I : M → Type*) :=
topological_vector_bundle_core.fiber.topological_vector_bundle
(tangent_bundle_core I M).to_topological_vector_bundle_core
end tangent_bundle_instances
variable (M)
/-- The tangent bundle projection on the basis is a continuous map. -/
lemma tangent_bundle_proj_continuous : continuous (tangent_bundle.proj I M) :=
((tangent_bundle_core I M).to_topological_vector_bundle_core).continuous_proj
/-- The tangent bundle projection on the basis is an open map. -/
lemma tangent_bundle_proj_open : is_open_map (tangent_bundle.proj I M) :=
((tangent_bundle_core I M).to_topological_vector_bundle_core).is_open_map_proj
/-- In the tangent bundle to the model space, the charts are just the canonical identification
between a product type and a sigma type, a.k.a. `equiv.sigma_equiv_prod`. -/
@[simp, mfld_simps] lemma tangent_bundle_model_space_chart_at (p : tangent_bundle I H) :
(chart_at (model_prod H E) p).to_local_equiv = (equiv.sigma_equiv_prod H E).to_local_equiv :=
begin
have A : ∀ x_fst, fderiv_within 𝕜 (I ∘ I.symm) (range I) (I x_fst) = continuous_linear_map.id 𝕜 E,
{ assume x_fst,
have : fderiv_within 𝕜 (I ∘ I.symm) (range I) (I x_fst)
= fderiv_within 𝕜 id (range I) (I x_fst),
{ refine fderiv_within_congr I.unique_diff_at_image (λ y hy, _) (by simp),
exact model_with_corners.right_inv _ hy },
rwa fderiv_within_id I.unique_diff_at_image at this },
ext x : 1,
show (chart_at (model_prod H E) p : tangent_bundle I H → model_prod H E) x =
(equiv.sigma_equiv_prod H E) x,
{ cases x,
simp only [chart_at, basic_smooth_vector_bundle_core.chart, tangent_bundle_core,
basic_smooth_vector_bundle_core.to_topological_vector_bundle_core, A, prod.mk.inj_iff,
continuous_linear_map.coe_id'] with mfld_simps,
exact (tangent_bundle_core I H).coord_change_self _ _ trivial x_snd, },
show ∀ x, ((chart_at (model_prod H E) p).to_local_equiv).symm x =
(equiv.sigma_equiv_prod H E).symm x,
{ rintros ⟨x_fst, x_snd⟩,
simp only [basic_smooth_vector_bundle_core.to_topological_vector_bundle_core,
tangent_bundle_core, A, continuous_linear_map.coe_id', basic_smooth_vector_bundle_core.chart,
chart_at, continuous_linear_map.coe_coe, sigma.mk.inj_iff] with mfld_simps, },
show ((chart_at (model_prod H E) p).to_local_equiv).source = univ,
by simp only [chart_at] with mfld_simps,
end
@[simp, mfld_simps] lemma tangent_bundle_model_space_coe_chart_at (p : tangent_bundle I H) :
⇑(chart_at (model_prod H E) p) = equiv.sigma_equiv_prod H E :=
by { unfold_coes, simp only with mfld_simps }
@[simp, mfld_simps] lemma tangent_bundle_model_space_coe_chart_at_symm (p : tangent_bundle I H) :
((chart_at (model_prod H E) p).symm : model_prod H E → tangent_bundle I H) =
(equiv.sigma_equiv_prod H E).symm :=
by { unfold_coes, simp only with mfld_simps }
variable (H)
/-- The canonical identification between the tangent bundle to the model space and the product,
as a homeomorphism -/
def tangent_bundle_model_space_homeomorph : tangent_bundle I H ≃ₜ model_prod H E :=
{ continuous_to_fun :=
begin
let p : tangent_bundle I H := ⟨I.symm (0 : E), (0 : E)⟩,
have : continuous (chart_at (model_prod H E) p),
{ rw continuous_iff_continuous_on_univ,
convert local_homeomorph.continuous_on _,
simp only with mfld_simps },
simpa only with mfld_simps using this,
end,
continuous_inv_fun :=
begin
let p : tangent_bundle I H := ⟨I.symm (0 : E), (0 : E)⟩,
have : continuous (chart_at (model_prod H E) p).symm,
{ rw continuous_iff_continuous_on_univ,
convert local_homeomorph.continuous_on _,
simp only with mfld_simps },
simpa only with mfld_simps using this,
end,
.. equiv.sigma_equiv_prod H E }
@[simp, mfld_simps] lemma tangent_bundle_model_space_homeomorph_coe :
(tangent_bundle_model_space_homeomorph H I : tangent_bundle I H → model_prod H E)
= equiv.sigma_equiv_prod H E :=
rfl
@[simp, mfld_simps] lemma tangent_bundle_model_space_homeomorph_coe_symm :
((tangent_bundle_model_space_homeomorph H I).symm : model_prod H E → tangent_bundle I H)
= (equiv.sigma_equiv_prod H E).symm :=
rfl
end tangent_bundle
|
000fbe3db71444817badb49f18943fa790efe8c5 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/nat_pp.lean | 010698c334bd02dd00b6144b9ebaef4200475b63 | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 69 | lean | #reduce nat.add 3 6
open nat
#reduce nat.add 3 6
#reduce (3:nat) + 6
|
c1a77ed57b68ceed9c33ad7222dea4a51c0c7c09 | a7602958ab456501ff85db8cf5553f7bcab201d7 | /Notes/Theorem_Proving_in_Lean/Chapter2/2.8.lean | 8e1de3d64b384fa621dbb8787c03a78f6359866e | [] | no_license | enlauren/math-logic | 081e2e737c8afb28dbb337968df95ead47321ba0 | 086b6935543d1841f1db92d0e49add1124054c37 | refs/heads/master | 1,594,506,621,950 | 1,558,634,976,000 | 1,558,634,976,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,367 | lean | -- 2.8 Dependent Types
-- The short explanation of what makes dependent type theory "dependent",
-- is that types can depend on parameters. For example, with list:
constant α: Type 0
#check list
#check list α
#check list ℕ
#check list bool
#check list Prop
#check list (Type 1)
-- The type `list α` depends on the parameter α. Another perhaps more
-- clear example is the function `cons`. We expect `cons` to be
-- polymorphic, i.e., work on any "type". It makes sense then, to have
-- the first argument to `cons`, be the type that it is working on.
-- "In other words, for ever α, cons α is the function that takes an
-- element a: α and a list l: list α, and returns a new list, so we
-- have `cons α a l`: list α".
-- Obviously `cons α` should have type `(α → list α) → list α`, since
-- the first concrete parameter should be some a: α to add to the second
-- parameter, a list `list α`. The result is a list. But what type should
-- just plain old `cons` have?
-- Since the parameter/argument types of `cons` are dependent on the
-- given type, we're experiencing a "Pi type", or "dependent function
-- type".
constant x: α
constant β: α → Type
constant Bn: ℕ → Type
def dep_type (x: α) : Type := β x -- Just a pass-through for β.
-- Checks and reduces for dep_type.
#check Π x: α, β x -- Just Type. Not α → Type.
#check dep_type -- Just a pass-through for β.
#check dep_type x -- Type.
#reduce dep_type x -- Reduces to (β x), not further-reducible.
-- As you can see, the type `Π x: α, β x` is not the same as the function
-- dep_type. Π x is a type dependent on its input (which is not really
-- a traditional "input" it seems? Not sure, this is kinda confusing). It
-- seems that the Pi type is an atomic type (not a function) that depends
-- on what is basically invisible input.
-- `α → β` is just notation for Π x: α, β, when β does not depend on x.
-- When β does depend on x, `Π x: α, β` denotes a dependent function
-- type.
namespace hidden
universe u
constant list: Type u → Type u
constant cons: Π type: Type u, type → list type → list type
constant prop_to_add: Prop
constant list_of_props: list Prop
#check cons
#check cons Prop -- First takes a type.
#check cons Prop prop_to_add -- Then takes an element
#check cons Prop prop_to_add list_of_props -- Then takes a list.
end hidden
-- A similar thing goes for vec operations.
namespace vector
universe u
-- A vector is defined by a type and a length.
constant vec: Type u → ℕ → Type u
constant empty: Π α: Type u, vec α 0
constant cons: Π (α: Type u) (n: ℕ), α → vec α n → vec α (n + 1)
constant append_vecs: Π (α: Type u) (m: ℕ) (n: ℕ),
vec α m → vec α n → vec α (m + n)
end vector
-- This might be confusing and abstract, but at the very least, you can
-- think of it like this: When you have a function whose parameter/return
-- types all depend on a given type, you'll want:
-- 1. The first parameter to be a "type"
-- 2. The rest of the types to "depend" on the type that was passed in
-- as the first parameter.
-- In this case, you'll use a dependent function type, or a Π type, of
-- the form Π (x: Type u), where x is the type that our other types
-- described in (2) can depend on. |
f8baabb659aa22daba456357a360252feb664ee6 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/data/psigma/order.lean | d148bc589dbbc568fee8a88e88eb75619279beef | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 7,350 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import data.sigma.lex
import order.bounded_order
/-!
# Lexicographic order on a sigma type
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file defines the lexicographic order on `Σₗ' i, α i`. `a` is less than `b` if its summand is
strictly less than the summand of `b` or they are in the same summand and `a` is less than `b`
there.
## Notation
* `Σₗ' i, α i`: Sigma type equipped with the lexicographic order. A type synonym of `Σ' i, α i`.
## See also
Related files are:
* `data.finset.colex`: Colexicographic order on finite sets.
* `data.list.lex`: Lexicographic order on lists.
* `data.pi.lex`: Lexicographic order on `Πₗ i, α i`.
* `data.sigma.order`: Lexicographic order on `Σₗ i, α i`. Basically a twin of this file.
* `data.prod.lex`: Lexicographic order on `α × β`.
## TODO
Define the disjoint order on `Σ' i, α i`, where `x ≤ y` only if `x.fst = y.fst`.
Prove that a sigma type is a `no_max_order`, `no_min_order`, `densely_ordered` when its summands
are.
-/
variables {ι : Type*} {α : ι → Type*}
namespace psigma
notation `Σₗ'` binders `, ` r:(scoped p, _root_.lex (psigma p)) := r
namespace lex
/-- The lexicographical `≤` on a sigma type. -/
instance has_le [has_lt ι] [Π i, has_le (α i)] : has_le (Σₗ' i, α i) := ⟨lex (<) (λ i, (≤))⟩
/-- The lexicographical `<` on a sigma type. -/
instance has_lt [has_lt ι] [Π i, has_lt (α i)] : has_lt (Σₗ' i, α i) := ⟨lex (<) (λ i, (<))⟩
instance preorder [preorder ι] [Π i, preorder (α i)] : preorder (Σₗ' i, α i) :=
{ le_refl := λ ⟨i, a⟩, lex.right _ le_rfl,
le_trans :=
begin
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⟨a₃, b₃⟩ ⟨h₁r⟩ ⟨h₂r⟩,
{ left, apply lt_trans, repeat { assumption } },
{ left, assumption },
{ left, assumption },
{ right, apply le_trans, repeat { assumption } }
end,
lt_iff_le_not_le :=
begin
refine λ a b, ⟨λ hab, ⟨hab.mono_right (λ i a b, le_of_lt), _⟩, _⟩,
{ rintro (⟨i, a, hji⟩ | ⟨i, hba⟩);
obtain (⟨_, _, hij⟩ | ⟨_, hab⟩) := hab,
{ exact hij.not_lt hji },
{ exact lt_irrefl _ hji },
{ exact lt_irrefl _ hij },
{ exact hab.not_le hba } },
{ rintro ⟨⟨j, b, hij⟩ | ⟨i, hab⟩, hba⟩,
{ exact lex.left _ _ hij },
{ exact lex.right _ (hab.lt_of_not_le $ λ h, hba $ lex.right _ h) } }
end,
.. lex.has_le,
.. lex.has_lt }
/-- Dictionary / lexicographic partial_order for dependent pairs. -/
instance partial_order [partial_order ι] [Π i, partial_order (α i)] :
partial_order (Σₗ' i, α i) :=
{ le_antisymm :=
begin
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (⟨_, _, hlt₁⟩ | ⟨_, hlt₁⟩) (⟨_, _, hlt₂⟩ | ⟨_, hlt₂⟩),
{ exact (lt_irrefl a₁ $ hlt₁.trans hlt₂).elim },
{ exact (lt_irrefl a₁ hlt₁).elim },
{ exact (lt_irrefl a₁ hlt₂).elim },
{ rw hlt₁.antisymm hlt₂ }
end
.. lex.preorder }
/-- Dictionary / lexicographic linear_order for pairs. -/
instance linear_order [linear_order ι] [Π i, linear_order (α i)] : linear_order (Σₗ' i, α i) :=
{ le_total :=
begin
rintro ⟨i, a⟩ ⟨j, b⟩,
obtain hij | rfl | hji := lt_trichotomy i j,
{ exact or.inl (lex.left _ _ hij) },
{ obtain hab | hba := le_total a b,
{ exact or.inl (lex.right _ hab) },
{ exact or.inr (lex.right _ hba) } },
{ exact or.inr (lex.left _ _ hji) }
end,
decidable_eq := psigma.decidable_eq,
decidable_le := lex.decidable _ _,
decidable_lt := lex.decidable _ _,
.. lex.partial_order }
/-- The lexicographical linear order on a sigma type. -/
instance order_bot [partial_order ι] [order_bot ι] [Π i, preorder (α i)] [order_bot (α ⊥)] :
order_bot (Σₗ' i, α i) :=
{ bot := ⟨⊥, ⊥⟩,
bot_le := λ ⟨a, b⟩, begin
obtain rfl | ha := eq_bot_or_bot_lt a,
{ exact lex.right _ bot_le },
{ exact lex.left _ _ ha }
end }
/-- The lexicographical linear order on a sigma type. -/
instance order_top [partial_order ι] [order_top ι] [Π i, preorder (α i)] [order_top (α ⊤)] :
order_top (Σₗ' i, α i) :=
{ top := ⟨⊤, ⊤⟩,
le_top := λ ⟨a, b⟩, begin
obtain rfl | ha := eq_top_or_lt_top a,
{ exact lex.right _ le_top },
{ exact lex.left _ _ ha }
end }
/-- The lexicographical linear order on a sigma type. -/
instance bounded_order [partial_order ι] [bounded_order ι] [Π i, preorder (α i)]
[order_bot (α ⊥)] [order_top (α ⊤)] :
bounded_order (Σₗ' i, α i) :=
{ .. lex.order_bot, .. lex.order_top }
instance densely_ordered [preorder ι] [densely_ordered ι] [Π i, nonempty (α i)]
[Π i, preorder (α i)] [Π i, densely_ordered (α i)] :
densely_ordered (Σₗ' i, α i) :=
⟨begin
rintro ⟨i, a⟩ ⟨j, b⟩ (⟨_, _, h⟩ | @⟨_, _, b, h⟩),
{ obtain ⟨k, hi, hj⟩ := exists_between h,
obtain ⟨c⟩ : nonempty (α k) := infer_instance,
exact ⟨⟨k, c⟩, left _ _ hi, left _ _ hj⟩ },
{ obtain ⟨c, ha, hb⟩ := exists_between h,
exact ⟨⟨i, c⟩, right _ ha, right _ hb⟩ }
end⟩
instance densely_ordered_of_no_max_order [preorder ι] [Π i, preorder (α i)]
[Π i, densely_ordered (α i)] [Π i, no_max_order (α i)] :
densely_ordered (Σₗ' i, α i) :=
⟨begin
rintro ⟨i, a⟩ ⟨j, b⟩ (⟨_, _, h⟩ | @⟨_, _, b, h⟩),
{ obtain ⟨c, ha⟩ := exists_gt a,
exact ⟨⟨i, c⟩, right _ ha, left _ _ h⟩ },
{ obtain ⟨c, ha, hb⟩ := exists_between h,
exact ⟨⟨i, c⟩, right _ ha, right _ hb⟩ }
end⟩
instance densely_ordered_of_no_min_order [preorder ι] [Π i, preorder (α i)]
[Π i, densely_ordered (α i)] [Π i, no_min_order (α i)] :
densely_ordered (Σₗ' i, α i) :=
⟨begin
rintro ⟨i, a⟩ ⟨j, b⟩ (⟨_, _, h⟩ | @⟨_, _, b, h⟩),
{ obtain ⟨c, hb⟩ := exists_lt b,
exact ⟨⟨j, c⟩, left _ _ h, right _ hb⟩ },
{ obtain ⟨c, ha, hb⟩ := exists_between h,
exact ⟨⟨i, c⟩, right _ ha, right _ hb⟩ }
end⟩
instance no_max_order_of_nonempty [preorder ι] [Π i, preorder (α i)] [no_max_order ι]
[Π i, nonempty (α i)] :
no_max_order (Σₗ' i, α i) :=
⟨begin
rintro ⟨i, a⟩,
obtain ⟨j, h⟩ := exists_gt i,
obtain ⟨b⟩ : nonempty (α j) := infer_instance,
exact ⟨⟨j, b⟩, left _ _ h⟩
end⟩
instance no_min_order_of_nonempty [preorder ι] [Π i, preorder (α i)] [no_max_order ι]
[Π i, nonempty (α i)] :
no_max_order (Σₗ' i, α i) :=
⟨begin
rintro ⟨i, a⟩,
obtain ⟨j, h⟩ := exists_gt i,
obtain ⟨b⟩ : nonempty (α j) := infer_instance,
exact ⟨⟨j, b⟩, left _ _ h⟩
end⟩
instance no_max_order [preorder ι] [Π i, preorder (α i)] [Π i, no_max_order (α i)] :
no_max_order (Σₗ' i, α i) :=
⟨by { rintro ⟨i, a⟩, obtain ⟨b, h⟩ := exists_gt a, exact ⟨⟨i, b⟩, right _ h⟩ }⟩
instance no_min_order [preorder ι] [Π i, preorder (α i)] [Π i, no_min_order (α i)] :
no_min_order (Σₗ' i, α i) :=
⟨by { rintro ⟨i, a⟩, obtain ⟨b, h⟩ := exists_lt a, exact ⟨⟨i, b⟩, right _ h⟩ }⟩
end lex
end psigma
|
f88be15b4eed2a85e5f18b3c71ad1af2355fb3e3 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/data/int/modeq.lean | 3fdca9184eea41dee47e7d37fd169700b76401fc | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 6,132 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import data.nat.modeq
import tactic.ring
namespace int
/-- `a ≡ b [ZMOD n]` when `a % n = b % n`. -/
def modeq (n a b : ℤ) := a % n = b % n
notation a ` ≡ `:50 b ` [ZMOD `:50 n `]`:0 := modeq n a b
namespace modeq
variables {n m a b c d : ℤ}
@[refl] protected theorem refl (a : ℤ) : a ≡ a [ZMOD n] := @rfl _ _
@[symm] protected theorem symm : a ≡ b [ZMOD n] → b ≡ a [ZMOD n] := eq.symm
@[trans] protected theorem trans : a ≡ b [ZMOD n] → b ≡ c [ZMOD n] → a ≡ c [ZMOD n] := eq.trans
lemma coe_nat_modeq_iff {a b n : ℕ} : a ≡ b [ZMOD n] ↔ a ≡ b [MOD n] :=
by unfold modeq nat.modeq; rw ← int.coe_nat_eq_coe_nat_iff; simp [int.coe_nat_mod]
instance : decidable (a ≡ b [ZMOD n]) := by unfold modeq; apply_instance
theorem modeq_zero_iff : a ≡ 0 [ZMOD n] ↔ n ∣ a :=
by rw [modeq, zero_mod, dvd_iff_mod_eq_zero]
theorem modeq_iff_dvd : a ≡ b [ZMOD n] ↔ (n:ℤ) ∣ b - a :=
by rw [modeq, eq_comm];
simp [int.mod_eq_mod_iff_mod_sub_eq_zero, int.dvd_iff_mod_eq_zero, -euclidean_domain.mod_eq_zero]
theorem modeq_of_dvd_of_modeq (d : m ∣ n) (h : a ≡ b [ZMOD n]) : a ≡ b [ZMOD m] :=
modeq_iff_dvd.2 $ dvd_trans d (modeq_iff_dvd.1 h)
theorem modeq_mul_left' (hc : 0 ≤ c) (h : a ≡ b [ZMOD n]) : c * a ≡ c * b [ZMOD (c * n)] :=
or.cases_on (lt_or_eq_of_le hc) (λ hc,
by unfold modeq;
simp [mul_mod_mul_of_pos _ _ hc, (show _ = _, from h)] )
(λ hc, by simp [hc.symm])
theorem modeq_mul_right' (hc : 0 ≤ c) (h : a ≡ b [ZMOD n]) : a * c ≡ b * c [ZMOD (n * c)] :=
by rw [mul_comm a, mul_comm b, mul_comm n]; exact modeq_mul_left' hc h
theorem modeq_add (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a + c ≡ b + d [ZMOD n] :=
modeq_iff_dvd.2 $ by {convert dvd_add (modeq_iff_dvd.1 h₁) (modeq_iff_dvd.1 h₂), ring}
theorem modeq_add_cancel_left (h₁ : a ≡ b [ZMOD n]) (h₂ : a + c ≡ b + d [ZMOD n]) :
c ≡ d [ZMOD n] :=
have d - c = b + d - (a + c) - (b - a) := by ring,
modeq_iff_dvd.2 $ by { rw [this], exact dvd_sub (modeq_iff_dvd.1 h₂) (modeq_iff_dvd.1 h₁) }
theorem modeq_add_cancel_right (h₁ : c ≡ d [ZMOD n]) (h₂ : a + c ≡ b + d [ZMOD n]) :
a ≡ b [ZMOD n] :=
by rw [add_comm a, add_comm b] at h₂; exact modeq_add_cancel_left h₁ h₂
theorem mod_modeq (a n) : a % n ≡ a [ZMOD n] := int.mod_mod _ _
theorem modeq_neg (h : a ≡ b [ZMOD n]) : -a ≡ -b [ZMOD n] :=
modeq_add_cancel_left h (by simp)
theorem modeq_sub (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a - c ≡ b - d [ZMOD n] :=
by rw [sub_eq_add_neg, sub_eq_add_neg]; exact modeq_add h₁ (modeq_neg h₂)
theorem modeq_mul_left (c : ℤ) (h : a ≡ b [ZMOD n]) : c * a ≡ c * b [ZMOD n] :=
or.cases_on (le_total 0 c)
(λ hc, modeq_of_dvd_of_modeq (dvd_mul_left _ _) (modeq_mul_left' hc h))
(λ hc, by rw [← neg_neg c, ← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul _ b];
exact modeq_neg (modeq_of_dvd_of_modeq (dvd_mul_left _ _)
(modeq_mul_left' (neg_nonneg.2 hc) h)))
theorem modeq_mul_right (c : ℤ) (h : a ≡ b [ZMOD n]) : a * c ≡ b * c [ZMOD n] :=
by rw [mul_comm a, mul_comm b]; exact modeq_mul_left c h
theorem modeq_mul (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a * c ≡ b * d [ZMOD n] :=
(modeq_mul_left _ h₂).trans (modeq_mul_right _ h₁)
theorem modeq_of_modeq_mul_left (m : ℤ) (h : a ≡ b [ZMOD m * n]) : a ≡ b [ZMOD n] :=
by rw [modeq_iff_dvd] at *; exact dvd.trans (dvd_mul_left n m) h
theorem modeq_of_modeq_mul_right (m : ℤ) : a ≡ b [ZMOD n * m] → a ≡ b [ZMOD n] :=
mul_comm m n ▸ modeq_of_modeq_mul_left _
lemma modeq_and_modeq_iff_modeq_mul {a b m n : ℤ} (hmn : nat.coprime m.nat_abs n.nat_abs) :
a ≡ b [ZMOD m] ∧ a ≡ b [ZMOD n] ↔ (a ≡ b [ZMOD m * n]) :=
⟨λ h, begin
rw [int.modeq.modeq_iff_dvd, int.modeq.modeq_iff_dvd] at h,
rw [int.modeq.modeq_iff_dvd, ← int.nat_abs_dvd, ← int.dvd_nat_abs,
int.coe_nat_dvd, int.nat_abs_mul],
refine hmn.mul_dvd_of_dvd_of_dvd _ _;
rw [← int.coe_nat_dvd, int.nat_abs_dvd, int.dvd_nat_abs]; tauto
end,
λ h, ⟨int.modeq.modeq_of_modeq_mul_right _ h, int.modeq.modeq_of_modeq_mul_left _ h⟩⟩
lemma gcd_a_modeq (a b : ℕ) : (a : ℤ) * nat.gcd_a a b ≡ nat.gcd a b [ZMOD b] :=
by rw [← add_zero ((a : ℤ) * _), nat.gcd_eq_gcd_ab];
exact int.modeq.modeq_add rfl (int.modeq.modeq_zero_iff.2 (dvd_mul_right _ _)).symm
theorem modeq_add_fac {a b n : ℤ} (c : ℤ) (ha : a ≡ b [ZMOD n]) : a + n*c ≡ b [ZMOD n] :=
calc a + n*c ≡ b + n*c [ZMOD n] : int.modeq.modeq_add ha (int.modeq.refl _)
... ≡ b + 0 [ZMOD n] : int.modeq.modeq_add (int.modeq.refl _)
(int.modeq.modeq_zero_iff.2 (dvd_mul_right _ _))
... ≡ b [ZMOD n] : by simp
open nat
lemma mod_coprime {a b : ℕ} (hab : coprime a b) : ∃ y : ℤ, a * y ≡ 1 [ZMOD b] :=
⟨ gcd_a a b,
have hgcd : nat.gcd a b = 1, from coprime.gcd_eq_one hab,
calc
↑a * gcd_a a b ≡ ↑a*gcd_a a b + ↑b*gcd_b a b [ZMOD ↑b] : int.modeq.symm $
modeq_add_fac _ $ int.modeq.refl _
... ≡ 1 [ZMOD ↑b] : by rw [←gcd_eq_gcd_ab, hgcd]; reflexivity ⟩
lemma exists_unique_equiv (a : ℤ) {b : ℤ} (hb : 0 < b) : ∃ z : ℤ, 0 ≤ z ∧ z < b ∧ z ≡ a [ZMOD b] :=
⟨ a % b, int.mod_nonneg _ (ne_of_gt hb),
have a % b < abs b, from int.mod_lt _ (ne_of_gt hb),
by rwa abs_of_pos hb at this,
by simp [int.modeq] ⟩
lemma exists_unique_equiv_nat (a : ℤ) {b : ℤ} (hb : 0 < b) : ∃ z : ℕ, ↑z < b ∧ ↑z ≡ a [ZMOD b] :=
let ⟨z, hz1, hz2, hz3⟩ := exists_unique_equiv a hb in
⟨z.nat_abs, by split; rw [←int.of_nat_eq_coe, int.of_nat_nat_abs_eq_of_nonneg hz1]; assumption⟩
end modeq
@[simp] lemma mod_mul_right_mod (a b c : ℤ) : a % (b * c) % b = a % b :=
int.modeq.modeq_of_modeq_mul_right _ (int.modeq.mod_modeq _ _)
@[simp] lemma mod_mul_left_mod (a b c : ℤ) : a % (b * c) % c = a % c :=
int.modeq.modeq_of_modeq_mul_left _ (int.modeq.mod_modeq _ _)
end int
|
14628cb56312a1d1de1252f43e320f7e5032cc7a | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Lean/Data/Position.lean | 23bcbb6a5f6721eac99c3fcb1a22cba4b30c0c26 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 2,688 | lean | /-
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Sebastian Ullrich
-/
import Lean.Data.Format
import Lean.ToExpr
namespace Lean
structure Position where
line : Nat
column : Nat
deriving Inhabited, DecidableEq, Repr
namespace Position
protected def lt : Position → Position → Bool
| ⟨l₁, c₁⟩, ⟨l₂, c₂⟩ => Prod.lexLt (l₁, c₁) (l₂, c₂)
instance : ToFormat Position :=
⟨fun ⟨l, c⟩ => "⟨" ++ format l ++ ", " ++ format c ++ "⟩"⟩
instance : ToString Position :=
⟨fun ⟨l, c⟩ => "⟨" ++ toString l ++ ", " ++ toString c ++ "⟩"⟩
instance : ToExpr Position where
toExpr p := mkAppN (mkConst ``Position.mk) #[toExpr p.line, toExpr p.column]
toTypeExpr := mkConst ``Position
end Position
structure FileMap where
source : String
positions : Array String.Pos
lines : Array Nat
deriving Inhabited
class MonadFileMap (m : Type → Type) where
getFileMap : m FileMap
export MonadFileMap (getFileMap)
namespace FileMap
partial def ofString (s : String) : FileMap :=
let rec loop (i : String.Pos) (line : Nat) (ps : Array String.Pos) (lines : Array Nat) : FileMap :=
if s.atEnd i then { source := s, positions := ps.push i, lines := lines.push line }
else
let c := s.get i
let i := s.next i
if c == '\n' then loop i (line+1) (ps.push i) (lines.push (line+1))
else loop i line ps lines
loop 0 1 (#[0]) (#[1])
partial def toPosition (fmap : FileMap) (pos : String.Pos) : Position :=
match fmap with
| { source := str, positions := ps, lines := lines } =>
if ps.size >= 2 && pos <= ps.back then
let rec toColumn (i : String.Pos) (c : Nat) : Nat :=
if i == pos || str.atEnd i then c
else toColumn (str.next i) (c+1)
let rec loop (b e : Nat) :=
let posB := ps[b]!
if e == b + 1 then { line := lines.get! b, column := toColumn posB 0 }
else
let m := (b + e) / 2;
let posM := ps.get! m;
if pos == posM then { line := lines.get! m, column := 0 }
else if pos > posM then loop m e
else loop b m
loop 0 (ps.size -1)
else if lines.isEmpty then
⟨0, 0⟩
else
-- Some systems like the delaborator use synthetic positions without an input file,
-- which would violate `toPositionAux`'s invariant.
-- Can also happen with EOF errors, which are not strictly inside the file.
⟨lines.back, (pos - ps.back).byteIdx⟩
end FileMap
end Lean
def String.toFileMap (s : String) : Lean.FileMap :=
Lean.FileMap.ofString s
|
72a5949e2c323123b187362edd5de7a94395e6a4 | bbecf0f1968d1fba4124103e4f6b55251d08e9c4 | /src/order/filter/bases.lean | e6c727a021d50d25d8be728cf8d8a76807855737 | [
"Apache-2.0"
] | permissive | waynemunro/mathlib | e3fd4ff49f4cb43d4a8ded59d17be407bc5ee552 | 065a70810b5480d584033f7bbf8e0409480c2118 | refs/heads/master | 1,693,417,182,397 | 1,634,644,781,000 | 1,634,644,781,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 39,401 | lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import order.filter.basic
import data.set.countable
import data.pprod
/-!
# Filter bases
A filter basis `B : filter_basis α` on a type `α` is a nonempty collection of sets of `α`
such that the intersection of two elements of this collection contains some element of
the collection. Compared to filters, filter bases do not require that any set containing
an element of `B` belongs to `B`.
A filter basis `B` can be used to construct `B.filter : filter α` such that a set belongs
to `B.filter` if and only if it contains an element of `B`.
Given an indexing type `ι`, a predicate `p : ι → Prop`, and a map `s : ι → set α`,
the proposition `h : filter.is_basis p s` makes sure the range of `s` bounded by `p`
(ie. `s '' set_of p`) defines a filter basis `h.filter_basis`.
If one already has a filter `l` on `α`, `filter.has_basis l p s` (where `p : ι → Prop`
and `s : ι → set α` as above) means that a set belongs to `l` if and
only if it contains some `s i` with `p i`. It implies `h : filter.is_basis p s`, and
`l = h.filter_basis.filter`. The point of this definition is that checking statements
involving elements of `l` often reduces to checking them on the basis elements.
We define a function `has_basis.index (h : filter.has_basis l p s) (t) (ht : t ∈ l)` that returns
some index `i` such that `p i` and `s i ⊆ t`. This function can be useful to avoid manual
destruction of `h.mem_iff.mpr ht` using `cases` or `let`.
This file also introduces more restricted classes of bases, involving monotonicity or
countability. In particular, for `l : filter α`, `l.is_countably_generated` means
there is a countable set of sets which generates `s`. This is reformulated in term of bases,
and consequences are derived.
## Main statements
* `has_basis.mem_iff`, `has_basis.mem_of_superset`, `has_basis.mem_of_mem` : restate `t ∈ f`
in terms of a basis;
* `basis_sets` : all sets of a filter form a basis;
* `has_basis.inf`, `has_basis.inf_principal`, `has_basis.prod`, `has_basis.prod_self`,
`has_basis.map`, `has_basis.comap` : combinators to construct filters of `l ⊓ l'`,
`l ⊓ 𝓟 t`, `l ×ᶠ l'`, `l ×ᶠ l`, `l.map f`, `l.comap f` respectively;
* `has_basis.le_iff`, `has_basis.ge_iff`, has_basis.le_basis_iff` : restate `l ≤ l'` in terms
of bases.
* `has_basis.tendsto_right_iff`, `has_basis.tendsto_left_iff`, `has_basis.tendsto_iff` : restate
`tendsto f l l'` in terms of bases.
* `is_countably_generated_iff_exists_antitone_basis` : proves a filter is
countably generated if and only if it admits a basis parametrized by a
decreasing sequence of sets indexed by `ℕ`.
* `tendsto_iff_seq_tendsto ` : an abstract version of "sequentially continuous implies continuous".
## Implementation notes
As with `Union`/`bUnion`/`sUnion`, there are three different approaches to filter bases:
* `has_basis l s`, `s : set (set α)`;
* `has_basis l s`, `s : ι → set α`;
* `has_basis l p s`, `p : ι → Prop`, `s : ι → set α`.
We use the latter one because, e.g., `𝓝 x` in an `emetric_space` or in a `metric_space` has a basis
of this form. The other two can be emulated using `s = id` or `p = λ _, true`.
With this approach sometimes one needs to `simp` the statement provided by the `has_basis`
machinery, e.g., `simp only [exists_prop, true_and]` or `simp only [forall_const]` can help
with the case `p = λ _, true`.
-/
open set filter
open_locale filter classical
section sort
variables {α β γ : Type*} {ι ι' : Sort*}
/-- A filter basis `B` on a type `α` is a nonempty collection of sets of `α`
such that the intersection of two elements of this collection contains some element
of the collection. -/
structure filter_basis (α : Type*) :=
(sets : set (set α))
(nonempty : sets.nonempty)
(inter_sets {x y} : x ∈ sets → y ∈ sets → ∃ z ∈ sets, z ⊆ x ∩ y)
instance filter_basis.nonempty_sets (B : filter_basis α) : nonempty B.sets := B.nonempty.to_subtype
/-- If `B` is a filter basis on `α`, and `U` a subset of `α` then we can write `U ∈ B` as
on paper. -/
@[reducible]
instance {α : Type*}: has_mem (set α) (filter_basis α) := ⟨λ U B, U ∈ B.sets⟩
-- For illustration purposes, the filter basis defining (at_top : filter ℕ)
instance : inhabited (filter_basis ℕ) :=
⟨{ sets := range Ici,
nonempty := ⟨Ici 0, mem_range_self 0⟩,
inter_sets := begin
rintros _ _ ⟨n, rfl⟩ ⟨m, rfl⟩,
refine ⟨Ici (max n m), mem_range_self _, _⟩,
rintros p p_in,
split ; rw mem_Ici at *,
exact le_of_max_le_left p_in,
exact le_of_max_le_right p_in,
end }⟩
/-- `is_basis p s` means the image of `s` bounded by `p` is a filter basis. -/
protected structure filter.is_basis (p : ι → Prop) (s : ι → set α) : Prop :=
(nonempty : ∃ i, p i)
(inter : ∀ {i j}, p i → p j → ∃ k, p k ∧ s k ⊆ s i ∩ s j)
namespace filter
namespace is_basis
/-- Constructs a filter basis from an indexed family of sets satisfying `is_basis`. -/
protected def filter_basis {p : ι → Prop} {s : ι → set α} (h : is_basis p s) : filter_basis α :=
{ sets := {t | ∃ i, p i ∧ s i = t},
nonempty := let ⟨i, hi⟩ := h.nonempty in ⟨s i, ⟨i, hi, rfl⟩⟩,
inter_sets := by { rintros _ _ ⟨i, hi, rfl⟩ ⟨j, hj, rfl⟩,
rcases h.inter hi hj with ⟨k, hk, hk'⟩,
exact ⟨_, ⟨k, hk, rfl⟩, hk'⟩ } }
variables {p : ι → Prop} {s : ι → set α} (h : is_basis p s)
lemma mem_filter_basis_iff {U : set α} : U ∈ h.filter_basis ↔ ∃ i, p i ∧ s i = U :=
iff.rfl
end is_basis
end filter
namespace filter_basis
/-- The filter associated to a filter basis. -/
protected def filter (B : filter_basis α) : filter α :=
{ sets := {s | ∃ t ∈ B, t ⊆ s},
univ_sets := let ⟨s, s_in⟩ := B.nonempty in ⟨s, s_in, s.subset_univ⟩,
sets_of_superset := λ x y ⟨s, s_in, h⟩ hxy, ⟨s, s_in, set.subset.trans h hxy⟩,
inter_sets := λ x y ⟨s, s_in, hs⟩ ⟨t, t_in, ht⟩,
let ⟨u, u_in, u_sub⟩ := B.inter_sets s_in t_in in
⟨u, u_in, set.subset.trans u_sub $ set.inter_subset_inter hs ht⟩ }
lemma mem_filter_iff (B : filter_basis α) {U : set α} : U ∈ B.filter ↔ ∃ s ∈ B, s ⊆ U :=
iff.rfl
lemma mem_filter_of_mem (B : filter_basis α) {U : set α} : U ∈ B → U ∈ B.filter:=
λ U_in, ⟨U, U_in, subset.refl _⟩
lemma eq_infi_principal (B : filter_basis α) : B.filter = ⨅ s : B.sets, 𝓟 s :=
begin
have : directed (≥) (λ (s : B.sets), 𝓟 (s : set α)),
{ rintros ⟨U, U_in⟩ ⟨V, V_in⟩,
rcases B.inter_sets U_in V_in with ⟨W, W_in, W_sub⟩,
use [W, W_in],
finish },
ext U,
simp [mem_filter_iff, mem_infi_of_directed this]
end
protected lemma generate (B : filter_basis α) : generate B.sets = B.filter :=
begin
apply le_antisymm,
{ intros U U_in,
rcases B.mem_filter_iff.mp U_in with ⟨V, V_in, h⟩,
exact generate_sets.superset (generate_sets.basic V_in) h },
{ rw sets_iff_generate,
apply mem_filter_of_mem }
end
end filter_basis
namespace filter
namespace is_basis
variables {p : ι → Prop} {s : ι → set α}
/-- Constructs a filter from an indexed family of sets satisfying `is_basis`. -/
protected def filter (h : is_basis p s) : filter α := h.filter_basis.filter
protected lemma mem_filter_iff (h : is_basis p s) {U : set α} :
U ∈ h.filter ↔ ∃ i, p i ∧ s i ⊆ U :=
begin
erw [h.filter_basis.mem_filter_iff],
simp only [mem_filter_basis_iff h, exists_prop],
split,
{ rintros ⟨_, ⟨i, pi, rfl⟩, h⟩,
tauto },
{ tauto }
end
lemma filter_eq_generate (h : is_basis p s) : h.filter = generate {U | ∃ i, p i ∧ s i = U} :=
by erw h.filter_basis.generate ; refl
end is_basis
/-- We say that a filter `l` has a basis `s : ι → set α` bounded by `p : ι → Prop`,
if `t ∈ l` if and only if `t` includes `s i` for some `i` such that `p i`. -/
protected structure has_basis (l : filter α) (p : ι → Prop) (s : ι → set α) : Prop :=
(mem_iff' : ∀ (t : set α), t ∈ l ↔ ∃ i (hi : p i), s i ⊆ t)
section same_type
variables {l l' : filter α} {p : ι → Prop} {s : ι → set α} {t : set α} {i : ι}
{p' : ι' → Prop} {s' : ι' → set α} {i' : ι'}
lemma has_basis_generate (s : set (set α)) :
(generate s).has_basis (λ t, finite t ∧ t ⊆ s) (λ t, ⋂₀ t) :=
⟨begin
intro U,
rw mem_generate_iff,
apply exists_congr,
tauto
end⟩
/-- The smallest filter basis containing a given collection of sets. -/
def filter_basis.of_sets (s : set (set α)) : filter_basis α :=
{ sets := sInter '' { t | finite t ∧ t ⊆ s},
nonempty := ⟨univ, ∅, ⟨⟨finite_empty, empty_subset s⟩, sInter_empty⟩⟩,
inter_sets := begin
rintros _ _ ⟨a, ⟨fina, suba⟩, rfl⟩ ⟨b, ⟨finb, subb⟩, rfl⟩,
exact ⟨⋂₀ (a ∪ b), mem_image_of_mem _ ⟨fina.union finb, union_subset suba subb⟩,
by rw sInter_union⟩,
end }
/-- Definition of `has_basis` unfolded with implicit set argument. -/
lemma has_basis.mem_iff (hl : l.has_basis p s) : t ∈ l ↔ ∃ i (hi : p i), s i ⊆ t :=
hl.mem_iff' t
lemma has_basis.eq_of_same_basis (hl : l.has_basis p s) (hl' : l'.has_basis p s) : l = l' :=
begin
ext t,
rw [hl.mem_iff, hl'.mem_iff]
end
lemma has_basis_iff : l.has_basis p s ↔ ∀ t, t ∈ l ↔ ∃ i (hi : p i), s i ⊆ t :=
⟨λ ⟨h⟩, h, λ h, ⟨h⟩⟩
lemma has_basis.ex_mem (h : l.has_basis p s) : ∃ i, p i :=
let ⟨i, pi, h⟩ := h.mem_iff.mp univ_mem in ⟨i, pi⟩
protected lemma has_basis.nonempty (h : l.has_basis p s) : nonempty ι :=
nonempty_of_exists h.ex_mem
protected lemma is_basis.has_basis (h : is_basis p s) : has_basis h.filter p s :=
⟨λ t, by simp only [h.mem_filter_iff, exists_prop]⟩
lemma has_basis.mem_of_superset (hl : l.has_basis p s) (hi : p i) (ht : s i ⊆ t) : t ∈ l :=
(hl.mem_iff).2 ⟨i, hi, ht⟩
lemma has_basis.mem_of_mem (hl : l.has_basis p s) (hi : p i) : s i ∈ l :=
hl.mem_of_superset hi $ subset.refl _
/-- Index of a basis set such that `s i ⊆ t` as an element of `subtype p`. -/
noncomputable def has_basis.index (h : l.has_basis p s) (t : set α) (ht : t ∈ l) :
{i : ι // p i} :=
⟨(h.mem_iff.1 ht).some, (h.mem_iff.1 ht).some_spec.fst⟩
lemma has_basis.property_index (h : l.has_basis p s) (ht : t ∈ l) : p (h.index t ht) :=
(h.index t ht).2
lemma has_basis.set_index_mem (h : l.has_basis p s) (ht : t ∈ l) : s (h.index t ht) ∈ l :=
h.mem_of_mem $ h.property_index _
lemma has_basis.set_index_subset (h : l.has_basis p s) (ht : t ∈ l) : s (h.index t ht) ⊆ t :=
(h.mem_iff.1 ht).some_spec.snd
lemma has_basis.is_basis (h : l.has_basis p s) : is_basis p s :=
{ nonempty := let ⟨i, hi, H⟩ := h.mem_iff.mp univ_mem in ⟨i, hi⟩,
inter := λ i j hi hj, by simpa [h.mem_iff]
using l.inter_sets (h.mem_of_mem hi) (h.mem_of_mem hj) }
lemma has_basis.filter_eq (h : l.has_basis p s) : h.is_basis.filter = l :=
by { ext U, simp [h.mem_iff, is_basis.mem_filter_iff] }
lemma has_basis.eq_generate (h : l.has_basis p s) : l = generate { U | ∃ i, p i ∧ s i = U } :=
by rw [← h.is_basis.filter_eq_generate, h.filter_eq]
lemma generate_eq_generate_inter (s : set (set α)) :
generate s = generate (sInter '' { t | finite t ∧ t ⊆ s}) :=
by erw [(filter_basis.of_sets s).generate, ← (has_basis_generate s).filter_eq] ; refl
lemma of_sets_filter_eq_generate (s : set (set α)) : (filter_basis.of_sets s).filter = generate s :=
by rw [← (filter_basis.of_sets s).generate, generate_eq_generate_inter s] ; refl
protected lemma _root_.filter_basis.has_basis {α : Type*} (B : filter_basis α) :
has_basis (B.filter) (λ s : set α, s ∈ B) id :=
⟨λ t, B.mem_filter_iff⟩
lemma has_basis.to_has_basis' (hl : l.has_basis p s) (h : ∀ i, p i → ∃ i', p' i' ∧ s' i' ⊆ s i)
(h' : ∀ i', p' i' → s' i' ∈ l) : l.has_basis p' s' :=
begin
refine ⟨λ t, ⟨λ ht, _, λ ⟨i', hi', ht⟩, mem_of_superset (h' i' hi') ht⟩⟩,
rcases hl.mem_iff.1 ht with ⟨i, hi, ht⟩,
rcases h i hi with ⟨i', hi', hs's⟩,
exact ⟨i', hi', subset.trans hs's ht⟩
end
lemma has_basis.to_has_basis (hl : l.has_basis p s) (h : ∀ i, p i → ∃ i', p' i' ∧ s' i' ⊆ s i)
(h' : ∀ i', p' i' → ∃ i, p i ∧ s i ⊆ s' i') : l.has_basis p' s' :=
hl.to_has_basis' h $ λ i' hi', let ⟨i, hi, hss'⟩ := h' i' hi' in hl.mem_iff.2 ⟨i, hi, hss'⟩
lemma has_basis.to_subset (hl : l.has_basis p s) {t : ι → set α} (h : ∀ i, p i → t i ⊆ s i)
(ht : ∀ i, p i → t i ∈ l) : l.has_basis p t :=
hl.to_has_basis' (λ i hi, ⟨i, hi, h i hi⟩) ht
lemma has_basis.eventually_iff (hl : l.has_basis p s) {q : α → Prop} :
(∀ᶠ x in l, q x) ↔ ∃ i, p i ∧ ∀ ⦃x⦄, x ∈ s i → q x :=
by simpa using hl.mem_iff
lemma has_basis.frequently_iff (hl : l.has_basis p s) {q : α → Prop} :
(∃ᶠ x in l, q x) ↔ ∀ i, p i → ∃ x ∈ s i, q x :=
by simp [filter.frequently, hl.eventually_iff]
lemma has_basis.exists_iff (hl : l.has_basis p s) {P : set α → Prop}
(mono : ∀ ⦃s t⦄, s ⊆ t → P t → P s) :
(∃ s ∈ l, P s) ↔ ∃ (i) (hi : p i), P (s i) :=
⟨λ ⟨s, hs, hP⟩, let ⟨i, hi, his⟩ := hl.mem_iff.1 hs in ⟨i, hi, mono his hP⟩,
λ ⟨i, hi, hP⟩, ⟨s i, hl.mem_of_mem hi, hP⟩⟩
lemma has_basis.forall_iff (hl : l.has_basis p s) {P : set α → Prop}
(mono : ∀ ⦃s t⦄, s ⊆ t → P s → P t) :
(∀ s ∈ l, P s) ↔ ∀ i, p i → P (s i) :=
⟨λ H i hi, H (s i) $ hl.mem_of_mem hi,
λ H s hs, let ⟨i, hi, his⟩ := hl.mem_iff.1 hs in mono his (H i hi)⟩
lemma has_basis.ne_bot_iff (hl : l.has_basis p s) :
ne_bot l ↔ (∀ {i}, p i → (s i).nonempty) :=
forall_mem_nonempty_iff_ne_bot.symm.trans $ hl.forall_iff $ λ _ _, nonempty.mono
lemma has_basis.eq_bot_iff (hl : l.has_basis p s) :
l = ⊥ ↔ ∃ i, p i ∧ s i = ∅ :=
not_iff_not.1 $ ne_bot_iff.symm.trans $ hl.ne_bot_iff.trans $
by simp only [not_exists, not_and, ← ne_empty_iff_nonempty]
lemma basis_sets (l : filter α) : l.has_basis (λ s : set α, s ∈ l) id :=
⟨λ t, exists_mem_subset_iff.symm⟩
lemma has_basis_self {l : filter α} {P : set α → Prop} :
has_basis l (λ s, s ∈ l ∧ P s) id ↔ ∀ t ∈ l, ∃ r ∈ l, P r ∧ r ⊆ t :=
begin
simp only [has_basis_iff, exists_prop, id, and_assoc],
exact forall_congr (λ s, ⟨λ h, h.1, λ h, ⟨h, λ ⟨t, hl, hP, hts⟩, mem_of_superset hl hts⟩⟩)
end
/-- If `{s i | p i}` is a basis of a filter `l` and each `s i` includes `s j` such that
`p j ∧ q j`, then `{s j | p j ∧ q j}` is a basis of `l`. -/
lemma has_basis.restrict (h : l.has_basis p s) {q : ι → Prop}
(hq : ∀ i, p i → ∃ j, p j ∧ q j ∧ s j ⊆ s i) :
l.has_basis (λ i, p i ∧ q i) s :=
begin
refine ⟨λ t, ⟨λ ht, _, λ ⟨i, hpi, hti⟩, h.mem_iff.2 ⟨i, hpi.1, hti⟩⟩⟩,
rcases h.mem_iff.1 ht with ⟨i, hpi, hti⟩,
rcases hq i hpi with ⟨j, hpj, hqj, hji⟩,
exact ⟨j, ⟨hpj, hqj⟩, subset.trans hji hti⟩
end
/-- If `{s i | p i}` is a basis of a filter `l` and `V ∈ l`, then `{s i | p i ∧ s i ⊆ V}`
is a basis of `l`. -/
lemma has_basis.restrict_subset (h : l.has_basis p s) {V : set α} (hV : V ∈ l) :
l.has_basis (λ i, p i ∧ s i ⊆ V) s :=
h.restrict $ λ i hi, (h.mem_iff.1 (inter_mem hV (h.mem_of_mem hi))).imp $
λ j hj, ⟨hj.fst, subset_inter_iff.1 hj.snd⟩
lemma has_basis.has_basis_self_subset {p : set α → Prop} (h : l.has_basis (λ s, s ∈ l ∧ p s) id)
{V : set α} (hV : V ∈ l) : l.has_basis (λ s, s ∈ l ∧ p s ∧ s ⊆ V) id :=
by simpa only [and_assoc] using h.restrict_subset hV
theorem has_basis.ge_iff (hl' : l'.has_basis p' s') : l ≤ l' ↔ ∀ i', p' i' → s' i' ∈ l :=
⟨λ h i' hi', h $ hl'.mem_of_mem hi',
λ h s hs, let ⟨i', hi', hs⟩ := hl'.mem_iff.1 hs in mem_of_superset (h _ hi') hs⟩
theorem has_basis.le_iff (hl : l.has_basis p s) : l ≤ l' ↔ ∀ t ∈ l', ∃ i (hi : p i), s i ⊆ t :=
by simp only [le_def, hl.mem_iff]
theorem has_basis.le_basis_iff (hl : l.has_basis p s) (hl' : l'.has_basis p' s') :
l ≤ l' ↔ ∀ i', p' i' → ∃ i (hi : p i), s i ⊆ s' i' :=
by simp only [hl'.ge_iff, hl.mem_iff]
lemma has_basis.ext (hl : l.has_basis p s) (hl' : l'.has_basis p' s')
(h : ∀ i, p i → ∃ i', p' i' ∧ s' i' ⊆ s i)
(h' : ∀ i', p' i' → ∃ i, p i ∧ s i ⊆ s' i') : l = l' :=
begin
apply le_antisymm,
{ rw hl.le_basis_iff hl',
simpa using h' },
{ rw hl'.le_basis_iff hl,
simpa using h },
end
lemma has_basis.inf' (hl : l.has_basis p s) (hl' : l'.has_basis p' s') :
(l ⊓ l').has_basis (λ i : pprod ι ι', p i.1 ∧ p' i.2) (λ i, s i.1 ∩ s' i.2) :=
⟨begin
intro t,
split,
{ simp only [mem_inf_iff, exists_prop, hl.mem_iff, hl'.mem_iff],
rintros ⟨t, ⟨i, hi, ht⟩, t', ⟨i', hi', ht'⟩, rfl⟩,
use [⟨i, i'⟩, ⟨hi, hi'⟩, inter_subset_inter ht ht'] },
{ rintros ⟨⟨i, i'⟩, ⟨hi, hi'⟩, H⟩,
exact mem_inf_of_inter (hl.mem_of_mem hi) (hl'.mem_of_mem hi') H }
end⟩
lemma has_basis.inf {ι ι' : Type*} {p : ι → Prop} {s : ι → set α} {p' : ι' → Prop}
{s' : ι' → set α} (hl : l.has_basis p s) (hl' : l'.has_basis p' s') :
(l ⊓ l').has_basis (λ i : ι × ι', p i.1 ∧ p' i.2) (λ i, s i.1 ∩ s' i.2) :=
(hl.inf' hl').to_has_basis (λ i hi, ⟨⟨i.1, i.2⟩, hi, subset.rfl⟩)
(λ i hi, ⟨⟨i.1, i.2⟩, hi, subset.rfl⟩)
lemma has_basis_principal (t : set α) : (𝓟 t).has_basis (λ i : unit, true) (λ i, t) :=
⟨λ U, by simp⟩
lemma has_basis_pure (x : α) : (pure x : filter α).has_basis (λ i : unit, true) (λ i, {x}) :=
by simp only [← principal_singleton, has_basis_principal]
lemma has_basis.sup' (hl : l.has_basis p s) (hl' : l'.has_basis p' s') :
(l ⊔ l').has_basis (λ i : pprod ι ι', p i.1 ∧ p' i.2) (λ i, s i.1 ∪ s' i.2) :=
⟨begin
intros t,
simp only [mem_sup, hl.mem_iff, hl'.mem_iff, pprod.exists, union_subset_iff, exists_prop,
and_assoc, exists_and_distrib_left],
simp only [← and_assoc, exists_and_distrib_right, and_comm]
end⟩
lemma has_basis.sup {ι ι' : Type*} {p : ι → Prop} {s : ι → set α} {p' : ι' → Prop}
{s' : ι' → set α} (hl : l.has_basis p s) (hl' : l'.has_basis p' s') :
(l ⊔ l').has_basis (λ i : ι × ι', p i.1 ∧ p' i.2) (λ i, s i.1 ∪ s' i.2) :=
(hl.sup' hl').to_has_basis (λ i hi, ⟨⟨i.1, i.2⟩, hi, subset.rfl⟩)
(λ i hi, ⟨⟨i.1, i.2⟩, hi, subset.rfl⟩)
lemma has_basis_supr {ι : Sort*} {ι' : ι → Type*} {l : ι → filter α}
{p : Π i, ι' i → Prop} {s : Π i, ι' i → set α} (hl : ∀ i, (l i).has_basis (p i) (s i)) :
(⨆ i, l i).has_basis (λ f : Π i, ι' i, ∀ i, p i (f i)) (λ f : Π i, ι' i, ⋃ i, s i (f i)) :=
has_basis_iff.mpr $ λ t, by simp only [has_basis_iff, (hl _).mem_iff, classical.skolem,
forall_and_distrib, Union_subset_iff, mem_supr]
lemma has_basis.sup_principal (hl : l.has_basis p s) (t : set α) :
(l ⊔ 𝓟 t).has_basis p (λ i, s i ∪ t) :=
⟨λ u, by simp only [(hl.sup' (has_basis_principal t)).mem_iff, pprod.exists, exists_prop, and_true,
unique.exists_iff]⟩
lemma has_basis.sup_pure (hl : l.has_basis p s) (x : α) :
(l ⊔ pure x).has_basis p (λ i, s i ∪ {x}) :=
by simp only [← principal_singleton, hl.sup_principal]
lemma has_basis.inf_principal (hl : l.has_basis p s) (s' : set α) :
(l ⊓ 𝓟 s').has_basis p (λ i, s i ∩ s') :=
⟨λ t, by simp only [mem_inf_principal, hl.mem_iff, subset_def, mem_set_of_eq,
mem_inter_iff, and_imp]⟩
lemma has_basis.inf_basis_ne_bot_iff (hl : l.has_basis p s) (hl' : l'.has_basis p' s') :
ne_bot (l ⊓ l') ↔ ∀ ⦃i⦄ (hi : p i) ⦃i'⦄ (hi' : p' i'), (s i ∩ s' i').nonempty :=
(hl.inf' hl').ne_bot_iff.trans $ by simp [@forall_swap _ ι']
lemma has_basis.inf_ne_bot_iff (hl : l.has_basis p s) :
ne_bot (l ⊓ l') ↔ ∀ ⦃i⦄ (hi : p i) ⦃s'⦄ (hs' : s' ∈ l'), (s i ∩ s').nonempty :=
hl.inf_basis_ne_bot_iff l'.basis_sets
lemma has_basis.inf_principal_ne_bot_iff (hl : l.has_basis p s) {t : set α} :
ne_bot (l ⊓ 𝓟 t) ↔ ∀ ⦃i⦄ (hi : p i), (s i ∩ t).nonempty :=
(hl.inf_principal t).ne_bot_iff
lemma inf_ne_bot_iff :
ne_bot (l ⊓ l') ↔ ∀ ⦃s : set α⦄ (hs : s ∈ l) ⦃s'⦄ (hs' : s' ∈ l'), (s ∩ s').nonempty :=
l.basis_sets.inf_ne_bot_iff
lemma inf_principal_ne_bot_iff {s : set α} :
ne_bot (l ⊓ 𝓟 s) ↔ ∀ U ∈ l, (U ∩ s).nonempty :=
l.basis_sets.inf_principal_ne_bot_iff
lemma inf_eq_bot_iff {f g : filter α} :
f ⊓ g = ⊥ ↔ ∃ (U ∈ f) (V ∈ g), U ∩ V = ∅ :=
not_iff_not.1 $ ne_bot_iff.symm.trans $ inf_ne_bot_iff.trans $
by simp [← ne_empty_iff_nonempty]
protected lemma disjoint_iff {f g : filter α} :
disjoint f g ↔ ∃ (U ∈ f) (V ∈ g), U ∩ V = ∅ :=
disjoint_iff.trans inf_eq_bot_iff
lemma mem_iff_inf_principal_compl {f : filter α} {s : set α} :
s ∈ f ↔ f ⊓ 𝓟 sᶜ = ⊥ :=
begin
refine not_iff_not.1 ((inf_principal_ne_bot_iff.trans _).symm.trans ne_bot_iff),
exact ⟨λ h hs, by simpa [empty_not_nonempty] using h s hs,
λ hs t ht, inter_compl_nonempty_iff.2 $ λ hts, hs $ mem_of_superset ht hts⟩,
end
lemma not_mem_iff_inf_principal_compl {f : filter α} {s : set α} :
s ∉ f ↔ ne_bot (f ⊓ 𝓟 sᶜ) :=
(not_congr mem_iff_inf_principal_compl).trans ne_bot_iff.symm
lemma mem_iff_disjoint_principal_compl {f : filter α} {s : set α} :
s ∈ f ↔ disjoint f (𝓟 sᶜ) :=
mem_iff_inf_principal_compl.trans disjoint_iff.symm
lemma le_iff_forall_disjoint_principal_compl {f g : filter α} :
f ≤ g ↔ ∀ V ∈ g, disjoint f (𝓟 Vᶜ) :=
forall_congr $ λ _, forall_congr $ λ _, mem_iff_disjoint_principal_compl
lemma le_iff_forall_inf_principal_compl {f g : filter α} :
f ≤ g ↔ ∀ V ∈ g, f ⊓ 𝓟 Vᶜ = ⊥ :=
forall_congr $ λ _, forall_congr $ λ _, mem_iff_inf_principal_compl
lemma inf_ne_bot_iff_frequently_left {f g : filter α} :
ne_bot (f ⊓ g) ↔ ∀ {p : α → Prop}, (∀ᶠ x in f, p x) → ∃ᶠ x in g, p x :=
by simpa only [inf_ne_bot_iff, frequently_iff, exists_prop, and_comm]
lemma inf_ne_bot_iff_frequently_right {f g : filter α} :
ne_bot (f ⊓ g) ↔ ∀ {p : α → Prop}, (∀ᶠ x in g, p x) → ∃ᶠ x in f, p x :=
by { rw inf_comm, exact inf_ne_bot_iff_frequently_left }
lemma has_basis.eq_binfi (h : l.has_basis p s) :
l = ⨅ i (_ : p i), 𝓟 (s i) :=
eq_binfi_of_mem_iff_exists_mem $ λ t, by simp only [h.mem_iff, mem_principal]
lemma has_basis.eq_infi (h : l.has_basis (λ _, true) s) :
l = ⨅ i, 𝓟 (s i) :=
by simpa only [infi_true] using h.eq_binfi
lemma has_basis_infi_principal {s : ι → set α} (h : directed (≥) s) [nonempty ι] :
(⨅ i, 𝓟 (s i)).has_basis (λ _, true) s :=
⟨begin
refine λ t, (mem_infi_of_directed (h.mono_comp _ _) t).trans $
by simp only [exists_prop, true_and, mem_principal],
exact λ _ _, principal_mono.2
end⟩
/-- If `s : ι → set α` is an indexed family of sets, then finite intersections of `s i` form a basis
of `⨅ i, 𝓟 (s i)`. -/
lemma has_basis_infi_principal_finite {ι : Type*} (s : ι → set α) :
(⨅ i, 𝓟 (s i)).has_basis (λ t : set ι, finite t) (λ t, ⋂ i ∈ t, s i) :=
begin
refine ⟨λ U, (mem_infi_finite _).trans _⟩,
simp only [infi_principal_finset, mem_Union, mem_principal, exists_prop,
exists_finite_iff_finset, finset.set_bInter_coe]
end
lemma has_basis_binfi_principal {s : β → set α} {S : set β} (h : directed_on (s ⁻¹'o (≥)) S)
(ne : S.nonempty) :
(⨅ i ∈ S, 𝓟 (s i)).has_basis (λ i, i ∈ S) s :=
⟨begin
refine λ t, (mem_binfi_of_directed _ ne).trans $ by simp only [mem_principal],
rw [directed_on_iff_directed, ← directed_comp, (∘)] at h ⊢,
apply h.mono_comp _ _,
exact λ _ _, principal_mono.2
end⟩
lemma has_basis_binfi_principal' {ι : Type*} {p : ι → Prop} {s : ι → set α}
(h : ∀ i, p i → ∀ j, p j → ∃ k (h : p k), s k ⊆ s i ∧ s k ⊆ s j) (ne : ∃ i, p i) :
(⨅ i (h : p i), 𝓟 (s i)).has_basis p s :=
filter.has_basis_binfi_principal h ne
lemma has_basis.map (f : α → β) (hl : l.has_basis p s) :
(l.map f).has_basis p (λ i, f '' (s i)) :=
⟨λ t, by simp only [mem_map, image_subset_iff, hl.mem_iff, preimage]⟩
lemma has_basis.comap (f : β → α) (hl : l.has_basis p s) :
(l.comap f).has_basis p (λ i, f ⁻¹' (s i)) :=
⟨begin
intro t,
simp only [mem_comap, exists_prop, hl.mem_iff],
split,
{ rintros ⟨t', ⟨i, hi, ht'⟩, H⟩,
exact ⟨i, hi, subset.trans (preimage_mono ht') H⟩ },
{ rintros ⟨i, hi, H⟩,
exact ⟨s i, ⟨i, hi, subset.refl _⟩, H⟩ }
end⟩
lemma comap_has_basis (f : α → β) (l : filter β) :
has_basis (comap f l) (λ s : set β, s ∈ l) (λ s, f ⁻¹' s) :=
⟨λ t, mem_comap⟩
lemma has_basis.prod_self (hl : l.has_basis p s) :
(l ×ᶠ l).has_basis p (λ i, (s i).prod (s i)) :=
⟨begin
intro t,
apply mem_prod_iff.trans,
split,
{ rintros ⟨t₁, ht₁, t₂, ht₂, H⟩,
rcases hl.mem_iff.1 (inter_mem ht₁ ht₂) with ⟨i, hi, ht⟩,
exact ⟨i, hi, λ p ⟨hp₁, hp₂⟩, H ⟨(ht hp₁).1, (ht hp₂).2⟩⟩ },
{ rintros ⟨i, hi, H⟩,
exact ⟨s i, hl.mem_of_mem hi, s i, hl.mem_of_mem hi, H⟩ }
end⟩
lemma mem_prod_self_iff {s} : s ∈ l ×ᶠ l ↔ ∃ t ∈ l, set.prod t t ⊆ s :=
l.basis_sets.prod_self.mem_iff
lemma has_basis.sInter_sets (h : has_basis l p s) :
⋂₀ l.sets = ⋂ i (hi : p i), s i :=
begin
ext x,
suffices : (∀ t ∈ l, x ∈ t) ↔ ∀ i, p i → x ∈ s i,
by simpa only [mem_Inter, mem_set_of_eq, mem_sInter],
simp_rw h.mem_iff,
split,
{ intros h i hi,
exact h (s i) ⟨i, hi, subset.refl _⟩ },
{ rintros h _ ⟨i, hi, sub⟩,
exact sub (h i hi) },
end
variables {ι'' : Type*} [preorder ι''] (l) (p'' : ι'' → Prop) (s'' : ι'' → set α)
/-- `is_antitone_basis p s` means the image of `s` bounded by `p` is a filter basis
such that `s` is decreasing and `p` is increasing, ie `i ≤ j → p i → p j`. -/
structure is_antitone_basis extends is_basis p'' s'' : Prop :=
(decreasing : ∀ {i j}, p'' i → p'' j → i ≤ j → s'' j ⊆ s'' i)
(mono : monotone p'')
/-- We say that a filter `l` has an antitone basis `s : ι → set α` bounded by `p : ι → Prop`,
if `t ∈ l` if and only if `t` includes `s i` for some `i` such that `p i`,
and `s` is decreasing and `p` is increasing, ie `i ≤ j → p i → p j`. -/
structure has_antitone_basis (l : filter α) (p : ι'' → Prop) (s : ι'' → set α)
extends has_basis l p s : Prop :=
(decreasing : ∀ {i j}, p i → p j → i ≤ j → s j ⊆ s i)
(mono : monotone p)
end same_type
section two_types
variables {la : filter α} {pa : ι → Prop} {sa : ι → set α}
{lb : filter β} {pb : ι' → Prop} {sb : ι' → set β} {f : α → β}
lemma has_basis.tendsto_left_iff (hla : la.has_basis pa sa) :
tendsto f la lb ↔ ∀ t ∈ lb, ∃ i (hi : pa i), maps_to f (sa i) t :=
by { simp only [tendsto, (hla.map f).le_iff, image_subset_iff], refl }
lemma has_basis.tendsto_right_iff (hlb : lb.has_basis pb sb) :
tendsto f la lb ↔ ∀ i (hi : pb i), ∀ᶠ x in la, f x ∈ sb i :=
by simpa only [tendsto, hlb.ge_iff, mem_map, filter.eventually]
lemma has_basis.tendsto_iff (hla : la.has_basis pa sa) (hlb : lb.has_basis pb sb) :
tendsto f la lb ↔ ∀ ib (hib : pb ib), ∃ ia (hia : pa ia), ∀ x ∈ sa ia, f x ∈ sb ib :=
by simp [hlb.tendsto_right_iff, hla.eventually_iff]
lemma tendsto.basis_left (H : tendsto f la lb) (hla : la.has_basis pa sa) :
∀ t ∈ lb, ∃ i (hi : pa i), maps_to f (sa i) t :=
hla.tendsto_left_iff.1 H
lemma tendsto.basis_right (H : tendsto f la lb) (hlb : lb.has_basis pb sb) :
∀ i (hi : pb i), ∀ᶠ x in la, f x ∈ sb i :=
hlb.tendsto_right_iff.1 H
lemma tendsto.basis_both (H : tendsto f la lb) (hla : la.has_basis pa sa)
(hlb : lb.has_basis pb sb) :
∀ ib (hib : pb ib), ∃ ia (hia : pa ia), ∀ x ∈ sa ia, f x ∈ sb ib :=
(hla.tendsto_iff hlb).1 H
lemma has_basis.prod'' (hla : la.has_basis pa sa) (hlb : lb.has_basis pb sb) :
(la ×ᶠ lb).has_basis (λ i : pprod ι ι', pa i.1 ∧ pb i.2) (λ i, (sa i.1).prod (sb i.2)) :=
(hla.comap prod.fst).inf' (hlb.comap prod.snd)
lemma has_basis.prod {ι ι' : Type*} {pa : ι → Prop} {sa : ι → set α} {pb : ι' → Prop}
{sb : ι' → set β} (hla : la.has_basis pa sa) (hlb : lb.has_basis pb sb) :
(la ×ᶠ lb).has_basis (λ i : ι × ι', pa i.1 ∧ pb i.2) (λ i, (sa i.1).prod (sb i.2)) :=
(hla.comap prod.fst).inf (hlb.comap prod.snd)
lemma has_basis.prod' {la : filter α} {lb : filter β} {ι : Type*} {p : ι → Prop}
{sa : ι → set α} {sb : ι → set β}
(hla : la.has_basis p sa) (hlb : lb.has_basis p sb)
(h_dir : ∀ {i j}, p i → p j → ∃ k, p k ∧ sa k ⊆ sa i ∧ sb k ⊆ sb j) :
(la ×ᶠ lb).has_basis p (λ i, (sa i).prod (sb i)) :=
begin
simp only [has_basis_iff, (hla.prod hlb).mem_iff],
refine λ t, ⟨_, _⟩,
{ rintros ⟨⟨i, j⟩, ⟨hi, hj⟩, hsub : (sa i).prod (sb j) ⊆ t⟩,
rcases h_dir hi hj with ⟨k, hk, ki, kj⟩,
exact ⟨k, hk, (set.prod_mono ki kj).trans hsub⟩ },
{ rintro ⟨i, hi, h⟩,
exact ⟨⟨i, i⟩, ⟨hi, hi⟩, h⟩ },
end
end two_types
end filter
end sort
namespace filter
variables {α β γ ι ι' : Type*}
/-- `is_countably_generated f` means `f = generate s` for some countable `s`. -/
def is_countably_generated (f : filter α) : Prop :=
∃ s : set (set α), countable s ∧ f = generate s
/-- `is_countable_basis p s` means the image of `s` bounded by `p` is a countable filter basis. -/
structure is_countable_basis (p : ι → Prop) (s : ι → set α) extends is_basis p s : Prop :=
(countable : countable $ set_of p)
/-- We say that a filter `l` has a countable basis `s : ι → set α` bounded by `p : ι → Prop`,
if `t ∈ l` if and only if `t` includes `s i` for some `i` such that `p i`, and the set
defined by `p` is countable. -/
structure has_countable_basis (l : filter α) (p : ι → Prop) (s : ι → set α)
extends has_basis l p s : Prop :=
(countable : countable $ set_of p)
/-- A countable filter basis `B` on a type `α` is a nonempty countable collection of sets of `α`
such that the intersection of two elements of this collection contains some element
of the collection. -/
structure countable_filter_basis (α : Type*) extends filter_basis α :=
(countable : countable sets)
-- For illustration purposes, the countable filter basis defining (at_top : filter ℕ)
instance nat.inhabited_countable_filter_basis : inhabited (countable_filter_basis ℕ) :=
⟨{ countable := countable_range (λ n, Ici n),
..(default $ filter_basis ℕ),}⟩
lemma antitone_seq_of_seq (s : ℕ → set α) :
∃ t : ℕ → set α, (∀ i j, i ≤ j → t j ⊆ t i) ∧ (⨅ i, 𝓟 $ s i) = ⨅ i, 𝓟 (t i) :=
begin
use λ n, ⋂ m ≤ n, s m, split,
{ exact λ i j hij, bInter_mono' (Iic_subset_Iic.2 hij) (λ n hn, subset.refl _) },
apply le_antisymm; rw le_infi_iff; intro i,
{ rw le_principal_iff, refine (bInter_mem (finite_le_nat _)).2 (λ j hji, _),
rw ← le_principal_iff, apply infi_le_of_le j _, apply le_refl _ },
{ apply infi_le_of_le i _, rw principal_mono, intro a, simp, intro h, apply h, refl },
end
lemma countable_binfi_eq_infi_seq [complete_lattice α] {B : set ι} (Bcbl : countable B)
(Bne : B.nonempty) (f : ι → α) :
∃ (x : ℕ → ι), (⨅ t ∈ B, f t) = ⨅ i, f (x i) :=
begin
rw countable_iff_exists_surjective_to_subtype Bne at Bcbl,
rcases Bcbl with ⟨g, gsurj⟩,
rw infi_subtype',
use (λ n, g n), apply le_antisymm; rw le_infi_iff,
{ intro i, apply infi_le_of_le (g i) _, apply le_refl _ },
{ intros a, rcases gsurj a with ⟨i, rfl⟩, apply infi_le }
end
lemma countable_binfi_eq_infi_seq' [complete_lattice α] {B : set ι} (Bcbl : countable B) (f : ι → α)
{i₀ : ι} (h : f i₀ = ⊤) :
∃ (x : ℕ → ι), (⨅ t ∈ B, f t) = ⨅ i, f (x i) :=
begin
cases B.eq_empty_or_nonempty with hB Bnonempty,
{ rw [hB, infi_emptyset],
use λ n, i₀,
simp [h] },
{ exact countable_binfi_eq_infi_seq Bcbl Bnonempty f }
end
lemma countable_binfi_principal_eq_seq_infi {B : set (set α)} (Bcbl : countable B) :
∃ (x : ℕ → set α), (⨅ t ∈ B, 𝓟 t) = ⨅ i, 𝓟 (x i) :=
countable_binfi_eq_infi_seq' Bcbl 𝓟 principal_univ
namespace is_countably_generated
/-- A set generating a countably generated filter. -/
def generating_set {f : filter α} (h : is_countably_generated f) :=
classical.some h
lemma countable_generating_set {f : filter α} (h : is_countably_generated f) :
countable h.generating_set :=
(classical.some_spec h).1
lemma eq_generate {f : filter α} (h : is_countably_generated f) :
f = generate h.generating_set :=
(classical.some_spec h).2
/-- A countable filter basis for a countably generated filter. -/
def countable_filter_basis {l : filter α} (h : is_countably_generated l) :
countable_filter_basis α :=
{ countable := (countable_set_of_finite_subset h.countable_generating_set).image _,
..filter_basis.of_sets (h.generating_set) }
lemma filter_basis_filter {l : filter α} (h : is_countably_generated l) :
h.countable_filter_basis.to_filter_basis.filter = l :=
begin
conv_rhs { rw h.eq_generate },
apply of_sets_filter_eq_generate,
end
lemma has_countable_basis {l : filter α} (h : is_countably_generated l) :
l.has_countable_basis (λ t, finite t ∧ t ⊆ h.generating_set) (λ t, ⋂₀ t) :=
⟨by convert has_basis_generate _ ; exact h.eq_generate,
countable_set_of_finite_subset h.countable_generating_set⟩
lemma exists_countable_infi_principal {f : filter α} (h : f.is_countably_generated) :
∃ s : set (set α), countable s ∧ f = ⨅ t ∈ s, 𝓟 t :=
begin
let B := h.countable_filter_basis,
use [B.sets, B.countable],
rw ← h.filter_basis_filter,
rw B.to_filter_basis.eq_infi_principal,
rw infi_subtype''
end
lemma exists_seq {f : filter α} (cblb : f.is_countably_generated) :
∃ x : ℕ → set α, f = ⨅ i, 𝓟 (x i) :=
begin
rcases cblb.exists_countable_infi_principal with ⟨B, Bcbl, rfl⟩,
exact countable_binfi_principal_eq_seq_infi Bcbl,
end
/-- If `f` is countably generated and `f.has_basis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
lemma exists_antitone_subbasis {f : filter α} (cblb : f.is_countably_generated)
{p : ι → Prop} {s : ι → set α} (hs : f.has_basis p s) :
∃ x : ℕ → ι, (∀ i, p (x i)) ∧ f.has_antitone_basis (λ _, true) (λ i, s (x i)) :=
begin
rcases cblb.exists_seq with ⟨x', hx'⟩,
have : ∀ i, x' i ∈ f := λ i, hx'.symm ▸ (infi_le (λ i, 𝓟 (x' i)) i) (mem_principal_self _),
let x : ℕ → {i : ι // p i} := λ n, nat.rec_on n (hs.index _ $ this 0)
(λ n xn, (hs.index _ $ inter_mem (this $ n + 1) (hs.mem_of_mem xn.coe_prop))),
have x_mono : ∀ n : ℕ, s (x n.succ) ⊆ s (x n) :=
λ n, subset.trans (hs.set_index_subset _) (inter_subset_right _ _),
replace x_mono : ∀ ⦃i j⦄, i ≤ j → s (x j) ≤ s (x i),
{ refine @monotone_nat_of_le_succ (order_dual $ set α) _ _ _,
exact x_mono },
have x_subset : ∀ i, s (x i) ⊆ x' i,
{ rintro (_|i),
exacts [hs.set_index_subset _, subset.trans (hs.set_index_subset _) (inter_subset_left _ _)] },
refine ⟨λ i, x i, λ i, (x i).2, _⟩,
have : (⨅ i, 𝓟 (s (x i))).has_antitone_basis (λ _, true) (λ i, s (x i)) :=
⟨has_basis_infi_principal (directed_of_sup x_mono), λ i j _ _ hij, x_mono hij, monotone_const⟩,
convert this,
exact le_antisymm (le_infi $ λ i, le_principal_iff.2 $ by cases i; apply hs.set_index_mem)
(hx'.symm ▸ le_infi (λ i, le_principal_iff.2 $
this.to_has_basis.mem_iff.2 ⟨i, trivial, x_subset i⟩))
end
/-- A countably generated filter admits a basis formed by an antitone sequence of sets. -/
lemma exists_antitone_basis {f : filter α} (cblb : f.is_countably_generated) :
∃ x : ℕ → set α, f.has_antitone_basis (λ _, true) x :=
let ⟨x, hxf, hx⟩ := cblb.exists_antitone_subbasis f.basis_sets in ⟨x, hx⟩
end is_countably_generated
lemma has_countable_basis.is_countably_generated {f : filter α} {p : ι → Prop} {s : ι → set α}
(h : f.has_countable_basis p s) :
f.is_countably_generated :=
⟨{t | ∃ i, p i ∧ s i = t}, h.countable.image s, h.to_has_basis.eq_generate⟩
lemma is_countably_generated_seq (x : ℕ → set α) : is_countably_generated (⨅ i, 𝓟 $ x i) :=
begin
obtain ⟨y, am, h⟩ := antitone_seq_of_seq x,
rw h,
use [range y, countable_range _],
rw (has_basis_infi_principal _).eq_generate,
{ simp [range] },
{ exact directed_of_sup am },
{ use 0 },
end
lemma is_countably_generated_of_seq {f : filter α} (h : ∃ x : ℕ → set α, f = ⨅ i, 𝓟 $ x i) :
f.is_countably_generated :=
let ⟨x, h⟩ := h in by rw h ; apply is_countably_generated_seq
lemma is_countably_generated_binfi_principal {B : set $ set α} (h : countable B) :
is_countably_generated (⨅ (s ∈ B), 𝓟 s) :=
is_countably_generated_of_seq (countable_binfi_principal_eq_seq_infi h)
lemma is_countably_generated_iff_exists_antitone_basis {f : filter α} :
is_countably_generated f ↔ ∃ x : ℕ → set α, f.has_antitone_basis (λ _, true) x :=
begin
split,
{ exact λ h, h.exists_antitone_basis },
{ rintros ⟨x, h⟩,
rw h.to_has_basis.eq_infi,
exact is_countably_generated_seq x },
end
lemma is_countably_generated_principal (s : set α) : is_countably_generated (𝓟 s) :=
begin
rw show 𝓟 s = ⨅ i : ℕ, 𝓟 s, by simp,
apply is_countably_generated_seq
end
namespace is_countably_generated
lemma inf {f g : filter α} (hf : is_countably_generated f) (hg : is_countably_generated g) :
is_countably_generated (f ⊓ g) :=
begin
rw is_countably_generated_iff_exists_antitone_basis at hf hg,
rcases hf with ⟨s, hs⟩,
rcases hg with ⟨t, ht⟩,
exact has_countable_basis.is_countably_generated
⟨hs.to_has_basis.inf ht.to_has_basis, set.countable_encodable _⟩
end
lemma inf_principal {f : filter α} (h : is_countably_generated f) (s : set α) :
is_countably_generated (f ⊓ 𝓟 s) :=
h.inf (filter.is_countably_generated_principal s)
lemma exists_antitone_seq' {f : filter α} (cblb : f.is_countably_generated) :
∃ x : ℕ → set α, (∀ i j, i ≤ j → x j ⊆ x i) ∧ ∀ {s}, (s ∈ f ↔ ∃ i, x i ⊆ s) :=
let ⟨x, hx⟩ := is_countably_generated_iff_exists_antitone_basis.mp cblb in
⟨x, λ i j, hx.decreasing trivial trivial, λ s, by simp [hx.to_has_basis.mem_iff]⟩
protected lemma comap {l : filter β} (h : l.is_countably_generated) (f : α → β) :
(comap f l).is_countably_generated :=
let ⟨x, hx_mono⟩ := h.exists_antitone_basis in
is_countably_generated_of_seq ⟨_, (hx_mono.to_has_basis.comap _).eq_infi⟩
end is_countably_generated
end filter
|
db46e274d00e4fc571b17c416061008b9bf67def | 4727251e0cd73359b15b664c3170e5d754078599 | /src/data/seq/seq.lean | 05f1631891f2ac21aaa8233ee1ab5b39e6a8a1fa | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 28,430 | lean | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import data.lazy_list
import data.nat.basic
import data.stream.init
import data.seq.computation
universes u v w
/-
coinductive seq (α : Type u) : Type u
| nil : seq α
| cons : α → seq α → seq α
-/
/--
A stream `s : option α` is a sequence if `s.nth n = none` implies `s.nth (n + 1) = none`.
-/
def stream.is_seq {α : Type u} (s : stream (option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
/-- `seq α` is the type of possibly infinite lists (referred here as sequences).
It is encoded as an infinite stream of options such that if `f n = none`, then
`f m = none` for all `m ≥ n`. -/
def seq (α : Type u) : Type u := { f : stream (option α) // f.is_seq }
/-- `seq1 α` is the type of nonempty sequences. -/
def seq1 (α) := α × seq α
namespace seq
variables {α : Type u} {β : Type v} {γ : Type w}
/-- The empty sequence -/
def nil : seq α := ⟨stream.const none, λn h, rfl⟩
instance : inhabited (seq α) := ⟨nil⟩
/-- Prepend an element to a sequence -/
def cons (a : α) : seq α → seq α
| ⟨f, al⟩ := ⟨some a :: f, λn h, by {cases n with n, contradiction, exact al h}⟩
/-- Get the nth element of a sequence (if it exists) -/
def nth : seq α → ℕ → option α := subtype.val
/-- A sequence has terminated at position `n` if the value at position `n` equals `none`. -/
def terminated_at (s : seq α) (n : ℕ) : Prop := s.nth n = none
/-- It is decidable whether a sequence terminates at a given position. -/
instance terminated_at_decidable (s : seq α) (n : ℕ) : decidable (s.terminated_at n) :=
decidable_of_iff' (s.nth n).is_none $ by unfold terminated_at; cases s.nth n; simp
/-- A sequence terminates if there is some position `n` at which it has terminated. -/
def terminates (s : seq α) : Prop := ∃ (n : ℕ), s.terminated_at n
/-- Functorial action of the functor `option (α × _)` -/
@[simp] def omap (f : β → γ) : option (α × β) → option (α × γ)
| none := none
| (some (a, b)) := some (a, f b)
/-- Get the first element of a sequence -/
def head (s : seq α) : option α := nth s 0
/-- Get the tail of a sequence (or `nil` if the sequence is `nil`) -/
def tail : seq α → seq α
| ⟨f, al⟩ := ⟨f.tail, λ n, al⟩
protected def mem (a : α) (s : seq α) := some a ∈ s.1
instance : has_mem α (seq α) :=
⟨seq.mem⟩
theorem le_stable (s : seq α) {m n} (h : m ≤ n) :
s.nth m = none → s.nth n = none :=
by {cases s with f al, induction h with n h IH, exacts [id, λ h2, al (IH h2)]}
/-- If a sequence terminated at position `n`, it also terminated at `m ≥ n `. -/
lemma terminated_stable (s : seq α) {m n : ℕ} (m_le_n : m ≤ n)
(terminated_at_m : s.terminated_at m) :
s.terminated_at n :=
le_stable s m_le_n terminated_at_m
/--
If `s.nth n = some aₙ` for some value `aₙ`, then there is also some value `aₘ` such
that `s.nth = some aₘ` for `m ≤ n`.
-/
lemma ge_stable (s : seq α) {aₙ : α} {n m : ℕ} (m_le_n : m ≤ n)
(s_nth_eq_some : s.nth n = some aₙ) :
∃ (aₘ : α), s.nth m = some aₘ :=
have s.nth n ≠ none, by simp [s_nth_eq_some],
have s.nth m ≠ none, from mt (s.le_stable m_le_n) this,
option.ne_none_iff_exists'.mp this
theorem not_mem_nil (a : α) : a ∉ @nil α :=
λ ⟨n, (h : some a = none)⟩, by injection h
theorem mem_cons (a : α) : ∀ (s : seq α), a ∈ cons a s
| ⟨f, al⟩ := stream.mem_cons (some a) _
theorem mem_cons_of_mem (y : α) {a : α} : ∀ {s : seq α}, a ∈ s → a ∈ cons y s
| ⟨f, al⟩ := stream.mem_cons_of_mem (some y)
theorem eq_or_mem_of_mem_cons {a b : α} : ∀ {s : seq α}, a ∈ cons b s → a = b ∨ a ∈ s
| ⟨f, al⟩ h := (stream.eq_or_mem_of_mem_cons h).imp_left (λh, by injection h)
@[simp] theorem mem_cons_iff {a b : α} {s : seq α} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=
⟨eq_or_mem_of_mem_cons, λo, by cases o with e m;
[{rw e, apply mem_cons}, exact mem_cons_of_mem _ m]⟩
/-- Destructor for a sequence, resulting in either `none` (for `nil`) or
`some (a, s)` (for `cons a s`). -/
def destruct (s : seq α) : option (seq1 α) :=
(λa', (a', s.tail)) <$> nth s 0
theorem destruct_eq_nil {s : seq α} : destruct s = none → s = nil :=
begin
dsimp [destruct],
induction f0 : nth s 0; intro h,
{ apply subtype.eq,
funext n,
induction n with n IH, exacts [f0, s.2 IH] },
{ contradiction }
end
theorem destruct_eq_cons {s : seq α} {a s'} : destruct s = some (a, s') → s = cons a s' :=
begin
dsimp [destruct],
induction f0 : nth s 0 with a'; intro h,
{ contradiction },
{ cases s with f al,
injections with _ h1 h2,
rw ←h2, apply subtype.eq, dsimp [tail, cons],
rw h1 at f0, rw ←f0,
exact (stream.eta f).symm }
end
@[simp] theorem destruct_nil : destruct (nil : seq α) = none := rfl
@[simp] theorem destruct_cons (a : α) : ∀ s, destruct (cons a s) = some (a, s)
| ⟨f, al⟩ := begin
unfold cons destruct functor.map,
apply congr_arg (λ s, some (a, s)),
apply subtype.eq, dsimp [tail], rw [stream.tail_cons]
end
theorem head_eq_destruct (s : seq α) : head s = prod.fst <$> destruct s :=
by unfold destruct head; cases nth s 0; refl
@[simp] theorem head_nil : head (nil : seq α) = none := rfl
@[simp] theorem head_cons (a : α) (s) : head (cons a s) = some a :=
by rw [head_eq_destruct, destruct_cons]; refl
@[simp] theorem tail_nil : tail (nil : seq α) = nil := rfl
@[simp] theorem tail_cons (a : α) (s) : tail (cons a s) = s :=
by cases s with f al; apply subtype.eq; dsimp [tail, cons]; rw [stream.tail_cons]
def cases_on {C : seq α → Sort v} (s : seq α)
(h1 : C nil) (h2 : ∀ x s, C (cons x s)) : C s := begin
induction H : destruct s with v v,
{ rw destruct_eq_nil H, apply h1 },
{ cases v with a s', rw destruct_eq_cons H, apply h2 }
end
theorem mem_rec_on {C : seq α → Prop} {a s} (M : a ∈ s)
(h1 : ∀ b s', (a = b ∨ C s') → C (cons b s')) : C s :=
begin
cases M with k e, unfold stream.nth at e,
induction k with k IH generalizing s,
{ have TH : s = cons a (tail s),
{ apply destruct_eq_cons,
unfold destruct nth functor.map, rw ←e, refl },
rw TH, apply h1 _ _ (or.inl rfl) },
revert e, apply s.cases_on _ (λ b s', _); intro e,
{ injection e },
{ have h_eq : (cons b s').val (nat.succ k) = s'.val k, { cases s'; refl },
rw [h_eq] at e,
apply h1 _ _ (or.inr (IH e)) }
end
def corec.F (f : β → option (α × β)) : option β → option α × option β
| none := (none, none)
| (some b) := match f b with none := (none, none) | some (a, b') := (some a, some b') end
/-- Corecursor for `seq α` as a coinductive type. Iterates `f` to produce new elements
of the sequence until `none` is obtained. -/
def corec (f : β → option (α × β)) (b : β) : seq α :=
begin
refine ⟨stream.corec' (corec.F f) (some b), λn h, _⟩,
rw stream.corec'_eq,
change stream.corec' (corec.F f) (corec.F f (some b)).2 n = none,
revert h, generalize : some b = o, revert o,
induction n with n IH; intro o,
{ change (corec.F f o).1 = none → (corec.F f (corec.F f o).2).1 = none,
cases o with b; intro h, { refl },
dsimp [corec.F] at h, dsimp [corec.F],
cases f b with s, { refl },
{ cases s with a b', contradiction } },
{ rw [stream.corec'_eq (corec.F f) (corec.F f o).2,
stream.corec'_eq (corec.F f) o],
exact IH (corec.F f o).2 }
end
@[simp] theorem corec_eq (f : β → option (α × β)) (b : β) :
destruct (corec f b) = omap (corec f) (f b) :=
begin
dsimp [corec, destruct, nth],
change stream.corec' (corec.F f) (some b) 0 with (corec.F f (some b)).1,
dsimp [corec.F],
induction h : f b with s, { refl },
cases s with a b', dsimp [corec.F],
apply congr_arg (λ b', some (a, b')),
apply subtype.eq,
dsimp [corec, tail],
rw [stream.corec'_eq, stream.tail_cons],
dsimp [corec.F], rw h, refl
end
/-- Embed a list as a sequence -/
def of_list (l : list α) : seq α :=
⟨list.nth l, λn h, begin
induction l with a l IH generalizing n, refl,
dsimp [list.nth], cases n with n; dsimp [list.nth] at h,
{ contradiction },
{ apply IH _ h }
end⟩
instance coe_list : has_coe (list α) (seq α) := ⟨of_list⟩
section bisim
variable (R : seq α → seq α → Prop)
local infix ` ~ `:50 := R
def bisim_o : option (seq1 α) → option (seq1 α) → Prop
| none none := true
| (some (a, s)) (some (a', s')) := a = a' ∧ R s s'
| _ _ := false
attribute [simp] bisim_o
def is_bisimulation := ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → bisim_o R (destruct s₁) (destruct s₂)
-- If two streams are bisimilar, then they are equal
theorem eq_of_bisim (bisim : is_bisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ :=
begin
apply subtype.eq,
apply stream.eq_of_bisim (λx y, ∃ s s' : seq α, s.1 = x ∧ s'.1 = y ∧ R s s'),
dsimp [stream.is_bisimulation],
intros t₁ t₂ e,
exact match t₁, t₂, e with ._, ._, ⟨s, s', rfl, rfl, r⟩ :=
suffices head s = head s' ∧ R (tail s) (tail s'), from
and.imp id (λr, ⟨tail s, tail s',
by cases s; refl, by cases s'; refl, r⟩) this,
begin
have := bisim r, revert r this,
apply cases_on s _ _; intros; apply cases_on s' _ _; intros; intros r this,
{ constructor, refl, assumption },
{ rw [destruct_nil, destruct_cons] at this,
exact false.elim this },
{ rw [destruct_nil, destruct_cons] at this,
exact false.elim this },
{ rw [destruct_cons, destruct_cons] at this,
rw [head_cons, head_cons, tail_cons, tail_cons],
cases this with h1 h2,
constructor, rw h1, exact h2 }
end
end,
exact ⟨s₁, s₂, rfl, rfl, r⟩
end
end bisim
theorem coinduction : ∀ {s₁ s₂ : seq α}, head s₁ = head s₂ →
(∀ (β : Type u) (fr : seq α → β),
fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂)) → s₁ = s₂
| ⟨f₁, a₁⟩ ⟨f₂, a₂⟩ hh ht :=
subtype.eq (stream.coinduction hh (λ β fr, ht β (λs, fr s.1)))
theorem coinduction2 (s) (f g : seq α → seq β)
(H : ∀ s, bisim_o (λ (s1 s2 : seq β), ∃ (s : seq α), s1 = f s ∧ s2 = g s)
(destruct (f s)) (destruct (g s)))
: f s = g s :=
begin
refine eq_of_bisim (λ s1 s2, ∃ s, s1 = f s ∧ s2 = g s) _ ⟨s, rfl, rfl⟩,
intros s1 s2 h, rcases h with ⟨s, h1, h2⟩,
rw [h1, h2], apply H
end
/-- Embed an infinite stream as a sequence -/
def of_stream (s : stream α) : seq α :=
⟨s.map some, λn h, by contradiction⟩
instance coe_stream : has_coe (stream α) (seq α) := ⟨of_stream⟩
/-- Embed a `lazy_list α` as a sequence. Note that even though this
is non-meta, it will produce infinite sequences if used with
cyclic `lazy_list`s created by meta constructions. -/
def of_lazy_list : lazy_list α → seq α :=
corec (λl, match l with
| lazy_list.nil := none
| lazy_list.cons a l' := some (a, l' ())
end)
instance coe_lazy_list : has_coe (lazy_list α) (seq α) := ⟨of_lazy_list⟩
/-- Translate a sequence into a `lazy_list`. Since `lazy_list` and `list`
are isomorphic as non-meta types, this function is necessarily meta. -/
meta def to_lazy_list : seq α → lazy_list α | s :=
match destruct s with
| none := lazy_list.nil
| some (a, s') := lazy_list.cons a (to_lazy_list s')
end
/-- Translate a sequence to a list. This function will run forever if
run on an infinite sequence. -/
meta def force_to_list (s : seq α) : list α := (to_lazy_list s).to_list
/-- The sequence of natural numbers some 0, some 1, ... -/
def nats : seq ℕ := stream.nats
@[simp]
lemma nats_nth (n : ℕ) : nats.nth n = some n := rfl
/-- Append two sequences. If `s₁` is infinite, then `s₁ ++ s₂ = s₁`,
otherwise it puts `s₂` at the location of the `nil` in `s₁`. -/
def append (s₁ s₂ : seq α) : seq α :=
@corec α (seq α × seq α) (λ⟨s₁, s₂⟩,
match destruct s₁ with
| none := omap (λs₂, (nil, s₂)) (destruct s₂)
| some (a, s₁') := some (a, s₁', s₂)
end) (s₁, s₂)
/-- Map a function over a sequence. -/
def map (f : α → β) : seq α → seq β | ⟨s, al⟩ :=
⟨s.map (option.map f),
λn, begin
dsimp [stream.map, stream.nth],
induction e : s n; intro,
{ rw al e, assumption }, { contradiction }
end⟩
/-- Flatten a sequence of sequences. (It is required that the
sequences be nonempty to ensure productivity; in the case
of an infinite sequence of `nil`, the first element is never
generated.) -/
def join : seq (seq1 α) → seq α :=
corec (λS, match destruct S with
| none := none
| some ((a, s), S') := some (a, match destruct s with
| none := S'
| some s' := cons s' S'
end)
end)
/-- Remove the first `n` elements from the sequence. -/
def drop (s : seq α) : ℕ → seq α
| 0 := s
| (n+1) := tail (drop n)
attribute [simp] drop
/-- Take the first `n` elements of the sequence (producing a list) -/
def take : ℕ → seq α → list α
| 0 s := []
| (n+1) s := match destruct s with
| none := []
| some (x, r) := list.cons x (take n r)
end
/-- Split a sequence at `n`, producing a finite initial segment
and an infinite tail. -/
def split_at : ℕ → seq α → list α × seq α
| 0 s := ([], s)
| (n+1) s := match destruct s with
| none := ([], nil)
| some (x, s') := let (l, r) := split_at n s' in (list.cons x l, r)
end
section zip_with
/-- Combine two sequences with a function -/
def zip_with (f : α → β → γ) : seq α → seq β → seq γ
| ⟨f₁, a₁⟩ ⟨f₂, a₂⟩ := ⟨λn,
match f₁ n, f₂ n with
| some a, some b := some (f a b)
| _, _ := none
end,
λn, begin
induction h1 : f₁ n,
{ intro H, simp only [(a₁ h1)], refl },
induction h2 : f₂ n; dsimp [seq.zip_with._match_1]; intro H,
{ rw (a₂ h2), cases f₁ (n + 1); refl },
{ rw [h1, h2] at H, contradiction }
end⟩
variables {s : seq α} {s' : seq β} {n : ℕ}
lemma zip_with_nth_some {a : α} {b : β} (s_nth_eq_some : s.nth n = some a)
(s_nth_eq_some' : s'.nth n = some b) (f : α → β → γ) :
(zip_with f s s').nth n = some (f a b) :=
begin
cases s with st,
have : st n = some a, from s_nth_eq_some,
cases s' with st',
have : st' n = some b, from s_nth_eq_some',
simp only [zip_with, seq.nth, *]
end
lemma zip_with_nth_none (s_nth_eq_none : s.nth n = none) (f : α → β → γ) :
(zip_with f s s').nth n = none :=
begin
cases s with st,
have : st n = none, from s_nth_eq_none,
cases s' with st',
cases st'_nth_eq : st' n;
simp only [zip_with, seq.nth, *]
end
lemma zip_with_nth_none' (s'_nth_eq_none : s'.nth n = none) (f : α → β → γ) :
(zip_with f s s').nth n = none :=
begin
cases s' with st',
have : st' n = none, from s'_nth_eq_none,
cases s with st,
cases st_nth_eq : st n;
simp only [zip_with, seq.nth, *]
end
end zip_with
/-- Pair two sequences into a sequence of pairs -/
def zip : seq α → seq β → seq (α × β) := zip_with prod.mk
/-- Separate a sequence of pairs into two sequences -/
def unzip (s : seq (α × β)) : seq α × seq β := (map prod.fst s, map prod.snd s)
/-- Convert a sequence which is known to terminate into a list -/
def to_list (s : seq α) (h : ∃ n, ¬ (nth s n).is_some) : list α :=
take (nat.find h) s
/-- Convert a sequence which is known not to terminate into a stream -/
def to_stream (s : seq α) (h : ∀ n, (nth s n).is_some) : stream α :=
λn, option.get (h n)
/-- Convert a sequence into either a list or a stream depending on whether
it is finite or infinite. (Without decidability of the infiniteness predicate,
this is not constructively possible.) -/
def to_list_or_stream (s : seq α) [decidable (∃ n, ¬ (nth s n).is_some)] :
list α ⊕ stream α :=
if h : ∃ n, ¬ (nth s n).is_some
then sum.inl (to_list s h)
else sum.inr (to_stream s (λn, decidable.by_contradiction (λ hn, h ⟨n, hn⟩)))
@[simp] theorem nil_append (s : seq α) : append nil s = s :=
begin
apply coinduction2, intro s,
dsimp [append], rw [corec_eq],
dsimp [append], apply cases_on s _ _,
{ trivial },
{ intros x s,
rw [destruct_cons], dsimp,
exact ⟨rfl, s, rfl, rfl⟩ }
end
@[simp] theorem cons_append (a : α) (s t) : append (cons a s) t = cons a (append s t) :=
destruct_eq_cons $ begin
dsimp [append], rw [corec_eq],
dsimp [append], rw [destruct_cons],
dsimp [append], refl
end
@[simp] theorem append_nil (s : seq α) : append s nil = s :=
begin
apply coinduction2 s, intro s,
apply cases_on s _ _,
{ trivial },
{ intros x s,
rw [cons_append, destruct_cons, destruct_cons], dsimp,
exact ⟨rfl, s, rfl, rfl⟩ }
end
@[simp] theorem append_assoc (s t u : seq α) :
append (append s t) u = append s (append t u) :=
begin
apply eq_of_bisim (λs1 s2, ∃ s t u,
s1 = append (append s t) u ∧ s2 = append s (append t u)),
{ intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, t, u, rfl, rfl⟩ := begin
apply cases_on s; simp,
{ apply cases_on t; simp,
{ apply cases_on u; simp,
{ intros x u, refine ⟨nil, nil, u, _, _⟩; simp } },
{ intros x t, refine ⟨nil, t, u, _, _⟩; simp } },
{ intros x s, exact ⟨s, t, u, rfl, rfl⟩ }
end end },
{ exact ⟨s, t, u, rfl, rfl⟩ }
end
@[simp] theorem map_nil (f : α → β) : map f nil = nil := rfl
@[simp] theorem map_cons (f : α → β) (a) : ∀ s, map f (cons a s) = cons (f a) (map f s)
| ⟨s, al⟩ := by apply subtype.eq; dsimp [cons, map]; rw stream.map_cons; refl
@[simp] theorem map_id : ∀ (s : seq α), map id s = s
| ⟨s, al⟩ := begin
apply subtype.eq; dsimp [map],
rw [option.map_id, stream.map_id]; refl
end
@[simp] theorem map_tail (f : α → β) : ∀ s, map f (tail s) = tail (map f s)
| ⟨s, al⟩ := by apply subtype.eq; dsimp [tail, map]; rw stream.map_tail; refl
theorem map_comp (f : α → β) (g : β → γ) : ∀ (s : seq α), map (g ∘ f) s = map g (map f s)
| ⟨s, al⟩ := begin
apply subtype.eq; dsimp [map],
rw stream.map_map,
apply congr_arg (λ f : _ → option γ, stream.map f s),
ext ⟨⟩; refl
end
@[simp] theorem map_append (f : α → β) (s t) : map f (append s t) = append (map f s) (map f t) :=
begin
apply eq_of_bisim (λs1 s2, ∃ s t,
s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)) _ ⟨s, t, rfl, rfl⟩,
intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, t, rfl, rfl⟩ := begin
apply cases_on s; simp,
{ apply cases_on t; simp,
{ intros x t, refine ⟨nil, t, _, _⟩; simp } },
{ intros x s, refine ⟨s, t, rfl, rfl⟩ }
end end
end
@[simp] theorem map_nth (f : α → β) : ∀ s n, nth (map f s) n = (nth s n).map f
| ⟨s, al⟩ n := rfl
instance : functor seq := {map := @map}
instance : is_lawful_functor seq :=
{ id_map := @map_id, comp_map := @map_comp }
@[simp] theorem join_nil : join nil = (nil : seq α) := destruct_eq_nil rfl
@[simp] theorem join_cons_nil (a : α) (S) :
join (cons (a, nil) S) = cons a (join S) :=
destruct_eq_cons $ by simp [join]
@[simp] theorem join_cons_cons (a b : α) (s S) :
join (cons (a, cons b s) S) = cons a (join (cons (b, s) S)) :=
destruct_eq_cons $ by simp [join]
@[simp, priority 990] theorem join_cons (a : α) (s S) :
join (cons (a, s) S) = cons a (append s (join S)) :=
begin
apply eq_of_bisim (λs1 s2, s1 = s2 ∨
∃ a s S, s1 = join (cons (a, s) S) ∧
s2 = cons a (append s (join S))) _ (or.inr ⟨a, s, S, rfl, rfl⟩),
intros s1 s2 h,
exact match s1, s2, h with
| _, _, (or.inl $ eq.refl s) := begin
apply cases_on s, { trivial },
{ intros x s, rw [destruct_cons], exact ⟨rfl, or.inl rfl⟩ }
end
| ._, ._, (or.inr ⟨a, s, S, rfl, rfl⟩) := begin
apply cases_on s,
{ simp },
{ intros x s, simp, refine or.inr ⟨x, s, S, rfl, rfl⟩ }
end
end
end
@[simp] theorem join_append (S T : seq (seq1 α)) :
join (append S T) = append (join S) (join T) :=
begin
apply eq_of_bisim (λs1 s2, ∃ s S T,
s1 = append s (join (append S T)) ∧
s2 = append s (append (join S) (join T))),
{ intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, S, T, rfl, rfl⟩ := begin
apply cases_on s; simp,
{ apply cases_on S; simp,
{ apply cases_on T, { simp },
{ intros s T, cases s with a s; simp,
refine ⟨s, nil, T, _, _⟩; simp } },
{ intros s S, cases s with a s; simp,
exact ⟨s, S, T, rfl, rfl⟩ } },
{ intros x s, exact ⟨s, S, T, rfl, rfl⟩ }
end end },
{ refine ⟨nil, S, T, _, _⟩; simp }
end
@[simp] theorem of_list_nil : of_list [] = (nil : seq α) := rfl
@[simp] theorem of_list_cons (a : α) (l) :
of_list (a :: l) = cons a (of_list l) :=
by ext (_|n) : 2; simp [of_list, cons, stream.nth, stream.cons]
@[simp] theorem of_stream_cons (a : α) (s) :
of_stream (a :: s) = cons a (of_stream s) :=
by apply subtype.eq; simp [of_stream, cons]; rw stream.map_cons
@[simp] theorem of_list_append (l l' : list α) :
of_list (l ++ l') = append (of_list l) (of_list l') :=
by induction l; simp [*]
@[simp] theorem of_stream_append (l : list α) (s : stream α) :
of_stream (l ++ₛ s) = append (of_list l) (of_stream s) :=
by induction l; simp [*, stream.nil_append_stream, stream.cons_append_stream]
/-- Convert a sequence into a list, embedded in a computation to allow for
the possibility of infinite sequences (in which case the computation
never returns anything). -/
def to_list' {α} (s : seq α) : computation (list α) :=
@computation.corec (list α) (list α × seq α) (λ⟨l, s⟩,
match destruct s with
| none := sum.inl l.reverse
| some (a, s') := sum.inr (a::l, s')
end) ([], s)
theorem dropn_add (s : seq α) (m) : ∀ n, drop s (m + n) = drop (drop s m) n
| 0 := rfl
| (n+1) := congr_arg tail (dropn_add n)
theorem dropn_tail (s : seq α) (n) : drop (tail s) n = drop s (n + 1) :=
by rw add_comm; symmetry; apply dropn_add
theorem nth_tail : ∀ (s : seq α) n, nth (tail s) n = nth s (n + 1)
| ⟨f, al⟩ n := rfl
@[ext]
protected lemma ext (s s': seq α) (hyp : ∀ (n : ℕ), s.nth n = s'.nth n) : s = s' :=
begin
let ext := (λ (s s' : seq α), ∀ n, s.nth n = s'.nth n),
apply seq.eq_of_bisim ext _ hyp,
-- we have to show that ext is a bisimulation
clear hyp s s',
assume s s' (hyp : ext s s'),
unfold seq.destruct,
rw (hyp 0),
cases (s'.nth 0),
{ simp [seq.bisim_o] }, -- option.none
{ -- option.some
suffices : ext s.tail s'.tail, by simpa,
assume n,
simp only [seq.nth_tail _ n, (hyp $ n + 1)] }
end
@[simp] theorem head_dropn (s : seq α) (n) : head (drop s n) = nth s n :=
begin
induction n with n IH generalizing s, { refl },
rw [nat.succ_eq_add_one, ←nth_tail, ←dropn_tail], apply IH
end
theorem mem_map (f : α → β) {a : α} : ∀ {s : seq α}, a ∈ s → f a ∈ map f s
| ⟨g, al⟩ := stream.mem_map (option.map f)
theorem exists_of_mem_map {f} {b : β} : ∀ {s : seq α}, b ∈ map f s → ∃ a, a ∈ s ∧ f a = b
| ⟨g, al⟩ h := let ⟨o, om, oe⟩ := stream.exists_of_mem_map h in
by cases o with a; injection oe with h'; exact ⟨a, om, h'⟩
theorem of_mem_append {s₁ s₂ : seq α} {a : α} (h : a ∈ append s₁ s₂) : a ∈ s₁ ∨ a ∈ s₂ :=
begin
have := h, revert this,
generalize e : append s₁ s₂ = ss, intro h, revert s₁,
apply mem_rec_on h _,
intros b s' o s₁,
apply s₁.cases_on _ (λ c t₁, _); intros m e;
have := congr_arg destruct e,
{ apply or.inr, simpa using m },
{ cases (show a = c ∨ a ∈ append t₁ s₂, by simpa using m) with e' m,
{ rw e', exact or.inl (mem_cons _ _) },
{ cases (show c = b ∧ append t₁ s₂ = s', by simpa) with i1 i2,
cases o with e' IH,
{ simp [i1, e'] },
{ exact or.imp_left (mem_cons_of_mem _) (IH m i2) } } }
end
theorem mem_append_left {s₁ s₂ : seq α} {a : α} (h : a ∈ s₁) : a ∈ append s₁ s₂ :=
by apply mem_rec_on h; intros; simp [*]
end seq
namespace seq1
variables {α : Type u} {β : Type v} {γ : Type w}
open seq
/-- Convert a `seq1` to a sequence. -/
def to_seq : seq1 α → seq α
| (a, s) := cons a s
instance coe_seq : has_coe (seq1 α) (seq α) := ⟨to_seq⟩
/-- Map a function on a `seq1` -/
def map (f : α → β) : seq1 α → seq1 β
| (a, s) := (f a, seq.map f s)
theorem map_id : ∀ (s : seq1 α), map id s = s | ⟨a, s⟩ := by simp [map]
/-- Flatten a nonempty sequence of nonempty sequences -/
def join : seq1 (seq1 α) → seq1 α
| ((a, s), S) := match destruct s with
| none := (a, seq.join S)
| some s' := (a, seq.join (cons s' S))
end
@[simp] theorem join_nil (a : α) (S) : join ((a, nil), S) = (a, seq.join S) := rfl
@[simp] theorem join_cons (a b : α) (s S) :
join ((a, cons b s), S) = (a, seq.join (cons (b, s) S)) :=
by dsimp [join]; rw [destruct_cons]; refl
/-- The `return` operator for the `seq1` monad,
which produces a singleton sequence. -/
def ret (a : α) : seq1 α := (a, nil)
instance [inhabited α] : inhabited (seq1 α) := ⟨ret default⟩
/-- The `bind` operator for the `seq1` monad,
which maps `f` on each element of `s` and appends the results together.
(Not all of `s` may be evaluated, because the first few elements of `s`
may already produce an infinite result.) -/
def bind (s : seq1 α) (f : α → seq1 β) : seq1 β :=
join (map f s)
@[simp] theorem join_map_ret (s : seq α) : seq.join (seq.map ret s) = s :=
by apply coinduction2 s; intro s; apply cases_on s; simp [ret]
@[simp] theorem bind_ret (f : α → β) : ∀ s, bind s (ret ∘ f) = map f s
| ⟨a, s⟩ := begin
dsimp [bind, map], change (λx, ret (f x)) with (ret ∘ f),
rw [map_comp], simp [function.comp, ret]
end
@[simp] theorem ret_bind (a : α) (f : α → seq1 β) : bind (ret a) f = f a :=
begin
simp [ret, bind, map],
cases f a with a s,
apply cases_on s; intros; simp
end
@[simp] theorem map_join' (f : α → β) (S) :
seq.map f (seq.join S) = seq.join (seq.map (map f) S) :=
begin
apply eq_of_bisim (λs1 s2,
∃ s S, s1 = append s (seq.map f (seq.join S)) ∧
s2 = append s (seq.join (seq.map (map f) S))),
{ intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, S, rfl, rfl⟩ := begin
apply cases_on s; simp,
{ apply cases_on S; simp,
{ intros x S, cases x with a s; simp [map],
exact ⟨_, _, rfl, rfl⟩ } },
{ intros x s, refine ⟨s, S, rfl, rfl⟩ }
end end },
{ refine ⟨nil, S, _, _⟩; simp }
end
@[simp] theorem map_join (f : α → β) : ∀ S, map f (join S) = join (map (map f) S)
| ((a, s), S) := by apply cases_on s; intros; simp [map]
@[simp] theorem join_join (SS : seq (seq1 (seq1 α))) :
seq.join (seq.join SS) = seq.join (seq.map join SS) :=
begin
apply eq_of_bisim (λs1 s2,
∃ s SS, s1 = seq.append s (seq.join (seq.join SS)) ∧
s2 = seq.append s (seq.join (seq.map join SS))),
{ intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, SS, rfl, rfl⟩ := begin
apply cases_on s; simp,
{ apply cases_on SS; simp,
{ intros S SS, cases S with s S; cases s with x s; simp [map],
apply cases_on s; simp,
{ exact ⟨_, _, rfl, rfl⟩ },
{ intros x s,
refine ⟨cons x (append s (seq.join S)), SS, _, _⟩; simp } } },
{ intros x s, exact ⟨s, SS, rfl, rfl⟩ }
end end },
{ refine ⟨nil, SS, _, _⟩; simp }
end
@[simp] theorem bind_assoc (s : seq1 α) (f : α → seq1 β) (g : β → seq1 γ) :
bind (bind s f) g = bind s (λ (x : α), bind (f x) g) :=
begin
cases s with a s,
simp [bind, map],
rw [←map_comp],
change (λ x, join (map g (f x))) with (join ∘ ((map g) ∘ f)),
rw [map_comp _ join],
generalize : seq.map (map g ∘ f) s = SS,
rcases map g (f a) with ⟨⟨a, s⟩, S⟩,
apply cases_on s; intros; apply cases_on S; intros; simp,
{ cases x with x t, apply cases_on t; intros; simp },
{ cases x_1 with y t; simp }
end
instance : monad seq1 :=
{ map := @map,
pure := @ret,
bind := @bind }
instance : is_lawful_monad seq1 :=
{ id_map := @map_id,
bind_pure_comp_eq_map := @bind_ret,
pure_bind := @ret_bind,
bind_assoc := @bind_assoc }
end seq1
|
efb5cb78f2eb9c57756b91eb47c0695238742a6b | 491068d2ad28831e7dade8d6dff871c3e49d9431 | /hott/arity.hlean | cd1d4d2b3290ffaf28ea4817a9c34784f162e2b0 | [
"Apache-2.0"
] | permissive | davidmueller13/lean | 65a3ed141b4088cd0a268e4de80eb6778b21a0e9 | c626e2e3c6f3771e07c32e82ee5b9e030de5b050 | refs/heads/master | 1,611,278,313,401 | 1,444,021,177,000 | 1,444,021,177,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 10,900 | hlean | /-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Theorems about functions with multiple arguments
-/
variables {A U V W X Y Z : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
{E : Πa b c, D a b c → Type} {F : Πa b c d, E a b c d → Type}
{G : Πa b c d e, F a b c d e → Type} {H : Πa b c d e f, G a b c d e f → Type}
variables {a a' : A} {u u' : U} {v v' : V} {w w' : W} {x x' x'' : X} {y y' : Y} {z z' : Z}
{b : B a} {b' : B a'}
{c : C a b} {c' : C a' b'}
{d : D a b c} {d' : D a' b' c'}
{e : E a b c d} {e' : E a' b' c' d'}
{ff : F a b c d e} {f' : F a' b' c' d' e'}
{g : G a b c d e ff} {g' : G a' b' c' d' e' f'}
{h : H a b c d e ff g} {h' : H a' b' c' d' e' f' g'}
namespace eq
/-
Naming convention:
The theorem which states how to construct an path between two function applications is
api₀i₁...iₙ.
Here i₀, ... iₙ are digits, n is the arity of the function(s),
and iⱼ specifies the dimension of the path between the jᵗʰ argument
(i₀ specifies the dimension of the path between the functions).
A value iⱼ ≡ 0 means that the jᵗʰ arguments are definitionally equal
The functions are non-dependent, except when the theorem name contains trailing zeroes
(where the function is dependent only in the arguments where it doesn't result in any
transports in the theorem statement).
For the fully-dependent versions (except that the conclusion doesn't contain a transport)
we write
apdi₀i₁...iₙ.
For versions where only some arguments depend on some other arguments,
or for versions with transport in the conclusion (like apd), we don't have a
consistent naming scheme (yet).
We don't prove each theorem systematically, but prove only the ones which we actually need.
-/
definition homotopy2 [reducible] (f g : Πa b, C a b) : Type :=
Πa b, f a b = g a b
definition homotopy3 [reducible] (f g : Πa b c, D a b c) : Type :=
Πa b c, f a b c = g a b c
definition homotopy4 [reducible] (f g : Πa b c d, E a b c d) : Type :=
Πa b c d, f a b c d = g a b c d
notation f `~2`:50 g := homotopy2 f g
notation f `~3`:50 g := homotopy3 f g
definition ap011 (f : U → V → W) (Hu : u = u') (Hv : v = v') : f u v = f u' v' :=
by cases Hu; congruence; repeat assumption
definition ap0111 (f : U → V → W → X) (Hu : u = u') (Hv : v = v') (Hw : w = w')
: f u v w = f u' v' w' :=
by cases Hu; congruence; repeat assumption
definition ap01111 (f : U → V → W → X → Y) (Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x')
: f u v w x = f u' v' w' x' :=
by cases Hu; congruence; repeat assumption
definition ap011111 (f : U → V → W → X → Y → Z)
(Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x') (Hy : y = y')
: f u v w x y = f u' v' w' x' y' :=
by cases Hu; congruence; repeat assumption
definition ap0111111 (f : U → V → W → X → Y → Z → A)
(Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x') (Hy : y = y') (Hz : z = z')
: f u v w x y z = f u' v' w' x' y' z' :=
by cases Hu; congruence; repeat assumption
definition ap010 (f : X → Πa, B a) (Hx : x = x') : f x ~ f x' :=
by intros; cases Hx; reflexivity
definition ap0100 (f : X → Πa b, C a b) (Hx : x = x') : f x ~2 f x' :=
by intros; cases Hx; reflexivity
definition ap01000 (f : X → Πa b c, D a b c) (Hx : x = x') : f x ~3 f x' :=
by intros; cases Hx; reflexivity
definition apd011 (f : Πa, B a → Z) (Ha : a = a') (Hb : transport B Ha b = b')
: f a b = f a' b' :=
by cases Ha; cases Hb; reflexivity
definition apd0111 (f : Πa b, C a b → Z) (Ha : a = a') (Hb : transport B Ha b = b')
(Hc : cast (apd011 C Ha Hb) c = c')
: f a b c = f a' b' c' :=
by cases Ha; cases Hb; cases Hc; reflexivity
definition apd01111 (f : Πa b c, D a b c → Z) (Ha : a = a') (Hb : transport B Ha b = b')
(Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
: f a b c d = f a' b' c' d' :=
by cases Ha; cases Hb; cases Hc; cases Hd; reflexivity
definition apd011111 (f : Πa b c d, E a b c d → Z) (Ha : a = a') (Hb : transport B Ha b = b')
(Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
(He : cast (apd01111 E Ha Hb Hc Hd) e = e')
: f a b c d e = f a' b' c' d' e' :=
by cases Ha; cases Hb; cases Hc; cases Hd; cases He; reflexivity
definition apd0111111 (f : Πa b c d e, F a b c d e → Z) (Ha : a = a') (Hb : transport B Ha b = b')
(Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
(He : cast (apd01111 E Ha Hb Hc Hd) e = e') (Hf : cast (apd011111 F Ha Hb Hc Hd He) ff = f')
: f a b c d e ff = f a' b' c' d' e' f' :=
begin cases Ha, cases Hb, cases Hc, cases Hd, cases He, cases Hf, reflexivity end
-- definition apd0111111 (f : Πa b c d e ff, G a b c d e ff → Z) (Ha : a = a') (Hb : transport B Ha b = b')
-- (Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
-- (He : cast (apd01111 E Ha Hb Hc Hd) e = e') (Hf : cast (apd011111 F Ha Hb Hc Hd He) ff = f')
-- (Hg : cast (apd0111111 G Ha Hb Hc Hd He Hf) g = g')
-- : f a b c d e ff g = f a' b' c' d' e' f' g' :=
-- by cases Ha; cases Hb; cases Hc; cases Hd; cases He; cases Hf; cases Hg; reflexivity
-- definition apd01111111 (f : Πa b c d e ff g, G a b c d e ff g → Z) (Ha : a = a') (Hb : transport B Ha b = b')
-- (Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
-- (He : cast (apd01111 E Ha Hb Hc Hd) e = e') (Hf : cast (apd011111 F Ha Hb Hc Hd He) ff = f')
-- (Hg : cast (apd0111111 G Ha Hb Hc Hd He Hf) g = g') (Hh : cast (apd01111111 H Ha Hb Hc Hd He Hf Hg) h = h')
-- : f a b c d e ff g h = f a' b' c' d' e' f' g' h' :=
-- by cases Ha; cases Hb; cases Hc; cases Hd; cases He; cases Hf; cases Hg; cases Hh; reflexivity
definition apd100 [unfold 6] {f g : Πa b, C a b} (p : f = g) : f ~2 g :=
λa b, apd10 (apd10 p a) b
definition apd1000 [unfold 7] {f g : Πa b c, D a b c} (p : f = g) : f ~3 g :=
λa b c, apd100 (apd10 p a) b c
/- some properties of these variants of ap -/
-- we only prove what we currently need
definition ap010_con (f : X → Πa, B a) (p : x = x') (q : x' = x'') :
ap010 f (p ⬝ q) a = ap010 f p a ⬝ ap010 f q a :=
eq.rec_on q (eq.rec_on p idp)
definition ap010_ap (f : X → Πa, B a) (g : Y → X) (p : y = y') :
ap010 f (ap g p) a = ap010 (λy, f (g y)) p a :=
eq.rec_on p idp
/- the following theorems are function extentionality for functions with multiple arguments -/
definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f ~2 g) : f = g :=
eq_of_homotopy (λa, eq_of_homotopy (H a))
definition eq_of_homotopy3 {f g : Πa b c, D a b c} (H : f ~3 g) : f = g :=
eq_of_homotopy (λa, eq_of_homotopy2 (H a))
definition eq_of_homotopy2_id (f : Πa b, C a b)
: eq_of_homotopy2 (λa b, idpath (f a b)) = idpath f :=
begin
transitivity eq_of_homotopy (λ a, idpath (f a)),
{apply (ap eq_of_homotopy), apply eq_of_homotopy, intros, apply eq_of_homotopy_idp},
apply eq_of_homotopy_idp
end
definition eq_of_homotopy3_id (f : Πa b c, D a b c)
: eq_of_homotopy3 (λa b c, idpath (f a b c)) = idpath f :=
begin
transitivity _,
{apply (ap eq_of_homotopy), apply eq_of_homotopy, intros, apply eq_of_homotopy2_id},
apply eq_of_homotopy_idp
end
definition eq_of_homotopy2_inv {f g : Πa b, C a b} (H : f ~2 g)
: eq_of_homotopy2 (λa b, (H a b)⁻¹) = (eq_of_homotopy2 H)⁻¹ :=
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy_inv)) ⬝ !eq_of_homotopy_inv
definition eq_of_homotopy3_inv {f g : Πa b c, D a b c} (H : f ~3 g)
: eq_of_homotopy3 (λa b c, (H a b c)⁻¹) = (eq_of_homotopy3 H)⁻¹ :=
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy2_inv)) ⬝ !eq_of_homotopy_inv
definition eq_of_homotopy2_con {f g h : Πa b, C a b} (H1 : f ~2 g) (H2 : g ~2 h)
: eq_of_homotopy2 (λa b, H1 a b ⬝ H2 a b) = eq_of_homotopy2 H1 ⬝ eq_of_homotopy2 H2 :=
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy_con)) ⬝ !eq_of_homotopy_con
definition eq_of_homotopy3_con {f g h : Πa b c, D a b c} (H1 : f ~3 g) (H2 : g ~3 h)
: eq_of_homotopy3 (λa b c, H1 a b c ⬝ H2 a b c) = eq_of_homotopy3 H1 ⬝ eq_of_homotopy3 H2 :=
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy2_con)) ⬝ !eq_of_homotopy_con
end eq
open eq is_equiv
namespace funext
definition is_equiv_apd100 [instance] (f g : Πa b, C a b) : is_equiv (@apd100 A B C f g) :=
adjointify _
eq_of_homotopy2
begin
intro H, esimp [apd100, eq_of_homotopy2],
apply eq_of_homotopy, intro a,
apply concat, apply (ap (λx, apd10 (x a))), apply (right_inv apd10),
apply (right_inv apd10)
end
begin
intro p, cases p, apply eq_of_homotopy2_id
end
definition is_equiv_apd1000 [instance] (f g : Πa b c, D a b c)
: is_equiv (@apd1000 A B C D f g) :=
adjointify _
eq_of_homotopy3
begin
intro H, esimp,
apply eq_of_homotopy, intro a,
transitivity apd100 (eq_of_homotopy2 (H a)),
{apply ap (λx, apd100 (x a)),
apply right_inv apd10},
apply right_inv apd100
end
begin
intro p, cases p, apply eq_of_homotopy3_id
end
end funext
namespace eq
open funext
local attribute funext.is_equiv_apd100 [instance]
protected definition homotopy2.rec_on {f g : Πa b, C a b} {P : (f ~2 g) → Type}
(p : f ~2 g) (H : Π(q : f = g), P (apd100 q)) : P p :=
right_inv apd100 p ▸ H (eq_of_homotopy2 p)
protected definition homotopy3.rec_on {f g : Πa b c, D a b c} {P : (f ~3 g) → Type}
(p : f ~3 g) (H : Π(q : f = g), P (apd1000 q)) : P p :=
right_inv apd1000 p ▸ H (eq_of_homotopy3 p)
definition apd10_ap (f : X → Πa, B a) (p : x = x')
: apd10 (ap f p) = ap010 f p :=
eq.rec_on p idp
definition eq_of_homotopy_ap010 (f : X → Πa, B a) (p : x = x')
: eq_of_homotopy (ap010 f p) = ap f p :=
inv_eq_of_eq !apd10_ap⁻¹
definition ap_eq_ap_of_homotopy {f : X → Πa, B a} {p q : x = x'} (H : ap010 f p ~ ap010 f q)
: ap f p = ap f q :=
calc
ap f p = eq_of_homotopy (ap010 f p) : eq_of_homotopy_ap010
... = eq_of_homotopy (ap010 f q) : eq_of_homotopy H
... = ap f q : eq_of_homotopy_ap010
end eq
|
6d5e6f72b87adcf2e0a30a13be4af8f12883a81e | 947fa6c38e48771ae886239b4edce6db6e18d0fb | /src/geometry/manifold/derivation_bundle.lean | 68308fc1b2656812cfe16ddf7a2aa878083baa18 | [
"Apache-2.0"
] | permissive | ramonfmir/mathlib | c5dc8b33155473fab97c38bd3aa6723dc289beaa | 14c52e990c17f5a00c0cc9e09847af16fabbed25 | refs/heads/master | 1,661,979,343,526 | 1,660,830,384,000 | 1,660,830,384,000 | 182,072,989 | 0 | 0 | null | 1,555,585,876,000 | 1,555,585,876,000 | null | UTF-8 | Lean | false | false | 6,364 | lean | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import geometry.manifold.algebra.smooth_functions
import ring_theory.derivation
/-!
# Derivation bundle
In this file we define the derivations at a point of a manifold on the algebra of smooth fuctions.
Moreover, we define the differential of a function in terms of derivations.
The content of this file is not meant to be regarded as an alternative definition to the current
tangent bundle but rather as a purely algebraic theory that provides a purely algebraic definition
of the Lie algebra for a Lie group.
-/
variables (𝕜 : Type*) [nontrivially_normed_field 𝕜]
{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
{H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H)
(M : Type*) [topological_space M] [charted_space H M] (n : with_top ℕ)
open_locale manifold
-- the following two instances prevent poorly understood type class inference timeout problems
instance smooth_functions_algebra : algebra 𝕜 C^∞⟮I, M; 𝕜⟯ := by apply_instance
instance smooth_functions_tower : is_scalar_tower 𝕜 C^∞⟮I, M; 𝕜⟯ C^∞⟮I, M; 𝕜⟯ := by apply_instance
/-- Type synonym, introduced to put a different `has_smul` action on `C^n⟮I, M; 𝕜⟯`
which is defined as `f • r = f(x) * r`. -/
@[nolint unused_arguments] def pointed_smooth_map (x : M) := C^n⟮I, M; 𝕜⟯
localized "notation `C^` n `⟮` I `,` M `;` 𝕜 `⟯⟨` x `⟩` :=
pointed_smooth_map 𝕜 I M n x" in derivation
variables {𝕜 M}
namespace pointed_smooth_map
instance {x : M} : has_coe_to_fun C^∞⟮I, M; 𝕜⟯⟨x⟩ (λ _, M → 𝕜) :=
cont_mdiff_map.has_coe_to_fun
instance {x : M} : comm_ring C^∞⟮I, M; 𝕜⟯⟨x⟩ := smooth_map.comm_ring
instance {x : M} : algebra 𝕜 C^∞⟮I, M; 𝕜⟯⟨x⟩ := smooth_map.algebra
instance {x : M} : inhabited C^∞⟮I, M; 𝕜⟯⟨x⟩ := ⟨0⟩
instance {x : M} : algebra C^∞⟮I, M; 𝕜⟯⟨x⟩ C^∞⟮I, M; 𝕜⟯ := algebra.id C^∞⟮I, M; 𝕜⟯
instance {x : M} : is_scalar_tower 𝕜 C^∞⟮I, M; 𝕜⟯⟨x⟩ C^∞⟮I, M; 𝕜⟯ := is_scalar_tower.right
variable {I}
/-- `smooth_map.eval_ring_hom` gives rise to an algebra structure of `C^∞⟮I, M; 𝕜⟯` on `𝕜`. -/
instance eval_algebra {x : M} : algebra C^∞⟮I, M; 𝕜⟯⟨x⟩ 𝕜 :=
(smooth_map.eval_ring_hom x : C^∞⟮I, M; 𝕜⟯⟨x⟩ →+* 𝕜).to_algebra
/-- With the `eval_algebra` algebra structure evaluation is actually an algebra morphism. -/
def eval (x : M) : C^∞⟮I, M; 𝕜⟯ →ₐ[C^∞⟮I, M; 𝕜⟯⟨x⟩] 𝕜 :=
algebra.of_id C^∞⟮I, M; 𝕜⟯⟨x⟩ 𝕜
lemma smul_def (x : M) (f : C^∞⟮I, M; 𝕜⟯⟨x⟩) (k : 𝕜) : f • k = f x * k := rfl
instance (x : M) : is_scalar_tower 𝕜 C^∞⟮I, M; 𝕜⟯⟨x⟩ 𝕜 :=
{ smul_assoc := λ k f h, by { simp only [smul_def, algebra.id.smul_eq_mul, smooth_map.coe_smul,
pi.smul_apply, mul_assoc]} }
end pointed_smooth_map
open_locale derivation
/-- The derivations at a point of a manifold. Some regard this as a possible definition of the
tangent space -/
@[reducible] def point_derivation (x : M) := derivation 𝕜 (C^∞⟮I, M; 𝕜⟯⟨x⟩) 𝕜
section
variables (I) {M} (X Y : derivation 𝕜 C^∞⟮I, M; 𝕜⟯ C^∞⟮I, M; 𝕜⟯) (f g : C^∞⟮I, M; 𝕜⟯) (r : 𝕜)
/-- Evaluation at a point gives rise to a `C^∞⟮I, M; 𝕜⟯`-linear map between `C^∞⟮I, M; 𝕜⟯` and `𝕜`.
-/
def smooth_function.eval_at (x : M) : C^∞⟮I, M; 𝕜⟯ →ₗ[C^∞⟮I, M; 𝕜⟯⟨x⟩] 𝕜 :=
(pointed_smooth_map.eval x).to_linear_map
namespace derivation
variable {I}
/-- The evaluation at a point as a linear map. -/
def eval_at (x : M) : (derivation 𝕜 C^∞⟮I, M; 𝕜⟯ C^∞⟮I, M; 𝕜⟯) →ₗ[𝕜] point_derivation I x :=
(smooth_function.eval_at I x).comp_der
lemma eval_at_apply (x : M) : eval_at x X f = (X f) x := rfl
end derivation
variables {I} {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E']
{H' : Type*} [topological_space H'] {I' : model_with_corners 𝕜 E' H'}
{M' : Type*} [topological_space M'] [charted_space H' M']
/-- The heterogeneous differential as a linear map. Instead of taking a function as an argument this
differential takes `h : f x = y`. It is particularly handy to deal with situations where the points
on where it has to be evaluated are equal but not definitionally equal. -/
def hfdifferential {f : C^∞⟮I, M; I', M'⟯} {x : M} {y : M'} (h : f x = y) :
point_derivation I x →ₗ[𝕜] point_derivation I' y :=
{ to_fun := λ v, derivation.mk'
{ to_fun := λ g, v (g.comp f),
map_add' := λ g g', by rw [smooth_map.add_comp, derivation.map_add],
map_smul' := λ k g,
by simp only [smooth_map.smul_comp, derivation.map_smul, ring_hom.id_apply], }
(λ g g', by simp only [derivation.leibniz, smooth_map.mul_comp, linear_map.coe_mk,
pointed_smooth_map.smul_def, cont_mdiff_map.comp_apply, h]),
map_smul' := λ k v, rfl,
map_add' := λ v w, rfl }
/-- The homogeneous differential as a linear map. -/
def fdifferential (f : C^∞⟮I, M; I', M'⟯) (x : M) :
point_derivation I x →ₗ[𝕜] point_derivation I' (f x) :=
hfdifferential (rfl : f x = f x)
/- Standard notation for the differential. The abbreviation is `MId`. -/
localized "notation `𝒅` := fdifferential" in manifold
/- Standard notation for the differential. The abbreviation is `MId`. -/
localized "notation `𝒅ₕ` := hfdifferential" in manifold
@[simp] lemma apply_fdifferential (f : C^∞⟮I, M; I', M'⟯) {x : M} (v : point_derivation I x)
(g : C^∞⟮I', M'; 𝕜⟯) : 𝒅f x v g = v (g.comp f) := rfl
@[simp] lemma apply_hfdifferential {f : C^∞⟮I, M; I', M'⟯} {x : M} {y : M'} (h : f x = y)
(v : point_derivation I x) (g : C^∞⟮I', M'; 𝕜⟯) : 𝒅ₕh v g = 𝒅f x v g := rfl
variables {E'' : Type*} [normed_add_comm_group E''] [normed_space 𝕜 E'']
{H'' : Type*} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type*} [topological_space M''] [charted_space H'' M'']
@[simp] lemma fdifferential_comp (g : C^∞⟮I', M'; I'', M''⟯) (f : C^∞⟮I, M; I', M'⟯) (x : M) :
𝒅(g.comp f) x = (𝒅g (f x)).comp (𝒅f x) := rfl
end
|
f6966ba24d2af24c2ce94a8054c4197d8bada9b9 | 9f1929110d7ce695e60c30a12c83161ea7eadc15 | /experiments/bench/pose1000.lean | a5991c245083450360c60f116ddcad9405cdf4c3 | [] | no_license | andres-erbsen/coq-experiments | 4820d05608c445eba0ea965e23bd9cf39a5c418f | 2018edd397a23c0429d316c96e86f9be7a9678f1 | refs/heads/master | 1,619,935,106,811 | 1,578,518,728,000 | 1,578,518,728,000 | 120,850,328 | 1 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 204 | lean | open tactic
lemma test : true :=
begin
iterate_exactly 1000 (do pose (`_) none `(trivial), skip),
exact trivial
end
-- 0.61user 0.04system 0:00.65elapsed 99%CPU (0avgtext+0avgdata 96756maxresident)k
|
ad966c02560998f4ffaeda7d2d8ea9a2f9c2233f | fe84e287c662151bb313504482b218a503b972f3 | /src/group_theory/perm_extra.lean | eb82b52493d72ca6fac942f082b2e6443637382c | [] | no_license | NeilStrickland/lean_lib | 91e163f514b829c42fe75636407138b5c75cba83 | 6a9563de93748ace509d9db4302db6cd77d8f92c | refs/heads/master | 1,653,408,198,261 | 1,652,996,419,000 | 1,652,996,419,000 | 181,006,067 | 4 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 2,990 | lean | import group_theory.perm.cycles
import data.vector
import data.fin_extra
import group_theory.cyclic
open int.modeq
variables {α : Type*} [decidable_eq α]
def finperm (n : ℕ) := equiv.perm (fin n)
namespace finperm
variable (n : ℕ)
instance : group (finperm n) := by { dsimp[finperm], apply_instance }
instance : decidable_eq (finperm n) := by { dsimp[finperm], apply_instance }
instance : has_coe_to_fun (finperm n) (λ _, (fin n) → (fin n)) := by { dsimp[finperm], apply_instance }
variable {n}
@[ext] lemma ext {s t : finperm n} : s = t ↔ ∀ (i : fin n), s i = t i :=
begin
split,
{ rintro ⟨_⟩ i, refl },
{ intro h, ext i, rw[h i] }
end
lemma ext' {s t : finperm n}
(h : ∀ (i : ℕ) (hi : i < n), ((s ⟨i,hi⟩) : ℕ) = ((t ⟨i,hi⟩) : ℕ)) : s = t :=
begin
ext i, rcases i with ⟨i,hi⟩, exact h i hi,
end
variable (n)
lemma fin_coe_int_inj {n : ℕ} {i₀ i₁ : fin n} (h : (i₀ : ℤ) = (i₁ : ℤ)) : i₀ = i₁ :=
fin.coe_inj (int.coe_nat_inj h)
def fin_of_int (i : ℤ) : fin (n + 1) :=
⟨i.nat_mod n.succ,
begin
cases i; let j := i % n.succ; have hj : j < n.succ := i.mod_lt n.succ_pos,
{ exact hj },
{ change int.to_nat (int.sub_nat_nat n.succ j.succ) < n.succ,
rw [int.sub_nat_nat_eq_coe, ← int.coe_nat_sub (hj)],
change n.succ - j.succ < n.succ,
rw [nat.succ_sub_succ],
exact lt_of_le_of_lt (n.sub_le j) n.lt_succ_self }
end⟩
lemma coe_fin_of_int (i : ℤ) : ((fin_of_int n i) : ℕ) = i.nat_mod n.succ := rfl
lemma coe_fin_of_int' (i : ℤ) : ((fin_of_int n i) : ℤ) = i % n.succ :=
begin
have : (n.succ : ℤ) ≠ 0 := λ h, by { cases h },
exact int.to_nat_of_nonneg (i.mod_nonneg this),
end
def shift : ℤ → (fin (n + 1)) → (fin (n + 1)) :=
λ k i, fin_of_int n (k + (i : ℤ))
lemma shift_zero (i : fin (n + 1)) : shift n 0 i = i :=
begin
apply fin_coe_int_inj,
dsimp [shift, fin_of_int, int.nat_mod],
have : (n + 1 : ℤ) = n.succ := rfl,
rw [this, zero_add, int.of_nat_mod],
congr' 1, exact nat.mod_eq_of_lt i.is_lt,
end
lemma shift_shift (k₀ k₁ : ℤ) (i : fin (n + 1)) :
shift n k₀ (shift n k₁ i) = shift n (k₀ + k₁) i :=
begin
apply fin_coe_int_inj,
simp only [shift, coe_fin_of_int'],
rw [add_assoc k₀ k₁],
change _ ≡ _ [ZMOD _],
apply int.modeq.add, refl, apply int.mod_modeq,
end
lemma shift_of_zero (i : fin (n + 1)) : shift n i 0 = i :=
begin
apply fin_coe_int_inj,
have : ((0 : fin (n + 1)) : ℤ) = 0 := rfl,
simp only [shift, coe_fin_of_int', this, add_zero],
change ((i : ℕ) : ℤ) % n.succ = (i : ℕ),
rw [int.of_nat_mod], rw[nat.mod_eq_of_lt i.is_lt], refl,
end
/-
def p {l₀ l₁ : list α} (h : list.perm l₀ l₁) : equiv.perm (fin l₀.length) :=
by {
let n := l₀.length, change equiv.perm (fin n),
induction l₀ with a₀ m₀ ih generalizing l₁ h,
{ exact equiv.refl _ },
{ rcases (list.cons_perm_iff_perm_erase.mp h) with ⟨hm,hp⟩ }
}
--/
end finperm |
fd39c20f11654057f488d4c58c8a5d036f7a2b76 | c8b4b578b2fe61d500fbca7480e506f6603ea698 | /src/flt_three/edwards.lean | 0311c109c7a47b05c1050ee50cae23507590c862 | [] | no_license | leanprover-community/flt-regular | aa7e564f2679dfd2e86015a5a9674a6e1197f7cc | 67fb3e176584bbc03616c221a7be6fa28c5ccd32 | refs/heads/master | 1,692,188,905,751 | 1,691,766,312,000 | 1,691,766,312,000 | 421,021,216 | 19 | 4 | null | 1,694,532,115,000 | 1,635,166,136,000 | Lean | UTF-8 | Lean | false | false | 22,331 | lean | /-
Copyright (c) 2020 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
-/
import .primes
import .spts
import .odd_prime_or_four
lemma zsqrtd.associated_norm_of_associated {d : ℤ} {f g : ℤ√d}
(h : associated f g) : associated f.norm g.norm :=
begin
obtain ⟨u, rfl⟩ := h,
have := (zsqrtd.is_unit_iff_norm_is_unit ↑u).mp u.is_unit,
rw [zsqrtd.norm_mul],
exact (associated_mul_unit_right (zsqrtd.norm f) _ this),
end
lemma odd_prime_or_four.im_ne_zero
{p : ℤ√-3}
(h: odd_prime_or_four p.norm)
(hcoprime: is_coprime p.re p.im) :
p.im ≠ 0 :=
begin
intro H,
rw [zsqrtd.norm_def, H, mul_zero, sub_zero, ←pow_two] at h,
obtain h|⟨hp, hodd⟩ := h,
{ rw [H, is_coprime_zero_right, int.is_unit_iff_abs_eq] at hcoprime,
rw [←sq_abs, hcoprime] at h,
norm_num at h },
{ exact pow_not_prime one_lt_two.ne' hp }
end
lemma zsqrt3.is_unit_iff {z : ℤ√-3} : is_unit z ↔ abs z.re = 1 ∧ z.im = 0 :=
begin
rw [←zsqrtd.norm_eq_one_iff, ←int.coe_nat_inj', int.coe_nat_one,
int.nat_abs_of_nonneg (zsqrtd.norm_nonneg (by norm_num) z)],
refine ⟨spts.eq_one, λ h, _⟩,
rw [zsqrtd.norm_def, h.2, mul_zero, sub_zero, ←sq, ←sq_abs, h.1, one_pow]
end
lemma zsqrt3.coe_of_is_unit {z : ℤ√-3} (h : is_unit z) : ∃ u : units ℤ, z = u :=
begin
obtain ⟨u, hu⟩ := zsqrt3.is_unit_iff.mp h,
obtain ⟨v, hv⟩ : is_unit z.re,
{ rwa int.is_unit_iff_abs_eq },
use v,
rw [zsqrtd.ext, hu, ←hv],
simp only [zsqrtd.coe_int_re, and_true, eq_self_iff_true, coe_coe, zsqrtd.coe_int_im],
end
lemma zsqrt3.coe_of_is_unit' {z : ℤ√-3} (h : is_unit z) : ∃ u : ℤ, z = u ∧ abs u = 1 :=
begin
obtain ⟨u, hu⟩ := zsqrt3.coe_of_is_unit h,
refine ⟨u, _, _⟩,
{ rw [hu, coe_coe] },
{ exact int.is_unit_iff_abs_eq.mp u.is_unit },
end
lemma odd_prime_or_four.factors
(a : ℤ√-3)
(x : ℤ)
(hcoprime : is_coprime a.re a.im)
(hx : odd_prime_or_four x)
(hfactor : x ∣ a.norm) :
∃ c : ℤ√-3, abs x = c.norm ∧ 0 ≤ c.re ∧ c.im ≠ 0 :=
begin
obtain rfl|⟨hprime, hodd⟩ := hx,
{ refine ⟨⟨1, 1⟩, _, zero_le_one, one_ne_zero⟩,
simp only [zsqrtd.norm_def, mul_one, abs_of_pos, zero_lt_four, sub_neg_eq_add],
norm_num },
{ obtain ⟨c, hc⟩ := factors a x hcoprime hodd hfactor,
rw hc,
apply zsqrtd.exists c,
intro H,
apply pow_not_prime one_lt_two.ne',
convert hprime.abs,
rwa [zsqrtd.norm_def, H, mul_zero, sub_zero, ←pow_two, eq_comm] at hc }
end
lemma step1a
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
(heven : even a.norm) :
odd a.re ∧ odd a.im :=
begin
rw [int.odd_iff_not_even, int.odd_iff_not_even],
have : even a.re ↔ even a.im,
{ simpa [zsqrtd.norm_def] with parity_simps using heven },
apply (iff_iff_and_or_not_and_not.mp this).resolve_left,
rw [even_iff_two_dvd, even_iff_two_dvd],
rintro ⟨hre, him⟩,
have := hcoprime.is_unit_of_dvd' hre him,
rw is_unit_iff_dvd_one at this,
norm_num at this,
end
lemma step1
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
(heven : even a.norm) :
∃ u : ℤ√-3, (a = ⟨1, 1⟩ * u ∨ a = ⟨1, -1⟩ * u) :=
begin
obtain ⟨ha, hb⟩ := step1a hcoprime heven,
obtain hdvd|hdvd := int.four_dvd_add_or_sub_of_odd ha hb,
{ obtain ⟨v, hv⟩ := hdvd,
rw ←eq_sub_iff_add_eq at hv,
use ⟨v - a.im, v⟩,
right,
rw [zsqrtd.ext, hv, zsqrtd.mul_re, zsqrtd.mul_im],
dsimp only,
split; ring },
{ obtain ⟨v, hv⟩ := hdvd,
rw sub_eq_iff_eq_add at hv,
use ⟨a.im + v, -v⟩,
left,
rw [zsqrtd.ext, hv, zsqrtd.mul_re, zsqrtd.mul_im],
dsimp only,
split; ring },
end
lemma step1'
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
(heven : even a.norm) :
∃ u : ℤ√-3,
is_coprime u.re u.im ∧
a.norm = 4 * u.norm :=
begin
obtain ⟨u', hu'⟩ := step1 hcoprime heven,
refine ⟨u', _, _⟩,
{ apply zsqrtd.coprime_of_dvd_coprime hcoprime,
obtain (rfl|rfl) := hu'; apply dvd_mul_left },
{ cases hu';
{ rw [hu', zsqrtd.norm_mul], congr } }
end
lemma step1'' {a p : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
(hp : p.norm = 4)
(hq : p.im ≠ 0)
(heven : even a.norm) :
∃ (u : ℤ√-3),
is_coprime u.re u.im ∧
(a = p * u ∨ a = star p * u) ∧ a.norm = 4 * u.norm :=
begin
obtain ⟨u, h2⟩ := step1 hcoprime heven,
set q : ℤ√-3 := ⟨1, 1⟩,
replace h2 : a = q * u ∨ a = star q * u := h2,
obtain ⟨hp', hq'⟩ := spts.four hp hq,
refine ⟨p.re * u, _, _, _⟩,
{ rw ←int.is_unit_iff_abs_eq at hp',
rwa [zsqrtd.smul_re, zsqrtd.smul_im, is_coprime_mul_unit_left hp'],
apply zsqrtd.coprime_of_dvd_coprime hcoprime,
obtain (rfl|rfl) := h2; apply dvd_mul_left },
{ rw (abs_eq $ zero_le_one' ℤ) at hp' hq',
cases hp',
{ have h4 : p = q ∨ p = star q,
{ apply or.imp _ _ hq';
{ intro h5, simp [zsqrtd.ext, hp', h5] } },
simp only [hp', one_mul, int.cast_one],
cases h4; cases h2; simp [h4, h2, or_comm], },
{ have h4 : p = -q ∨ p = -star q,
{ apply or.imp _ _ hq'.symm,
{ intro h5, simp [zsqrtd.ext, hp', h5] },
{ intro h5, simp [zsqrtd.ext, hp', h5] } },
simp only [hp', one_mul, zsqrtd.norm_neg, int.cast_one, int.cast_neg, neg_one_mul],
cases h4; cases h2;
simp [h4, h2] } },
{ rw [zsqrtd.norm_mul, zsqrtd.norm_int_cast, ←sq, ←sq_abs, hp', one_pow, one_mul],
cases h2;
{ rw [h2, zsqrtd.norm_mul], congr } },
end
lemma step2
{a p : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
(hdvd : p.norm ∣ a.norm)
(hpprime : prime p.norm) :
∃ u : ℤ√-3,
is_coprime u.re u.im ∧
(a = p * u ∨ a = star p * u) ∧
a.norm = p.norm * u.norm :=
begin
obtain ⟨u', h, h'⟩ := spts.mul_of_dvd'' hdvd hpprime,
refine ⟨u', _, h, h'⟩,
apply zsqrtd.coprime_of_dvd_coprime hcoprime,
obtain (rfl|rfl) := h; apply dvd_mul_left
end
lemma step1_2
{a p : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
(hdvd : p.norm ∣ a.norm)
(hp : odd_prime_or_four p.norm)
(hq : p.im ≠ 0) :
∃ u : ℤ√-3,
is_coprime u.re u.im ∧
(a = p * u ∨ a = star p * u) ∧
a.norm = p.norm * u.norm :=
begin
obtain hp|⟨hpprime, hpodd⟩ := hp,
{ rw hp at hdvd ⊢,
have heven : even a.norm,
{ apply even_iff_two_dvd.mpr (dvd_trans _ hdvd),
norm_num },
exact step1'' hcoprime hp hq heven },
{ apply step2 hcoprime hdvd hpprime }
end
lemma odd_prime_or_four.factors'
{a : ℤ√-3}
(h2 : a.norm ≠ 1)
(hcoprime : is_coprime a.re a.im) :
∃ (u q : ℤ√-3),
0 ≤ q.re ∧
q.im ≠ 0 ∧
odd_prime_or_four q.norm ∧
a = q * u ∧
is_coprime u.re u.im ∧
u.norm < a.norm :=
begin
have h2 : 2 < a.norm,
{ apply lt_of_le_of_ne _ (spts.not_two _).symm,
rw [←one_mul (2 : ℤ), mul_two, int.add_one_le_iff],
apply lt_of_le_of_ne _ h2.symm,
rw [←int.sub_one_lt_iff, sub_self],
exact spts.pos_of_coprime' hcoprime },
obtain ⟨p, hpdvd, hp⟩ := odd_prime_or_four.exists_and_dvd h2,
obtain ⟨q', hcd, hc, hd⟩ := odd_prime_or_four.factors a p hcoprime hp hpdvd,
rw [←abs_dvd, hcd] at hpdvd,
have hp' := hp.abs,
rw hcd at hp',
obtain ⟨u, huvcoprime, huv, huvdvd⟩ := step1_2 hcoprime hpdvd hp' hd,
use u,
cases huv; [use q', use star q'];
simp only [is_coprime, zsqrtd.star_re, zsqrtd.star_im, ne.def, neg_eq_zero, zsqrtd.norm_conj];
use [hc, hd, hp', huv, huvcoprime];
{ rw [huvdvd, lt_mul_iff_one_lt_left (spts.pos_of_coprime' huvcoprime), ←hcd],
exact hp.one_lt_abs, },
end
lemma step3
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
:
∃ f : multiset ℤ√-3,
(a = f.prod ∨ a = - f.prod) ∧
∀ pq : ℤ√-3, pq ∈ f →
0 ≤ pq.re ∧
pq.im ≠ 0 ∧
odd_prime_or_four pq.norm
:=
begin
suffices : ∀ n : ℕ, a.norm.nat_abs = n →
∃ f : multiset ℤ√-3,
(a = f.prod ∨ a = - f.prod) ∧
∀ pq : ℤ√-3, pq ∈ f →
0 ≤ pq.re ∧
pq.im ≠ 0 ∧
odd_prime_or_four pq.norm,
{ exact this a.norm.nat_abs rfl },
intros n hn,
induction n using nat.strong_induction_on with n ih generalizing a,
dsimp only at ih,
cases eq_or_ne a.norm 1 with h h,
{ use 0,
split,
{ simp only [multiset.prod_zero, spts.eq_one' h] },
{ intros pq hpq,
exact absurd hpq (multiset.not_mem_zero _) } },
{ obtain ⟨U, q, hc, hd, hp, huv, huvcoprime, descent⟩ := odd_prime_or_four.factors'
h hcoprime,
replace descent := int.nat_abs_lt_nat_abs_of_nonneg_of_lt (zsqrtd.norm_nonneg (by norm_num) _) descent,
rw [hn] at descent,
obtain ⟨g, hgprod, hgfactors⟩ := ih U.norm.nat_abs descent huvcoprime rfl,
refine ⟨q ::ₘ g, _, _⟩,
{ simp only [huv, multiset.prod_cons, ←mul_neg],
exact or.imp (congr_arg _) (congr_arg _) hgprod },
{ rintros pq hpq,
rw multiset.mem_cons at hpq,
obtain rfl|ind := hpq,
{ exact ⟨hc, hd, hp⟩ },
{ exact hgfactors pq ind } } },
end
lemma step4_3
{p p' : ℤ√-3}
(hcoprime : is_coprime p.re p.im)
(hcoprime' : is_coprime p'.re p'.im)
(h : odd_prime_or_four p.norm)
(heq : p.norm = p'.norm) :
abs p.re = abs p'.re ∧ abs p.im = abs p'.im :=
begin
have hdvd : p'.norm ∣ p.norm,
{ rw heq },
have him : p'.im ≠ 0,
{ apply odd_prime_or_four.im_ne_zero _ hcoprime',
rwa [←heq] },
obtain ⟨u, hcoprime'', (H|H), h1⟩ := step1_2 hcoprime hdvd (by rwa ←heq) him;
{ rw heq at h1,
have := int.eq_one_of_mul_eq_self_right (spts.ne_zero_of_coprime' _ hcoprime') h1.symm,
obtain ⟨ha, hb⟩ := spts.eq_one this,
simp only [hb, zsqrtd.ext, zsqrtd.mul_re, zsqrtd.mul_im, add_zero, zero_add, mul_zero] at H,
rw [H.1, H.2, abs_mul, abs_mul, ha, mul_one, mul_one],
try { rw [zsqrtd.conj_re, zsqrtd.conj_im, abs_neg] },
simp },
end
lemma prod_map_norm {d : ℤ} {s : multiset ℤ√d} :
(s.map zsqrtd.norm).prod = zsqrtd.norm s.prod :=
multiset.prod_hom s (zsqrtd.norm_monoid_hom : ℤ√d →* ℤ)
lemma factors_unique_prod : ∀{f g : multiset ℤ√-3},
(∀x∈f, odd_prime_or_four (zsqrtd.norm x)) →
(∀x∈g, odd_prime_or_four (zsqrtd.norm x)) →
(associated f.prod.norm g.prod.norm) →
multiset.rel associated (f.map zsqrtd.norm) (g.map zsqrtd.norm) :=
begin
intros f g hf hg hassoc,
apply factors_unique_prod',
{ intros x hx,
rw multiset.mem_map at hx,
obtain ⟨y, hy, rfl⟩ := hx,
exact hf y hy },
{ intros x hx,
rw multiset.mem_map at hx,
obtain ⟨y, hy, rfl⟩ := hx,
exact hg y hy },
{ rwa [prod_map_norm, prod_map_norm] },
end
/-- The multiset of factors. -/
noncomputable def factorization'
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
: multiset ℤ√-3
:= classical.some (step3 hcoprime)
lemma factorization'_prop
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im) :
(a = (factorization' hcoprime).prod ∨ a = - (factorization' hcoprime).prod) ∧
∀ pq : ℤ√-3, pq ∈ (factorization' hcoprime) →
0 ≤ pq.re ∧
pq.im ≠ 0 ∧
odd_prime_or_four pq.norm :=
classical.some_spec (step3 hcoprime)
lemma factorization'.coprime_of_mem
{a b : ℤ√-3} (h : is_coprime a.re a.im) (hmem : b ∈ factorization' h) :
is_coprime b.re b.im :=
begin
obtain ⟨h1, -⟩ := factorization'_prop h,
set f := factorization' h,
apply zsqrtd.coprime_of_dvd_coprime h,
apply (multiset.dvd_prod hmem).trans,
cases h1; simp only [h1, dvd_neg],
end
lemma no_conj
(s : multiset ℤ√-3)
{p : ℤ√-3}
(hp : ¬ is_unit p)
(hcoprime : is_coprime s.prod.re s.prod.im) :
¬(p ∈ s ∧ star p ∈ s) :=
begin
contrapose! hp,
obtain ⟨h1, h2⟩ := hp,
by_cases him : p.im = 0,
{ obtain ⟨t, rfl⟩ := multiset.exists_cons_of_mem h1,
rw multiset.prod_cons at hcoprime,
simp only [him, add_zero, zero_mul, zsqrtd.mul_im, zsqrtd.mul_re, mul_zero] at hcoprime,
rw zsqrt3.is_unit_iff,
simp only [him, and_true, eq_self_iff_true],
rw ←int.is_unit_iff_abs_eq,
apply is_coprime.is_unit_of_dvd' hcoprime; apply dvd_mul_right },
{ have : star p ≠ p,
{ rw [ne.def, zsqrtd.ext],
rintro ⟨-, H⟩,
apply him,
apply eq_zero_of_neg_eq,
simpa using H },
obtain ⟨t1, rfl⟩ := multiset.exists_cons_of_mem h1,
have : star p ∈ t1,
{ rw multiset.mem_cons at h2,
exact h2.resolve_left this },
obtain ⟨t2, rfl⟩ := multiset.exists_cons_of_mem this,
rw [multiset.prod_cons, multiset.prod_cons, ←mul_assoc, ←zsqrtd.norm_eq_mul_conj] at hcoprime,
rw zsqrtd.is_unit_iff_norm_is_unit,
apply is_coprime.is_unit_of_dvd' hcoprime;
simp only [add_zero, zsqrtd.coe_int_re, zero_mul, dvd_mul_right, zsqrtd.mul_re, mul_zero,
zsqrtd.mul_im, zsqrtd.coe_int_im] },
end
/-- Associated elements in `ℤ√-3`. -/
def associated' (x y : ℤ√-3) : Prop := associated x y ∨ associated x (star y)
@[refl] theorem associated'.refl (x : ℤ√-3) : associated' x x := or.inl (by refl)
lemma associated'.norm_eq
{x y : ℤ√-3}
(h : associated' x y) :
x.norm = y.norm :=
begin
cases h; simp only [zsqrtd.norm_eq_of_associated (by norm_num) h, zsqrtd.norm_conj],
end
lemma associated'_of_abs_eq {x y : ℤ√-3} (hre : abs x.re = abs y.re) (him : abs x.im = abs y.im) :
associated' x y :=
begin
rw abs_eq_abs at hre him,
cases hre with h1 h1;
cases him with h2 h2;
[{ left, use 1}, {right, use 1}, {right, use -1}, {left, use -1}];
simp [zsqrtd.ext, h1, h2],
end
lemma associated'_of_associated_norm {x y : ℤ√-3} (h : associated (zsqrtd.norm x) (zsqrtd.norm y))
(hx : is_coprime x.re x.im)
(hy : is_coprime y.re y.im)
(h' : odd_prime_or_four x.norm) :
associated' x y :=
begin
have heq : x.norm = y.norm,
{ have hd : (-3 : ℤ) ≤ 0,
{ norm_num },
exact int.eq_of_associated_of_nonneg h (zsqrtd.norm_nonneg hd _) (zsqrtd.norm_nonneg hd _) },
obtain ⟨hre, him⟩ := step4_3 hx hy h' heq,
exact associated'_of_abs_eq hre him,
end
lemma factorization'.associated'_of_norm_eq
{a b c : ℤ√-3} (h : is_coprime a.re a.im)
(hbmem : b ∈ factorization' h) (hcmem : c ∈ factorization' h)
(hnorm : b.norm = c.norm) :
associated' b c :=
begin
apply associated'_of_associated_norm,
{ rw hnorm },
{ exact factorization'.coprime_of_mem h hbmem },
{ exact factorization'.coprime_of_mem h hcmem },
{ exact ((factorization'_prop h).2 b hbmem).2.2 },
end
lemma factors_unique
{f g : multiset ℤ√-3}
(hf : ∀x∈f, odd_prime_or_four (zsqrtd.norm x))
(hf' : ∀x∈f, is_coprime (zsqrtd.re x) (zsqrtd.im x))
(hg : ∀x∈g, odd_prime_or_four (zsqrtd.norm x))
(hg' : ∀x∈g, is_coprime (zsqrtd.re x) (zsqrtd.im x))
(h : associated f.prod g.prod) :
multiset.rel associated' f g :=
begin
have p : ∀ (x : ℤ√-3) (hx : x ∈ f) (y : ℤ√-3) (hy : y ∈ g),
associated x.norm y.norm → associated' x y,
{ intros x hx y hy hxy,
apply associated'_of_associated_norm hxy,
{ exact hf' x hx },
{ exact hg' y hy },
{ exact hf x hx } },
refine multiset.rel.mono _ p,
rw ←multiset.rel_map,
apply factors_unique_prod hf hg,
exact zsqrtd.associated_norm_of_associated h,
end
lemma factors_2_even'
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im) :
even (even_factor_exp a.norm) :=
begin
induction hn : a.norm.nat_abs using nat.strong_induction_on with n ih generalizing a,
dsimp only at ih,
by_cases hparity : even a.norm,
{ obtain ⟨u, huvcoprime, huvprod⟩ := step1' hcoprime hparity,
have huv := spts.ne_zero_of_coprime' _ huvcoprime,
rw [huvprod, factors_2_even huv, nat.even_add],
apply iff_of_true even_two,
apply ih _ _ huvcoprime rfl,
rw [←hn, huvprod, int.nat_abs_mul, lt_mul_iff_one_lt_left (int.nat_abs_pos_of_ne_zero huv)],
norm_num },
{ convert even_zero,
simp only [even_factor_exp, multiset.count_eq_zero, hn],
contrapose! hparity with hfactor,
rw even_iff_two_dvd,
exact unique_factorization_monoid.dvd_of_mem_normalized_factors hfactor }
end
lemma factors_odd_prime_or_four.unique
{a : ℤ√-3}
(hcoprime : is_coprime a.re a.im)
{f : multiset ℤ}
(hf : ∀x∈f, odd_prime_or_four x)
(hf' : ∀x∈f, (0 : ℤ) ≤ x)
(hassoc : associated f.prod a.norm) :
f = (factors_odd_prime_or_four a.norm) :=
factors_odd_prime_or_four.unique' hf hf' (spts.pos_of_coprime' hcoprime) (factors_2_even' hcoprime)
hassoc
lemma eq_or_eq_conj_of_associated_of_re_zero
{x A : ℤ√-3}
(hx : x.re = 0)
(h : associated x A) :
x = A ∨ x = star A :=
begin
obtain ⟨u, hu⟩ := h,
obtain ⟨v, hv1, hv2⟩ := zsqrt3.coe_of_is_unit' u.is_unit,
have hA : A.re = 0,
{ simp only [←hu, hv1, hx, add_zero, zero_mul, zsqrtd.mul_re, mul_zero, zsqrtd.coe_int_im] },
rw (abs_eq $ zero_le_one' ℤ) at hv2,
cases hv2 with hv2 hv2,
{ left,
simpa only [hv1, hv2, mul_one, int.cast_one] using hu },
{ right,
simp only [hv1, hv2, mul_one, int.cast_one, mul_neg, int.cast_neg] at hu,
simp [←hu, hx, zsqrtd.ext] }
end
lemma eq_or_eq_conj_iff_associated'_of_nonneg
{x A : ℤ√-3}
(hx : 0 ≤ x.re)
(hA : 0 ≤ A.re) :
associated' x A ↔ (x = A ∨ x = star A) :=
begin
split,
{ rintro (⟨u, hu⟩|⟨u, hu⟩); obtain ⟨v, hv1, hv2⟩ := zsqrt3.coe_of_is_unit' u.is_unit,
-- associated x A
{ by_cases hxre : x.re = 0,
{ apply eq_or_eq_conj_of_associated_of_re_zero hxre ⟨u, hu⟩ },
{ rw hv1 at hu,
rw (abs_eq $ zero_le_one' ℤ) at hv2,
cases hv2 with habsv habsv,
{ left,
rw [←hu, habsv, int.cast_one, mul_one] },
{ exfalso,
simp only [habsv, mul_one, int.cast_one, mul_neg, int.cast_neg] at hu,
apply lt_irrefl (0 : ℤ),
calc 0 ≤ A.re : hA
... = -x.re : _
... < 0 : _,
{ simp only [←hu, zsqrtd.neg_re] },
{ simp only [neg_neg_iff_pos],
exact lt_of_le_of_ne hx (ne.symm hxre) } } } },
-- associated x A.conj
{ by_cases hxre : x.re = 0,
{ convert (eq_or_eq_conj_of_associated_of_re_zero hxre ⟨u, hu⟩).symm,
rw star_star },
{ rw hv1 at hu,
rw (abs_eq $ zero_le_one' ℤ) at hv2,
cases hv2 with habsv habsv,
{ right,
rw [←hu, habsv, int.cast_one, mul_one] },
{ exfalso,
simp only [habsv, mul_one, int.cast_one, mul_neg, int.cast_neg] at hu,
apply lt_irrefl (0 : ℤ),
calc 0 ≤ A.re : hA
... = -x.re : _
... < 0 : _,
{ rw [←star_star A, ←hu],
simp only [zsqrtd.neg_re, zsqrtd.star_re]},
{ simp only [neg_neg_iff_pos],
apply lt_of_le_of_ne hx (ne.symm hxre) } } } } },
{ rintro (rfl|rfl),
{ refl },
{ right, refl } },
end
lemma step5' -- lemma page 54
(a : ℤ√-3)
(r : ℤ)
(hcoprime : is_coprime a.re a.im)
(hcube : r ^ 3 = a.norm) :
∃ g : multiset ℤ√-3, factorization' hcoprime = 3 • g :=
begin
classical,
obtain ⟨h1, h2⟩ := factorization'_prop hcoprime,
set f := factorization' hcoprime with hf,
apply multiset.exists_smul_of_dvd_count,
intros x hx,
convert dvd_mul_right _ _,
have : even (even_factor_exp r),
{ suffices : even (3 * even_factor_exp r),
{ rw nat.even_mul at this,
apply this.resolve_left,
norm_num },
rw [←even_factor_exp.pow r 3, hcube],
exact factors_2_even' hcoprime },
calc multiset.count x f
= multiset.card (multiset.filter (eq x) f) :
by rw [multiset.count, multiset.countp_eq_card_filter]
... = multiset.card (multiset.filter (λ (a : ℤ√-3), eq x.norm a.norm) f) :
congr_arg _ (multiset.filter_congr (λ A HA, _))
... = multiset.countp (eq x.norm) (multiset.map zsqrtd.norm f) :
(multiset.countp_map zsqrtd.norm f (eq x.norm)).symm
... = multiset.countp (eq x.norm) (factors_odd_prime_or_four a.norm) : _
... = multiset.count x.norm (factors_odd_prime_or_four a.norm) : by rw multiset.count
... = multiset.count x.norm (factors_odd_prime_or_four (r ^ 3)) : by rw hcube
... = multiset.count x.norm (3 • _) : congr_arg _ $ factors_odd_prime_or_four.pow _ _ this
... = 3 * _ : multiset.count_nsmul x.norm _ _,
show multiset.countp (eq x.norm) (multiset.map zsqrtd.norm f) =
multiset.countp (eq x.norm) (factors_odd_prime_or_four a.norm),
congr',
{ apply factors_odd_prime_or_four.unique hcoprime,
{ intros x hx,
obtain ⟨y, hy, rfl⟩ := multiset.mem_map.mp hx,
exact (h2 y hy).2.2 },
{ intros x hx,
obtain ⟨y, hy, rfl⟩ := multiset.mem_map.mp hx,
apply zsqrtd.norm_nonneg,
norm_num },
{ rw [prod_map_norm],
cases h1; rw [h1],
rw zsqrtd.norm_neg } },
have h2x := h2 x hx,
refine ⟨congr_arg _, λ h, _⟩,
have hassoc := factorization'.associated'_of_norm_eq hcoprime hx HA h,
have eq_or_eq_conj := (eq_or_eq_conj_iff_associated'_of_nonneg h2x.1 (h2 A HA).1).mp hassoc,
refine eq_or_eq_conj.resolve_right (λ H, _),
apply no_conj f,
{ intro HH,
apply h2x.2.1,
rw [zsqrt3.is_unit_iff] at HH,
exact HH.2 },
{ cases h1; rw h1 at hcoprime,
{ exact hcoprime },
{ rwa [←is_coprime.neg_neg_iff, ←zsqrtd.neg_im, ←zsqrtd.neg_re] } },
{ refine ⟨hx, _⟩,
rwa [H, star_star] },
end
lemma step5 -- lemma page 54
(a : ℤ√-3)
(r : ℤ)
(hcoprime : is_coprime a.re a.im)
(hcube : r ^ 3 = a.norm) :
∃ p : ℤ√-3, a = p ^ 3 :=
begin
obtain ⟨f, hf⟩ := step5' a r hcoprime hcube,
obtain ⟨h1, -⟩ := factorization'_prop hcoprime,
cases h1,
{ use f.prod,
rw [h1, hf, multiset.prod_nsmul] },
{ use -f.prod,
rw [h1, hf, multiset.prod_nsmul, neg_pow_bit1] },
end
lemma step6
(a b r : ℤ)
(hcoprime : is_coprime a b)
(hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2)
:
∃ p q,
a = p ^ 3 - 9 * p * q ^ 2 ∧
b = 3 * p ^ 2 * q - 3 * q ^ 3
:=
begin
set A : ℤ√-3 := ⟨a, b⟩,
have hcube' : r ^ 3 = A.norm,
{ simp only [hcube, zsqrtd.norm_def, A], ring },
obtain ⟨p, hp⟩ := step5 A r hcoprime hcube',
use [p.re, p.im],
simp only [zsqrtd.ext, pow_succ', pow_two, zsqrtd.mul_re, zsqrtd.mul_im] at hp,
convert hp; ring,
end
|
5e2d3fc0ff117ae01d1a2d2b3531714743920226 | 80cc5bf14c8ea85ff340d1d747a127dcadeb966f | /src/topology/instances/ennreal.lean | 312107b09fbdf4b429d9116aa2f6581f0221f69f | [
"Apache-2.0"
] | permissive | lacker/mathlib | f2439c743c4f8eb413ec589430c82d0f73b2d539 | ddf7563ac69d42cfa4a1bfe41db1fed521bd795f | refs/heads/master | 1,671,948,326,773 | 1,601,479,268,000 | 1,601,479,268,000 | 298,686,743 | 0 | 0 | Apache-2.0 | 1,601,070,794,000 | 1,601,070,794,000 | null | UTF-8 | Lean | false | false | 40,850 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Johannes Hölzl
-/
import topology.instances.nnreal
/-!
# Extended non-negative reals
-/
noncomputable theory
open classical set filter metric
open_locale classical
open_locale topological_space
variables {α : Type*} {β : Type*} {γ : Type*}
open_locale ennreal big_operators filter
namespace ennreal
variables {a b c d : ennreal} {r p q : nnreal}
variables {x y z : ennreal} {ε ε₁ ε₂ : ennreal} {s : set ennreal}
section topological_space
open topological_space
/-- Topology on `ennreal`.
Note: this is different from the `emetric_space` topology. The `emetric_space` topology has
`is_open {⊤}`, while this topology doesn't have singleton elements. -/
instance : topological_space ennreal := preorder.topology ennreal
instance : order_topology ennreal := ⟨rfl⟩
instance : t2_space ennreal := by apply_instance -- short-circuit type class inference
instance : second_countable_topology ennreal :=
⟨⟨⋃q ≥ (0:ℚ), {{a : ennreal | a < nnreal.of_real q}, {a : ennreal | ↑(nnreal.of_real q) < a}},
(countable_encodable _).bUnion $ assume a ha, (countable_singleton _).insert _,
le_antisymm
(le_generate_from $ by simp [or_imp_distrib, is_open_lt', is_open_gt'] {contextual := tt})
(le_generate_from $ λ s h, begin
rcases h with ⟨a, hs | hs⟩;
[ rw show s = ⋃q∈{q:ℚ | 0 ≤ q ∧ a < nnreal.of_real q}, {b | ↑(nnreal.of_real q) < b},
from set.ext (assume b, by simp [hs, @ennreal.lt_iff_exists_rat_btwn a b, and_assoc]),
rw show s = ⋃q∈{q:ℚ | 0 ≤ q ∧ ↑(nnreal.of_real q) < a}, {b | b < ↑(nnreal.of_real q)},
from set.ext (assume b, by simp [hs, @ennreal.lt_iff_exists_rat_btwn b a, and_comm, and_assoc])];
{ apply is_open_Union, intro q,
apply is_open_Union, intro hq,
exact generate_open.basic _ (mem_bUnion hq.1 $ by simp) }
end)⟩⟩
lemma embedding_coe : embedding (coe : nnreal → ennreal) :=
⟨⟨begin
refine le_antisymm _ _,
{ rw [@order_topology.topology_eq_generate_intervals ennreal _,
← coinduced_le_iff_le_induced],
refine le_generate_from (assume s ha, _),
rcases ha with ⟨a, rfl | rfl⟩,
show is_open {b : nnreal | a < ↑b},
{ cases a; simp [none_eq_top, some_eq_coe, is_open_lt'] },
show is_open {b : nnreal | ↑b < a},
{ cases a; simp [none_eq_top, some_eq_coe, is_open_gt', is_open_const] } },
{ rw [@order_topology.topology_eq_generate_intervals nnreal _],
refine le_generate_from (assume s ha, _),
rcases ha with ⟨a, rfl | rfl⟩,
exact ⟨Ioi a, is_open_Ioi, by simp [Ioi]⟩,
exact ⟨Iio a, is_open_Iio, by simp [Iio]⟩ }
end⟩,
assume a b, coe_eq_coe.1⟩
lemma is_open_ne_top : is_open {a : ennreal | a ≠ ⊤} := is_open_ne
lemma is_open_Ico_zero : is_open (Ico 0 b) := by { rw ennreal.Ico_eq_Iio, exact is_open_Iio}
lemma coe_range_mem_nhds : range (coe : nnreal → ennreal) ∈ 𝓝 (r : ennreal) :=
have {a : ennreal | a ≠ ⊤} = range (coe : nnreal → ennreal),
from set.ext $ assume a, by cases a; simp [none_eq_top, some_eq_coe],
this ▸ mem_nhds_sets is_open_ne_top coe_ne_top
@[norm_cast] lemma tendsto_coe {f : filter α} {m : α → nnreal} {a : nnreal} :
tendsto (λa, (m a : ennreal)) f (𝓝 ↑a) ↔ tendsto m f (𝓝 a) :=
embedding_coe.tendsto_nhds_iff.symm
lemma continuous_coe {α} [topological_space α] {f : α → nnreal} :
continuous (λa, (f a : ennreal)) ↔ continuous f :=
embedding_coe.continuous_iff.symm
lemma nhds_coe {r : nnreal} : 𝓝 (r : ennreal) = (𝓝 r).map coe :=
by rw [embedding_coe.induced, map_nhds_induced_eq coe_range_mem_nhds]
lemma nhds_coe_coe {r p : nnreal} : 𝓝 ((r : ennreal), (p : ennreal)) =
(𝓝 (r, p)).map (λp:nnreal×nnreal, (p.1, p.2)) :=
begin
rw [(embedding_coe.prod_mk embedding_coe).map_nhds_eq],
rw [← prod_range_range_eq],
exact prod_mem_nhds_sets coe_range_mem_nhds coe_range_mem_nhds
end
lemma continuous_of_real : continuous ennreal.of_real :=
(continuous_coe.2 continuous_id).comp nnreal.continuous_of_real
lemma tendsto_of_real {f : filter α} {m : α → ℝ} {a : ℝ} (h : tendsto m f (𝓝 a)) :
tendsto (λa, ennreal.of_real (m a)) f (𝓝 (ennreal.of_real a)) :=
tendsto.comp (continuous.tendsto continuous_of_real _) h
lemma tendsto_to_nnreal {a : ennreal} : a ≠ ⊤ →
tendsto (ennreal.to_nnreal) (𝓝 a) (𝓝 a.to_nnreal) :=
begin
cases a; simp [some_eq_coe, none_eq_top, nhds_coe, tendsto_map'_iff, (∘)],
exact tendsto_id
end
lemma continuous_on_to_nnreal : continuous_on ennreal.to_nnreal {a | a ≠ ∞} :=
continuous_on_iff_continuous_restrict.2 $ continuous_iff_continuous_at.2 $ λ x,
(tendsto_to_nnreal x.2).comp continuous_at_subtype_coe
lemma tendsto_to_real {a : ennreal} : a ≠ ⊤ → tendsto (ennreal.to_real) (𝓝 a) (𝓝 a.to_real) :=
λ ha, tendsto.comp ((@nnreal.tendsto_coe _ (𝓝 a.to_nnreal) id (a.to_nnreal)).2 tendsto_id)
(tendsto_to_nnreal ha)
lemma tendsto_nhds_top {m : α → ennreal} {f : filter α}
(h : ∀ n : ℕ, ∀ᶠ a in f, ↑n < m a) : tendsto m f (𝓝 ⊤) :=
tendsto_nhds_generate_from $ assume s hs,
match s, hs with
| _, ⟨none, or.inl rfl⟩, hr := (lt_irrefl ⊤ hr).elim
| _, ⟨some r, or.inl rfl⟩, hr :=
let ⟨n, hrn⟩ := exists_nat_gt r in
mem_sets_of_superset (h n) $ assume a hnma, show ↑r < m a, from
lt_trans (show (r : ennreal) < n, from (coe_nat n) ▸ coe_lt_coe.2 hrn) hnma
| _, ⟨a, or.inr rfl⟩, hr := (not_top_lt $ show ⊤ < a, from hr).elim
end
lemma tendsto_nat_nhds_top : tendsto (λ n : ℕ, ↑n) at_top (𝓝 ∞) :=
tendsto_nhds_top $ λ n, mem_at_top_sets.2
⟨n+1, λ m hm, ennreal.coe_nat_lt_coe_nat.2 $ nat.lt_of_succ_le hm⟩
lemma nhds_top : 𝓝 ∞ = ⨅a ≠ ∞, 𝓟 (Ioi a) :=
nhds_top_order.trans $ by simp [lt_top_iff_ne_top, Ioi]
lemma nhds_zero : 𝓝 (0 : ennreal) = ⨅a ≠ 0, 𝓟 (Iio a) :=
nhds_bot_order.trans $ by simp [bot_lt_iff_ne_bot, Iio]
/-- The set of finite `ennreal` numbers is homeomorphic to `nnreal`. -/
def ne_top_homeomorph_nnreal : {a | a ≠ ∞} ≃ₜ nnreal :=
{ to_fun := λ x, ennreal.to_nnreal x,
inv_fun := λ x, ⟨x, coe_ne_top⟩,
left_inv := λ ⟨x, hx⟩, subtype.eq $ coe_to_nnreal hx,
right_inv := λ x, to_nnreal_coe,
continuous_to_fun := continuous_on_iff_continuous_restrict.1 continuous_on_to_nnreal,
continuous_inv_fun := continuous_subtype_mk _ (continuous_coe.2 continuous_id) }
/-- The set of finite `ennreal` numbers is homeomorphic to `nnreal`. -/
def lt_top_homeomorph_nnreal : {a | a < ∞} ≃ₜ nnreal :=
by refine (homeomorph.set_congr $ set.ext $ λ x, _).trans ne_top_homeomorph_nnreal;
simp only [mem_set_of_eq, lt_top_iff_ne_top]
-- using Icc because
-- • don't have 'Ioo (x - ε) (x + ε) ∈ 𝓝 x' unless x > 0
-- • (x - y ≤ ε ↔ x ≤ ε + y) is true, while (x - y < ε ↔ x < ε + y) is not
lemma Icc_mem_nhds : x ≠ ⊤ → 0 < ε → Icc (x - ε) (x + ε) ∈ 𝓝 x :=
begin
assume xt ε0, rw mem_nhds_sets_iff,
by_cases x0 : x = 0,
{ use Iio (x + ε),
have : Iio (x + ε) ⊆ Icc (x - ε) (x + ε), assume a, rw x0, simpa using le_of_lt,
use this, exact ⟨is_open_Iio, mem_Iio_self_add xt ε0⟩ },
{ use Ioo (x - ε) (x + ε), use Ioo_subset_Icc_self,
exact ⟨is_open_Ioo, mem_Ioo_self_sub_add xt x0 ε0 ε0 ⟩ }
end
lemma nhds_of_ne_top : x ≠ ⊤ → 𝓝 x = ⨅ε > 0, 𝓟 (Icc (x - ε) (x + ε)) :=
begin
assume xt, refine le_antisymm _ _,
-- first direction
simp only [le_infi_iff, le_principal_iff], assume ε ε0, exact Icc_mem_nhds xt ε0,
-- second direction
rw nhds_generate_from, refine le_infi (assume s, le_infi $ assume hs, _),
simp only [mem_set_of_eq] at hs, rcases hs with ⟨xs, ⟨a, ha⟩⟩,
cases ha,
{ rw ha at *,
rcases dense xs with ⟨b, ⟨ab, bx⟩⟩,
have xb_pos : x - b > 0 := zero_lt_sub_iff_lt.2 bx,
have xxb : x - (x - b) = b := sub_sub_cancel (by rwa lt_top_iff_ne_top) (le_of_lt bx),
refine infi_le_of_le (x - b) (infi_le_of_le xb_pos _),
simp only [mem_principal_sets, le_principal_iff],
assume y, rintros ⟨h₁, h₂⟩, rw xxb at h₁, calc a < b : ab ... ≤ y : h₁ },
{ rw ha at *,
rcases dense xs with ⟨b, ⟨xb, ba⟩⟩,
have bx_pos : b - x > 0 := zero_lt_sub_iff_lt.2 xb,
have xbx : x + (b - x) = b := add_sub_cancel_of_le (le_of_lt xb),
refine infi_le_of_le (b - x) (infi_le_of_le bx_pos _),
simp only [mem_principal_sets, le_principal_iff],
assume y, rintros ⟨h₁, h₂⟩, rw xbx at h₂, calc y ≤ b : h₂ ... < a : ba },
end
/-- Characterization of neighborhoods for `ennreal` numbers. See also `tendsto_order`
for a version with strict inequalities. -/
protected theorem tendsto_nhds {f : filter α} {u : α → ennreal} {a : ennreal} (ha : a ≠ ⊤) :
tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, (u x) ∈ Icc (a - ε) (a + ε) :=
by simp only [nhds_of_ne_top ha, tendsto_infi, tendsto_principal, mem_Icc]
protected lemma tendsto_at_top [nonempty β] [semilattice_sup β] {f : β → ennreal} {a : ennreal}
(ha : a ≠ ⊤) : tendsto f at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n≥N, (f n) ∈ Icc (a - ε) (a + ε) :=
by simp only [ennreal.tendsto_nhds ha, mem_at_top_sets, mem_set_of_eq, filter.eventually]
lemma tendsto_coe_nnreal_nhds_top {α} {l : filter α} {f : α → nnreal} (h : tendsto f l at_top) :
tendsto (λa, (f a : ennreal)) l (𝓝 ∞) :=
tendsto_nhds_top $ assume n,
have ∀ᶠ a in l, ↑(n+1) ≤ f a := h $ mem_at_top _,
mem_sets_of_superset this $ assume a (ha : ↑(n+1) ≤ f a),
begin
rw [← coe_nat],
dsimp,
exact coe_lt_coe.2 (lt_of_lt_of_le (nat.cast_lt.2 (nat.lt_succ_self _)) ha)
end
instance : has_continuous_add ennreal :=
⟨ continuous_iff_continuous_at.2 $
have hl : ∀a:ennreal, tendsto (λ (p : ennreal × ennreal), p.fst + p.snd) (𝓝 (⊤, a)) (𝓝 ⊤), from
assume a, tendsto_nhds_top $ assume n,
have set.prod {a | ↑n < a } univ ∈ 𝓝 ((⊤:ennreal), a), from
prod_mem_nhds_sets (lt_mem_nhds $ coe_nat n ▸ coe_lt_top) univ_mem_sets,
show {a : ennreal × ennreal | ↑n < a.fst + a.snd} ∈ 𝓝 (⊤, a),
begin filter_upwards [this] assume ⟨a₁, a₂⟩ ⟨h₁, h₂⟩, lt_of_lt_of_le h₁ (le_add_right $ le_refl _) end,
begin
rintro ⟨a₁, a₂⟩,
cases a₁, { simp [continuous_at, none_eq_top, hl a₂], },
cases a₂, { simp [continuous_at, none_eq_top, some_eq_coe, nhds_swap (a₁ : ennreal) ⊤,
tendsto_map'_iff, (∘)], convert hl a₁, simp [add_comm] },
simp [continuous_at, some_eq_coe, nhds_coe_coe, tendsto_map'_iff, (∘)],
simp only [coe_add.symm, tendsto_coe, tendsto_add]
end ⟩
protected lemma tendsto_mul (ha : a ≠ 0 ∨ b ≠ ⊤) (hb : b ≠ 0 ∨ a ≠ ⊤) :
tendsto (λp:ennreal×ennreal, p.1 * p.2) (𝓝 (a, b)) (𝓝 (a * b)) :=
have ht : ∀b:ennreal, b ≠ 0 → tendsto (λp:ennreal×ennreal, p.1 * p.2) (𝓝 ((⊤:ennreal), b)) (𝓝 ⊤),
begin
refine assume b hb, tendsto_nhds_top $ assume n, _,
rcases dense (zero_lt_iff_ne_zero.2 hb) with ⟨ε', hε', hεb'⟩,
rcases ennreal.lt_iff_exists_coe.1 hεb' with ⟨ε, rfl, h⟩,
rcases exists_nat_gt (↑n / ε) with ⟨m, hm⟩,
have hε : ε > 0, from coe_lt_coe.1 hε',
refine mem_sets_of_superset (prod_mem_nhds_sets (lt_mem_nhds $ @coe_lt_top m) (lt_mem_nhds $ h)) _,
rintros ⟨a₁, a₂⟩ ⟨h₁, h₂⟩,
dsimp at h₁ h₂ ⊢,
calc (n:ennreal) = ↑(((n:nnreal) / ε) * ε) :
begin
norm_cast,
simp [nnreal.div_def, mul_assoc, nnreal.inv_mul_cancel (ne_of_gt hε)]
end
... < (↑m * ε : nnreal) : coe_lt_coe.2 $ mul_lt_mul hm (le_refl _) hε (nat.cast_nonneg _)
... ≤ a₁ * a₂ : by rw [coe_mul]; exact canonically_ordered_semiring.mul_le_mul
(le_of_lt h₁)
(le_of_lt h₂)
end,
begin
cases a, {simp [none_eq_top] at hb, simp [none_eq_top, ht b hb, top_mul, hb] },
cases b, {
simp [none_eq_top] at ha,
simp [*, nhds_swap (a : ennreal) ⊤, none_eq_top, some_eq_coe, top_mul, tendsto_map'_iff, (∘), mul_comm] },
simp [some_eq_coe, nhds_coe_coe, tendsto_map'_iff, (∘)],
simp only [coe_mul.symm, tendsto_coe, tendsto_mul]
end
protected lemma tendsto.mul {f : filter α} {ma : α → ennreal} {mb : α → ennreal} {a b : ennreal}
(hma : tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ⊤) (hmb : tendsto mb f (𝓝 b)) (hb : b ≠ 0 ∨ a ≠ ⊤) :
tendsto (λa, ma a * mb a) f (𝓝 (a * b)) :=
show tendsto ((λp:ennreal×ennreal, p.1 * p.2) ∘ (λa, (ma a, mb a))) f (𝓝 (a * b)), from
tendsto.comp (ennreal.tendsto_mul ha hb) (hma.prod_mk_nhds hmb)
protected lemma tendsto.const_mul {f : filter α} {m : α → ennreal} {a b : ennreal}
(hm : tendsto m f (𝓝 b)) (hb : b ≠ 0 ∨ a ≠ ⊤) : tendsto (λb, a * m b) f (𝓝 (a * b)) :=
by_cases
(assume : a = 0, by simp [this, tendsto_const_nhds])
(assume ha : a ≠ 0, ennreal.tendsto.mul tendsto_const_nhds (or.inl ha) hm hb)
protected lemma tendsto.mul_const {f : filter α} {m : α → ennreal} {a b : ennreal}
(hm : tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ⊤) : tendsto (λx, m x * b) f (𝓝 (a * b)) :=
by simpa only [mul_comm] using ennreal.tendsto.const_mul hm ha
protected lemma continuous_at_const_mul {a b : ennreal} (h : a ≠ ⊤ ∨ b ≠ 0) :
continuous_at ((*) a) b :=
tendsto.const_mul tendsto_id h.symm
protected lemma continuous_at_mul_const {a b : ennreal} (h : a ≠ ⊤ ∨ b ≠ 0) :
continuous_at (λ x, x * a) b :=
tendsto.mul_const tendsto_id h.symm
protected lemma continuous_const_mul {a : ennreal} (ha : a ≠ ⊤) : continuous ((*) a) :=
continuous_iff_continuous_at.2 $ λ x, ennreal.continuous_at_const_mul (or.inl ha)
protected lemma continuous_mul_const {a : ennreal} (ha : a ≠ ⊤) : continuous (λ x, x * a) :=
continuous_iff_continuous_at.2 $ λ x, ennreal.continuous_at_mul_const (or.inl ha)
lemma infi_mul_left {ι} [nonempty ι] {f : ι → ennreal} {a : ennreal}
(h : a = ⊤ → (⨅ i, f i) = 0 → ∃ i, f i = 0) :
(⨅ i, a * f i) = a * ⨅ i, f i :=
begin
by_cases H : a = ⊤ ∧ (⨅ i, f i) = 0,
{ rcases h H.1 H.2 with ⟨i, hi⟩,
rw [H.2, mul_zero, ← bot_eq_zero, infi_eq_bot],
exact λ b hb, ⟨i, by rwa [hi, mul_zero, ← bot_eq_zero]⟩ },
{ rw not_and_distrib at H,
exact (map_infi_of_continuous_at_of_monotone' (ennreal.continuous_at_const_mul H)
ennreal.mul_left_mono).symm }
end
lemma infi_mul_right {ι} [nonempty ι] {f : ι → ennreal} {a : ennreal}
(h : a = ⊤ → (⨅ i, f i) = 0 → ∃ i, f i = 0) :
(⨅ i, f i * a) = (⨅ i, f i) * a :=
by simpa only [mul_comm a] using infi_mul_left h
protected lemma continuous_inv : continuous (has_inv.inv : ennreal → ennreal) :=
continuous_iff_continuous_at.2 $ λ a, tendsto_order.2
⟨begin
assume b hb,
simp only [@ennreal.lt_inv_iff_lt_inv b],
exact gt_mem_nhds (ennreal.lt_inv_iff_lt_inv.1 hb),
end,
begin
assume b hb,
simp only [gt_iff_lt, @ennreal.inv_lt_iff_inv_lt _ b],
exact lt_mem_nhds (ennreal.inv_lt_iff_inv_lt.1 hb)
end⟩
@[simp] protected lemma tendsto_inv_iff {f : filter α} {m : α → ennreal} {a : ennreal} :
tendsto (λ x, (m x)⁻¹) f (𝓝 a⁻¹) ↔ tendsto m f (𝓝 a) :=
⟨λ h, by simpa only [function.comp, ennreal.inv_inv]
using (ennreal.continuous_inv.tendsto a⁻¹).comp h,
(ennreal.continuous_inv.tendsto a).comp⟩
protected lemma tendsto.div {f : filter α} {ma : α → ennreal} {mb : α → ennreal} {a b : ennreal}
(hma : tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) (hmb : tendsto mb f (𝓝 b)) (hb : b ≠ ⊤ ∨ a ≠ ⊤) :
tendsto (λa, ma a / mb a) f (𝓝 (a / b)) :=
by { apply tendsto.mul hma _ (ennreal.tendsto_inv_iff.2 hmb) _; simp [ha, hb] }
protected lemma tendsto.const_div {f : filter α} {m : α → ennreal} {a b : ennreal}
(hm : tendsto m f (𝓝 b)) (hb : b ≠ ⊤ ∨ a ≠ ⊤) : tendsto (λb, a / m b) f (𝓝 (a / b)) :=
by { apply tendsto.const_mul (ennreal.tendsto_inv_iff.2 hm), simp [hb] }
protected lemma tendsto.div_const {f : filter α} {m : α → ennreal} {a b : ennreal}
(hm : tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) : tendsto (λx, m x / b) f (𝓝 (a / b)) :=
by { apply tendsto.mul_const hm, simp [ha] }
protected lemma tendsto_inv_nat_nhds_zero : tendsto (λ n : ℕ, (n : ennreal)⁻¹) at_top (𝓝 0) :=
ennreal.inv_top ▸ ennreal.tendsto_inv_iff.2 tendsto_nat_nhds_top
lemma Sup_add {s : set ennreal} (hs : s.nonempty) : Sup s + a = ⨆b∈s, b + a :=
have Sup ((λb, b + a) '' s) = Sup s + a,
from is_lub.Sup_eq (is_lub_of_is_lub_of_tendsto
(assume x _ y _ h, add_le_add h (le_refl _))
(is_lub_Sup s)
hs
(tendsto.add (tendsto_id' inf_le_left) tendsto_const_nhds)),
by simp [Sup_image, -add_comm] at this; exact this.symm
lemma supr_add {ι : Sort*} {s : ι → ennreal} [h : nonempty ι] : supr s + a = ⨆b, s b + a :=
let ⟨x⟩ := h in
calc supr s + a = Sup (range s) + a : by simp [Sup_range]
... = (⨆b∈range s, b + a) : Sup_add ⟨s x, x, rfl⟩
... = _ : supr_range
lemma add_supr {ι : Sort*} {s : ι → ennreal} [h : nonempty ι] : a + supr s = ⨆b, a + s b :=
by rw [add_comm, supr_add]; simp [add_comm]
lemma supr_add_supr {ι : Sort*} {f g : ι → ennreal} (h : ∀i j, ∃k, f i + g j ≤ f k + g k) :
supr f + supr g = (⨆ a, f a + g a) :=
begin
by_cases hι : nonempty ι,
{ letI := hι,
refine le_antisymm _ (supr_le $ λ a, add_le_add (le_supr _ _) (le_supr _ _)),
simpa [add_supr, supr_add] using
λ i j:ι, show f i + g j ≤ ⨆ a, f a + g a, from
let ⟨k, hk⟩ := h i j in le_supr_of_le k hk },
{ have : ∀f:ι → ennreal, (⨆i, f i) = 0 := assume f, bot_unique (supr_le $ assume i, (hι ⟨i⟩).elim),
rw [this, this, this, zero_add] }
end
lemma supr_add_supr_of_monotone {ι : Sort*} [semilattice_sup ι]
{f g : ι → ennreal} (hf : monotone f) (hg : monotone g) :
supr f + supr g = (⨆ a, f a + g a) :=
supr_add_supr $ assume i j, ⟨i ⊔ j, add_le_add (hf $ le_sup_left) (hg $ le_sup_right)⟩
lemma finset_sum_supr_nat {α} {ι} [semilattice_sup ι] {s : finset α} {f : α → ι → ennreal}
(hf : ∀a, monotone (f a)) :
∑ a in s, supr (f a) = (⨆ n, ∑ a in s, f a n) :=
begin
refine finset.induction_on s _ _,
{ simp,
exact (bot_unique $ supr_le $ assume i, le_refl ⊥).symm },
{ assume a s has ih,
simp only [finset.sum_insert has],
rw [ih, supr_add_supr_of_monotone (hf a)],
assume i j h,
exact (finset.sum_le_sum $ assume a ha, hf a h) }
end
section priority
-- for some reason the next proof fails without changing the priority of this instance
local attribute [instance, priority 1000] classical.prop_decidable
lemma mul_Sup {s : set ennreal} {a : ennreal} : a * Sup s = ⨆i∈s, a * i :=
begin
by_cases hs : ∀x∈s, x = (0:ennreal),
{ have h₁ : Sup s = 0 := (bot_unique $ Sup_le $ assume a ha, (hs a ha).symm ▸ le_refl 0),
have h₂ : (⨆i ∈ s, a * i) = 0 :=
(bot_unique $ supr_le $ assume a, supr_le $ assume ha, by simp [hs a ha]),
rw [h₁, h₂, mul_zero] },
{ simp only [not_forall] at hs,
rcases hs with ⟨x, hx, hx0⟩,
have s₁ : Sup s ≠ 0 :=
zero_lt_iff_ne_zero.1 (lt_of_lt_of_le (zero_lt_iff_ne_zero.2 hx0) (le_Sup hx)),
have : Sup ((λb, a * b) '' s) = a * Sup s :=
is_lub.Sup_eq (is_lub_of_is_lub_of_tendsto
(assume x _ y _ h, canonically_ordered_semiring.mul_le_mul (le_refl _) h)
(is_lub_Sup _)
⟨x, hx⟩
(ennreal.tendsto.const_mul (tendsto_id' inf_le_left) (or.inl s₁))),
rw [this.symm, Sup_image] }
end
end priority
lemma mul_supr {ι : Sort*} {f : ι → ennreal} {a : ennreal} : a * supr f = ⨆i, a * f i :=
by rw [← Sup_range, mul_Sup, supr_range]
lemma supr_mul {ι : Sort*} {f : ι → ennreal} {a : ennreal} : supr f * a = ⨆i, f i * a :=
by rw [mul_comm, mul_supr]; congr; funext; rw [mul_comm]
protected lemma tendsto_coe_sub : ∀{b:ennreal}, tendsto (λb:ennreal, ↑r - b) (𝓝 b) (𝓝 (↑r - b)) :=
begin
refine (forall_ennreal.2 $ and.intro (assume a, _) _),
{ simp [@nhds_coe a, tendsto_map'_iff, (∘), tendsto_coe, coe_sub.symm],
exact nnreal.tendsto.sub tendsto_const_nhds tendsto_id },
simp,
exact (tendsto.congr' (mem_sets_of_superset (lt_mem_nhds $ @coe_lt_top r) $
by simp [le_of_lt] {contextual := tt})) tendsto_const_nhds
end
lemma sub_supr {ι : Sort*} [hι : nonempty ι] {b : ι → ennreal} (hr : a < ⊤) :
a - (⨆i, b i) = (⨅i, a - b i) :=
let ⟨i⟩ := hι in
let ⟨r, eq, _⟩ := lt_iff_exists_coe.mp hr in
have Inf ((λb, ↑r - b) '' range b) = ↑r - (⨆i, b i),
from is_glb.Inf_eq $ is_glb_of_is_lub_of_tendsto
(assume x _ y _, sub_le_sub (le_refl _))
is_lub_supr
⟨_, i, rfl⟩
(tendsto.comp ennreal.tendsto_coe_sub (tendsto_id' inf_le_left)),
by rw [eq, ←this]; simp [Inf_image, infi_range, -mem_range]; exact le_refl _
end topological_space
section tsum
variables {f g : α → ennreal}
@[norm_cast] protected lemma has_sum_coe {f : α → nnreal} {r : nnreal} :
has_sum (λa, (f a : ennreal)) ↑r ↔ has_sum f r :=
have (λs:finset α, ∑ a in s, ↑(f a)) = (coe : nnreal → ennreal) ∘ (λs:finset α, ∑ a in s, f a),
from funext $ assume s, ennreal.coe_finset_sum.symm,
by unfold has_sum; rw [this, tendsto_coe]
protected lemma tsum_coe_eq {f : α → nnreal} (h : has_sum f r) : (∑'a, (f a : ennreal)) = r :=
(ennreal.has_sum_coe.2 h).tsum_eq
protected lemma coe_tsum {f : α → nnreal} : summable f → ↑(tsum f) = (∑'a, (f a : ennreal))
| ⟨r, hr⟩ := by rw [hr.tsum_eq, ennreal.tsum_coe_eq hr]
protected lemma has_sum : has_sum f (⨆s:finset α, ∑ a in s, f a) :=
tendsto_order.2
⟨assume a' ha',
let ⟨s, hs⟩ := lt_supr_iff.mp ha' in
mem_at_top_sets.mpr ⟨s, assume t ht, lt_of_lt_of_le hs $ finset.sum_le_sum_of_subset ht⟩,
assume a' ha',
univ_mem_sets' $ assume s,
have ∑ a in s, f a ≤ ⨆(s : finset α), ∑ a in s, f a,
from le_supr (λ(s : finset α), ∑ a in s, f a) s,
lt_of_le_of_lt this ha'⟩
@[simp] protected lemma summable : summable f := ⟨_, ennreal.has_sum⟩
lemma tsum_coe_ne_top_iff_summable {f : β → nnreal} :
(∑' b, (f b:ennreal)) ≠ ∞ ↔ summable f :=
begin
refine ⟨λ h, _, λ h, ennreal.coe_tsum h ▸ ennreal.coe_ne_top⟩,
lift (∑' b, (f b:ennreal)) to nnreal using h with a ha,
refine ⟨a, ennreal.has_sum_coe.1 _⟩,
rw ha,
exact ennreal.summable.has_sum
end
protected lemma tsum_eq_supr_sum : (∑'a, f a) = (⨆s:finset α, ∑ a in s, f a) :=
ennreal.has_sum.tsum_eq
protected lemma tsum_eq_supr_sum' {ι : Type*} (s : ι → finset α) (hs : ∀ t, ∃ i, t ⊆ s i) :
(∑' a, f a) = ⨆ i, ∑ a in s i, f a :=
begin
rw [ennreal.tsum_eq_supr_sum],
symmetry,
change (⨆i:ι, (λ t : finset α, ∑ a in t, f a) (s i)) = ⨆s:finset α, ∑ a in s, f a,
exact (finset.sum_mono_set f).supr_comp_eq hs
end
protected lemma tsum_sigma {β : α → Type*} (f : Πa, β a → ennreal) :
(∑'p:Σa, β a, f p.1 p.2) = (∑'a b, f a b) :=
tsum_sigma' (assume b, ennreal.summable) ennreal.summable
protected lemma tsum_sigma' {β : α → Type*} (f : (Σ a, β a) → ennreal) :
(∑'p:(Σa, β a), f p) = (∑'a b, f ⟨a, b⟩) :=
tsum_sigma' (assume b, ennreal.summable) ennreal.summable
protected lemma tsum_prod {f : α → β → ennreal} : (∑'p:α×β, f p.1 p.2) = (∑'a, ∑'b, f a b) :=
tsum_prod' ennreal.summable $ λ _, ennreal.summable
protected lemma tsum_comm {f : α → β → ennreal} : (∑'a, ∑'b, f a b) = (∑'b, ∑'a, f a b) :=
tsum_comm' ennreal.summable (λ _, ennreal.summable) (λ _, ennreal.summable)
protected lemma tsum_add : (∑'a, f a + g a) = (∑'a, f a) + (∑'a, g a) :=
tsum_add ennreal.summable ennreal.summable
protected lemma tsum_le_tsum (h : ∀a, f a ≤ g a) : (∑'a, f a) ≤ (∑'a, g a) :=
tsum_le_tsum h ennreal.summable ennreal.summable
protected lemma sum_le_tsum {f : α → ennreal} (s : finset α) : ∑ x in s, f x ≤ ∑' x, f x :=
sum_le_tsum s (λ x hx, zero_le _) ennreal.summable
protected lemma tsum_eq_supr_nat {f : ℕ → ennreal} :
(∑'i:ℕ, f i) = (⨆i:ℕ, ∑ a in finset.range i, f a) :=
ennreal.tsum_eq_supr_sum' _ finset.exists_nat_subset_range
protected lemma le_tsum (a : α) : f a ≤ (∑'a, f a) :=
le_tsum' ennreal.summable a
protected lemma tsum_eq_top_of_eq_top : (∃ a, f a = ∞) → (∑' a, f a) = ∞
| ⟨a, ha⟩ := top_unique $ ha ▸ ennreal.le_tsum a
protected lemma ne_top_of_tsum_ne_top (h : (∑' a, f a) ≠ ∞) (a : α) : f a ≠ ∞ :=
λ ha, h $ ennreal.tsum_eq_top_of_eq_top ⟨a, ha⟩
protected lemma tsum_mul_left : (∑'i, a * f i) = a * (∑'i, f i) :=
if h : ∀i, f i = 0 then by simp [h] else
let ⟨i, (hi : f i ≠ 0)⟩ := not_forall.mp h in
have sum_ne_0 : (∑'i, f i) ≠ 0, from ne_of_gt $
calc 0 < f i : lt_of_le_of_ne (zero_le _) hi.symm
... ≤ (∑'i, f i) : ennreal.le_tsum _,
have tendsto (λs:finset α, ∑ j in s, a * f j) at_top (𝓝 (a * (∑'i, f i))),
by rw [← show (*) a ∘ (λs:finset α, ∑ j in s, f j) = λs, ∑ j in s, a * f j,
from funext $ λ s, finset.mul_sum];
exact ennreal.tendsto.const_mul ennreal.summable.has_sum (or.inl sum_ne_0),
has_sum.tsum_eq this
protected lemma tsum_mul_right : (∑'i, f i * a) = (∑'i, f i) * a :=
by simp [mul_comm, ennreal.tsum_mul_left]
@[simp] lemma tsum_supr_eq {α : Type*} (a : α) {f : α → ennreal} :
(∑'b:α, ⨆ (h : a = b), f b) = f a :=
le_antisymm
(by rw [ennreal.tsum_eq_supr_sum]; exact supr_le (assume s,
calc (∑ b in s, ⨆ (h : a = b), f b) ≤ ∑ b in {a}, ⨆ (h : a = b), f b :
finset.sum_le_sum_of_ne_zero $ assume b _ hb,
suffices a = b, by simpa using this.symm,
classical.by_contradiction $ assume h,
by simpa [h] using hb
... = f a : by simp))
(calc f a ≤ (⨆ (h : a = a), f a) : le_supr (λh:a=a, f a) rfl
... ≤ (∑'b:α, ⨆ (h : a = b), f b) : ennreal.le_tsum _)
lemma has_sum_iff_tendsto_nat {f : ℕ → ennreal} (r : ennreal) :
has_sum f r ↔ tendsto (λn:ℕ, ∑ i in finset.range n, f i) at_top (𝓝 r) :=
begin
refine ⟨has_sum.tendsto_sum_nat, assume h, _⟩,
rw [← supr_eq_of_tendsto _ h, ← ennreal.tsum_eq_supr_nat],
{ exact ennreal.summable.has_sum },
{ exact assume s t hst, finset.sum_le_sum_of_subset (finset.range_subset.2 hst) }
end
lemma to_nnreal_apply_of_tsum_ne_top {α : Type*} {f : α → ennreal} (hf : (∑' i, f i) ≠ ∞) (x : α) :
(((ennreal.to_nnreal ∘ f) x : nnreal) : ennreal) = f x :=
coe_to_nnreal $ ennreal.ne_top_of_tsum_ne_top hf _
lemma summable_to_nnreal_of_tsum_ne_top {α : Type*} {f : α → ennreal} (hf : (∑' i, f i) ≠ ∞) :
summable (ennreal.to_nnreal ∘ f) :=
by simpa only [←tsum_coe_ne_top_iff_summable, to_nnreal_apply_of_tsum_ne_top hf] using hf
end tsum
end ennreal
namespace nnreal
lemma exists_le_has_sum_of_le {f g : β → nnreal} {r : nnreal}
(hgf : ∀b, g b ≤ f b) (hfr : has_sum f r) : ∃p≤r, has_sum g p :=
have (∑'b, (g b : ennreal)) ≤ r,
begin
refine has_sum_le (assume b, _) ennreal.summable.has_sum (ennreal.has_sum_coe.2 hfr),
exact ennreal.coe_le_coe.2 (hgf _)
end,
let ⟨p, eq, hpr⟩ := ennreal.le_coe_iff.1 this in
⟨p, hpr, ennreal.has_sum_coe.1 $ eq ▸ ennreal.summable.has_sum⟩
lemma summable_of_le {f g : β → nnreal} (hgf : ∀b, g b ≤ f b) : summable f → summable g
| ⟨r, hfr⟩ := let ⟨p, _, hp⟩ := exists_le_has_sum_of_le hgf hfr in hp.summable
lemma has_sum_iff_tendsto_nat {f : ℕ → nnreal} (r : nnreal) :
has_sum f r ↔ tendsto (λn:ℕ, ∑ i in finset.range n, f i) at_top (𝓝 r) :=
begin
rw [← ennreal.has_sum_coe, ennreal.has_sum_iff_tendsto_nat],
simp only [ennreal.coe_finset_sum.symm],
exact ennreal.tendsto_coe
end
lemma tsum_comp_le_tsum_of_inj {β : Type*} {f : α → nnreal} (hf : summable f)
{i : β → α} (hi : function.injective i) : tsum (f ∘ i) ≤ tsum f :=
tsum_le_tsum_of_inj i hi (λ c hc, zero_le _) (λ b, le_refl _) (summable_comp_injective hf hi) hf
open finset
/-- If `f : ℕ → ℝ≥0` and `∑' f` exists, then `∑' k, f (k + i)` tends to zero. -/
lemma tendsto_sum_nat_add (f : ℕ → nnreal) (hf : summable f) :
tendsto (λ i, ∑' k, f (k + i)) at_top (𝓝 0) :=
begin
have h₀ : (λ i, (∑' i, f i) - ∑ j in range i, f j) = λ i, ∑' (k : ℕ), f (k + i),
{ ext1 i,
rw [sub_eq_iff_eq_add, sum_add_tsum_nat_add i hf, add_comm],
exact sum_le_tsum _ (λ _ _, zero_le _) hf },
have h₁ : tendsto (λ i : ℕ, ∑' i, f i) at_top (𝓝 (∑' i, f i)) := tendsto_const_nhds,
simpa only [h₀, sub_self] using tendsto.sub h₁ hf.has_sum.tendsto_sum_nat
end
end nnreal
namespace ennreal
lemma tendsto_sum_nat_add (f : ℕ → ennreal) (hf : (∑' i, f i) ≠ ∞) :
tendsto (λ i, ∑' k, f (k + i)) at_top (𝓝 0) :=
begin
have : ∀ i, (∑' k, (((ennreal.to_nnreal ∘ f) (k + i) : nnreal) : ennreal)) =
(∑' k, (ennreal.to_nnreal ∘ f) (k + i) : nnreal) :=
λ i, (ennreal.coe_tsum (nnreal.summable_nat_add _ (summable_to_nnreal_of_tsum_ne_top hf) _)).symm,
simp only [λ x, (to_nnreal_apply_of_tsum_ne_top hf x).symm, ←ennreal.coe_zero,
this, ennreal.tendsto_coe] { single_pass := tt },
exact nnreal.tendsto_sum_nat_add _ (summable_to_nnreal_of_tsum_ne_top hf)
end
end ennreal
lemma tsum_comp_le_tsum_of_inj {β : Type*} {f : α → ℝ} (hf : summable f) (hn : ∀ a, 0 ≤ f a)
{i : β → α} (hi : function.injective i) : tsum (f ∘ i) ≤ tsum f :=
begin
let g : α → nnreal := λ a, ⟨f a, hn a⟩,
have hg : summable g, by rwa ← nnreal.summable_coe,
convert nnreal.coe_le_coe.2 (nnreal.tsum_comp_le_tsum_of_inj hg hi);
{ rw nnreal.coe_tsum, congr }
end
lemma summable_of_nonneg_of_le {f g : β → ℝ}
(hg : ∀b, 0 ≤ g b) (hgf : ∀b, g b ≤ f b) (hf : summable f) : summable g :=
let f' (b : β) : nnreal := ⟨f b, le_trans (hg b) (hgf b)⟩ in
let g' (b : β) : nnreal := ⟨g b, hg b⟩ in
have summable f', from nnreal.summable_coe.1 hf,
have summable g', from
nnreal.summable_of_le (assume b, (@nnreal.coe_le_coe (g' b) (f' b)).2 $ hgf b) this,
show summable (λb, g' b : β → ℝ), from nnreal.summable_coe.2 this
lemma has_sum_iff_tendsto_nat_of_nonneg {f : ℕ → ℝ} (hf : ∀i, 0 ≤ f i) (r : ℝ) :
has_sum f r ↔ tendsto (λn:ℕ, ∑ i in finset.range n, f i) at_top (𝓝 r) :=
⟨has_sum.tendsto_sum_nat,
assume hfr,
have 0 ≤ r := ge_of_tendsto hfr $ univ_mem_sets' $ assume i,
show 0 ≤ ∑ j in finset.range i, f j, from finset.sum_nonneg $ assume i _, hf i,
let f' (n : ℕ) : nnreal := ⟨f n, hf n⟩, r' : nnreal := ⟨r, this⟩ in
have f_eq : f = (λi:ℕ, (f' i : ℝ)) := rfl,
have r_eq : r = r' := rfl,
begin
rw [f_eq, r_eq, nnreal.has_sum_coe, nnreal.has_sum_iff_tendsto_nat, ← nnreal.tendsto_coe],
simp only [nnreal.coe_sum],
exact hfr
end⟩
lemma infi_real_pos_eq_infi_nnreal_pos {α : Type*} [complete_lattice α] {f : ℝ → α} :
(⨅(n:ℝ) (h : 0 < n), f n) = (⨅(n:nnreal) (h : 0 < n), f n) :=
le_antisymm
(le_infi $ assume n, le_infi $ assume hn, infi_le_of_le n $ infi_le _ (nnreal.coe_pos.2 hn))
(le_infi $ assume r, le_infi $ assume hr, infi_le_of_le ⟨r, le_of_lt hr⟩ $ infi_le _ hr)
section
variables [emetric_space β]
open ennreal filter emetric
/-- In an emetric ball, the distance between points is everywhere finite -/
lemma edist_ne_top_of_mem_ball {a : β} {r : ennreal} (x y : ball a r) : edist x.1 y.1 ≠ ⊤ :=
lt_top_iff_ne_top.1 $
calc edist x y ≤ edist a x + edist a y : edist_triangle_left x.1 y.1 a
... < r + r : by rw [edist_comm a x, edist_comm a y]; exact add_lt_add x.2 y.2
... ≤ ⊤ : le_top
/-- Each ball in an extended metric space gives us a metric space, as the edist
is everywhere finite. -/
def metric_space_emetric_ball (a : β) (r : ennreal) : metric_space (ball a r) :=
emetric_space.to_metric_space edist_ne_top_of_mem_ball
local attribute [instance] metric_space_emetric_ball
lemma nhds_eq_nhds_emetric_ball (a x : β) (r : ennreal) (h : x ∈ ball a r) :
𝓝 x = map (coe : ball a r → β) (𝓝 ⟨x, h⟩) :=
(map_nhds_subtype_coe_eq _ $ mem_nhds_sets emetric.is_open_ball h).symm
end
section
variable [emetric_space α]
open emetric
lemma tendsto_iff_edist_tendsto_0 {l : filter β} {f : β → α} {y : α} :
tendsto f l (𝓝 y) ↔ tendsto (λ x, edist (f x) y) l (𝓝 0) :=
by simp only [emetric.nhds_basis_eball.tendsto_right_iff, emetric.mem_ball,
@tendsto_order ennreal β _ _, forall_prop_of_false ennreal.not_lt_zero, forall_const, true_and]
/-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. -/
lemma emetric.cauchy_seq_iff_le_tendsto_0 [nonempty β] [semilattice_sup β] {s : β → α} :
cauchy_seq s ↔ (∃ (b: β → ennreal), (∀ n m N : β, N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N)
∧ (tendsto b at_top (𝓝 0))) :=
⟨begin
assume hs,
rw emetric.cauchy_seq_iff at hs,
/- `s` is Cauchy sequence. The sequence `b` will be constructed by taking
the supremum of the distances between `s n` and `s m` for `n m ≥ N`-/
let b := λN, Sup ((λ(p : β × β), edist (s p.1) (s p.2))''{p | p.1 ≥ N ∧ p.2 ≥ N}),
--Prove that it bounds the distances of points in the Cauchy sequence
have C : ∀ n m N, N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N,
{ refine λm n N hm hn, le_Sup _,
use (prod.mk m n),
simp only [and_true, eq_self_iff_true, set.mem_set_of_eq],
exact ⟨hm, hn⟩ },
--Prove that it tends to `0`, by using the Cauchy property of `s`
have D : tendsto b at_top (𝓝 0),
{ refine tendsto_order.2 ⟨λa ha, absurd ha (ennreal.not_lt_zero), λε εpos, _⟩,
rcases dense εpos with ⟨δ, δpos, δlt⟩,
rcases hs δ δpos with ⟨N, hN⟩,
refine filter.mem_at_top_sets.2 ⟨N, λn hn, _⟩,
have : b n ≤ δ := Sup_le begin
simp only [and_imp, set.mem_image, set.mem_set_of_eq, exists_imp_distrib, prod.exists],
intros d p q hp hq hd,
rw ← hd,
exact le_of_lt (hN p q (le_trans hn hp) (le_trans hn hq))
end,
simpa using lt_of_le_of_lt this δlt },
-- Conclude
exact ⟨b, ⟨C, D⟩⟩
end,
begin
rintros ⟨b, ⟨b_bound, b_lim⟩⟩,
/-b : ℕ → ℝ, b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N,
b_lim : tendsto b at_top (𝓝 0)-/
refine emetric.cauchy_seq_iff.2 (λε εpos, _),
have : ∀ᶠ n in at_top, b n < ε := (tendsto_order.1 b_lim ).2 _ εpos,
rcases filter.mem_at_top_sets.1 this with ⟨N, hN⟩,
exact ⟨N, λm n hm hn, calc
edist (s m) (s n) ≤ b N : b_bound m n N hm hn
... < ε : (hN _ (le_refl N)) ⟩
end⟩
lemma continuous_of_le_add_edist {f : α → ennreal} (C : ennreal)
(hC : C ≠ ⊤) (h : ∀x y, f x ≤ f y + C * edist x y) : continuous f :=
begin
refine continuous_iff_continuous_at.2 (λx, tendsto_order.2 ⟨_, _⟩),
show ∀e, e < f x → ∀ᶠ y in 𝓝 x, e < f y,
{ assume e he,
let ε := min (f x - e) 1,
have : ε < ⊤ := lt_of_le_of_lt (min_le_right _ _) (by simp [lt_top_iff_ne_top]),
have : 0 < ε := by simp [ε, hC, he, ennreal.zero_lt_one],
have : 0 < C⁻¹ * (ε/2) := bot_lt_iff_ne_bot.2 (by simp [hC, (ne_of_lt this).symm, mul_eq_zero]),
have I : C * (C⁻¹ * (ε/2)) < ε,
{ by_cases C_zero : C = 0,
{ simp [C_zero, ‹0 < ε›] },
{ calc C * (C⁻¹ * (ε/2)) = (C * C⁻¹) * (ε/2) : by simp [mul_assoc]
... = ε/2 : by simp [ennreal.mul_inv_cancel C_zero hC]
... < ε : ennreal.half_lt_self (bot_lt_iff_ne_bot.1 ‹0 < ε›) (lt_top_iff_ne_top.1 ‹ε < ⊤›) }},
have : ball x (C⁻¹ * (ε/2)) ⊆ {y : α | e < f y},
{ rintros y hy,
by_cases htop : f y = ⊤,
{ simp [htop, lt_top_iff_ne_top, ne_top_of_lt he] },
{ simp at hy,
have : e + ε < f y + ε := calc
e + ε ≤ e + (f x - e) : add_le_add_left (min_le_left _ _) _
... = f x : by simp [le_of_lt he]
... ≤ f y + C * edist x y : h x y
... = f y + C * edist y x : by simp [edist_comm]
... ≤ f y + C * (C⁻¹ * (ε/2)) :
add_le_add_left (canonically_ordered_semiring.mul_le_mul (le_refl _) (le_of_lt hy)) _
... < f y + ε : (ennreal.add_lt_add_iff_left (lt_top_iff_ne_top.2 htop)).2 I,
show e < f y, from
(ennreal.add_lt_add_iff_right ‹ε < ⊤›).1 this }},
apply filter.mem_sets_of_superset (ball_mem_nhds _ (‹0 < C⁻¹ * (ε/2)›)) this },
show ∀e, f x < e → ∀ᶠ y in 𝓝 x, f y < e,
{ assume e he,
let ε := min (e - f x) 1,
have : ε < ⊤ := lt_of_le_of_lt (min_le_right _ _) (by simp [lt_top_iff_ne_top]),
have : 0 < ε := by simp [ε, he, ennreal.zero_lt_one],
have : 0 < C⁻¹ * (ε/2) := bot_lt_iff_ne_bot.2 (by simp [hC, (ne_of_lt this).symm, mul_eq_zero]),
have I : C * (C⁻¹ * (ε/2)) < ε,
{ by_cases C_zero : C = 0,
simp [C_zero, ‹0 < ε›],
calc C * (C⁻¹ * (ε/2)) = (C * C⁻¹) * (ε/2) : by simp [mul_assoc]
... = ε/2 : by simp [ennreal.mul_inv_cancel C_zero hC]
... < ε : ennreal.half_lt_self (bot_lt_iff_ne_bot.1 ‹0 < ε›) (lt_top_iff_ne_top.1 ‹ε < ⊤›) },
have : ball x (C⁻¹ * (ε/2)) ⊆ {y : α | f y < e},
{ rintros y hy,
have htop : f x ≠ ⊤ := ne_top_of_lt he,
show f y < e, from calc
f y ≤ f x + C * edist y x : h y x
... ≤ f x + C * (C⁻¹ * (ε/2)) :
add_le_add_left (canonically_ordered_semiring.mul_le_mul (le_refl _) (le_of_lt hy)) _
... < f x + ε : (ennreal.add_lt_add_iff_left (lt_top_iff_ne_top.2 htop)).2 I
... ≤ f x + (e - f x) : add_le_add_left (min_le_left _ _) _
... = e : by simp [le_of_lt he] },
apply filter.mem_sets_of_superset (ball_mem_nhds _ (‹0 < C⁻¹ * (ε/2)›)) this },
end
theorem continuous_edist : continuous (λp:α×α, edist p.1 p.2) :=
begin
apply continuous_of_le_add_edist 2 (by norm_num),
rintros ⟨x, y⟩ ⟨x', y'⟩,
calc edist x y ≤ edist x x' + edist x' y' + edist y' y : edist_triangle4 _ _ _ _
... = edist x' y' + (edist x x' + edist y y') : by simp [edist_comm]; cc
... ≤ edist x' y' + (edist (x, y) (x', y') + edist (x, y) (x', y')) :
add_le_add_left (add_le_add (le_max_left _ _) (le_max_right _ _)) _
... = edist x' y' + 2 * edist (x, y) (x', y') : by rw [← mul_two, mul_comm]
end
theorem continuous.edist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, edist (f b) (g b)) :=
continuous_edist.comp (hf.prod_mk hg)
theorem filter.tendsto.edist {f g : β → α} {x : filter β} {a b : α}
(hf : tendsto f x (𝓝 a)) (hg : tendsto g x (𝓝 b)) :
tendsto (λx, edist (f x) (g x)) x (𝓝 (edist a b)) :=
(continuous_edist.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
lemma cauchy_seq_of_edist_le_of_tsum_ne_top {f : ℕ → α} (d : ℕ → ennreal)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : tsum d ≠ ∞) :
cauchy_seq f :=
begin
lift d to (ℕ → nnreal) using (λ i, ennreal.ne_top_of_tsum_ne_top hd i),
rw ennreal.tsum_coe_ne_top_iff_summable at hd,
exact cauchy_seq_of_edist_le_of_summable d hf hd
end
lemma emetric.is_closed_ball {a : α} {r : ennreal} : is_closed (closed_ball a r) :=
is_closed_le (continuous_id.edist continuous_const) continuous_const
/-- If `edist (f n) (f (n+1))` is bounded above by a function `d : ℕ → ennreal`,
then the distance from `f n` to the limit is bounded by `∑'_{k=n}^∞ d k`. -/
lemma edist_le_tsum_of_edist_le_of_tendsto {f : ℕ → α} (d : ℕ → ennreal)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n)
{a : α} (ha : tendsto f at_top (𝓝 a)) (n : ℕ) :
edist (f n) a ≤ ∑' m, d (n + m) :=
begin
refine le_of_tendsto (tendsto_const_nhds.edist ha)
(mem_at_top_sets.2 ⟨n, λ m hnm, _⟩),
refine le_trans (edist_le_Ico_sum_of_edist_le hnm (λ k _ _, hf k)) _,
rw [finset.sum_Ico_eq_sum_range],
exact sum_le_tsum _ (λ _ _, zero_le _) ennreal.summable
end
/-- If `edist (f n) (f (n+1))` is bounded above by a function `d : ℕ → ennreal`,
then the distance from `f 0` to the limit is bounded by `∑'_{k=0}^∞ d k`. -/
lemma edist_le_tsum_of_edist_le_of_tendsto₀ {f : ℕ → α} (d : ℕ → ennreal)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n)
{a : α} (ha : tendsto f at_top (𝓝 a)) :
edist (f 0) a ≤ ∑' m, d m :=
by simpa using edist_le_tsum_of_edist_le_of_tendsto d hf ha 0
end --section
|
695bc3bd99d2b137580e5df84db7b658e609a9c4 | e00ea76a720126cf9f6d732ad6216b5b824d20a7 | /src/ring_theory/adjoin_root.lean | 3ec46fcb13aa203494dbb8875830b77d30a4343a | [
"Apache-2.0"
] | permissive | vaibhavkarve/mathlib | a574aaf68c0a431a47fa82ce0637f0f769826bfe | 17f8340912468f49bdc30acdb9a9fa02eeb0473a | refs/heads/master | 1,621,263,802,637 | 1,585,399,588,000 | 1,585,399,588,000 | 250,833,447 | 0 | 0 | Apache-2.0 | 1,585,410,341,000 | 1,585,410,341,000 | null | UTF-8 | Lean | false | false | 4,176 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
Adjoining roots of polynomials
-/
import data.polynomial ring_theory.principal_ideal_domain
/-!
# Adjoining roots of polynomials
This file defines the commutative ring `adjoin_root f`, the ring R[X]/(f) obtained from a
commutative ring `R` and a polynomial `f : R[X]`. If furthermore `R` is a field and `f` is
irreducible, the field structure on `adjoin_root f` is constructed.
## Main definitions and results
The main definitions are in the `adjoin_root` namespace.
* `mk f : polynomial R →+* adjoin_root f`, the natural ring homomorphism.
* `of f : R →+* adjoin_root f`, the natural ring homomorphism.
* `root f : adjoin_root f`, the image of X in R[X]/(f).
* `lift (i : R →+* S) (x : S) (h : f.eval₂ i x = 0) : (adjoin_root f) →+* S`, the ring
homomorphism from R[X]/(f) to S extending `i : R →+* S` and sending `X` to `x`.
-/
noncomputable theory
universes u v w
variables {R : Type u} {S : Type v} {K : Type w}
open polynomial ideal
/-- Adjoin a root of a polynomial `f` to a commutative ring `R`. We define the new ring
as the quotient of `R` by the principal ideal of `f`. -/
def adjoin_root [comm_ring R] (f : polynomial R) : Type u :=
ideal.quotient (span {f} : ideal (polynomial R))
namespace adjoin_root
section comm_ring
variables [comm_ring R] (f : polynomial R)
instance : comm_ring (adjoin_root f) := ideal.quotient.comm_ring _
instance : inhabited (adjoin_root f) := ⟨0⟩
instance : decidable_eq (adjoin_root f) := classical.dec_eq _
/-- Ring homomorphism from `R[x]` to `adjoin_root f` sending `X` to the `root`. -/
def mk : polynomial R →+* adjoin_root f := ideal.quotient.mk_hom _
/-- Embedding of the original ring `R` into `adjoin_root f`. -/
def of : R →+* adjoin_root f := (mk f).comp (ring_hom.of C)
/-- The adjoined root. -/
def root : adjoin_root f := mk f X
variables {f}
instance adjoin_root.has_coe_t : has_coe_t R (adjoin_root f) := ⟨of f⟩
@[simp] lemma mk_self : mk f f = 0 :=
quotient.sound' (mem_span_singleton.2 $ by simp)
instance : is_ring_hom (coe : R → adjoin_root f) := (of f).is_ring_hom
@[simp] lemma mk_C (x : R) : mk f (C x) = x := rfl
@[simp] lemma eval₂_root (f : polynomial R) : f.eval₂ coe (root f) = 0 :=
quotient.induction_on' (root f)
(λ (g : polynomial R) (hg : mk f g = mk f X),
show finsupp.sum f (λ (e : ℕ) (a : R), mk f (C a) * mk f g ^ e) = 0,
by simp only [hg, ((mk f).map_pow _ _).symm, ((mk f).map_mul _ _).symm];
rw [finsupp.sum, ← (mk f).map_sum,
show finset.sum _ _ = _, from sum_C_mul_X_eq _, mk_self])
(show (root f) = mk f X, from rfl)
lemma is_root_root (f : polynomial R) : is_root (f.map coe) (root f) :=
by rw [is_root, eval_map, eval₂_root]
variables [comm_ring S]
/-- Lift a ring homomorphism `i : R →+* S` to `adjoin_root f →+* S`. -/
def lift (i : R →+* S) (x : S) (h : f.eval₂ i x = 0) : (adjoin_root f) →+* S :=
begin
apply ideal.quotient.lift _ (ring_hom.of (eval₂ i x)),
intros g H,
rcases mem_span_singleton.1 H with ⟨y, hy⟩,
rw [hy, ring_hom.map_mul, ring_hom.coe_of, h, zero_mul]
end
variables {i : R →+* S} {a : S} {h : f.eval₂ i a = 0}
@[simp] lemma lift_mk {g : polynomial R} : lift i a h (mk f g) = g.eval₂ i a :=
ideal.quotient.lift_mk _ _ _
@[simp] lemma lift_root : lift i a h (root f) = a := by simp [root, h]
@[simp] lemma lift_of {x : R} : lift i a h x = i x :=
by rw [← mk_C x, lift_mk, eval₂_C]
end comm_ring
variables [field K] {f : polynomial K} [irreducible f]
instance is_maximal_span : is_maximal (span {f} : ideal (polynomial K)) :=
principal_ideal_domain.is_maximal_of_irreducible ‹irreducible f›
noncomputable instance field : field (adjoin_root f) :=
ideal.quotient.field (span {f} : ideal (polynomial K))
lemma coe_injective : function.injective (coe : K → adjoin_root f) :=
(of f).injective
variable (f)
lemma mul_div_root_cancel :
(X - C (root f)) * (f.map coe / (X - C (root f))) = f.map coe :=
mul_div_eq_iff_is_root.2 $ is_root_root _
end adjoin_root
|
21d03792020bb6fabe2f1051b8a9beb3e8bd3b10 | d6124c8dbe5661dcc5b8c9da0a56fbf1f0480ad6 | /test/run/ir/types.lean | 2bac5f777b1e37afe4181f7fcae4d6bc37a3c16c | [
"Apache-2.0"
] | permissive | xubaiw/lean4-papyrus | c3fbbf8ba162eb5f210155ae4e20feb2d32c8182 | 02e82973a5badda26fc0f9fd15b3d37e2eb309e0 | refs/heads/master | 1,691,425,756,824 | 1,632,122,825,000 | 1,632,123,075,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,048 | lean | import Papyrus
open Papyrus
def checkTypeRoundtrips (type : «Type») : IO PUnit :=
LlvmM.run do
let ref ← type.getRef
let actual ← ref.purify
unless type == actual do
ref.dump
IO.println s!"expected {repr type}, got {repr actual}"
throw <| IO.userError "did not round trip"
macro tk:"#check_type" x:term : command => do
Script.mkEvalAt tk <| ← ``(checkTypeRoundtrips $x)
#check_type voidType
#check_type labelType
#check_type metadataType
#check_type tokenType
#check_type x86MMXType
#check_type x86AMXType
#check_type halfType
#check_type bfloatType
#check_type floatType
#check_type doubleType
#check_type x86FP80Type
#check_type fp128Type
#check_type ppcFP128Type
#check_type integerType 100
#check_type functionType voidType #[int8Type.pointerType] true
#check_type pointerType fp128Type
#check_type structType "foo" #[integerType 24] true
#check_type arrayType halfType 6
#check_type vectorType int32Type 4 true
#check_type fixedVectorType doubleType 8
#check_type scalableVectorType int1Type 16
|
cb526e541f213a1e51c3566ea55a2cd55d49f085 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /test/set.lean | e264d107075472ebe015f5028d4782fe5a7c5c10 | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 1,110 | lean | import data.matrix.notation
example (x : ℕ) (h : x = 3) : x + x + x = 9 :=
begin
set y := x with ←h_xy,
guard_hyp y : ℕ := x,
guard_hyp h_xy : x = y,
guard_hyp h : y = 3,
guard_target y + y + y = 9,
set! z : ℕ := y,
guard_target y + y + y = 9,
simp [h]
end
example : true :=
let X1 := (![![1, 0], ![0, 0]] : matrix (fin 2) (fin 2) ℕ), -- X1 : fin 1.succ → fin 2 → ℕ
X2 : matrix (fin 2) (fin 2) ℕ := ![![1, 0], ![0, 0]] in -- X2 : matrix (fin 2) (fin 2) ℕ
begin
set Y1 := (![![1, 0], ![0, 0]] : matrix (fin 2) (fin 2) ℕ),
set Y2 : matrix (fin 2) (fin 2) ℕ := ![![1, 0], ![0, 0]],
let Z1 := (![![1, 0], ![0, 0]] : matrix (fin 2) (fin 2) ℕ),
let Z2 : matrix (fin 2) (fin 2) ℕ := ![![1, 0], ![0, 0]],
guard_hyp Y2 : matrix (fin 2) (fin 2) ℕ := (![![1, 0], ![0, 0]] : matrix (fin 2) (fin 2) ℕ),
trivial
end
def T {α : Type} := ℕ
def v : @T ℕ := nat.zero
def S := @T ℕ
def u : S := nat.zero
def p : true :=
begin
set a : T := v,
set b : T := u, -- the type `T` can't be fully elaborated without the body but this is fine
trivial
end
|
d412687f147a240a2d26a3a9fe6e064f1df7d1dc | 46125763b4dbf50619e8846a1371029346f4c3db | /src/category/monad/writer.lean | 51e580c7c57a7c45a49bc288c73bfd68a48edc4b | [
"Apache-2.0"
] | permissive | thjread/mathlib | a9d97612cedc2c3101060737233df15abcdb9eb1 | 7cffe2520a5518bba19227a107078d83fa725ddc | refs/heads/master | 1,615,637,696,376 | 1,583,953,063,000 | 1,583,953,063,000 | 246,680,271 | 0 | 0 | Apache-2.0 | 1,583,960,875,000 | 1,583,960,875,000 | null | UTF-8 | Lean | false | false | 7,371 | lean |
/-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
The writer monad transformer for passing immutable state.
-/
import tactic.basic category.monad.basic
universes u v w
structure writer_t (ω : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
(run : m (α × ω))
@[reducible] def writer (ω : Type u) := writer_t ω id
attribute [pp_using_anonymous_constructor] writer_t
namespace writer_t
section
variable {ω : Type u}
variable {m : Type u → Type v}
variable [monad m]
variables {α β : Type u}
open function
@[ext]
protected lemma ext (x x' : writer_t ω m α)
(h : x.run = x'.run) :
x = x' := by cases x; cases x'; congr; apply h
@[inline] protected def tell (w : ω) : writer_t ω m punit :=
⟨pure (punit.star, w)⟩
@[inline] protected def listen : writer_t ω m α → writer_t ω m (α × ω)
| ⟨ cmd ⟩ := ⟨ (λ x : α × ω, ((x.1,x.2),x.2)) <$> cmd ⟩
@[inline] protected def pass : writer_t ω m (α × (ω → ω)) → writer_t ω m α
| ⟨ cmd ⟩ := ⟨ uncurry (uncurry $ λ x (f : ω → ω) w, (x,f w)) <$> cmd ⟩
@[inline] protected def pure [has_one ω] (a : α) : writer_t ω m α :=
⟨ pure (a,1) ⟩
@[inline] protected def bind [has_mul ω] (x : writer_t ω m α) (f : α → writer_t ω m β) : writer_t ω m β :=
⟨ do x ← x.run,
x' ← (f x.1).run,
pure (x'.1,x.2 * x'.2) ⟩
instance [has_one ω] [has_mul ω] : monad (writer_t ω m) :=
{ pure := λ α, writer_t.pure, bind := λ α β, writer_t.bind }
instance [monoid ω] [is_lawful_monad m] : is_lawful_monad (writer_t ω m) :=
{ id_map := by { intros, cases x, simp [(<$>),writer_t.bind,writer_t.pure] },
pure_bind := by { intros, simp [has_pure.pure,writer_t.pure,(>>=),writer_t.bind], ext; refl },
bind_assoc := by { intros, simp [(>>=),writer_t.bind,mul_assoc] with functor_norm } }
@[inline] protected def lift [has_one ω] (a : m α) : writer_t ω m α :=
⟨ flip prod.mk 1 <$> a ⟩
instance (m) [monad m] [has_one ω] : has_monad_lift m (writer_t ω m) :=
⟨ λ α, writer_t.lift ⟩
@[inline] protected def monad_map {m m'} [monad m] [monad m'] {α} (f : Π {α}, m α → m' α) : writer_t ω m α → writer_t ω m' α :=
λ x, ⟨ f x.run ⟩
instance (m m') [monad m] [monad m'] : monad_functor m m' (writer_t ω m) (writer_t ω m') :=
⟨@writer_t.monad_map ω m m' _ _⟩
@[inline] protected def adapt {ω' : Type u} {α : Type u} (f : ω → ω') : writer_t ω m α → writer_t ω' m α :=
λ x, ⟨prod.map id f <$> x.run⟩
instance (ε) [has_one ω] [monad m] [monad_except ε m] : monad_except ε (writer_t ω m) :=
{ throw := λ α, writer_t.lift ∘ throw,
catch := λ α x c, ⟨catch x.run (λ e, (c e).run)⟩ }
end
end writer_t
/-- An implementation of [MonadReader](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Reader-Class.html#t:MonadReader).
It does not contain `local` because this function cannot be lifted using `monad_lift`.
Instead, the `monad_reader_adapter` class provides the more general `adapt_reader` function.
Note: This class can be seen as a simplification of the more "principled" definition
```
class monad_reader (ρ : out_param (Type u)) (n : Type u → Type u) :=
(lift {} {α : Type u} : (∀ {m : Type u → Type u} [monad m], reader_t ρ m α) → n α)
```
-/
class monad_writer (ω : out_param (Type u)) (m : Type u → Type v) :=
(tell {} (w : ω) : m punit)
(listen {α} : m α → m (α × ω))
(pass {α : Type u} : m (α × (ω → ω)) → m α)
export monad_writer
instance {ω : Type u} {m : Type u → Type v} [monad m] : monad_writer ω (writer_t ω m) :=
{ tell := writer_t.tell,
listen := λ α, writer_t.listen,
pass := λ α, writer_t.pass }
instance {ω ρ : Type u} {m : Type u → Type v} [monad m] [monad_writer ω m] : monad_writer ω (reader_t ρ m) :=
{ tell := λ x, monad_lift (tell x : m punit),
listen := λ α ⟨ cmd ⟩, ⟨ λ r, listen (cmd r) ⟩,
pass := λ α ⟨ cmd ⟩, ⟨ λ r, pass (cmd r) ⟩ }
def swap_right {α β γ} : (α × β) × γ → (α × γ) × β
| ⟨⟨x,y⟩,z⟩ := ((x,z),y)
instance {ω σ : Type u} {m : Type u → Type v} [monad m] [monad_writer ω m] : monad_writer ω (state_t σ m) :=
{ tell := λ x, monad_lift (tell x : m punit),
listen := λ α ⟨ cmd ⟩, ⟨ λ r, swap_right <$> listen (cmd r) ⟩,
pass := λ α ⟨ cmd ⟩, ⟨ λ r, pass (swap_right <$> cmd r) ⟩ }
open function
def except_t.pass_aux {ε α ω} : except ε (α × (ω → ω)) → except ε α × (ω → ω)
| (except.error a) := (except.error a,id)
| (except.ok (x,y)) := (except.ok x,y)
instance {ω ε : Type u} {m : Type u → Type v} [monad m] [monad_writer ω m] : monad_writer ω (except_t ε m) :=
{ tell := λ x, monad_lift (tell x : m punit),
listen := λ α ⟨ cmd ⟩, ⟨ uncurry (λ x y, flip prod.mk y <$> x) <$> listen cmd ⟩,
pass := λ α ⟨ cmd ⟩, ⟨ pass (except_t.pass_aux <$> cmd) ⟩ }
def option_t.pass_aux {α ω} : option (α × (ω → ω)) → option α × (ω → ω)
| none := (none ,id)
| (some (x,y)) := (some x,y)
instance {ω : Type u} {m : Type u → Type v} [monad m] [monad_writer ω m] : monad_writer ω (option_t m) :=
{ tell := λ x, monad_lift (tell x : m punit),
listen := λ α ⟨ cmd ⟩, ⟨ uncurry (λ x y, flip prod.mk y <$> x) <$> listen cmd ⟩,
pass := λ α ⟨ cmd ⟩, ⟨ pass (option_t.pass_aux <$> cmd) ⟩ }
/-- Adapt a monad stack, changing the type of its top-most environment.
This class is comparable to [Control.Lens.Magnify](https://hackage.haskell.org/package/lens-4.15.4/docs/Control-Lens-Zoom.html#t:Magnify), but does not use lenses (why would it), and is derived automatically for any transformer implementing `monad_functor`.
Note: This class can be seen as a simplification of the more "principled" definition
```
class monad_reader_functor (ρ ρ' : out_param (Type u)) (n n' : Type u → Type u) :=
(map {} {α : Type u} : (∀ {m : Type u → Type u} [monad m], reader_t ρ m α → reader_t ρ' m α) → n α → n' α)
```
-/
class monad_writer_adapter (ω ω' : out_param (Type u)) (m m' : Type u → Type v) :=
(adapt_writer {} {α : Type u} : (ω → ω') → m α → m' α)
export monad_writer_adapter (adapt_writer)
section
variables {ω ω' : Type u} {m m' : Type u → Type v}
/-- Transitivity.
This instance generates the type-class problem with a metavariable argument (which is why this is marked as
`[nolint dangerous_instance]`).
Currently that is not a problem, as there are almost no instances of `monad_functor` or `monad_writer_adapter`.
see Note [lower instance priority] -/
@[nolint dangerous_instance, priority 100]
instance monad_writer_adapter_trans {n n' : Type u → Type v} [monad_functor m m' n n']
[monad_writer_adapter ω ω' m m'] : monad_writer_adapter ω ω' n n' :=
⟨λ α f, monad_map (λ α, (adapt_writer f : m α → m' α))⟩
instance [monad m] : monad_writer_adapter ω ω' (writer_t ω m) (writer_t ω' m) :=
⟨λ α, writer_t.adapt⟩
end
instance (ω : Type u) (m out) [monad_run out m] : monad_run (λ α, out (α × ω)) (writer_t ω m) :=
⟨λ α x, run $ x.run ⟩
|
0e2e55d6991920e5d2043153b9381ebd396339d0 | d9d511f37a523cd7659d6f573f990e2a0af93c6f | /src/analysis/normed_space/bounded_linear_maps.lean | 819f0c571e5a0fab0758ba44fe4fc1d6d9ef7837 | [
"Apache-2.0"
] | permissive | hikari0108/mathlib | b7ea2b7350497ab1a0b87a09d093ecc025a50dfa | a9e7d333b0cfd45f13a20f7b96b7d52e19fa2901 | refs/heads/master | 1,690,483,608,260 | 1,631,541,580,000 | 1,631,541,580,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 22,060 | lean | /-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import analysis.normed_space.multilinear
/-!
# Continuous linear maps
This file defines a class stating that a map between normed vector spaces is (bi)linear and
continuous.
Instead of asking for continuity, the definition takes the equivalent condition (because the space
is normed) that `∥f x∥` is bounded by a multiple of `∥x∥`. Hence the "bounded" in the name refers to
`∥f x∥/∥x∥` rather than `∥f x∥` itself.
## Main declarations
* `is_bounded_linear_map`: Class stating that a map `f : E → F` is linear and has `∥f x∥` bounded
by a multiple of `∥x∥`.
* `is_bounded_bilinear_map`: Class stating that a map `f : E × F → G` is bilinear and continuous,
but through the simpler to provide statement that `∥f (x, y)∥` is bounded by a multiple of
`∥x∥ * ∥y∥`
* `is_bounded_bilinear_map.linear_deriv`: Derivative of a continuous bilinear map as a linear map.
* `is_bounded_bilinear_map.deriv`: Derivative of a continuous bilinear map as a continuous linear
map. The proof that it is indeed the derivative is `is_bounded_bilinear_map.has_fderiv_at` in
`analysis.calculus.fderiv`.
* `linear_map.norm_apply_of_isometry`: A linear isometry preserves the norm.
## Notes
The main use of this file is `is_bounded_bilinear_map`. The file `analysis.normed_space.multilinear`
already expounds the theory of multilinear maps, but the `2`-variables case is sufficiently simpler
to currently deserve its own treatment.
`is_bounded_linear_map` is effectively an unbundled version of `continuous_linear_map` (defined
in `topology.algebra.module`, theory over normed spaces developed in
`analysis.normed_space.operator_norm`), albeit the name disparity. A bundled
`continuous_linear_map` is to be preferred over a `is_bounded_linear_map` hypothesis. Historical
artifact, really.
-/
noncomputable theory
open_locale classical big_operators topological_space
open filter (tendsto) metric
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]
{G : Type*} [normed_group G] [normed_space 𝕜 G]
/-- A function `f` satisfies `is_bounded_linear_map 𝕜 f` if it is linear and satisfies the
inequality `∥f x∥ ≤ M * ∥x∥` for some positive constant `M`. -/
structure is_bounded_linear_map (𝕜 : Type*) [normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E]
{F : Type*} [normed_group F] [normed_space 𝕜 F] (f : E → F)
extends is_linear_map 𝕜 f : Prop :=
(bound : ∃ M, 0 < M ∧ ∀ x : E, ∥f x∥ ≤ M * ∥x∥)
lemma is_linear_map.with_bound
{f : E → F} (hf : is_linear_map 𝕜 f) (M : ℝ) (h : ∀ x : E, ∥f x∥ ≤ M * ∥x∥) :
is_bounded_linear_map 𝕜 f :=
⟨ hf, classical.by_cases
(assume : M ≤ 0, ⟨1, zero_lt_one, λ x,
(h x).trans $ mul_le_mul_of_nonneg_right (this.trans zero_le_one) (norm_nonneg x)⟩)
(assume : ¬ M ≤ 0, ⟨M, lt_of_not_ge this, h⟩)⟩
/-- A continuous linear map satisfies `is_bounded_linear_map` -/
lemma continuous_linear_map.is_bounded_linear_map (f : E →L[𝕜] F) : is_bounded_linear_map 𝕜 f :=
{ bound := f.bound,
..f.to_linear_map.is_linear }
namespace is_bounded_linear_map
/-- Construct a linear map from a function `f` satisfying `is_bounded_linear_map 𝕜 f`. -/
def to_linear_map (f : E → F) (h : is_bounded_linear_map 𝕜 f) : E →ₗ[𝕜] F :=
(is_linear_map.mk' _ h.to_is_linear_map)
/-- Construct a continuous linear map from is_bounded_linear_map -/
def to_continuous_linear_map {f : E → F} (hf : is_bounded_linear_map 𝕜 f) : E →L[𝕜] F :=
{ cont := let ⟨C, Cpos, hC⟩ := hf.bound in linear_map.continuous_of_bound _ C hC,
..to_linear_map f hf}
lemma zero : is_bounded_linear_map 𝕜 (λ (x:E), (0:F)) :=
(0 : E →ₗ[𝕜] F).is_linear.with_bound 0 $ by simp [le_refl]
lemma id : is_bounded_linear_map 𝕜 (λ (x:E), x) :=
linear_map.id.is_linear.with_bound 1 $ by simp [le_refl]
lemma fst : is_bounded_linear_map 𝕜 (λ x : E × F, x.1) :=
begin
refine (linear_map.fst 𝕜 E F).is_linear.with_bound 1 (λ x, _),
rw one_mul,
exact le_max_left _ _
end
lemma snd : is_bounded_linear_map 𝕜 (λ x : E × F, x.2) :=
begin
refine (linear_map.snd 𝕜 E F).is_linear.with_bound 1 (λ x, _),
rw one_mul,
exact le_max_right _ _
end
variables {f g : E → F}
lemma smul (c : 𝕜) (hf : is_bounded_linear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (c • f) :=
let ⟨hlf, M, hMp, hM⟩ := hf in
(c • hlf.mk' f).is_linear.with_bound (∥c∥ * M) $ λ x,
calc ∥c • f x∥ = ∥c∥ * ∥f x∥ : norm_smul c (f x)
... ≤ ∥c∥ * (M * ∥x∥) : mul_le_mul_of_nonneg_left (hM _) (norm_nonneg _)
... = (∥c∥ * M) * ∥x∥ : (mul_assoc _ _ _).symm
lemma neg (hf : is_bounded_linear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (λ e, -f e) :=
begin
rw show (λ e, -f e) = (λ e, (-1 : 𝕜) • f e), { funext, simp },
exact smul (-1) hf
end
lemma add (hf : is_bounded_linear_map 𝕜 f) (hg : is_bounded_linear_map 𝕜 g) :
is_bounded_linear_map 𝕜 (λ e, f e + g e) :=
let ⟨hlf, Mf, hMfp, hMf⟩ := hf in
let ⟨hlg, Mg, hMgp, hMg⟩ := hg in
(hlf.mk' _ + hlg.mk' _).is_linear.with_bound (Mf + Mg) $ λ x,
calc ∥f x + g x∥ ≤ Mf * ∥x∥ + Mg * ∥x∥ : norm_add_le_of_le (hMf x) (hMg x)
... ≤ (Mf + Mg) * ∥x∥ : by rw add_mul
lemma sub (hf : is_bounded_linear_map 𝕜 f) (hg : is_bounded_linear_map 𝕜 g) :
is_bounded_linear_map 𝕜 (λ e, f e - g e) :=
by simpa [sub_eq_add_neg] using add hf (neg hg)
lemma comp {g : F → G}
(hg : is_bounded_linear_map 𝕜 g) (hf : is_bounded_linear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (g ∘ f) :=
(hg.to_continuous_linear_map.comp hf.to_continuous_linear_map).is_bounded_linear_map
protected lemma tendsto (x : E) (hf : is_bounded_linear_map 𝕜 f) :
tendsto f (𝓝 x) (𝓝 (f x)) :=
let ⟨hf, M, hMp, hM⟩ := hf in
tendsto_iff_norm_tendsto_zero.2 $
squeeze_zero (λ e, norm_nonneg _)
(λ e,
calc ∥f e - f x∥ = ∥hf.mk' f (e - x)∥ : by rw (hf.mk' _).map_sub e x; refl
... ≤ M * ∥e - x∥ : hM (e - x))
(suffices tendsto (λ (e : E), M * ∥e - x∥) (𝓝 x) (𝓝 (M * 0)), by simpa,
tendsto_const_nhds.mul (tendsto_norm_sub_self _))
lemma continuous (hf : is_bounded_linear_map 𝕜 f) : continuous f :=
continuous_iff_continuous_at.2 $ λ _, hf.tendsto _
lemma lim_zero_bounded_linear_map (hf : is_bounded_linear_map 𝕜 f) :
tendsto f (𝓝 0) (𝓝 0) :=
(hf.1.mk' _).map_zero ▸ continuous_iff_continuous_at.1 hf.continuous 0
section
open asymptotics filter
theorem is_O_id {f : E → F} (h : is_bounded_linear_map 𝕜 f) (l : filter E) :
is_O f (λ x, x) l :=
let ⟨M, hMp, hM⟩ := h.bound in is_O.of_bound _ (mem_of_superset univ_mem (λ x _, hM x))
theorem is_O_comp {E : Type*} {g : F → G} (hg : is_bounded_linear_map 𝕜 g)
{f : E → F} (l : filter E) : is_O (λ x', g (f x')) f l :=
(hg.is_O_id ⊤).comp_tendsto le_top
theorem is_O_sub {f : E → F} (h : is_bounded_linear_map 𝕜 f)
(l : filter E) (x : E) : is_O (λ x', f (x' - x)) (λ x', x' - x) l :=
is_O_comp h l
end
end is_bounded_linear_map
section
variables {ι : Type*} [decidable_eq ι] [fintype ι]
/-- Taking the cartesian product of two continuous multilinear maps
is a bounded linear operation. -/
lemma is_bounded_linear_map_prod_multilinear
{E : ι → Type*} [∀ i, normed_group (E i)] [∀ i, normed_space 𝕜 (E i)] :
is_bounded_linear_map 𝕜
(λ p : (continuous_multilinear_map 𝕜 E F) × (continuous_multilinear_map 𝕜 E G), p.1.prod p.2) :=
{ map_add := λ p₁ p₂, by { ext1 m, refl },
map_smul := λ c p, by { ext1 m, refl },
bound := ⟨1, zero_lt_one, λ p, begin
rw one_mul,
apply continuous_multilinear_map.op_norm_le_bound _ (norm_nonneg _) (λ m, _),
rw [continuous_multilinear_map.prod_apply, norm_prod_le_iff],
split,
{ exact (p.1.le_op_norm m).trans
(mul_le_mul_of_nonneg_right (norm_fst_le p) (finset.prod_nonneg (λ i hi, norm_nonneg _))) },
{ exact (p.2.le_op_norm m).trans
(mul_le_mul_of_nonneg_right (norm_snd_le p) (finset.prod_nonneg (λ i hi, norm_nonneg _))) },
end⟩ }
/-- Given a fixed continuous linear map `g`, associating to a continuous multilinear map `f` the
continuous multilinear map `f (g m₁, ..., g mₙ)` is a bounded linear operation. -/
lemma is_bounded_linear_map_continuous_multilinear_map_comp_linear (g : G →L[𝕜] E) :
is_bounded_linear_map 𝕜 (λ f : continuous_multilinear_map 𝕜 (λ (i : ι), E) F,
f.comp_continuous_linear_map (λ _, g)) :=
begin
refine is_linear_map.with_bound ⟨λ f₁ f₂, by { ext m, refl }, λ c f, by { ext m, refl }⟩
(∥g∥ ^ (fintype.card ι)) (λ f, _),
apply continuous_multilinear_map.op_norm_le_bound _ _ (λ m, _),
{ apply_rules [mul_nonneg, pow_nonneg, norm_nonneg] },
calc ∥f (g ∘ m)∥ ≤
∥f∥ * ∏ i, ∥g (m i)∥ : f.le_op_norm _
... ≤ ∥f∥ * ∏ i, (∥g∥ * ∥m i∥) : begin
apply mul_le_mul_of_nonneg_left _ (norm_nonneg _),
exact finset.prod_le_prod (λ i hi, norm_nonneg _) (λ i hi, g.le_op_norm _)
end
... = ∥g∥ ^ fintype.card ι * ∥f∥ * ∏ i, ∥m i∥ :
by { simp [finset.prod_mul_distrib, finset.card_univ], ring }
end
end
section bilinear_map
variable (𝕜)
/-- A map `f : E × F → G` satisfies `is_bounded_bilinear_map 𝕜 f` if it is bilinear and
continuous. -/
structure is_bounded_bilinear_map (f : E × F → G) : Prop :=
(add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y))
(smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y))
(add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂))
(smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x,y))
(bound : ∃ C > 0, ∀ (x : E) (y : F), ∥f (x, y)∥ ≤ C * ∥x∥ * ∥y∥)
variable {𝕜}
variable {f : E × F → G}
lemma continuous_linear_map.is_bounded_bilinear_map (f : E →L[𝕜] F →L[𝕜] G) :
is_bounded_bilinear_map 𝕜 (λ x : E × F, f x.1 x.2) :=
{ add_left := λ x₁ x₂ y, by rw [f.map_add, continuous_linear_map.add_apply],
smul_left := λ c x y, by rw [f.map_smul _, continuous_linear_map.smul_apply],
add_right := λ x, (f x).map_add,
smul_right := λ c x y, (f x).map_smul c y,
bound := ⟨max ∥f∥ 1, zero_lt_one.trans_le (le_max_right _ _),
λ x y, (f.le_op_norm₂ x y).trans $
by apply_rules [mul_le_mul_of_nonneg_right, norm_nonneg, le_max_left]⟩ }
protected lemma is_bounded_bilinear_map.is_O (h : is_bounded_bilinear_map 𝕜 f) :
asymptotics.is_O f (λ p : E × F, ∥p.1∥ * ∥p.2∥) ⊤ :=
let ⟨C, Cpos, hC⟩ := h.bound in asymptotics.is_O.of_bound _ $
filter.eventually_of_forall $ λ ⟨x, y⟩, by simpa [mul_assoc] using hC x y
lemma is_bounded_bilinear_map.is_O_comp {α : Type*} (H : is_bounded_bilinear_map 𝕜 f)
{g : α → E} {h : α → F} {l : filter α} :
asymptotics.is_O (λ x, f (g x, h x)) (λ x, ∥g x∥ * ∥h x∥) l :=
H.is_O.comp_tendsto le_top
protected lemma is_bounded_bilinear_map.is_O' (h : is_bounded_bilinear_map 𝕜 f) :
asymptotics.is_O f (λ p : E × F, ∥p∥ * ∥p∥) ⊤ :=
h.is_O.trans (asymptotics.is_O_fst_prod'.norm_norm.mul asymptotics.is_O_snd_prod'.norm_norm)
lemma is_bounded_bilinear_map.map_sub_left (h : is_bounded_bilinear_map 𝕜 f) {x y : E} {z : F} :
f (x - y, z) = f (x, z) - f(y, z) :=
calc f (x - y, z) = f (x + (-1 : 𝕜) • y, z) : by simp [sub_eq_add_neg]
... = f (x, z) + (-1 : 𝕜) • f (y, z) : by simp only [h.add_left, h.smul_left]
... = f (x, z) - f (y, z) : by simp [sub_eq_add_neg]
lemma is_bounded_bilinear_map.map_sub_right (h : is_bounded_bilinear_map 𝕜 f) {x : E} {y z : F} :
f (x, y - z) = f (x, y) - f (x, z) :=
calc f (x, y - z) = f (x, y + (-1 : 𝕜) • z) : by simp [sub_eq_add_neg]
... = f (x, y) + (-1 : 𝕜) • f (x, z) : by simp only [h.add_right, h.smul_right]
... = f (x, y) - f (x, z) : by simp [sub_eq_add_neg]
lemma is_bounded_bilinear_map.is_bounded_linear_map_left (h : is_bounded_bilinear_map 𝕜 f) (y : F) :
is_bounded_linear_map 𝕜 (λ x, f (x, y)) :=
{ map_add := λ x x', h.add_left _ _ _,
map_smul := λ c x, h.smul_left _ _ _,
bound := begin
rcases h.bound with ⟨C, C_pos, hC⟩,
refine ⟨C * (∥y∥ + 1), mul_pos C_pos (lt_of_lt_of_le (zero_lt_one) (by simp)), λ x, _⟩,
have : ∥y∥ ≤ ∥y∥ + 1, by simp [zero_le_one],
calc ∥f (x, y)∥ ≤ C * ∥x∥ * ∥y∥ : hC x y
... ≤ C * ∥x∥ * (∥y∥ + 1) :
by apply_rules [norm_nonneg, mul_le_mul_of_nonneg_left, le_of_lt C_pos, mul_nonneg]
... = C * (∥y∥ + 1) * ∥x∥ : by ring
end }
lemma is_bounded_bilinear_map.is_bounded_linear_map_right
(h : is_bounded_bilinear_map 𝕜 f) (x : E) :
is_bounded_linear_map 𝕜 (λ y, f (x, y)) :=
{ map_add := λ y y', h.add_right _ _ _,
map_smul := λ c y, h.smul_right _ _ _,
bound := begin
rcases h.bound with ⟨C, C_pos, hC⟩,
refine ⟨C * (∥x∥ + 1), mul_pos C_pos (lt_of_lt_of_le (zero_lt_one) (by simp)), λ y, _⟩,
have : ∥x∥ ≤ ∥x∥ + 1, by simp [zero_le_one],
calc ∥f (x, y)∥ ≤ C * ∥x∥ * ∥y∥ : hC x y
... ≤ C * (∥x∥ + 1) * ∥y∥ :
by apply_rules [mul_le_mul_of_nonneg_right, norm_nonneg, mul_le_mul_of_nonneg_left,
le_of_lt C_pos]
end }
lemma is_bounded_bilinear_map_smul {𝕜' : Type*} [normed_field 𝕜']
[normed_algebra 𝕜 𝕜'] {E : Type*} [normed_group E] [normed_space 𝕜 E] [normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E] :
is_bounded_bilinear_map 𝕜 (λ (p : 𝕜' × E), p.1 • p.2) :=
{ add_left := add_smul,
smul_left := λ c x y, by simp [smul_assoc],
add_right := smul_add,
smul_right := λ c x y, by simp [smul_assoc, smul_algebra_smul_comm],
bound := ⟨1, zero_lt_one, λ x y, by simp [norm_smul] ⟩ }
lemma is_bounded_bilinear_map_mul :
is_bounded_bilinear_map 𝕜 (λ (p : 𝕜 × 𝕜), p.1 * p.2) :=
by simp_rw ← smul_eq_mul; exact is_bounded_bilinear_map_smul
lemma is_bounded_bilinear_map_comp :
is_bounded_bilinear_map 𝕜 (λ (p : (E →L[𝕜] F) × (F →L[𝕜] G)), p.2.comp p.1) :=
{ add_left := λ x₁ x₂ y, begin
ext z,
change y (x₁ z + x₂ z) = y (x₁ z) + y (x₂ z),
rw y.map_add
end,
smul_left := λ c x y, begin
ext z,
change y (c • (x z)) = c • y (x z),
rw continuous_linear_map.map_smul
end,
add_right := λ x y₁ y₂, rfl,
smul_right := λ c x y, rfl,
bound := ⟨1, zero_lt_one, λ x y, calc
∥continuous_linear_map.comp ((x, y).snd) ((x, y).fst)∥
≤ ∥y∥ * ∥x∥ : continuous_linear_map.op_norm_comp_le _ _
... = 1 * ∥x∥ * ∥ y∥ : by ring ⟩ }
lemma continuous_linear_map.is_bounded_linear_map_comp_left (g : F →L[𝕜] G) :
is_bounded_linear_map 𝕜 (λ (f : E →L[𝕜] F), continuous_linear_map.comp g f) :=
is_bounded_bilinear_map_comp.is_bounded_linear_map_left _
lemma continuous_linear_map.is_bounded_linear_map_comp_right (f : E →L[𝕜] F) :
is_bounded_linear_map 𝕜 (λ (g : F →L[𝕜] G), continuous_linear_map.comp g f) :=
is_bounded_bilinear_map_comp.is_bounded_linear_map_right _
lemma is_bounded_bilinear_map_apply :
is_bounded_bilinear_map 𝕜 (λ p : (E →L[𝕜] F) × E, p.1 p.2) :=
{ add_left := by simp,
smul_left := by simp,
add_right := by simp,
smul_right := by simp,
bound := ⟨1, zero_lt_one, by simp [continuous_linear_map.le_op_norm]⟩ }
/-- The function `continuous_linear_map.smul_right`, associating to a continuous linear map
`f : E → 𝕜` and a scalar `c : F` the tensor product `f ⊗ c` as a continuous linear map from `E` to
`F`, is a bounded bilinear map. -/
lemma is_bounded_bilinear_map_smul_right :
is_bounded_bilinear_map 𝕜
(λ p, (continuous_linear_map.smul_right : (E →L[𝕜] 𝕜) → F → (E →L[𝕜] F)) p.1 p.2) :=
{ add_left := λ m₁ m₂ f, by { ext z, simp [add_smul] },
smul_left := λ c m f, by { ext z, simp [mul_smul] },
add_right := λ m f₁ f₂, by { ext z, simp [smul_add] },
smul_right := λ c m f, by { ext z, simp [smul_smul, mul_comm] },
bound := ⟨1, zero_lt_one, λ m f, by simp⟩ }
/-- The composition of a continuous linear map with a continuous multilinear map is a bounded
bilinear operation. -/
lemma is_bounded_bilinear_map_comp_multilinear {ι : Type*} {E : ι → Type*}
[decidable_eq ι] [fintype ι] [∀ i, normed_group (E i)] [∀ i, normed_space 𝕜 (E i)] :
is_bounded_bilinear_map 𝕜 (λ p : (F →L[𝕜] G) × (continuous_multilinear_map 𝕜 E F),
p.1.comp_continuous_multilinear_map p.2) :=
{ add_left := λ g₁ g₂ f, by { ext m, refl },
smul_left := λ c g f, by { ext m, refl },
add_right := λ g f₁ f₂, by { ext m, simp },
smul_right := λ c g f, by { ext m, simp },
bound := ⟨1, zero_lt_one, λ g f, begin
apply continuous_multilinear_map.op_norm_le_bound _ _ (λ m, _),
{ apply_rules [mul_nonneg, zero_le_one, norm_nonneg] },
calc ∥g (f m)∥ ≤ ∥g∥ * ∥f m∥ : g.le_op_norm _
... ≤ ∥g∥ * (∥f∥ * ∏ i, ∥m i∥) :
mul_le_mul_of_nonneg_left (f.le_op_norm _) (norm_nonneg _)
... = 1 * ∥g∥ * ∥f∥ * ∏ i, ∥m i∥ : by ring
end⟩ }
/-- Definition of the derivative of a bilinear map `f`, given at a point `p` by
`q ↦ f(p.1, q.2) + f(q.1, p.2)` as in the standard formula for the derivative of a product.
We define this function here as a linear map `E × F →ₗ[𝕜] G`, then `is_bounded_bilinear_map.deriv`
strengthens it to a continuous linear map `E × F →L[𝕜] G`.
``. -/
def is_bounded_bilinear_map.linear_deriv (h : is_bounded_bilinear_map 𝕜 f) (p : E × F) :
E × F →ₗ[𝕜] G :=
{ to_fun := λ q, f (p.1, q.2) + f (q.1, p.2),
map_add' := λ q₁ q₂, begin
change f (p.1, q₁.2 + q₂.2) + f (q₁.1 + q₂.1, p.2) =
f (p.1, q₁.2) + f (q₁.1, p.2) + (f (p.1, q₂.2) + f (q₂.1, p.2)),
simp [h.add_left, h.add_right], abel
end,
map_smul' := λ c q, begin
change f (p.1, c • q.2) + f (c • q.1, p.2) = c • (f (p.1, q.2) + f (q.1, p.2)),
simp [h.smul_left, h.smul_right, smul_add]
end }
/-- The derivative of a bounded bilinear map at a point `p : E × F`, as a continuous linear map
from `E × F` to `G`. The statement that this is indeed the derivative of `f` is
`is_bounded_bilinear_map.has_fderiv_at` in `analysis.calculus.fderiv`. -/
def is_bounded_bilinear_map.deriv (h : is_bounded_bilinear_map 𝕜 f) (p : E × F) : E × F →L[𝕜] G :=
(h.linear_deriv p).mk_continuous_of_exists_bound $ begin
rcases h.bound with ⟨C, Cpos, hC⟩,
refine ⟨C * ∥p.1∥ + C * ∥p.2∥, λ q, _⟩,
calc ∥f (p.1, q.2) + f (q.1, p.2)∥
≤ C * ∥p.1∥ * ∥q.2∥ + C * ∥q.1∥ * ∥p.2∥ : norm_add_le_of_le (hC _ _) (hC _ _)
... ≤ C * ∥p.1∥ * ∥q∥ + C * ∥q∥ * ∥p.2∥ : begin
apply add_le_add,
exact mul_le_mul_of_nonneg_left
(le_max_right _ _) (mul_nonneg (le_of_lt Cpos) (norm_nonneg _)),
apply mul_le_mul_of_nonneg_right _ (norm_nonneg _),
exact mul_le_mul_of_nonneg_left (le_max_left _ _) (le_of_lt Cpos),
end
... = (C * ∥p.1∥ + C * ∥p.2∥) * ∥q∥ : by ring
end
@[simp] lemma is_bounded_bilinear_map_deriv_coe (h : is_bounded_bilinear_map 𝕜 f) (p q : E × F) :
h.deriv p q = f (p.1, q.2) + f (q.1, p.2) := rfl
variables (𝕜)
/-- The function `lmul_left_right : 𝕜' × 𝕜' → (𝕜' →L[𝕜] 𝕜')` is a bounded bilinear map. -/
lemma continuous_linear_map.lmul_left_right_is_bounded_bilinear
(𝕜' : Type*) [normed_ring 𝕜'] [normed_algebra 𝕜 𝕜'] :
is_bounded_bilinear_map 𝕜 (λ p : 𝕜' × 𝕜', continuous_linear_map.lmul_left_right 𝕜 𝕜' p.1 p.2) :=
(continuous_linear_map.lmul_left_right 𝕜 𝕜').is_bounded_bilinear_map
variables {𝕜}
/-- Given a bounded bilinear map `f`, the map associating to a point `p` the derivative of `f` at
`p` is itself a bounded linear map. -/
lemma is_bounded_bilinear_map.is_bounded_linear_map_deriv (h : is_bounded_bilinear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (λ p : E × F, h.deriv p) :=
begin
rcases h.bound with ⟨C, Cpos : 0 < C, hC⟩,
refine is_linear_map.with_bound ⟨λ p₁ p₂, _, λ c p, _⟩ (C + C) (λ p, _),
{ ext; simp [h.add_left, h.add_right]; abel },
{ ext; simp [h.smul_left, h.smul_right, smul_add] },
{ refine continuous_linear_map.op_norm_le_bound _
(mul_nonneg (add_nonneg Cpos.le Cpos.le) (norm_nonneg _)) (λ q, _),
calc ∥f (p.1, q.2) + f (q.1, p.2)∥
≤ C * ∥p.1∥ * ∥q.2∥ + C * ∥q.1∥ * ∥p.2∥ : norm_add_le_of_le (hC _ _) (hC _ _)
... ≤ C * ∥p∥ * ∥q∥ + C * ∥q∥ * ∥p∥ : by apply_rules [add_le_add, mul_le_mul, norm_nonneg,
Cpos.le, le_refl, le_max_left, le_max_right, mul_nonneg]
... = (C + C) * ∥p∥ * ∥q∥ : by ring },
end
end bilinear_map
/-- A linear isometry preserves the norm. -/
lemma linear_map.norm_apply_of_isometry (f : E →ₗ[𝕜] F) {x : E} (hf : isometry f) : ∥f x∥ = ∥x∥ :=
by { simp_rw [←dist_zero_right, ←f.map_zero], exact isometry.dist_eq hf _ _ }
/-- Construct a continuous linear equiv from
a linear map that is also an isometry with full range. -/
def continuous_linear_equiv.of_isometry (f : E →ₗ[𝕜] F) (hf : isometry f) (hfr : f.range = ⊤) :
E ≃L[𝕜] F :=
continuous_linear_equiv.of_homothety
(linear_equiv.of_bijective f (isometry.injective hf) (linear_map.range_eq_top.mp hfr))
1 zero_lt_one (λ _, by simp [one_mul, f.norm_apply_of_isometry hf])
|
2e3441f862b3757c74c7140bc5ba9788636c0463 | df7bb3acd9623e489e95e85d0bc55590ab0bc393 | /lean/love05_inductive_predicates_homework_sheet.lean | e98fb71f75f8a62d52b58cd0c471b54c188f74d5 | [] | no_license | MaschavanderMarel/logical_verification_2020 | a41c210b9237c56cb35f6cd399e3ac2fe42e775d | 7d562ef174cc6578ca6013f74db336480470b708 | refs/heads/master | 1,692,144,223,196 | 1,634,661,675,000 | 1,634,661,675,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,999 | lean | import .lovelib
/- # LoVe Homework 5: Inductive Predicates
Homework must be done individually. -/
set_option pp.beta true
set_option pp.generalized_field_notation false
namespace LoVe
/- ## Question 1 (3 points): A Type of λ-Terms
Recall the following type of λ-terms from question 3 of exercise 4. -/
inductive term : Type
| var : string → term
| lam : string → term → term
| app : term → term → term
/- 1.1 (1 point). Define an inductive predicate `is_app` that returns true if
and only if its argument has an `app` constructor at the top level. -/
-- enter your definition here
/- 1.2 (2 points). Define an inductive predicate `is_abs_free` that is true if
and only if its argument is a λ-term that contains no λ-expressions. -/
-- enter your definition here
/- ## Question 2 (6 points): Even and Odd
Consider the following inductive definition of even numbers: -/
inductive even : ℕ → Prop
| zero : even 0
| add_two {n : ℕ} : even n → even (n + 2)
/- 2.1 (1 point). Define a similar predicate for odd numbers, by completing the
Lean definition below. The definition should distinguish two cases, like `even`,
and should not rely on `even`. -/
inductive odd : ℕ → Prop
-- supply the missing cases here
/- 2.2 (1 point). Give proof terms for the following propositions, based on
your answer to question 2.1. -/
lemma odd_3 :
odd 3 :=
sorry
lemma odd_5 :
odd 5 :=
sorry
/- 2.3 (2 points). Prove the following lemma by rule induction, as a "paper"
proof. This is a good exercise to develop a deeper understanding of how rule
induction works (and is good practice for the final exam).
lemma even_odd {n : ℕ} (heven : even n) :
odd (n + 1) :=
Guidelines for paper proofs:
We expect detailed, rigorous, mathematical proofs. You are welcome to use
standard mathematical notation or Lean structured commands (e.g., `assume`,
`have`, `show`, `calc`). You can also use tactical proofs (e.g., `intro`,
`apply`), but then please indicate some of the intermediate goals, so that we
can follow the chain of reasoning.
Major proof steps, including applications of induction and invocation of the
induction hypothesis, must be stated explicitly. For each case of a proof by
induction, you must list the inductive hypotheses assumed (if any) and the goal
to be proved. Minor proof steps corresponding to `refl`, `simp`, or `linarith`
need not be justified if you think they are obvious (to humans), but you should
say which key lemmas they depend on. You should be explicit whenever you use a
function definition or an introduction rule for an inductive predicate. -/
-- enter your paper proof here
/- 2.4 (1 point). Prove the same lemma again, but this time in Lean: -/
lemma even_odd {n : ℕ} (heven : even n) :
odd (n + 1) :=
sorry
/- 2.5 (1 point). Prove the following lemma in Lean.
Hint: Recall that `¬ a` is defined as `a → false`. -/
lemma even_not_odd {n : ℕ} (heven : even n) :
¬ odd n :=
sorry
end LoVe
|
e2f918d2fb7006969dae27f03ab4ac7ea8376dc6 | 2eab05920d6eeb06665e1a6df77b3157354316ad | /src/analysis/fourier.lean | 3ef9081a630d572511b45645c2385b453b5c5124 | [
"Apache-2.0"
] | permissive | ayush1801/mathlib | 78949b9f789f488148142221606bf15c02b960d2 | ce164e28f262acbb3de6281b3b03660a9f744e3c | refs/heads/master | 1,692,886,907,941 | 1,635,270,866,000 | 1,635,270,866,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 8,200 | lean | /-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import measure_theory.function.continuous_map_dense
import measure_theory.function.l2_space
import measure_theory.measure.haar
import analysis.complex.circle
import topology.metric_space.emetric_paracompact
import topology.continuous_function.stone_weierstrass
/-!
# Fourier analysis on the circle
This file contains basic technical results for a development of Fourier series.
## Main definitions
* `haar_circle`, Haar measure on the circle, normalized to have total measure `1`
* instances `measure_space`, `is_probability_measure` for the circle with respect to this measure
* for `n : ℤ`, `fourier n` is the monomial `λ z, z ^ n`, bundled as a continuous map from `circle`
to `ℂ`
* for `n : ℤ` and `p : ℝ≥0∞`, `fourier_Lp p n` is an abbreviation for the monomial `fourier n`
considered as an element of the Lᵖ-space `Lp ℂ p haar_circle`, via the embedding
`continuous_map.to_Lp`
## Main statements
The theorem `span_fourier_closure_eq_top` states that the span of the monomials `fourier n` is
dense in `C(circle, ℂ)`, i.e. that its `submodule.topological_closure` is `⊤`. This follows from
the Stone-Weierstrass theorem after checking that it is a subalgebra, closed under conjugation, and
separates points.
The theorem `span_fourier_Lp_closure_eq_top` states that for `1 ≤ p < ∞` the span of the monomials
`fourier_Lp` is dense in `Lp ℂ p haar_circle`, i.e. that its `submodule.topological_closure` is
`⊤`. This follows from the previous theorem using general theory on approximation of Lᵖ functions
by continuous functions.
The theorem `orthonormal_fourier` states that the monomials `fourier_Lp 2 n` form an orthonormal
set (in the L² space of the circle).
By definition, a Hilbert basis for an inner product space is an orthonormal set whose span is
dense. Thus, the last two results together establish that the functions `fourier_Lp 2 n` form a
Hilbert basis for L².
## TODO
Once mathlib has general theory showing that a Hilbert basis of an inner product space induces a
unitary equivalence with L², the results in this file will give Fourier series applications such
as Parseval's formula.
-/
noncomputable theory
open_locale ennreal complex_conjugate
open topological_space continuous_map measure_theory measure_theory.measure algebra submodule set
local attribute [instance] fact_one_le_two_ennreal
/-! ### Choice of measure on the circle -/
section haar_circle
/-! We make the circle into a measure space, using the Haar measure normalized to have total
measure 1. -/
instance : measurable_space circle := borel circle
instance : borel_space circle := ⟨rfl⟩
/-- Haar measure on the circle, normalized to have total measure 1. -/
def haar_circle : measure circle := haar_measure positive_compacts_univ
instance : is_probability_measure haar_circle := ⟨haar_measure_self⟩
instance : measure_space circle :=
{ volume := haar_circle,
.. circle.measurable_space }
end haar_circle
/-! ### Monomials on the circle -/
section fourier
/-- The family of monomials `λ z, z ^ n`, parametrized by `n : ℤ` and considered as bundled
continuous maps from `circle` to `ℂ`. -/
@[simps] def fourier (n : ℤ) : C(circle, ℂ) :=
{ to_fun := λ z, z ^ n,
continuous_to_fun := continuous_subtype_coe.fpow n $ λ z, or.inl (nonzero_of_mem_circle z) }
@[simp] lemma fourier_zero {z : circle} : fourier 0 z = 1 := rfl
@[simp] lemma fourier_neg {n : ℤ} {z : circle} : fourier (-n) z = conj (fourier n z) :=
by simp [← coe_inv_circle_eq_conj z]
@[simp] lemma fourier_add {m n : ℤ} {z : circle} :
fourier (m + n) z = (fourier m z) * (fourier n z) :=
by simp [fpow_add (nonzero_of_mem_circle z)]
/-- The subalgebra of `C(circle, ℂ)` generated by `z ^ n` for `n ∈ ℤ`; equivalently, polynomials in
`z` and `conj z`. -/
def fourier_subalgebra : subalgebra ℂ C(circle, ℂ) := algebra.adjoin ℂ (range fourier)
/-- The subalgebra of `C(circle, ℂ)` generated by `z ^ n` for `n ∈ ℤ` is in fact the linear span of
these functions. -/
lemma fourier_subalgebra_coe : fourier_subalgebra.to_submodule = span ℂ (range fourier) :=
begin
apply adjoin_eq_span_of_subset,
refine subset.trans _ submodule.subset_span,
intros x hx,
apply submonoid.closure_induction hx (λ _, id) ⟨0, rfl⟩,
rintros _ _ ⟨m, rfl⟩ ⟨n, rfl⟩,
refine ⟨m + n, _⟩,
ext1 z,
exact fourier_add,
end
/-- The subalgebra of `C(circle, ℂ)` generated by `z ^ n` for `n ∈ ℤ` separates points. -/
lemma fourier_subalgebra_separates_points : fourier_subalgebra.separates_points :=
begin
intros x y hxy,
refine ⟨_, ⟨fourier 1, _, rfl⟩, _⟩,
{ exact subset_adjoin ⟨1, rfl⟩ },
{ simp [hxy] }
end
/-- The subalgebra of `C(circle, ℂ)` generated by `z ^ n` for `n ∈ ℤ` is invariant under complex
conjugation. -/
lemma fourier_subalgebra_conj_invariant :
conj_invariant_subalgebra (fourier_subalgebra.restrict_scalars ℝ) :=
begin
rintros _ ⟨f, hf, rfl⟩,
change _ ∈ fourier_subalgebra,
change _ ∈ fourier_subalgebra at hf,
apply adjoin_induction hf,
{ rintros _ ⟨n, rfl⟩,
suffices : fourier (-n) ∈ fourier_subalgebra,
{ convert this,
ext1,
simp },
exact subset_adjoin ⟨-n, rfl⟩ },
{ intros c,
exact fourier_subalgebra.algebra_map_mem (conj c) },
{ intros f g hf hg,
convert fourier_subalgebra.add_mem hf hg,
exact alg_hom.map_add _ f g, },
{ intros f g hf hg,
convert fourier_subalgebra.mul_mem hf hg,
exact alg_hom.map_mul _ f g, }
end
/-- The subalgebra of `C(circle, ℂ)` generated by `z ^ n` for `n ∈ ℤ` is dense. -/
lemma fourier_subalgebra_closure_eq_top : fourier_subalgebra.topological_closure = ⊤ :=
continuous_map.subalgebra_complex_topological_closure_eq_top_of_separates_points
fourier_subalgebra
fourier_subalgebra_separates_points
fourier_subalgebra_conj_invariant
/-- The linear span of the monomials `z ^ n` is dense in `C(circle, ℂ)`. -/
lemma span_fourier_closure_eq_top : (span ℂ (range fourier)).topological_closure = ⊤ :=
begin
rw ← fourier_subalgebra_coe,
exact congr_arg subalgebra.to_submodule fourier_subalgebra_closure_eq_top,
end
/-- The family of monomials `λ z, z ^ n`, parametrized by `n : ℤ` and considered as elements of
the `Lp` space of functions on `circle` taking values in `ℂ`. -/
abbreviation fourier_Lp (p : ℝ≥0∞) [fact (1 ≤ p)] (n : ℤ) : Lp ℂ p haar_circle :=
to_Lp p haar_circle ℂ (fourier n)
/-- For each `1 ≤ p < ∞`, the linear span of the monomials `z ^ n` is dense in
`Lp ℂ p haar_circle`. -/
lemma span_fourier_Lp_closure_eq_top {p : ℝ≥0∞} [fact (1 ≤ p)] (hp : p ≠ ∞) :
(span ℂ (range (fourier_Lp p))).topological_closure = ⊤ :=
begin
convert (continuous_map.to_Lp_dense_range ℂ hp haar_circle ℂ).topological_closure_map_submodule
span_fourier_closure_eq_top,
rw [map_span, range_comp],
simp
end
/-- For `n ≠ 0`, a rotation by `n⁻¹ * real.pi` negates the monomial `z ^ n`. -/
lemma fourier_add_half_inv_index {n : ℤ} (hn : n ≠ 0) (z : circle) :
fourier n ((exp_map_circle (n⁻¹ * real.pi) * z)) = - fourier n z :=
begin
have : ↑n * ((↑n)⁻¹ * ↑real.pi * complex.I) = ↑real.pi * complex.I,
{ have : (n:ℂ) ≠ 0 := by exact_mod_cast hn,
field_simp,
ring },
simp [mul_fpow, ← complex.exp_int_mul, complex.exp_pi_mul_I, this]
end
/-- The monomials `z ^ n` are an orthonormal set with respect to Haar measure on the circle. -/
lemma orthonormal_fourier : orthonormal ℂ (fourier_Lp 2) :=
begin
rw orthonormal_iff_ite,
intros i j,
rw continuous_map.inner_to_Lp haar_circle (fourier i) (fourier j),
split_ifs,
{ simp [h, is_probability_measure.measure_univ, ← fourier_neg, ← fourier_add, -fourier_to_fun] },
simp only [← fourier_add, ← fourier_neg],
have hij : -i + j ≠ 0,
{ rw add_comm,
exact sub_ne_zero.mpr (ne.symm h) },
exact integral_zero_of_mul_left_eq_neg (is_mul_left_invariant_haar_measure _)
(fourier_add_half_inv_index hij)
end
end fourier
|
c2fe52cf72d445fe5f77486cd24e0e337ee2d5dc | a45212b1526d532e6e83c44ddca6a05795113ddc | /src/algebra/gcd_domain.lean | 772892fee9c3cbbd33336c5e30bcc96f42d2c287 | [
"Apache-2.0"
] | permissive | fpvandoorn/mathlib | b21ab4068db079cbb8590b58fda9cc4bc1f35df4 | b3433a51ea8bc07c4159c1073838fc0ee9b8f227 | refs/heads/master | 1,624,791,089,608 | 1,556,715,231,000 | 1,556,715,231,000 | 165,722,980 | 5 | 0 | Apache-2.0 | 1,552,657,455,000 | 1,547,494,646,000 | Lean | UTF-8 | Lean | false | false | 26,084 | lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
GCD domain and integral domains with normalization functions
TODO: abstract the domains to to semi domains (i.e. domains on semirings) to include ℕ and ℕ[X] etc.
-/
import algebra.associated data.int.gcd
variables {α : Type*}
set_option old_structure_cmd true
/-- Normalization domain: multiplying with `norm_unit` gives a normal form for associated elements. -/
class normalization_domain (α : Type*) extends integral_domain α :=
(norm_unit : α → units α)
(norm_unit_zero : norm_unit 0 = 1)
(norm_unit_mul : ∀{a b}, a ≠ 0 → b ≠ 0 → norm_unit (a * b) = norm_unit a * norm_unit b)
(norm_unit_coe_units : ∀(u : units α), norm_unit u = u⁻¹)
export normalization_domain (norm_unit norm_unit_zero norm_unit_mul norm_unit_coe_units)
attribute [simp] norm_unit_coe_units norm_unit_zero norm_unit_mul
section normalization_domain
variable [normalization_domain α]
def normalize (x : α) : α :=
x * norm_unit x
theorem associated_normalize {x : α} : associated x (normalize x) :=
⟨_, rfl⟩
theorem normalize_associated {x : α} : associated (normalize x) x :=
associated_normalize.symm
@[simp] theorem norm_unit_one : norm_unit (1:α) = 1 :=
norm_unit_coe_units 1
@[simp] lemma normalize_zero : normalize (0 : α) = 0 :=
by rw [normalize, zero_mul]
@[simp] lemma normalize_one : normalize (1 : α) = 1 :=
by rw [normalize, norm_unit_one, units.coe_one, mul_one]
lemma normalize_coe_units (u : units α) : normalize (u : α) = 1 :=
by rw [normalize, norm_unit_coe_units, ← units.coe_mul, mul_inv_self, units.coe_one]
theorem normalize_mul (x y : α) : normalize (x * y) = normalize x * normalize y :=
classical.by_cases (λ hx : x = 0, by rw [hx, zero_mul, normalize_zero, zero_mul]) $ λ hx,
classical.by_cases (λ hy : y = 0, by rw [hy, mul_zero, normalize_zero, mul_zero]) $ λ hy,
by simp only [normalize, norm_unit_mul hx hy, units.coe_mul]; simp only [mul_assoc, mul_left_comm y]
lemma normalize_eq_zero {x : α} : normalize x = 0 ↔ x = 0 :=
⟨λ hx, (associated_zero_iff_eq_zero x).1 $ hx ▸ associated_normalize, by rintro rfl; exact normalize_zero⟩
lemma normalize_eq_one {x : α} : normalize x = 1 ↔ is_unit x :=
⟨λ hx, is_unit_iff_exists_inv.2 ⟨_, hx⟩, λ ⟨u, hu⟩, hu.symm ▸ normalize_coe_units u⟩
theorem norm_unit_mul_norm_unit (a : α) : norm_unit (a * norm_unit a) = 1 :=
classical.by_cases (assume : a = 0, by simp only [this, norm_unit_zero, zero_mul]) $
assume h, by rw [norm_unit_mul h (units.ne_zero _), norm_unit_coe_units, mul_inv_eq_one]
theorem normalize_idem (x : α) : normalize (normalize x) = normalize x :=
by rw [normalize, normalize, norm_unit_mul_norm_unit, units.coe_one, mul_one]
theorem normalize_eq_normalize {a b : α}
(hab : a ∣ b) (hba : b ∣ a) : normalize a = normalize b :=
begin
rcases associated_of_dvd_dvd hab hba with ⟨u, rfl⟩,
refine classical.by_cases (by rintro rfl; simp only [zero_mul]) (assume ha : a ≠ 0, _),
suffices : a * ↑(norm_unit a) = a * ↑u * ↑(norm_unit a) * ↑u⁻¹,
by simpa only [normalize, mul_assoc, norm_unit_mul ha u.ne_zero, norm_unit_coe_units],
calc a * ↑(norm_unit a) = a * ↑(norm_unit a) * ↑u * ↑u⁻¹:
(units.mul_inv_cancel_right _ _).symm
... = a * ↑u * ↑(norm_unit a) * ↑u⁻¹ : by rw mul_right_comm a
end
lemma normalize_eq_normalize_iff {x y : α} : normalize x = normalize y ↔ x ∣ y ∧ y ∣ x :=
⟨λ h, ⟨dvd_mul_unit_iff.1 ⟨_, h.symm⟩, dvd_mul_unit_iff.1 ⟨_, h⟩⟩,
λ ⟨hxy, hyx⟩, normalize_eq_normalize hxy hyx⟩
theorem dvd_antisymm_of_normalize_eq {a b : α}
(ha : normalize a = a) (hb : normalize b = b) (hab : a ∣ b) (hba : b ∣ a) :
a = b :=
ha ▸ hb ▸ normalize_eq_normalize hab hba
@[simp] lemma dvd_normalize_iff {a b : α} : a ∣ normalize b ↔ a ∣ b :=
dvd_mul_unit_iff
@[simp] lemma normalize_dvd_iff {a b : α} : normalize a ∣ b ↔ a ∣ b :=
mul_unit_dvd_iff
end normalization_domain
namespace associates
variable [normalization_domain α]
local attribute [instance] associated.setoid
protected def out : associates α → α :=
quotient.lift (normalize : α → α) $ λ a b ⟨u, hu⟩, hu ▸
normalize_eq_normalize ⟨_, rfl⟩ (mul_unit_dvd_iff.2 $ dvd_refl a)
lemma out_mk (a : α) : (associates.mk a).out = normalize a := rfl
@[simp] lemma out_one : (1 : associates α).out = 1 :=
normalize_one
lemma out_mul (a b : associates α) : (a * b).out = a.out * b.out :=
quotient.induction_on₂ a b $ assume a b,
by simp only [associates.quotient_mk_eq_mk, out_mk, mk_mul_mk, normalize_mul]
lemma dvd_out_iff (a : α) (b : associates α) : a ∣ b.out ↔ associates.mk a ≤ b :=
quotient.induction_on b $ by simp [associates.out_mk, associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]
lemma out_dvd_iff (a : α) (b : associates α) : b.out ∣ a ↔ b ≤ associates.mk a :=
quotient.induction_on b $ by simp [associates.out_mk, associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]
@[simp] lemma out_top : (⊤ : associates α).out = 0 :=
normalize_zero
@[simp] lemma normalize_out (a : associates α) : normalize a.out = a.out :=
quotient.induction_on a normalize_idem
end associates
/-- GCD domain: an integral domain with normalization and `gcd` (greatest common divisor) and
`lcm` (least common multiple) operations. In this setting `gcd` and `lcm` form a bounded lattice on
the associated elements where `gcd` is the infimum, `lcm` is the supremum, `1` is bottom, and
`0` is top. The type class focuses on `gcd` and we derive the correpsonding `lcm` facts from `gcd`.
-/
class gcd_domain (α : Type*) extends normalization_domain α :=
(gcd : α → α → α)
(lcm : α → α → α)
(gcd_dvd_left : ∀a b, gcd a b ∣ a)
(gcd_dvd_right : ∀a b, gcd a b ∣ b)
(dvd_gcd : ∀{a b c}, a ∣ c → a ∣ b → a ∣ gcd c b)
(normalize_gcd : ∀a b, normalize (gcd a b) = gcd a b)
(gcd_mul_lcm : ∀a b, gcd a b * lcm a b = normalize (a * b))
(lcm_zero_left : ∀a, lcm 0 a = 0)
(lcm_zero_right : ∀a, lcm a 0 = 0)
export gcd_domain (gcd lcm gcd_dvd_left gcd_dvd_right dvd_gcd lcm_zero_left lcm_zero_right)
attribute [simp] lcm_zero_left lcm_zero_right
section gcd_domain
variables [gcd_domain α]
@[simp] theorem normalize_gcd : ∀a b:α, normalize (gcd a b) = gcd a b :=
gcd_domain.normalize_gcd
@[simp] theorem gcd_mul_lcm : ∀a b:α, gcd a b * lcm a b = normalize (a * b) :=
gcd_domain.gcd_mul_lcm
section gcd
theorem dvd_gcd_iff (a b c : α) : a ∣ gcd b c ↔ (a ∣ b ∧ a ∣ c) :=
iff.intro
(assume h, ⟨dvd_trans h (gcd_dvd_left _ _), dvd_trans h (gcd_dvd_right _ _)⟩)
(assume ⟨hab, hac⟩, dvd_gcd hab hac)
theorem gcd_comm (a b : α) : gcd a b = gcd b a :=
dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
(dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
(dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
theorem gcd_assoc (m n k : α) : gcd (gcd m n) k = gcd m (gcd n k) :=
dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
(dvd_gcd
(dvd.trans (gcd_dvd_left (gcd m n) k) (gcd_dvd_left m n))
(dvd_gcd (dvd.trans (gcd_dvd_left (gcd m n) k) (gcd_dvd_right m n)) (gcd_dvd_right (gcd m n) k)))
(dvd_gcd
(dvd_gcd (gcd_dvd_left m (gcd n k)) (dvd.trans (gcd_dvd_right m (gcd n k)) (gcd_dvd_left n k)))
(dvd.trans (gcd_dvd_right m (gcd n k)) (gcd_dvd_right n k)))
instance : is_commutative α gcd := ⟨gcd_comm⟩
instance : is_associative α gcd := ⟨gcd_assoc⟩
theorem gcd_eq_normalize {a b c : α} (habc : gcd a b ∣ c) (hcab : c ∣ gcd a b) :
gcd a b = normalize c :=
normalize_gcd a b ▸ normalize_eq_normalize habc hcab
@[simp] theorem gcd_zero_left (a : α) : gcd 0 a = normalize a :=
gcd_eq_normalize (gcd_dvd_right 0 a) (dvd_gcd (dvd_zero _) (dvd_refl a))
@[simp] theorem gcd_zero_right (a : α) : gcd a 0 = normalize a :=
gcd_eq_normalize (gcd_dvd_left a 0) (dvd_gcd (dvd_refl a) (dvd_zero _))
@[simp] theorem gcd_eq_zero_iff (a b : α) : gcd a b = 0 ↔ a = 0 ∧ b = 0 :=
iff.intro
(assume h, let ⟨ca, ha⟩ := gcd_dvd_left a b, ⟨cb, hb⟩ := gcd_dvd_right a b in
by rw [h, zero_mul] at ha hb; exact ⟨ha, hb⟩)
(assume ⟨ha, hb⟩, by rw [ha, hb, gcd_zero_left, normalize_zero])
@[simp] theorem gcd_one_left (a : α) : gcd 1 a = 1 :=
dvd_antisymm_of_normalize_eq (normalize_gcd _ _) normalize_one (gcd_dvd_left _ _) (one_dvd _)
@[simp] theorem gcd_one_right (a : α) : gcd a 1 = 1 :=
dvd_antisymm_of_normalize_eq (normalize_gcd _ _) normalize_one (gcd_dvd_right _ _) (one_dvd _)
theorem gcd_dvd_gcd {a b c d: α} (hab : a ∣ b) (hcd : c ∣ d) : gcd a c ∣ gcd b d :=
dvd_gcd (dvd.trans (gcd_dvd_left _ _) hab) (dvd.trans (gcd_dvd_right _ _) hcd)
@[simp] theorem gcd_same (a : α) : gcd a a = normalize a :=
gcd_eq_normalize (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) (dvd_refl a))
@[simp] theorem gcd_mul_left (a b c : α) : gcd (a * b) (a * c) = normalize a * gcd b c :=
classical.by_cases (by rintro rfl; simp only [zero_mul, gcd_zero_left, normalize_zero]) $ assume ha : a ≠ 0,
suffices gcd (a * b) (a * c) = normalize (a * gcd b c),
by simpa only [normalize_mul, normalize_gcd],
let ⟨d, eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c) in
gcd_eq_normalize
(eq.symm ▸ mul_dvd_mul_left a $ show d ∣ gcd b c, from
dvd_gcd
((mul_dvd_mul_iff_left ha).1 $ eq ▸ gcd_dvd_left _ _)
((mul_dvd_mul_iff_left ha).1 $ eq ▸ gcd_dvd_right _ _))
(dvd_gcd
(mul_dvd_mul_left a $ gcd_dvd_left _ _)
(mul_dvd_mul_left a $ gcd_dvd_right _ _))
@[simp] theorem gcd_mul_right (a b c : α) : gcd (b * a) (c * a) = gcd b c * normalize a :=
by simp only [mul_comm, gcd_mul_left]
theorem gcd_eq_left_iff (a b : α) (h : normalize a = a) : gcd a b = a ↔ a ∣ b :=
iff.intro (assume eq, eq ▸ gcd_dvd_right _ _) $
assume hab, dvd_antisymm_of_normalize_eq (normalize_gcd _ _) h (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) hab)
theorem gcd_eq_right_iff (a b : α) (h : normalize b = b) : gcd a b = b ↔ b ∣ a :=
by simpa only [gcd_comm a b] using gcd_eq_left_iff b a h
theorem gcd_dvd_gcd_mul_left (m n k : α) : gcd m n ∣ gcd (k * m) n :=
gcd_dvd_gcd (dvd_mul_left _ _) (dvd_refl _)
theorem gcd_dvd_gcd_mul_right (m n k : α) : gcd m n ∣ gcd (m * k) n :=
gcd_dvd_gcd (dvd_mul_right _ _) (dvd_refl _)
theorem gcd_dvd_gcd_mul_left_right (m n k : α) : gcd m n ∣ gcd m (k * n) :=
gcd_dvd_gcd (dvd_refl _) (dvd_mul_left _ _)
theorem gcd_dvd_gcd_mul_right_right (m n k : α) : gcd m n ∣ gcd m (n * k) :=
gcd_dvd_gcd (dvd_refl _) (dvd_mul_right _ _)
end gcd
section lcm
lemma lcm_dvd_iff {a b c : α} : lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c :=
classical.by_cases
(assume : a = 0 ∨ b = 0, by rcases this with rfl | rfl;
simp only [iff_def, lcm_zero_left, lcm_zero_right, zero_dvd_iff, dvd_zero,
eq_self_iff_true, and_true, imp_true_iff] {contextual:=tt})
(assume this : ¬ (a = 0 ∨ b = 0),
let ⟨h1, h2⟩ := not_or_distrib.1 this in
have h : gcd a b ≠ 0, from λ H, h1 ((gcd_eq_zero_iff _ _).1 H).1,
by rw [← mul_dvd_mul_iff_left h, gcd_mul_lcm, normalize_dvd_iff, ← dvd_normalize_iff,
normalize_mul, normalize_gcd, ← gcd_mul_right, dvd_gcd_iff,
mul_comm b c, mul_dvd_mul_iff_left h1, mul_dvd_mul_iff_right h2, and_comm])
lemma dvd_lcm_left (a b : α) : a ∣ lcm a b := (lcm_dvd_iff.1 (dvd_refl _)).1
lemma dvd_lcm_right (a b : α) : b ∣ lcm a b := (lcm_dvd_iff.1 (dvd_refl _)).2
lemma lcm_dvd {a b c : α} (hab : a ∣ b) (hcb : c ∣ b) : lcm a c ∣ b :=
lcm_dvd_iff.2 ⟨hab, hcb⟩
@[simp] theorem lcm_eq_zero_iff (a b : α) : lcm a b = 0 ↔ a = 0 ∨ b = 0 :=
iff.intro
(assume h : lcm a b = 0,
have normalize (a * b) = 0,
by rw [← gcd_mul_lcm _ _, h, mul_zero],
by simpa only [normalize_eq_zero, mul_eq_zero, units.ne_zero, or_false])
(by rintro (rfl | rfl); [apply lcm_zero_left, apply lcm_zero_right])
@[simp] lemma normalize_lcm (a b : α) : normalize (lcm a b) = lcm a b :=
classical.by_cases (assume : lcm a b = 0, by rw [this, normalize_zero]) $
assume h_lcm : lcm a b ≠ 0,
have h1 : gcd a b ≠ 0, from mt (by rw [gcd_eq_zero_iff, lcm_eq_zero_iff];
rintros ⟨rfl, rfl⟩; left; refl) h_lcm,
have h2 : normalize (gcd a b * lcm a b) = gcd a b * lcm a b,
by rw [gcd_mul_lcm, normalize_idem],
by simpa only [normalize_mul, normalize_gcd, one_mul, domain.mul_left_inj h1] using h2
theorem lcm_comm (a b : α) : lcm a b = lcm b a :=
dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
(lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
(lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
theorem lcm_assoc (m n k : α) : lcm (lcm m n) k = lcm m (lcm n k) :=
dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
(lcm_dvd
(lcm_dvd (dvd_lcm_left _ _) (dvd.trans (dvd_lcm_left _ _) (dvd_lcm_right _ _)))
(dvd.trans (dvd_lcm_right _ _) (dvd_lcm_right _ _)))
(lcm_dvd
(dvd.trans (dvd_lcm_left _ _) (dvd_lcm_left _ _))
(lcm_dvd (dvd.trans (dvd_lcm_right _ _) (dvd_lcm_left _ _)) (dvd_lcm_right _ _)))
instance : is_commutative α lcm := ⟨lcm_comm⟩
instance : is_associative α lcm := ⟨lcm_assoc⟩
lemma lcm_eq_normalize {a b c : α} (habc : lcm a b ∣ c) (hcab : c ∣ lcm a b) :
lcm a b = normalize c :=
normalize_lcm a b ▸ normalize_eq_normalize habc hcab
theorem lcm_dvd_lcm {a b c d : α} (hab : a ∣ b) (hcd : c ∣ d) : lcm a c ∣ lcm b d :=
lcm_dvd (dvd.trans hab (dvd_lcm_left _ _)) (dvd.trans hcd (dvd_lcm_right _ _))
@[simp] theorem lcm_units_coe_left (u : units α) (a : α) : lcm ↑u a = normalize a :=
lcm_eq_normalize (lcm_dvd (units.coe_dvd _ _) (dvd_refl _)) (dvd_lcm_right _ _)
@[simp] theorem lcm_units_coe_right (a : α) (u : units α) : lcm a ↑u = normalize a :=
(lcm_comm a u).trans $ lcm_units_coe_left _ _
@[simp] theorem lcm_one_left (a : α) : lcm 1 a = normalize a :=
lcm_units_coe_left 1 a
@[simp] theorem lcm_one_right (a : α) : lcm a 1 = normalize a :=
lcm_units_coe_right a 1
@[simp] theorem lcm_same (a : α) : lcm a a = normalize a :=
lcm_eq_normalize (lcm_dvd (dvd_refl _) (dvd_refl _)) (dvd_lcm_left _ _)
@[simp] theorem lcm_eq_one_iff (a b : α) : lcm a b = 1 ↔ a ∣ 1 ∧ b ∣ 1 :=
iff.intro
(assume eq, eq ▸ ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩)
(assume ⟨⟨c, hc⟩, ⟨d, hd⟩⟩,
show lcm (units.mk_of_mul_eq_one a c hc.symm : α) (units.mk_of_mul_eq_one b d hd.symm) = 1,
by rw [lcm_units_coe_left, normalize_coe_units])
@[simp] theorem lcm_mul_left (a b c : α) : lcm (a * b) (a * c) = normalize a * lcm b c :=
classical.by_cases (by rintro rfl; simp only [zero_mul, lcm_zero_left, normalize_zero]) $ assume ha : a ≠ 0,
suffices lcm (a * b) (a * c) = normalize (a * lcm b c),
by simpa only [normalize_mul, normalize_lcm],
have a ∣ lcm (a * b) (a * c), from dvd.trans (dvd_mul_right _ _) (dvd_lcm_left _ _),
let ⟨d, eq⟩ := this in
lcm_eq_normalize
(lcm_dvd (mul_dvd_mul_left a (dvd_lcm_left _ _)) (mul_dvd_mul_left a (dvd_lcm_right _ _)))
(eq.symm ▸ (mul_dvd_mul_left a $ lcm_dvd
((mul_dvd_mul_iff_left ha).1 $ eq ▸ dvd_lcm_left _ _)
((mul_dvd_mul_iff_left ha).1 $ eq ▸ dvd_lcm_right _ _)))
@[simp] theorem lcm_mul_right (a b c : α) : lcm (b * a) (c * a) = lcm b c * normalize a :=
by simp only [mul_comm, lcm_mul_left]
theorem lcm_eq_left_iff (a b : α) (h : normalize a = a) : lcm a b = a ↔ b ∣ a :=
iff.intro (assume eq, eq ▸ dvd_lcm_right _ _) $
assume hab, dvd_antisymm_of_normalize_eq (normalize_lcm _ _) h (lcm_dvd (dvd_refl a) hab) (dvd_lcm_left _ _)
theorem lcm_eq_right_iff (a b : α) (h : normalize b = b) : lcm a b = b ↔ a ∣ b :=
by simpa only [lcm_comm b a] using lcm_eq_left_iff b a h
theorem lcm_dvd_lcm_mul_left (m n k : α) : lcm m n ∣ lcm (k * m) n :=
lcm_dvd_lcm (dvd_mul_left _ _) (dvd_refl _)
theorem lcm_dvd_lcm_mul_right (m n k : α) : lcm m n ∣ lcm (m * k) n :=
lcm_dvd_lcm (dvd_mul_right _ _) (dvd_refl _)
theorem lcm_dvd_lcm_mul_left_right (m n k : α) : lcm m n ∣ lcm m (k * n) :=
lcm_dvd_lcm (dvd_refl _) (dvd_mul_left _ _)
theorem lcm_dvd_lcm_mul_right_right (m n k : α) : lcm m n ∣ lcm m (n * k) :=
lcm_dvd_lcm (dvd_refl _) (dvd_mul_right _ _)
end lcm
end gcd_domain
namespace int
section normalization_domain
instance : normalization_domain ℤ :=
{ norm_unit := λa:ℤ, if 0 ≤ a then 1 else -1,
norm_unit_zero := if_pos (le_refl _),
norm_unit_mul := assume a b hna hnb,
begin
by_cases ha : 0 ≤ a; by_cases hb : 0 ≤ b; simp [ha, hb],
exact if_pos (mul_nonneg ha hb),
exact if_neg (assume h, hb $ nonneg_of_mul_nonneg_left h $ lt_of_le_of_ne ha hna.symm),
exact if_neg (assume h, ha $ nonneg_of_mul_nonneg_right h $ lt_of_le_of_ne hb hnb.symm),
exact if_pos (mul_nonneg_of_nonpos_of_nonpos (le_of_not_ge ha) (le_of_not_ge hb))
end,
norm_unit_coe_units := assume u, (units_eq_one_or u).elim
(assume eq, eq.symm ▸ if_pos zero_le_one)
(assume eq, eq.symm ▸ if_neg (not_le_of_gt $ show (-1:ℤ) < 0, by simp [@neg_lt ℤ _ 1 0])),
.. (infer_instance : integral_domain ℤ) }
lemma normalize_of_nonneg {z : ℤ} (h : 0 ≤ z) : normalize z = z :=
show z * ↑(ite _ _ _) = z, by rw [if_pos h, units.coe_one, mul_one]
lemma normalize_of_neg {z : ℤ} (h : z < 0) : normalize z = -z :=
show z * ↑(ite _ _ _) = -z, by rw [if_neg (not_le_of_gt h), units.coe_neg, units.coe_one, mul_neg_one]
lemma normalize_coe_nat (n : ℕ) : normalize (n : ℤ) = n :=
normalize_of_nonneg (coe_nat_le_coe_nat_of_le $ nat.zero_le n)
theorem coe_nat_abs_eq_normalize (z : ℤ) : (z.nat_abs : ℤ) = normalize z :=
begin
by_cases 0 ≤ z,
{ simp [nat_abs_of_nonneg h, normalize_of_nonneg h] },
{ simp [of_nat_nat_abs_of_nonpos (le_of_not_ge h), normalize_of_neg (lt_of_not_ge h)] }
end
end normalization_domain
/-- ℤ specific version of least common multiple. -/
def lcm (i j : ℤ) : ℕ := nat.lcm (nat_abs i) (nat_abs j)
theorem lcm_def (i j : ℤ) : lcm i j = nat.lcm (nat_abs i) (nat_abs j) := rfl
section gcd_domain
theorem gcd_dvd_left (i j : ℤ) : (gcd i j : ℤ) ∣ i :=
dvd_nat_abs.mp $ coe_nat_dvd.mpr $ nat.gcd_dvd_left _ _
theorem gcd_dvd_right (i j : ℤ) : (gcd i j : ℤ) ∣ j :=
dvd_nat_abs.mp $ coe_nat_dvd.mpr $ nat.gcd_dvd_right _ _
theorem dvd_gcd {i j k : ℤ} (h1 : k ∣ i) (h2 : k ∣ j) : k ∣ gcd i j :=
nat_abs_dvd.1 $ coe_nat_dvd.2 $ nat.dvd_gcd (nat_abs_dvd_abs_iff.2 h1) (nat_abs_dvd_abs_iff.2 h2)
theorem gcd_mul_lcm (i j : ℤ) : gcd i j * lcm i j = nat_abs (i * j) :=
by rw [int.gcd, int.lcm, nat.gcd_mul_lcm, nat_abs_mul]
instance : gcd_domain ℤ :=
{ gcd := λa b, int.gcd a b,
lcm := λa b, int.lcm a b,
gcd_dvd_left := assume a b, int.gcd_dvd_left _ _,
gcd_dvd_right := assume a b, int.gcd_dvd_right _ _,
dvd_gcd := assume a b c, dvd_gcd,
normalize_gcd := assume a b, normalize_coe_nat _,
gcd_mul_lcm := by intros; rw [← int.coe_nat_mul, gcd_mul_lcm, coe_nat_abs_eq_normalize],
lcm_zero_left := assume a, coe_nat_eq_zero.2 $ nat.lcm_zero_left _,
lcm_zero_right := assume a, coe_nat_eq_zero.2 $ nat.lcm_zero_right _,
.. int.normalization_domain }
lemma coe_gcd (i j : ℤ) : ↑(int.gcd i j) = gcd_domain.gcd i j := rfl
lemma coe_lcm (i j : ℤ) : ↑(int.lcm i j) = gcd_domain.lcm i j := rfl
lemma nat_abs_gcd (i j : ℤ) : nat_abs (gcd_domain.gcd i j) = int.gcd i j := rfl
lemma nat_abs_lcm (i j : ℤ) : nat_abs (gcd_domain.lcm i j) = int.lcm i j := rfl
end gcd_domain
theorem gcd_comm (i j : ℤ) : gcd i j = gcd j i := nat.gcd_comm _ _
theorem gcd_assoc (i j k : ℤ) : gcd (gcd i j) k = gcd i (gcd j k) := nat.gcd_assoc _ _ _
@[simp] theorem gcd_self (i : ℤ) : gcd i i = nat_abs i := by simp [gcd]
@[simp] theorem gcd_zero_left (i : ℤ) : gcd 0 i = nat_abs i := by simp [gcd]
@[simp] theorem gcd_zero_right (i : ℤ) : gcd i 0 = nat_abs i := by simp [gcd]
@[simp] theorem gcd_one_left (i : ℤ) : gcd 1 i = 1 := nat.gcd_one_left _
@[simp] theorem gcd_one_right (i : ℤ) : gcd i 1 = 1 := nat.gcd_one_right _
theorem gcd_mul_left (i j k : ℤ) : gcd (i * j) (i * k) = nat_abs i * gcd j k :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_gcd, coe_nat_abs_eq_normalize]
theorem gcd_mul_right (i j k : ℤ) : gcd (i * j) (k * j) = gcd i k * nat_abs j :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_gcd, coe_nat_abs_eq_normalize]
theorem gcd_pos_of_non_zero_left {i : ℤ} (j : ℤ) (i_non_zero : i ≠ 0) : gcd i j > 0 :=
nat.gcd_pos_of_pos_left (nat_abs j) (nat_abs_pos_of_ne_zero i_non_zero)
theorem gcd_pos_of_non_zero_right (i : ℤ) {j : ℤ} (j_non_zero : j ≠ 0) : gcd i j > 0 :=
nat.gcd_pos_of_pos_right (nat_abs i) (nat_abs_pos_of_ne_zero j_non_zero)
theorem gcd_eq_zero_iff {i j : ℤ} : gcd i j = 0 ↔ i = 0 ∧ j = 0 :=
by rw [← int.coe_nat_eq_coe_nat_iff, int.coe_nat_zero, coe_gcd, gcd_eq_zero_iff]
theorem gcd_div {i j k : ℤ} (H1 : k ∣ i) (H2 : k ∣ j) :
gcd (i / k) (j / k) = gcd i j / nat_abs k :=
by rw [gcd, nat_abs_div i k H1, nat_abs_div j k H2];
exact nat.gcd_div (nat_abs_dvd_abs_iff.mpr H1) (nat_abs_dvd_abs_iff.mpr H2)
theorem gcd_dvd_gcd_of_dvd_left {i k : ℤ} (j : ℤ) (H : i ∣ k) : gcd i j ∣ gcd k j :=
int.coe_nat_dvd.1 $ dvd_gcd (dvd.trans (gcd_dvd_left i j) H) (gcd_dvd_right i j)
theorem gcd_dvd_gcd_of_dvd_right {i k : ℤ} (j : ℤ) (H : i ∣ k) : gcd j i ∣ gcd j k :=
int.coe_nat_dvd.1 $ dvd_gcd (gcd_dvd_left j i) (dvd.trans (gcd_dvd_right j i) H)
theorem gcd_dvd_gcd_mul_left (i j k : ℤ) : gcd i j ∣ gcd (k * i) j :=
gcd_dvd_gcd_of_dvd_left _ (dvd_mul_left _ _)
theorem gcd_dvd_gcd_mul_right (i j k : ℤ) : gcd i j ∣ gcd (i * k) j :=
gcd_dvd_gcd_of_dvd_left _ (dvd_mul_right _ _)
theorem gcd_dvd_gcd_mul_left_right (i j k : ℤ) : gcd i j ∣ gcd i (k * j) :=
gcd_dvd_gcd_of_dvd_right _ (dvd_mul_left _ _)
theorem gcd_dvd_gcd_mul_right_right (i j k : ℤ) : gcd i j ∣ gcd i (j * k) :=
gcd_dvd_gcd_of_dvd_right _ (dvd_mul_right _ _)
theorem gcd_eq_left {i j : ℤ} (H : i ∣ j) : gcd i j = nat_abs i :=
nat.dvd_antisymm (by unfold gcd; exact nat.gcd_dvd_left _ _)
(by unfold gcd; exact nat.dvd_gcd (dvd_refl _) (nat_abs_dvd_abs_iff.mpr H))
theorem gcd_eq_right {i j : ℤ} (H : j ∣ i) : gcd i j = nat_abs j :=
by rw [gcd_comm, gcd_eq_left H]
/- lcm -/
theorem lcm_comm (i j : ℤ) : lcm i j = lcm j i :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, lcm_comm]
theorem lcm_assoc (i j k : ℤ) : lcm (lcm i j) k = lcm i (lcm j k) :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, lcm_assoc]
@[simp] theorem lcm_zero_left (i : ℤ) : lcm 0 i = 0 :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm]
@[simp] theorem lcm_zero_right (i : ℤ) : lcm i 0 = 0 :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm]
@[simp] theorem lcm_one_left (i : ℤ) : lcm 1 i = nat_abs i :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, coe_nat_abs_eq_normalize]
@[simp] theorem lcm_one_right (i : ℤ) : lcm i 1 = nat_abs i :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, coe_nat_abs_eq_normalize]
@[simp] theorem lcm_self (i : ℤ) : lcm i i = nat_abs i :=
by simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, coe_nat_abs_eq_normalize]
theorem dvd_lcm_left (i j : ℤ) : i ∣ lcm i j :=
by rw [coe_lcm]; exact dvd_lcm_left _ _
theorem dvd_lcm_right (i j : ℤ) : j ∣ lcm i j :=
by rw [coe_lcm]; exact dvd_lcm_right _ _
theorem lcm_dvd {i j k : ℤ} : i ∣ k → j ∣ k → (lcm i j : ℤ) ∣ k :=
by rw [coe_lcm]; exact lcm_dvd
end int
theorem irreducible_iff_nat_prime : ∀(a : ℕ), irreducible a ↔ nat.prime a
| 0 := by simp [nat.not_prime_zero]
| 1 := by simp [nat.prime, one_lt_two]
| (n + 2) :=
have h₁ : ¬n + 2 = 1, from dec_trivial,
begin
simp [h₁, nat.prime, irreducible, (≥), nat.le_add_left 2 n, (∣)],
refine forall_congr (assume a, forall_congr $ assume b, forall_congr $ assume hab, _),
by_cases a = 1; simp [h],
split,
{ assume hb, simpa [hb] using hab.symm },
{ assume ha, subst ha,
have : n + 2 > 0, from dec_trivial,
refine nat.eq_of_mul_eq_mul_left this _,
rw [← hab, mul_one] }
end
lemma nat.prime_iff_prime {p : ℕ} : p.prime ↔ _root_.prime (p : ℕ) :=
⟨λ hp, ⟨nat.pos_iff_ne_zero.1 hp.pos, mt is_unit_iff_dvd_one.1 hp.not_dvd_one,
λ a b, hp.dvd_mul.1⟩,
λ hp, ⟨nat.one_lt_iff_ne_zero_and_ne_one.2 ⟨hp.1, λ h1, hp.2.1 $ h1.symm ▸ is_unit_one⟩,
λ a h, let ⟨b, hab⟩ := h in
(hp.2.2 a b (hab ▸ dvd_refl _)).elim
(λ ha, or.inr (nat.dvd_antisymm h ha))
(λ hb, or.inl (have hpb : p = b, from nat.dvd_antisymm hb
(hab.symm ▸ dvd_mul_left _ _),
(nat.mul_left_inj (show 0 < p, from
nat.pos_of_ne_zero hp.1)).1 $
by rw [hpb, mul_comm, ← hab, hpb, mul_one]))⟩⟩
lemma nat.prime_iff_prime_int {p : ℕ} : p.prime ↔ _root_.prime (p : ℤ) :=
⟨λ hp, ⟨int.coe_nat_ne_zero_iff_pos.2 hp.pos, mt is_unit_int.1 hp.ne_one,
λ a b h, by rw [← int.dvd_nat_abs, int.coe_nat_dvd, int.nat_abs_mul, hp.dvd_mul] at h;
rwa [← int.dvd_nat_abs, int.coe_nat_dvd, ← int.dvd_nat_abs, int.coe_nat_dvd]⟩,
λ hp, nat.prime_iff_prime.2 ⟨int.coe_nat_ne_zero.1 hp.1,
mt is_unit_nat.1 $ λ h, by simpa [h, not_prime_one] using hp,
λ a b, by simpa only [int.coe_nat_dvd, (int.coe_nat_mul _ _).symm] using hp.2.2 a b⟩⟩
def associates_int_equiv_nat : (associates ℤ) ≃ ℕ :=
begin
refine ⟨λz, z.out.nat_abs, λn, associates.mk n, _, _⟩,
{ refine (assume a, quotient.induction_on' a $ assume a,
associates.mk_eq_mk_iff_associated.2 $ associated.symm $ ⟨norm_unit a, _⟩),
show normalize a = int.nat_abs (normalize a),
rw [int.coe_nat_abs_eq_normalize, normalize_idem] },
{ assume n, show int.nat_abs (normalize n) = n,
rw [← int.coe_nat_abs_eq_normalize, int.nat_abs_of_nat, int.nat_abs_of_nat] }
end
|
f6cf19fa076228e76e6543413729ddb3c9e11253 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/category_theory/single_obj.lean | 1279f34f75d2bffb93e7c0b5d47d0938ff0e8287 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,969 | lean | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.category_theory.endomorphism
import Mathlib.category_theory.category.Cat
import Mathlib.algebra.category.Mon.basic
import Mathlib.PostPort
universes u v w u_1
namespace Mathlib
/-!
# Single-object category
Single object category with a given monoid of endomorphisms. It is defined to facilitate transfering
some definitions and lemmas (e.g., conjugacy etc.) from category theory to monoids and groups.
## Main definitions
Given a type `α` with a monoid structure, `single_obj α` is `unit` type with `category` structure
such that `End (single_obj α).star` is the monoid `α`. This can be extended to a functor `Mon ⥤
Cat`.
If `α` is a group, then `single_obj α` is a groupoid.
An element `x : α` can be reinterpreted as an element of `End (single_obj.star α)` using
`single_obj.to_End`.
## Implementation notes
- `category_struct.comp` on `End (single_obj.star α)` is `flip (*)`, not `(*)`. This way
multiplication on `End` agrees with the multiplication on `α`.
- By default, Lean puts instances into `category_theory` namespace instead of
`category_theory.single_obj`, so we give all names explicitly.
-/
namespace category_theory
/-- Type tag on `unit` used to define single-object categories and groupoids. -/
def single_obj (α : Type u) :=
Unit
namespace single_obj
/-- One and `flip (*)` become `id` and `comp` for morphisms of the single object category. -/
protected instance category_struct (α : Type u) [HasOne α] [Mul α] : category_struct (single_obj α) :=
category_struct.mk (fun (_x : single_obj α) => 1)
fun (_x _x_1 _x_2 : single_obj α) (x : _x ⟶ _x_1) (y : _x_1 ⟶ _x_2) => y * x
/-- Monoid laws become category laws for the single object category. -/
protected instance category (α : Type u) [monoid α] : category (single_obj α) :=
category.mk
/--
Groupoid structure on `single_obj α`.
See https://stacks.math.columbia.edu/tag/0019.
-/
protected instance groupoid (α : Type u) [group α] : groupoid (single_obj α) :=
groupoid.mk fun (_x _x_1 : single_obj α) (x : _x ⟶ _x_1) => x⁻¹
/-- The single object in `single_obj α`. -/
protected def star (α : Type u) : single_obj α :=
Unit.unit
/-- The endomorphisms monoid of the only object in `single_obj α` is equivalent to the original
monoid α. -/
def to_End (α : Type u) [monoid α] : α ≃* End (single_obj.star α) :=
mul_equiv.mk (equiv.to_fun (equiv.refl α)) (equiv.inv_fun (equiv.refl α)) sorry sorry sorry
theorem to_End_def (α : Type u) [monoid α] (x : α) : coe_fn (to_End α) x = x :=
rfl
/-- There is a 1-1 correspondence between monoid homomorphisms `α → β` and functors between the
corresponding single-object categories. It means that `single_obj` is a fully faithful
functor.
See https://stacks.math.columbia.edu/tag/001F --
although we do not characterize when the functor is full or faithful.
-/
def map_hom (α : Type u) (β : Type v) [monoid α] [monoid β] : (α →* β) ≃ single_obj α ⥤ single_obj β :=
equiv.mk (fun (f : α →* β) => functor.mk id fun (_x _x : single_obj α) => ⇑f)
(fun (f : single_obj α ⥤ single_obj β) => monoid_hom.mk (functor.map f) sorry sorry) sorry sorry
theorem map_hom_id (α : Type u) [monoid α] : coe_fn (map_hom α α) (monoid_hom.id α) = 𝟭 :=
rfl
theorem map_hom_comp {α : Type u} {β : Type v} [monoid α] [monoid β] (f : α →* β) {γ : Type w} [monoid γ] (g : β →* γ) : coe_fn (map_hom α γ) (monoid_hom.comp g f) = coe_fn (map_hom α β) f ⋙ coe_fn (map_hom β γ) g :=
rfl
end single_obj
end category_theory
namespace monoid_hom
/-- Reinterpret a monoid homomorphism `f : α → β` as a functor `(single_obj α) ⥤ (single_obj β)`.
See also `category_theory.single_obj.map_hom` for an equivalence between these types. -/
def to_functor {α : Type u} {β : Type v} [monoid α] [monoid β] (f : α →* β) : category_theory.single_obj α ⥤ category_theory.single_obj β :=
coe_fn (category_theory.single_obj.map_hom α β) f
@[simp] theorem id_to_functor (α : Type u) [monoid α] : to_functor (id α) = 𝟭 :=
rfl
@[simp] theorem comp_to_functor {α : Type u} {β : Type v} [monoid α] [monoid β] (f : α →* β) {γ : Type w} [monoid γ] (g : β →* γ) : to_functor (comp g f) = to_functor f ⋙ to_functor g :=
rfl
end monoid_hom
namespace units
/--
The units in a monoid are (multiplicatively) equivalent to
the automorphisms of `star` when we think of the monoid as a single-object category. -/
def to_Aut (α : Type u) [monoid α] : units α ≃* category_theory.Aut (category_theory.single_obj.star α) :=
mul_equiv.trans (map_equiv (category_theory.single_obj.to_End α))
(category_theory.Aut.units_End_equiv_Aut (category_theory.single_obj.star α))
@[simp] theorem to_Aut_hom (α : Type u) [monoid α] (x : units α) : category_theory.iso.hom (coe_fn (to_Aut α) x) = coe_fn (category_theory.single_obj.to_End α) ↑x :=
rfl
@[simp] theorem to_Aut_inv (α : Type u) [monoid α] (x : units α) : category_theory.iso.inv (coe_fn (to_Aut α) x) = coe_fn (category_theory.single_obj.to_End α) ↑(x⁻¹) :=
rfl
end units
namespace Mon
/-- The fully faithful functor from `Mon` to `Cat`. -/
def to_Cat : Mon ⥤ category_theory.Cat :=
category_theory.functor.mk (fun (x : Mon) => category_theory.Cat.of (category_theory.single_obj ↥x))
fun (x y : Mon) (f : x ⟶ y) => coe_fn (category_theory.single_obj.map_hom ↥x ↥y) f
protected instance to_Cat_full : category_theory.full to_Cat :=
category_theory.full.mk fun (x y : Mon) => equiv.inv_fun (category_theory.single_obj.map_hom ↥x ↥y)
protected instance to_Cat_faithful : category_theory.faithful to_Cat :=
category_theory.faithful.mk
|
5f2a118a6d076d5a2c8c053b3b0d4242033110b8 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/lift.lean | 147c7fdf82a3a94293ce54dfec9bda6eb780a7fc | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 449 | lean | open nat
namespace test
inductive {u} lift (A : Type u) : Type (u+1)
| up : A → lift
namespace lift
definition down {A : Type} (a : lift A) : A :=
lift.rec (λ a, a) a
theorem down_up {A : Type} (a : A) : down (up a) = a :=
rfl
theorem up_down {A : Type} (a' : lift A) : up (down a') = a' :=
lift.cases_on a' (λ a, rfl)
end lift
set_option pp.universes true
#check nat
#check lift nat
open lift
definition one1 : lift nat := up 1
end test
|
7fb31649d08c2382830de33c0c6be6c3a2093d59 | c56b090bd37dca8992124f0b09c22340ef7e27bc | /src/lecture6.lean | 276e66d86d69f494cf9d5fd01d1409827228bdcd | [] | no_license | stjordanis/math135 | df113ed7ae9f81b27316cc5b000f88b385391a81 | e270f3a9cae435c066c0d2574f03a8adbe40b7b5 | refs/heads/master | 1,677,069,951,721 | 1,611,905,122,000 | 1,611,905,122,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 904 | lean | import tactic
import data.nat.parity
import data.set
import data.set.basic
namespace lecture6
open set
example : {n | 4 ∣ n + 1} ⊆ {k | odd k} :=
begin
rw set.subset_def,
intros x hx,
simp at hx,
simp,
cases nat.even_or_odd x,
{
apply nat.even_succ.symm.mpr,
refine dvd_of_mul_left_dvd _,
use 2,
exact hx,
},
exact nat.odd_iff_not_even.mp h,
end
example : Π {α : Type} (s t : set α), s = t ↔ s ∪ t = s ∩ t :=
begin
intros a s t,
split,
{
intro s_eq_t,
rw [s_eq_t, union_self, inter_self],
},
intro union_eq_intersect,
apply le_antisymm,
{
calc s ⊆ s ∪ t : subset_union_left s t
... = s ∩ t : union_eq_intersect
... ⊆ t : inter_subset_right s t,
},
calc t ⊆ s ∪ t : subset_union_right s t
... = s ∩ t : union_eq_intersect
... ⊆ s : inter_subset_left s t,
end
end lecture6
|
c83bd4c4608726de3ec6d629cd60640983d35a30 | 4bcaca5dc83d49803f72b7b5920b75b6e7d9de2d | /src/Lean/Elab/Command.lean | df6e42f5b91d91336c947e8b0c71d89c86cd21bb | [
"Apache-2.0"
] | permissive | subfish-zhou/leanprover-zh_CN.github.io | 30b9fba9bd790720bd95764e61ae796697d2f603 | 8b2985d4a3d458ceda9361ac454c28168d920d3f | refs/heads/master | 1,689,709,967,820 | 1,632,503,056,000 | 1,632,503,056,000 | 409,962,097 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 18,381 | lean | /-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Parser.Command
import Lean.ResolveName
import Lean.Meta.Reduce
import Lean.Elab.Log
import Lean.Elab.Term
import Lean.Elab.Binders
import Lean.Elab.SyntheticMVars
import Lean.Elab.DeclModifiers
import Lean.Elab.InfoTree
import Lean.Elab.SetOption
namespace Lean.Elab.Command
structure Scope where
header : String
opts : Options := {}
currNamespace : Name := Name.anonymous
openDecls : List OpenDecl := []
levelNames : List Name := []
/-- section variables as `bracketedBinder`s -/
varDecls : Array Syntax := #[]
/-- Globally unique internal identifiers for the `varDecls` -/
varUIds : Array Name := #[]
deriving Inhabited
structure State where
env : Environment
messages : MessageLog := {}
scopes : List Scope := [{ header := "" }]
nextMacroScope : Nat := firstFrontendMacroScope + 1
maxRecDepth : Nat
nextInstIdx : Nat := 1 -- for generating anonymous instance names
ngen : NameGenerator := {}
infoState : InfoState := {}
traceState : TraceState := {}
deriving Inhabited
structure Context where
fileName : String
fileMap : FileMap
currRecDepth : Nat := 0
cmdPos : String.Pos := 0
macroStack : MacroStack := []
currMacroScope : MacroScope := firstFrontendMacroScope
ref : Syntax := Syntax.missing
abbrev CommandElabCoreM (ε) := ReaderT Context $ StateRefT State $ EIO ε
abbrev CommandElabM := CommandElabCoreM Exception
abbrev CommandElab := Syntax → CommandElabM Unit
abbrev Linter := Syntax → CommandElabM Unit
-- Make the compiler generate specialized `pure`/`bind` so we do not have to optimize through the
-- whole monad stack at every use site. May eventually be covered by `deriving`.
instance : Monad CommandElabM := let i := inferInstanceAs (Monad CommandElabM); { pure := i.pure, bind := i.bind }
def mkState (env : Environment) (messages : MessageLog := {}) (opts : Options := {}) : State := {
env := env
messages := messages
scopes := [{ header := "", opts := opts }]
maxRecDepth := maxRecDepth.get opts
}
/- Linters should be loadable as plugins, so store in a global IO ref instead of an attribute managed by the
environment (which only contains `import`ed objects). -/
builtin_initialize lintersRef : IO.Ref (Array Linter) ← IO.mkRef #[]
def addLinter (l : Linter) : IO Unit := do
let ls ← lintersRef.get
lintersRef.set (ls.push l)
instance : MonadInfoTree CommandElabM where
getInfoState := return (← get).infoState
modifyInfoState f := modify fun s => { s with infoState := f s.infoState }
instance : MonadEnv CommandElabM where
getEnv := do pure (← get).env
modifyEnv f := modify fun s => { s with env := f s.env }
instance : MonadOptions CommandElabM where
getOptions := do pure (← get).scopes.head!.opts
protected def getRef : CommandElabM Syntax :=
return (← read).ref
instance : AddMessageContext CommandElabM where
addMessageContext := addMessageContextPartial
instance : MonadRef CommandElabM where
getRef := Command.getRef
withRef ref x := withReader (fun ctx => { ctx with ref := ref }) x
instance : MonadTrace CommandElabM where
getTraceState := return (← get).traceState
modifyTraceState f := modify fun s => { s with traceState := f s.traceState }
instance : AddErrorMessageContext CommandElabM where
add ref msg := do
let ctx ← read
let ref := getBetterRef ref ctx.macroStack
let msg ← addMessageContext msg
let msg ← addMacroStack msg ctx.macroStack
return (ref, msg)
def mkMessageAux (ctx : Context) (ref : Syntax) (msgData : MessageData) (severity : MessageSeverity) : Message :=
mkMessageCore ctx.fileName ctx.fileMap msgData severity (ref.getPos?.getD ctx.cmdPos)
private def mkCoreContext (ctx : Context) (s : State) (heartbeats : Nat) : Core.Context :=
let scope := s.scopes.head!
{ options := scope.opts
currRecDepth := ctx.currRecDepth
maxRecDepth := s.maxRecDepth
ref := ctx.ref
currNamespace := scope.currNamespace
openDecls := scope.openDecls
initHeartbeats := heartbeats }
def liftCoreM {α} (x : CoreM α) : CommandElabM α := do
let s ← get
let ctx ← read
let heartbeats ← IO.getNumHeartbeats (ε := Exception)
let Eα := Except Exception α
let x : CoreM Eα := try let a ← x; pure <| Except.ok a catch ex => pure <| Except.error ex
let x : EIO Exception (Eα × Core.State) := (ReaderT.run x (mkCoreContext ctx s heartbeats)).run { env := s.env, ngen := s.ngen, traceState := s.traceState }
let (ea, coreS) ← liftM x
modify fun s => { s with env := coreS.env, ngen := coreS.ngen, traceState := coreS.traceState }
match ea with
| Except.ok a => pure a
| Except.error e => throw e
private def ioErrorToMessage (ctx : Context) (ref : Syntax) (err : IO.Error) : Message :=
let ref := getBetterRef ref ctx.macroStack
mkMessageAux ctx ref (toString err) MessageSeverity.error
@[inline] def liftEIO {α} (x : EIO Exception α) : CommandElabM α := liftM x
@[inline] def liftIO {α} (x : IO α) : CommandElabM α := do
let ctx ← read
IO.toEIO (fun (ex : IO.Error) => Exception.error ctx.ref ex.toString) x
instance : MonadLiftT IO CommandElabM where
monadLift := liftIO
def getScope : CommandElabM Scope := do pure (← get).scopes.head!
instance : MonadResolveName CommandElabM where
getCurrNamespace := return (← getScope).currNamespace
getOpenDecls := return (← getScope).openDecls
instance : MonadLog CommandElabM where
getRef := getRef
getFileMap := return (← read).fileMap
getFileName := return (← read).fileName
logMessage msg := do
let currNamespace ← getCurrNamespace
let openDecls ← getOpenDecls
let msg := { msg with data := MessageData.withNamingContext { currNamespace := currNamespace, openDecls := openDecls } msg.data }
modify fun s => { s with messages := s.messages.add msg }
def runLinters (stx : Syntax) : CommandElabM Unit := do
let linters ← lintersRef.get
unless linters.isEmpty do
for linter in linters do
let savedState ← get
try
linter stx
catch ex =>
logException ex
finally
modify fun s => { savedState with messages := s.messages }
protected def getCurrMacroScope : CommandElabM Nat := do pure (← read).currMacroScope
protected def getMainModule : CommandElabM Name := do pure (← getEnv).mainModule
protected def withFreshMacroScope {α} (x : CommandElabM α) : CommandElabM α := do
let fresh ← modifyGet (fun st => (st.nextMacroScope, { st with nextMacroScope := st.nextMacroScope + 1 }))
withReader (fun ctx => { ctx with currMacroScope := fresh }) x
instance : MonadQuotation CommandElabM where
getCurrMacroScope := Command.getCurrMacroScope
getMainModule := Command.getMainModule
withFreshMacroScope := Command.withFreshMacroScope
unsafe def mkCommandElabAttributeUnsafe : IO (KeyedDeclsAttribute CommandElab) :=
mkElabAttribute CommandElab `Lean.Elab.Command.commandElabAttribute `builtinCommandElab `commandElab `Lean.Parser.Command `Lean.Elab.Command.CommandElab "command"
@[implementedBy mkCommandElabAttributeUnsafe]
constant mkCommandElabAttribute : IO (KeyedDeclsAttribute CommandElab)
builtin_initialize commandElabAttribute : KeyedDeclsAttribute CommandElab ← mkCommandElabAttribute
private def addTraceAsMessagesCore (ctx : Context) (log : MessageLog) (traceState : TraceState) : MessageLog :=
traceState.traces.foldl (init := log) fun (log : MessageLog) traceElem =>
let ref := replaceRef traceElem.ref ctx.ref;
let pos := ref.getPos?.getD 0;
log.add (mkMessageCore ctx.fileName ctx.fileMap traceElem.msg MessageSeverity.information pos)
private def addTraceAsMessages : CommandElabM Unit := do
let ctx ← read
modify fun s => { s with
messages := addTraceAsMessagesCore ctx s.messages s.traceState
traceState.traces := {}
}
private def mkInfoTree (elaborator : Name) (stx : Syntax) (trees : Std.PersistentArray InfoTree) : CommandElabM InfoTree := do
let ctx ← read
let s ← get
let scope := s.scopes.head!
let tree := InfoTree.node (Info.ofCommandInfo { elaborator, stx }) trees
return InfoTree.context {
env := s.env, fileMap := ctx.fileMap, mctx := {}, currNamespace := scope.currNamespace, openDecls := scope.openDecls, options := scope.opts
} tree
private def elabCommandUsing (s : State) (stx : Syntax) : List (KeyedDeclsAttribute.AttributeEntry CommandElab) → CommandElabM Unit
| [] => throwError "unexpected syntax{indentD stx}"
| (elabFn::elabFns) =>
catchInternalId unsupportedSyntaxExceptionId
(withInfoTreeContext (mkInfoTree := mkInfoTree elabFn.declName stx) <| elabFn.value stx)
(fun _ => do set s; elabCommandUsing s stx elabFns)
/- Elaborate `x` with `stx` on the macro stack -/
def withMacroExpansion {α} (beforeStx afterStx : Syntax) (x : CommandElabM α) : CommandElabM α :=
withReader (fun ctx => { ctx with macroStack := { before := beforeStx, after := afterStx } :: ctx.macroStack }) x
instance : MonadMacroAdapter CommandElabM where
getCurrMacroScope := getCurrMacroScope
getNextMacroScope := return (← get).nextMacroScope
setNextMacroScope next := modify fun s => { s with nextMacroScope := next }
instance : MonadRecDepth CommandElabM where
withRecDepth d x := withReader (fun ctx => { ctx with currRecDepth := d }) x
getRecDepth := return (← read).currRecDepth
getMaxRecDepth := return (← get).maxRecDepth
register_builtin_option showPartialSyntaxErrors : Bool := {
defValue := false
descr := "show elaboration errors from partial syntax trees (i.e. after parser recovery)"
}
def withLogging (x : CommandElabM Unit) : CommandElabM Unit := do
try
x
catch ex => match ex with
| Exception.error _ _ => logException ex
| Exception.internal id _ =>
if isAbortExceptionId id then
pure ()
else
let idName ← liftIO <| id.getName;
logError m!"internal exception {idName}"
builtin_initialize registerTraceClass `Elab.command
partial def elabCommand (stx : Syntax) : CommandElabM Unit := do
withLogging <| withRef stx <| withIncRecDepth <| withFreshMacroScope do
match stx with
| Syntax.node k args =>
if k == nullKind then
-- list of commands => elaborate in order
-- The parser will only ever return a single command at a time, but syntax quotations can return multiple ones
args.forM elabCommand
else do
trace `Elab.command fun _ => stx;
let s ← get
match (← liftMacroM <| expandMacroImpl? s.env stx) with
| some (decl, stxNew) =>
withInfoTreeContext (mkInfoTree := mkInfoTree decl stx) do
withMacroExpansion stx stxNew do
elabCommand stxNew
| _ =>
match commandElabAttribute.getEntries s.env k with
| [] => throwError "elaboration function for '{k}' has not been implemented"
| elabFns => elabCommandUsing s stx elabFns
| _ => throwError "unexpected command"
/--
`elabCommand` wrapper that should be used for the initial invocation, not for recursive calls after
macro expansion etc.
-/
def elabCommandTopLevel (stx : Syntax) : CommandElabM Unit := withRef stx do
let initMsgs ← modifyGet fun st => (st.messages, { st with messages := {} })
let initInfoTrees ← getResetInfoTrees
withLogging do
runLinters stx
-- We should *not* factor out `elabCommand`'s `withLogging` to here since it would make its error
-- recovery more coarse. In particular, If `c` in `set_option ... in $c` fails, the remaining
-- `end` command of the `in` macro would be skipped and the option would be leaked to the outside!
elabCommand stx
-- note the order: first process current messages & info trees, then add back old messages & trees,
-- then convert new traces to messages
let mut msgs ← (← get).messages
-- `stx.hasMissing` should imply `initMsgs.hasErrors`, but the latter should be cheaper to check in general
if !showPartialSyntaxErrors.get (← getOptions) && initMsgs.hasErrors && stx.hasMissing then
-- discard elaboration errors, except for a few important and unlikely misleading ones, on parse error
msgs := ⟨msgs.msgs.filter fun msg =>
msg.data.hasTag `Elab.synthPlaceholder || msg.data.hasTag `Tactic.unsolvedGoals⟩
for tree in (← getInfoTrees) do
trace[Elab.info] (← tree.format)
modify fun st => { st with
messages := initMsgs ++ msgs
infoState := { st.infoState with trees := initInfoTrees ++ st.infoState.trees }
}
addTraceAsMessages
/-- Adapt a syntax transformation to a regular, command-producing elaborator. -/
def adaptExpander (exp : Syntax → CommandElabM Syntax) : CommandElab := fun stx => do
let stx' ← exp stx
withMacroExpansion stx stx' <| elabCommand stx'
private def getVarDecls (s : State) : Array Syntax :=
s.scopes.head!.varDecls
instance {α} : Inhabited (CommandElabM α) where
default := throw arbitrary
private def mkMetaContext : Meta.Context := {
config := { foApprox := true, ctxApprox := true, quasiPatternApprox := true }
}
def getBracketedBinderIds : Syntax → Array Name
| `(bracketedBinder|($ids* $[: $ty?]? $(annot?)?)) => ids.map Syntax.getId
| `(bracketedBinder|{$ids* $[: $ty?]?}) => ids.map Syntax.getId
| `(bracketedBinder|[$id : $ty]) => #[id.getId]
| `(bracketedBinder|[$ty]) => #[Name.anonymous]
| _ => #[]
private def mkTermContext (ctx : Context) (s : State) (declName? : Option Name) : Term.Context := do
let scope := s.scopes.head!
let mut sectionVars := {}
for id in scope.varDecls.concatMap getBracketedBinderIds, uid in scope.varUIds do
sectionVars := sectionVars.insert id uid
{ macroStack := ctx.macroStack
fileName := ctx.fileName
fileMap := ctx.fileMap
currMacroScope := ctx.currMacroScope
declName? := declName?
sectionVars := sectionVars }
private def mkTermState (scope : Scope) (s : State) : Term.State := {
messages := {}
levelNames := scope.levelNames
infoState.enabled := s.infoState.enabled
}
def liftTermElabM {α} (declName? : Option Name) (x : TermElabM α) : CommandElabM α := do
let ctx ← read
let s ← get
let heartbeats ← IO.getNumHeartbeats (ε := Exception)
-- dbg_trace "heartbeats: {heartbeats}"
let scope := s.scopes.head!
-- We execute `x` with an empty message log. Thus, `x` cannot modify/view messages produced by previous commands.
-- This is useful for implementing `runTermElabM` where we use `Term.resetMessageLog`
let x : TermElabM _ := withSaveInfoContext x
let x : MetaM _ := (observing x).run (mkTermContext ctx s declName?) (mkTermState scope s)
let x : CoreM _ := x.run mkMetaContext {}
let x : EIO _ _ := x.run (mkCoreContext ctx s heartbeats) { env := s.env, ngen := s.ngen, nextMacroScope := s.nextMacroScope }
let (((ea, termS), metaS), coreS) ← liftEIO x
modify fun s => { s with
env := coreS.env
messages := addTraceAsMessagesCore ctx (s.messages ++ termS.messages) coreS.traceState
nextMacroScope := coreS.nextMacroScope
ngen := coreS.ngen
infoState.trees := s.infoState.trees.append termS.infoState.trees
}
match ea with
| Except.ok a => pure a
| Except.error ex => throw ex
@[inline] def runTermElabM {α} (declName? : Option Name) (elabFn : Array Expr → TermElabM α) : CommandElabM α := do
let scope ← getScope
liftTermElabM declName? <|
Term.withAutoBoundImplicit <|
Term.elabBinders scope.varDecls fun xs => do
-- We need to synthesize postponed terms because this is a checkpoint for the auto-bound implicit feature
-- If we don't use this checkpoint here, then auto-bound implicits in the postponed terms will not be handled correctly.
Term.synthesizeSyntheticMVarsNoPostponing
let mut sectionFVars := {}
for uid in scope.varUIds, x in xs do
sectionFVars := sectionFVars.insert uid x
withReader ({ · with sectionFVars := sectionFVars }) do
-- We don't want to store messages produced when elaborating `(getVarDecls s)` because they have already been saved when we elaborated the `variable`(s) command.
-- So, we use `Term.resetMessageLog`.
Term.resetMessageLog
let xs ← Term.addAutoBoundImplicits xs
Term.withoutAutoBoundImplicit <| elabFn xs
@[inline] def catchExceptions (x : CommandElabM Unit) : CommandElabCoreM Empty Unit := fun ctx ref =>
EIO.catchExceptions (withLogging x ctx ref) (fun _ => pure ())
private def liftAttrM {α} (x : AttrM α) : CommandElabM α := do
liftCoreM x
def getScopes : CommandElabM (List Scope) := do
pure (← get).scopes
def modifyScope (f : Scope → Scope) : CommandElabM Unit :=
modify fun s => { s with
scopes := match s.scopes with
| h::t => f h :: t
| [] => unreachable!
}
def withScope (f : Scope → Scope) (x : CommandElabM α) : CommandElabM α := do
match (← get).scopes with
| [] => x
| h :: t =>
try
modify fun s => { s with scopes := f h :: t }
x
finally
modify fun s => { s with scopes := h :: t }
def getLevelNames : CommandElabM (List Name) :=
return (← getScope).levelNames
def addUnivLevel (idStx : Syntax) : CommandElabM Unit := withRef idStx do
let id := idStx.getId
let levelNames ← getLevelNames
if levelNames.elem id then
throwAlreadyDeclaredUniverseLevel id
else
modifyScope fun scope => { scope with levelNames := id :: scope.levelNames }
def expandDeclId (declId : Syntax) (modifiers : Modifiers) : CommandElabM ExpandDeclIdResult := do
let currNamespace ← getCurrNamespace
let currLevelNames ← getLevelNames
Lean.Elab.expandDeclId currNamespace currLevelNames declId modifiers
end Elab.Command
export Elab.Command (Linter addLinter)
end Lean
|
c9cbb8417930c5a55459c37d9f94f808fe693f32 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/data/sym/basic.lean | 556b140d654e5ccee7c34491c462d3e849180729 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 17,929 | lean | /-
Copyright (c) 2020 Kyle Miller All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import data.multiset.basic
import data.vector.basic
import data.setoid.basic
import tactic.apply_fun
/-!
# Symmetric powers
This file defines symmetric powers of a type. The nth symmetric power
consists of homogeneous n-tuples modulo permutations by the symmetric
group.
The special case of 2-tuples is called the symmetric square, which is
addressed in more detail in `data.sym.sym2`.
TODO: This was created as supporting material for `sym2`; it
needs a fleshed-out interface.
## Tags
symmetric powers
-/
open function
/--
The nth symmetric power is n-tuples up to permutation. We define it
as a subtype of `multiset` since these are well developed in the
library. We also give a definition `sym.sym'` in terms of vectors, and we
show these are equivalent in `sym.sym_equiv_sym'`.
-/
def sym (α : Type*) (n : ℕ) := {s : multiset α // s.card = n}
instance sym.has_coe (α : Type*) (n : ℕ) : has_coe (sym α n) (multiset α) := coe_subtype
/--
This is the `list.perm` setoid lifted to `vector`.
See note [reducible non-instances].
-/
@[reducible]
def vector.perm.is_setoid (α : Type*) (n : ℕ) : setoid (vector α n) :=
(list.is_setoid α).comap subtype.val
local attribute [instance] vector.perm.is_setoid
namespace sym
variables {α β : Type*} {n n' m : ℕ} {s : sym α n} {a b : α}
lemma coe_injective : injective (coe : sym α n → multiset α) := subtype.coe_injective
@[simp, norm_cast] lemma coe_inj {s₁ s₂ : sym α n} : (s₁ : multiset α) = s₂ ↔ s₁ = s₂ :=
coe_injective.eq_iff
/--
Construct an element of the `n`th symmetric power from a multiset of cardinality `n`.
-/
@[simps, pattern]
abbreviation mk (m : multiset α) (h : m.card = n) : sym α n := ⟨m, h⟩
/--
The unique element in `sym α 0`.
-/
@[pattern] def nil : sym α 0 := ⟨0, multiset.card_zero⟩
@[simp] lemma coe_nil : (coe (@sym.nil α)) = (0 : multiset α) := rfl
/--
Inserts an element into the term of `sym α n`, increasing the length by one.
-/
@[pattern] def cons (a : α) (s : sym α n) : sym α n.succ :=
⟨a ::ₘ s.1, by rw [multiset.card_cons, s.2]⟩
infixr ` ::ₛ `:67 := cons
@[simp]
lemma cons_inj_right (a : α) (s s' : sym α n) : a ::ₛ s = a ::ₛ s' ↔ s = s' :=
subtype.ext_iff.trans $ (multiset.cons_inj_right _).trans subtype.ext_iff.symm
@[simp]
lemma cons_inj_left (a a' : α) (s : sym α n) : a ::ₛ s = a' ::ₛ s ↔ a = a' :=
subtype.ext_iff.trans $ multiset.cons_inj_left _
lemma cons_swap (a b : α) (s : sym α n) : a ::ₛ b ::ₛ s = b ::ₛ a ::ₛ s :=
subtype.ext $ multiset.cons_swap a b s.1
lemma coe_cons (s : sym α n) (a : α) : (a ::ₛ s : multiset α) = a ::ₘ s := rfl
/--
This is the quotient map that takes a list of n elements as an n-tuple and produces an nth
symmetric power.
-/
instance : has_lift (vector α n) (sym α n) :=
{ lift := λ x, ⟨↑x.val, (multiset.coe_card _).trans x.2⟩ }
@[simp] lemma of_vector_nil : ↑(vector.nil : vector α 0) = (sym.nil : sym α 0) := rfl
@[simp] lemma of_vector_cons (a : α) (v : vector α n) :
↑(vector.cons a v) = a ::ₛ (↑v : sym α n) := by { cases v, refl }
/--
`α ∈ s` means that `a` appears as one of the factors in `s`.
-/
instance : has_mem α (sym α n) := ⟨λ a s, a ∈ s.1⟩
instance decidable_mem [decidable_eq α] (a : α) (s : sym α n) : decidable (a ∈ s) :=
s.1.decidable_mem _
@[simp]
lemma mem_mk (a : α) (s : multiset α) (h : s.card = n) : a ∈ mk s h ↔ a ∈ s := iff.rfl
@[simp] lemma mem_cons : a ∈ b ::ₛ s ↔ a = b ∨ a ∈ s :=
multiset.mem_cons
@[simp] lemma mem_coe : a ∈ (s : multiset α) ↔ a ∈ s := iff.rfl
lemma mem_cons_of_mem (h : a ∈ s) : a ∈ b ::ₛ s :=
multiset.mem_cons_of_mem h
@[simp] lemma mem_cons_self (a : α) (s : sym α n) : a ∈ a ::ₛ s :=
multiset.mem_cons_self a s.1
lemma cons_of_coe_eq (a : α) (v : vector α n) : a ::ₛ (↑v : sym α n) = ↑(a ::ᵥ v) :=
subtype.ext $ by { cases v, refl }
lemma sound {a b : vector α n} (h : a.val ~ b.val) : (↑a : sym α n) = ↑b :=
subtype.ext $ quotient.sound h
/-- `erase s a h` is the sym that subtracts 1 from the
multiplicity of `a` if a is present in the sym. -/
def erase [decidable_eq α] (s : sym α (n + 1)) (a : α) (h : a ∈ s) : sym α n :=
⟨s.val.erase a, (multiset.card_erase_of_mem h).trans $ s.property.symm ▸ n.pred_succ⟩
@[simp] lemma erase_mk [decidable_eq α] (m : multiset α) (hc : m.card = n + 1) (a : α) (h : a ∈ m) :
(mk m hc).erase a h = mk (m.erase a) (by { rw [multiset.card_erase_of_mem h, hc], refl }) := rfl
@[simp] lemma coe_erase [decidable_eq α] {s : sym α n.succ} {a : α} (h : a ∈ s) :
(s.erase a h : multiset α) = multiset.erase s a := rfl
@[simp] lemma cons_erase [decidable_eq α] {s : sym α n.succ} {a : α} (h : a ∈ s) :
a ::ₛ s.erase a h = s :=
coe_injective $ multiset.cons_erase h
@[simp] lemma erase_cons_head [decidable_eq α] (s : sym α n) (a : α)
(h : a ∈ a ::ₛ s := mem_cons_self a s) : (a ::ₛ s).erase a h = s :=
coe_injective $ multiset.erase_cons_head a s.1
/--
Another definition of the nth symmetric power, using vectors modulo permutations. (See `sym`.)
-/
def sym' (α : Type*) (n : ℕ) := quotient (vector.perm.is_setoid α n)
/--
This is `cons` but for the alternative `sym'` definition.
-/
def cons' {α : Type*} {n : ℕ} : α → sym' α n → sym' α (nat.succ n) :=
λ a, quotient.map (vector.cons a) (λ ⟨l₁, h₁⟩ ⟨l₂, h₂⟩ h, list.perm.cons _ h)
notation (name := sym.cons') a :: b := cons' a b
/--
Multisets of cardinality n are equivalent to length-n vectors up to permutations.
-/
def sym_equiv_sym' {α : Type*} {n : ℕ} : sym α n ≃ sym' α n :=
equiv.subtype_quotient_equiv_quotient_subtype _ _ (λ _, by refl) (λ _ _, by refl)
lemma cons_equiv_eq_equiv_cons (α : Type*) (n : ℕ) (a : α) (s : sym α n) :
a :: sym_equiv_sym' s = sym_equiv_sym' (a ::ₛ s) :=
by { rcases s with ⟨⟨l⟩, _⟩, refl, }
instance : has_zero (sym α 0) := ⟨⟨0, rfl⟩⟩
instance : has_emptyc (sym α 0) := ⟨0⟩
lemma eq_nil_of_card_zero (s : sym α 0) : s = nil :=
subtype.ext $ multiset.card_eq_zero.1 s.2
instance unique_zero : unique (sym α 0) :=
⟨⟨nil⟩, eq_nil_of_card_zero⟩
/-- `repeat a n` is the sym containing only `a` with multiplicity `n`. -/
def repeat (a : α) (n : ℕ) : sym α n := ⟨multiset.repeat a n, multiset.card_repeat _ _⟩
lemma repeat_succ {a : α} {n : ℕ} : repeat a n.succ = a ::ₛ repeat a n := rfl
lemma coe_repeat : (repeat a n : multiset α) = multiset.repeat a n := rfl
@[simp] lemma mem_repeat : b ∈ repeat a n ↔ n ≠ 0 ∧ b = a := multiset.mem_repeat
lemma eq_repeat_iff : s = repeat a n ↔ ∀ b ∈ s, b = a :=
begin
rw [subtype.ext_iff, coe_repeat],
convert multiset.eq_repeat',
exact s.2.symm,
end
lemma exists_mem (s : sym α n.succ) : ∃ a, a ∈ s :=
multiset.card_pos_iff_exists_mem.1 $ s.2.symm ▸ n.succ_pos
lemma exists_eq_cons_of_succ (s : sym α n.succ) : ∃ (a : α) (s' : sym α n), s = a ::ₛ s' :=
begin
obtain ⟨a, ha⟩ := exists_mem s,
classical,
exact ⟨a, s.erase a ha, (cons_erase ha).symm⟩,
end
lemma eq_repeat {a : α} {n : ℕ} {s : sym α n} : s = repeat a n ↔ ∀ b ∈ s, b = a :=
subtype.ext_iff.trans $ multiset.eq_repeat.trans $ and_iff_right s.prop
lemma eq_repeat_of_subsingleton [subsingleton α] (a : α) {n : ℕ} (s : sym α n) : s = repeat a n :=
eq_repeat.2 $ λ b hb, subsingleton.elim _ _
instance [subsingleton α] (n : ℕ) : subsingleton (sym α n) :=
⟨begin
cases n,
{ simp, },
{ intros s s',
obtain ⟨b, -⟩ := exists_mem s,
rw [eq_repeat_of_subsingleton b s', eq_repeat_of_subsingleton b s], },
end⟩
instance inhabited_sym [inhabited α] (n : ℕ) : inhabited (sym α n) :=
⟨repeat default n⟩
instance inhabited_sym' [inhabited α] (n : ℕ) : inhabited (sym' α n) :=
⟨quotient.mk' (vector.repeat default n)⟩
instance (n : ℕ) [is_empty α] : is_empty (sym α n.succ) :=
⟨λ s, by { obtain ⟨a, -⟩ := exists_mem s, exact is_empty_elim a }⟩
instance (n : ℕ) [unique α] : unique (sym α n) := unique.mk' _
lemma repeat_left_inj {a b : α} {n : ℕ} (h : n ≠ 0) : repeat a n = repeat b n ↔ a = b :=
subtype.ext_iff.trans (multiset.repeat_left_inj h)
lemma repeat_left_injective {n : ℕ} (h : n ≠ 0) : function.injective (λ x : α, repeat x n) :=
λ a b, (repeat_left_inj h).1
instance (n : ℕ) [nontrivial α] : nontrivial (sym α (n + 1)) :=
(repeat_left_injective n.succ_ne_zero).nontrivial
/-- A function `α → β` induces a function `sym α n → sym β n` by applying it to every element of
the underlying `n`-tuple. -/
def map {n : ℕ} (f : α → β) (x : sym α n) : sym β n :=
⟨x.val.map f, by simpa [multiset.card_map] using x.property⟩
@[simp] lemma mem_map {n : ℕ} {f : α → β} {b : β} {l : sym α n} :
b ∈ sym.map f l ↔ ∃ a, a ∈ l ∧ f a = b := multiset.mem_map
/-- Note: `sym.map_id` is not simp-normal, as simp ends up unfolding `id` with `sym.map_congr` -/
@[simp] lemma map_id' {α : Type*} {n : ℕ} (s : sym α n) : sym.map (λ (x : α), x) s = s :=
by simp [sym.map]
lemma map_id {α : Type*} {n : ℕ} (s : sym α n) : sym.map id s = s :=
by simp [sym.map]
@[simp] lemma map_map {α β γ : Type*} {n : ℕ} (g : β → γ) (f : α → β) (s : sym α n) :
sym.map g (sym.map f s) = sym.map (g ∘ f) s :=
by simp [sym.map]
@[simp] lemma map_zero (f : α → β) :
sym.map f (0 : sym α 0) = (0 : sym β 0) := rfl
@[simp] lemma map_cons {n : ℕ} (f : α → β) (a : α) (s : sym α n) :
(a ::ₛ s).map f = (f a) ::ₛ s.map f :=
by simp [map, cons]
@[congr] lemma map_congr {f g : α → β} {s : sym α n} (h : ∀ x ∈ s, f x = g x) :
map f s = map g s := subtype.ext $ multiset.map_congr rfl h
@[simp] lemma map_mk {f : α → β} {m : multiset α} {hc : m.card = n} :
map f (mk m hc) = mk (m.map f) (by simp [hc]) := rfl
@[simp] lemma coe_map (s : sym α n) (f : α → β) : ↑(s.map f) = multiset.map f s := rfl
lemma map_injective {f : α → β} (hf : injective f) (n : ℕ) :
injective (map f : sym α n → sym β n) :=
λ s t h, coe_injective $ multiset.map_injective hf $ coe_inj.2 h
/-- Mapping an equivalence `α ≃ β` using `sym.map` gives an equivalence between `sym α n` and
`sym β n`. -/
@[simps]
def equiv_congr (e : α ≃ β) : sym α n ≃ sym β n :=
{ to_fun := map e,
inv_fun := map e.symm,
left_inv := λ x, by rw [map_map, equiv.symm_comp_self, map_id],
right_inv := λ x, by rw [map_map, equiv.self_comp_symm, map_id] }
/-- "Attach" a proof that `a ∈ s` to each element `a` in `s` to produce
an element of the symmetric power on `{x // x ∈ s}`. -/
def attach (s : sym α n) : sym {x // x ∈ s} n := ⟨s.val.attach, by rw [multiset.card_attach, s.2]⟩
@[simp] lemma attach_mk {m : multiset α} {hc : m.card = n} :
attach (mk m hc) = mk m.attach (multiset.card_attach.trans hc) := rfl
@[simp] lemma coe_attach (s : sym α n) : (s.attach : multiset {a // a ∈ s}) = multiset.attach s :=
rfl
lemma attach_map_coe (s : sym α n) : s.attach.map coe = s :=
coe_injective $ multiset.attach_map_val _
@[simp] lemma mem_attach (s : sym α n) (x : {x // x ∈ s}) : x ∈ s.attach :=
multiset.mem_attach _ _
@[simp] lemma attach_nil : (nil : sym α 0).attach = nil := rfl
@[simp] lemma attach_cons (x : α) (s : sym α n) :
(cons x s).attach = cons ⟨x, mem_cons_self _ _⟩ (s.attach.map (λ x, ⟨x, mem_cons_of_mem x.prop⟩))
:=
coe_injective $ multiset.attach_cons _ _
/-- Change the length of a `sym` using an equality.
The simp-normal form is for the `cast` to be pushed outward. -/
protected def cast {n m : ℕ} (h : n = m) : sym α n ≃ sym α m :=
{ to_fun := λ s, ⟨s.val, s.2.trans h⟩,
inv_fun := λ s, ⟨s.val, s.2.trans h.symm⟩,
left_inv := λ s, subtype.ext rfl,
right_inv := λ s, subtype.ext rfl }
@[simp] lemma cast_rfl : sym.cast rfl s = s := subtype.ext rfl
@[simp] lemma cast_cast {n'' : ℕ} (h : n = n') (h' : n' = n'') :
sym.cast h' (sym.cast h s) = sym.cast (h.trans h') s := rfl
@[simp] lemma coe_cast (h : n = m) : (sym.cast h s : multiset α) = s := rfl
@[simp] lemma mem_cast (h : n = m) : a ∈ sym.cast h s ↔ a ∈ s := iff.rfl
/-- Append a pair of `sym` terms. -/
def append (s : sym α n) (s' : sym α n') : sym α (n + n') :=
⟨s.1 + s'.1, by simp_rw [← s.2, ← s'.2, map_add]⟩
@[simp] lemma append_inj_right (s : sym α n) {t t' : sym α n'} :
s.append t = s.append t' ↔ t = t' :=
subtype.ext_iff.trans $ (add_right_inj _).trans subtype.ext_iff.symm
@[simp] lemma append_inj_left {s s' : sym α n} (t : sym α n') :
s.append t = s'.append t ↔ s = s' :=
subtype.ext_iff.trans $ (add_left_inj _).trans subtype.ext_iff.symm
lemma append_comm (s : sym α n') (s' : sym α n') :
s.append s' = sym.cast (add_comm _ _) (s'.append s) :=
by { ext, simp [append, add_comm], }
@[simp, norm_cast] lemma coe_append (s : sym α n) (s' : sym α n') :
(s.append s' : multiset α) = s + s' := rfl
lemma mem_append_iff {s' : sym α m} : a ∈ s.append s' ↔ a ∈ s ∨ a ∈ s' := multiset.mem_add
/-- Fill a term `m : sym α (n - i)` with `i` copies of `a` to obtain a term of `sym α n`.
This is a convenience wrapper for `m.append (repeat a i)` that adjusts the term using `sym.cast`. -/
def fill (a : α) (i : fin (n + 1)) (m : sym α (n - i)) : sym α n :=
sym.cast (nat.sub_add_cancel i.is_le) (m.append (repeat a i))
lemma coe_fill {a : α} {i : fin (n + 1)} {m : sym α (n - i)} :
(fill a i m : multiset α) = m + repeat a i := rfl
lemma mem_fill_iff {a b : α} {i : fin (n + 1)} {s : sym α (n - i)} :
a ∈ sym.fill b i s ↔ ((i : ℕ) ≠ 0 ∧ a = b) ∨ a ∈ s :=
by rw [fill, mem_cast, mem_append_iff, or_comm, mem_repeat]
open multiset
/-- Remove every `a` from a given `sym α n`.
Yields the number of copies `i` and a term of `sym α (n - i)`. -/
def filter_ne [decidable_eq α] (a : α) (m : sym α n) : Σ i : fin (n + 1), sym α (n - i) :=
⟨⟨m.1.count a, (count_le_card _ _).trans_lt $ by rw [m.2, nat.lt_succ_iff]⟩,
m.1.filter ((≠) a), eq_tsub_of_add_eq $ eq.trans begin
rw [← countp_eq_card_filter, add_comm],
exact (card_eq_countp_add_countp _ _).symm,
end m.2⟩
lemma sigma_sub_ext {m₁ m₂ : Σ i : fin (n + 1), sym α (n - i)}
(h : (m₁.2 : multiset α) = m₂.2) : m₁ = m₂ :=
sigma.subtype_ext (fin.ext $ by rw [← nat.sub_sub_self m₁.1.is_le, ← nat.sub_sub_self m₂.1.is_le,
← m₁.2.2, ← m₂.2.2, subtype.val_eq_coe, subtype.val_eq_coe, h]) h
lemma fill_filter_ne [decidable_eq α] (a : α) (m : sym α n) :
(m.filter_ne a).2.fill a (m.filter_ne a).1 = m :=
subtype.ext begin
dsimp only [coe_fill, filter_ne, subtype.coe_mk, fin.coe_mk],
ext b, rw [count_add, count_filter, sym.coe_repeat, count_repeat],
obtain rfl | h := eq_or_ne a b,
{ rw [if_pos rfl, if_neg (not_not.2 rfl), zero_add], refl },
{ rw [if_pos h, if_neg h.symm, add_zero], refl },
end
lemma filter_ne_fill [decidable_eq α] (a : α) (m : Σ i : fin (n + 1), sym α (n - i)) (h : a ∉ m.2) :
(m.2.fill a m.1).filter_ne a = m :=
sigma_sub_ext begin
dsimp only [filter_ne, subtype.coe_mk, subtype.val_eq_coe, coe_fill],
rw [filter_add, filter_eq_self.2, add_right_eq_self, eq_zero_iff_forall_not_mem],
{ intros b hb, rw [mem_filter, sym.mem_coe, mem_repeat] at hb, exact hb.2 hb.1.2.symm },
{ exact λ b hb, (hb.ne_of_not_mem h).symm },
end
end sym
section equiv
/-! ### Combinatorial equivalences -/
variables {α : Type*} {n : ℕ}
open sym
namespace sym_option_succ_equiv
/-- Function from the symmetric product over `option` splitting on whether or not
it contains a `none`. -/
def encode [decidable_eq α] (s : sym (option α) n.succ) : sym (option α) n ⊕ sym α n.succ :=
if h : none ∈ s
then sum.inl (s.erase none h)
else sum.inr (s.attach.map $ λ o,
option.get $ option.ne_none_iff_is_some.1 $ ne_of_mem_of_not_mem o.2 h)
@[simp] lemma encode_of_none_mem [decidable_eq α] (s : sym (option α) n.succ) (h : none ∈ s) :
encode s = sum.inl (s.erase none h) := dif_pos h
@[simp] lemma encode_of_not_none_mem [decidable_eq α] (s : sym (option α) n.succ) (h : ¬ none ∈ s) :
encode s = sum.inr (s.attach.map $ λ o,
option.get $ option.ne_none_iff_is_some.1 $ ne_of_mem_of_not_mem o.2 h) := dif_neg h
/-- Inverse of `sym_option_succ_equiv.decode`. -/
@[simp] def decode : sym (option α) n ⊕ sym α n.succ → sym (option α) n.succ
| (sum.inl s) := none ::ₛ s
| (sum.inr s) := s.map embedding.coe_option
@[simp] lemma decode_encode [decidable_eq α] (s : sym (option α) n.succ) :
decode (encode s) = s :=
begin
by_cases h : none ∈ s,
{ simp [h] },
{ simp only [h, decode, not_false_iff, subtype.val_eq_coe, encode_of_not_none_mem,
embedding.coe_option_apply, map_map, comp_app, option.coe_get],
convert s.attach_map_coe }
end
@[simp] lemma encode_decode [decidable_eq α] (s : sym (option α) n ⊕ sym α n.succ) :
encode (decode s) = s :=
begin
obtain (s | s) := s,
{ simp },
{ unfold sym_option_succ_equiv.encode,
split_ifs,
{ obtain ⟨a, _, ha⟩ := multiset.mem_map.mp h,
exact option.some_ne_none _ ha },
{ refine map_injective (option.some_injective _) _ _,
convert eq.trans _ (sym_option_succ_equiv.decode (sum.inr s)).attach_map_coe,
simp } }
end
end sym_option_succ_equiv
/-- The symmetric product over `option` is a disjoint union over simpler symmetric products. -/
@[simps] def sym_option_succ_equiv [decidable_eq α] :
sym (option α) n.succ ≃ sym (option α) n ⊕ sym α n.succ :=
{ to_fun := sym_option_succ_equiv.encode,
inv_fun := sym_option_succ_equiv.decode,
left_inv := sym_option_succ_equiv.decode_encode,
right_inv := sym_option_succ_equiv.encode_decode }
end equiv
|
e5979915fe5384daf29ca0ff452989b6cea8876a | e030b0259b777fedcdf73dd966f3f1556d392178 | /tests/lean/run/simplifier_custom_relations.lean | 6eac102dfd56b6fcbf75fe08d1729afd7098424d | [
"Apache-2.0"
] | permissive | fgdorais/lean | 17b46a095b70b21fa0790ce74876658dc5faca06 | c3b7c54d7cca7aaa25328f0a5660b6b75fe26055 | refs/heads/master | 1,611,523,590,686 | 1,484,412,902,000 | 1,484,412,902,000 | 38,489,734 | 0 | 0 | null | 1,435,923,380,000 | 1,435,923,379,000 | null | UTF-8 | Lean | false | false | 1,022 | lean | open tactic
universe l
constants (A : Type.{l}) (rel : A → A → Prop)
(rel.refl : ∀ a, rel a a)
(rel.symm : ∀ a b, rel a b → rel b a)
(rel.trans : ∀ a b c, rel a b → rel b c → rel a c)
attribute [refl] rel.refl
attribute [symm] rel.symm
attribute [trans] rel.trans
constants (x y z : A) (f g h : A → A)
(H₁ : rel (f x) (g y))
(H₂ : rel (h (g y)) z)
(Hf : ∀ (a b : A), rel a b → rel (f a) (f b))
(Hg : ∀ (a b : A), rel a b → rel (g a) (g b))
(Hh : ∀ (a b : A), rel a b → rel (h a) (h b))
attribute [simp] H₁ H₂
attribute [congr] Hf Hg Hh
print [simp] default
print [congr] default
meta definition relsimp_core (e : expr) : tactic (expr × expr) :=
do S ← simp_lemmas.mk_default,
e_type ← infer_type e >>= whnf,
simplify_core default_simplify_config S `rel e
example : rel (h (f x)) z :=
by do e₁ ← to_expr `(h (f x)),
(e₁', pf) ← relsimp_core e₁,
exact pf
|
2cf64659c2371bc2689eacc684c63d31a832a527 | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/analysis/special_functions/trigonometric/inverse_deriv.lean | 8897aaace3c24e9cecc8ebe655b29a7e72f7783e | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 7,953 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import analysis.special_functions.trigonometric.inverse
import analysis.special_functions.trigonometric.deriv
/-!
# derivatives of the inverse trigonometric functions
Derivatives of `arcsin` and `arccos`.
-/
noncomputable theory
open_locale classical topological_space filter
open set filter
open_locale real
namespace real
section arcsin
lemma deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_strict_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x ∧ times_cont_diff_at ℝ ⊤ arcsin x :=
begin
cases h₁.lt_or_lt with h₁ h₁,
{ have : 1 - x ^ 2 < 0, by nlinarith [h₁],
rw [sqrt_eq_zero'.2 this.le, div_zero],
have : arcsin =ᶠ[𝓝 x] λ _, -(π / 2) :=
(gt_mem_nhds h₁).mono (λ y hy, arcsin_of_le_neg_one hy.le),
exact ⟨(has_strict_deriv_at_const _ _).congr_of_eventually_eq this.symm,
times_cont_diff_at_const.congr_of_eventually_eq this⟩ },
cases h₂.lt_or_lt with h₂ h₂,
{ have : 0 < sqrt (1 - x ^ 2) := sqrt_pos.2 (by nlinarith [h₁, h₂]),
simp only [← cos_arcsin h₁.le h₂.le, one_div] at this ⊢,
exact ⟨sin_local_homeomorph.has_strict_deriv_at_symm ⟨h₁, h₂⟩ this.ne'
(has_strict_deriv_at_sin _),
sin_local_homeomorph.times_cont_diff_at_symm_deriv this.ne' ⟨h₁, h₂⟩
(has_deriv_at_sin _) times_cont_diff_sin.times_cont_diff_at⟩ },
{ have : 1 - x ^ 2 < 0, by nlinarith [h₂],
rw [sqrt_eq_zero'.2 this.le, div_zero],
have : arcsin =ᶠ[𝓝 x] λ _, π / 2 := (lt_mem_nhds h₂).mono (λ y hy, arcsin_of_one_le hy.le),
exact ⟨(has_strict_deriv_at_const _ _).congr_of_eventually_eq this.symm,
times_cont_diff_at_const.congr_of_eventually_eq this⟩ }
end
lemma has_strict_deriv_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_strict_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x :=
(deriv_arcsin_aux h₁ h₂).1
lemma has_deriv_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x :=
(has_strict_deriv_at_arcsin h₁ h₂).has_deriv_at
lemma times_cont_diff_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : with_top ℕ} :
times_cont_diff_at ℝ n arcsin x :=
(deriv_arcsin_aux h₁ h₂).2.of_le le_top
lemma has_deriv_within_at_arcsin_Ici {x : ℝ} (h : x ≠ -1) :
has_deriv_within_at arcsin (1 / sqrt (1 - x ^ 2)) (Ici x) x :=
begin
rcases em (x = 1) with (rfl|h'),
{ convert (has_deriv_within_at_const _ _ (π / 2)).congr _ _;
simp [arcsin_of_one_le] { contextual := tt } },
{ exact (has_deriv_at_arcsin h h').has_deriv_within_at }
end
lemma has_deriv_within_at_arcsin_Iic {x : ℝ} (h : x ≠ 1) :
has_deriv_within_at arcsin (1 / sqrt (1 - x ^ 2)) (Iic x) x :=
begin
rcases em (x = -1) with (rfl|h'),
{ convert (has_deriv_within_at_const _ _ (-(π / 2))).congr _ _;
simp [arcsin_of_le_neg_one] { contextual := tt } },
{ exact (has_deriv_at_arcsin h' h).has_deriv_within_at }
end
lemma differentiable_within_at_arcsin_Ici {x : ℝ} :
differentiable_within_at ℝ arcsin (Ici x) x ↔ x ≠ -1 :=
begin
refine ⟨_, λ h, (has_deriv_within_at_arcsin_Ici h).differentiable_within_at⟩,
rintro h rfl,
have : sin ∘ arcsin =ᶠ[𝓝[≥] (-1 : ℝ)] id,
{ filter_upwards [Icc_mem_nhds_within_Ici ⟨le_rfl, neg_lt_self (@zero_lt_one ℝ _ _)⟩]
with x using sin_arcsin', },
have := h.has_deriv_within_at.sin.congr_of_eventually_eq this.symm (by simp),
simpa using (unique_diff_on_Ici _ _ left_mem_Ici).eq_deriv _ this (has_deriv_within_at_id _ _)
end
lemma differentiable_within_at_arcsin_Iic {x : ℝ} :
differentiable_within_at ℝ arcsin (Iic x) x ↔ x ≠ 1 :=
begin
refine ⟨λ h, _, λ h, (has_deriv_within_at_arcsin_Iic h).differentiable_within_at⟩,
rw [← neg_neg x, ← image_neg_Ici] at h,
have := (h.comp (-x) differentiable_within_at_id.neg (maps_to_image _ _)).neg,
simpa [(∘), differentiable_within_at_arcsin_Ici] using this
end
lemma differentiable_at_arcsin {x : ℝ} :
differentiable_at ℝ arcsin x ↔ x ≠ -1 ∧ x ≠ 1 :=
⟨λ h, ⟨differentiable_within_at_arcsin_Ici.1 h.differentiable_within_at,
differentiable_within_at_arcsin_Iic.1 h.differentiable_within_at⟩,
λ h, (has_deriv_at_arcsin h.1 h.2).differentiable_at⟩
@[simp] lemma deriv_arcsin : deriv arcsin = λ x, 1 / sqrt (1 - x ^ 2) :=
begin
funext x,
by_cases h : x ≠ -1 ∧ x ≠ 1,
{ exact (has_deriv_at_arcsin h.1 h.2).deriv },
{ rw [deriv_zero_of_not_differentiable_at (mt differentiable_at_arcsin.1 h)],
simp only [not_and_distrib, ne.def, not_not] at h,
rcases h with (rfl|rfl); simp }
end
lemma differentiable_on_arcsin : differentiable_on ℝ arcsin {-1, 1}ᶜ :=
λ x hx, (differentiable_at_arcsin.2
⟨λ h, hx (or.inl h), λ h, hx (or.inr h)⟩).differentiable_within_at
lemma times_cont_diff_on_arcsin {n : with_top ℕ} :
times_cont_diff_on ℝ n arcsin {-1, 1}ᶜ :=
λ x hx, (times_cont_diff_at_arcsin (mt or.inl hx) (mt or.inr hx)).times_cont_diff_within_at
lemma times_cont_diff_at_arcsin_iff {x : ℝ} {n : with_top ℕ} :
times_cont_diff_at ℝ n arcsin x ↔ n = 0 ∨ (x ≠ -1 ∧ x ≠ 1) :=
⟨λ h, or_iff_not_imp_left.2 $ λ hn, differentiable_at_arcsin.1 $ h.differentiable_at $
with_top.one_le_iff_pos.2 (pos_iff_ne_zero.2 hn),
λ h, h.elim (λ hn, hn.symm ▸ (times_cont_diff_zero.2 continuous_arcsin).times_cont_diff_at) $
λ hx, times_cont_diff_at_arcsin hx.1 hx.2⟩
end arcsin
section arccos
lemma has_strict_deriv_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_strict_deriv_at arccos (-(1 / sqrt (1 - x ^ 2))) x :=
(has_strict_deriv_at_arcsin h₁ h₂).const_sub (π / 2)
lemma has_deriv_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_deriv_at arccos (-(1 / sqrt (1 - x ^ 2))) x :=
(has_deriv_at_arcsin h₁ h₂).const_sub (π / 2)
lemma times_cont_diff_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : with_top ℕ} :
times_cont_diff_at ℝ n arccos x :=
times_cont_diff_at_const.sub (times_cont_diff_at_arcsin h₁ h₂)
lemma has_deriv_within_at_arccos_Ici {x : ℝ} (h : x ≠ -1) :
has_deriv_within_at arccos (-(1 / sqrt (1 - x ^ 2))) (Ici x) x :=
(has_deriv_within_at_arcsin_Ici h).const_sub _
lemma has_deriv_within_at_arccos_Iic {x : ℝ} (h : x ≠ 1) :
has_deriv_within_at arccos (-(1 / sqrt (1 - x ^ 2))) (Iic x) x :=
(has_deriv_within_at_arcsin_Iic h).const_sub _
lemma differentiable_within_at_arccos_Ici {x : ℝ} :
differentiable_within_at ℝ arccos (Ici x) x ↔ x ≠ -1 :=
(differentiable_within_at_const_sub_iff _).trans differentiable_within_at_arcsin_Ici
lemma differentiable_within_at_arccos_Iic {x : ℝ} :
differentiable_within_at ℝ arccos (Iic x) x ↔ x ≠ 1 :=
(differentiable_within_at_const_sub_iff _).trans differentiable_within_at_arcsin_Iic
lemma differentiable_at_arccos {x : ℝ} :
differentiable_at ℝ arccos x ↔ x ≠ -1 ∧ x ≠ 1 :=
(differentiable_at_const_sub_iff _).trans differentiable_at_arcsin
@[simp] lemma deriv_arccos : deriv arccos = λ x, -(1 / sqrt (1 - x ^ 2)) :=
funext $ λ x, (deriv_const_sub _).trans $ by simp only [deriv_arcsin]
lemma differentiable_on_arccos : differentiable_on ℝ arccos {-1, 1}ᶜ :=
differentiable_on_arcsin.const_sub _
lemma times_cont_diff_on_arccos {n : with_top ℕ} :
times_cont_diff_on ℝ n arccos {-1, 1}ᶜ :=
times_cont_diff_on_const.sub times_cont_diff_on_arcsin
lemma times_cont_diff_at_arccos_iff {x : ℝ} {n : with_top ℕ} :
times_cont_diff_at ℝ n arccos x ↔ n = 0 ∨ (x ≠ -1 ∧ x ≠ 1) :=
by refine iff.trans ⟨λ h, _, λ h, _⟩ times_cont_diff_at_arcsin_iff;
simpa [arccos] using (@times_cont_diff_at_const _ _ _ _ _ _ _ _ _ _ (π / 2)).sub h
end arccos
end real
|
0a802a63cbebe57b633bc3a7bb4e5808d421e3fe | 4bcaca5dc83d49803f72b7b5920b75b6e7d9de2d | /stage0/src/Lean/DocString.lean | 2e7555f81f55dee56a92eba6c7c6ed17f16a8c3e | [
"Apache-2.0"
] | permissive | subfish-zhou/leanprover-zh_CN.github.io | 30b9fba9bd790720bd95764e61ae796697d2f603 | 8b2985d4a3d458ceda9361ac454c28168d920d3f | refs/heads/master | 1,689,709,967,820 | 1,632,503,056,000 | 1,632,503,056,000 | 409,962,097 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,493 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.MonadEnv
namespace Lean
private builtin_initialize docStringExt : MapDeclarationExtension String ← mkMapDeclarationExtension `docstring
def addDocString [MonadEnv m] (declName : Name) (docString : String) : m Unit :=
modifyEnv fun env => docStringExt.insert env declName docString
def addDocString' [Monad m] [MonadEnv m] (declName : Name) (docString? : Option String) : m Unit :=
match docString? with
| some docString => addDocString declName docString
| none => return ()
def findDocString? [Monad m] [MonadEnv m] (declName : Name) : m (Option String) :=
return docStringExt.find? (← getEnv) declName
private builtin_initialize moduleDocExt : SimplePersistentEnvExtension String (Std.PersistentArray String) ← registerSimplePersistentEnvExtension {
name := `moduleDocExt
addImportedFn := fun _ => {}
addEntryFn := fun s e => s.push e
toArrayFn := fun es => es.toArray
}
def addMainModuleDoc (env : Environment) (doc : String) : Environment :=
moduleDocExt.addEntry env doc
def getMainModuleDoc (env : Environment) : Std.PersistentArray String :=
moduleDocExt.getState env
def getModuleDoc? (env : Environment) (moduleName : Name) : Option (Array String) :=
env.getModuleIdx? moduleName |>.map fun modIdx => moduleDocExt.getModuleEntries env modIdx
end Lean
|
8c12b5197d0c2436897ba9a13618c509697a21a3 | 94e33a31faa76775069b071adea97e86e218a8ee | /src/data/buffer/parser/numeral.lean | db45fd8dee01c483a1d89de5c1b83169389a1805 | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 5,106 | lean | /-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import data.buffer.parser.basic
/-!
# Numeral parsers
This file expands on the existing `nat : parser ℕ` to provide parsers into any type `α` that
can be represented by a numeral, which relies on `α` having a 0, 1, and addition operation.
There are also convenience parsers that ensure that the numeral parsed in is not larger than
the cardinality of the type `α` , if it is known that `fintype α`. Additionally, there are
convenience parsers that parse in starting from "1", which can be useful for positive ordinals;
or parser from a given character or character range.
## Main definitions
* `parser.numeral` : The parser which uses `nat.cast`
to map the result of `parser.nat` to the desired `α`
* `parser.numeral.of_fintype` : The parser which `guard`s to make sure the parsed
numeral is within the cardinality of the target `fintype` type `α`.
## Implementation details
When the `parser.numeral` or related parsers are invoked, the desired type is provided explicitly.
In many cases, it can be inferred, so one can write, for example
```lean
def get_fin : string → fin 5 :=
sum.elim (λ _, 0) id ∘ parser.run_string (parser.numeral.of_fintype _)
```
In the definitions of the parsers (except for `parser.numeral`), there is an
explicit `nat.bin_cast` instead an explicit or implicit `nat.cast`.
-/
open parser parse_result
namespace parser
variables (α : Type) [has_zero α] [has_one α] [has_add α]
/--
Parse a string of digits as a numeral while casting it to target type `α`.
-/
@[derive [mono, bounded, prog]]
def numeral : parser α :=
nat.bin_cast <$> nat
/--
Parse a string of digits as a numeral while casting it to target type `α`,
which has a `[fintype α]` constraint. The parser ensures that the numeral parsed in
is within the cardinality of the type `α`.
-/
@[derive [mono, bounded, prog]]
def numeral.of_fintype [fintype α] : parser α :=
do
c ← nat,
decorate_error (sformat!"<numeral less than {to_string (fintype.card α)}>")
(guard (c < fintype.card α)),
pure $ nat.bin_cast c
/--
Parse a string of digits as a numeral while casting it to target type `α`. The parsing starts
at "1", so `"1"` is parsed in as `nat.cast 0`. Providing `"0"` to the parser causes a failure.
-/
@[derive [mono, bounded, prog]]
def numeral.from_one : parser α :=
do
c ← nat,
decorate_error ("<positive numeral>")
(guard (0 < c)),
pure $ nat.bin_cast (c - 1)
/--
Parse a string of digits as a numeral while casting it to target type `α`,
which has a `[fintype α]` constraint. The parser ensures that the numeral parsed in
is within the cardinality of the type `α`. The parsing starts
at "1", so `"1"` is parsed in as `nat.cast 0`. Providing `"0"` to the parser causes a failure.
-/
@[derive [mono, bounded, prog]]
def numeral.from_one.of_fintype [fintype α] : parser α :=
do
c ← nat,
decorate_error (sformat!"<positive numeral less than or equal to {to_string (fintype.card α)}>")
(guard (0 < c ∧ c ≤ fintype.card α)),
pure $ nat.bin_cast (c - 1)
/--
Parse a character as a numeral while casting it to target type `α`,
The parser ensures that the character parsed in is within the bounds set by `fromc` and `toc`,
and subtracts the value of `fromc` from the parsed in character.
-/
@[derive [mono, bounded, err_static, step]]
def numeral.char (fromc toc : char) : parser α :=
do
c ← decorate_error
(sformat!"<char between '{fromc.to_string}' to '{toc.to_string}' inclusively>")
(sat (λ c, fromc ≤ c ∧ c ≤ toc)),
pure $ nat.bin_cast (c.to_nat - fromc.to_nat)
/--
Parse a character as a numeral while casting it to target type `α`,
which has a `[fintype α]` constraint.
The parser ensures that the character parsed in is greater or equal to `fromc` and
and subtracts the value of `fromc` from the parsed in character. There is also a check
that the resulting value is within the cardinality of the type `α`.
-/
@[derive [mono, bounded, err_static, step]]
def numeral.char.of_fintype [fintype α] (fromc : char) : parser α :=
do
c ← decorate_error
(sformat!"<char from '{fromc.to_string}' to '
{ (char.of_nat (fromc.to_nat + fintype.card α - 1)).to_string}' inclusively>")
(sat (λ c, fromc ≤ c ∧ c.to_nat - fintype.card α < fromc.to_nat)),
pure $ nat.bin_cast (c.to_nat - fromc.to_nat)
/-! ## Specific numeral types -/
/--
Matches an integer, like `43` or `-2`.
Large numbers may cause performance issues, so don't run this parser on untrusted input.
-/
meta def int : parser int :=
(coe <$> nat) <|> (ch '-' >> has_neg.neg <$> coe <$> nat)
/--
Matches an rational number, like `43/1` or `-2/3`.
Requires that the negation is in the numerator,
and that both a numerator and denominator are provided (e.g. will not match `43`).
Large numbers may cause performance issues, so don't run this parser on untrusted input.
-/
meta def rat : parser rat :=
(λ x y, ↑x / ↑y) <$> int <*> (ch '/' >> nat)
end parser
|
c0eff348fa904843115756586d64872f1c397cd9 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/change_tac.lean | e27165fb1ad9f2ff0fcfc705b3be67f9b256c017 | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 884 | lean | example : 1 + 2 = 3 :=
begin
change 2 + 1 = 3,
trace_state,
refl
end
example : 1 + 2 = 3 :=
begin
change 2 + 2 = 3 -- ERROR
end
example (h : 1 + 2 = 3) : 2 + 2 = 4 :=
begin
change 2 + 1 = 3 at h,
trace_state,
refl
end
example (h : 1 + 2 = 3) : 2 + 2 = 4 :=
begin
change 2 + 1 = 3 at h h, -- ERROR
end
example (h : 1 + 2 = 3) : 2 + 2 = 4 :=
begin
change 2 + 1 = 3 at *, -- ERROR
end
example (h : 1 + 2 = 3) : 1 + 2 = 3 :=
begin
change 1 + 2 with 2 + 1 at h,
trace_state,
refl
end
example (h : 1 + 2 = 1 + 2 + 1) : 1 + 2 = 3 :=
begin
change 1 + 2 with 3 at *,
trace_state,
refl
end
example (h : 1 + 2 = 1 + 2 + 1) : 1 + 2 = 3 :=
begin
change 1 + 2 with 4 at *, -- ERROR
end
example : true :=
begin
have : ∀ x : ℕ, x = x,
change ∀ y : ℕ, y = y,
trace_state,
admit,
change ∀ y : ℕ, y = y at this,
trace_state,
trivial
end
|
3e2edd78b3cf4d0eeecdef795e42c2031568818c | 02005f45e00c7ecf2c8ca5db60251bd1e9c860b5 | /src/algebra/ordered_group.lean | d2a10ba9edb5c5ebd276bb8d6e3f9132f04740de | [
"Apache-2.0"
] | permissive | anthony2698/mathlib | 03cd69fe5c280b0916f6df2d07c614c8e1efe890 | 407615e05814e98b24b2ff322b14e8e3eb5e5d67 | refs/heads/master | 1,678,792,774,873 | 1,614,371,563,000 | 1,614,371,563,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 30,826 | lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import algebra.ordered_monoid
/-!
# Ordered groups
This file develops the basics of ordered groups.
## Implementation details
Unfortunately, the number of `'` appended to lemmas in this file
may differ between the multiplicative and the additive version of a lemma.
The reason is that we did not want to change existing names in the library.
-/
set_option old_structure_cmd true
universe u
variable {α : Type u}
/-- An ordered additive commutative group is an additive commutative group
with a partial order in which addition is strictly monotone. -/
@[protect_proj, ancestor add_comm_group partial_order]
class ordered_add_comm_group (α : Type u) extends add_comm_group α, partial_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
/-- An ordered commutative group is an commutative group
with a partial order in which multiplication is strictly monotone. -/
@[protect_proj, ancestor comm_group partial_order]
class ordered_comm_group (α : Type u) extends comm_group α, partial_order α :=
(mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b)
attribute [to_additive] ordered_comm_group
/--The units of an ordered commutative monoid form an ordered commutative group. -/
@[to_additive]
instance units.ordered_comm_group [ordered_comm_monoid α] : ordered_comm_group (units α) :=
{ mul_le_mul_left := λ a b h c, mul_le_mul_left' h _,
.. units.partial_order,
.. (infer_instance : comm_group (units α)) }
section ordered_comm_group
variables [ordered_comm_group α] {a b c d : α}
@[to_additive ordered_add_comm_group.add_lt_add_left]
lemma ordered_comm_group.mul_lt_mul_left' (a b : α) (h : a < b) (c : α) : c * a < c * b :=
begin
rw lt_iff_le_not_le at h ⊢,
split,
{ apply ordered_comm_group.mul_le_mul_left _ _ h.1 },
{ intro w,
replace w : c⁻¹ * (c * b) ≤ c⁻¹ * (c * a) := ordered_comm_group.mul_le_mul_left _ _ w _,
simp only [mul_one, mul_comm, mul_left_inv, mul_left_comm] at w,
exact h.2 w },
end
@[to_additive ordered_add_comm_group.le_of_add_le_add_left]
lemma ordered_comm_group.le_of_mul_le_mul_left (h : a * b ≤ a * c) : b ≤ c :=
have a⁻¹ * (a * b) ≤ a⁻¹ * (a * c), from ordered_comm_group.mul_le_mul_left _ _ h _,
begin simp [inv_mul_cancel_left] at this, assumption end
@[to_additive]
lemma ordered_comm_group.lt_of_mul_lt_mul_left (h : a * b < a * c) : b < c :=
have a⁻¹ * (a * b) < a⁻¹ * (a * c), from ordered_comm_group.mul_lt_mul_left' _ _ h _,
begin simp [inv_mul_cancel_left] at this, assumption end
@[priority 100, to_additive] -- see Note [lower instance priority]
instance ordered_comm_group.to_ordered_cancel_comm_monoid (α : Type u)
[s : ordered_comm_group α] : ordered_cancel_comm_monoid α :=
{ mul_left_cancel := @mul_left_cancel α _,
mul_right_cancel := @mul_right_cancel α _,
le_of_mul_le_mul_left := @ordered_comm_group.le_of_mul_le_mul_left α _,
..s }
@[to_additive neg_le_neg]
lemma inv_le_inv' (h : a ≤ b) : b⁻¹ ≤ a⁻¹ :=
have 1 ≤ a⁻¹ * b, from mul_left_inv a ▸ mul_le_mul_left' h _,
have 1 * b⁻¹ ≤ a⁻¹ * b * b⁻¹, from mul_le_mul_right' this _,
by rwa [mul_inv_cancel_right, one_mul] at this
@[to_additive]
lemma le_of_inv_le_inv (h : b⁻¹ ≤ a⁻¹) : a ≤ b :=
suffices (a⁻¹)⁻¹ ≤ (b⁻¹)⁻¹, from
begin simp [inv_inv] at this, assumption end,
inv_le_inv' h
@[to_additive]
lemma one_le_of_inv_le_one (h : a⁻¹ ≤ 1) : 1 ≤ a :=
have a⁻¹ ≤ 1⁻¹, by rwa one_inv,
le_of_inv_le_inv this
@[to_additive]
lemma inv_le_one_of_one_le (h : 1 ≤ a) : a⁻¹ ≤ 1 :=
have a⁻¹ ≤ 1⁻¹, from inv_le_inv' h,
by rwa one_inv at this
@[to_additive nonpos_of_neg_nonneg]
lemma le_one_of_one_le_inv (h : 1 ≤ a⁻¹) : a ≤ 1 :=
have 1⁻¹ ≤ a⁻¹, by rwa one_inv,
le_of_inv_le_inv this
@[to_additive neg_nonneg_of_nonpos]
lemma one_le_inv_of_le_one (h : a ≤ 1) : 1 ≤ a⁻¹ :=
have 1⁻¹ ≤ a⁻¹, from inv_le_inv' h,
by rwa one_inv at this
@[to_additive neg_lt_neg]
lemma inv_lt_inv' (h : a < b) : b⁻¹ < a⁻¹ :=
have 1 < a⁻¹ * b, from mul_left_inv a ▸ mul_lt_mul_left' h (a⁻¹),
have 1 * b⁻¹ < a⁻¹ * b * b⁻¹, from mul_lt_mul_right' this (b⁻¹),
by rwa [mul_inv_cancel_right, one_mul] at this
@[to_additive]
lemma lt_of_inv_lt_inv (h : b⁻¹ < a⁻¹) : a < b :=
inv_inv a ▸ inv_inv b ▸ inv_lt_inv' h
@[to_additive]
lemma one_lt_of_inv_inv (h : a⁻¹ < 1) : 1 < a :=
have a⁻¹ < 1⁻¹, by rwa one_inv,
lt_of_inv_lt_inv this
@[to_additive]
lemma inv_inv_of_one_lt (h : 1 < a) : a⁻¹ < 1 :=
have a⁻¹ < 1⁻¹, from inv_lt_inv' h,
by rwa one_inv at this
@[to_additive neg_of_neg_pos]
lemma inv_of_one_lt_inv (h : 1 < a⁻¹) : a < 1 :=
have 1⁻¹ < a⁻¹, by rwa one_inv,
lt_of_inv_lt_inv this
@[to_additive neg_pos_of_neg]
lemma one_lt_inv_of_inv (h : a < 1) : 1 < a⁻¹ :=
have 1⁻¹ < a⁻¹, from inv_lt_inv' h,
by rwa one_inv at this
@[to_additive]
lemma le_inv_of_le_inv (h : a ≤ b⁻¹) : b ≤ a⁻¹ :=
begin
have h := inv_le_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma inv_le_of_inv_le (h : a⁻¹ ≤ b) : b⁻¹ ≤ a :=
begin
have h := inv_le_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma lt_inv_of_lt_inv (h : a < b⁻¹) : b < a⁻¹ :=
begin
have h := inv_lt_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma inv_lt_of_inv_lt (h : a⁻¹ < b) : b⁻¹ < a :=
begin
have h := inv_lt_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma mul_le_of_le_inv_mul (h : b ≤ a⁻¹ * c) : a * b ≤ c :=
begin
have h := mul_le_mul_left' h a,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma le_inv_mul_of_mul_le (h : a * b ≤ c) : b ≤ a⁻¹ * c :=
begin
have h := mul_le_mul_left' h a⁻¹,
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma le_mul_of_inv_mul_le (h : b⁻¹ * a ≤ c) : a ≤ b * c :=
begin
have h := mul_le_mul_left' h b,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma inv_mul_le_of_le_mul (h : a ≤ b * c) : b⁻¹ * a ≤ c :=
begin
have h := mul_le_mul_left' h b⁻¹,
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma le_mul_of_inv_mul_le_left (h : b⁻¹ * a ≤ c) : a ≤ b * c :=
le_mul_of_inv_mul_le h
@[to_additive]
lemma inv_mul_le_left_of_le_mul (h : a ≤ b * c) : b⁻¹ * a ≤ c :=
inv_mul_le_of_le_mul h
@[to_additive]
lemma le_mul_of_inv_mul_le_right (h : c⁻¹ * a ≤ b) : a ≤ b * c :=
by { rw mul_comm, exact le_mul_of_inv_mul_le h }
@[to_additive]
lemma inv_mul_le_right_of_le_mul (h : a ≤ b * c) : c⁻¹ * a ≤ b :=
by { rw mul_comm at h, apply inv_mul_le_left_of_le_mul h }
@[to_additive]
lemma mul_lt_of_lt_inv_mul (h : b < a⁻¹ * c) : a * b < c :=
begin
have h := mul_lt_mul_left' h a,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma lt_inv_mul_of_mul_lt (h : a * b < c) : b < a⁻¹ * c :=
begin
have h := mul_lt_mul_left' h (a⁻¹),
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma lt_mul_of_inv_mul_lt (h : b⁻¹ * a < c) : a < b * c :=
begin
have h := mul_lt_mul_left' h b,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma inv_mul_lt_of_lt_mul (h : a < b * c) : b⁻¹ * a < c :=
begin
have h := mul_lt_mul_left' h (b⁻¹),
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma lt_mul_of_inv_mul_lt_left (h : b⁻¹ * a < c) : a < b * c :=
lt_mul_of_inv_mul_lt h
@[to_additive]
lemma inv_mul_lt_left_of_lt_mul (h : a < b * c) : b⁻¹ * a < c :=
inv_mul_lt_of_lt_mul h
@[to_additive]
lemma lt_mul_of_inv_mul_lt_right (h : c⁻¹ * a < b) : a < b * c :=
by { rw mul_comm, exact lt_mul_of_inv_mul_lt h }
@[to_additive]
lemma inv_mul_lt_right_of_lt_mul (h : a < b * c) : c⁻¹ * a < b :=
by { rw mul_comm at h, exact inv_mul_lt_of_lt_mul h }
@[simp, to_additive]
lemma inv_lt_one_iff_one_lt : a⁻¹ < 1 ↔ 1 < a :=
⟨ one_lt_of_inv_inv, inv_inv_of_one_lt ⟩
@[simp, to_additive]
lemma inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
have a * b * a⁻¹ ≤ a * b * b⁻¹ ↔ a⁻¹ ≤ b⁻¹, from mul_le_mul_iff_left _,
by { rw [mul_inv_cancel_right, mul_comm a, mul_inv_cancel_right] at this, rw [this] }
@[to_additive neg_le]
lemma inv_le' : a⁻¹ ≤ b ↔ b⁻¹ ≤ a :=
have a⁻¹ ≤ (b⁻¹)⁻¹ ↔ b⁻¹ ≤ a, from inv_le_inv_iff,
by rwa inv_inv at this
@[to_additive le_neg]
lemma le_inv' : a ≤ b⁻¹ ↔ b ≤ a⁻¹ :=
have (a⁻¹)⁻¹ ≤ b⁻¹ ↔ b ≤ a⁻¹, from inv_le_inv_iff,
by rwa inv_inv at this
@[to_additive neg_le_iff_add_nonneg]
lemma inv_le_iff_one_le_mul : a⁻¹ ≤ b ↔ 1 ≤ b * a :=
(mul_le_mul_iff_right a).symm.trans $ by rw inv_mul_self
@[to_additive neg_le_iff_add_nonneg']
lemma inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b :=
(mul_le_mul_iff_left a).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < b * a :=
(mul_lt_mul_iff_right a).symm.trans $ by rw inv_mul_self
@[to_additive]
lemma inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < a * b :=
(mul_lt_mul_iff_left a).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma le_inv_iff_mul_le_one : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_self
@[to_additive]
lemma le_inv_iff_mul_le_one' : a ≤ b⁻¹ ↔ b * a ≤ 1 :=
(mul_le_mul_iff_left b).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 :=
(mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_self
@[to_additive]
lemma lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 :=
(mul_lt_mul_iff_left b).symm.trans $ by rw mul_inv_self
@[simp, to_additive neg_nonpos]
lemma inv_le_one' : a⁻¹ ≤ 1 ↔ 1 ≤ a :=
have a⁻¹ ≤ 1⁻¹ ↔ 1 ≤ a, from inv_le_inv_iff,
by rwa one_inv at this
@[simp, to_additive neg_nonneg]
lemma one_le_inv' : 1 ≤ a⁻¹ ↔ a ≤ 1 :=
have 1⁻¹ ≤ a⁻¹ ↔ a ≤ 1, from inv_le_inv_iff,
by rwa one_inv at this
@[to_additive]
lemma inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a :=
le_trans (inv_le_one'.2 h) h
@[to_additive]
lemma self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ :=
le_trans h (one_le_inv'.2 h)
@[simp, to_additive]
lemma inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a :=
have a * b * a⁻¹ < a * b * b⁻¹ ↔ a⁻¹ < b⁻¹, from mul_lt_mul_iff_left _,
by { rw [mul_inv_cancel_right, mul_comm a, mul_inv_cancel_right] at this, rw [this] }
@[to_additive neg_lt_zero]
lemma inv_lt_one' : a⁻¹ < 1 ↔ 1 < a :=
have a⁻¹ < 1⁻¹ ↔ 1 < a, from inv_lt_inv_iff,
by rwa one_inv at this
@[to_additive neg_pos]
lemma one_lt_inv' : 1 < a⁻¹ ↔ a < 1 :=
have 1⁻¹ < a⁻¹ ↔ a < 1, from inv_lt_inv_iff,
by rwa one_inv at this
@[to_additive neg_lt]
lemma inv_lt' : a⁻¹ < b ↔ b⁻¹ < a :=
have a⁻¹ < (b⁻¹)⁻¹ ↔ b⁻¹ < a, from inv_lt_inv_iff,
by rwa inv_inv at this
@[to_additive lt_neg]
lemma lt_inv' : a < b⁻¹ ↔ b < a⁻¹ :=
have (a⁻¹)⁻¹ < b⁻¹ ↔ b < a⁻¹, from inv_lt_inv_iff,
by rwa inv_inv at this
@[to_additive]
lemma inv_lt_self (h : 1 < a) : a⁻¹ < a :=
(inv_lt_one'.2 h).trans h
@[to_additive]
lemma le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c :=
have a⁻¹ * (a * b) ≤ a⁻¹ * c ↔ a * b ≤ c, from mul_le_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[simp, to_additive]
lemma inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c :=
have b⁻¹ * a ≤ b⁻¹ * (b * c) ↔ a ≤ b * c, from mul_le_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[to_additive]
lemma mul_inv_le_iff_le_mul : a * c⁻¹ ≤ b ↔ a ≤ b * c :=
by rw [mul_comm a, mul_comm b, inv_mul_le_iff_le_mul]
@[simp, to_additive]
lemma mul_inv_le_iff_le_mul' : a * b⁻¹ ≤ c ↔ a ≤ b * c :=
by rw [← inv_mul_le_iff_le_mul, mul_comm]
@[to_additive]
lemma inv_mul_le_iff_le_mul' : c⁻¹ * a ≤ b ↔ a ≤ b * c :=
by rw [inv_mul_le_iff_le_mul, mul_comm]
@[simp, to_additive]
lemma lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c :=
have a⁻¹ * (a * b) < a⁻¹ * c ↔ a * b < c, from mul_lt_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[simp, to_additive]
lemma inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c :=
have b⁻¹ * a < b⁻¹ * (b * c) ↔ a < b * c, from mul_lt_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[to_additive]
lemma inv_mul_lt_iff_lt_mul_right : c⁻¹ * a < b ↔ a < b * c :=
by rw [inv_mul_lt_iff_lt_mul, mul_comm]
@[to_additive add_neg_le_add_neg_iff]
lemma div_le_div_iff' : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b :=
begin
split ; intro h,
have := mul_le_mul_right' (mul_le_mul_right' h b) d,
rwa [inv_mul_cancel_right, mul_assoc _ _ b, mul_comm _ b, ← mul_assoc, inv_mul_cancel_right]
at this,
have := mul_le_mul_right' (mul_le_mul_right' h d⁻¹) b⁻¹,
rwa [mul_inv_cancel_right, _root_.mul_assoc, _root_.mul_comm d⁻¹ b⁻¹, ← mul_assoc,
mul_inv_cancel_right] at this,
end
end ordered_comm_group
section ordered_add_comm_group
variables [ordered_add_comm_group α] {a b c d : α}
lemma sub_le_sub (hab : a ≤ b) (hcd : c ≤ d) : a - d ≤ b - c :=
by simpa only [sub_eq_add_neg] using add_le_add hab (neg_le_neg hcd)
lemma sub_lt_sub (hab : a < b) (hcd : c < d) : a - d < b - c :=
by simpa only [sub_eq_add_neg] using add_lt_add hab (neg_lt_neg hcd)
@[simp] lemma sub_le_self_iff (a : α) {b : α} : a - b ≤ a ↔ 0 ≤ b :=
by simp [sub_eq_add_neg]
alias sub_le_self_iff ↔ _ sub_le_self
@[simp] lemma sub_lt_self_iff (a : α) {b : α} : a - b < a ↔ 0 < b :=
by simp [sub_eq_add_neg]
alias sub_lt_self_iff ↔ _ sub_lt_self
lemma sub_le_sub_iff : a - b ≤ c - d ↔ a + d ≤ c + b :=
by simpa only [sub_eq_add_neg] using add_neg_le_add_neg_iff
@[simp]
lemma sub_le_sub_iff_left (a : α) {b c : α} : a - b ≤ a - c ↔ c ≤ b :=
by rw [sub_eq_add_neg, sub_eq_add_neg, add_le_add_iff_left, neg_le_neg_iff]
lemma sub_le_sub_left (h : a ≤ b) (c : α) : c - b ≤ c - a :=
(sub_le_sub_iff_left c).2 h
@[simp]
lemma sub_le_sub_iff_right (c : α) : a - c ≤ b - c ↔ a ≤ b :=
by simpa only [sub_eq_add_neg] using add_le_add_iff_right _
lemma sub_le_sub_right (h : a ≤ b) (c : α) : a - c ≤ b - c :=
(sub_le_sub_iff_right c).2 h
@[simp]
lemma sub_lt_sub_iff_left (a : α) {b c : α} : a - b < a - c ↔ c < b :=
by rw [sub_eq_add_neg, sub_eq_add_neg, add_lt_add_iff_left, neg_lt_neg_iff]
lemma sub_lt_sub_left (h : a < b) (c : α) : c - b < c - a :=
(sub_lt_sub_iff_left c).2 h
@[simp]
lemma sub_lt_sub_iff_right (c : α) : a - c < b - c ↔ a < b :=
by simpa only [sub_eq_add_neg] using add_lt_add_iff_right _
lemma sub_lt_sub_right (h : a < b) (c : α) : a - c < b - c :=
(sub_lt_sub_iff_right c).2 h
@[simp] lemma sub_nonneg : 0 ≤ a - b ↔ b ≤ a :=
by rw [← sub_self a, sub_le_sub_iff_left]
alias sub_nonneg ↔ le_of_sub_nonneg sub_nonneg_of_le
@[simp] lemma sub_nonpos : a - b ≤ 0 ↔ a ≤ b :=
by rw [← sub_self b, sub_le_sub_iff_right]
alias sub_nonpos ↔ le_of_sub_nonpos sub_nonpos_of_le
@[simp] lemma sub_pos : 0 < a - b ↔ b < a :=
by rw [← sub_self a, sub_lt_sub_iff_left]
alias sub_pos ↔ lt_of_sub_pos sub_pos_of_lt
@[simp] lemma sub_lt_zero : a - b < 0 ↔ a < b :=
by rw [← sub_self b, sub_lt_sub_iff_right]
alias sub_lt_zero ↔ lt_of_sub_neg sub_neg_of_lt
lemma le_sub_iff_add_le' : b ≤ c - a ↔ a + b ≤ c :=
by rw [sub_eq_add_neg, add_comm, le_neg_add_iff_add_le]
lemma le_sub_iff_add_le : a ≤ c - b ↔ a + b ≤ c :=
by rw [le_sub_iff_add_le', add_comm]
alias le_sub_iff_add_le ↔ add_le_of_le_sub_right le_sub_right_of_add_le
lemma sub_le_iff_le_add' : a - b ≤ c ↔ a ≤ b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_le_iff_le_add]
alias le_sub_iff_add_le' ↔ add_le_of_le_sub_left le_sub_left_of_add_le
lemma sub_le_iff_le_add : a - c ≤ b ↔ a ≤ b + c :=
by rw [sub_le_iff_le_add', add_comm]
@[simp] lemma neg_le_sub_iff_le_add : -b ≤ a - c ↔ c ≤ a + b :=
le_sub_iff_add_le.trans neg_add_le_iff_le_add'
lemma neg_le_sub_iff_le_add' : -a ≤ b - c ↔ c ≤ a + b :=
by rw [neg_le_sub_iff_le_add, add_comm]
lemma sub_le : a - b ≤ c ↔ a - c ≤ b :=
sub_le_iff_le_add'.trans sub_le_iff_le_add.symm
theorem le_sub : a ≤ b - c ↔ c ≤ b - a :=
le_sub_iff_add_le'.trans le_sub_iff_add_le.symm
lemma lt_sub_iff_add_lt' : b < c - a ↔ a + b < c :=
by rw [sub_eq_add_neg, add_comm, lt_neg_add_iff_add_lt]
alias lt_sub_iff_add_lt' ↔ add_lt_of_lt_sub_left lt_sub_left_of_add_lt
lemma lt_sub_iff_add_lt : a < c - b ↔ a + b < c :=
by rw [lt_sub_iff_add_lt', add_comm]
alias lt_sub_iff_add_lt ↔ add_lt_of_lt_sub_right lt_sub_right_of_add_lt
lemma sub_lt_iff_lt_add' : a - b < c ↔ a < b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_lt_iff_lt_add]
alias sub_lt_iff_lt_add' ↔ lt_add_of_sub_left_lt sub_left_lt_of_lt_add
lemma sub_lt_iff_lt_add : a - c < b ↔ a < b + c :=
by rw [sub_lt_iff_lt_add', add_comm]
alias sub_lt_iff_lt_add ↔ lt_add_of_sub_right_lt sub_right_lt_of_lt_add
@[simp] lemma neg_lt_sub_iff_lt_add : -b < a - c ↔ c < a + b :=
lt_sub_iff_add_lt.trans neg_add_lt_iff_lt_add_right
lemma neg_lt_sub_iff_lt_add' : -a < b - c ↔ c < a + b :=
by rw [neg_lt_sub_iff_lt_add, add_comm]
lemma sub_lt : a - b < c ↔ a - c < b :=
sub_lt_iff_lt_add'.trans sub_lt_iff_lt_add.symm
theorem lt_sub : a < b - c ↔ c < b - a :=
lt_sub_iff_add_lt'.trans lt_sub_iff_add_lt.symm
end ordered_add_comm_group
/-!
### Linearly ordered commutative groups
-/
/-- A linearly ordered additive commutative group is an
additive commutative group with a linear order in which
addition is monotone. -/
@[protect_proj] class linear_ordered_add_comm_group (α : Type u)
extends add_comm_group α, linear_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
/-- A linearly ordered commutative group is a
commutative group with a linear order in which
multiplication is monotone. -/
@[protect_proj, to_additive] class linear_ordered_comm_group (α : Type u)
extends comm_group α, linear_order α :=
(mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b)
@[to_additive, priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_group.to_ordered_comm_group (α : Type u)
[s : linear_ordered_comm_group α] : ordered_comm_group α :=
{ ..s }
section linear_ordered_add_comm_group
variables [linear_ordered_add_comm_group α] {a b c : α}
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_add_comm_group.to_linear_ordered_cancel_add_comm_monoid :
linear_ordered_cancel_add_comm_monoid α :=
{ le_of_add_le_add_left := λ x y z, le_of_add_le_add_left,
add_left_cancel := λ x y z, add_left_cancel,
add_right_cancel := λ x y z, add_right_cancel,
..‹linear_ordered_add_comm_group α› }
lemma linear_ordered_add_comm_group.add_lt_add_left
(a b : α) (h : a < b) (c : α) : c + a < c + b :=
ordered_add_comm_group.add_lt_add_left a b h c
lemma le_of_forall_pos_le_add [densely_ordered α] (h : ∀ ε : α, 0 < ε → a ≤ b + ε) : a ≤ b :=
le_of_forall_le_of_dense $ λ c hc,
calc a ≤ b + (c - b) : h _ (sub_pos_of_lt hc)
... = c : add_sub_cancel'_right _ _
lemma min_neg_neg (a b : α) : min (-a) (-b) = -max a b :=
eq.symm $ @monotone.map_max α (order_dual α) _ _ has_neg.neg a b $ λ a b, neg_le_neg
lemma max_neg_neg (a b : α) : max (-a) (-b) = -min a b :=
eq.symm $ @monotone.map_min α (order_dual α) _ _ has_neg.neg a b $ λ a b, neg_le_neg
lemma min_sub_sub_right (a b c : α) : min (a - c) (b - c) = min a b - c :=
by simpa only [sub_eq_add_neg] using min_add_add_right a b (-c)
lemma max_sub_sub_right (a b c : α) : max (a - c) (b - c) = max a b - c :=
by simpa only [sub_eq_add_neg] using max_add_add_right a b (-c)
lemma min_sub_sub_left (a b c : α) : min (a - b) (a - c) = a - max b c :=
by simp only [sub_eq_add_neg, min_add_add_left, min_neg_neg]
lemma max_sub_sub_left (a b c : α) : max (a - b) (a - c) = a - min b c :=
by simp only [sub_eq_add_neg, max_add_add_left, max_neg_neg]
lemma max_zero_sub_eq_self (a : α) : max a 0 - max (-a) 0 = a :=
begin
rcases le_total a 0,
{ rw [max_eq_right h, max_eq_left, zero_sub, neg_neg], { rwa [le_neg, neg_zero] } },
{ rw [max_eq_left, max_eq_right, sub_zero], { rwa [neg_le, neg_zero] }, exact h }
end
/-- `abs a` is the absolute value of `a`. -/
def abs (a : α) : α := max a (-a)
lemma abs_of_nonneg (h : 0 ≤ a) : abs a = a :=
max_eq_left $ (neg_nonpos.2 h).trans h
lemma abs_of_pos (h : 0 < a) : abs a = a :=
abs_of_nonneg h.le
lemma abs_of_nonpos (h : a ≤ 0) : abs a = -a :=
max_eq_right $ h.trans (neg_nonneg.2 h)
lemma abs_of_neg (h : a < 0) : abs a = -a :=
abs_of_nonpos h.le
@[simp] lemma abs_zero : abs 0 = (0:α) :=
abs_of_nonneg le_rfl
@[simp] lemma abs_neg (a : α) : abs (-a) = abs a :=
begin unfold abs, rw [max_comm, neg_neg] end
@[simp] lemma abs_pos : 0 < abs a ↔ a ≠ 0 :=
begin
rcases lt_trichotomy a 0 with (ha|rfl|ha),
{ simp [abs_of_neg ha, neg_pos, ha.ne, ha] },
{ simp },
{ simp [abs_of_pos ha, ha, ha.ne.symm] }
end
lemma abs_pos_of_pos (h : 0 < a) : 0 < abs a := abs_pos.2 h.ne.symm
lemma abs_pos_of_neg (h : a < 0) : 0 < abs a := abs_pos.2 h.ne
lemma abs_sub (a b : α) : abs (a - b) = abs (b - a) :=
by rw [← neg_sub, abs_neg]
lemma abs_le' : abs a ≤ b ↔ a ≤ b ∧ -a ≤ b := max_le_iff
lemma abs_le : abs a ≤ b ↔ - b ≤ a ∧ a ≤ b :=
by rw [abs_le', and.comm, neg_le]
lemma neg_le_of_abs_le (h : abs a ≤ b) : -b ≤ a := (abs_le.mp h).1
lemma le_of_abs_le (h : abs a ≤ b) : a ≤ b := (abs_le.mp h).2
lemma le_abs : a ≤ abs b ↔ a ≤ b ∨ a ≤ -b := le_max_iff
lemma le_abs_self (a : α) : a ≤ abs a := le_max_left _ _
lemma neg_le_abs_self (a : α) : -a ≤ abs a := le_max_right _ _
lemma abs_nonneg (a : α) : 0 ≤ abs a :=
(le_total 0 a).elim (λ h, h.trans (le_abs_self a)) (λ h, (neg_nonneg.2 h).trans $ neg_le_abs_self a)
@[simp] lemma abs_abs (a : α) : abs (abs a) = abs a :=
abs_of_nonneg $ abs_nonneg a
@[simp] lemma abs_eq_zero : abs a = 0 ↔ a = 0 :=
not_iff_not.1 $ ne_comm.trans $ (abs_nonneg a).lt_iff_ne.symm.trans abs_pos
@[simp] lemma abs_nonpos_iff {a : α} : abs a ≤ 0 ↔ a = 0 :=
(abs_nonneg a).le_iff_eq.trans abs_eq_zero
lemma abs_lt : abs a < b ↔ - b < a ∧ a < b :=
max_lt_iff.trans $ and.comm.trans $ by rw [neg_lt]
lemma neg_lt_of_abs_lt (h : abs a < b) : -b < a := (abs_lt.mp h).1
lemma lt_of_abs_lt (h : abs a < b) : a < b := (abs_lt.mp h).2
lemma lt_abs : a < abs b ↔ a < b ∨ a < -b := lt_max_iff
lemma max_sub_min_eq_abs' (a b : α) : max a b - min a b = abs (a - b) :=
begin
cases le_total a b with ab ba,
{ rw [max_eq_right ab, min_eq_left ab, abs_of_nonpos, neg_sub], rwa sub_nonpos },
{ rw [max_eq_left ba, min_eq_right ba, abs_of_nonneg], rwa sub_nonneg }
end
lemma max_sub_min_eq_abs (a b : α) : max a b - min a b = abs (b - a) :=
by { rw [abs_sub], exact max_sub_min_eq_abs' _ _ }
lemma abs_add (a b : α) : abs (a + b) ≤ abs a + abs b :=
abs_le.2 ⟨(neg_add (abs a) (abs b)).symm ▸
add_le_add (neg_le.2 $ neg_le_abs_self _) (neg_le.2 $ neg_le_abs_self _),
add_le_add (le_abs_self _) (le_abs_self _)⟩
lemma abs_sub_le_iff : abs (a - b) ≤ c ↔ a - b ≤ c ∧ b - a ≤ c :=
by rw [abs_le, neg_le_sub_iff_le_add, @sub_le_iff_le_add' _ _ b, and_comm]
lemma abs_sub_lt_iff : abs (a - b) < c ↔ a - b < c ∧ b - a < c :=
by rw [abs_lt, neg_lt_sub_iff_lt_add, @sub_lt_iff_lt_add' _ _ b, and_comm]
lemma sub_le_of_abs_sub_le_left (h : abs (a - b) ≤ c) : b - c ≤ a :=
sub_le.1 $ (abs_sub_le_iff.1 h).2
lemma sub_le_of_abs_sub_le_right (h : abs (a - b) ≤ c) : a - c ≤ b :=
sub_le_of_abs_sub_le_left (abs_sub a b ▸ h)
lemma sub_lt_of_abs_sub_lt_left (h : abs (a - b) < c) : b - c < a :=
sub_lt.1 $ (abs_sub_lt_iff.1 h).2
lemma sub_lt_of_abs_sub_lt_right (h : abs (a - b) < c) : a - c < b :=
sub_lt_of_abs_sub_lt_left (abs_sub a b ▸ h)
lemma abs_sub_abs_le_abs_sub (a b : α) : abs a - abs b ≤ abs (a - b) :=
sub_le_iff_le_add.2 $
calc abs a = abs (a - b + b) : by rw [sub_add_cancel]
... ≤ abs (a - b) + abs b : abs_add _ _
lemma abs_abs_sub_abs_le_abs_sub (a b : α) : abs (abs a - abs b) ≤ abs (a - b) :=
abs_sub_le_iff.2 ⟨abs_sub_abs_le_abs_sub _ _, by rw abs_sub; apply abs_sub_abs_le_abs_sub⟩
lemma abs_eq (hb : 0 ≤ b) : abs a = b ↔ a = b ∨ a = -b :=
iff.intro
begin
cases le_total a 0 with a_nonpos a_nonneg,
{ rw [abs_of_nonpos a_nonpos, neg_eq_iff_neg_eq, eq_comm], exact or.inr },
{ rw [abs_of_nonneg a_nonneg, eq_comm], exact or.inl }
end
(by intro h; cases h; subst h; try { rw abs_neg }; exact abs_of_nonneg hb)
lemma abs_le_max_abs_abs (hab : a ≤ b) (hbc : b ≤ c) : abs b ≤ max (abs a) (abs c) :=
abs_le'.2
⟨by simp [hbc.trans (le_abs_self c)],
by simp [(neg_le_neg hab).trans (neg_le_abs_self a)]⟩
theorem abs_le_abs (h₀ : a ≤ b) (h₁ : -a ≤ b) : abs a ≤ abs b :=
(abs_le'.2 ⟨h₀, h₁⟩).trans (le_abs_self b)
lemma abs_max_sub_max_le_abs (a b c : α) : abs (max a c - max b c) ≤ abs (a - b) :=
begin
simp_rw [abs_le, le_sub_iff_add_le, sub_le_iff_le_add, ← max_add_add_left],
split; apply max_le_max; simp only [← le_sub_iff_add_le, ← sub_le_iff_le_add, sub_self, neg_le,
neg_le_abs_self, neg_zero, abs_nonneg, le_abs_self]
end
lemma eq_zero_of_neg_eq {a : α} (h : -a = a) : a = 0 :=
match lt_trichotomy a 0 with
| or.inl h₁ :=
have 0 < a, from h ▸ neg_pos_of_neg h₁,
absurd h₁ this.asymm
| or.inr (or.inl h₁) := h₁
| or.inr (or.inr h₁) :=
have a < 0, from h ▸ neg_neg_of_pos h₁,
absurd h₁ this.asymm
end
lemma eq_of_abs_sub_eq_zero {a b : α} (h : abs (a - b) = 0) : a = b :=
sub_eq_zero.1 $ abs_eq_zero.1 h
lemma abs_by_cases (P : α → Prop) {a : α} (h1 : P a) (h2 : P (-a)) : P (abs a) :=
sup_ind _ _ h1 h2
lemma abs_sub_le (a b c : α) : abs (a - c) ≤ abs (a - b) + abs (b - c) :=
calc
abs (a - c) = abs (a - b + (b - c)) : by rw [sub_add_sub_cancel]
... ≤ abs (a - b) + abs (b - c) : abs_add _ _
lemma abs_add_three (a b c : α) : abs (a + b + c) ≤ abs a + abs b + abs c :=
(abs_add _ _).trans (add_le_add_right (abs_add _ _) _)
lemma dist_bdd_within_interval {a b lb ub : α} (hal : lb ≤ a) (hau : a ≤ ub)
(hbl : lb ≤ b) (hbu : b ≤ ub) : abs (a - b) ≤ ub - lb :=
abs_sub_le_iff.2 ⟨sub_le_sub hau hbl, sub_le_sub hbu hal⟩
lemma eq_of_abs_sub_nonpos (h : abs (a - b) ≤ 0) : a = b :=
eq_of_abs_sub_eq_zero (le_antisymm h (abs_nonneg (a - b)))
lemma exists_zero_lt [nontrivial α] : ∃ (a:α), 0 < a :=
begin
obtain ⟨y, hy⟩ := exists_ne (0 : α),
cases hy.lt_or_lt,
{ exact ⟨- y, neg_pos.mpr h⟩ },
{ exact ⟨y, h⟩ }
end
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_add_comm_group.to_no_top_order [nontrivial α] :
no_top_order α :=
⟨ begin
obtain ⟨y, hy⟩ : ∃ (a:α), 0 < a := exists_zero_lt,
exact λ a, ⟨a + y, lt_add_of_pos_right a hy⟩
end ⟩
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_add_comm_group.to_no_bot_order [nontrivial α] : no_bot_order α :=
⟨ begin
obtain ⟨y, hy⟩ : ∃ (a:α), 0 < a := exists_zero_lt,
exact λ a, ⟨a - y, sub_lt_self a hy⟩
end ⟩
end linear_ordered_add_comm_group
/-- This is not so much a new structure as a construction mechanism
for ordered groups, by specifying only the "positive cone" of the group. -/
class nonneg_add_comm_group (α : Type*) extends add_comm_group α :=
(nonneg : α → Prop)
(pos : α → Prop := λ a, nonneg a ∧ ¬ nonneg (neg a))
(pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬ nonneg (-a) . order_laws_tac)
(zero_nonneg : nonneg 0)
(add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b))
(nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0)
namespace nonneg_add_comm_group
variable [s : nonneg_add_comm_group α]
include s
@[reducible, priority 100] -- see Note [lower instance priority]
instance to_ordered_add_comm_group : ordered_add_comm_group α :=
{ le := λ a b, nonneg (b - a),
lt := λ a b, pos (b - a),
lt_iff_le_not_le := λ a b, by simp; rw [pos_iff]; simp,
le_refl := λ a, by simp [zero_nonneg],
le_trans := λ a b c nab nbc, by simp [-sub_eq_add_neg];
rw ← sub_add_sub_cancel; exact add_nonneg nbc nab,
le_antisymm := λ a b nab nba, eq_of_sub_eq_zero $
nonneg_antisymm nba (by rw neg_sub; exact nab),
add_le_add_left := λ a b nab c, by simpa [(≤), preorder.le] using nab,
..s }
theorem nonneg_def {a : α} : nonneg a ↔ 0 ≤ a :=
show _ ↔ nonneg _, by simp
theorem pos_def {a : α} : pos a ↔ 0 < a :=
show _ ↔ pos _, by simp
theorem not_zero_pos : ¬ pos (0 : α) :=
mt pos_def.1 (lt_irrefl _)
theorem zero_lt_iff_nonneg_nonneg {a : α} :
0 < a ↔ nonneg a ∧ ¬ nonneg (-a) :=
pos_def.symm.trans (pos_iff _)
theorem nonneg_total_iff :
(∀ a : α, nonneg a ∨ nonneg (-a)) ↔
(∀ a b : α, a ≤ b ∨ b ≤ a) :=
⟨λ h a b, by have := h (b - a); rwa [neg_sub] at this,
λ h a, by rw [nonneg_def, nonneg_def, neg_nonneg]; apply h⟩
/--
A `nonneg_add_comm_group` is a `linear_ordered_add_comm_group`
if `nonneg` is total and decidable.
-/
def to_linear_ordered_add_comm_group
[decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: linear_ordered_add_comm_group α :=
{ le := (≤),
lt := (<),
le_total := nonneg_total_iff.1 nonneg_total,
decidable_le := by apply_instance,
decidable_lt := by apply_instance,
..@nonneg_add_comm_group.to_ordered_add_comm_group _ s }
end nonneg_add_comm_group
namespace order_dual
instance [ordered_add_comm_group α] : ordered_add_comm_group (order_dual α) :=
{ add_left_neg := λ a : α, add_left_neg a,
sub := λ a b, (a - b : α),
..order_dual.ordered_add_comm_monoid,
..show add_comm_group α, by apply_instance }
instance [linear_ordered_add_comm_group α] :
linear_ordered_add_comm_group (order_dual α) :=
{ add_le_add_left := λ a b h c, @add_le_add_left α _ b a h _,
..order_dual.linear_order α,
..show add_comm_group α, by apply_instance }
end order_dual
namespace prod
variables {G H : Type*}
@[to_additive]
instance [ordered_comm_group G] [ordered_comm_group H] :
ordered_comm_group (G × H) :=
{ .. prod.comm_group, .. prod.partial_order G H, .. prod.ordered_cancel_comm_monoid }
end prod
section type_tags
instance [ordered_add_comm_group α] : ordered_comm_group (multiplicative α) :=
{ ..multiplicative.comm_group,
..multiplicative.ordered_comm_monoid }
instance [ordered_comm_group α] : ordered_add_comm_group (additive α) :=
{ ..additive.add_comm_group,
..additive.ordered_add_comm_monoid }
end type_tags
|
4cdf74aef7b0e2ac0a22e3d55af682e350935a24 | 785b41b0993f39cbfa9b02fe0940ce3f2f51a57d | /conf/all4.lean | 5cb2d19a17bc2725001dc1698e6138f56553f98b | [
"MIT"
] | permissive | loso3000/OpenWrt-DIY-1 | 75b0d70314d703203508218a29acefc3b914d32d | 5858be81ee44199908cbaa1a752b17505c9834e8 | refs/heads/main | 1,690,532,461,283 | 1,631,008,241,000 | 1,631,008,241,000 | 354,817,508 | 1 | 0 | MIT | 1,617,623,493,000 | 1,617,623,492,000 | null | UTF-8 | Lean | false | false | 13,115 | lean | CONFIG_TARGET_x86=y
CONFIG_TARGET_x86_64=y
CONFIG_TARGET_x86_64_DEVICE_generic=y
# 设置固件大小
CONFIG_TARGET_KERNEL_PARTSIZE=64
CONFIG_TARGET_ROOTFS_PARTSIZE=1516
# EFI支持:
CONFIG_GRUB_IMAGES=y
CONFIG_EFI_IMAGES=y
# CONFIG_VMDK_IMAGES is not set
# 不压缩efi
CONFIG_TARGET_ROOTFS_TARGZ=n
# CONFIG_TARGET_IMAGES_GZIP is not set
# Wireless
CONFIG_PACKAGE_wpad-basic-wolfssl=y
#ipv6
# CONFIG_PACKAGE_ipv6helper is not set
# CONFIG_PACKAGE_dnsmasq_full_dhcpv6 is not set
#添加SD卡支持
CONFIG_PACKAGE_kmod-mmc=y
CONFIG_PACKAGE_kmod-sdhci=y
#添加USB扩展支持
CONFIG_PACKAGE_block-mount=y
CONFIG_PACKAGE_librt=y
# x86
CONFIG_PACKAGE_kmod-usb-hid=y
CONFIG_PACKAGE_qemu-ga=y
CONFIG_PACKAGE_lm-sensors-detect=y
CONFIG_PACKAGE_kmod-bonding=y
CONFIG_PACKAGE_kmod-mmc-spi=y
CONFIG_PACKAGE_ppp-mod-pptp=y #VPN客户端
CONFIG_PACKAGE_kmod-vmxnet3=y
CONFIG_PACKAGE_kmod-igbvf=y
CONFIG_PACKAGE_kmod-ixgbe=y
CONFIG_PACKAGE_kmod-pcnet32=y
CONFIG_PACKAGE_kmod-r8125=y
CONFIG_PACKAGE_kmod-8139cp=y
CONFIG_PACKAGE_kmod-8139too=y
CONFIG_PACKAGE_kmod-rtl8xxxu=y
CONFIG_PACKAGE_kmod-i40e=y
CONFIG_PACKAGE_kmod-i40evf=y
CONFIG_PACKAGE_kmod-ath5k=y
CONFIG_PACKAGE_kmod-ath9k=y
CONFIG_PACKAGE_kmod-ath9k-htc=y
CONFIG_PACKAGE_kmod-ath10k=y
CONFIG_PACKAGE_kmod-rt2800-usb=y
CONFIG_PACKAGE_kmod-mlx4-core=y
CONFIG_PACKAGE_kmod-mlx5-core=y
CONFIG_PACKAGE_kmod-alx=y
CONFIG_PACKAGE_kmod-tulip=y
CONFIG_PACKAGE_kmod-tg3=y
CONFIG_PACKAGE_ntfs-3g=y
# CONFIG_PACKAGE_kmod-fs-antfs is not set
# CONFIG_PACKAGE_kmod-fs-ntfs is not set
CONFIG_PACKAGE_ath10k-firmware-qca9888=y
CONFIG_PACKAGE_ath10k-firmware-qca988x=y
CONFIG_PACKAGE_ath10k-firmware-qca9984=y
CONFIG_PACKAGE_brcmfmac-firmware-43602a1-pcie=y
CONFIG_PACKAGE_kmod-ac97=y
CONFIG_PACKAGE_kmod-sound-via82xx=y
CONFIG_PACKAGE_alsa-utils=y
CONFIG_PACKAGE_kmod-iwlwifi=y
# 工具
CONFIG_PACKAGE_acpid=y
CONFIG_PACKAGE_blkid=y
CONFIG_PACKAGE_smartmontools=y
CONFIG_PACKAGE_open-vm-tools=n # 虚拟机支持管理性能更好
CONFIG_PACKAGE_ethtool=y # 网卡工具
CONFIG_PACKAGE_iperf3=y # 局域网测速
CONFIG_PACKAGE_snmpd=y # 旁路由穿透显示真机器MAC
# CONFIG_PACKAGE_parted=n # 128个区分区工具z
CONFIG_PACKAGE_fdisk=y # 分区工具
CONFIG_PACKAGE_hdparm=y # 移动硬盘设置
CONFIG_PACKAGE_curl=y
# nfs
CONFIG_PACKAGE_kmod-fs-nfsd=y
CONFIG_PACKAGE_kmod-fs-nfs=y
CONFIG_PACKAGE_kmod-fs-nfs-v4=y
# Sound Support
CONFIG_PACKAGE_kmod-sound-core=y
CONFIG_PACKAGE_kmod-sound-hda-core=y
CONFIG_PACKAGE_kmod-sound-hda-codec-realtek=y
CONFIG_PACKAGE_kmod-sound-hda-codec-via=y
CONFIG_PACKAGE_kmod-sound-hda-intel=y
CONFIG_PACKAGE_kmod-sound-hda-codec-hdmi=y
# L2TP
CONFIG_PACKAGE_kmod-pppol2tp=y
# ipsec-vpnd
CONFIG_PACKAGE_kmod-crypto-authenc=y
CONFIG_PACKAGE_kmod-ipsec=y
CONFIG_PACKAGE_kmod-ipsec4=y
CONFIG_PACKAGE_kmod-ipsec6=y
CONFIG_PACKAGE_kmod-ipt-ipsec=y
# cifsmount
CONFIG_PACKAGE_kmod-fs-cifs=y
CONFIG_PACKAGE_kmod-nls-utf8=y
CONFIG_PACKAGE_kmod-crypto-misc=y
# eqos
CONFIG_PACKAGE_kmod-ifb=y
# map
CONFIG_PACKAGE_kmod-ip6-tunnel=y
CONFIG_PACKAGE_kmod-nat46=y
# ebtables
CONFIG_PACKAGE_kmod-ebtables=y
CONFIG_PACKAGE_kmod-ebtables-ipv4=y
CONFIG_PACKAGE_kmod-ebtables-ipv6=y
#add upnp
# CONFIG_PACKAGE_irqbalance=y
CONFIG_PACKAGE_luci-app-upnp=y
CONFIG_PACKAGE_luci-app-boostupnp=n
# CONFIG_PACKAGE_luci-app-wol is not set
CONFIG_PACKAGE_luci-app-wolplus=y
# base插件
CONFIG_PACKAGE_ddns-scripts_cloudflare.com-v4=y
CONFIG_PACKAGE_ddns-scripts=y
CONFIG_PACKAGE_ddns-scripts_freedns_42_pl=y
CONFIG_PACKAGE_ddns-scripts_godaddy.com-v1=y
CONFIG_PACKAGE_ddns-scripts_no-ip_com=y
CONFIG_PACKAGE_ddns-scripts_nsupdate=y
CONFIG_PACKAGE_ddns-scripts_route53-v1=y
# CONFIG_PACKAGE_autosamba is not set
CONFIG_PACKAGE_autosamba-ksmbd=n
CONFIG_PACKAGE_autosamba-samba4=y
# CONFIG_PACKAGE_luci-app-accesscontrol is not set
# CONFIG_PACKAGE_luci-app-adbyby-plus is not set
CONFIG_PACKAGE_luci-app-adguardhome=y
CONFIG_PACKAGE_luci-app-advanced=y
CONFIG_PACKAGE_luci-app-autotimeset=n
CONFIG_PACKAGE_luci-app-rebootschedule=y
# CONFIG_PACKAGE_luci-app-autoreboot is not set
CONFIG_PACKAGE_luci-app-control-timewol=y
CONFIG_PACKAGE_luci-app-control-weburl=y
CONFIG_PACKAGE_luci-app-control-webrestriction=n
CONFIG_PACKAGE_luci-app-control-speedlimit=y
CONFIG_PACKAGE_luci-app-timecontrol=y
CONFIG_PACKAGE_luci-app-webadmin=y
CONFIG_PACKAGE_luci-app-cpulimit=y
CONFIG_PACKAGE_luci-app-diskman=y
CONFIG_PACKAGE_luci-app-diskman_INCLUDE_kmod_md_linear=n
CONFIG_PACKAGE_luci-app-diskman_INCLUDE_kmod_md_raid456=n
CONFIG_PACKAGE_luci-app-diskman_INCLUDE_mdadm=y
CONFIG_PACKAGE_luci-app-eqos=n
CONFIG_PACKAGE_luci-app-hd-idle=y
CONFIG_PACKAGE_luci-app-jd-dailybonus=y
CONFIG_PACKAGE_luci-app-koolproxyR=y
CONFIG_PACKAGE_luci-app-netdata=y
CONFIG_PACKAGE_luci-app-onliner=n
CONFIG_PACKAGE_luci-app-openclash=y
# CONFIG_PACKAGE_luci-app-samba is not set
CONFIG_PACKAGE_luci-app-samba4=y
CONFIG_PACKAGE_luci-app-serverchan=y
# CONFIG_PACKAGE_luci-app-sfe is not set
CONFIG_PACKAGE_luci-app-flowoffload=n
# CONFIG_PACKAGE_luci-app-filetransfer is not set
CONFIG_PACKAGE_luci-app-smartdns=y
CONFIG_PACKAGE_luci-app-passwall=y
# CONFIG_PACKAGE_luci-app-ssr-plus is not set
CONFIG_PACKAGE_luci-app-ssrpro=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Kcptun=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_NaiveProxy=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Redsocks2=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Shadowsocks_Libev_Client=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Shadowsocks_Libev_Server=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_ShadowsocksR_Libev_Client=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_ShadowsocksR_Libev_Server=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Simple_Obfs=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Trojan=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_V2ray_Plugin=y
CONFIG_PACKAGE_luci-app-ssrpro_INCLUDE_Xray=y
CONFIG_PACKAGE_luci-app-ttyd=y
# CONFIG_PACKAGE_luci-app-turboacc is not set
# CONFIG_PACKAGE_luci-app-turboacc_INCLUDE_flow-offload=n
# CONFIG_PACKAGE_luci-app-turboacc_INCLUDE_shortcut-fe=y
CONFIG_PACKAGE_luci-app-vssr=y
CONFIG_PACKAGE_luci-app-wrtbwmon=y
CONFIG_PACKAGE_luci-app-nlbwmon=y
CONFIG_PACKAGE_luci-app-netspeedtest=y
CONFIG_PACKAGE_luci-app-bypass=n
# CONFIG_PACKAGE_luci-app-bypass_INCLUDE_NaiveProxy=n
# CONFIG_PACKAGE_luci-app-bypass_INCLUDE_V2ray=n
CONFIG_PACKAGE_luci-app-dnsfilter=y
CONFIG_PACKAGE_luci-app-dnsto=y
CONFIG_PACKAGE_luci-app-vsftpd=y
CONFIG_PACKAGE_luci-app-kodexplorer=y
CONFIG_PACKAGE_luci-app-uhttpd=y
CONFIG_PACKAGE_luci-app-n2n_v2=y
CONFIG_PACKAGE_luci-app-adblock-plus=n
# 主题
CONFIG_PACKAGE_luci-theme-argon_new=y
CONFIG_PACKAGE_luci-theme-btmod=n
CONFIG_PACKAGE_luci-theme-opentomcat=n
CONFIG_PACKAGE_luci-theme-opentopd=y
# 增加其它插件
CONFIG_PACKAGE_luci-app-ksmbd=n
CONFIG_PACKAGE_luci-app-cifsd=n
CONFIG_PACKAGE_luci-app-cifs-mount=y
CONFIG_PACKAGE_luci-app-xlnetacc=y
CONFIG_PACKAGE_luci-app-zerotier=y
CONFIG_PACKAGE_luci-app-mwan3=n
CONFIG_PACKAGE_luci-app-unblockneteasemusic=y
CONFIG_PACKAGE_luci-app-unblockmusic=y
CONFIG_UnblockNeteaseMusic_Go=y
CONFIG_UnblockNeteaseMusic_NodeJS=y
CONFIG_PACKAGE_luci-app-minidlna=y
CONFIG_PACKAGE_luci-app-rclone=y
CONFIG_PACKAGE_luci-app-rclone_INCLUDE_fuse-utils=y
CONFIG_PACKAGE_luci-app-rclone_INCLUDE_rclone-ng=y
CONFIG_PACKAGE_luci-app-rclone_INCLUDE_rclone-webui=y
CONFIG_PACKAGE_luci-app-pptp-server=y
CONFIG_PACKAGE_luci-app-pppoe-server=n
CONFIG_PACKAGE_luci-app-ipsec-server=n
CONFIG_PACKAGE_luci-app-ipsec-vpnd=y
CONFIG_PACKAGE_luci-app-ipsec-vpnserver-manyusers=n
CONFIG_PACKAGE_luci-app-docker=n
CONFIG_PACKAGE_luci-app-dockerman=n
CONFIG_PACKAGE_luci-app-koolddns=y
CONFIG_PACKAGE_luci-app-syncdial=n
CONFIG_PACKAGE_luci-app-softethervpn=y
CONFIG_PACKAGE_luci-app-uugamebooster=y
CONFIG_DEFAULT_luci-app-cpufreq=n
CONFIG_PACKAGE_luci-app-udpxy=n
CONFIG_PACKAGE_luci-app-socat=n
CONFIG_PACKAGE_luci-app-oaf=n
CONFIG_PACKAGE_luci-app-transmission=y
CONFIG_PACKAGE_luci-app-usb-printer=y
CONFIG_PACKAGE_luci-app-mwan3helper=n
CONFIG_PACKAGE_luci-app-qbittorrent=y
CONFIG_PACKAGE_luci-app-familycloud=n
CONFIG_PACKAGE_luci-app-nps=y
CONFIG_PACKAGE_luci-app-frpc=y
# CONFIG_PACKAGE_luci-app-nfs=n
CONFIG_PACKAGE_luci-app-openvpn-server=y
CONFIG_PACKAGE_luci-app-aria2=y
CONFIG_PACKAGE_luci-app-openvpn=y
CONFIG_PACKAGE_luci-app-ttnode=n
CONFIG_PACKAGE_luci-app-easymesh=n
CONFIG_PACKAGE_luci-app-amule=n
# network
# CONFIG_PACKAGE_r8169-firmware=n
CONFIG_PACKAGE_bnx2x-firmware=y
CONFIG_PACKAGE_e100-firmware=y
CONFIG_PACKAGE_kmod-3c59x=y
CONFIG_PACKAGE_kmod-alx=y
CONFIG_PACKAGE_kmod-atl1=y
CONFIG_PACKAGE_kmod-atl1c=y
CONFIG_PACKAGE_kmod-atl1e=y
CONFIG_PACKAGE_kmod-atl2=y
CONFIG_PACKAGE_kmod-atm=y
CONFIG_PACKAGE_kmod-b44=y
CONFIG_PACKAGE_kmod-be2net=y
CONFIG_PACKAGE_kmod-bnx2x=y
CONFIG_PACKAGE_kmod-dm9000=y
CONFIG_PACKAGE_kmod-dummy=y
CONFIG_PACKAGE_kmod-e100=y
CONFIG_PACKAGE_kmod-et131x=y
CONFIG_PACKAGE_kmod-ethoc=y
CONFIG_PACKAGE_kmod-hfcmulti=y
CONFIG_PACKAGE_kmod-hfcpci=y
CONFIG_PACKAGE_kmod-iavf=y
CONFIG_PACKAGE_kmod-ifb=y
CONFIG_PACKAGE_kmod-ixgbevf=y
CONFIG_PACKAGE_kmod-lib-crc32c=y
CONFIG_PACKAGE_kmod-mdio-gpio=y
CONFIG_PACKAGE_kmod-misdn=y
CONFIG_PACKAGE_kmod-natsemi=y
CONFIG_PACKAGE_kmod-ne2k-pci=y
CONFIG_PACKAGE_kmod-niu=y
CONFIG_PACKAGE_kmod-of-mdio=y
CONFIG_PACKAGE_kmod-phy-bcm84881=y
CONFIG_PACKAGE_kmod-phy-broadcom=y
CONFIG_PACKAGE_kmod-phy-realtek=y
CONFIG_PACKAGE_kmod-phylib-broadcom=y
CONFIG_PACKAGE_kmod-phylink=y
CONFIG_PACKAGE_kmod-r8169=y
CONFIG_PACKAGE_kmod-r8168=y
CONFIG_PACKAGE_kmod-r8125=y
CONFIG_PACKAGE_kmod-random-core=y
CONFIG_PACKAGE_kmod-sfp=y
CONFIG_PACKAGE_kmod-siit=y
CONFIG_PACKAGE_kmod-sis190=y
CONFIG_PACKAGE_kmod-sis900=y
CONFIG_PACKAGE_kmod-skge=y
CONFIG_PACKAGE_kmod-sky2=y
CONFIG_PACKAGE_kmod-solos-pci=y
CONFIG_PACKAGE_kmod-spi-ks8995=y
CONFIG_PACKAGE_kmod-ssb=y
CONFIG_PACKAGE_kmod-swconfig=y
CONFIG_PACKAGE_kmod-switch-bcm53xx=y
CONFIG_PACKAGE_kmod-switch-bcm53xx-mdio=y
CONFIG_PACKAGE_kmod-switch-ip17xx=y
CONFIG_PACKAGE_kmod-switch-mvsw61xx=y
CONFIG_PACKAGE_kmod-switch-rtl8306=y
CONFIG_PACKAGE_kmod-switch-rtl8366-smi=y
CONFIG_PACKAGE_kmod-switch-rtl8366rb=y
CONFIG_PACKAGE_kmod-switch-rtl8366s=y
CONFIG_PACKAGE_kmod-switch-rtl8367b=y
CONFIG_PACKAGE_kmod-tg3=y
CONFIG_PACKAGE_kmod-usb-acm=y
CONFIG_PACKAGE_kmod-usb-atm=y
CONFIG_PACKAGE_kmod-usb-atm-cxacru=y
CONFIG_PACKAGE_kmod-usb-atm-speedtouch=y
CONFIG_PACKAGE_kmod-usb-atm-ueagle=y
CONFIG_PACKAGE_kmod-usb-cm109=y
CONFIG_PACKAGE_kmod-usb-dwc2=y
CONFIG_PACKAGE_kmod-usb-dwc3=y
CONFIG_PACKAGE_kmod-usb-ehci=y
CONFIG_PACKAGE_kmod-usb-ledtrig-usbport=y
CONFIG_PACKAGE_kmod-usb-net-cdc-eem=y
CONFIG_PACKAGE_kmod-usb-net-cdc-ether=y
CONFIG_PACKAGE_kmod-usb-net-cdc-mbim=y
CONFIG_PACKAGE_kmod-usb-net-cdc-ncm=y
CONFIG_PACKAGE_kmod-usb-net-cdc-subset=y
CONFIG_PACKAGE_kmod-usb-net-dm9601-ether=y
CONFIG_PACKAGE_kmod-usb-net-hso=y
CONFIG_PACKAGE_kmod-usb-net-huawei-cdc-ncm=y
CONFIG_PACKAGE_kmod-usb-net-ipheth=y
CONFIG_PACKAGE_kmod-usb-net-kalmia=y
CONFIG_PACKAGE_kmod-usb-net-kaweth=y
CONFIG_PACKAGE_kmod-usb-net-mcs7830=y
CONFIG_PACKAGE_kmod-usb-net-pegasus=y
CONFIG_PACKAGE_kmod-usb-net-pl=y
CONFIG_PACKAGE_kmod-usb-net-qmi-wwan=y
CONFIG_PACKAGE_kmod-usb-net-rndis=y
CONFIG_PACKAGE_kmod-usb-net-sierrawireless=y
CONFIG_PACKAGE_kmod-usb-net-smsc95xx=y
CONFIG_PACKAGE_kmod-usb-net-sr9700=y
CONFIG_PACKAGE_kmod-usb-ohci=y
CONFIG_PACKAGE_kmod-usb-ohci-pci=y
CONFIG_PACKAGE_kmod-usb-printer=y
CONFIG_PACKAGE_kmod-usb-uhci=y
CONFIG_PACKAGE_kmod-usb-wdm=y
CONFIG_PACKAGE_kmod-usb-yealink=y
CONFIG_PACKAGE_kmod-usb2=y
CONFIG_PACKAGE_kmod-usb2-pci=y
CONFIG_PACKAGE_kmod-usb3=y
CONFIG_PACKAGE_kmod-usbip=y
CONFIG_PACKAGE_kmod-usbip-client=y
CONFIG_PACKAGE_kmod-usbip-server=y
CONFIG_PACKAGE_kmod-usbmon=y
CONFIG_PACKAGE_kmod-via-rhine=y
CONFIG_PACKAGE_kmod-via-velocity=y
# add drive
CONFIG_PACKAGE_kmod-usb-net-rtl8150=y
CONFIG_PACKAGE_kmod-usb-net-rtl8152-vendor=y
CONFIG_PACKAGE_kmod-usb-net-asix-ax88179=y
CONFIG_PACKAGE_kmod-usb-storage=y
# Block Devices
CONFIG_PACKAGE_kmod-ata-core=y
CONFIG_PACKAGE_kmod-block2mtd=y
CONFIG_PACKAGE_kmod-scsi-core=y
CONFIG_PACKAGE_kmod-scsi-generic=y
CONFIG_PACKAGE_blockd=y
CONFIG_PACKAGE_kmod-ata-ahci=y
CONFIG_PACKAGE_kmod-ata-artop=y
CONFIG_PACKAGE_kmod-ata-marvell-sata=y
CONFIG_PACKAGE_kmod-ata-nvidia-sata=y
CONFIG_PACKAGE_kmod-ata-pdc202xx-old=y
CONFIG_PACKAGE_kmod-ata-piix=y
CONFIG_PACKAGE_kmod-ata-sil=y
CONFIG_PACKAGE_kmod-ata-sil24=y
CONFIG_PACKAGE_kmod-ata-via-sata=y
CONFIG_PACKAGE_kmod-dax=y
CONFIG_PACKAGE_kmod-dm=y
CONFIG_PACKAGE_kmod-dm-raid=y
CONFIG_PACKAGE_kmod-fs-autofs4=y
CONFIG_PACKAGE_kmod-lib-crc32c=y
CONFIG_PACKAGE_kmod-lib-raid6=y
CONFIG_PACKAGE_kmod-lib-xor=y
CONFIG_PACKAGE_kmod-md-mod=y
CONFIG_PACKAGE_kmod-md-raid0=y
CONFIG_PACKAGE_kmod-md-raid1=y
CONFIG_PACKAGE_kmod-md-raid10=y
CONFIG_PACKAGE_kmod-md-raid456=y
# 3G/4G Support
CONFIG_PACKAGE_kmod-usb-serial=y
CONFIG_PACKAGE_kmod-usb-serial-option=y
CONFIG_PACKAGE_kmod-usb-serial-wwan=y
CONFIG_PACKAGE_kmod-mii=y
CONFIG_PACKAGE_kmod-usb-acm=y
# docker
CONFIG_PACKAGE_kmod-fs-btrfs=y
CONFIG_PACKAGE_kmod-dm=y
CONFIG_PACKAGE_kmod-br-netfilter=y
CONFIG_PACKAGE_kmod-ikconfig=y
CONFIG_PACKAGE_kmod-nf-conntrack-netlink=y
CONFIG_PACKAGE_kmod-nf-ipvs=y
CONFIG_PACKAGE_kmod-veth=y
CONFIG_PACKAGE_kmod-dummy=y
|
f221f3b4d2aa8c9a6e7e431bc0218fbdc108d7a2 | d29d82a0af640c937e499f6be79fc552eae0aa13 | /src/data/matrix/basic.lean | f856492dfbc12e7ca7f59b8be851d08ab81fce73 | [
"Apache-2.0"
] | permissive | AbdulMajeedkhurasani/mathlib | 835f8a5c5cf3075b250b3737172043ab4fa1edf6 | 79bc7323b164aebd000524ebafd198eb0e17f956 | refs/heads/master | 1,688,003,895,660 | 1,627,788,521,000 | 1,627,788,521,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 49,300 | lean | /-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import algebra.big_operators.pi
import algebra.module.pi
import algebra.module.linear_map
import algebra.big_operators.ring
import algebra.star.pi
import data.equiv.ring
import data.fintype.card
import data.matrix.dmatrix
/-!
# Matrices
-/
universes u u' v w
open_locale big_operators
open dmatrix
/-- `matrix m n` is the type of matrices whose rows are indexed by the fintype `m`
and whose columns are indexed by the fintype `n`. -/
@[nolint unused_arguments]
def matrix (m : Type u) (n : Type u') [fintype m] [fintype n] (α : Type v) : Type (max u u' v) :=
m → n → α
variables {l m n o : Type*} [fintype l] [fintype m] [fintype n] [fintype o]
variables {m' : o → Type*} [∀ i, fintype (m' i)]
variables {n' : o → Type*} [∀ i, fintype (n' i)]
variables {R : Type*} {S : Type*} {α : Type v} {β : Type w}
namespace matrix
section ext
variables {M N : matrix m n α}
theorem ext_iff : (∀ i j, M i j = N i j) ↔ M = N :=
⟨λ h, funext $ λ i, funext $ h i, λ h, by simp [h]⟩
@[ext] theorem ext : (∀ i j, M i j = N i j) → M = N :=
ext_iff.mp
end ext
/-- `M.map f` is the matrix obtained by applying `f` to each entry of the matrix `M`. -/
def map (M : matrix m n α) (f : α → β) : matrix m n β := λ i j, f (M i j)
@[simp]
lemma map_apply {M : matrix m n α} {f : α → β} {i : m} {j : n} :
M.map f i j = f (M i j) := rfl
@[simp]
lemma map_map {M : matrix m n α} {β γ : Type*} {f : α → β} {g : β → γ} :
(M.map f).map g = M.map (g ∘ f) :=
by { ext, simp, }
/-- The transpose of a matrix. -/
def transpose (M : matrix m n α) : matrix n m α
| x y := M y x
localized "postfix `ᵀ`:1500 := matrix.transpose" in matrix
/-- The conjugate transpose of a matrix defined in term of `star`. -/
def conj_transpose [has_star α] (M : matrix m n α) : matrix n m α :=
M.transpose.map star
localized "postfix `ᴴ`:1500 := matrix.conj_transpose" in matrix
/-- `matrix.col u` is the column matrix whose entries are given by `u`. -/
def col (w : m → α) : matrix m unit α
| x y := w x
/-- `matrix.row u` is the row matrix whose entries are given by `u`. -/
def row (v : n → α) : matrix unit n α
| x y := v y
instance [inhabited α] : inhabited (matrix m n α) := pi.inhabited _
instance [has_add α] : has_add (matrix m n α) := pi.has_add
instance [add_semigroup α] : add_semigroup (matrix m n α) := pi.add_semigroup
instance [add_comm_semigroup α] : add_comm_semigroup (matrix m n α) := pi.add_comm_semigroup
instance [has_zero α] : has_zero (matrix m n α) := pi.has_zero
instance [add_monoid α] : add_monoid (matrix m n α) := pi.add_monoid
instance [add_comm_monoid α] : add_comm_monoid (matrix m n α) := pi.add_comm_monoid
instance [has_neg α] : has_neg (matrix m n α) := pi.has_neg
instance [has_sub α] : has_sub (matrix m n α) := pi.has_sub
instance [add_group α] : add_group (matrix m n α) := pi.add_group
instance [add_comm_group α] : add_comm_group (matrix m n α) := pi.add_comm_group
instance [unique α] : unique (matrix m n α) := pi.unique
instance [subsingleton α] : subsingleton (matrix m n α) := pi.subsingleton
instance [nonempty m] [nonempty n] [nontrivial α] : nontrivial (matrix m n α) :=
function.nontrivial
instance [has_scalar R α] : has_scalar R (matrix m n α) := pi.has_scalar
instance [has_scalar R α] [has_scalar S α] [smul_comm_class R S α] :
smul_comm_class R S (matrix m n α) := pi.smul_comm_class
instance [has_scalar R S] [has_scalar R α] [has_scalar S α] [is_scalar_tower R S α] :
is_scalar_tower R S (matrix m n α) := pi.is_scalar_tower
instance [monoid R] [mul_action R α] :
mul_action R (matrix m n α) := pi.mul_action _
instance [monoid R] [add_monoid α] [distrib_mul_action R α] :
distrib_mul_action R (matrix m n α) := pi.distrib_mul_action _
instance [semiring R] [add_comm_monoid α] [module R α] :
module R (matrix m n α) := pi.module _ _ _
@[simp] lemma map_zero [has_zero α] [has_zero β] {f : α → β} (h : f 0 = 0) :
(0 : matrix m n α).map f = 0 :=
by { ext, simp [h], }
lemma map_add [add_monoid α] [add_monoid β] (f : α →+ β)
(M N : matrix m n α) : (M + N).map f = M.map f + N.map f :=
by { ext, simp, }
lemma map_sub [add_group α] [add_group β] (f : α →+ β)
(M N : matrix m n α) : (M - N).map f = M.map f - N.map f :=
by { ext, simp }
lemma map_smul [has_scalar R α] [has_scalar R β] (f : α →[R] β) (r : R)
(M : matrix m n α) : (r • M).map f = r • (M.map f) :=
by { ext, simp, }
-- TODO[gh-6025]: make this an instance once safe to do so
lemma subsingleton_of_empty_left [is_empty m] : subsingleton (matrix m n α) :=
⟨λ M N, by { ext, exact is_empty_elim i }⟩
-- TODO[gh-6025]: make this an instance once safe to do so
lemma subsingleton_of_empty_right [is_empty n] : subsingleton (matrix m n α) :=
⟨λ M N, by { ext, exact is_empty_elim j }⟩
end matrix
/-- The `add_monoid_hom` between spaces of matrices induced by an `add_monoid_hom` between their
coefficients. -/
def add_monoid_hom.map_matrix [add_monoid α] [add_monoid β] (f : α →+ β) :
matrix m n α →+ matrix m n β :=
{ to_fun := λ M, M.map f,
map_zero' := by simp,
map_add' := matrix.map_add f, }
@[simp] lemma add_monoid_hom.map_matrix_apply [add_monoid α] [add_monoid β]
(f : α →+ β) (M : matrix m n α) : f.map_matrix M = M.map f := rfl
/-- The `linear_map` between spaces of matrices induced by a `linear_map` between their
coefficients. -/
@[simps]
def linear_map.map_matrix [semiring R] [add_comm_monoid α] [add_comm_monoid β]
[module R α] [module R β] (f : α →ₗ[R] β) : matrix m n α →ₗ[R] matrix m n β :=
{ to_fun := λ M, M.map f,
map_add' := matrix.map_add f.to_add_monoid_hom,
map_smul' := matrix.map_smul f.to_mul_action_hom, }
open_locale matrix
namespace matrix
section diagonal
variables [decidable_eq n]
/-- `diagonal d` is the square matrix such that `(diagonal d) i i = d i` and `(diagonal d) i j = 0`
if `i ≠ j`. -/
def diagonal [has_zero α] (d : n → α) : matrix n n α := λ i j, if i = j then d i else 0
@[simp] theorem diagonal_apply_eq [has_zero α] {d : n → α} (i : n) : (diagonal d) i i = d i :=
by simp [diagonal]
@[simp] theorem diagonal_apply_ne [has_zero α] {d : n → α} {i j : n} (h : i ≠ j) :
(diagonal d) i j = 0 := by simp [diagonal, h]
theorem diagonal_apply_ne' [has_zero α] {d : n → α} {i j : n} (h : j ≠ i) :
(diagonal d) i j = 0 := diagonal_apply_ne h.symm
@[simp] theorem diagonal_zero [has_zero α] : (diagonal (λ _, 0) : matrix n n α) = 0 :=
by simp [diagonal]; refl
@[simp] lemma diagonal_transpose [has_zero α] (v : n → α) :
(diagonal v)ᵀ = diagonal v :=
begin
ext i j,
by_cases h : i = j,
{ simp [h, transpose] },
{ simp [h, transpose, diagonal_apply_ne' h] }
end
@[simp] theorem diagonal_add [add_monoid α] (d₁ d₂ : n → α) :
diagonal d₁ + diagonal d₂ = diagonal (λ i, d₁ i + d₂ i) :=
by ext i j; by_cases h : i = j; simp [h]
@[simp] lemma diagonal_map [has_zero α] [has_zero β] {f : α → β} (h : f 0 = 0) {d : n → α} :
(diagonal d).map f = diagonal (λ m, f (d m)) :=
by { ext, simp only [diagonal, map_apply], split_ifs; simp [h], }
section one
variables [has_zero α] [has_one α]
instance : has_one (matrix n n α) := ⟨diagonal (λ _, 1)⟩
@[simp] theorem diagonal_one : (diagonal (λ _, 1) : matrix n n α) = 1 := rfl
theorem one_apply {i j} : (1 : matrix n n α) i j = if i = j then 1 else 0 := rfl
@[simp] theorem one_apply_eq (i) : (1 : matrix n n α) i i = 1 := diagonal_apply_eq i
@[simp] theorem one_apply_ne {i j} : i ≠ j → (1 : matrix n n α) i j = 0 :=
diagonal_apply_ne
theorem one_apply_ne' {i j} : j ≠ i → (1 : matrix n n α) i j = 0 :=
diagonal_apply_ne'
@[simp] lemma one_map [has_zero β] [has_one β]
{f : α → β} (h₀ : f 0 = 0) (h₁ : f 1 = 1) :
(1 : matrix n n α).map f = (1 : matrix n n β) :=
by { ext, simp only [one_apply, map_apply], split_ifs; simp [h₀, h₁], }
end one
section numeral
@[simp] lemma bit0_apply [has_add α] (M : matrix m m α) (i : m) (j : m) :
(bit0 M) i j = bit0 (M i j) := rfl
variables [add_monoid α] [has_one α]
lemma bit1_apply (M : matrix n n α) (i : n) (j : n) :
(bit1 M) i j = if i = j then bit1 (M i j) else bit0 (M i j) :=
by dsimp [bit1]; by_cases h : i = j; simp [h]
@[simp]
lemma bit1_apply_eq (M : matrix n n α) (i : n) :
(bit1 M) i i = bit1 (M i i) :=
by simp [bit1_apply]
@[simp]
lemma bit1_apply_ne (M : matrix n n α) {i j : n} (h : i ≠ j) :
(bit1 M) i j = bit0 (M i j) :=
by simp [bit1_apply, h]
end numeral
end diagonal
section dot_product
/-- `dot_product v w` is the sum of the entrywise products `v i * w i` -/
def dot_product [has_mul α] [add_comm_monoid α] (v w : m → α) : α :=
∑ i, v i * w i
lemma dot_product_assoc [non_unital_semiring α] (u : m → α) (v : m → n → α) (w : n → α) :
dot_product (λ j, dot_product u (λ i, v i j)) w = dot_product u (λ i, dot_product (v i) w) :=
by simpa [dot_product, finset.mul_sum, finset.sum_mul, mul_assoc] using finset.sum_comm
lemma dot_product_comm [comm_semiring α] (v w : m → α) :
dot_product v w = dot_product w v :=
by simp_rw [dot_product, mul_comm]
@[simp] lemma dot_product_punit [add_comm_monoid α] [has_mul α] (v w : punit → α) :
dot_product v w = v ⟨⟩ * w ⟨⟩ :=
by simp [dot_product]
section non_unital_non_assoc_semiring
variables [non_unital_non_assoc_semiring α] (u v w : m → α)
@[simp] lemma dot_product_zero : dot_product v 0 = 0 := by simp [dot_product]
@[simp] lemma dot_product_zero' : dot_product v (λ _, 0) = 0 := dot_product_zero v
@[simp] lemma zero_dot_product : dot_product 0 v = 0 := by simp [dot_product]
@[simp] lemma zero_dot_product' : dot_product (λ _, (0 : α)) v = 0 := zero_dot_product v
@[simp] lemma add_dot_product : dot_product (u + v) w = dot_product u w + dot_product v w :=
by simp [dot_product, add_mul, finset.sum_add_distrib]
@[simp] lemma dot_product_add : dot_product u (v + w) = dot_product u v + dot_product u w :=
by simp [dot_product, mul_add, finset.sum_add_distrib]
end non_unital_non_assoc_semiring
section non_unital_non_assoc_semiring_decidable
variables [decidable_eq m] [non_unital_non_assoc_semiring α] (u v w : m → α)
@[simp] lemma diagonal_dot_product (i : m) : dot_product (diagonal v i) w = v i * w i :=
have ∀ j ≠ i, diagonal v i j * w j = 0 := λ j hij, by simp [diagonal_apply_ne' hij],
by convert finset.sum_eq_single i (λ j _, this j) _ using 1; simp
@[simp] lemma dot_product_diagonal (i : m) : dot_product v (diagonal w i) = v i * w i :=
have ∀ j ≠ i, v j * diagonal w i j = 0 := λ j hij, by simp [diagonal_apply_ne' hij],
by convert finset.sum_eq_single i (λ j _, this j) _ using 1; simp
@[simp] lemma dot_product_diagonal' (i : m) : dot_product v (λ j, diagonal w j i) = v i * w i :=
have ∀ j ≠ i, v j * diagonal w j i = 0 := λ j hij, by simp [diagonal_apply_ne hij],
by convert finset.sum_eq_single i (λ j _, this j) _ using 1; simp
end non_unital_non_assoc_semiring_decidable
section ring
variables [ring α] (u v w : m → α)
@[simp] lemma neg_dot_product : dot_product (-v) w = - dot_product v w := by simp [dot_product]
@[simp] lemma dot_product_neg : dot_product v (-w) = - dot_product v w := by simp [dot_product]
@[simp] lemma sub_dot_product : dot_product (u - v) w = dot_product u w - dot_product v w :=
by simp [sub_eq_add_neg]
@[simp] lemma dot_product_sub : dot_product u (v - w) = dot_product u v - dot_product u w :=
by simp [sub_eq_add_neg]
end ring
section distrib_mul_action
variables [monoid R] [has_mul α] [add_comm_monoid α] [distrib_mul_action R α]
@[simp] lemma smul_dot_product [is_scalar_tower R α α] (x : R) (v w : m → α) :
dot_product (x • v) w = x • dot_product v w :=
by simp [dot_product, finset.smul_sum, smul_mul_assoc]
@[simp] lemma dot_product_smul [smul_comm_class R α α] (x : R) (v w : m → α) :
dot_product v (x • w) = x • dot_product v w :=
by simp [dot_product, finset.smul_sum, mul_smul_comm]
end distrib_mul_action
section star_ring
variables [semiring α] [star_ring α] (v w : m → α)
lemma star_dot_product_star : dot_product (star v) (star w) = star (dot_product w v) :=
by simp [dot_product]
lemma star_dot_product : dot_product (star v) w = star (dot_product (star w) v) :=
by simp [dot_product]
lemma dot_product_star : dot_product v (star w) = star (dot_product w (star v)) :=
by simp [dot_product]
end star_ring
end dot_product
/-- `M ⬝ N` is the usual product of matrices `M` and `N`, i.e. we have that
`(M ⬝ N) i k` is the dot product of the `i`-th row of `M` by the `k`-th column of `Ǹ`. -/
protected def mul [has_mul α] [add_comm_monoid α] (M : matrix l m α) (N : matrix m n α) :
matrix l n α :=
λ i k, dot_product (λ j, M i j) (λ j, N j k)
localized "infixl ` ⬝ `:75 := matrix.mul" in matrix
theorem mul_apply [has_mul α] [add_comm_monoid α] {M : matrix l m α} {N : matrix m n α} {i k} :
(M ⬝ N) i k = ∑ j, M i j * N j k := rfl
instance [has_mul α] [add_comm_monoid α] : has_mul (matrix n n α) := ⟨matrix.mul⟩
@[simp] theorem mul_eq_mul [has_mul α] [add_comm_monoid α] (M N : matrix n n α) :
M * N = M ⬝ N := rfl
theorem mul_apply' [has_mul α] [add_comm_monoid α] {M : matrix l m α} {N : matrix m n α} {i k} :
(M ⬝ N) i k = dot_product (λ j, M i j) (λ j, N j k) := rfl
@[simp] theorem diagonal_neg [decidable_eq n] [add_group α] (d : n → α) :
-diagonal d = diagonal (λ i, -d i) :=
by ext i j; by_cases i = j; simp [h]
lemma sum_apply [add_comm_monoid α] (i : m) (j : n)
(s : finset β) (g : β → matrix m n α) :
(∑ c in s, g c) i j = ∑ c in s, g c i j :=
(congr_fun (s.sum_apply i g) j).trans (s.sum_apply j _)
section non_unital_non_assoc_semiring
variables [non_unital_non_assoc_semiring α]
@[simp] protected theorem mul_zero (M : matrix m n α) : M ⬝ (0 : matrix n o α) = 0 :=
by { ext i j, apply dot_product_zero }
@[simp] protected theorem zero_mul (M : matrix m n α) : (0 : matrix l m α) ⬝ M = 0 :=
by { ext i j, apply zero_dot_product }
protected theorem mul_add (L : matrix m n α) (M N : matrix n o α) : L ⬝ (M + N) = L ⬝ M + L ⬝ N :=
by { ext i j, apply dot_product_add }
protected theorem add_mul (L M : matrix l m α) (N : matrix m n α) : (L + M) ⬝ N = L ⬝ N + M ⬝ N :=
by { ext i j, apply add_dot_product }
instance : non_unital_non_assoc_semiring (matrix n n α) :=
{ mul := (*),
add := (+),
zero := 0,
mul_zero := matrix.mul_zero,
zero_mul := matrix.zero_mul,
left_distrib := matrix.mul_add,
right_distrib := matrix.add_mul,
.. matrix.add_comm_monoid}
@[simp] theorem diagonal_mul [decidable_eq m]
(d : m → α) (M : matrix m n α) (i j) : (diagonal d).mul M i j = d i * M i j :=
diagonal_dot_product _ _ _
@[simp] theorem mul_diagonal [decidable_eq n]
(d : n → α) (M : matrix m n α) (i j) : (M ⬝ diagonal d) i j = M i j * d j :=
by { rw ← diagonal_transpose, apply dot_product_diagonal }
@[simp] theorem diagonal_mul_diagonal [decidable_eq n] (d₁ d₂ : n → α) :
(diagonal d₁) ⬝ (diagonal d₂) = diagonal (λ i, d₁ i * d₂ i) :=
by ext i j; by_cases i = j; simp [h]
theorem diagonal_mul_diagonal' [decidable_eq n] (d₁ d₂ : n → α) :
diagonal d₁ * diagonal d₂ = diagonal (λ i, d₁ i * d₂ i) :=
diagonal_mul_diagonal _ _
lemma is_add_monoid_hom_mul_left (M : matrix l m α) :
is_add_monoid_hom (λ x : matrix m n α, M ⬝ x) :=
{ to_is_add_hom := ⟨matrix.mul_add _⟩, map_zero := matrix.mul_zero _ }
lemma is_add_monoid_hom_mul_right (M : matrix m n α) :
is_add_monoid_hom (λ x : matrix l m α, x ⬝ M) :=
{ to_is_add_hom := ⟨λ _ _, matrix.add_mul _ _ _⟩, map_zero := matrix.zero_mul _ }
protected lemma sum_mul (s : finset β) (f : β → matrix l m α)
(M : matrix m n α) : (∑ a in s, f a) ⬝ M = ∑ a in s, f a ⬝ M :=
(@finset.sum_hom _ _ _ _ _ s f (λ x, x ⬝ M) M.is_add_monoid_hom_mul_right).symm
protected lemma mul_sum (s : finset β) (f : β → matrix m n α)
(M : matrix l m α) : M ⬝ ∑ a in s, f a = ∑ a in s, M ⬝ f a :=
(@finset.sum_hom _ _ _ _ _ s f (λ x, M ⬝ x) M.is_add_monoid_hom_mul_left).symm
end non_unital_non_assoc_semiring
section non_assoc_semiring
variables [non_assoc_semiring α]
@[simp] protected theorem one_mul [decidable_eq m] (M : matrix m n α) :
(1 : matrix m m α) ⬝ M = M :=
by ext i j; rw [← diagonal_one, diagonal_mul, one_mul]
@[simp] protected theorem mul_one [decidable_eq n] (M : matrix m n α) :
M ⬝ (1 : matrix n n α) = M :=
by ext i j; rw [← diagonal_one, mul_diagonal, mul_one]
instance [decidable_eq n] : non_assoc_semiring (matrix n n α) :=
{ one := 1,
one_mul := matrix.one_mul,
mul_one := matrix.mul_one,
.. matrix.non_unital_non_assoc_semiring }
@[simp]
lemma map_mul {L : matrix m n α} {M : matrix n o α} [non_assoc_semiring β] {f : α →+* β} :
(L ⬝ M).map f = L.map f ⬝ M.map f :=
by { ext, simp [mul_apply, ring_hom.map_sum], }
end non_assoc_semiring
section non_unital_semiring
variables [non_unital_semiring α]
protected theorem mul_assoc (L : matrix l m α) (M : matrix m n α) (N : matrix n o α) :
(L ⬝ M) ⬝ N = L ⬝ (M ⬝ N) :=
by { ext, apply dot_product_assoc }
instance : non_unital_semiring (matrix n n α) :=
{ mul_assoc := matrix.mul_assoc, ..matrix.non_unital_non_assoc_semiring }
end non_unital_semiring
section semiring
variables [semiring α]
instance [decidable_eq n] : semiring (matrix n n α) :=
{ ..matrix.non_unital_semiring, ..matrix.non_assoc_semiring }
end semiring
section homs
-- TODO: there should be a way to avoid restating these for each `foo_hom`.
/-- A version of `one_map` where `f` is a ring hom. -/
@[simp] lemma ring_hom_map_one [decidable_eq n] [semiring α] [semiring β] (f : α →+* β) :
(1 : matrix n n α).map f = 1 :=
one_map f.map_zero f.map_one
/-- A version of `one_map` where `f` is a `ring_equiv`. -/
@[simp] lemma ring_equiv_map_one [decidable_eq n] [semiring α] [semiring β] (f : α ≃+* β) :
(1 : matrix n n α).map f = 1 :=
one_map f.map_zero f.map_one
/-- A version of `map_zero` where `f` is a `zero_hom`. -/
@[simp] lemma zero_hom_map_zero [has_zero α] [has_zero β] (f : zero_hom α β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `add_monoid_hom`. -/
@[simp] lemma add_monoid_hom_map_zero [add_monoid α] [add_monoid β] (f : α →+ β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `add_equiv`. -/
@[simp] lemma add_equiv_map_zero [add_monoid α] [add_monoid β] (f : α ≃+ β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `linear_map`. -/
@[simp] lemma linear_map_map_zero [semiring R] [add_comm_monoid α] [add_comm_monoid β]
[module R α] [module R β] (f : α →ₗ[R] β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `linear_equiv`. -/
@[simp] lemma linear_equiv_map_zero [semiring R] [add_comm_monoid α] [add_comm_monoid β]
[module R α] [module R β] (f : α ≃ₗ[R] β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `ring_hom`. -/
@[simp] lemma ring_hom_map_zero [semiring α] [semiring β] (f : α →+* β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `ring_equiv`. -/
@[simp] lemma ring_equiv_map_zero [semiring α] [semiring β] (f : α ≃+* β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
end homs
end matrix
/-- The `ring_hom` between spaces of square matrices induced by a `ring_hom` between their
coefficients. -/
@[simps]
def ring_hom.map_matrix [decidable_eq m] [semiring α] [semiring β] (f : α →+* β) :
matrix m m α →+* matrix m m β :=
{ to_fun := λ M, M.map f,
map_one' := by simp,
map_mul' := λ L M, matrix.map_mul,
..(f.to_add_monoid_hom).map_matrix }
open_locale matrix
namespace matrix
section ring
variables [ring α]
@[simp] theorem neg_mul (M : matrix m n α) (N : matrix n o α) :
(-M) ⬝ N = -(M ⬝ N) :=
by { ext, apply neg_dot_product }
@[simp] theorem mul_neg (M : matrix m n α) (N : matrix n o α) :
M ⬝ (-N) = -(M ⬝ N) :=
by { ext, apply dot_product_neg }
protected theorem sub_mul (M M' : matrix m n α) (N : matrix n o α) :
(M - M') ⬝ N = M ⬝ N - M' ⬝ N :=
by rw [sub_eq_add_neg, matrix.add_mul, neg_mul, sub_eq_add_neg]
protected theorem mul_sub (M : matrix m n α) (N N' : matrix n o α) :
M ⬝ (N - N') = M ⬝ N - M ⬝ N' :=
by rw [sub_eq_add_neg, matrix.mul_add, mul_neg, sub_eq_add_neg]
end ring
instance [decidable_eq n] [ring α] : ring (matrix n n α) :=
{ ..matrix.semiring, ..matrix.add_comm_group }
section semiring
variables [semiring α]
lemma smul_eq_diagonal_mul [decidable_eq m] (M : matrix m n α) (a : α) :
a • M = diagonal (λ _, a) ⬝ M :=
by { ext, simp }
@[simp] lemma smul_mul [monoid R] [distrib_mul_action R α] [is_scalar_tower R α α]
(a : R) (M : matrix m n α) (N : matrix n l α) :
(a • M) ⬝ N = a • M ⬝ N :=
by { ext, apply smul_dot_product }
/-- This instance enables use with `smul_mul_assoc`. -/
instance semiring.is_scalar_tower [monoid R] [distrib_mul_action R α] [is_scalar_tower R α α] :
is_scalar_tower R (matrix n n α) (matrix n n α) :=
⟨λ r m n, matrix.smul_mul r m n⟩
@[simp] lemma mul_smul [monoid R] [distrib_mul_action R α] [smul_comm_class R α α]
(M : matrix m n α) (a : R) (N : matrix n l α) : M ⬝ (a • N) = a • M ⬝ N :=
by { ext, apply dot_product_smul }
/-- This instance enables use with `mul_smul_comm`. -/
instance semiring.smul_comm_class [monoid R] [distrib_mul_action R α] [smul_comm_class R α α] :
smul_comm_class R (matrix n n α) (matrix n n α) :=
⟨λ r m n, (matrix.mul_smul m r n).symm⟩
@[simp] lemma mul_mul_left (M : matrix m n α) (N : matrix n o α) (a : α) :
(λ i j, a * M i j) ⬝ N = a • (M ⬝ N) :=
smul_mul a M N
/--
The ring homomorphism `α →+* matrix n n α`
sending `a` to the diagonal matrix with `a` on the diagonal.
-/
def scalar (n : Type u) [decidable_eq n] [fintype n] : α →+* matrix n n α :=
{ to_fun := λ a, a • 1,
map_one' := by simp,
map_mul' := by { intros, ext, simp [mul_assoc], },
.. (smul_add_hom α _).flip (1 : matrix n n α) }
section scalar
variable [decidable_eq n]
@[simp] lemma coe_scalar : (scalar n : α → matrix n n α) = λ a, a • 1 := rfl
lemma scalar_apply_eq (a : α) (i : n) :
scalar n a i i = a :=
by simp only [coe_scalar, smul_eq_mul, mul_one, one_apply_eq, pi.smul_apply]
lemma scalar_apply_ne (a : α) (i j : n) (h : i ≠ j) :
scalar n a i j = 0 :=
by simp only [h, coe_scalar, one_apply_ne, ne.def, not_false_iff, pi.smul_apply, smul_zero]
lemma scalar_inj [nonempty n] {r s : α} : scalar n r = scalar n s ↔ r = s :=
begin
split,
{ intro h,
inhabit n,
rw [← scalar_apply_eq r (arbitrary n), ← scalar_apply_eq s (arbitrary n), h] },
{ rintro rfl, refl }
end
end scalar
end semiring
section comm_semiring
variables [comm_semiring α]
lemma smul_eq_mul_diagonal [decidable_eq n] (M : matrix m n α) (a : α) :
a • M = M ⬝ diagonal (λ _, a) :=
by { ext, simp [mul_comm] }
@[simp] lemma mul_mul_right (M : matrix m n α) (N : matrix n o α) (a : α) :
M ⬝ (λ i j, a * N i j) = a • (M ⬝ N) :=
mul_smul M a N
lemma scalar.commute [decidable_eq n] (r : α) (M : matrix n n α) : commute (scalar n r) M :=
by simp [commute, semiconj_by]
end comm_semiring
/-- For two vectors `w` and `v`, `vec_mul_vec w v i j` is defined to be `w i * v j`.
Put another way, `vec_mul_vec w v` is exactly `col w ⬝ row v`. -/
def vec_mul_vec [has_mul α] (w : m → α) (v : n → α) : matrix m n α
| x y := w x * v y
section non_unital_non_assoc_semiring
variables [non_unital_non_assoc_semiring α]
/-- `mul_vec M v` is the matrix-vector product of `M` and `v`, where `v` is seen as a column matrix.
Put another way, `mul_vec M v` is the vector whose entries
are those of `M ⬝ col v` (see `col_mul_vec`). -/
def mul_vec (M : matrix m n α) (v : n → α) : m → α
| i := dot_product (λ j, M i j) v
/-- `vec_mul v M` is the vector-matrix product of `v` and `M`, where `v` is seen as a row matrix.
Put another way, `vec_mul v M` is the vector whose entries
are those of `row v ⬝ M` (see `row_vec_mul`). -/
def vec_mul (v : m → α) (M : matrix m n α) : n → α
| j := dot_product v (λ i, M i j)
instance mul_vec.is_add_monoid_hom_left (v : n → α) :
is_add_monoid_hom (λM:matrix m n α, mul_vec M v) :=
{ map_zero := by ext; simp [mul_vec]; refl,
map_add :=
begin
intros x y,
ext m,
apply add_dot_product
end }
lemma mul_vec_diagonal [decidable_eq m] (v w : m → α) (x : m) :
mul_vec (diagonal v) w x = v x * w x :=
diagonal_dot_product v w x
lemma vec_mul_diagonal [decidable_eq m] (v w : m → α) (x : m) :
vec_mul v (diagonal w) x = v x * w x :=
dot_product_diagonal' v w x
@[simp] lemma mul_vec_zero (A : matrix m n α) : mul_vec A 0 = 0 :=
by { ext, simp [mul_vec] }
@[simp] lemma zero_vec_mul (A : matrix m n α) : vec_mul 0 A = 0 :=
by { ext, simp [vec_mul] }
@[simp] lemma zero_mul_vec (v : n → α) : mul_vec (0 : matrix m n α) v = 0 :=
by { ext, simp [mul_vec] }
@[simp] lemma vec_mul_zero (v : m → α) : vec_mul v (0 : matrix m n α) = 0 :=
by { ext, simp [vec_mul] }
lemma vec_mul_vec_eq (w : m → α) (v : n → α) :
vec_mul_vec w v = (col w) ⬝ (row v) :=
by { ext i j, simp [vec_mul_vec, mul_apply], refl }
lemma smul_mul_vec_assoc [monoid R] [distrib_mul_action R α] [is_scalar_tower R α α]
(a : R) (A : matrix m n α) (b : n → α) :
(a • A).mul_vec b = a • (A.mul_vec b) :=
by { ext, apply smul_dot_product, }
lemma mul_vec_add (A : matrix m n α) (x y : n → α) :
A.mul_vec (x + y) = A.mul_vec x + A.mul_vec y :=
by { ext, apply dot_product_add }
lemma add_mul_vec (A B : matrix m n α) (x : n → α) :
(A + B).mul_vec x = A.mul_vec x + B.mul_vec x :=
by { ext, apply add_dot_product }
lemma vec_mul_add (A B : matrix m n α) (x : m → α) :
vec_mul x (A + B) = vec_mul x A + vec_mul x B :=
by { ext, apply dot_product_add }
lemma add_vec_mul (A : matrix m n α) (x y : m → α) :
vec_mul (x + y) A = vec_mul x A + vec_mul y A :=
by { ext, apply add_dot_product }
end non_unital_non_assoc_semiring
section non_unital_semiring
variables [non_unital_semiring α]
@[simp] lemma vec_mul_vec_mul (v : m → α) (M : matrix m n α) (N : matrix n o α) :
vec_mul (vec_mul v M) N = vec_mul v (M ⬝ N) :=
by { ext, apply dot_product_assoc }
@[simp] lemma mul_vec_mul_vec (v : o → α) (M : matrix m n α) (N : matrix n o α) :
mul_vec M (mul_vec N v) = mul_vec (M ⬝ N) v :=
by { ext, symmetry, apply dot_product_assoc }
end non_unital_semiring
section non_assoc_semiring
variables [non_assoc_semiring α]
@[simp] lemma one_mul_vec [decidable_eq m] (v : m → α) : mul_vec 1 v = v :=
by { ext, rw [←diagonal_one, mul_vec_diagonal, one_mul] }
@[simp] lemma vec_mul_one [decidable_eq m] (v : m → α) : vec_mul v 1 = v :=
by { ext, rw [←diagonal_one, vec_mul_diagonal, mul_one] }
end non_assoc_semiring
section semiring
variables [semiring α]
variables [decidable_eq m] [decidable_eq n]
/--
`std_basis_matrix i j a` is the matrix with `a` in the `i`-th row, `j`-th column,
and zeroes elsewhere.
-/
def std_basis_matrix (i : m) (j : n) (a : α) : matrix m n α :=
(λ i' j', if i' = i ∧ j' = j then a else 0)
@[simp] lemma smul_std_basis_matrix (i : m) (j : n) (a b : α) :
b • std_basis_matrix i j a = std_basis_matrix i j (b • a) :=
by { unfold std_basis_matrix, ext, simp }
@[simp] lemma std_basis_matrix_zero (i : m) (j : n) :
std_basis_matrix i j (0 : α) = 0 :=
by { unfold std_basis_matrix, ext, simp }
lemma std_basis_matrix_add (i : m) (j : n) (a b : α) :
std_basis_matrix i j (a + b) = std_basis_matrix i j a + std_basis_matrix i j b :=
begin
unfold std_basis_matrix, ext,
split_ifs with h; simp [h],
end
lemma matrix_eq_sum_std_basis (x : matrix n m α) :
x = ∑ (i : n) (j : m), std_basis_matrix i j (x i j) :=
begin
ext, symmetry,
iterate 2 { rw finset.sum_apply },
convert fintype.sum_eq_single i _,
{ simp [std_basis_matrix] },
{ intros j hj,
simp [std_basis_matrix, hj.symm] }
end
-- TODO: tie this up with the `basis` machinery of linear algebra
-- this is not completely trivial because we are indexing by two types, instead of one
-- TODO: add `std_basis_vec`
lemma std_basis_eq_basis_mul_basis (i : m) (j : n) :
std_basis_matrix i j 1 = vec_mul_vec (λ i', ite (i = i') 1 0) (λ j', ite (j = j') 1 0) :=
begin
ext, norm_num [std_basis_matrix, vec_mul_vec],
split_ifs; tauto,
end
@[elab_as_eliminator] protected lemma induction_on'
{X : Type*} [semiring X] {M : matrix n n X → Prop} (m : matrix n n X)
(h_zero : M 0)
(h_add : ∀p q, M p → M q → M (p + q))
(h_std_basis : ∀ i j x, M (std_basis_matrix i j x)) :
M m :=
begin
rw [matrix_eq_sum_std_basis m, ← finset.sum_product'],
apply finset.sum_induction _ _ h_add h_zero,
{ intros, apply h_std_basis, }
end
@[elab_as_eliminator] protected lemma induction_on
[nonempty n] {X : Type*} [semiring X] {M : matrix n n X → Prop} (m : matrix n n X)
(h_add : ∀p q, M p → M q → M (p + q))
(h_std_basis : ∀ i j x, M (std_basis_matrix i j x)) :
M m :=
matrix.induction_on' m
begin
have i : n := classical.choice (by assumption),
simpa using h_std_basis i i 0,
end
h_add h_std_basis
end semiring
section ring
variables [ring α]
lemma neg_vec_mul (v : m → α) (A : matrix m n α) : vec_mul (-v) A = - vec_mul v A :=
by { ext, apply neg_dot_product }
lemma vec_mul_neg (v : m → α) (A : matrix m n α) : vec_mul v (-A) = - vec_mul v A :=
by { ext, apply dot_product_neg }
lemma neg_mul_vec (v : n → α) (A : matrix m n α) : mul_vec (-A) v = - mul_vec A v :=
by { ext, apply neg_dot_product }
lemma mul_vec_neg (v : n → α) (A : matrix m n α) : mul_vec A (-v) = - mul_vec A v :=
by { ext, apply dot_product_neg }
end ring
section comm_semiring
variables [comm_semiring α]
lemma mul_vec_smul_assoc (A : matrix m n α) (b : n → α) (a : α) :
A.mul_vec (a • b) = a • (A.mul_vec b) :=
by { ext, apply dot_product_smul }
lemma mul_vec_transpose (A : matrix m n α) (x : m → α) :
mul_vec Aᵀ x = vec_mul x A :=
by { ext, apply dot_product_comm }
lemma vec_mul_transpose (A : matrix m n α) (x : n → α) :
vec_mul x Aᵀ = mul_vec A x :=
by { ext, apply dot_product_comm }
end comm_semiring
section transpose
open_locale matrix
/--
Tell `simp` what the entries are in a transposed matrix.
Compare with `mul_apply`, `diagonal_apply_eq`, etc.
-/
@[simp] lemma transpose_apply (M : matrix m n α) (i j) : M.transpose j i = M i j := rfl
@[simp] lemma transpose_transpose (M : matrix m n α) :
Mᵀᵀ = M :=
by ext; refl
@[simp] lemma transpose_zero [has_zero α] : (0 : matrix m n α)ᵀ = 0 :=
by ext i j; refl
@[simp] lemma transpose_one [decidable_eq n] [has_zero α] [has_one α] : (1 : matrix n n α)ᵀ = 1 :=
begin
ext i j,
unfold has_one.one transpose,
by_cases i = j,
{ simp only [h, diagonal_apply_eq] },
{ simp only [diagonal_apply_ne h, diagonal_apply_ne (λ p, h (symm p))] }
end
@[simp] lemma transpose_add [has_add α] (M : matrix m n α) (N : matrix m n α) :
(M + N)ᵀ = Mᵀ + Nᵀ :=
by { ext i j, simp }
@[simp] lemma transpose_sub [add_group α] (M : matrix m n α) (N : matrix m n α) :
(M - N)ᵀ = Mᵀ - Nᵀ :=
by { ext i j, simp }
@[simp] lemma transpose_mul [comm_semiring α] (M : matrix m n α) (N : matrix n l α) :
(M ⬝ N)ᵀ = Nᵀ ⬝ Mᵀ :=
begin
ext i j,
apply dot_product_comm
end
@[simp] lemma transpose_smul [semiring α] (c : α) (M : matrix m n α) :
(c • M)ᵀ = c • Mᵀ :=
by { ext i j, refl }
@[simp] lemma transpose_neg [has_neg α] (M : matrix m n α) :
(- M)ᵀ = - Mᵀ :=
by ext i j; refl
lemma transpose_map {f : α → β} {M : matrix m n α} : Mᵀ.map f = (M.map f)ᵀ :=
by { ext, refl }
end transpose
section conj_transpose
open_locale matrix
/--
Tell `simp` what the entries are in a conjugate transposed matrix.
Compare with `mul_apply`, `diagonal_apply_eq`, etc.
-/
@[simp] lemma conj_transpose_apply [has_star α] (M : matrix m n α) (i j) :
M.conj_transpose j i = star (M i j) := rfl
@[simp] lemma conj_transpose_conj_transpose [has_involutive_star α] (M : matrix m n α) :
Mᴴᴴ = M :=
by ext; simp
@[simp] lemma conj_transpose_zero [semiring α] [star_ring α] : (0 : matrix m n α)ᴴ = 0 :=
by ext i j; simp
@[simp] lemma conj_transpose_one [decidable_eq n] [semiring α] [star_ring α]:
(1 : matrix n n α)ᴴ = 1 :=
by simp [conj_transpose]
@[simp] lemma conj_transpose_add
[semiring α] [star_ring α] (M : matrix m n α) (N : matrix m n α) :
(M + N)ᴴ = Mᴴ + Nᴴ := by ext i j; simp
@[simp] lemma conj_transpose_sub [ring α] [star_ring α] (M : matrix m n α) (N : matrix m n α) :
(M - N)ᴴ = Mᴴ - Nᴴ := by ext i j; simp
@[simp] lemma conj_transpose_smul [comm_monoid α] [star_monoid α] (c : α) (M : matrix m n α) :
(c • M)ᴴ = (star c) • Mᴴ :=
by ext i j; simp [mul_comm]
@[simp] lemma conj_transpose_mul [semiring α] [star_ring α] (M : matrix m n α) (N : matrix n l α) :
(M ⬝ N)ᴴ = Nᴴ ⬝ Mᴴ := by ext i j; simp [mul_apply]
@[simp] lemma conj_transpose_neg [ring α] [star_ring α] (M : matrix m n α) :
(- M)ᴴ = - Mᴴ := by ext i j; simp
end conj_transpose
section star
/-- When `α` has a star operation, square matrices `matrix n n α` have a star
operation equal to `matrix.conj_transpose`. -/
instance [has_star α] : has_star (matrix n n α) := {star := conj_transpose}
lemma star_eq_conj_transpose [has_star α] (M : matrix m m α) : star M = Mᴴ := rfl
@[simp] lemma star_apply [has_star α] (M : matrix n n α) (i j) :
(star M) i j = star (M j i) := rfl
instance [has_involutive_star α] : has_involutive_star (matrix n n α) :=
{ star_involutive := conj_transpose_conj_transpose }
/-- When `α` is a `*`-(semi)ring, `matrix.has_star` is also a `*`-(semi)ring. -/
instance [decidable_eq n] [semiring α] [star_ring α] : star_ring (matrix n n α) :=
{ star_add := conj_transpose_add,
star_mul := conj_transpose_mul, }
/-- A version of `star_mul` for `⬝` instead of `*`. -/
lemma star_mul [semiring α] [star_ring α] (M N : matrix n n α) :
star (M ⬝ N) = star N ⬝ star M := conj_transpose_mul _ _
end star
/-- Given maps `(r_reindex : l → m)` and `(c_reindex : o → n)` reindexing the rows and columns of
a matrix `M : matrix m n α`, the matrix `M.minor r_reindex c_reindex : matrix l o α` is defined
by `(M.minor r_reindex c_reindex) i j = M (r_reindex i) (c_reindex j)` for `(i,j) : l × o`.
Note that the total number of row and columns does not have to be preserved. -/
def minor (A : matrix m n α) (r_reindex : l → m) (c_reindex : o → n) : matrix l o α :=
λ i j, A (r_reindex i) (c_reindex j)
@[simp] lemma minor_apply (A : matrix m n α) (r_reindex : l → m) (c_reindex : o → n) (i j) :
A.minor r_reindex c_reindex i j = A (r_reindex i) (c_reindex j) := rfl
@[simp] lemma minor_id_id (A : matrix m n α) :
A.minor id id = A :=
ext $ λ _ _, rfl
@[simp] lemma minor_minor {l₂ o₂ : Type*} [fintype l₂] [fintype o₂] (A : matrix m n α)
(r₁ : l → m) (c₁ : o → n) (r₂ : l₂ → l) (c₂ : o₂ → o) :
(A.minor r₁ c₁).minor r₂ c₂ = A.minor (r₁ ∘ r₂) (c₁ ∘ c₂) :=
ext $ λ _ _, rfl
@[simp] lemma transpose_minor (A : matrix m n α) (r_reindex : l → m) (c_reindex : o → n) :
(A.minor r_reindex c_reindex)ᵀ = Aᵀ.minor c_reindex r_reindex :=
ext $ λ _ _, rfl
@[simp] lemma conj_transpose_minor
[has_star α] (A : matrix m n α) (r_reindex : l → m) (c_reindex : o → n) :
(A.minor r_reindex c_reindex)ᴴ = Aᴴ.minor c_reindex r_reindex :=
ext $ λ _ _, rfl
lemma minor_add [has_add α] (A B : matrix m n α) :
((A + B).minor : (l → m) → (o → n) → matrix l o α) = A.minor + B.minor := rfl
lemma minor_neg [has_neg α] (A : matrix m n α) :
((-A).minor : (l → m) → (o → n) → matrix l o α) = -A.minor := rfl
lemma minor_sub [has_sub α] (A B : matrix m n α) :
((A - B).minor : (l → m) → (o → n) → matrix l o α) = A.minor - B.minor := rfl
@[simp]
lemma minor_zero [has_zero α] :
((0 : matrix m n α).minor : (l → m) → (o → n) → matrix l o α) = 0 := rfl
lemma minor_smul {R : Type*} [semiring R] [add_comm_monoid α] [module R α] (r : R)
(A : matrix m n α) :
((r • A : matrix m n α).minor : (l → m) → (o → n) → matrix l o α) = r • A.minor := rfl
lemma minor_map (f : α → β) (e₁ : l → m) (e₂ : o → n) (A : matrix m n α) :
(A.map f).minor e₁ e₂ = (A.minor e₁ e₂).map f := rfl
/-- Given a `(m × m)` diagonal matrix defined by a map `d : m → α`, if the reindexing map `e` is
injective, then the resulting matrix is again diagonal. -/
lemma minor_diagonal [has_zero α] [decidable_eq m] [decidable_eq l] (d : m → α) (e : l → m)
(he : function.injective e) :
(diagonal d).minor e e = diagonal (d ∘ e) :=
ext $ λ i j, begin
rw minor_apply,
by_cases h : i = j,
{ rw [h, diagonal_apply_eq, diagonal_apply_eq], },
{ rw [diagonal_apply_ne h, diagonal_apply_ne (he.ne h)], },
end
lemma minor_one [has_zero α] [has_one α] [decidable_eq m] [decidable_eq l] (e : l → m)
(he : function.injective e) :
(1 : matrix m m α).minor e e = 1 :=
minor_diagonal _ e he
lemma minor_mul [semiring α] {p q : Type*} [fintype p] [fintype q]
(M : matrix m n α) (N : matrix n p α)
(e₁ : l → m) (e₂ : o → n) (e₃ : q → p) (he₂ : function.bijective e₂) :
(M ⬝ N).minor e₁ e₃ = (M.minor e₁ e₂) ⬝ (N.minor e₂ e₃) :=
ext $ λ _ _, (he₂.sum_comp _).symm
/-! `simp` lemmas for `matrix.minor`s interaction with `matrix.diagonal`, `1`, and `matrix.mul` for
when the mappings are bundled. -/
@[simp]
lemma minor_diagonal_embedding [has_zero α] [decidable_eq m] [decidable_eq l] (d : m → α)
(e : l ↪ m) :
(diagonal d).minor e e = diagonal (d ∘ e) :=
minor_diagonal d e e.injective
@[simp]
lemma minor_diagonal_equiv [has_zero α] [decidable_eq m] [decidable_eq l] (d : m → α)
(e : l ≃ m) :
(diagonal d).minor e e = diagonal (d ∘ e) :=
minor_diagonal d e e.injective
@[simp]
lemma minor_one_embedding [has_zero α] [has_one α] [decidable_eq m] [decidable_eq l] (e : l ↪ m) :
(1 : matrix m m α).minor e e = 1 :=
minor_one e e.injective
@[simp]
lemma minor_one_equiv [has_zero α] [has_one α] [decidable_eq m] [decidable_eq l] (e : l ≃ m) :
(1 : matrix m m α).minor e e = 1 :=
minor_one e e.injective
lemma minor_mul_equiv [semiring α] {p q : Type*} [fintype p] [fintype q]
(M : matrix m n α) (N : matrix n p α) (e₁ : l → m) (e₂ : o ≃ n) (e₃ : q → p) :
(M ⬝ N).minor e₁ e₃ = (M.minor e₁ e₂) ⬝ (N.minor e₂ e₃) :=
minor_mul M N e₁ e₂ e₃ e₂.bijective
lemma mul_minor_one [semiring α] [decidable_eq o] (e₁ : n ≃ o) (e₂ : l → o) (M : matrix m n α) :
M ⬝ (1 : matrix o o α).minor e₁ e₂ = minor M id (e₁.symm ∘ e₂) :=
begin
let A := M.minor id e₁.symm,
have : M = A.minor id e₁,
{ simp only [minor_minor, function.comp.right_id, minor_id_id, equiv.symm_comp_self], },
rw [this, ←minor_mul_equiv],
simp only [matrix.mul_one, minor_minor, function.comp.right_id, minor_id_id,
equiv.symm_comp_self],
end
lemma one_minor_mul [semiring α] [decidable_eq o] (e₁ : l → o) (e₂ : m ≃ o) (M : matrix m n α) :
((1 : matrix o o α).minor e₁ e₂).mul M = minor M (e₂.symm ∘ e₁) id :=
begin
let A := M.minor e₂.symm id,
have : M = A.minor e₂ id,
{ simp only [minor_minor, function.comp.right_id, minor_id_id, equiv.symm_comp_self], },
rw [this, ←minor_mul_equiv],
simp only [matrix.one_mul, minor_minor, function.comp.right_id, minor_id_id,
equiv.symm_comp_self],
end
/-- The natural map that reindexes a matrix's rows and columns with equivalent types is an
equivalence. -/
def reindex (eₘ : m ≃ l) (eₙ : n ≃ o) : matrix m n α ≃ matrix l o α :=
{ to_fun := λ M, M.minor eₘ.symm eₙ.symm,
inv_fun := λ M, M.minor eₘ eₙ,
left_inv := λ M, by simp,
right_inv := λ M, by simp, }
@[simp] lemma reindex_apply (eₘ : m ≃ l) (eₙ : n ≃ o) (M : matrix m n α) :
reindex eₘ eₙ M = M.minor eₘ.symm eₙ.symm :=
rfl
@[simp] lemma reindex_refl_refl (A : matrix m n α) :
reindex (equiv.refl _) (equiv.refl _) A = A :=
A.minor_id_id
@[simp] lemma reindex_symm (eₘ : m ≃ l) (eₙ : n ≃ o) :
(reindex eₘ eₙ).symm = (reindex eₘ.symm eₙ.symm : matrix l o α ≃ _) :=
rfl
@[simp] lemma reindex_trans {l₂ o₂ : Type*} [fintype l₂] [fintype o₂]
(eₘ : m ≃ l) (eₙ : n ≃ o) (eₘ₂ : l ≃ l₂) (eₙ₂ : o ≃ o₂) :
(reindex eₘ eₙ).trans (reindex eₘ₂ eₙ₂) =
(reindex (eₘ.trans eₘ₂) (eₙ.trans eₙ₂) : matrix m n α ≃ _) :=
equiv.ext $ λ A, (A.minor_minor eₘ.symm eₙ.symm eₘ₂.symm eₙ₂.symm : _)
lemma transpose_reindex (eₘ : m ≃ l) (eₙ : n ≃ o) (M : matrix m n α) :
(reindex eₘ eₙ M)ᵀ = (reindex eₙ eₘ Mᵀ) :=
rfl
lemma conj_transpose_reindex [has_star α] (eₘ : m ≃ l) (eₙ : n ≃ o) (M : matrix m n α) :
(reindex eₘ eₙ M)ᴴ = (reindex eₙ eₘ Mᴴ) :=
rfl
/-- The left `n × l` part of a `n × (l+r)` matrix. -/
@[reducible]
def sub_left {m l r : nat} (A : matrix (fin m) (fin (l + r)) α) : matrix (fin m) (fin l) α :=
minor A id (fin.cast_add r)
/-- The right `n × r` part of a `n × (l+r)` matrix. -/
@[reducible]
def sub_right {m l r : nat} (A : matrix (fin m) (fin (l + r)) α) : matrix (fin m) (fin r) α :=
minor A id (fin.nat_add l)
/-- The top `u × n` part of a `(u+d) × n` matrix. -/
@[reducible]
def sub_up {d u n : nat} (A : matrix (fin (u + d)) (fin n) α) : matrix (fin u) (fin n) α :=
minor A (fin.cast_add d) id
/-- The bottom `d × n` part of a `(u+d) × n` matrix. -/
@[reducible]
def sub_down {d u n : nat} (A : matrix (fin (u + d)) (fin n) α) : matrix (fin d) (fin n) α :=
minor A (fin.nat_add u) id
/-- The top-right `u × r` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_up_right {d u l r : nat} (A: matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin r) α :=
sub_up (sub_right A)
/-- The bottom-right `d × r` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_down_right {d u l r : nat} (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin r) α :=
sub_down (sub_right A)
/-- The top-left `u × l` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_up_left {d u l r : nat} (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin (l)) α :=
sub_up (sub_left A)
/-- The bottom-left `d × l` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_down_left {d u l r : nat} (A: matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin (l)) α :=
sub_down (sub_left A)
section row_col
/-!
### `row_col` section
Simplification lemmas for `matrix.row` and `matrix.col`.
-/
open_locale matrix
@[simp] lemma col_add [has_add α] (v w : m → α) : col (v + w) = col v + col w := by { ext, refl }
@[simp] lemma col_smul [has_scalar R α] (x : R) (v : m → α) : col (x • v) = x • col v :=
by { ext, refl }
@[simp] lemma row_add [has_add α] (v w : m → α) : row (v + w) = row v + row w := by { ext, refl }
@[simp] lemma row_smul [has_scalar R α] (x : R) (v : m → α) : row (x • v) = x • row v :=
by { ext, refl }
@[simp] lemma col_apply (v : m → α) (i j) : matrix.col v i j = v i := rfl
@[simp] lemma row_apply (v : m → α) (i j) : matrix.row v i j = v j := rfl
@[simp]
lemma transpose_col (v : m → α) : (matrix.col v)ᵀ = matrix.row v := by { ext, refl }
@[simp]
lemma transpose_row (v : m → α) : (matrix.row v)ᵀ = matrix.col v := by { ext, refl }
@[simp]
lemma conj_transpose_col [has_star α] (v : m → α) : (col v)ᴴ = row (star v) := by { ext, refl }
@[simp]
lemma conj_transpose_row [has_star α] (v : m → α) : (row v)ᴴ = col (star v) := by { ext, refl }
lemma row_vec_mul [semiring α] (M : matrix m n α) (v : m → α) :
matrix.row (matrix.vec_mul v M) = matrix.row v ⬝ M := by {ext, refl}
lemma col_vec_mul [semiring α] (M : matrix m n α) (v : m → α) :
matrix.col (matrix.vec_mul v M) = (matrix.row v ⬝ M)ᵀ := by {ext, refl}
lemma col_mul_vec [semiring α] (M : matrix m n α) (v : n → α) :
matrix.col (matrix.mul_vec M v) = M ⬝ matrix.col v := by {ext, refl}
lemma row_mul_vec [semiring α] (M : matrix m n α) (v : n → α) :
matrix.row (matrix.mul_vec M v) = (M ⬝ matrix.col v)ᵀ := by {ext, refl}
@[simp]
lemma row_mul_col_apply [has_mul α] [add_comm_monoid α] (v w : m → α) (i j) :
(row v ⬝ col w) i j = dot_product v w :=
rfl
end row_col
section update
/-- Update, i.e. replace the `i`th row of matrix `A` with the values in `b`. -/
def update_row [decidable_eq n] (M : matrix n m α) (i : n) (b : m → α) : matrix n m α :=
function.update M i b
/-- Update, i.e. replace the `j`th column of matrix `A` with the values in `b`. -/
def update_column [decidable_eq m] (M : matrix n m α) (j : m) (b : n → α) : matrix n m α :=
λ i, function.update (M i) j (b i)
variables {M : matrix n m α} {i : n} {j : m} {b : m → α} {c : n → α}
@[simp] lemma update_row_self [decidable_eq n] : update_row M i b i = b :=
function.update_same i b M
@[simp] lemma update_column_self [decidable_eq m] : update_column M j c i j = c i :=
function.update_same j (c i) (M i)
@[simp] lemma update_row_ne [decidable_eq n] {i' : n} (i_ne : i' ≠ i) :
update_row M i b i' = M i' := function.update_noteq i_ne b M
@[simp] lemma update_column_ne [decidable_eq m] {j' : m} (j_ne : j' ≠ j) :
update_column M j c i j' = M i j' := function.update_noteq j_ne (c i) (M i)
lemma update_row_apply [decidable_eq n] {i' : n} :
update_row M i b i' j = if i' = i then b j else M i' j :=
begin
by_cases i' = i,
{ rw [h, update_row_self, if_pos rfl] },
{ rwa [update_row_ne h, if_neg h] }
end
lemma update_column_apply [decidable_eq m] {j' : m} :
update_column M j c i j' = if j' = j then c i else M i j' :=
begin
by_cases j' = j,
{ rw [h, update_column_self, if_pos rfl] },
{ rwa [update_column_ne h, if_neg h] }
end
@[simp] lemma update_column_subsingleton [subsingleton m] (A : matrix n m R)
(i : m) (b : n → R) :
A.update_column i b = (col b).minor id (function.const m ()) :=
begin
ext x y,
simp [update_column_apply, subsingleton.elim i y]
end
@[simp] lemma update_row_subsingleton [subsingleton n] (A : matrix n m R)
(i : n) (b : m → R) :
A.update_row i b = (row b).minor (function.const n ()) id :=
begin
ext x y,
simp [update_column_apply, subsingleton.elim i x]
end
lemma map_update_row [decidable_eq n] (f : α → β) :
map (update_row M i b) f = update_row (M.map f) i (f ∘ b) :=
begin
ext i' j',
rw [update_row_apply, map_apply, map_apply, update_row_apply],
exact apply_ite f _ _ _,
end
lemma map_update_column [decidable_eq m] (f : α → β) :
map (update_column M j c) f = update_column (M.map f) j (f ∘ c) :=
begin
ext i' j',
rw [update_column_apply, map_apply, map_apply, update_column_apply],
exact apply_ite f _ _ _,
end
lemma update_row_transpose [decidable_eq m] : update_row Mᵀ j c = (update_column M j c)ᵀ :=
begin
ext i' j,
rw [transpose_apply, update_row_apply, update_column_apply],
refl
end
lemma update_column_transpose [decidable_eq n] : update_column Mᵀ i b = (update_row M i b)ᵀ :=
begin
ext i' j,
rw [transpose_apply, update_row_apply, update_column_apply],
refl
end
lemma update_row_conj_transpose [decidable_eq m] [has_star α] :
update_row Mᴴ j (star c) = (update_column M j c)ᴴ :=
begin
rw [conj_transpose, conj_transpose, transpose_map, transpose_map, update_row_transpose,
map_update_column],
refl,
end
lemma update_column_conj_transpose [decidable_eq n] [has_star α] :
update_column Mᴴ i (star b) = (update_row M i b)ᴴ :=
begin
rw [conj_transpose, conj_transpose, transpose_map, transpose_map, update_column_transpose,
map_update_row],
refl,
end
@[simp] lemma update_row_eq_self [decidable_eq m]
(A : matrix m n α) {i : m} :
A.update_row i (A i) = A :=
function.update_eq_self i A
@[simp] lemma update_column_eq_self [decidable_eq n]
(A : matrix m n α) {i : n} :
A.update_column i (λ j, A j i) = A :=
funext $ λ j, function.update_eq_self i (A j)
end update
end matrix
namespace ring_hom
variables [semiring α] [semiring β]
lemma map_matrix_mul (M : matrix m n α) (N : matrix n o α) (i : m) (j : o) (f : α →+* β) :
f (matrix.mul M N i j) = matrix.mul (λ i j, f (M i j)) (λ i j, f (N i j)) i j :=
by simp [matrix.mul_apply, ring_hom.map_sum]
end ring_hom
|
d39c6cd787e789ccf1434d7a606e1bb879f5b5b8 | 618003631150032a5676f229d13a079ac875ff77 | /src/algebra/group/default.lean | 054de35a2b1dadfc04d42a7037838aebcf42eaaa | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 463 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Michael Howes
-/
import algebra.group.type_tags
import algebra.group.conj
import algebra.group.with_one
import algebra.group.anti_hom
import algebra.group.units_hom
/-!
# Various multiplicative and additive structures.
This file `import`s all files in this subdirectory except for `prod`.
-/
|
12309ec99090c3866cb3dded61cd571ac5dddea0 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/algebra/group_ring_action.lean | 520c11f19a127d31f5213bf39357d36dd34161c5 | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 4,408 | lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.ring.equiv
import group_theory.group_action.group
import ring_theory.subring.basic
/-!
# Group action on rings
This file defines the typeclass of monoid acting on semirings `mul_semiring_action M R`,
and the corresponding typeclass of invariant subrings.
Note that `algebra` does not satisfy the axioms of `mul_semiring_action`.
## Implementation notes
There is no separate typeclass for group acting on rings, group acting on fields, etc.
They are all grouped under `mul_semiring_action`.
## Tags
group action, invariant subring
-/
universes u v
open_locale big_operators
/-- Typeclass for multiplicative actions by monoids on semirings.
This combines `distrib_mul_action` with `mul_distrib_mul_action`. -/
class mul_semiring_action (M : Type u) (R : Type v) [monoid M] [semiring R]
extends distrib_mul_action M R :=
(smul_one : ∀ (g : M), (g • 1 : R) = 1)
(smul_mul : ∀ (g : M) (x y : R), g • (x * y) = (g • x) * (g • y))
section semiring
variables (M G : Type u) [monoid M] [group G]
variables (A R S F : Type v) [add_monoid A] [semiring R] [comm_semiring S] [division_ring F]
-- note we could not use `extends` since these typeclasses are made with `old_structure_cmd`
@[priority 100]
instance mul_semiring_action.to_mul_distrib_mul_action [h : mul_semiring_action M R] :
mul_distrib_mul_action M R :=
{ ..h }
/-- Each element of the monoid defines a semiring homomorphism. -/
@[simps]
def mul_semiring_action.to_ring_hom [mul_semiring_action M R] (x : M) : R →+* R :=
{ .. mul_distrib_mul_action.to_monoid_hom R x,
.. distrib_mul_action.to_add_monoid_hom R x }
theorem to_ring_hom_injective [mul_semiring_action M R] [has_faithful_smul M R] :
function.injective (mul_semiring_action.to_ring_hom M R) :=
λ m₁ m₂ h, eq_of_smul_eq_smul $ λ r, ring_hom.ext_iff.1 h r
/-- Each element of the group defines a semiring isomorphism. -/
@[simps]
def mul_semiring_action.to_ring_equiv [mul_semiring_action G R] (x : G) : R ≃+* R :=
{ .. distrib_mul_action.to_add_equiv R x,
.. mul_semiring_action.to_ring_hom G R x }
section
variables {M G R}
/-- A stronger version of `submonoid.distrib_mul_action`. -/
instance submonoid.mul_semiring_action [mul_semiring_action M R] (H : submonoid M) :
mul_semiring_action H R :=
{ smul := (•),
.. H.mul_distrib_mul_action,
.. H.distrib_mul_action }
/-- A stronger version of `subgroup.distrib_mul_action`. -/
instance subgroup.mul_semiring_action [mul_semiring_action G R] (H : subgroup G) :
mul_semiring_action H R :=
H.to_submonoid.mul_semiring_action
/-- A stronger version of `subsemiring.distrib_mul_action`. -/
instance subsemiring.mul_semiring_action {R'} [semiring R'] [mul_semiring_action R' R]
(H : subsemiring R') :
mul_semiring_action H R :=
H.to_submonoid.mul_semiring_action
/-- A stronger version of `subring.distrib_mul_action`. -/
instance subring.mul_semiring_action {R'} [ring R'] [mul_semiring_action R' R]
(H : subring R') :
mul_semiring_action H R :=
H.to_subsemiring.mul_semiring_action
end
section simp_lemmas
variables {M G A R F}
attribute [simp] smul_one smul_mul' smul_zero smul_add
/-- Note that `smul_inv'` refers to the group case, and `smul_inv` has an additional inverse
on `x`. -/
@[simp] lemma smul_inv'' [mul_semiring_action M F] (x : M) (m : F) : x • m⁻¹ = (x • m)⁻¹ :=
map_inv₀ (mul_semiring_action.to_ring_hom M F x) _
end simp_lemmas
end semiring
section ring
variables (M : Type u) [monoid M] {R : Type v} [ring R] [mul_semiring_action M R]
variables (S : subring R)
open mul_action
/-- A typeclass for subrings invariant under a `mul_semiring_action`. -/
class is_invariant_subring : Prop :=
(smul_mem : ∀ (m : M) {x : R}, x ∈ S → m • x ∈ S)
instance is_invariant_subring.to_mul_semiring_action [is_invariant_subring M S] :
mul_semiring_action M S :=
{ smul := λ m x, ⟨m • x, is_invariant_subring.smul_mem m x.2⟩,
one_smul := λ s, subtype.eq $ one_smul M s,
mul_smul := λ m₁ m₂ s, subtype.eq $ mul_smul m₁ m₂ s,
smul_add := λ m s₁ s₂, subtype.eq $ smul_add m s₁ s₂,
smul_zero := λ m, subtype.eq $ smul_zero m,
smul_one := λ m, subtype.eq $ smul_one m,
smul_mul := λ m s₁ s₂, subtype.eq $ smul_mul' m s₁ s₂ }
end ring
|
a03d7a62579d3482861952c32a0fb3d3c94db215 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/structure_default_value_issue.lean | d3f638c94d05afa81f9136c1b0914a720ff750ae | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 690 | lean | namespace test
universes u v w
structure lens (α : Type u) (β : Type v) :=
(get : α → β)
(modify : α → (β → β) → α)
(set : α → β → α := λ a b, modify a (λ _, b))
def lens.compose {α : Type u} {β : Type v} {σ : Type w} (t : lens β σ) (s : lens α β) : lens α σ :=
{ get := t^.get ∘ s^.get,
modify := λ a f, s^.modify a $ λ b, t^.modify b f,
set := λ a v, s^.modify a $ λ b, t^.set b v }
infix `∙`:1 := lens.compose
def fst {α β} : lens (α × β) α :=
{ get := prod.fst,
modify := λ ⟨a, b⟩ f, (f a, b) }
def snd {α β} : lens (α × β) β :=
{ get := prod.snd,
modify := λ ⟨a, b⟩ f, (a, f b) }
end test
|
e12f8835f455fbe7a783000181e816cd136d5912 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/field_theory/cardinality.lean | 2d48f1aa45b252e806d84070a48535c65952fd81 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 3,379 | lean | /-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import algebra.ring.ulift
import data.mv_polynomial.cardinal
import data.rat.denumerable
import field_theory.finite.galois_field
import logic.equiv.transfer_instance
import ring_theory.localization.cardinality
import set_theory.cardinal.divisibility
/-!
# Cardinality of Fields
In this file we show all the possible cardinalities of fields. All infinite cardinals can harbour
a field structure, and so can all types with prime power cardinalities, and this is sharp.
## Main statements
* `fintype.nonempty_field_iff`: A `fintype` can be given a field structure iff its cardinality is a
prime power.
* `infinite.nonempty_field` : Any infinite type can be endowed a field structure.
* `field.nonempty_iff` : There is a field structure on type iff its cardinality is a prime power.
-/
local notation `‖` x `‖` := fintype.card x
open_locale cardinal non_zero_divisors
universe u
/-- A finite field has prime power cardinality. -/
lemma fintype.is_prime_pow_card_of_field {α} [fintype α] [field α] : is_prime_pow (‖α‖) :=
begin
casesI char_p.exists α with p _,
haveI hp := fact.mk (char_p.char_is_prime α p),
let b := is_noetherian.finset_basis (zmod p) α,
rw [module.card_fintype b, zmod.card, is_prime_pow_pow_iff],
{ exact hp.1.is_prime_pow },
rw ←finite_dimensional.finrank_eq_card_basis b,
exact finite_dimensional.finrank_pos.ne'
end
/-- A `fintype` can be given a field structure iff its cardinality is a prime power. -/
lemma fintype.nonempty_field_iff {α} [fintype α] : nonempty (field α) ↔ is_prime_pow (‖α‖) :=
begin
refine ⟨λ ⟨h⟩, by exactI fintype.is_prime_pow_card_of_field, _⟩,
rintros ⟨p, n, hp, hn, hα⟩,
haveI := fact.mk (nat.prime_iff.mpr hp),
exact ⟨(fintype.equiv_of_card_eq ((galois_field.card p n hn.ne').trans hα)).symm.field⟩,
end
lemma fintype.not_is_field_of_card_not_prime_pow {α} [fintype α] [ring α] :
¬ is_prime_pow (‖α‖) → ¬ is_field α :=
mt $ λ h, fintype.nonempty_field_iff.mp ⟨h.to_field⟩
/-- Any infinite type can be endowed a field structure. -/
lemma infinite.nonempty_field {α : Type u} [infinite α] : nonempty (field α) :=
begin
letI K := fraction_ring (mv_polynomial α $ ulift.{u} ℚ),
suffices : #α = #K,
{ obtain ⟨e⟩ := cardinal.eq.1 this,
exact ⟨e.field⟩ },
rw ←is_localization.card (mv_polynomial α $ ulift.{u} ℚ)⁰ K le_rfl,
apply le_antisymm,
{ refine ⟨⟨λ a, mv_polynomial.monomial (finsupp.single a 1) (1 : ulift.{u} ℚ), λ x y h, _⟩⟩,
simpa [mv_polynomial.monomial_eq_monomial_iff, finsupp.single_eq_single_iff] using h },
{ simpa using @mv_polynomial.cardinal_mk_le_max α (ulift.{u} ℚ) _ }
end
/-- There is a field structure on type if and only if its cardinality is a prime power. -/
lemma field.nonempty_iff {α : Type u} : nonempty (field α) ↔ is_prime_pow (#α) :=
begin
rw cardinal.is_prime_pow_iff,
casesI fintype_or_infinite α with h h,
{ simpa only [cardinal.mk_fintype, nat.cast_inj, exists_eq_left',
(cardinal.nat_lt_omega _).not_le, false_or]
using fintype.nonempty_field_iff },
{ simpa only [← cardinal.infinite_iff, h, true_or, iff_true]
using infinite.nonempty_field },
end
|
ab0b951d26eb5b81416f8f52f40460e26868d2f0 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/listLength.lean | 9b45b6d8e4462167b3018f2e2ece17663eba4fcb | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 80 | lean | set_option trace.compiler.ir.init true
def f (xs : List Nat) :=
2 * xs.length
|
3d03a9a75613be82580722cb2b7f0cc2be10e1bd | 35677d2df3f081738fa6b08138e03ee36bc33cad | /src/order/lattice.lean | 8daa58e55f13d7d106edefa10d26beab34b7879f | [
"Apache-2.0"
] | permissive | gebner/mathlib | eab0150cc4f79ec45d2016a8c21750244a2e7ff0 | cc6a6edc397c55118df62831e23bfbd6e6c6b4ab | refs/heads/master | 1,625,574,853,976 | 1,586,712,827,000 | 1,586,712,827,000 | 99,101,412 | 1 | 0 | Apache-2.0 | 1,586,716,389,000 | 1,501,667,958,000 | Lean | UTF-8 | Lean | false | false | 16,600 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Defines the inf/sup (semi)-lattice with optionally top/bot type class hierarchy.
-/
import order.basic
set_option old_structure_cmd true
universes u v w
variables {α : Type u} {β : Type v}
-- TODO: move this eventually, if we decide to use them
attribute [ematch] le_trans lt_of_le_of_lt lt_of_lt_of_le lt_trans
section
-- TODO: this seems crazy, but it also seems to work reasonably well
@[ematch] theorem le_antisymm' [partial_order α] : ∀ {a b : α}, (: a ≤ b :) → b ≤ a → a = b :=
@le_antisymm _ _
end
/- TODO: automatic construction of dual definitions / theorems -/
reserve infixl ` ⊓ `:70
reserve infixl ` ⊔ `:65
/-- Typeclass for the `⊔` (`\lub`) notation -/
class has_sup (α : Type u) := (sup : α → α → α)
/-- Typeclass for the `⊓` (`\glb`) notation -/
class has_inf (α : Type u) := (inf : α → α → α)
infix ⊔ := has_sup.sup
infix ⊓ := has_inf.inf
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A `semilattice_sup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class semilattice_sup (α : Type u) extends has_sup α, partial_order α :=
(le_sup_left : ∀ a b : α, a ≤ a ⊔ b)
(le_sup_right : ∀ a b : α, b ≤ a ⊔ b)
(sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c)
end prio
section semilattice_sup
variables [semilattice_sup α] {a b c d : α}
@[simp] theorem le_sup_left : a ≤ a ⊔ b :=
semilattice_sup.le_sup_left a b
@[ematch] theorem le_sup_left' : a ≤ (: a ⊔ b :) :=
semilattice_sup.le_sup_left a b
@[simp] theorem le_sup_right : b ≤ a ⊔ b :=
semilattice_sup.le_sup_right a b
@[ematch] theorem le_sup_right' : b ≤ (: a ⊔ b :) :=
semilattice_sup.le_sup_right a b
theorem le_sup_left_of_le (h : c ≤ a) : c ≤ a ⊔ b :=
by finish
theorem le_sup_right_of_le (h : c ≤ b) : c ≤ a ⊔ b :=
by finish
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
semilattice_sup.sup_le a b c
@[simp] theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨assume h : a ⊔ b ≤ c, ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
assume ⟨h₁, h₂⟩, sup_le h₁ h₂⟩
-- TODO: if we just write le_antisymm, Lean doesn't know which ≤ we want to use
-- Can we do anything about that?
theorem sup_of_le_left (h : b ≤ a) : a ⊔ b = a :=
by apply le_antisymm; finish
theorem sup_of_le_right (h : a ≤ b) : a ⊔ b = b :=
by apply le_antisymm; finish
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
by finish
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
by finish
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
by finish
theorem le_of_sup_eq (h : a ⊔ b = b) : a ≤ b :=
by finish
/-- A monotone function on a sup-semilattice is directed. -/
lemma directed_of_mono (f : α → β) {r : β → β → Prop}
(H : ∀ ⦃i j⦄, i ≤ j → r (f i) (f j)) : directed r f :=
λ a b, ⟨a ⊔ b, H le_sup_left, H le_sup_right⟩
@[simp] lemma sup_lt_iff [is_total α (≤)] {a b c : α} : b ⊔ c < a ↔ b < a ∧ c < a :=
begin
cases (@is_total.total _ (≤) _ b c) with h,
{ simp [sup_of_le_right h], exact ⟨λI, ⟨lt_of_le_of_lt h I, I⟩, λH, H.2⟩ },
{ simp [sup_of_le_left h], exact ⟨λI, ⟨I, lt_of_le_of_lt h I⟩, λH, H.1⟩ }
end
@[simp] theorem sup_idem : a ⊔ a = a :=
by apply le_antisymm; finish
instance sup_is_idempotent : is_idempotent α (⊔) := ⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a :=
by apply le_antisymm; finish
instance sup_is_commutative : is_commutative α (⊔) := ⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
by apply le_antisymm; finish
instance sup_is_associative : is_associative α (⊔) := ⟨@sup_assoc _ _⟩
lemma sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) :=
by rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
lemma forall_le_or_exists_lt_sup (a : α) : (∀b, b ≤ a) ∨ (∃b, a < b) :=
suffices (∃b, ¬b ≤ a) → (∃b, a < b),
by rwa [classical.or_iff_not_imp_left, classical.not_forall],
assume ⟨b, hb⟩,
have a ≠ a ⊔ b, from assume eq, hb $ eq.symm ▸ le_sup_right,
⟨a ⊔ b, lt_of_le_of_ne le_sup_left ‹a ≠ a ⊔ b›⟩
theorem semilattice_sup.ext_sup {α} {A B : semilattice_sup α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y)
(x y : α) : (by haveI := A; exact (x ⊔ y)) = x ⊔ y :=
eq_of_forall_ge_iff $ λ c,
by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
theorem semilattice_sup.ext {α} {A B : semilattice_sup α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
haveI this := partial_order.ext H,
have ss := funext (λ x, funext $ semilattice_sup.ext_sup H x),
cases A; cases B; injection this; congr'
end
lemma directed_of_sup {β : Type*} {r : β → β → Prop} {f : α → β}
(hf : ∀a₁ a₂, a₁ ≤ a₂ → r (f a₁) (f a₂)) : directed r f :=
assume x y, ⟨x ⊔ y, hf _ _ le_sup_left, hf _ _ le_sup_right⟩
end semilattice_sup
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A `semilattice_inf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class semilattice_inf (α : Type u) extends has_inf α, partial_order α :=
(inf_le_left : ∀ a b : α, a ⊓ b ≤ a)
(inf_le_right : ∀ a b : α, a ⊓ b ≤ b)
(le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c)
end prio
section semilattice_inf
variables [semilattice_inf α] {a b c d : α}
@[simp] theorem inf_le_left : a ⊓ b ≤ a :=
semilattice_inf.inf_le_left a b
@[ematch] theorem inf_le_left' : (: a ⊓ b :) ≤ a :=
semilattice_inf.inf_le_left a b
@[simp] theorem inf_le_right : a ⊓ b ≤ b :=
semilattice_inf.inf_le_right a b
@[ematch] theorem inf_le_right' : (: a ⊓ b :) ≤ b :=
semilattice_inf.inf_le_right a b
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
semilattice_inf.le_inf a b c
theorem inf_le_left_of_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
theorem inf_le_right_of_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
@[simp] theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
⟨assume h : a ≤ b ⊓ c, ⟨le_trans h inf_le_left, le_trans h inf_le_right⟩,
assume ⟨h₁, h₂⟩, le_inf h₁ h₂⟩
theorem inf_of_le_left (h : a ≤ b) : a ⊓ b = a :=
by apply le_antisymm; finish
theorem inf_of_le_right (h : b ≤ a) : a ⊓ b = b :=
by apply le_antisymm; finish
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
by finish
lemma inf_le_inf_left (a : α) {b c: α} (h : b ≤ c): b ⊓ a ≤ c ⊓ a :=
by finish
lemma inf_le_inf_right (a : α) {b c: α} (h : b ≤ c): a ⊓ b ≤ a ⊓ c :=
by finish
theorem le_of_inf_eq (h : a ⊓ b = a) : a ≤ b :=
by finish
@[simp] lemma lt_inf_iff [is_total α (≤)] {a b c : α} : a < b ⊓ c ↔ a < b ∧ a < c :=
begin
cases (@is_total.total _ (≤) _ b c) with h,
{ simp [inf_of_le_left h], exact ⟨λI, ⟨I, lt_of_lt_of_le I h⟩, λH, H.1⟩ },
{ simp [inf_of_le_right h], exact ⟨λI, ⟨lt_of_lt_of_le I h, I⟩, λH, H.2⟩ }
end
@[simp] theorem inf_idem : a ⊓ a = a :=
by apply le_antisymm; finish
instance inf_is_idempotent : is_idempotent α (⊓) := ⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
by apply le_antisymm; finish
instance inf_is_commutative : is_commutative α (⊓) := ⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
by apply le_antisymm; finish
instance inf_is_associative : is_associative α (⊓) := ⟨@inf_assoc _ _⟩
lemma inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
by rw [← inf_assoc, ← inf_assoc, @inf_comm α _ a]
lemma forall_le_or_exists_lt_inf (a : α) : (∀b, a ≤ b) ∨ (∃b, b < a) :=
suffices (∃b, ¬a ≤ b) → (∃b, b < a),
by rwa [classical.or_iff_not_imp_left, classical.not_forall],
assume ⟨b, hb⟩,
have a ⊓ b ≠ a, from assume eq, hb $ eq ▸ inf_le_right,
⟨a ⊓ b, lt_of_le_of_ne inf_le_left ‹a ⊓ b ≠ a›⟩
theorem semilattice_inf.ext_inf {α} {A B : semilattice_inf α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y)
(x y : α) : (by haveI := A; exact (x ⊓ y)) = x ⊓ y :=
eq_of_forall_le_iff $ λ c,
by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
theorem semilattice_inf.ext {α} {A B : semilattice_inf α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
haveI this := partial_order.ext H,
have ss := funext (λ x, funext $ semilattice_inf.ext_inf H x),
cases A; cases B; injection this; congr'
end
/-- An antimonotone function on an inf-semilattice is directed. -/
lemma directed_of_inf {r : β → β → Prop} {f : α → β}
(hf : ∀a₁ a₂, a₁ ≤ a₂ → r (f a₂) (f a₁)) : directed r f :=
assume x y, ⟨x ⊓ y, hf _ _ inf_le_left, hf _ _ inf_le_right⟩
end semilattice_inf
/- Lattices -/
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class lattice (α : Type u) extends semilattice_sup α, semilattice_inf α
end prio
section lattice
variables [lattice α] {a b c d : α}
/- Distributivity laws -/
/- TODO: better names? -/
theorem sup_inf_le : a ⊔ (b ⊓ c) ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
by finish
theorem le_inf_sup : (a ⊓ b) ⊔ (a ⊓ c) ≤ a ⊓ (b ⊔ c) :=
by finish
theorem inf_sup_self : a ⊓ (a ⊔ b) = a :=
le_antisymm (by finish) (by finish)
theorem sup_inf_self : a ⊔ (a ⊓ b) = a :=
le_antisymm (by finish) (by finish)
theorem lattice.ext {α} {A B : lattice α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
have SS : @lattice.to_semilattice_sup α A =
@lattice.to_semilattice_sup α B := semilattice_sup.ext H,
have II := semilattice_inf.ext H,
resetI, cases A; cases B; injection SS; injection II; congr'
end
end lattice
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distribution properties (of sup over inf or inf over sup,
on the left or right). A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class distrib_lattice α extends lattice α :=
(le_sup_inf : ∀x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z))
end prio
section distrib_lattice
variables [distrib_lattice α] {x y z : α}
theorem le_sup_inf : ∀{x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z) :=
distrib_lattice.le_sup_inf
theorem sup_inf_left : x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
theorem sup_inf_right : (y ⊓ z) ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) :=
by simp only [sup_inf_left, λy:α, @sup_comm α _ y x, eq_self_iff_true]
theorem inf_sup_left : x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z) :=
calc x ⊓ (y ⊔ z) = (x ⊓ (x ⊔ z)) ⊓ (y ⊔ z) : by rw [inf_sup_self]
... = x ⊓ ((x ⊓ y) ⊔ z) : by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
... = (x ⊔ (x ⊓ y)) ⊓ ((x ⊓ y) ⊔ z) : by rw [sup_inf_self]
... = ((x ⊓ y) ⊔ x) ⊓ ((x ⊓ y) ⊔ z) : by rw [sup_comm]
... = (x ⊓ y) ⊔ (x ⊓ z) : by rw [sup_inf_left]
theorem inf_sup_right : (y ⊔ z) ⊓ x = (y ⊓ x) ⊔ (z ⊓ x) :=
by simp only [inf_sup_left, λy:α, @inf_comm α _ y x, eq_self_iff_true]
lemma eq_of_sup_eq_inf_eq {α : Type u} [distrib_lattice α] {a b c : α}
(h₁ : b ⊓ a = c ⊓ a) (h₂ : b ⊔ a = c ⊔ a) : b = c :=
le_antisymm
(calc b ≤ (c ⊓ a) ⊔ b : le_sup_right
... = (c ⊔ b) ⊓ (a ⊔ b) : sup_inf_right
... = c ⊔ (c ⊓ a) : by rw [←h₁, sup_inf_left, ←h₂]; simp only [sup_comm, eq_self_iff_true]
... = c : sup_inf_self)
(calc c ≤ (b ⊓ a) ⊔ c : le_sup_right
... = (b ⊔ c) ⊓ (a ⊔ c) : sup_inf_right
... = b ⊔ (b ⊓ a) : by rw [h₁, sup_inf_left, h₂]; simp only [sup_comm, eq_self_iff_true]
... = b : sup_inf_self)
end distrib_lattice
/- Lattices derived from linear orders -/
@[priority 100] -- see Note [lower instance priority]
instance lattice_of_decidable_linear_order {α : Type u} [o : decidable_linear_order α] : lattice α :=
{ sup := max,
le_sup_left := le_max_left,
le_sup_right := le_max_right,
sup_le := assume a b c, max_le,
inf := min,
inf_le_left := min_le_left,
inf_le_right := min_le_right,
le_inf := assume a b c, le_min,
..o }
theorem sup_eq_max [decidable_linear_order α] {x y : α} : x ⊔ y = max x y := rfl
theorem inf_eq_min [decidable_linear_order α] {x y : α} : x ⊓ y = min x y := rfl
@[priority 100] -- see Note [lower instance priority]
instance distrib_lattice_of_decidable_linear_order {α : Type u} [o : decidable_linear_order α] :
distrib_lattice α :=
{ le_sup_inf := assume a b c,
match le_total b c with
| or.inl h := inf_le_left_of_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| or.inr h := inf_le_right_of_le $ sup_le_sup_left (le_inf h (le_refl c)) _
end,
..lattice_of_decidable_linear_order }
instance nat.distrib_lattice : distrib_lattice ℕ :=
by apply_instance
namespace monotone
lemma le_map_sup [semilattice_sup α] [semilattice_sup β]
{f : α → β} (h : monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
lemma map_inf_le [semilattice_inf α] [semilattice_inf β]
{f : α → β} (h : monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
end monotone
namespace order_dual
variable (α)
instance [has_inf α] : has_sup (order_dual α) := ⟨((⊓) : α → α → α)⟩
instance [has_sup α] : has_inf (order_dual α) := ⟨((⊔) : α → α → α)⟩
instance [semilattice_inf α] : semilattice_sup (order_dual α) :=
{ le_sup_left := @inf_le_left α _,
le_sup_right := @inf_le_right α _,
sup_le := assume a b c hca hcb, @le_inf α _ _ _ _ hca hcb,
.. order_dual.partial_order α, .. order_dual.has_sup α }
instance [semilattice_sup α] : semilattice_inf (order_dual α) :=
{ inf_le_left := @le_sup_left α _,
inf_le_right := @le_sup_right α _,
le_inf := assume a b c hca hcb, @sup_le α _ _ _ _ hca hcb,
.. order_dual.partial_order α, .. order_dual.has_inf α }
instance [lattice α] : lattice (order_dual α) :=
{ .. order_dual.semilattice_sup α, .. order_dual.semilattice_inf α }
instance [distrib_lattice α] : distrib_lattice (order_dual α) :=
{ le_sup_inf := assume x y z, le_of_eq inf_sup_left.symm,
.. order_dual.lattice α }
end order_dual
namespace prod
variables (α β)
instance [has_sup α] [has_sup β] : has_sup (α × β) := ⟨λp q, ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [has_inf α] [has_inf β] : has_inf (α × β) := ⟨λp q, ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
instance [semilattice_sup α] [semilattice_sup β] : semilattice_sup (α × β) :=
{ sup_le := assume a b c h₁ h₂, ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩,
le_sup_left := assume a b, ⟨le_sup_left, le_sup_left⟩,
le_sup_right := assume a b, ⟨le_sup_right, le_sup_right⟩,
.. prod.partial_order α β, .. prod.has_sup α β }
instance [semilattice_inf α] [semilattice_inf β] : semilattice_inf (α × β) :=
{ le_inf := assume a b c h₁ h₂, ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩,
inf_le_left := assume a b, ⟨inf_le_left, inf_le_left⟩,
inf_le_right := assume a b, ⟨inf_le_right, inf_le_right⟩,
.. prod.partial_order α β, .. prod.has_inf α β }
instance [lattice α] [lattice β] : lattice (α × β) :=
{ .. prod.semilattice_inf α β, .. prod.semilattice_sup α β }
instance [distrib_lattice α] [distrib_lattice β] : distrib_lattice (α × β) :=
{ le_sup_inf := assume a b c, ⟨le_sup_inf, le_sup_inf⟩,
.. prod.lattice α β }
end prod
|
73782982ead37261eeb1a1bb0e64efa099acdc36 | e4e5bde6f14c01a8a34267a9d7bb45e137735696 | /src/exercises/logic_and_proof/16.lean | 74c9b56c69551a1f99c76b6022b2b44b7fcee55b | [] | no_license | jamesdabbs/proofs | fb5dab6f3c4f3f5f952fca033ec649888ae787c6 | 00baf355b08e7aec00de34208e1b2cb4a8d7b701 | refs/heads/master | 1,645,645,735,797 | 1,569,559,636,000 | 1,569,559,636,000 | 211,238,170 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,734 | lean | /-
1. Fill in the sorry’s in the last three proofs below.
-/
import data.set
section ex1
open function int algebra
def f (x : ℤ) : ℤ := x + 3
def g (x : ℤ) : ℤ := -x
def h (x : ℤ) : ℤ := 2 * x + 3
example : injective f :=
assume x1 x2,
assume h1 : x1 + 3 = x2 + 3, -- Lean knows this is the same as f x1 = f x2
show x1 = x2, from eq_of_add_eq_add_right h1
example : surjective f :=
assume y,
have h1 : f (y - 3) = y, from calc
f (y - 3) = (y - 3) + 3 : rfl
... = y : by rw sub_add_cancel,
show ∃ x, f x = y, from exists.intro (y - 3) h1
example (x y : ℤ) (h : 2 * x = 2 * y) : x = y :=
have h1 : 2 ≠ (0 : ℤ), from dec_trivial, -- this tells Lean to figure it out itself
show x = y, from eq_of_mul_eq_mul_left h1 h
example (x : ℤ) : -(-x) = x := neg_neg x
example (A B : Type) (u : A → B) (v : B → A) (h : left_inverse u v) :
∀ x, u (v x) = x :=
h
example (A B : Type) (u : A → B) (v : B → A) (h : left_inverse u v) :
right_inverse v u :=
h
-- fill in the sorry's in the following proofs
example : injective h :=
assume x y : ℤ,
assume q : 2 * x + 3 = 2 * y + 3,
have h : 2 * x = 2 * y, from
calc 2 * x = (2 * x + 3) - 3 : by simp
... = (2 * y + 3) - 3 : by rw q
... = 2 * y : by simp,
have h1 : 2 ≠ (0 : ℤ), from dec_trivial,
show x = y, from eq_of_mul_eq_mul_left h1 h
example : surjective g :=
assume y,
have h1 : g (-y) = y, from calc
g (-y) = -(-y) : rfl
... = y : neg_neg y,
show ∃ x, g x = y, from ⟨-y, h1⟩
example (A B : Type) (u : A → B) (v1 : B → A) (v2 : B → A)
(h1 : left_inverse v1 u) (h2 : right_inverse v2 u) : v1 = v2 :=
funext
(assume x,
calc
v1 x = v1 (u (v2 x)) : by rw h2
... = v2 x : by rw h1)
end ex1
/-
2. Fill in the sorry in the proof below.
-/
section ex2
open function set
variables {X Y : Type}
variable f : X → Y
variables A B : set X
example : f '' (A ∪ B) = f '' A ∪ f '' B :=
eq_of_subset_of_subset
(assume y,
assume h1 : y ∈ f '' (A ∪ B),
exists.elim h1 $
assume x h,
have h2 : x ∈ A ∪ B, from h.left,
have h3 : f x = y, from h.right,
or.elim h2
(assume h4 : x ∈ A,
have h5 : y ∈ f '' A, from ⟨x, h4, h3⟩,
show y ∈ f '' A ∪ f '' B, from or.inl h5)
(assume h4 : x ∈ B,
have h5 : y ∈ f '' B, from ⟨x, h4, h3⟩,
show y ∈ f '' A ∪ f '' B, from or.inr h5))
(assume y,
assume h2 : y ∈ f '' A ∪ f '' B,
or.elim h2
(assume h3 : y ∈ f '' A,
exists.elim h3 $
assume x h,
have h4 : x ∈ A, from h.left,
have h5 : f x = y, from h.right,
have h6 : x ∈ A ∪ B, from or.inl h4,
show y ∈ f '' (A ∪ B), from ⟨x, h6, h5⟩)
(assume h3 : y ∈ f '' B,
exists.elim h3 $
assume x h,
have h4 : x ∈ B, from h.left,
have h5 : f x = y, from h.right,
have h6 : x ∈ A ∪ B, from or.inr h4,
show y ∈ f '' (A ∪ B), from ⟨x, h6, h5⟩))
-- remember, x ∈ A ∩ B is the same as x ∈ A ∧ x ∈ B
example (x : X) (h1 : x ∈ A) (h2 : x ∈ B) : x ∈ A ∩ B :=
and.intro h1 h2
example (x : X) (h1 : x ∈ A ∩ B) : x ∈ A :=
and.left h1
-- Fill in the proof below.
-- (It should take about 8 lines.)
example : f '' (A ∩ B) ⊆ f '' A ∩ f '' B :=
assume y,
assume h1 : y ∈ f '' (A ∩ B),
show y ∈ f '' A ∩ f '' B, from (
exists.elim h1 $
assume (x : X) (h : x ∈ A ∩ B ∧ f x = y),
have ha : x ∈ A, from h.left.left,
have hb : x ∈ B, from h.left.right,
have he : f x = y, from h.right,
show y ∈ f '' A ∩ f '' B, from ⟨⟨x, ha, he⟩, ⟨x, hb, he⟩⟩
)
end ex2 |
26dbbdfc043b4d8fea63d22918377f4f69d3743a | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/tactic/equiv_rw.lean | 6fcceabd49fd985902ab376bf84ba5097bd05da8 | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 12,104 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import tactic.simp_result
import tactic.clear
import control.equiv_functor.instances
/-!
# The `equiv_rw` tactic transports goals or hypotheses along equivalences.
The basic syntax is `equiv_rw e`, where `e : α ≃ β` is an equivalence.
This will try to replace occurrences of `α` in the goal with `β`, for example
transforming
* `⊢ α` to `⊢ β`,
* `⊢ option α` to `⊢ option β`
* `⊢ {a // P}` to `{b // P (⇑(equiv.symm e) b)}`
The tactic can also be used to rewrite hypotheses, using the syntax `equiv_rw e at h`.
## Implementation details
The main internal function is `equiv_rw_type e t`,
which attempts to turn an expression `e : α ≃ β` into a new equivalence with left hand side `t`.
As an example, with `t = option α`, it will generate `functor.map_equiv option e`.
This is achieved by generating a new synthetic goal `%%t ≃ _`,
and calling `solve_by_elim` with an appropriate set of congruence lemmas.
To avoid having to specify the relevant congruence lemmas by hand,
we mostly rely on `equiv_functor.map_equiv` and `bifunctor.map_equiv`
along with some structural congruence lemmas such as
* `equiv.arrow_congr'`,
* `equiv.subtype_equiv_of_subtype'`,
* `equiv.sigma_congr_left'`, and
* `equiv.Pi_congr_left'`.
The main `equiv_rw` function, when operating on the goal, simply generates a new equivalence `e'`
with left hand side matching the target, and calls `apply e'.inv_fun`.
When operating on a hypothesis `x : α`, we introduce a new fact `h : x = e.symm (e x)`,
revert this, and then attempt to `generalize`, replacing all occurrences of `e x` with a new constant `y`,
before `intro`ing and `subst`ing `h`, and renaming `y` back to `x`.
## Future improvements
In a future PR I anticipate that `derive equiv_functor` should work on many examples,
(internally using `transport`, which is in turn based on `equiv_rw`)
and we can incrementally bootstrap the strength of `equiv_rw`.
An ambitious project might be to add `equiv_rw!`,
a tactic which, when failing to find appropriate `equiv_functor` instances,
attempts to `derive` them on the spot.
For now `equiv_rw` is entirely based on `equiv`,
but the framework can readily be generalised to also work with other types of equivalences,
for example specific notations such as ring equivalence (`≃+*`),
or general categorical isomorphisms (`≅`).
This will allow us to transport across more general types of equivalences,
but this will wait for another subsequent PR.
-/
namespace tactic
/-- A list of lemmas used for constructing congruence equivalences. -/
-- Although this looks 'hard-coded', in fact the lemma `equiv_functor.map_equiv`
-- allows us to extend `equiv_rw` simply by constructing new instance so `equiv_functor`.
-- TODO: We should also use `category_theory.functorial` and `category_theory.hygienic` instances.
-- (example goal: we could rewrite along an isomorphism of rings (either as `R ≅ S` or `R ≃+* S`)
-- and turn an `x : mv_polynomial σ R` into an `x : mv_polynomial σ S`.).
meta def equiv_congr_lemmas : tactic (list expr) :=
do exprs ←
[
`equiv.of_iff,
-- TODO decide what to do with this; it's an equiv_bifunctor?
`equiv.equiv_congr,
-- The function arrow is technically a bifunctor `Typeᵒᵖ → Type → Type`,
-- but the pattern matcher will never see this.
`equiv.arrow_congr',
-- Allow rewriting in subtypes:
`equiv.subtype_equiv_of_subtype',
-- Allow rewriting in the first component of a sigma-type:
`equiv.sigma_congr_left',
-- Allow rewriting ∀s:
-- (You might think that repeated application of `equiv.forall_congr'
-- would handle the higher arity cases, but unfortunately unification is not clever enough.)
`equiv.forall₃_congr',
`equiv.forall₂_congr',
`equiv.forall_congr',
-- Allow rewriting in argument of Pi types:
`equiv.Pi_congr_left',
-- Handles `sum` and `prod`, and many others:
`bifunctor.map_equiv,
-- Handles `list`, `option`, `unique`, and many others:
`equiv_functor.map_equiv,
-- We have to filter results to ensure we don't cheat and use exclusively `equiv.refl` and `iff.refl`!
`equiv.refl,
`iff.refl
].mmap (λ n, try_core (mk_const n)),
return (exprs.map option.to_list).join -- TODO: implement `.mfilter_map mk_const`?
declare_trace equiv_rw_type
/--
Configuration structure for `equiv_rw`.
* `max_depth` bounds the search depth for equivalences to rewrite along.
The default value is 10.
(e.g., if you're rewriting along `e : α ≃ β`, and `max_depth := 2`,
you can rewrite `option (option α))` but not `option (option (option α))`.
-/
meta structure equiv_rw_cfg :=
(max_depth : ℕ := 10)
/--
Implementation of `equiv_rw_type`, using `solve_by_elim`.
Expects a goal of the form `t ≃ _`,
and tries to solve it using `eq : α ≃ β` and congruence lemmas.
-/
meta def equiv_rw_type_core (eq : expr) (cfg : equiv_rw_cfg) : tactic unit :=
do
-- Assemble the relevant lemmas.
equiv_congr_lemmas ← equiv_congr_lemmas,
/-
We now call `solve_by_elim` to try to generate the requested equivalence.
There are a few subtleties!
* We make sure that `eq` is the first lemma, so it is applied whenever possible.
* In `equiv_congr_lemmas`, we put `equiv.refl` last so it is only used when it is not possible
to descend further.
* Since some congruence lemmas generate subgoals with `∀` statements,
we use the `pre_apply` subtactic of `solve_by_elim` to preprocess each new goal with `intros`.
-/
solve_by_elim
{ use_symmetry := false,
use_exfalso := false,
lemmas := some (eq :: equiv_congr_lemmas),
max_depth := cfg.max_depth,
-- Subgoals may contain function types,
-- and we want to continue trying to construct equivalences after the binders.
pre_apply := tactic.intros >> skip,
-- If solve_by_elim gets stuck, make sure it isn't because there's a later `≃` or `↔` goal
-- that we should still attempt.
discharger := `[show _ ≃ _] <|> `[show _ ↔ _] <|>
trace_if_enabled `equiv_rw_type "Failed, no congruence lemma applied!" >> failed,
-- We use the `accept` tactic in `solve_by_elim` to provide tracing.
accept := λ goals, lock_tactic_state (do
when_tracing `equiv_rw_type (do
goals.mmap pp >>= λ goals, trace format!"So far, we've built: {goals}"),
done <|>
when_tracing `equiv_rw_type (do
gs ← get_goals,
gs ← gs.mmap (λ g, infer_type g >>= pp),
trace format!"Attempting to adapt to {gs}")) }
/--
`equiv_rw_type e t` rewrites the type `t` using the equivalence `e : α ≃ β`,
returning a new equivalence `t ≃ t'`.
-/
meta def equiv_rw_type (eqv : expr) (ty : expr) (cfg : equiv_rw_cfg) : tactic expr :=
do
when_tracing `equiv_rw_type (do
ty_pp ← pp ty,
eqv_pp ← pp eqv,
eqv_ty_pp ← infer_type eqv >>= pp,
trace format!"Attempting to rewrite the type `{ty_pp}` using `{eqv_pp} : {eqv_ty_pp}`."),
`(_ ≃ _) ← infer_type eqv | fail format!"{eqv} must be an `equiv`",
-- We prepare a synthetic goal of type `(%%ty ≃ _)`, for some placeholder right hand side.
equiv_ty ← to_expr ``(%%ty ≃ _),
-- Now call `equiv_rw_type_core`.
new_eqv ← prod.snd <$> (solve_aux equiv_ty $ equiv_rw_type_core eqv cfg),
-- Check that we actually used the equivalence `eq`
-- (`equiv_rw_type_core` will always find `equiv.refl`, but hopefully only after all other possibilities)
new_eqv ← instantiate_mvars new_eqv,
-- We previously had `guard (eqv.occurs new_eqv)` here, but `kdepends_on` is more reliable.
kdepends_on new_eqv eqv >>= guardb <|> (do
eqv_pp ← pp eqv,
ty_pp ← pp ty,
fail format!"Could not construct an equivalence from {eqv_pp} of the form: {ty_pp} ≃ _"),
-- Finally we simplify the resulting equivalence,
-- to compress away some `map_equiv equiv.refl` subexpressions.
prod.fst <$> new_eqv.simp {fail_if_unchanged := ff}
mk_simp_attribute equiv_rw_simp "The simpset `equiv_rw_simp` is used by the tactic `equiv_rw` to
simplify applications of equivalences and their inverses."
attribute [equiv_rw_simp] equiv.symm_symm equiv.apply_symm_apply equiv.symm_apply_apply
/--
Attempt to replace the hypothesis with name `x`
by transporting it along the equivalence in `e : α ≃ β`.
-/
meta def equiv_rw_hyp (x : name) (e : expr) (cfg : equiv_rw_cfg := {}) : tactic unit :=
-- We call `dsimp_result` to perform the beta redex introduced by `revert`
dsimp_result (do
x' ← get_local x,
x_ty ← infer_type x',
-- Adapt `e` to an equivalence with left-hand-side `x_ty`.
e ← equiv_rw_type e x_ty cfg,
eq ← to_expr ``(%%x' = equiv.symm %%e (equiv.to_fun %%e %%x')),
prf ← to_expr ``((equiv.symm_apply_apply %%e %%x').symm),
h ← note_anon eq prf,
-- Revert the new hypothesis, so it is also part of the goal.
revert h,
ex ← to_expr ``(equiv.to_fun %%e %%x'),
-- Now call `generalize`,
-- attempting to replace all occurrences of `e x`,
-- calling it for now `j : β`, with `k : x = e.symm j`.
generalize ex (by apply_opt_param) transparency.none,
-- Reintroduce `x` (now of type `b`), and the hypothesis `h`.
intro x,
h ← intro1,
-- Finally, if we're working on properties, substitute along `h`, then do some cleanup,
-- and if we're working on data, just throw out the old `x`.
b ← target >>= is_prop,
if b then do
subst h,
`[try { simp only with equiv_rw_simp }]
else
-- We may need to unfreeze `x` before we can `clear` it.
unfreezing_hyp x' (clear' tt [x']) <|>
fail format!"equiv_rw expected to be able to clear the original hypothesis {x}, but couldn't.",
skip)
{fail_if_unchanged := ff} tt -- call `dsimp_result` with `no_defaults := tt`.
/-- Rewrite the goal using an equiv `e`. -/
meta def equiv_rw_target (e : expr) (cfg : equiv_rw_cfg := {}) : tactic unit :=
do
t ← target,
e ← equiv_rw_type e t cfg,
s ← to_expr ``(equiv.inv_fun %%e),
tactic.eapply s,
skip
end tactic
namespace tactic.interactive
open lean.parser
open interactive interactive.types
open tactic
local postfix `?`:9001 := optional
/--
`equiv_rw e at h`, where `h : α` is a hypothesis, and `e : α ≃ β`,
will attempt to transport `h` along `e`, producing a new hypothesis `h : β`,
with all occurrences of `h` in other hypotheses and the goal replaced with `e.symm h`.
`equiv_rw e` will attempt to transport the goal along an equivalence `e : α ≃ β`.
In its minimal form it replaces the goal `⊢ α` with `⊢ β` by calling `apply e.inv_fun`.
`equiv_rw` will also try rewriting under (equiv_)functors, so can turn
a hypothesis `h : list α` into `h : list β` or
a goal `⊢ unique α` into `⊢ unique β`.
The maximum search depth for rewriting in subexpressions is controlled by
`equiv_rw e {max_depth := n}`.
-/
meta def equiv_rw (e : parse texpr) (loc : parse $ (tk "at" *> ident)?) (cfg : equiv_rw_cfg := {}) : itactic :=
do e ← to_expr e,
match loc with
| (some hyp) := equiv_rw_hyp hyp e cfg
| none := equiv_rw_target e cfg
end
add_tactic_doc
{ name := "equiv_rw",
category := doc_category.tactic,
decl_names := [`tactic.interactive.equiv_rw],
tags := ["rewriting", "equiv", "transport"] }
/--
Solve a goal of the form `t ≃ _`,
by constructing an equivalence from `e : α ≃ β`.
This is the same equivalence that `equiv_rw` would use to rewrite a term of type `t`.
A typical usage might be:
```
have e' : option α ≃ option β := by equiv_rw_type e
```
-/
meta def equiv_rw_type (e : parse texpr) (cfg : equiv_rw_cfg := {}) : itactic :=
do
`(%%t ≃ _) ← target | fail "`equiv_rw_type` solves goals of the form `t ≃ _`.",
e ← to_expr e,
tactic.equiv_rw_type e t cfg >>= tactic.exact
add_tactic_doc
{ name := "equiv_rw_type",
category := doc_category.tactic,
decl_names := [`tactic.interactive.equiv_rw_type],
tags := ["rewriting", "equiv", "transport"] }
end tactic.interactive
|
e84b755ae502fb9ae6e473e4e93ae6b02c4b3b4b | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/data/real/nnreal.lean | 15685b7067bbac7b74b9cb92a6a8f8f7b6cb2231 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 36,143 | lean | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import order.conditionally_complete_lattice.group
import algebra.algebra.basic
import algebra.order.nonneg.field
import algebra.order.field.canonical.basic
import data.real.pointwise
import tactic.positivity
/-!
# Nonnegative real numbers
In this file we define `nnreal` (notation: `ℝ≥0`) to be the type of non-negative real numbers,
a.k.a. the interval `[0, ∞)`. We also define the following operations and structures on `ℝ≥0`:
* the order on `ℝ≥0` is the restriction of the order on `ℝ`; these relations define a conditionally
complete linear order with a bottom element, `conditionally_complete_linear_order_bot`;
* `a + b` and `a * b` are the restrictions of addition and multiplication of real numbers to `ℝ≥0`;
these operations together with `0 = ⟨0, _⟩` and `1 = ⟨1, _⟩` turn `ℝ≥0` into a conditionally
complete linear ordered archimedean commutative semifield; we have no typeclass for this in
`mathlib` yet, so we define the following instances instead:
- `linear_ordered_semiring ℝ≥0`;
- `ordered_comm_semiring ℝ≥0`;
- `canonically_ordered_comm_semiring ℝ≥0`;
- `linear_ordered_comm_group_with_zero ℝ≥0`;
- `canonically_linear_ordered_add_monoid ℝ≥0`;
- `archimedean ℝ≥0`;
- `conditionally_complete_linear_order_bot ℝ≥0`.
These instances are derived from corresponding instances about the type `{x : α // 0 ≤ x}` in an
appropriate ordered field/ring/group/monoid `α`. See `algebra/order/nonneg`.
* `real.to_nnreal x` is defined as `⟨max x 0, _⟩`, i.e. `↑(real.to_nnreal x) = x` when `0 ≤ x` and
`↑(real.to_nnreal x) = 0` otherwise.
We also define an instance `can_lift ℝ ℝ≥0`. This instance can be used by the `lift` tactic to
replace `x : ℝ` and `hx : 0 ≤ x` in the proof context with `x : ℝ≥0` while replacing all occurences
of `x` with `↑x`. This tactic also works for a function `f : α → ℝ` with a hypothesis
`hf : ∀ x, 0 ≤ f x`.
## Notations
This file defines `ℝ≥0` as a localized notation for `nnreal`.
-/
open_locale classical big_operators
/-- Nonnegative real numbers. -/
@[derive [
strict_ordered_semiring, comm_monoid_with_zero, -- to ensure these instances are computable
floor_semiring, comm_semiring, semiring,
semilattice_inf, semilattice_sup,
distrib_lattice, densely_ordered, order_bot,
canonically_linear_ordered_semifield, linear_ordered_comm_group_with_zero, archimedean,
linear_ordered_semiring, ordered_comm_semiring, canonically_ordered_comm_semiring,
has_sub, has_ordered_sub, has_div, inhabited]]
def nnreal := {r : ℝ // 0 ≤ r}
localized "notation (name := nnreal) `ℝ≥0` := nnreal" in nnreal
namespace nnreal
instance : has_coe ℝ≥0 ℝ := ⟨subtype.val⟩
/- Simp lemma to put back `n.val` into the normal form given by the coercion. -/
@[simp] lemma val_eq_coe (n : ℝ≥0) : n.val = n := rfl
instance can_lift : can_lift ℝ ℝ≥0 coe (λ r, 0 ≤ r) := subtype.can_lift _
protected lemma eq {n m : ℝ≥0} : (n : ℝ) = (m : ℝ) → n = m := subtype.eq
protected lemma eq_iff {n m : ℝ≥0} : (n : ℝ) = (m : ℝ) ↔ n = m :=
iff.intro nnreal.eq (congr_arg coe)
lemma ne_iff {x y : ℝ≥0} : (x : ℝ) ≠ (y : ℝ) ↔ x ≠ y :=
not_iff_not_of_iff $ nnreal.eq_iff
protected lemma «forall» {p : ℝ≥0 → Prop} : (∀ x : ℝ≥0, p x) ↔ ∀ (x : ℝ) (hx : 0 ≤ x), p ⟨x, hx⟩ :=
subtype.forall
protected lemma «exists» {p : ℝ≥0 → Prop} : (∃ x : ℝ≥0, p x) ↔ ∃ (x : ℝ) (hx : 0 ≤ x), p ⟨x, hx⟩ :=
subtype.exists
/-- Reinterpret a real number `r` as a non-negative real number. Returns `0` if `r < 0`. -/
noncomputable def _root_.real.to_nnreal (r : ℝ) : ℝ≥0 := ⟨max r 0, le_max_right _ _⟩
lemma _root_.real.coe_to_nnreal (r : ℝ) (hr : 0 ≤ r) : (real.to_nnreal r : ℝ) = r :=
max_eq_left hr
lemma _root_.real.to_nnreal_of_nonneg {r : ℝ} (hr : 0 ≤ r) : r.to_nnreal = ⟨r, hr⟩ :=
by simp_rw [real.to_nnreal, max_eq_left hr]
lemma _root_.real.le_coe_to_nnreal (r : ℝ) : r ≤ real.to_nnreal r :=
le_max_left r 0
lemma coe_nonneg (r : ℝ≥0) : (0 : ℝ) ≤ r := r.2
@[norm_cast]
theorem coe_mk (a : ℝ) (ha) : ((⟨a, ha⟩ : ℝ≥0) : ℝ) = a := rfl
example : has_zero ℝ≥0 := by apply_instance
example : has_one ℝ≥0 := by apply_instance
example : has_add ℝ≥0 := by apply_instance
noncomputable example : has_sub ℝ≥0 := by apply_instance
example : has_mul ℝ≥0 := by apply_instance
noncomputable example : has_inv ℝ≥0 := by apply_instance
noncomputable example : has_div ℝ≥0 := by apply_instance
example : has_le ℝ≥0 := by apply_instance
example : has_bot ℝ≥0 := by apply_instance
example : inhabited ℝ≥0 := by apply_instance
example : nontrivial ℝ≥0 := by apply_instance
protected lemma coe_injective : function.injective (coe : ℝ≥0 → ℝ) := subtype.coe_injective
@[simp, norm_cast] protected lemma coe_eq {r₁ r₂ : ℝ≥0} : (r₁ : ℝ) = r₂ ↔ r₁ = r₂ :=
nnreal.coe_injective.eq_iff
protected lemma coe_zero : ((0 : ℝ≥0) : ℝ) = 0 := rfl
protected lemma coe_one : ((1 : ℝ≥0) : ℝ) = 1 := rfl
protected lemma coe_add (r₁ r₂ : ℝ≥0) : ((r₁ + r₂ : ℝ≥0) : ℝ) = r₁ + r₂ := rfl
protected lemma coe_mul (r₁ r₂ : ℝ≥0) : ((r₁ * r₂ : ℝ≥0) : ℝ) = r₁ * r₂ := rfl
protected lemma coe_inv (r : ℝ≥0) : ((r⁻¹ : ℝ≥0) : ℝ) = r⁻¹ := rfl
protected lemma coe_div (r₁ r₂ : ℝ≥0) : ((r₁ / r₂ : ℝ≥0) : ℝ) = r₁ / r₂ := rfl
@[simp, norm_cast] protected lemma coe_bit0 (r : ℝ≥0) : ((bit0 r : ℝ≥0) : ℝ) = bit0 r := rfl
@[simp, norm_cast] protected lemma coe_bit1 (r : ℝ≥0) : ((bit1 r : ℝ≥0) : ℝ) = bit1 r := rfl
protected lemma coe_two : ((2 : ℝ≥0) : ℝ) = 2 := rfl
@[simp, norm_cast] protected lemma coe_sub {r₁ r₂ : ℝ≥0} (h : r₂ ≤ r₁) :
((r₁ - r₂ : ℝ≥0) : ℝ) = r₁ - r₂ :=
max_eq_left $ le_sub_comm.2 $ by simp [show (r₂ : ℝ) ≤ r₁, from h]
@[simp, norm_cast] protected lemma coe_eq_zero (r : ℝ≥0) : ↑r = (0 : ℝ) ↔ r = 0 :=
by rw [← nnreal.coe_zero, nnreal.coe_eq]
@[simp, norm_cast] protected lemma coe_eq_one (r : ℝ≥0) : ↑r = (1 : ℝ) ↔ r = 1 :=
by rw [← nnreal.coe_one, nnreal.coe_eq]
lemma coe_ne_zero {r : ℝ≥0} : (r : ℝ) ≠ 0 ↔ r ≠ 0 := by norm_cast
example : comm_semiring ℝ≥0 := by apply_instance
/-- Coercion `ℝ≥0 → ℝ` as a `ring_hom`. -/
def to_real_hom : ℝ≥0 →+* ℝ :=
⟨coe, nnreal.coe_one, nnreal.coe_mul, nnreal.coe_zero, nnreal.coe_add⟩
@[simp] lemma coe_to_real_hom : ⇑to_real_hom = coe := rfl
section actions
/-- A `mul_action` over `ℝ` restricts to a `mul_action` over `ℝ≥0`. -/
instance {M : Type*} [mul_action ℝ M] : mul_action ℝ≥0 M :=
mul_action.comp_hom M to_real_hom.to_monoid_hom
lemma smul_def {M : Type*} [mul_action ℝ M] (c : ℝ≥0) (x : M) :
c • x = (c : ℝ) • x := rfl
instance {M N : Type*} [mul_action ℝ M] [mul_action ℝ N] [has_smul M N]
[is_scalar_tower ℝ M N] : is_scalar_tower ℝ≥0 M N :=
{ smul_assoc := λ r, (smul_assoc (r : ℝ) : _)}
instance smul_comm_class_left {M N : Type*} [mul_action ℝ N] [has_smul M N]
[smul_comm_class ℝ M N] : smul_comm_class ℝ≥0 M N :=
{ smul_comm := λ r, (smul_comm (r : ℝ) : _)}
instance smul_comm_class_right {M N : Type*} [mul_action ℝ N] [has_smul M N]
[smul_comm_class M ℝ N] : smul_comm_class M ℝ≥0 N :=
{ smul_comm := λ m r, (smul_comm m (r : ℝ) : _)}
/-- A `distrib_mul_action` over `ℝ` restricts to a `distrib_mul_action` over `ℝ≥0`. -/
instance {M : Type*} [add_monoid M] [distrib_mul_action ℝ M] : distrib_mul_action ℝ≥0 M :=
distrib_mul_action.comp_hom M to_real_hom.to_monoid_hom
/-- A `module` over `ℝ` restricts to a `module` over `ℝ≥0`. -/
instance {M : Type*} [add_comm_monoid M] [module ℝ M] : module ℝ≥0 M :=
module.comp_hom M to_real_hom
/-- An `algebra` over `ℝ` restricts to an `algebra` over `ℝ≥0`. -/
instance {A : Type*} [semiring A] [algebra ℝ A] : algebra ℝ≥0 A :=
{ smul := (•),
commutes' := λ r x, by simp [algebra.commutes],
smul_def' := λ r x, by simp [←algebra.smul_def (r : ℝ) x, smul_def],
to_ring_hom := ((algebra_map ℝ A).comp (to_real_hom : ℝ≥0 →+* ℝ)) }
-- verify that the above produces instances we might care about
example : algebra ℝ≥0 ℝ := by apply_instance
example : distrib_mul_action ℝ≥0ˣ ℝ := by apply_instance
end actions
example : monoid_with_zero ℝ≥0 := by apply_instance
example : comm_monoid_with_zero ℝ≥0 := by apply_instance
noncomputable example : comm_group_with_zero ℝ≥0 := by apply_instance
@[simp, norm_cast] lemma coe_indicator {α} (s : set α) (f : α → ℝ≥0) (a : α) :
((s.indicator f a : ℝ≥0) : ℝ) = s.indicator (λ x, f x) a :=
(to_real_hom : ℝ≥0 →+ ℝ).map_indicator _ _ _
@[simp, norm_cast] lemma coe_pow (r : ℝ≥0) (n : ℕ) : ((r^n : ℝ≥0) : ℝ) = r^n :=
to_real_hom.map_pow r n
@[simp, norm_cast] lemma coe_zpow (r : ℝ≥0) (n : ℤ) : ((r^n : ℝ≥0) : ℝ) = r^n :=
by cases n; simp
@[norm_cast] lemma coe_list_sum (l : list ℝ≥0) :
((l.sum : ℝ≥0) : ℝ) = (l.map coe).sum :=
to_real_hom.map_list_sum l
@[norm_cast] lemma coe_list_prod (l : list ℝ≥0) :
((l.prod : ℝ≥0) : ℝ) = (l.map coe).prod :=
to_real_hom.map_list_prod l
@[norm_cast] lemma coe_multiset_sum (s : multiset ℝ≥0) :
((s.sum : ℝ≥0) : ℝ) = (s.map coe).sum :=
to_real_hom.map_multiset_sum s
@[norm_cast] lemma coe_multiset_prod (s : multiset ℝ≥0) :
((s.prod : ℝ≥0) : ℝ) = (s.map coe).prod :=
to_real_hom.map_multiset_prod s
@[norm_cast] lemma coe_sum {α} {s : finset α} {f : α → ℝ≥0} :
↑(∑ a in s, f a) = ∑ a in s, (f a : ℝ) :=
to_real_hom.map_sum _ _
lemma _root_.real.to_nnreal_sum_of_nonneg {α} {s : finset α} {f : α → ℝ}
(hf : ∀ a, a ∈ s → 0 ≤ f a) :
real.to_nnreal (∑ a in s, f a) = ∑ a in s, real.to_nnreal (f a) :=
begin
rw [←nnreal.coe_eq, nnreal.coe_sum, real.coe_to_nnreal _ (finset.sum_nonneg hf)],
exact finset.sum_congr rfl (λ x hxs, by rw real.coe_to_nnreal _ (hf x hxs)),
end
@[norm_cast] lemma coe_prod {α} {s : finset α} {f : α → ℝ≥0} :
↑(∏ a in s, f a) = ∏ a in s, (f a : ℝ) :=
to_real_hom.map_prod _ _
lemma _root_.real.to_nnreal_prod_of_nonneg {α} {s : finset α} {f : α → ℝ}
(hf : ∀ a, a ∈ s → 0 ≤ f a) :
real.to_nnreal (∏ a in s, f a) = ∏ a in s, real.to_nnreal (f a) :=
begin
rw [←nnreal.coe_eq, nnreal.coe_prod, real.coe_to_nnreal _ (finset.prod_nonneg hf)],
exact finset.prod_congr rfl (λ x hxs, by rw real.coe_to_nnreal _ (hf x hxs)),
end
lemma nsmul_coe (r : ℝ≥0) (n : ℕ) : ↑(n • r) = n • (r:ℝ) :=
by norm_cast
@[simp, norm_cast] protected lemma coe_nat_cast (n : ℕ) : (↑(↑n : ℝ≥0) : ℝ) = n :=
map_nat_cast to_real_hom n
noncomputable example : linear_order ℝ≥0 := by apply_instance
@[simp, norm_cast] protected lemma coe_le_coe {r₁ r₂ : ℝ≥0} : (r₁ : ℝ) ≤ r₂ ↔ r₁ ≤ r₂ := iff.rfl
@[simp, norm_cast] protected lemma coe_lt_coe {r₁ r₂ : ℝ≥0} : (r₁ : ℝ) < r₂ ↔ r₁ < r₂ := iff.rfl
@[simp, norm_cast] protected lemma coe_pos {r : ℝ≥0} : (0 : ℝ) < r ↔ 0 < r := iff.rfl
protected lemma coe_mono : monotone (coe : ℝ≥0 → ℝ) := λ _ _, nnreal.coe_le_coe.2
protected lemma _root_.real.to_nnreal_mono : monotone real.to_nnreal :=
λ x y h, max_le_max h (le_refl 0)
@[simp] lemma _root_.real.to_nnreal_coe {r : ℝ≥0} : real.to_nnreal r = r :=
nnreal.eq $ max_eq_left r.2
@[simp] lemma mk_coe_nat (n : ℕ) : @eq ℝ≥0 (⟨(n : ℝ), n.cast_nonneg⟩ : ℝ≥0) n :=
nnreal.eq (nnreal.coe_nat_cast n).symm
@[simp] lemma to_nnreal_coe_nat (n : ℕ) : real.to_nnreal n = n :=
nnreal.eq $ by simp [real.coe_to_nnreal]
/-- `real.to_nnreal` and `coe : ℝ≥0 → ℝ` form a Galois insertion. -/
noncomputable def gi : galois_insertion real.to_nnreal coe :=
galois_insertion.monotone_intro nnreal.coe_mono real.to_nnreal_mono
real.le_coe_to_nnreal (λ _, real.to_nnreal_coe)
-- note that anything involving the (decidability of the) linear order,
-- will be noncomputable, everything else should not be.
example : order_bot ℝ≥0 := by apply_instance
example : partial_order ℝ≥0 := by apply_instance
noncomputable example : canonically_linear_ordered_add_monoid ℝ≥0 := by apply_instance
noncomputable example : linear_ordered_add_comm_monoid ℝ≥0 := by apply_instance
example : distrib_lattice ℝ≥0 := by apply_instance
example : semilattice_inf ℝ≥0 := by apply_instance
example : semilattice_sup ℝ≥0 := by apply_instance
noncomputable example : linear_ordered_semiring ℝ≥0 := by apply_instance
example : ordered_comm_semiring ℝ≥0 := by apply_instance
noncomputable example : linear_ordered_comm_monoid ℝ≥0 := by apply_instance
noncomputable example : linear_ordered_comm_monoid_with_zero ℝ≥0 := by apply_instance
noncomputable example : linear_ordered_comm_group_with_zero ℝ≥0 := by apply_instance
example : canonically_ordered_comm_semiring ℝ≥0 := by apply_instance
example : densely_ordered ℝ≥0 := by apply_instance
example : no_max_order ℝ≥0 := by apply_instance
/-- If `a` is a nonnegative real number, then the closed interval `[0, a]` in `ℝ` is order
isomorphic to the interval `set.Iic a`. -/
@[simps apply_coe_coe] def order_iso_Icc_zero_coe (a : ℝ≥0) : set.Icc (0 : ℝ) a ≃o set.Iic a :=
{ to_equiv := equiv.set.sep (set.Ici 0) (λ x, x ≤ a),
map_rel_iff' := λ x y, iff.rfl }
@[simp] lemma order_iso_Icc_zero_coe_symm_apply_coe (a : ℝ≥0) (b : set.Iic a) :
((order_iso_Icc_zero_coe a).symm b : ℝ) = b :=
rfl
-- note we need the `@` to make the `has_mem.mem` have a sensible type
lemma coe_image {s : set ℝ≥0} : coe '' s = {x : ℝ | ∃ h : 0 ≤ x, @has_mem.mem (ℝ≥0) _ _ ⟨x, h⟩ s} :=
subtype.coe_image
lemma bdd_above_coe {s : set ℝ≥0} : bdd_above ((coe : ℝ≥0 → ℝ) '' s) ↔ bdd_above s :=
iff.intro
(assume ⟨b, hb⟩, ⟨real.to_nnreal b, assume ⟨y, hy⟩ hys, show y ≤ max b 0, from
le_max_of_le_left $ hb $ set.mem_image_of_mem _ hys⟩)
(assume ⟨b, hb⟩, ⟨b, assume y ⟨x, hx, eq⟩, eq ▸ hb hx⟩)
lemma bdd_below_coe (s : set ℝ≥0) : bdd_below ((coe : ℝ≥0 → ℝ) '' s) :=
⟨0, assume r ⟨q, _, eq⟩, eq ▸ q.2⟩
noncomputable instance : conditionally_complete_linear_order_bot ℝ≥0 :=
nonneg.conditionally_complete_linear_order_bot real.Sup_empty.le
@[norm_cast] lemma coe_Sup (s : set ℝ≥0) : (↑(Sup s) : ℝ) = Sup ((coe : ℝ≥0 → ℝ) '' s) :=
eq.symm $ @subset_Sup_of_within ℝ (set.Ici 0) _ ⟨(0 : ℝ≥0)⟩ s $
real.Sup_nonneg _ $ λ y ⟨x, _, hy⟩, hy ▸ x.2
@[norm_cast] lemma coe_supr {ι : Sort*} (s : ι → ℝ≥0) : (↑(⨆ i, s i) : ℝ) = ⨆ i, (s i) :=
by rw [supr, supr, coe_Sup, set.range_comp]
@[norm_cast] lemma coe_Inf (s : set ℝ≥0) : (↑(Inf s) : ℝ) = Inf ((coe : ℝ≥0 → ℝ) '' s) :=
eq.symm $ @subset_Inf_of_within ℝ (set.Ici 0) _ ⟨(0 : ℝ≥0)⟩ s $
real.Inf_nonneg _ $ λ y ⟨x, _, hy⟩, hy ▸ x.2
@[simp] lemma Inf_empty : Inf (∅ : set ℝ≥0) = 0 :=
by rw [← nnreal.coe_eq_zero, coe_Inf, set.image_empty, real.Inf_empty]
@[norm_cast] lemma coe_infi {ι : Sort*} (s : ι → ℝ≥0) : (↑(⨅ i, s i) : ℝ) = ⨅ i, (s i) :=
by rw [infi, infi, coe_Inf, set.range_comp]
lemma le_infi_add_infi {ι ι' : Sort*} [nonempty ι] [nonempty ι'] {f : ι → ℝ≥0} {g : ι' → ℝ≥0}
{a : ℝ≥0} (h : ∀ i j, a ≤ f i + g j) : a ≤ (⨅ i, f i) + ⨅ j, g j :=
begin
rw [← nnreal.coe_le_coe, nnreal.coe_add, coe_infi, coe_infi],
exact le_cinfi_add_cinfi h
end
example : archimedean ℝ≥0 := by apply_instance
-- TODO: why are these three instances necessary? why aren't they inferred?
instance covariant_add : covariant_class ℝ≥0 ℝ≥0 (+) (≤) :=
ordered_add_comm_monoid.to_covariant_class_left ℝ≥0
instance contravariant_add : contravariant_class ℝ≥0 ℝ≥0 (+) (<) :=
ordered_cancel_add_comm_monoid.to_contravariant_class_left ℝ≥0
instance covariant_mul : covariant_class ℝ≥0 ℝ≥0 (*) (≤) :=
ordered_comm_monoid.to_covariant_class_left ℝ≥0
-- Why isn't `nnreal.contravariant_add` inferred?
lemma le_of_forall_pos_le_add {a b : ℝ≥0} (h : ∀ε, 0 < ε → a ≤ b + ε) : a ≤ b :=
@le_of_forall_pos_le_add _ _ _ _ _ _ nnreal.contravariant_add _ _ h
lemma lt_iff_exists_rat_btwn (a b : ℝ≥0) :
a < b ↔ (∃q:ℚ, 0 ≤ q ∧ a < real.to_nnreal q ∧ real.to_nnreal q < b) :=
iff.intro
(assume (h : (↑a:ℝ) < (↑b:ℝ)),
let ⟨q, haq, hqb⟩ := exists_rat_btwn h in
have 0 ≤ (q : ℝ), from le_trans a.2 $ le_of_lt haq,
⟨q, rat.cast_nonneg.1 this,
by simp [real.coe_to_nnreal _ this, nnreal.coe_lt_coe.symm, haq, hqb]⟩)
(assume ⟨q, _, haq, hqb⟩, lt_trans haq hqb)
lemma bot_eq_zero : (⊥ : ℝ≥0) = 0 := rfl
lemma mul_sup (a b c : ℝ≥0) : a * (b ⊔ c) = (a * b) ⊔ (a * c) :=
mul_max_of_nonneg _ _ $ zero_le a
lemma sup_mul (a b c : ℝ≥0) : (a ⊔ b) * c = (a * c) ⊔ (b * c) :=
max_mul_of_nonneg _ _ $ zero_le c
lemma mul_finset_sup {α} (r : ℝ≥0) (s : finset α) (f : α → ℝ≥0) :
r * s.sup f = s.sup (λ a, r * f a) :=
(finset.comp_sup_eq_sup_comp _ (nnreal.mul_sup r) (mul_zero r))
lemma finset_sup_mul {α} (s : finset α) (f : α → ℝ≥0) (r : ℝ≥0) :
s.sup f * r = s.sup (λ a, f a * r) :=
(finset.comp_sup_eq_sup_comp (* r) (λ x y, nnreal.sup_mul x y r) (zero_mul r))
lemma finset_sup_div {α} {f : α → ℝ≥0} {s : finset α} (r : ℝ≥0) :
s.sup f / r = s.sup (λ a, f a / r) :=
by simp only [div_eq_inv_mul, mul_finset_sup]
@[simp, norm_cast] lemma coe_max (x y : ℝ≥0) :
((max x y : ℝ≥0) : ℝ) = max (x : ℝ) (y : ℝ) :=
nnreal.coe_mono.map_max
@[simp, norm_cast] lemma coe_min (x y : ℝ≥0) :
((min x y : ℝ≥0) : ℝ) = min (x : ℝ) (y : ℝ) :=
nnreal.coe_mono.map_min
@[simp] lemma zero_le_coe {q : ℝ≥0} : 0 ≤ (q : ℝ) := q.2
end nnreal
namespace real
section to_nnreal
@[simp] lemma to_nnreal_zero : real.to_nnreal 0 = 0 :=
by simp [real.to_nnreal]; refl
@[simp] lemma to_nnreal_one : real.to_nnreal 1 = 1 :=
by simp [real.to_nnreal, max_eq_left (zero_le_one : (0 :ℝ) ≤ 1)]; refl
@[simp] lemma to_nnreal_pos {r : ℝ} : 0 < real.to_nnreal r ↔ 0 < r :=
by simp [real.to_nnreal, nnreal.coe_lt_coe.symm, lt_irrefl]
@[simp] lemma to_nnreal_eq_zero {r : ℝ} : real.to_nnreal r = 0 ↔ r ≤ 0 :=
by simpa [-to_nnreal_pos] using (not_iff_not.2 (@to_nnreal_pos r))
lemma to_nnreal_of_nonpos {r : ℝ} : r ≤ 0 → real.to_nnreal r = 0 :=
to_nnreal_eq_zero.2
@[simp] lemma coe_to_nnreal' (r : ℝ) : (real.to_nnreal r : ℝ) = max r 0 := rfl
@[simp] lemma to_nnreal_le_to_nnreal_iff {r p : ℝ} (hp : 0 ≤ p) :
real.to_nnreal r ≤ real.to_nnreal p ↔ r ≤ p :=
by simp [nnreal.coe_le_coe.symm, real.to_nnreal, hp]
@[simp] lemma to_nnreal_eq_to_nnreal_iff {r p : ℝ} (hr : 0 ≤ r) (hp : 0 ≤ p) :
real.to_nnreal r = real.to_nnreal p ↔ r = p :=
by simp [← nnreal.coe_eq, coe_to_nnreal, hr, hp]
@[simp] lemma to_nnreal_lt_to_nnreal_iff' {r p : ℝ} :
real.to_nnreal r < real.to_nnreal p ↔ r < p ∧ 0 < p :=
nnreal.coe_lt_coe.symm.trans max_lt_max_left_iff
lemma to_nnreal_lt_to_nnreal_iff {r p : ℝ} (h : 0 < p) :
real.to_nnreal r < real.to_nnreal p ↔ r < p :=
to_nnreal_lt_to_nnreal_iff'.trans (and_iff_left h)
lemma to_nnreal_lt_to_nnreal_iff_of_nonneg {r p : ℝ} (hr : 0 ≤ r) :
real.to_nnreal r < real.to_nnreal p ↔ r < p :=
to_nnreal_lt_to_nnreal_iff'.trans ⟨and.left, λ h, ⟨h, lt_of_le_of_lt hr h⟩⟩
@[simp] lemma to_nnreal_add {r p : ℝ} (hr : 0 ≤ r) (hp : 0 ≤ p) :
real.to_nnreal (r + p) = real.to_nnreal r + real.to_nnreal p :=
nnreal.eq $ by simp [real.to_nnreal, hr, hp, add_nonneg]
lemma to_nnreal_add_to_nnreal {r p : ℝ} (hr : 0 ≤ r) (hp : 0 ≤ p) :
real.to_nnreal r + real.to_nnreal p = real.to_nnreal (r + p) :=
(real.to_nnreal_add hr hp).symm
lemma to_nnreal_le_to_nnreal {r p : ℝ} (h : r ≤ p) :
real.to_nnreal r ≤ real.to_nnreal p :=
real.to_nnreal_mono h
lemma to_nnreal_add_le {r p : ℝ} :
real.to_nnreal (r + p) ≤ real.to_nnreal r + real.to_nnreal p :=
nnreal.coe_le_coe.1 $ max_le (add_le_add (le_max_left _ _) (le_max_left _ _)) nnreal.zero_le_coe
lemma to_nnreal_le_iff_le_coe {r : ℝ} {p : ℝ≥0} : real.to_nnreal r ≤ p ↔ r ≤ ↑p :=
nnreal.gi.gc r p
lemma le_to_nnreal_iff_coe_le {r : ℝ≥0} {p : ℝ} (hp : 0 ≤ p) : r ≤ real.to_nnreal p ↔ ↑r ≤ p :=
by rw [← nnreal.coe_le_coe, real.coe_to_nnreal p hp]
lemma le_to_nnreal_iff_coe_le' {r : ℝ≥0} {p : ℝ} (hr : 0 < r) : r ≤ real.to_nnreal p ↔ ↑r ≤ p :=
(le_or_lt 0 p).elim le_to_nnreal_iff_coe_le $ λ hp,
by simp only [(hp.trans_le r.coe_nonneg).not_le, to_nnreal_eq_zero.2 hp.le, hr.not_le]
lemma to_nnreal_lt_iff_lt_coe {r : ℝ} {p : ℝ≥0} (ha : 0 ≤ r) : real.to_nnreal r < p ↔ r < ↑p :=
by rw [← nnreal.coe_lt_coe, real.coe_to_nnreal r ha]
lemma lt_to_nnreal_iff_coe_lt {r : ℝ≥0} {p : ℝ} : r < real.to_nnreal p ↔ ↑r < p :=
begin
cases le_total 0 p,
{ rw [← nnreal.coe_lt_coe, real.coe_to_nnreal p h] },
{ rw [to_nnreal_eq_zero.2 h], split,
{ intro, have := not_lt_of_le (zero_le r), contradiction },
{ intro rp, have : ¬(p ≤ 0) := not_le_of_lt (lt_of_le_of_lt (nnreal.coe_nonneg _) rp),
contradiction } }
end
@[simp] lemma to_nnreal_bit0 (r : ℝ) : real.to_nnreal (bit0 r) = bit0 (real.to_nnreal r) :=
begin
cases le_total r 0 with hr hr,
{ rw [to_nnreal_of_nonpos hr, to_nnreal_of_nonpos, bit0_zero],
exact add_nonpos hr hr },
{ exact to_nnreal_add hr hr }
end
@[simp] lemma to_nnreal_bit1 {r : ℝ} (hr : 0 ≤ r) :
real.to_nnreal (bit1 r) = bit1 (real.to_nnreal r) :=
(real.to_nnreal_add (by simp [hr]) zero_le_one).trans (by simp [bit1])
lemma to_nnreal_pow {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : (x ^ n).to_nnreal = (x.to_nnreal) ^ n :=
by rw [← nnreal.coe_eq, nnreal.coe_pow, real.coe_to_nnreal _ (pow_nonneg hx _),
real.coe_to_nnreal x hx]
end to_nnreal
end real
open real
namespace nnreal
section mul
lemma mul_eq_mul_left {a b c : ℝ≥0} (h : a ≠ 0) : (a * b = a * c ↔ b = c) :=
by rw [mul_eq_mul_left_iff, or_iff_left h]
lemma _root_.real.to_nnreal_mul {p q : ℝ} (hp : 0 ≤ p) :
real.to_nnreal (p * q) = real.to_nnreal p * real.to_nnreal q :=
begin
cases le_total 0 q with hq hq,
{ apply nnreal.eq,
simp [real.to_nnreal, hp, hq, max_eq_left, mul_nonneg] },
{ have hpq := mul_nonpos_of_nonneg_of_nonpos hp hq,
rw [to_nnreal_eq_zero.2 hq, to_nnreal_eq_zero.2 hpq, mul_zero] }
end
end mul
section pow
lemma pow_antitone_exp {a : ℝ≥0} (m n : ℕ) (mn : m ≤ n) (a1 : a ≤ 1) :
a ^ n ≤ a ^ m :=
pow_le_pow_of_le_one (zero_le a) a1 mn
lemma exists_pow_lt_of_lt_one {a b : ℝ≥0} (ha : 0 < a) (hb : b < 1) : ∃ n : ℕ, b ^ n < a :=
by simpa only [← coe_pow, nnreal.coe_lt_coe]
using exists_pow_lt_of_lt_one (nnreal.coe_pos.2 ha) (nnreal.coe_lt_coe.2 hb)
lemma exists_mem_Ico_zpow
{x : ℝ≥0} {y : ℝ≥0} (hx : x ≠ 0) (hy : 1 < y) :
∃ n : ℤ, x ∈ set.Ico (y ^ n) (y ^ (n + 1)) :=
begin
obtain ⟨n, hn, h'n⟩ : ∃ n : ℤ, (y : ℝ) ^ n ≤ x ∧ (x : ℝ) < y ^ (n + 1) :=
exists_mem_Ico_zpow (bot_lt_iff_ne_bot.mpr hx) hy,
rw ← nnreal.coe_zpow at hn h'n,
exact ⟨n, hn, h'n⟩,
end
lemma exists_mem_Ioc_zpow
{x : ℝ≥0} {y : ℝ≥0} (hx : x ≠ 0) (hy : 1 < y) :
∃ n : ℤ, x ∈ set.Ioc (y ^ n) (y ^ (n + 1)) :=
begin
obtain ⟨n, hn, h'n⟩ : ∃ n : ℤ, (y : ℝ) ^ n < x ∧ (x : ℝ) ≤ y ^ (n + 1) :=
exists_mem_Ioc_zpow (bot_lt_iff_ne_bot.mpr hx) hy,
rw ← nnreal.coe_zpow at hn h'n,
exact ⟨n, hn, h'n⟩,
end
end pow
section sub
/-!
### Lemmas about subtraction
In this section we provide a few lemmas about subtraction that do not fit well into any other
typeclass. For lemmas about subtraction and addition see lemmas
about `has_ordered_sub` in the file `algebra.order.sub`. See also `mul_tsub` and `tsub_mul`. -/
lemma sub_def {r p : ℝ≥0} : r - p = real.to_nnreal (r - p) := rfl
lemma coe_sub_def {r p : ℝ≥0} : ↑(r - p) = max (r - p : ℝ) 0 := rfl
noncomputable example : has_ordered_sub ℝ≥0 := by apply_instance
lemma sub_div (a b c : ℝ≥0) : (a - b) / c = a / c - b / c := tsub_div _ _ _
end sub
section inv
lemma sum_div {ι} (s : finset ι) (f : ι → ℝ≥0) (b : ℝ≥0) :
(∑ i in s, f i) / b = ∑ i in s, (f i / b) :=
finset.sum_div
@[simp] lemma inv_pos {r : ℝ≥0} : 0 < r⁻¹ ↔ 0 < r := inv_pos
lemma div_pos {r p : ℝ≥0} (hr : 0 < r) (hp : 0 < p) : 0 < r / p := div_pos hr hp
lemma div_self_le (r : ℝ≥0) : r / r ≤ 1 := div_self_le_one r
@[simp] lemma inv_le {r p : ℝ≥0} (h : r ≠ 0) : r⁻¹ ≤ p ↔ 1 ≤ r * p :=
by rw [← mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h]
lemma inv_le_of_le_mul {r p : ℝ≥0} (h : 1 ≤ r * p) : r⁻¹ ≤ p :=
by by_cases r = 0; simp [*, inv_le]
@[simp] lemma le_inv_iff_mul_le {r p : ℝ≥0} (h : p ≠ 0) : (r ≤ p⁻¹ ↔ r * p ≤ 1) :=
by rw [← mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]
@[simp] lemma lt_inv_iff_mul_lt {r p : ℝ≥0} (h : p ≠ 0) : (r < p⁻¹ ↔ r * p < 1) :=
by rw [← mul_lt_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]
lemma mul_le_iff_le_inv {a b r : ℝ≥0} (hr : r ≠ 0) : r * a ≤ b ↔ a ≤ r⁻¹ * b :=
have 0 < r, from lt_of_le_of_ne (zero_le r) hr.symm,
by rw [← mul_le_mul_left (inv_pos.mpr this), ← mul_assoc, inv_mul_cancel hr, one_mul]
lemma le_div_iff_mul_le {a b r : ℝ≥0} (hr : r ≠ 0) : a ≤ b / r ↔ a * r ≤ b := le_div_iff₀ hr
lemma div_le_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a / r ≤ b ↔ a ≤ b * r := div_le_iff₀ hr
lemma div_le_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a / r ≤ b ↔ a ≤ r * b :=
@div_le_iff' ℝ _ a r b $ pos_iff_ne_zero.2 hr
lemma div_le_of_le_mul {a b c : ℝ≥0} (h : a ≤ b * c) : a / c ≤ b :=
if h0 : c = 0 then by simp [h0] else (div_le_iff h0).2 h
lemma div_le_of_le_mul' {a b c : ℝ≥0} (h : a ≤ b * c) : a / b ≤ c :=
div_le_of_le_mul $ mul_comm b c ▸ h
lemma le_div_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a ≤ b / r ↔ a * r ≤ b :=
@le_div_iff ℝ _ a b r $ pos_iff_ne_zero.2 hr
lemma le_div_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a ≤ b / r ↔ r * a ≤ b :=
@le_div_iff' ℝ _ a b r $ pos_iff_ne_zero.2 hr
lemma div_lt_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a / r < b ↔ a < b * r :=
lt_iff_lt_of_le_iff_le (le_div_iff hr)
lemma div_lt_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a / r < b ↔ a < r * b :=
lt_iff_lt_of_le_iff_le (le_div_iff' hr)
lemma lt_div_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a < b / r ↔ a * r < b :=
lt_iff_lt_of_le_iff_le (div_le_iff hr)
lemma lt_div_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a < b / r ↔ r * a < b :=
lt_iff_lt_of_le_iff_le (div_le_iff' hr)
lemma mul_lt_of_lt_div {a b r : ℝ≥0} (h : a < b / r) : a * r < b :=
begin
refine (lt_div_iff $ λ hr, false.elim _).1 h,
subst r,
simpa using h
end
lemma div_le_div_left_of_le {a b c : ℝ≥0} (b0 : 0 < b) (c0 : 0 < c) (cb : c ≤ b) :
a / b ≤ a / c :=
begin
by_cases a0 : a = 0,
{ rw [a0, zero_div, zero_div] },
{ cases a with a ha,
replace a0 : 0 < a := lt_of_le_of_ne ha (ne_of_lt (zero_lt_iff.mpr a0)),
exact (div_le_div_left a0 b0 c0).mpr cb }
end
lemma div_le_div_left {a b c : ℝ≥0} (a0 : 0 < a) (b0 : 0 < b) (c0 : 0 < c) :
a / b ≤ a / c ↔ c ≤ b :=
div_le_div_left a0 b0 c0
lemma le_of_forall_lt_one_mul_le {x y : ℝ≥0} (h : ∀a<1, a * x ≤ y) : x ≤ y :=
le_of_forall_ge_of_dense $ assume a ha,
have hx : x ≠ 0 := pos_iff_ne_zero.1 (lt_of_le_of_lt (zero_le _) ha),
have hx' : x⁻¹ ≠ 0, by rwa [(≠), inv_eq_zero],
have a * x⁻¹ < 1, by rwa [← lt_inv_iff_mul_lt hx', inv_inv],
have (a * x⁻¹) * x ≤ y, from h _ this,
by rwa [mul_assoc, inv_mul_cancel hx, mul_one] at this
lemma div_add_div_same (a b c : ℝ≥0) : a / c + b / c = (a + b) / c := div_add_div_same _ _ _
lemma half_pos {a : ℝ≥0} (h : 0 < a) : 0 < a / 2 := half_pos h
lemma add_halves (a : ℝ≥0) : a / 2 + a / 2 = a := add_halves _
lemma half_le_self (a : ℝ≥0) : a / 2 ≤ a := half_le_self bot_le
lemma half_lt_self {a : ℝ≥0} (h : a ≠ 0) : a / 2 < a := half_lt_self h.bot_lt
lemma two_inv_lt_one : (2⁻¹:ℝ≥0) < 1 := two_inv_lt_one
lemma div_lt_one_of_lt {a b : ℝ≥0} (h : a < b) : a / b < 1 :=
begin
rwa [div_lt_iff, one_mul],
exact ne_of_gt (lt_of_le_of_lt (zero_le _) h)
end
@[field_simps] lemma div_add_div (a : ℝ≥0) {b : ℝ≥0} (c : ℝ≥0) {d : ℝ≥0}
(hb : b ≠ 0) (hd : d ≠ 0) : a / b + c / d = (a * d + b * c) / (b * d) :=
div_add_div _ _ hb hd
@[field_simps] lemma add_div' (a b c : ℝ≥0) (hc : c ≠ 0) :
b + a / c = (b * c + a) / c :=
add_div' _ _ _ hc
@[field_simps] lemma div_add' (a b c : ℝ≥0) (hc : c ≠ 0) :
a / c + b = (a + b * c) / c :=
div_add' _ _ _ hc
lemma _root_.real.to_nnreal_inv {x : ℝ} :
real.to_nnreal x⁻¹ = (real.to_nnreal x)⁻¹ :=
begin
by_cases hx : 0 ≤ x,
{ nth_rewrite 0 ← real.coe_to_nnreal x hx,
rw [←nnreal.coe_inv, real.to_nnreal_coe], },
{ have hx' := le_of_not_ge hx,
rw [to_nnreal_eq_zero.mpr hx', inv_zero, to_nnreal_eq_zero.mpr (inv_nonpos.mpr hx')], },
end
lemma _root_.real.to_nnreal_div {x y : ℝ} (hx : 0 ≤ x) :
real.to_nnreal (x / y) = real.to_nnreal x / real.to_nnreal y :=
by rw [div_eq_mul_inv, div_eq_mul_inv, ← real.to_nnreal_inv, ← real.to_nnreal_mul hx]
lemma _root_.real.to_nnreal_div' {x y : ℝ} (hy : 0 ≤ y) :
real.to_nnreal (x / y) = real.to_nnreal x / real.to_nnreal y :=
by rw [div_eq_inv_mul, div_eq_inv_mul, real.to_nnreal_mul (inv_nonneg.2 hy), real.to_nnreal_inv]
lemma inv_lt_one_iff {x : ℝ≥0} (hx : x ≠ 0) : x⁻¹ < 1 ↔ 1 < x :=
by rwa [← one_div, div_lt_iff hx, one_mul]
lemma inv_lt_one {x : ℝ≥0} (hx : 1 < x) : x⁻¹ < 1 := inv_lt_one hx
lemma zpow_pos {x : ℝ≥0} (hx : x ≠ 0) (n : ℤ) : 0 < x ^ n :=
begin
cases n,
{ simp [pow_pos hx.bot_lt _] },
{ simp [pow_pos hx.bot_lt _] }
end
lemma inv_lt_inv_iff {x y : ℝ≥0} (hx : x ≠ 0) (hy : y ≠ 0) : y⁻¹ < x⁻¹ ↔ x < y := inv_lt_inv₀ hy hx
lemma inv_lt_inv {x y : ℝ≥0} (hx : x ≠ 0) (h : x < y) : y⁻¹ < x⁻¹ :=
(inv_lt_inv_iff hx ((bot_le.trans_lt h).ne')).2 h
end inv
@[simp] lemma abs_eq (x : ℝ≥0) : |(x : ℝ)| = x :=
abs_of_nonneg x.property
section csupr
open set
variables {ι : Sort*} {f : ι → ℝ≥0}
lemma le_to_nnreal_of_coe_le {x : ℝ≥0} {y : ℝ} (h : ↑x ≤ y) : x ≤ y.to_nnreal :=
(le_to_nnreal_iff_coe_le $ x.2.trans h).2 h
lemma Sup_of_not_bdd_above {s : set ℝ≥0} (hs : ¬bdd_above s) : has_Sup.Sup s = 0 :=
begin
rw [← bdd_above_coe] at hs,
rw [← nnreal.coe_eq, coe_Sup],
exact Sup_of_not_bdd_above hs,
end
lemma supr_of_not_bdd_above (hf : ¬ bdd_above (range f)) : (⨆ i, f i) = 0 :=
Sup_of_not_bdd_above hf
lemma infi_empty [is_empty ι] (f : ι → ℝ≥0) : (⨅ i, f i) = 0 :=
by { rw [← nnreal.coe_eq, coe_infi], exact real.cinfi_empty _, }
@[simp] lemma infi_const_zero {α : Sort*} : (⨅ i : α, (0 : ℝ≥0)) = 0 :=
by { rw [← nnreal.coe_eq, coe_infi], exact real.cinfi_const_zero, }
lemma infi_mul (f : ι → ℝ≥0) (a : ℝ≥0) : infi f * a = ⨅ i, f i * a :=
begin
rw [← nnreal.coe_eq, nnreal.coe_mul, coe_infi, coe_infi],
exact real.infi_mul_of_nonneg (nnreal.coe_nonneg _) _,
end
lemma mul_infi (f : ι → ℝ≥0) (a : ℝ≥0) : a * infi f = ⨅ i, a * f i :=
by simpa only [mul_comm] using infi_mul f a
lemma mul_supr (f : ι → ℝ≥0) (a : ℝ≥0) : a * (⨆ i, f i) = ⨆ i, a * f i :=
begin
rw [← nnreal.coe_eq, nnreal.coe_mul, nnreal.coe_supr, nnreal.coe_supr],
exact real.mul_supr_of_nonneg (nnreal.coe_nonneg _) _,
end
lemma supr_mul (f : ι → ℝ≥0) (a : ℝ≥0) : (⨆ i, f i) * a = ⨆ i, f i * a :=
by { rw [mul_comm, mul_supr], simp_rw [mul_comm] }
lemma supr_div (f : ι → ℝ≥0) (a : ℝ≥0) : (⨆ i, f i) / a = ⨆ i, f i / a :=
by simp only [div_eq_mul_inv, supr_mul]
variable [nonempty ι]
lemma le_mul_infi {a : ℝ≥0} {g : ℝ≥0} {h : ι → ℝ≥0} (H : ∀ j, a ≤ g * h j) : a ≤ g * infi h :=
by { rw [mul_infi], exact le_cinfi H }
lemma mul_supr_le {a : ℝ≥0} {g : ℝ≥0} {h : ι → ℝ≥0} (H : ∀ j, g * h j ≤ a) : g * supr h ≤ a :=
by { rw [mul_supr], exact csupr_le H }
lemma le_infi_mul {a : ℝ≥0} {g : ι → ℝ≥0} {h : ℝ≥0} (H : ∀ i, a ≤ g i * h) : a ≤ infi g * h :=
by { rw infi_mul, exact le_cinfi H }
lemma supr_mul_le {a : ℝ≥0} {g : ι → ℝ≥0} {h : ℝ≥0} (H : ∀ i, g i * h ≤ a) : supr g * h ≤ a :=
by { rw supr_mul, exact csupr_le H }
lemma le_infi_mul_infi {a : ℝ≥0} {g h : ι → ℝ≥0} (H : ∀ i j, a ≤ g i * h j) :
a ≤ infi g * infi h :=
le_infi_mul $ λ i, le_mul_infi $ H i
lemma supr_mul_supr_le {a : ℝ≥0} {g h : ι → ℝ≥0} (H : ∀ i j, g i * h j ≤ a) :
supr g * supr h ≤ a :=
supr_mul_le $ λ i, mul_supr_le $ H _
end csupr
end nnreal
namespace set
namespace ord_connected
variables {s : set ℝ} {t : set ℝ≥0}
lemma preimage_coe_nnreal_real (h : s.ord_connected) : (coe ⁻¹' s : set ℝ≥0).ord_connected :=
h.preimage_mono nnreal.coe_mono
lemma image_coe_nnreal_real (h : t.ord_connected) : (coe '' t : set ℝ).ord_connected :=
⟨ball_image_iff.2 $ λ x hx, ball_image_iff.2 $ λ y hy z hz,
⟨⟨z, x.2.trans hz.1⟩, h.out hx hy hz, rfl⟩⟩
lemma image_real_to_nnreal (h : s.ord_connected) : (real.to_nnreal '' s).ord_connected :=
begin
refine ⟨ball_image_iff.2 $ λ x hx, ball_image_iff.2 $ λ y hy z hz, _⟩,
cases le_total y 0 with hy₀ hy₀,
{ rw [mem_Icc, real.to_nnreal_of_nonpos hy₀, nonpos_iff_eq_zero] at hz,
exact ⟨y, hy, (to_nnreal_of_nonpos hy₀).trans hz.2.symm⟩ },
{ lift y to ℝ≥0 using hy₀,
rw [to_nnreal_coe] at hz,
exact ⟨z, h.out hx hy ⟨to_nnreal_le_iff_le_coe.1 hz.1, hz.2⟩, to_nnreal_coe⟩ }
end
lemma preimage_real_to_nnreal (h : t.ord_connected) : (real.to_nnreal ⁻¹' t).ord_connected :=
h.preimage_mono real.to_nnreal_mono
end ord_connected
end set
namespace real
/-- The absolute value on `ℝ` as a map to `ℝ≥0`. -/
@[pp_nodot] def nnabs : ℝ →*₀ ℝ≥0 :=
{ to_fun := λ x, ⟨|x|, abs_nonneg x⟩,
map_zero' := by { ext, simp },
map_one' := by { ext, simp },
map_mul' := λ x y, by { ext, simp [abs_mul] } }
@[norm_cast, simp] lemma coe_nnabs (x : ℝ) : (nnabs x : ℝ) = |x| :=
rfl
@[simp] lemma nnabs_of_nonneg {x : ℝ} (h : 0 ≤ x) : nnabs x = to_nnreal x :=
by { ext, simp [coe_to_nnreal x h, abs_of_nonneg h] }
lemma nnabs_coe (x : ℝ≥0) : nnabs x = x := by simp
lemma coe_to_nnreal_le (x : ℝ) : (to_nnreal x : ℝ) ≤ |x| :=
max_le (le_abs_self _) (abs_nonneg _)
lemma cast_nat_abs_eq_nnabs_cast (n : ℤ) :
(n.nat_abs : ℝ≥0) = nnabs n :=
by { ext, rw [nnreal.coe_nat_cast, int.cast_nat_abs, real.coe_nnabs] }
end real
namespace tactic
open positivity
private lemma nnreal_coe_pos {r : ℝ≥0} : 0 < r → 0 < (r : ℝ) := nnreal.coe_pos.2
/-- Extension for the `positivity` tactic: cast from `ℝ≥0` to `ℝ`. -/
@[positivity]
meta def positivity_coe_nnreal_real : expr → tactic strictness
| `(@coe _ _ %%inst %%a) := do
unify inst `(@coe_to_lift _ _ $ @coe_base _ _ nnreal.real.has_coe),
strictness_a ← core a,
match strictness_a with
| positive p := positive <$> mk_app ``nnreal_coe_pos [p]
| _ := nonnegative <$> mk_app ``nnreal.coe_nonneg [a]
end
| e := pp e >>= fail ∘ format.bracket "The expression "
" is not of the form `(r : ℝ)` for `r : ℝ≥0`"
end tactic
|
11b7b6d231faea22beaa0f1070a70c64b7c8d1ed | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/category_theory/limits/functor_category.lean | 3f9a96a0f049be2d2e43a436c0426d5aad137609 | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 4,791 | lean | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import category_theory.limits.preserves
open category_theory category_theory.category
namespace category_theory.limits
universes v u -- declare the `v`'s first; see `category_theory.category` for an explanation
variables {C : Type u} [category.{v} C]
variables {J K : Type v} [small_category J] [small_category K]
@[simp] lemma cone.functor_w {F : J ⥤ (K ⥤ C)} (c : cone F) {j j' : J} (f : j ⟶ j') (k : K) :
(c.π.app j).app k ≫ (F.map f).app k = (c.π.app j').app k :=
by convert ←nat_trans.congr_app (c.π.naturality f).symm k; apply id_comp
@[simp] lemma cocone.functor_w {F : J ⥤ (K ⥤ C)} (c : cocone F) {j j' : J} (f : j ⟶ j') (k : K) :
(F.map f).app k ≫ (c.ι.app j').app k = (c.ι.app j).app k :=
by convert ←nat_trans.congr_app (c.ι.naturality f) k; apply comp_id
@[simps] def functor_category_limit_cone [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
cone F :=
{ X := F.flip ⋙ lim,
π :=
{ app := λ j,
{ app := λ k, limit.π (F.flip.obj k) j },
naturality' := λ j j' f,
by ext k; convert (limit.w (F.flip.obj k) _).symm using 1; apply id_comp } }
@[simps] def functor_category_colimit_cocone [has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) :
cocone F :=
{ X := F.flip ⋙ colim,
ι :=
{ app := λ j,
{ app := λ k, colimit.ι (F.flip.obj k) j },
naturality' := λ j j' f,
by ext k; convert (colimit.w (F.flip.obj k) _) using 1; apply comp_id } }
@[simp] def evaluate_functor_category_limit_cone
[has_limits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
((evaluation K C).obj k).map_cone (functor_category_limit_cone F) ≅
limit.cone (F.flip.obj k) :=
cones.ext (iso.refl _) (by tidy)
@[simp] def evaluate_functor_category_colimit_cocone
[has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
((evaluation K C).obj k).map_cocone (functor_category_colimit_cocone F) ≅
colimit.cocone (F.flip.obj k) :=
cocones.ext (iso.refl _) (by tidy)
def functor_category_is_limit_cone [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
is_limit (functor_category_limit_cone F) :=
{ lift := λ s,
{ app := λ k, limit.lift (F.flip.obj k) (((evaluation K C).obj k).map_cone s) },
uniq' := λ s m w,
begin
ext1, ext1 k,
exact is_limit.uniq _
(((evaluation K C).obj k).map_cone s) (m.app k) (λ j, nat_trans.congr_app (w j) k)
end }
def functor_category_is_colimit_cocone [has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) :
is_colimit (functor_category_colimit_cocone F) :=
{ desc := λ s,
{ app := λ k, colimit.desc (F.flip.obj k) (((evaluation K C).obj k).map_cocone s) },
uniq' := λ s m w,
begin
ext1, ext1 k,
exact is_colimit.uniq _
(((evaluation K C).obj k).map_cocone s) (m.app k) (λ j, nat_trans.congr_app (w j) k)
end }
instance functor_category_has_limits_of_shape
[has_limits_of_shape J C] : has_limits_of_shape J (K ⥤ C) :=
{ has_limit := λ F,
{ cone := functor_category_limit_cone F,
is_limit := functor_category_is_limit_cone F } }
instance functor_category_has_colimits_of_shape
[has_colimits_of_shape J C] : has_colimits_of_shape J (K ⥤ C) :=
{ has_colimit := λ F,
{ cocone := functor_category_colimit_cocone F,
is_colimit := functor_category_is_colimit_cocone F } }
instance functor_category_has_limits [has_limits C] : has_limits (K ⥤ C) :=
{ has_limits_of_shape := λ J 𝒥, by resetI; apply_instance }
instance functor_category_has_colimits [has_colimits C] : has_colimits (K ⥤ C) :=
{ has_colimits_of_shape := λ J 𝒥, by resetI; apply_instance }
instance evaluation_preserves_limits_of_shape [has_limits_of_shape J C] (k : K) :
preserves_limits_of_shape J ((evaluation K C).obj k) :=
{ preserves_limit :=
λ F, preserves_limit_of_preserves_limit_cone (limit.is_limit _) $
is_limit.of_iso_limit (limit.is_limit _)
(evaluate_functor_category_limit_cone F k).symm }
instance evaluation_preserves_colimits_of_shape [has_colimits_of_shape J C] (k : K) :
preserves_colimits_of_shape J ((evaluation K C).obj k) :=
{ preserves_colimit :=
λ F, preserves_colimit_of_preserves_colimit_cocone (colimit.is_colimit _) $
is_colimit.of_iso_colimit (colimit.is_colimit _)
(evaluate_functor_category_colimit_cocone F k).symm }
instance evaluation_preserves_limits [has_limits C] (k : K) :
preserves_limits ((evaluation K C).obj k) :=
{ preserves_limits_of_shape := λ J 𝒥, by resetI; apply_instance }
instance evaluation_preserves_colimits [has_colimits C] (k : K) :
preserves_colimits ((evaluation K C).obj k) :=
{ preserves_colimits_of_shape := λ J 𝒥, by resetI; apply_instance }
end category_theory.limits
|
ec197c9cf037fa3721be70bb7ccbf047ea658d55 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/data/set/prod.lean | 4084dc924d3001e382261030ca631db596fcf3cb | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 18,631 | lean | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import data.set.basic
/-!
# Sets in product and pi types
This file defines the product of sets in `α × β` and in `Π i, α i` along with the diagonal of a
type.
## Main declarations
* `set.prod`: Binary product of sets. For `s : set α`, `t : set β`, we have
`s.prod t : set (α × β)`.
* `set.diagonal`: Diagonal of a type. `set.diagonal α = {(x, x) | x : α}`.
* `set.pi`: Arbitrary product of sets.
-/
open function
/-- Notation class for product of subobjects (sets, submonoids, subgroups, etc). -/
class has_set_prod (α β : Type*) (γ : out_param Type*) :=
(prod : α → β → γ)
/- This notation binds more strongly than (pre)images, unions and intersections. -/
infixr ` ×ˢ `:82 := has_set_prod.prod
namespace set
/-! ### Cartesian binary product of sets -/
section prod
variables {α β γ δ : Type*} {s s₁ s₂ : set α} {t t₁ t₂ : set β} {a : α} {b : β}
/-- The cartesian product `prod s t` is the set of `(a, b)`
such that `a ∈ s` and `b ∈ t`. -/
instance : has_set_prod (set α) (set β) (set (α × β)) :=
⟨λ s t, {p | p.1 ∈ s ∧ p.2 ∈ t}⟩
lemma prod_eq (s : set α) (t : set β) : s ×ˢ t = prod.fst ⁻¹' s ∩ prod.snd ⁻¹' t := rfl
lemma mem_prod_eq {p : α × β} : p ∈ s ×ˢ t = (p.1 ∈ s ∧ p.2 ∈ t) := rfl
@[simp] lemma mem_prod {p : α × β} : p ∈ s ×ˢ t ↔ p.1 ∈ s ∧ p.2 ∈ t := iff.rfl
@[simp] lemma prod_mk_mem_set_prod_eq : (a, b) ∈ s ×ˢ t = (a ∈ s ∧ b ∈ t) := rfl
lemma mk_mem_prod (ha : a ∈ s) (hb : b ∈ t) : (a, b) ∈ s ×ˢ t := ⟨ha, hb⟩
lemma prod_mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : s₁ ×ˢ t₁ ⊆ s₂ ×ˢ t₂ :=
λ x ⟨h₁, h₂⟩, ⟨hs h₁, ht h₂⟩
lemma prod_subset_iff {P : set (α × β)} : s ×ˢ t ⊆ P ↔ ∀ (x ∈ s) (y ∈ t), (x, y) ∈ P :=
⟨λ h _ hx _ hy, h (mk_mem_prod hx hy), λ h ⟨_, _⟩ hp, h _ hp.1 _ hp.2⟩
lemma forall_prod_set {p : α × β → Prop} : (∀ x ∈ s ×ˢ t, p x) ↔ ∀ (x ∈ s) (y ∈ t), p (x, y) :=
prod_subset_iff
lemma exists_prod_set {p : α × β → Prop} : (∃ x ∈ s ×ˢ t, p x) ↔ ∃ (x ∈ s) (y ∈ t), p (x, y) :=
by simp [and_assoc]
@[simp] lemma prod_empty : s ×ˢ (∅ : set β) = ∅ := by { ext, exact and_false _ }
@[simp] lemma empty_prod : (∅ : set α) ×ˢ t = ∅ := by { ext, exact false_and _ }
@[simp] lemma univ_prod_univ : @univ α ×ˢ @univ β = univ := by { ext, exact true_and _ }
lemma univ_prod {t : set β} : (univ : set α) ×ˢ t = prod.snd ⁻¹' t := by simp [prod_eq]
lemma prod_univ {s : set α} : s ×ˢ (univ : set β) = prod.fst ⁻¹' s := by simp [prod_eq]
@[simp] lemma singleton_prod : ({a} : set α) ×ˢ t = prod.mk a '' t :=
by { ext ⟨x, y⟩, simp [and.left_comm, eq_comm] }
@[simp] lemma prod_singleton : s ×ˢ ({b} : set β) = (λ a, (a, b)) '' s :=
by { ext ⟨x, y⟩, simp [and.left_comm, eq_comm] }
lemma singleton_prod_singleton : ({a} : set α) ×ˢ ({b} : set β) = {(a, b)} :=by simp
@[simp] lemma union_prod : (s₁ ∪ s₂) ×ˢ t = s₁ ×ˢ t ∪ s₂ ×ˢ t :=
by { ext ⟨x, y⟩, simp [or_and_distrib_right] }
@[simp] lemma prod_union : s ×ˢ (t₁ ∪ t₂) = s ×ˢ t₁ ∪ s ×ˢ t₂ :=
by { ext ⟨x, y⟩, simp [and_or_distrib_left] }
lemma prod_inter_prod : s₁ ×ˢ t₁ ∩ s₂ ×ˢ t₂ = (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂) :=
by { ext ⟨x, y⟩, simp [and_assoc, and.left_comm] }
lemma insert_prod : insert a s ×ˢ t = (prod.mk a '' t) ∪ s ×ˢ t :=
by { ext ⟨x, y⟩, simp [image, iff_def, or_imp_distrib, imp.swap] {contextual := tt} }
lemma prod_insert : s ×ˢ (insert b t) = ((λa, (a, b)) '' s) ∪ s ×ˢ t :=
by { ext ⟨x, y⟩, simp [image, iff_def, or_imp_distrib, imp.swap] {contextual := tt} }
lemma prod_preimage_eq {f : γ → α} {g : δ → β} :
(f ⁻¹' s) ×ˢ (g ⁻¹' t) = (λ p : γ × δ, (f p.1, g p.2)) ⁻¹' s ×ˢ t := rfl
lemma prod_preimage_left {f : γ → α} :
(f ⁻¹' s) ×ˢ t = (λ p : γ × β, (f p.1, p.2)) ⁻¹' s ×ˢ t := rfl
lemma prod_preimage_right {g : δ → β} :
s ×ˢ (g ⁻¹' t) = (λ p : α × δ, (p.1, g p.2)) ⁻¹' s ×ˢ t := rfl
lemma preimage_prod_map_prod (f : α → β) (g : γ → δ) (s : set β) (t : set δ) :
prod.map f g ⁻¹' s ×ˢ t = (f ⁻¹' s) ×ˢ (g ⁻¹' t) :=
rfl
lemma mk_preimage_prod (f : γ → α) (g : γ → β) :
(λ x, (f x, g x)) ⁻¹' s ×ˢ t = f ⁻¹' s ∩ g ⁻¹' t := rfl
@[simp] lemma mk_preimage_prod_left (hb : b ∈ t) : (λ a, (a, b)) ⁻¹' s ×ˢ t = s :=
by { ext a, simp [hb] }
@[simp] lemma mk_preimage_prod_right (ha : a ∈ s) : prod.mk a ⁻¹' s ×ˢ t = t :=
by { ext b, simp [ha] }
@[simp] lemma mk_preimage_prod_left_eq_empty (hb : b ∉ t) : (λ a, (a, b)) ⁻¹' s ×ˢ t = ∅ :=
by { ext a, simp [hb] }
@[simp] lemma mk_preimage_prod_right_eq_empty (ha : a ∉ s) : prod.mk a ⁻¹' s ×ˢ t = ∅ :=
by { ext b, simp [ha] }
lemma mk_preimage_prod_left_eq_if [decidable_pred (∈ t)] :
(λ a, (a, b)) ⁻¹' s ×ˢ t = if b ∈ t then s else ∅ :=
by split_ifs; simp [h]
lemma mk_preimage_prod_right_eq_if [decidable_pred (∈ s)] :
prod.mk a ⁻¹' s ×ˢ t = if a ∈ s then t else ∅ :=
by split_ifs; simp [h]
lemma mk_preimage_prod_left_fn_eq_if [decidable_pred (∈ t)] (f : γ → α) :
(λ a, (f a, b)) ⁻¹' s ×ˢ t = if b ∈ t then f ⁻¹' s else ∅ :=
by rw [← mk_preimage_prod_left_eq_if, prod_preimage_left, preimage_preimage]
lemma mk_preimage_prod_right_fn_eq_if [decidable_pred (∈ s)] (g : δ → β) :
(λ b, (a, g b)) ⁻¹' s ×ˢ t = if a ∈ s then g ⁻¹' t else ∅ :=
by rw [← mk_preimage_prod_right_eq_if, prod_preimage_right, preimage_preimage]
lemma preimage_swap_prod {s : set α} {t : set β} : prod.swap ⁻¹' t ×ˢ s = s ×ˢ t :=
by { ext ⟨x, y⟩, simp [and_comm] }
lemma image_swap_prod : prod.swap '' t ×ˢ s = s ×ˢ t :=
by rw [image_swap_eq_preimage_swap, preimage_swap_prod]
lemma prod_image_image_eq {m₁ : α → γ} {m₂ : β → δ} :
(m₁ '' s) ×ˢ (m₂ '' t) = (λ p : α × β, (m₁ p.1, m₂ p.2)) '' s ×ˢ t :=
ext $ by simp [-exists_and_distrib_right, exists_and_distrib_right.symm, and.left_comm,
and.assoc, and.comm]
lemma prod_range_range_eq {m₁ : α → γ} {m₂ : β → δ} :
(range m₁) ×ˢ (range m₂) = range (λ p : α × β, (m₁ p.1, m₂ p.2)) :=
ext $ by simp [range]
@[simp] lemma range_prod_map {m₁ : α → γ} {m₂ : β → δ} :
range (prod.map m₁ m₂) = (range m₁) ×ˢ (range m₂) :=
prod_range_range_eq.symm
lemma prod_range_univ_eq {m₁ : α → γ} :
(range m₁) ×ˢ (univ : set β) = range (λ p : α × β, (m₁ p.1, p.2)) :=
ext $ by simp [range]
lemma prod_univ_range_eq {m₂ : β → δ} :
(univ : set α) ×ˢ (range m₂) = range (λ p : α × β, (p.1, m₂ p.2)) :=
ext $ by simp [range]
lemma range_pair_subset (f : α → β) (g : α → γ) :
range (λ x, (f x, g x)) ⊆ (range f) ×ˢ (range g) :=
have (λ x, (f x, g x)) = prod.map f g ∘ (λ x, (x, x)), from funext (λ x, rfl),
by { rw [this, ← range_prod_map], apply range_comp_subset_range }
lemma nonempty.prod : s.nonempty → t.nonempty → (s ×ˢ t : set _).nonempty :=
λ ⟨x, hx⟩ ⟨y, hy⟩, ⟨(x, y), ⟨hx, hy⟩⟩
lemma nonempty.fst : (s ×ˢ t : set _).nonempty → s.nonempty := λ ⟨x, hx⟩, ⟨x.1, hx.1⟩
lemma nonempty.snd : (s ×ˢ t : set _).nonempty → t.nonempty := λ ⟨x, hx⟩, ⟨x.2, hx.2⟩
lemma prod_nonempty_iff : (s ×ˢ t : set _).nonempty ↔ s.nonempty ∧ t.nonempty :=
⟨λ h, ⟨h.fst, h.snd⟩, λ h, h.1.prod h.2⟩
lemma prod_eq_empty_iff : s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅ :=
by simp only [not_nonempty_iff_eq_empty.symm, prod_nonempty_iff, not_and_distrib]
lemma prod_sub_preimage_iff {W : set γ} {f : α × β → γ} :
s ×ˢ t ⊆ f ⁻¹' W ↔ ∀ a b, a ∈ s → b ∈ t → f (a, b) ∈ W :=
by simp [subset_def]
lemma image_prod_mk_subset_prod_left (hb : b ∈ t) : (λ a, (a, b)) '' s ⊆ s ×ˢ t :=
by { rintro _ ⟨a, ha, rfl⟩, exact ⟨ha, hb⟩ }
lemma image_prod_mk_subset_prod_right (ha : a ∈ s) : prod.mk a '' t ⊆ s ×ˢ t :=
by { rintro _ ⟨b, hb, rfl⟩, exact ⟨ha, hb⟩ }
lemma prod_subset_preimage_fst (s : set α) (t : set β) : s ×ˢ t ⊆ prod.fst ⁻¹' s :=
inter_subset_left _ _
lemma fst_image_prod_subset (s : set α) (t : set β) : prod.fst '' s ×ˢ t ⊆ s :=
image_subset_iff.2 $ prod_subset_preimage_fst s t
lemma fst_image_prod (s : set β) {t : set α} (ht : t.nonempty) : prod.fst '' s ×ˢ t = s :=
(fst_image_prod_subset _ _).antisymm $ λ y hy, let ⟨x, hx⟩ := ht in ⟨(y, x), ⟨hy, hx⟩, rfl⟩
lemma prod_subset_preimage_snd (s : set α) (t : set β) : s ×ˢ t ⊆ prod.snd ⁻¹' t :=
inter_subset_right _ _
lemma snd_image_prod_subset (s : set α) (t : set β) : prod.snd '' s ×ˢ t ⊆ t :=
image_subset_iff.2 $ prod_subset_preimage_snd s t
lemma snd_image_prod {s : set α} (hs : s.nonempty) (t : set β) : prod.snd '' s ×ˢ t = t :=
(snd_image_prod_subset _ _).antisymm $ λ y y_in, let ⟨x, x_in⟩ := hs in ⟨(x, y), ⟨x_in, y_in⟩, rfl⟩
lemma prod_diff_prod : s ×ˢ t \ s₁ ×ˢ t₁ = s ×ˢ (t \ t₁) ∪ (s \ s₁) ×ˢ t :=
by { ext x, by_cases h₁ : x.1 ∈ s₁; by_cases h₂ : x.2 ∈ t₁; simp * }
/-- A product set is included in a product set if and only factors are included, or a factor of the
first set is empty. -/
lemma prod_subset_prod_iff : s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ :=
begin
cases (s ×ˢ t : set _).eq_empty_or_nonempty with h h,
{ simp [h, prod_eq_empty_iff.1 h] },
have st : s.nonempty ∧ t.nonempty, by rwa [prod_nonempty_iff] at h,
refine ⟨λ H, or.inl ⟨_, _⟩, _⟩,
{ have := image_subset (prod.fst : α × β → α) H,
rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this },
{ have := image_subset (prod.snd : α × β → β) H,
rwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this },
{ intro H,
simp only [st.1.ne_empty, st.2.ne_empty, or_false] at H,
exact prod_mono H.1 H.2 }
end
@[simp] lemma prod_eq_iff_eq (ht : t.nonempty) : s ×ˢ t = s₁ ×ˢ t ↔ s = s₁ :=
begin
obtain ⟨b, hb⟩ := ht,
split,
{ simp only [set.ext_iff],
intros h a,
simpa [hb, set.mem_prod] using h (a, b) },
{ rintros rfl,
refl },
end
@[simp] lemma image_prod (f : α → β → γ) : (λ x : α × β, f x.1 x.2) '' s ×ˢ t = image2 f s t :=
set.ext $ λ a,
⟨ by { rintro ⟨_, _, rfl⟩, exact ⟨_, _, (mem_prod.mp ‹_›).1, (mem_prod.mp ‹_›).2, rfl⟩ },
by { rintro ⟨_, _, _, _, rfl⟩, exact ⟨(_, _), mem_prod.mpr ⟨‹_›, ‹_›⟩, rfl⟩ }⟩
end prod
/-! ### Diagonal -/
section diagonal
variables {α : Type*}
/-- `diagonal α` is the set of `α × α` consisting of all pairs of the form `(a, a)`. -/
def diagonal (α : Type*) : set (α × α) := {p | p.1 = p.2}
@[simp] lemma mem_diagonal (x : α) : (x, x) ∈ diagonal α := by simp [diagonal]
lemma preimage_coe_coe_diagonal (s : set α) : (prod.map coe coe) ⁻¹' (diagonal α) = diagonal s :=
by { ext ⟨⟨x, hx⟩, ⟨y, hy⟩⟩, simp [set.diagonal] }
lemma diagonal_eq_range : diagonal α = range (λ x, (x, x)) :=
by { ext ⟨x, y⟩, simp [diagonal, eq_comm] }
end diagonal
/-! ### Cartesian set-indexed product of sets -/
section pi
variables {ι : Type*} {α β : ι → Type*} {s s₁ s₂ : set ι} {t t₁ t₂ : Π i, set (α i)} {i : ι}
/-- Given an index set `ι` and a family of sets `t : Π i, set (α i)`, `pi s t`
is the set of dependent functions `f : Πa, π a` such that `f a` belongs to `t a`
whenever `a ∈ s`. -/
def pi (s : set ι) (t : Π i, set (α i)) : set (Π i, α i) := {f | ∀ i ∈ s, f i ∈ t i}
@[simp] lemma mem_pi {f : Π i, α i} : f ∈ s.pi t ↔ ∀ i ∈ s, f i ∈ t i := iff.rfl
@[simp] lemma mem_univ_pi {f : Π i, α i} : f ∈ pi univ t ↔ ∀ i, f i ∈ t i := by simp
@[simp] lemma empty_pi (s : Π i, set (α i)) : pi ∅ s = univ := by { ext, simp [pi] }
@[simp] lemma pi_univ (s : set ι) : pi s (λ i, (univ : set (α i))) = univ :=
eq_univ_of_forall $ λ f i hi, mem_univ _
lemma pi_mono (h : ∀ i ∈ s, t₁ i ⊆ t₂ i) : pi s t₁ ⊆ pi s t₂ :=
λ x hx i hi, (h i hi $ hx i hi)
lemma pi_inter_distrib : s.pi (λ i, t i ∩ t₁ i) = s.pi t ∩ s.pi t₁ :=
ext $ λ x, by simp only [forall_and_distrib, mem_pi, mem_inter_eq]
lemma pi_congr (h : s₁ = s₂) (h' : ∀ i ∈ s₁, t₁ i = t₂ i) : s₁.pi t₁ = s₂.pi t₂ :=
h ▸ (ext $ λ x, forall₂_congr $ λ i hi, h' i hi ▸ iff.rfl)
lemma pi_eq_empty (hs : i ∈ s) (ht : t i = ∅) : s.pi t = ∅ :=
by { ext f, simp only [mem_empty_eq, not_forall, iff_false, mem_pi, not_imp],
exact ⟨i, hs, by simp [ht]⟩ }
lemma univ_pi_eq_empty (ht : t i = ∅) : pi univ t = ∅ := pi_eq_empty (mem_univ i) ht
lemma pi_nonempty_iff : (s.pi t).nonempty ↔ ∀ i, ∃ x, i ∈ s → x ∈ t i :=
by simp [classical.skolem, set.nonempty]
lemma univ_pi_nonempty_iff : (pi univ t).nonempty ↔ ∀ i, (t i).nonempty :=
by simp [classical.skolem, set.nonempty]
lemma pi_eq_empty_iff : s.pi t = ∅ ↔ ∃ i, is_empty (α i) ∨ i ∈ s ∧ t i = ∅ :=
begin
rw [← not_nonempty_iff_eq_empty, pi_nonempty_iff],
push_neg,
refine exists_congr (λ i, ⟨λ h, (is_empty_or_nonempty (α i)).imp_right _, _⟩),
{ rintro ⟨x⟩,
exact ⟨(h x).1, by simp [eq_empty_iff_forall_not_mem, h]⟩ },
{ rintro (h | h) x,
{ exact h.elim' x },
{ simp [h] } }
end
lemma univ_pi_eq_empty_iff : pi univ t = ∅ ↔ ∃ i, t i = ∅ :=
by simp [← not_nonempty_iff_eq_empty, univ_pi_nonempty_iff]
@[simp] lemma univ_pi_empty [h : nonempty ι] : pi univ (λ i, ∅ : Π i, set (α i)) = ∅ :=
univ_pi_eq_empty_iff.2 $ h.elim $ λ x, ⟨x, rfl⟩
@[simp] lemma range_dcomp (f : Π i, α i → β i) :
range (λ (g : Π i, α i), (λ i, f i (g i))) = pi univ (λ i, range (f i)) :=
begin
apply subset.antisymm _ (λ x hx, _),
{ rintro _ ⟨x, rfl⟩ i -,
exact ⟨x i, rfl⟩ },
{ choose y hy using hx,
exact ⟨λ i, y i trivial, funext $ λ i, hy i trivial⟩ }
end
@[simp] lemma insert_pi (i : ι) (s : set ι) (t : Π i, set (α i)) :
pi (insert i s) t = (eval i ⁻¹' t i) ∩ pi s t :=
by { ext, simp [pi, or_imp_distrib, forall_and_distrib] }
@[simp] lemma singleton_pi (i : ι) (t : Π i, set (α i)) : pi {i} t = (eval i ⁻¹' t i) :=
by { ext, simp [pi] }
lemma singleton_pi' (i : ι) (t : Π i, set (α i)) : pi {i} t = {x | x i ∈ t i} := singleton_pi i t
lemma pi_if {p : ι → Prop} [h : decidable_pred p] (s : set ι) (t₁ t₂ : Π i, set (α i)) :
pi s (λ i, if p i then t₁ i else t₂ i) = pi {i ∈ s | p i} t₁ ∩ pi {i ∈ s | ¬ p i} t₂ :=
begin
ext f,
refine ⟨λ h, _, _⟩,
{ split; { rintro i ⟨his, hpi⟩, simpa [*] using h i } },
{ rintro ⟨ht₁, ht₂⟩ i his,
by_cases p i; simp * at * }
end
lemma union_pi : (s₁ ∪ s₂).pi t = s₁.pi t ∩ s₂.pi t :=
by simp [pi, or_imp_distrib, forall_and_distrib, set_of_and]
@[simp] lemma pi_inter_compl (s : set ι) : pi s t ∩ pi sᶜ t = pi univ t :=
by rw [← union_pi, union_compl_self]
lemma pi_update_of_not_mem [decidable_eq ι] (hi : i ∉ s) (f : Π j, α j) (a : α i)
(t : Π j, α j → set (β j)) :
s.pi (λ j, t j (update f i a j)) = s.pi (λ j, t j (f j)) :=
pi_congr rfl $ λ j hj, by { rw update_noteq, exact λ h, hi (h ▸ hj) }
lemma pi_update_of_mem [decidable_eq ι] (hi : i ∈ s) (f : Π j, α j) (a : α i)
(t : Π j, α j → set (β j)) :
s.pi (λ j, t j (update f i a j)) = {x | x i ∈ t i a} ∩ (s \ {i}).pi (λ j, t j (f j)) :=
calc s.pi (λ j, t j (update f i a j)) = ({i} ∪ s \ {i}).pi (λ j, t j (update f i a j)) :
by rw [union_diff_self, union_eq_self_of_subset_left (singleton_subset_iff.2 hi)]
... = {x | x i ∈ t i a} ∩ (s \ {i}).pi (λ j, t j (f j)) :
by { rw [union_pi, singleton_pi', update_same, pi_update_of_not_mem], simp }
lemma univ_pi_update [decidable_eq ι] {β : Π i, Type*} (i : ι) (f : Π j, α j) (a : α i)
(t : Π j, α j → set (β j)) :
pi univ (λ j, t j (update f i a j)) = {x | x i ∈ t i a} ∩ pi {i}ᶜ (λ j, t j (f j)) :=
by rw [compl_eq_univ_diff, ← pi_update_of_mem (mem_univ _)]
lemma univ_pi_update_univ [decidable_eq ι] (i : ι) (s : set (α i)) :
pi univ (update (λ j : ι, (univ : set (α j))) i s) = eval i ⁻¹' s :=
by rw [univ_pi_update i (λ j, (univ : set (α j))) s (λ j t, t), pi_univ, inter_univ, preimage]
lemma eval_image_pi_subset (hs : i ∈ s) : eval i '' s.pi t ⊆ t i :=
image_subset_iff.2 $ λ f hf, hf i hs
lemma eval_image_univ_pi_subset : eval i '' pi univ t ⊆ t i :=
eval_image_pi_subset (mem_univ i)
lemma eval_image_pi (hs : i ∈ s) (ht : (s.pi t).nonempty) : eval i '' s.pi t = t i :=
begin
refine (eval_image_pi_subset hs).antisymm _,
classical,
obtain ⟨f, hf⟩ := ht,
refine λ y hy, ⟨update f i y, λ j hj, _, update_same _ _ _⟩,
obtain rfl | hji := eq_or_ne j i; simp [*, hf _ hj]
end
@[simp] lemma eval_image_univ_pi (ht : (pi univ t).nonempty) :
(λ f : Π i, α i, f i) '' pi univ t = t i :=
eval_image_pi (mem_univ i) ht
lemma eval_preimage [decidable_eq ι] {s : set (α i)} :
eval i ⁻¹' s = pi univ (update (λ i, univ) i s) :=
by { ext x, simp [@forall_update_iff _ (λ i, set (α i)) _ _ _ _ (λ i' y, x i' ∈ y)] }
lemma eval_preimage' [decidable_eq ι] {s : set (α i)} :
eval i ⁻¹' s = pi {i} (update (λ i, univ) i s) :=
by { ext, simp }
lemma update_preimage_pi [decidable_eq ι] {f : Π i, α i} (hi : i ∈ s)
(hf : ∀ j ∈ s, j ≠ i → f j ∈ t j) :
(update f i) ⁻¹' s.pi t = t i :=
begin
ext x,
refine ⟨λ h, _, λ hx j hj, _⟩,
{ convert h i hi,
simp },
{ obtain rfl | h := eq_or_ne j i,
{ simpa },
{ rw update_noteq h,
exact hf j hj h } }
end
lemma update_preimage_univ_pi [decidable_eq ι] {f : Π i, α i} (hf : ∀ j ≠ i, f j ∈ t j) :
(update f i) ⁻¹' pi univ t = t i :=
update_preimage_pi (mem_univ i) (λ j _, hf j)
lemma subset_pi_eval_image (s : set ι) (u : set (Π i, α i)) : u ⊆ pi s (λ i, eval i '' u) :=
λ f hf i hi, ⟨f, hf, rfl⟩
lemma univ_pi_ite (s : set ι) [decidable_pred (∈ s)] (t : Π i, set (α i)) :
pi univ (λ i, if i ∈ s then t i else univ) = s.pi t :=
by { ext, simp_rw [mem_univ_pi], refine forall_congr (λ i, _), split_ifs; simp [h] }
end pi
end set
|
896ce9a59fbe8e6eb732d8b11a011fb1b19f3f52 | 94e33a31faa76775069b071adea97e86e218a8ee | /src/topology/metric_space/completion.lean | 5f3269b118b4cd88a40d8a7586e507b5bb2825b2 | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 8,126 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import topology.uniform_space.completion
import topology.metric_space.isometry
import topology.instances.real
/-!
# The completion of a metric space
Completion of uniform spaces are already defined in `topology.uniform_space.completion`. We show
here that the uniform space completion of a metric space inherits a metric space structure,
by extending the distance to the completion and checking that it is indeed a distance, and that
it defines the same uniformity as the already defined uniform structure on the completion
-/
open set filter uniform_space metric
open_locale filter topological_space uniformity
noncomputable theory
universes u v
variables {α : Type u} {β : Type v} [pseudo_metric_space α]
namespace uniform_space.completion
/-- The distance on the completion is obtained by extending the distance on the original space,
by uniform continuity. -/
instance : has_dist (completion α) :=
⟨completion.extension₂ dist⟩
/-- The new distance is uniformly continuous. -/
protected lemma uniform_continuous_dist :
uniform_continuous (λp:completion α × completion α, dist p.1 p.2) :=
uniform_continuous_extension₂ dist
/-- The new distance is continuous. -/
protected lemma continuous_dist [topological_space β] {f g : β → completion α} (hf : continuous f)
(hg : continuous g) :
continuous (λ x, dist (f x) (g x)) :=
completion.uniform_continuous_dist.continuous.comp (hf.prod_mk hg : _)
/-- The new distance is an extension of the original distance. -/
@[simp] protected lemma dist_eq (x y : α) : dist (x : completion α) y = dist x y :=
completion.extension₂_coe_coe uniform_continuous_dist _ _
/- Let us check that the new distance satisfies the axioms of a distance, by starting from the
properties on α and extending them to `completion α` by continuity. -/
protected lemma dist_self (x : completion α) : dist x x = 0 :=
begin
apply induction_on x,
{ refine is_closed_eq _ continuous_const,
exact completion.continuous_dist continuous_id continuous_id },
{ assume a,
rw [completion.dist_eq, dist_self] }
end
protected lemma dist_comm (x y : completion α) : dist x y = dist y x :=
begin
apply induction_on₂ x y,
{ exact is_closed_eq (completion.continuous_dist continuous_fst continuous_snd)
(completion.continuous_dist continuous_snd continuous_fst) },
{ assume a b,
rw [completion.dist_eq, completion.dist_eq, dist_comm] }
end
protected lemma dist_triangle (x y z : completion α) : dist x z ≤ dist x y + dist y z :=
begin
apply induction_on₃ x y z,
{ refine is_closed_le _ (continuous.add _ _);
apply_rules [completion.continuous_dist, continuous.fst, continuous.snd, continuous_id] },
{ assume a b c,
rw [completion.dist_eq, completion.dist_eq, completion.dist_eq],
exact dist_triangle a b c }
end
/-- Elements of the uniformity (defined generally for completions) can be characterized in terms
of the distance. -/
protected lemma mem_uniformity_dist (s : set (completion α × completion α)) :
s ∈ 𝓤 (completion α) ↔ (∃ε>0, ∀{a b}, dist a b < ε → (a, b) ∈ s) :=
begin
split,
{ /- Start from an entourage `s`. It contains a closed entourage `t`. Its pullback in `α` is an
entourage, so it contains an `ε`-neighborhood of the diagonal by definition of the entourages
in metric spaces. Then `t` contains an `ε`-neighborhood of the diagonal in `completion α`, as
closed properties pass to the completion. -/
assume hs,
rcases mem_uniformity_is_closed hs with ⟨t, ht, ⟨tclosed, ts⟩⟩,
have A : {x : α × α | (coe (x.1), coe (x.2)) ∈ t} ∈ uniformity α :=
uniform_continuous_def.1 (uniform_continuous_coe α) t ht,
rcases mem_uniformity_dist.1 A with ⟨ε, εpos, hε⟩,
refine ⟨ε, εpos, λx y hxy, _⟩,
have : ε ≤ dist x y ∨ (x, y) ∈ t,
{ apply induction_on₂ x y,
{ have : {x : completion α × completion α | ε ≤ dist (x.fst) (x.snd) ∨ (x.fst, x.snd) ∈ t}
= {p : completion α × completion α | ε ≤ dist p.1 p.2} ∪ t, by ext; simp,
rw this,
apply is_closed.union _ tclosed,
exact is_closed_le continuous_const completion.uniform_continuous_dist.continuous },
{ assume x y,
rw completion.dist_eq,
by_cases h : ε ≤ dist x y,
{ exact or.inl h },
{ have Z := hε (not_le.1 h),
simp only [set.mem_set_of_eq] at Z,
exact or.inr Z }}},
simp only [not_le.mpr hxy, false_or, not_le] at this,
exact ts this },
{ /- Start from a set `s` containing an ε-neighborhood of the diagonal in `completion α`. To show
that it is an entourage, we use the fact that `dist` is uniformly continuous on
`completion α × completion α` (this is a general property of the extension of uniformly
continuous functions). Therefore, the preimage of the ε-neighborhood of the diagonal in ℝ
is an entourage in `completion α × completion α`. Massaging this property, it follows that
the ε-neighborhood of the diagonal is an entourage in `completion α`, and therefore this is
also the case of `s`. -/
rintros ⟨ε, εpos, hε⟩,
let r : set (ℝ × ℝ) := {p | dist p.1 p.2 < ε},
have : r ∈ uniformity ℝ := metric.dist_mem_uniformity εpos,
have T := uniform_continuous_def.1 (@completion.uniform_continuous_dist α _) r this,
simp only [uniformity_prod_eq_prod, mem_prod_iff, exists_prop,
filter.mem_map, set.mem_set_of_eq] at T,
rcases T with ⟨t1, ht1, t2, ht2, ht⟩,
refine mem_of_superset ht1 _,
have A : ∀a b : completion α, (a, b) ∈ t1 → dist a b < ε,
{ assume a b hab,
have : ((a, b), (a, a)) ∈ t1 ×ˢ t2 := ⟨hab, refl_mem_uniformity ht2⟩,
have I := ht this,
simp [completion.dist_self, real.dist_eq, completion.dist_comm] at I,
exact lt_of_le_of_lt (le_abs_self _) I },
show t1 ⊆ s,
{ rintros ⟨a, b⟩ hp,
have : dist a b < ε := A a b hp,
exact hε this }}
end
/-- If two points are at distance 0, then they coincide. -/
protected lemma eq_of_dist_eq_zero (x y : completion α) (h : dist x y = 0) : x = y :=
begin
/- This follows from the separation of `completion α` and from the description of
entourages in terms of the distance. -/
have : separated_space (completion α) := by apply_instance,
refine separated_def.1 this x y (λs hs, _),
rcases (completion.mem_uniformity_dist s).1 hs with ⟨ε, εpos, hε⟩,
rw ← h at εpos,
exact hε εpos
end
/-- Reformulate `completion.mem_uniformity_dist` in terms that are suitable for the definition
of the metric space structure. -/
protected lemma uniformity_dist' :
𝓤 (completion α) = (⨅ε:{ε : ℝ // 0 < ε}, 𝓟 {p | dist p.1 p.2 < ε.val}) :=
begin
ext s, rw mem_infi_of_directed,
{ simp [completion.mem_uniformity_dist, subset_def] },
{ rintro ⟨r, hr⟩ ⟨p, hp⟩, use ⟨min r p, lt_min hr hp⟩,
simp [lt_min_iff, (≥)] {contextual := tt} }
end
protected lemma uniformity_dist :
𝓤 (completion α) = (⨅ ε>0, 𝓟 {p | dist p.1 p.2 < ε}) :=
by simpa [infi_subtype] using @completion.uniformity_dist' α _
/-- Metric space structure on the completion of a pseudo_metric space. -/
instance : metric_space (completion α) :=
{ dist_self := completion.dist_self,
eq_of_dist_eq_zero := completion.eq_of_dist_eq_zero,
dist_comm := completion.dist_comm,
dist_triangle := completion.dist_triangle,
dist := dist,
to_uniform_space := by apply_instance,
uniformity_dist := completion.uniformity_dist }
/-- The embedding of a metric space in its completion is an isometry. -/
lemma coe_isometry : isometry (coe : α → completion α) :=
isometry_emetric_iff_metric.2 completion.dist_eq
@[simp] protected lemma edist_eq (x y : α) : edist (x : completion α) y = edist x y :=
coe_isometry x y
end uniform_space.completion
|
c09425310e9189e70a2331de2158ce265a582c58 | 35677d2df3f081738fa6b08138e03ee36bc33cad | /src/topology/category/Top/open_nhds.lean | 8d0a1d791540eac09a160c229d5e179df96f2136 | [
"Apache-2.0"
] | permissive | gebner/mathlib | eab0150cc4f79ec45d2016a8c21750244a2e7ff0 | cc6a6edc397c55118df62831e23bfbd6e6c6b4ab | refs/heads/master | 1,625,574,853,976 | 1,586,712,827,000 | 1,586,712,827,000 | 99,101,412 | 1 | 0 | Apache-2.0 | 1,586,716,389,000 | 1,501,667,958,000 | Lean | UTF-8 | Lean | false | false | 1,807 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import topology.category.Top.opens
import category_theory.full_subcategory
open category_theory
open topological_space
open opposite
universe u
variables {X Y : Top.{u}} (f : X ⟶ Y)
namespace topological_space
def open_nhds (x : X.α) := { U : opens X // x ∈ U }
namespace open_nhds
instance open_nhds_category (x : X.α) : category.{u} (open_nhds x) := by {unfold open_nhds, apply_instance}
def inclusion (x : X.α) : open_nhds x ⥤ opens X :=
full_subcategory_inclusion _
@[simp] lemma inclusion_obj (x : X.α) (U) (p) : (inclusion x).obj ⟨U,p⟩ = U := rfl
def map (x : X) : open_nhds (f x) ⥤ open_nhds x :=
{ obj := λ U, ⟨(opens.map f).obj U.1, by tidy⟩,
map := λ U V i, (opens.map f).map i }
@[simp] lemma map_obj (x : X) (U) (q) : (map f x).obj ⟨U, q⟩ = ⟨(opens.map f).obj U, by tidy⟩ :=
rfl
@[simp] lemma map_id_obj (x : X) (U) : (map (𝟙 X) x).obj U = U :=
by tidy
@[simp] lemma map_id_obj' (x : X) (U) (p) (q) : (map (𝟙 X) x).obj ⟨⟨U, p⟩, q⟩ = ⟨⟨U, p⟩, q⟩ :=
rfl
@[simp] lemma map_id_obj_unop (x : X) (U : (open_nhds x)ᵒᵖ) : (map (𝟙 X) x).obj (unop U) = unop U :=
by simp
@[simp] lemma op_map_id_obj (x : X) (U : (open_nhds x)ᵒᵖ) : (map (𝟙 X) x).op.obj U = U :=
by simp
def inclusion_map_iso (x : X) : inclusion (f x) ⋙ opens.map f ≅ map f x ⋙ inclusion x :=
nat_iso.of_components
(λ U, begin split, exact 𝟙 _, exact 𝟙 _ end)
(by tidy)
@[simp] lemma inclusion_map_iso_hom (x : X) : (inclusion_map_iso f x).hom = 𝟙 _ := rfl
@[simp] lemma inclusion_map_iso_inv (x : X) : (inclusion_map_iso f x).inv = 𝟙 _ := rfl
end open_nhds
end topological_space
|
da338b948f7ca7f724b1af4a350959ab27f95974 | 624f6f2ae8b3b1adc5f8f67a365c51d5126be45a | /src/Init/Data/Array.lean | 6e7adb07e52b31e1e7c20161125e58eac7614844 | [
"Apache-2.0"
] | permissive | mhuisi/lean4 | 28d35a4febc2e251c7f05492e13f3b05d6f9b7af | dda44bc47f3e5d024508060dac2bcb59fd12e4c0 | refs/heads/master | 1,621,225,489,283 | 1,585,142,689,000 | 1,585,142,689,000 | 250,590,438 | 0 | 2 | Apache-2.0 | 1,602,443,220,000 | 1,585,327,814,000 | C | UTF-8 | Lean | false | false | 259 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner
-/
prelude
import Init.Data.Array.Basic
import Init.Data.Array.QSort
import Init.Data.Array.BinSearch
|
ec8bf5b060ce074957a3527ec76441f9c1f5cea6 | b32d3853770e6eaf06817a1b8c52064baaed0ef1 | /test/super_examples.lean | 35498db1997519c37ec69ddee45c6d16deb338e9 | [] | no_license | gebner/super2 | 4d58b7477b6f7d945d5d866502982466db33ab0b | 9bc5256c31750021ab97d6b59b7387773e54b384 | refs/heads/master | 1,635,021,682,021 | 1,634,886,326,000 | 1,634,886,326,000 | 225,600,688 | 4 | 2 | null | 1,598,209,306,000 | 1,575,371,550,000 | Lean | UTF-8 | Lean | false | false | 3,176 | lean | import super
open tactic
set_option trace.super true
set_option profiler true
def prime (n : ℕ) := ∀ d, d ∣ n → d = 1 ∨ d = n
set_option trace.check true
lemma nat_mul_cancel_one {m n : ℕ} : m ≠ 0 → m * n = m → n = 1 :=
by cases m; super [gt, nat.zero_lt_succ,
nat.eq_of_mul_eq_mul_left, mul_one, zero_mul, nat.not_lt_zero]
lemma not_prime_zero : ¬ prime 0 :=
by intro h; cases h 2 ⟨0, by simp⟩; cases h_1
/-
example {m n : ℕ} : prime (m * n) → m = 1 ∨ n = 1 :=
by super [prime, dvd_refl, dvd_mul_right, dvd_mul_left,
nat_mul_cancel_one, not_prime_zero, mul_zero, zero_mul]
-/
example : nat.zero ≠ nat.succ nat.zero := by super [nat.zero_lt_succ, ne_of_lt]
example (x y : ℕ) : nat.succ x = nat.succ y → x = y := by super [nat.succ.inj]
example (i) (a b c : i) : [a,b,c] = [b,c,a] -> a = b ∧ b = c := by super [list.cons.inj]
definition is_positive (n : ℕ) := n > 0
example (n : ℕ) : n > 0 ↔ is_positive n := by super [is_positive]
example (m n : ℕ) : 0 + m = 0 + n → m = n :=
by super [zero_add]
example : ∀x y : ℕ, x + y = y + x :=
begin intros, have h : nat.zero = 0 := rfl, induction x,
super [add_zero, zero_add],
super [*, nat.add_succ, nat.succ_add] end
example (i) [inhabited i] : nonempty i := by super *
example (i) [nonempty i] : ¬(inhabited i → false) :=
by super [classical.inhabited_of_nonempty]
example : nonempty ℕ := by super
example : ¬(inhabited ℕ → false) := by super
example {a b} : ¬(b ∨ ¬a) ∨ (a → b) := by super
example {a} : a ∨ ¬a := by super
example {a} : (a ∧ a) ∨ (¬a ∧ ¬a) := by super
example {a} : ¬a → (¬a ∧ ¬a) := by super
example {a} : a ∨ a → a := by super
example (i) (c : i) (p : i → Prop) (f : i → i) :
p c → (∀x, p x → p (f x)) → p (f (f (f c))) := by super
example (i : Type) (p : i → Prop) : ∀x, p x → ∃x, p x := by super
example (i) [nonempty i] (p : i → i → Prop) : (∀x y, p x y) → ∃x, ∀z, p x z :=
by super
example (i) [nonempty i] (p : i → Prop) : (∀x, p x) → ¬¬∀x, p x := by super *
-- Requires non-empty domain.
example {i} [nonempty i] (p : i → Prop) :
(∀x y, p x ∨ p y) → ∃x y, p x ∧ p y := by super *
example (i) (a b : i) (p : i → Prop) (H : a = b) : p b → p a :=
by super *
example (i) (a b : i) (p : i → Prop) (H : a = b) : p a → p b :=
by super *
example (i) (a b : i) (p : i → Prop) (H : a = b) : p b = p a :=
by super *
example (i) (c : i) (p : i → Prop) (f g : i → i) :
p c → (∀x, p x → p (f x)) → (∀x, p x → f x = g x) → f (f c) = g (g c) :=
by super
-- This example from Davis-Putnam actually requires a non-empty domain
example (i) [nonempty i] (f g : i → i → Prop) :
∃x y, ∀z, (f x y → f y z ∧ f z z) ∧ (f x y ∧ g x y → g x z ∧ g z z) :=
by super
example (person) [nonempty person] (drinks : person → Prop) :
∃canary, drinks canary → ∀other, drinks other := by super
example {p q : ℕ → Prop} {r} : (∀x y, p x ∧ q y ∧ r) -> ∀x, (p x ∧ r ∧ q x) :=
by super
example {α} [add_group α] (x : α) : 0 + 0 + x + 0 + 0 + 0 = x :=
by super [add_zero, zero_add]
|
897b572a0aaaf60d3532533f9408d6bce45e0fa7 | 432d948a4d3d242fdfb44b81c9e1b1baacd58617 | /src/topology/order.lean | f8130767c1cae70dd25dddc8665ae6a65852d154 | [
"Apache-2.0"
] | permissive | JLimperg/aesop3 | 306cc6570c556568897ed2e508c8869667252e8a | a4a116f650cc7403428e72bd2e2c4cda300fe03f | refs/heads/master | 1,682,884,916,368 | 1,620,320,033,000 | 1,620,320,033,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 29,025 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import topology.tactic
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixed type `α` are ordered, by reverse inclusion.
That is, for topologies `t₁` and `t₂` on `α`, we write `t₁ ≤ t₂`
if every set open in `t₂` is also open in `t₁`.
(One also calls `t₁` finer than `t₂`, and `t₂` coarser than `t₁`.)
Any function `f : α → β` induces
`induced f : topological_space β → topological_space α`
and `coinduced f : topological_space α → topological_space β`.
Continuity, the ordering on topologies and (co)induced topologies are
related as follows:
* The identity map (α, t₁) → (α, t₂) is continuous iff t₁ ≤ t₂.
* A map f : (α, t) → (β, u) is continuous
iff t ≤ induced f u (`continuous_iff_le_induced`)
iff coinduced f t ≤ u (`continuous_iff_coinduced_le`).
Topologies on α form a complete lattice, with ⊥ the discrete topology
and ⊤ the indiscrete topology.
For a function f : α → β, (coinduced f, induced f) is a Galois connection
between topologies on α and topologies on β.
## Implementation notes
There is a Galois insertion between topologies on α (with the inclusion ordering)
and all collections of sets in α. The complete lattice structure on topologies
on α is defined as the reverse of the one obtained via this Galois insertion.
## Tags
finer, coarser, induced topology, coinduced topology
-/
open set filter classical
open_locale classical topological_space filter
universes u v w
namespace topological_space
variables {α : Type u}
/-- The open sets of the least topology containing a collection of basic sets. -/
inductive generate_open (g : set (set α)) : set α → Prop
| basic : ∀s∈g, generate_open s
| univ : generate_open univ
| inter : ∀s t, generate_open s → generate_open t → generate_open (s ∩ t)
| sUnion : ∀k, (∀s∈k, generate_open s) → generate_open (⋃₀ k)
/-- The smallest topological space containing the collection `g` of basic sets -/
def generate_from (g : set (set α)) : topological_space α :=
{ is_open := generate_open g,
is_open_univ := generate_open.univ,
is_open_inter := generate_open.inter,
is_open_sUnion := generate_open.sUnion }
lemma nhds_generate_from {g : set (set α)} {a : α} :
@nhds α (generate_from g) a = (⨅s∈{s | a ∈ s ∧ s ∈ g}, 𝓟 s) :=
by rw nhds_def; exact le_antisymm
(infi_le_infi $ assume s, infi_le_infi_const $ assume ⟨as, sg⟩, ⟨as, generate_open.basic _ sg⟩)
(le_infi $ assume s, le_infi $ assume ⟨as, hs⟩,
begin
revert as, clear_, induction hs,
case generate_open.basic : s hs
{ exact assume as, infi_le_of_le s $ infi_le _ ⟨as, hs⟩ },
case generate_open.univ
{ rw [principal_univ],
exact assume _, le_top },
case generate_open.inter : s t hs' ht' hs ht
{ exact assume ⟨has, hat⟩, calc _ ≤ 𝓟 s ⊓ 𝓟 t : le_inf (hs has) (ht hat)
... = _ : inf_principal },
case generate_open.sUnion : k hk' hk
{ exact λ ⟨t, htk, hat⟩, calc _ ≤ 𝓟 t : hk t htk hat
... ≤ _ : le_principal_iff.2 $ subset_sUnion_of_mem htk }
end)
lemma tendsto_nhds_generate_from {β : Type*} {m : α → β} {f : filter α} {g : set (set β)} {b : β}
(h : ∀s∈g, b ∈ s → m ⁻¹' s ∈ f) : tendsto m f (@nhds β (generate_from g) b) :=
by rw [nhds_generate_from]; exact
(tendsto_infi.2 $ assume s, tendsto_infi.2 $ assume ⟨hbs, hsg⟩, tendsto_principal.2 $ h s hsg hbs)
/-- Construct a topology on α given the filter of neighborhoods of each point of α. -/
protected def mk_of_nhds (n : α → filter α) : topological_space α :=
{ is_open := λs, ∀a∈s, s ∈ n a,
is_open_univ := assume x h, univ_mem_sets,
is_open_inter := assume s t hs ht x ⟨hxs, hxt⟩, inter_mem_sets (hs x hxs) (ht x hxt),
is_open_sUnion := assume s hs a ⟨x, hx, hxa⟩,
mem_sets_of_superset (hs x hx _ hxa) (set.subset_sUnion_of_mem hx) }
lemma nhds_mk_of_nhds (n : α → filter α) (a : α)
(h₀ : pure ≤ n) (h₁ : ∀{a s}, s ∈ n a → ∃ t ∈ n a, t ⊆ s ∧ ∀a' ∈ t, s ∈ n a') :
@nhds α (topological_space.mk_of_nhds n) a = n a :=
begin
letI := topological_space.mk_of_nhds n,
refine le_antisymm (assume s hs, _) (assume s hs, _),
{ have h₀ : {b | s ∈ n b} ⊆ s := assume b hb, mem_pure_sets.1 $ h₀ b hb,
have h₁ : {b | s ∈ n b} ∈ 𝓝 a,
{ refine mem_nhds_sets (assume b (hb : s ∈ n b), _) hs,
rcases h₁ hb with ⟨t, ht, hts, h⟩,
exact mem_sets_of_superset ht h },
exact mem_sets_of_superset h₁ h₀ },
{ rcases (@mem_nhds_sets_iff α (topological_space.mk_of_nhds n) _ _).1 hs with ⟨t, hts, ht, hat⟩,
exact (n a).sets_of_superset (ht _ hat) hts },
end
end topological_space
section lattice
variables {α : Type u} {β : Type v}
/-- The inclusion ordering on topologies on α. We use it to get a complete
lattice instance via the Galois insertion method, but the partial order
that we will eventually impose on `topological_space α` is the reverse one. -/
def tmp_order : partial_order (topological_space α) :=
{ le := λt s, t.is_open ≤ s.is_open,
le_antisymm := assume t s h₁ h₂, topological_space_eq $ le_antisymm h₁ h₂,
le_refl := assume t, le_refl t.is_open,
le_trans := assume a b c h₁ h₂, @le_trans _ _ a.is_open b.is_open c.is_open h₁ h₂ }
local attribute [instance] tmp_order
/- We'll later restate this lemma in terms of the correct order on `topological_space α`. -/
private lemma generate_from_le_iff_subset_is_open {g : set (set α)} {t : topological_space α} :
topological_space.generate_from g ≤ t ↔ g ⊆ {s | t.is_open s} :=
iff.intro
(assume ht s hs, ht _ $ topological_space.generate_open.basic s hs)
(assume hg s hs, hs.rec_on (assume v hv, hg hv)
t.is_open_univ (assume u v _ _, t.is_open_inter u v) (assume k _, t.is_open_sUnion k))
/-- If `s` equals the collection of open sets in the topology it generates,
then `s` defines a topology. -/
protected def mk_of_closure (s : set (set α))
(hs : {u | (topological_space.generate_from s).is_open u} = s) : topological_space α :=
{ is_open := λu, u ∈ s,
is_open_univ := hs ▸ topological_space.generate_open.univ,
is_open_inter := hs ▸ topological_space.generate_open.inter,
is_open_sUnion := hs ▸ topological_space.generate_open.sUnion }
lemma mk_of_closure_sets {s : set (set α)}
{hs : {u | (topological_space.generate_from s).is_open u} = s} :
mk_of_closure s hs = topological_space.generate_from s :=
topological_space_eq hs.symm
/-- The Galois insertion between `set (set α)` and `topological_space α` whose lower part
sends a collection of subsets of α to the topology they generate, and whose upper part
sends a topology to its collection of open subsets. -/
def gi_generate_from (α : Type*) :
galois_insertion topological_space.generate_from (λt:topological_space α, {s | t.is_open s}) :=
{ gc := assume g t, generate_from_le_iff_subset_is_open,
le_l_u := assume ts s hs, topological_space.generate_open.basic s hs,
choice := λg hg, mk_of_closure g
(subset.antisymm hg $ generate_from_le_iff_subset_is_open.1 $ le_refl _),
choice_eq := assume s hs, mk_of_closure_sets }
lemma generate_from_mono {α} {g₁ g₂ : set (set α)} (h : g₁ ⊆ g₂) :
topological_space.generate_from g₁ ≤ topological_space.generate_from g₂ :=
(gi_generate_from _).gc.monotone_l h
/-- The complete lattice of topological spaces, but built on the inclusion ordering. -/
def tmp_complete_lattice {α : Type u} : complete_lattice (topological_space α) :=
(gi_generate_from α).lift_complete_lattice
/-- The ordering on topologies on the type `α`.
`t ≤ s` if every set open in `s` is also open in `t` (`t` is finer than `s`). -/
instance : partial_order (topological_space α) :=
{ le := λ t s, s.is_open ≤ t.is_open,
le_antisymm := assume t s h₁ h₂, topological_space_eq $ le_antisymm h₂ h₁,
le_refl := assume t, le_refl t.is_open,
le_trans := assume a b c h₁ h₂, le_trans h₂ h₁ }
lemma le_generate_from_iff_subset_is_open {g : set (set α)} {t : topological_space α} :
t ≤ topological_space.generate_from g ↔ g ⊆ {s | t.is_open s} :=
generate_from_le_iff_subset_is_open
/-- Topologies on `α` form a complete lattice, with `⊥` the discrete topology
and `⊤` the indiscrete topology. The infimum of a collection of topologies
is the topology generated by all their open sets, while the supremem is the
topology whose open sets are those sets open in every member of the collection. -/
instance : complete_lattice (topological_space α) :=
@order_dual.complete_lattice _ tmp_complete_lattice
/-- A topological space is discrete if every set is open, that is,
its topology equals the discrete topology `⊥`. -/
class discrete_topology (α : Type*) [t : topological_space α] : Prop :=
(eq_bot [] : t = ⊥)
@[priority 100]
instance discrete_topology_bot (α : Type*) : @discrete_topology α ⊥ :=
{ eq_bot := rfl }
@[simp] lemma is_open_discrete [topological_space α] [discrete_topology α] (s : set α) :
is_open s :=
(discrete_topology.eq_bot α).symm ▸ trivial
@[simp] lemma is_closed_discrete [topological_space α] [discrete_topology α] (s : set α) :
is_closed s :=
is_open_compl_iff.1 $ (discrete_topology.eq_bot α).symm ▸ trivial
lemma continuous_of_discrete_topology [topological_space α] [discrete_topology α]
[topological_space β] {f : α → β} : continuous f :=
continuous_def.2 $ λs hs, is_open_discrete _
lemma nhds_bot (α : Type*) : (@nhds α ⊥) = pure :=
begin
refine le_antisymm _ (@pure_le_nhds α ⊥),
assume a s hs,
exact @mem_nhds_sets α ⊥ a s trivial hs
end
lemma nhds_discrete (α : Type*) [topological_space α] [discrete_topology α] : (@nhds α _) = pure :=
(discrete_topology.eq_bot α).symm ▸ nhds_bot α
lemma le_of_nhds_le_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₁ x ≤ @nhds α t₂ x) :
t₁ ≤ t₂ :=
assume s, show @is_open α t₂ s → @is_open α t₁ s,
by { simp only [is_open_iff_nhds, le_principal_iff], exact assume hs a ha, h _ $ hs _ ha }
lemma eq_of_nhds_eq_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₁ x = @nhds α t₂ x) :
t₁ = t₂ :=
le_antisymm
(le_of_nhds_le_nhds $ assume x, le_of_eq $ h x)
(le_of_nhds_le_nhds $ assume x, le_of_eq $ (h x).symm)
lemma eq_bot_of_singletons_open {t : topological_space α} (h : ∀ x, t.is_open {x}) : t = ⊥ :=
bot_unique $ λ s hs, bUnion_of_singleton s ▸ is_open_bUnion (λ x _, h x)
lemma forall_open_iff_discrete {X : Type*} [topological_space X] :
(∀ s : set X, is_open s) ↔ discrete_topology X :=
⟨λ h, ⟨by { ext U , show is_open U ↔ true, simp [h U] }⟩, λ a, @is_open_discrete _ _ a⟩
lemma singletons_open_iff_discrete {X : Type*} [topological_space X] :
(∀ a : X, is_open ({a} : set X)) ↔ discrete_topology X :=
⟨λ h, ⟨eq_bot_of_singletons_open h⟩, λ a _, @is_open_discrete _ _ a _⟩
end lattice
section galois_connection
variables {α : Type*} {β : Type*} {γ : Type*}
/-- Given `f : α → β` and a topology on `β`, the induced topology on `α` is the collection of
sets that are preimages of some open set in `β`. This is the coarsest topology that
makes `f` continuous. -/
def topological_space.induced {α : Type u} {β : Type v} (f : α → β) (t : topological_space β) :
topological_space α :=
{ is_open := λs, ∃s', t.is_open s' ∧ f ⁻¹' s' = s,
is_open_univ := ⟨univ, t.is_open_univ, preimage_univ⟩,
is_open_inter := by rintro s₁ s₂ ⟨s'₁, hs₁, rfl⟩ ⟨s'₂, hs₂, rfl⟩;
exact ⟨s'₁ ∩ s'₂, t.is_open_inter _ _ hs₁ hs₂, preimage_inter⟩,
is_open_sUnion := assume s h,
begin
simp only [classical.skolem] at h,
cases h with f hf,
apply exists.intro (⋃(x : set α) (h : x ∈ s), f x h),
simp only [sUnion_eq_bUnion, preimage_Union, (λx h, (hf x h).right)], refine ⟨_, rfl⟩,
exact (@is_open_Union β _ t _ $ assume i,
show is_open (⋃h, f i h), from @is_open_Union β _ t _ $ assume h, (hf i h).left)
end }
lemma is_open_induced_iff [t : topological_space β] {s : set α} {f : α → β} :
@is_open α (t.induced f) s ↔ (∃t, is_open t ∧ f ⁻¹' t = s) :=
iff.rfl
lemma is_closed_induced_iff [t : topological_space β] {s : set α} {f : α → β} :
@is_closed α (t.induced f) s ↔ (∃t, is_closed t ∧ f ⁻¹' t = s) :=
begin
simp only [← is_open_compl_iff, is_open_induced_iff],
exact ⟨λ ⟨t, ht, heq⟩, ⟨tᶜ, by rwa compl_compl, by simp [preimage_compl, heq, compl_compl]⟩,
λ ⟨t, ht, heq⟩, ⟨tᶜ, ht, by simp only [preimage_compl, heq.symm]⟩⟩
end
/-- Given `f : α → β` and a topology on `α`, the coinduced topology on `β` is defined
such that `s:set β` is open if the preimage of `s` is open. This is the finest topology that
makes `f` continuous. -/
def topological_space.coinduced {α : Type u} {β : Type v} (f : α → β) (t : topological_space α) :
topological_space β :=
{ is_open := λs, t.is_open (f ⁻¹' s),
is_open_univ := by rw preimage_univ; exact t.is_open_univ,
is_open_inter := assume s₁ s₂ h₁ h₂, by rw preimage_inter; exact t.is_open_inter _ _ h₁ h₂,
is_open_sUnion := assume s h, by rw [preimage_sUnion]; exact (@is_open_Union _ _ t _ $ assume i,
show is_open (⋃ (H : i ∈ s), f ⁻¹' i), from
@is_open_Union _ _ t _ $ assume hi, h i hi) }
lemma is_open_coinduced {t : topological_space α} {s : set β} {f : α → β} :
@is_open β (topological_space.coinduced f t) s ↔ is_open (f ⁻¹' s) :=
iff.rfl
variables {t t₁ t₂ : topological_space α} {t' : topological_space β} {f : α → β} {g : β → α}
lemma continuous.coinduced_le (h : @continuous α β t t' f) :
t.coinduced f ≤ t' :=
λ s hs, (continuous_def.1 h s hs : _)
lemma coinduced_le_iff_le_induced {f : α → β} {tα : topological_space α}
{tβ : topological_space β} :
tα.coinduced f ≤ tβ ↔ tα ≤ tβ.induced f :=
iff.intro
(assume h s ⟨t, ht, hst⟩, hst ▸ h _ ht)
(assume h s hs, show tα.is_open (f ⁻¹' s), from h _ ⟨s, hs, rfl⟩)
lemma continuous.le_induced (h : @continuous α β t t' f) :
t ≤ t'.induced f :=
coinduced_le_iff_le_induced.1 h.coinduced_le
lemma gc_coinduced_induced (f : α → β) :
galois_connection (topological_space.coinduced f) (topological_space.induced f) :=
assume f g, coinduced_le_iff_le_induced
lemma induced_mono (h : t₁ ≤ t₂) : t₁.induced g ≤ t₂.induced g :=
(gc_coinduced_induced g).monotone_u h
lemma coinduced_mono (h : t₁ ≤ t₂) : t₁.coinduced f ≤ t₂.coinduced f :=
(gc_coinduced_induced f).monotone_l h
@[simp] lemma induced_top : (⊤ : topological_space α).induced g = ⊤ :=
(gc_coinduced_induced g).u_top
@[simp] lemma induced_inf : (t₁ ⊓ t₂).induced g = t₁.induced g ⊓ t₂.induced g :=
(gc_coinduced_induced g).u_inf
@[simp] lemma induced_infi {ι : Sort w} {t : ι → topological_space α} :
(⨅i, t i).induced g = (⨅i, (t i).induced g) :=
(gc_coinduced_induced g).u_infi
@[simp] lemma coinduced_bot : (⊥ : topological_space α).coinduced f = ⊥ :=
(gc_coinduced_induced f).l_bot
@[simp] lemma coinduced_sup : (t₁ ⊔ t₂).coinduced f = t₁.coinduced f ⊔ t₂.coinduced f :=
(gc_coinduced_induced f).l_sup
@[simp] lemma coinduced_supr {ι : Sort w} {t : ι → topological_space α} :
(⨆i, t i).coinduced f = (⨆i, (t i).coinduced f) :=
(gc_coinduced_induced f).l_supr
lemma induced_id [t : topological_space α] : t.induced id = t :=
topological_space_eq $ funext $ assume s, propext $
⟨assume ⟨s', hs, h⟩, h ▸ hs, assume hs, ⟨s, hs, rfl⟩⟩
lemma induced_compose [tγ : topological_space γ]
{f : α → β} {g : β → γ} : (tγ.induced g).induced f = tγ.induced (g ∘ f) :=
topological_space_eq $ funext $ assume s, propext $
⟨assume ⟨s', ⟨s, hs, h₂⟩, h₁⟩, h₁ ▸ h₂ ▸ ⟨s, hs, rfl⟩,
assume ⟨s, hs, h⟩, ⟨preimage g s, ⟨s, hs, rfl⟩, h ▸ rfl⟩⟩
lemma induced_const [t : topological_space α] {x : α} :
t.induced (λ y : β, x) = ⊤ :=
le_antisymm le_top (@continuous_const β α ⊤ t x).le_induced
lemma coinduced_id [t : topological_space α] : t.coinduced id = t :=
topological_space_eq rfl
lemma coinduced_compose [tα : topological_space α]
{f : α → β} {g : β → γ} : (tα.coinduced f).coinduced g = tα.coinduced (g ∘ f) :=
topological_space_eq rfl
end galois_connection
/- constructions using the complete lattice structure -/
section constructions
open topological_space
variables {α : Type u} {β : Type v}
instance inhabited_topological_space {α : Type u} : inhabited (topological_space α) :=
⟨⊤⟩
@[priority 100]
instance subsingleton.unique_topological_space [subsingleton α] :
unique (topological_space α) :=
{ default := ⊥,
uniq := λ t, eq_bot_of_singletons_open $ λ x, subsingleton.set_cases
(@is_open_empty _ t) (@is_open_univ _ t) ({x} : set α) }
@[priority 100]
instance subsingleton.discrete_topology [t : topological_space α] [subsingleton α] :
discrete_topology α :=
⟨unique.eq_default t⟩
instance : topological_space empty := ⊥
instance : discrete_topology empty := ⟨rfl⟩
instance : topological_space pempty := ⊥
instance : discrete_topology pempty := ⟨rfl⟩
instance : topological_space punit := ⊥
instance : discrete_topology punit := ⟨rfl⟩
instance : topological_space bool := ⊥
instance : discrete_topology bool := ⟨rfl⟩
instance : topological_space ℕ := ⊥
instance : discrete_topology ℕ := ⟨rfl⟩
instance : topological_space ℤ := ⊥
instance : discrete_topology ℤ := ⟨rfl⟩
instance sierpinski_space : topological_space Prop :=
generate_from {{true}}
lemma le_generate_from {t : topological_space α} { g : set (set α) } (h : ∀s∈g, is_open s) :
t ≤ generate_from g :=
le_generate_from_iff_subset_is_open.2 h
lemma induced_generate_from_eq {α β} {b : set (set β)} {f : α → β} :
(generate_from b).induced f = topological_space.generate_from (preimage f '' b) :=
le_antisymm
(le_generate_from $ ball_image_iff.2 $ assume s hs, ⟨s, generate_open.basic _ hs, rfl⟩)
(coinduced_le_iff_le_induced.1 $ le_generate_from $ assume s hs,
generate_open.basic _ $ mem_image_of_mem _ hs)
/-- This construction is left adjoint to the operation sending a topology on `α`
to its neighborhood filter at a fixed point `a : α`. -/
protected def topological_space.nhds_adjoint (a : α) (f : filter α) : topological_space α :=
{ is_open := λs, a ∈ s → s ∈ f,
is_open_univ := assume s, univ_mem_sets,
is_open_inter := assume s t hs ht ⟨has, hat⟩, inter_mem_sets (hs has) (ht hat),
is_open_sUnion := assume k hk ⟨u, hu, hau⟩, mem_sets_of_superset (hk u hu hau)
(subset_sUnion_of_mem hu) }
lemma gc_nhds (a : α) :
galois_connection (topological_space.nhds_adjoint a) (λt, @nhds α t a) :=
assume f t, by { rw le_nhds_iff, exact ⟨λ H s hs has, H _ has hs, λ H s has hs, H _ hs has⟩ }
lemma nhds_mono {t₁ t₂ : topological_space α} {a : α} (h : t₁ ≤ t₂) :
@nhds α t₁ a ≤ @nhds α t₂ a := (gc_nhds a).monotone_u h
lemma nhds_infi {ι : Sort*} {t : ι → topological_space α} {a : α} :
@nhds α (infi t) a = (⨅i, @nhds α (t i) a) := (gc_nhds a).u_infi
lemma nhds_Inf {s : set (topological_space α)} {a : α} :
@nhds α (Inf s) a = (⨅t∈s, @nhds α t a) := (gc_nhds a).u_Inf
lemma nhds_inf {t₁ t₂ : topological_space α} {a : α} :
@nhds α (t₁ ⊓ t₂) a = @nhds α t₁ a ⊓ @nhds α t₂ a := (gc_nhds a).u_inf
lemma nhds_top {a : α} : @nhds α ⊤ a = ⊤ := (gc_nhds a).u_top
local notation `cont` := @continuous _ _
local notation `tspace` := topological_space
open topological_space
variables {γ : Type*} {f : α → β} {ι : Sort*}
lemma continuous_iff_coinduced_le {t₁ : tspace α} {t₂ : tspace β} :
cont t₁ t₂ f ↔ coinduced f t₁ ≤ t₂ :=
continuous_def.trans iff.rfl
lemma continuous_iff_le_induced {t₁ : tspace α} {t₂ : tspace β} :
cont t₁ t₂ f ↔ t₁ ≤ induced f t₂ :=
iff.trans continuous_iff_coinduced_le (gc_coinduced_induced f _ _)
theorem continuous_generated_from {t : tspace α} {b : set (set β)}
(h : ∀s∈b, is_open (f ⁻¹' s)) : cont t (generate_from b) f :=
continuous_iff_coinduced_le.2 $ le_generate_from h
@[continuity]
lemma continuous_induced_dom {t : tspace β} : cont (induced f t) t f :=
by { rw continuous_def, assume s h, exact ⟨_, h, rfl⟩ }
lemma continuous_induced_rng {g : γ → α} {t₂ : tspace β} {t₁ : tspace γ}
(h : cont t₁ t₂ (f ∘ g)) : cont t₁ (induced f t₂) g :=
begin
rw continuous_def,
rintros s ⟨t, ht, s_eq⟩,
simpa [← s_eq] using continuous_def.1 h t ht,
end
lemma continuous_induced_rng' [topological_space α] [topological_space β] [topological_space γ]
{g : γ → α} (f : α → β) (H : ‹topological_space α› = ‹topological_space β›.induced f)
(h : continuous (f ∘ g)) : continuous g :=
H.symm ▸ continuous_induced_rng h
lemma continuous_coinduced_rng {t : tspace α} : cont t (coinduced f t) f :=
by { rw continuous_def, assume s h, exact h }
lemma continuous_coinduced_dom {g : β → γ} {t₁ : tspace α} {t₂ : tspace γ}
(h : cont t₁ t₂ (g ∘ f)) : cont (coinduced f t₁) t₂ g :=
begin
rw continuous_def at h ⊢,
assume s hs,
exact h _ hs
end
lemma continuous_le_dom {t₁ t₂ : tspace α} {t₃ : tspace β}
(h₁ : t₂ ≤ t₁) (h₂ : cont t₁ t₃ f) : cont t₂ t₃ f :=
begin
rw continuous_def at h₂ ⊢,
assume s h,
exact h₁ _ (h₂ s h)
end
lemma continuous_le_rng {t₁ : tspace α} {t₂ t₃ : tspace β}
(h₁ : t₂ ≤ t₃) (h₂ : cont t₁ t₂ f) : cont t₁ t₃ f :=
begin
rw continuous_def at h₂ ⊢,
assume s h,
exact h₂ s (h₁ s h)
end
lemma continuous_sup_dom {t₁ t₂ : tspace α} {t₃ : tspace β}
(h₁ : cont t₁ t₃ f) (h₂ : cont t₂ t₃ f) : cont (t₁ ⊔ t₂) t₃ f :=
begin
rw continuous_def at h₁ h₂ ⊢,
assume s h,
exact ⟨h₁ s h, h₂ s h⟩
end
lemma continuous_sup_rng_left {t₁ : tspace α} {t₃ t₂ : tspace β} :
cont t₁ t₂ f → cont t₁ (t₂ ⊔ t₃) f :=
continuous_le_rng le_sup_left
lemma continuous_sup_rng_right {t₁ : tspace α} {t₃ t₂ : tspace β} :
cont t₁ t₃ f → cont t₁ (t₂ ⊔ t₃) f :=
continuous_le_rng le_sup_right
lemma continuous_Sup_dom {t₁ : set (tspace α)} {t₂ : tspace β}
(h : ∀t∈t₁, cont t t₂ f) : cont (Sup t₁) t₂ f :=
continuous_iff_le_induced.2 $ Sup_le $ assume t ht, continuous_iff_le_induced.1 $ h t ht
lemma continuous_Sup_rng {t₁ : tspace α} {t₂ : set (tspace β)} {t : tspace β}
(h₁ : t ∈ t₂) (hf : cont t₁ t f) : cont t₁ (Sup t₂) f :=
continuous_iff_coinduced_le.2 $ le_Sup_of_le h₁ $ continuous_iff_coinduced_le.1 hf
lemma continuous_supr_dom {t₁ : ι → tspace α} {t₂ : tspace β}
(h : ∀i, cont (t₁ i) t₂ f) : cont (supr t₁) t₂ f :=
continuous_Sup_dom $ assume t ⟨i, (t_eq : t₁ i = t)⟩, t_eq ▸ h i
lemma continuous_supr_rng {t₁ : tspace α} {t₂ : ι → tspace β} {i : ι}
(h : cont t₁ (t₂ i) f) : cont t₁ (supr t₂) f :=
continuous_Sup_rng ⟨i, rfl⟩ h
lemma continuous_inf_rng {t₁ : tspace α} {t₂ t₃ : tspace β}
(h₁ : cont t₁ t₂ f) (h₂ : cont t₁ t₃ f) : cont t₁ (t₂ ⊓ t₃) f :=
continuous_iff_coinduced_le.2 $ le_inf
(continuous_iff_coinduced_le.1 h₁)
(continuous_iff_coinduced_le.1 h₂)
lemma continuous_inf_dom_left {t₁ t₂ : tspace α} {t₃ : tspace β} :
cont t₁ t₃ f → cont (t₁ ⊓ t₂) t₃ f :=
continuous_le_dom inf_le_left
lemma continuous_inf_dom_right {t₁ t₂ : tspace α} {t₃ : tspace β} :
cont t₂ t₃ f → cont (t₁ ⊓ t₂) t₃ f :=
continuous_le_dom inf_le_right
lemma continuous_Inf_dom {t₁ : set (tspace α)} {t₂ : tspace β} {t : tspace α} (h₁ : t ∈ t₁) :
cont t t₂ f → cont (Inf t₁) t₂ f :=
continuous_le_dom $ Inf_le h₁
lemma continuous_Inf_rng {t₁ : tspace α} {t₂ : set (tspace β)}
(h : ∀t∈t₂, cont t₁ t f) : cont t₁ (Inf t₂) f :=
continuous_iff_coinduced_le.2 $ le_Inf $ assume b hb, continuous_iff_coinduced_le.1 $ h b hb
lemma continuous_infi_dom {t₁ : ι → tspace α} {t₂ : tspace β} {i : ι} :
cont (t₁ i) t₂ f → cont (infi t₁) t₂ f :=
continuous_le_dom $ infi_le _ _
lemma continuous_infi_rng {t₁ : tspace α} {t₂ : ι → tspace β}
(h : ∀i, cont t₁ (t₂ i) f) : cont t₁ (infi t₂) f :=
continuous_iff_coinduced_le.2 $ le_infi $ assume i, continuous_iff_coinduced_le.1 $ h i
@[continuity] lemma continuous_bot {t : tspace β} : cont ⊥ t f :=
continuous_iff_le_induced.2 $ bot_le
@[continuity] lemma continuous_top {t : tspace α} : cont t ⊤ f :=
continuous_iff_coinduced_le.2 $ le_top
/- 𝓝 in the induced topology -/
theorem mem_nhds_induced [T : topological_space α] (f : β → α) (a : β) (s : set β) :
s ∈ @nhds β (topological_space.induced f T) a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s :=
begin
simp only [mem_nhds_sets_iff, is_open_induced_iff, exists_prop, set.mem_set_of_eq],
split,
{ rintros ⟨u, usub, ⟨v, openv, ueq⟩, au⟩,
exact ⟨v, ⟨v, set.subset.refl v, openv, by rwa ←ueq at au⟩, by rw ueq; exact usub⟩ },
rintros ⟨u, ⟨v, vsubu, openv, amem⟩, finvsub⟩,
exact ⟨f ⁻¹' v, set.subset.trans (set.preimage_mono vsubu) finvsub, ⟨⟨v, openv, rfl⟩, amem⟩⟩
end
theorem nhds_induced [T : topological_space α] (f : β → α) (a : β) :
@nhds β (topological_space.induced f T) a = comap f (𝓝 (f a)) :=
by { ext s, rw [mem_nhds_induced, mem_comap_sets] }
lemma induced_iff_nhds_eq [tα : topological_space α] [tβ : topological_space β] (f : β → α) :
tβ = tα.induced f ↔ ∀ b, 𝓝 b = comap f (𝓝 $ f b) :=
⟨λ h a, h.symm ▸ nhds_induced f a, λ h, eq_of_nhds_eq_nhds $ λ x, by rw [h, nhds_induced]⟩
theorem map_nhds_induced_of_surjective [T : topological_space α]
{f : β → α} (hf : function.surjective f) (a : β) :
map f (@nhds β (topological_space.induced f T) a) = 𝓝 (f a) :=
by rw [nhds_induced, map_comap_of_surjective hf]
end constructions
section induced
open topological_space
variables {α : Type*} {β : Type*}
variables [t : topological_space β] {f : α → β}
theorem is_open_induced_eq {s : set α} :
@is_open _ (induced f t) s ↔ s ∈ preimage f '' {s | is_open s} :=
iff.rfl
theorem is_open_induced {s : set β} (h : is_open s) : (induced f t).is_open (f ⁻¹' s) :=
⟨s, h, rfl⟩
lemma map_nhds_induced_eq (a : α) : map f (@nhds α (induced f t) a) = 𝓝[range f] (f a) :=
by rw [nhds_induced, filter.map_comap, nhds_within]
lemma map_nhds_induced_of_mem {a : α} (h : range f ∈ 𝓝 (f a)) :
map f (@nhds α (induced f t) a) = 𝓝 (f a) :=
by rw [nhds_induced, filter.map_comap_of_mem h]
lemma closure_induced [t : topological_space β] {f : α → β} {a : α} {s : set α} :
a ∈ @closure α (topological_space.induced f t) s ↔ f a ∈ closure (f '' s) :=
by simp only [mem_closure_iff_frequently, nhds_induced, frequently_comap, mem_image, and_comm]
end induced
section sierpinski
variables {α : Type*} [topological_space α]
@[simp] lemma is_open_singleton_true : is_open ({true} : set Prop) :=
topological_space.generate_open.basic _ (by simp)
lemma continuous_Prop {p : α → Prop} : continuous p ↔ is_open {x | p x} :=
⟨assume h : continuous p,
have is_open (p ⁻¹' {true}),
from is_open_singleton_true.preimage h,
by simp [preimage, eq_true] at this; assumption,
assume h : is_open {x | p x},
continuous_generated_from $ assume s (hs : s ∈ {{true}}),
by simp at hs; simp [hs, preimage, eq_true, h]⟩
lemma is_open_iff_continuous_mem {s : set α} : is_open s ↔ continuous (λ x, x ∈ s) :=
continuous_Prop.symm
end sierpinski
section infi
variables {α : Type u} {ι : Type v} {t : ι → topological_space α}
lemma is_open_supr_iff {s : set α} : @is_open _ (⨆ i, t i) s ↔ ∀ i, @is_open _ (t i) s :=
begin
-- s defines a map from α to Prop, which is continuous iff s is open.
suffices : @continuous _ _ (⨆ i, t i) _ s ↔ ∀ i, @continuous _ _ (t i) _ s,
{ simpa only [continuous_Prop] using this },
simp only [continuous_iff_le_induced, supr_le_iff]
end
lemma is_closed_infi_iff {s : set α} : @is_closed _ (⨆ i, t i) s ↔ ∀ i, @is_closed _ (t i) s :=
by simp [← is_open_compl_iff, is_open_supr_iff]
end infi
|
24a67cba52012d2f711a8ab325119a567eefb55a | 60bf3fa4185ec5075eaea4384181bfbc7e1dc319 | /src/game/sup_inf/unbdd_iff.lean | 4b961cc170cd69d9463cb5c0cb5fab6db032d8be | [
"Apache-2.0"
] | permissive | anrddh/real-number-game | 660f1127d03a78fd35986c771d65c3132c5f4025 | c708c4e02ec306c657e1ea67862177490db041b0 | refs/heads/master | 1,668,214,277,092 | 1,593,105,075,000 | 1,593,105,075,000 | 264,269,218 | 0 | 0 | null | 1,589,567,264,000 | 1,589,567,264,000 | null | UTF-8 | Lean | false | false | 1,516 | lean | import data.real.basic
namespace xena -- hide
/-
# Chapter 3 : Sup and Inf
## Level 12
-/
def unboundedAbove (A : set ℝ) := ∀ x : ℝ, x > 0 → ∃ a ∈ A, x < a
-- Might want to make this into an axiom to be placed on the left
def archimPrinciple := ∀ x : ℝ, x > 0 → ∃ n : ℕ, n > 0 ∧ (1/n : ℝ) < x
/- Lemma
The Archimedean principle is equivalent to the set of natural numbers being unbounded above.
-/
lemma nats_unbounded_iff :
unboundedAbove {x : ℝ | ∃ n : ℕ, x = n ∧ n > 0} ↔ archimPrinciple :=
begin
split,
-- left-right implication
intros unb x hx,
set A := {x : ℝ | ∃ n : ℕ, x = n ∧ x > 0} with hA,
have h1x : (1/x) > 0, from one_div_pos_of_pos hx,
have h1 := unb (1/x) h1x,
-- rcases h1 with ⟨nx, ⟨n, rfl⟩, h2⟩,
cases h1 with nx hnx,
cases hnx with h1 h2,
cases h1 with xn h12,
existsi xn, rw h12.left at h2,
have h0 : 0 < (1:ℝ), norm_num,
have h3 := div_lt_div_of_pos_of_lt_of_pos h1x h2 h0,
split, exact h12.right,
simp at h3, simp,
exact h3,
-- right-left implication
intros arc x hx,
have h1x : (1/x) > 0, from one_div_pos_of_pos hx,
have h1 := arc (1/x) h1x,
cases h1 with n hn,
use n, split,
existsi n, split, refl, exact hn.left,
set xn : ℝ := ↑n with hxn,
have h2n : 0 < xn,
rw hxn,
have hn0 : 0 < n, from hn.left, simp, assumption,
exact lt_of_one_div_lt_one_div h2n hn.right,
done
end
end xena -- hide
|
bfd0ef6725110e9decee3f472f3c44de7662db07 | 36938939954e91f23dec66a02728db08a7acfcf9 | /lean/deps/galois_stdlib/src/galois/data/sigma.lean | 976bf9ad6e82ab215fa60cf522a67d3dbb05d065 | [
"Apache-2.0"
] | permissive | pnwamk/reopt-vcg | f8b56dd0279392a5e1c6aee721be8138e6b558d3 | c9f9f185fbefc25c36c4b506bbc85fd1a03c3b6d | refs/heads/master | 1,631,145,017,772 | 1,593,549,019,000 | 1,593,549,143,000 | 254,191,418 | 0 | 0 | null | 1,586,377,077,000 | 1,586,377,077,000 | null | UTF-8 | Lean | false | false | 656 | lean |
namespace galois
namespace sigma
variables {α : Type*} {β : α → Type*}
-- from matlib
instance [h₁ : decidable_eq α] [h₂ : ∀a, decidable_eq (β a)] : decidable_eq (sigma β)
| ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ := match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, is_true (eq.refl a) :=
match b₁, b₂, h₂ a b₁ b₂ with
| _, _, is_true (eq.refl b) := is_true rfl
| b₁, b₂, is_false n := is_false (assume h, sigma.no_confusion h (λe₁ e₂, n $ eq_of_heq e₂))
end
| a₁, _, a₂, _, is_false n := is_false (assume h, sigma.no_confusion h (λe₁ e₂, n e₁))
end
end sigma
end galois
|
3e39a1d9e0e3c3d322afc42c45a5649ddbf28a88 | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/group_theory/monoid_localization.lean | 065cf9e382305504fc79249d59e009de1f0ed21f | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 49,353 | lean | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import group_theory.congruence
import algebra.group.units
import algebra.punit_instances
/-!
# Localizations of commutative monoids
Localizing a commutative ring at one of its submonoids does not rely on the ring's addition, so
we can generalize localizations to commutative monoids.
We characterize the localization of a commutative monoid `M` at a submonoid `S` up to
isomorphism; that is, a commutative monoid `N` is the localization of `M` at `S` iff we can find a
monoid homomorphism `f : M →* N` satisfying 3 properties:
1. For all `y ∈ S`, `f y` is a unit;
2. For all `z : N`, there exists `(x, y) : M × S` such that `z * f y = f x`;
3. For all `x, y : M`, `f x = f y` iff there exists `c ∈ S` such that `x * c = y * c`.
Given such a localization map `f : M →* N`, we can define the surjection
`localization_map.mk'` sending `(x, y) : M × S` to `f x * (f y)⁻¹`, and
`localization_map.lift`, the homomorphism from `N` induced by a homomorphism from `M` which maps
elements of `S` to invertible elements of the codomain. Similarly, given commutative monoids
`P, Q`, a submonoid `T` of `P` and a localization map for `T` from `P` to `Q`, then a homomorphism
`g : M →* P` such that `g(S) ⊆ T` induces a homomorphism of localizations,
`localization_map.map`, from `N` to `Q`.
We also define the quotient of `M × S` by the unique congruence relation (equivalence relation
preserving a binary operation) `r` such that for any other congruence relation `s` on `M × S`
satisfying '`∀ y ∈ S`, `(1, 1) ∼ (y, y)` under `s`', we have that `(x₁, y₁) ∼ (x₂, y₂)` by `s`
whenever `(x₁, y₁) ∼ (x₂, y₂)` by `r`. We show this relation is equivalent to the standard
localization relation.
This defines the localization as a quotient type, `localization`, but the majority of
subsequent lemmas in the file are given in terms of localizations up to isomorphism, using maps
which satisfy the characteristic predicate.
## Implementation notes
In maths it is natural to reason up to isomorphism, but in Lean we cannot naturally `rewrite` one
structure with an isomorphic one; one way around this is to isolate a predicate characterizing
a structure up to isomorphism, and reason about things that satisfy the predicate.
The infimum form of the localization congruence relation is chosen as 'canonical' here, since it
shortens some proofs.
To apply a localization map `f` as a function, we use `f.to_map`, as coercions don't work well for
this structure.
To reason about the localization as a quotient type, use `mk_eq_monoid_of_mk'` and associated
lemmas. These show the quotient map `mk : M → S → localization S` equals the
surjection `localization_map.mk'` induced by the map
`monoid_of : localization_map S (localization S)` (where `of` establishes the
localization as a quotient type satisfies the characteristic predicate). The lemma
`mk_eq_monoid_of_mk'` hence gives you access to the results in the rest of the file, which are
about the `localization_map.mk'` induced by any localization map.
## Tags
localization, monoid localization, quotient monoid, congruence relation, characteristic predicate,
commutative monoid
-/
set_option old_structure_cmd true
namespace add_submonoid
variables {M : Type*} [add_comm_monoid M] (S : add_submonoid M) (N : Type*) [add_comm_monoid N]
/-- The type of add_monoid homomorphisms satisfying the characteristic predicate: if `f : M →+ N`
satisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/
@[nolint has_inhabited_instance] structure localization_map
extends add_monoid_hom M N :=
(map_add_units' : ∀ y : S, is_add_unit (to_fun y))
(surj' : ∀ z : N, ∃ x : M × S, z + to_fun x.2 = to_fun x.1)
(eq_iff_exists' : ∀ x y, to_fun x = to_fun y ↔ ∃ c : S, x + c = y + c)
/-- The add_monoid hom underlying a `localization_map` of `add_comm_monoid`s. -/
add_decl_doc localization_map.to_add_monoid_hom
end add_submonoid
variables {M : Type*} [comm_monoid M] (S : submonoid M) (N : Type*) [comm_monoid N]
{P : Type*} [comm_monoid P]
namespace submonoid
/-- The type of monoid homomorphisms satisfying the characteristic predicate: if `f : M →* N`
satisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/
@[nolint has_inhabited_instance] structure localization_map
extends monoid_hom M N :=
(map_units' : ∀ y : S, is_unit (to_fun y))
(surj' : ∀ z : N, ∃ x : M × S, z * to_fun x.2 = to_fun x.1)
(eq_iff_exists' : ∀ x y, to_fun x = to_fun y ↔ ∃ c : S, x * c = y * c)
attribute [to_additive add_submonoid.localization_map] submonoid.localization_map
attribute [to_additive add_submonoid.localization_map.to_add_monoid_hom]
submonoid.localization_map.to_monoid_hom
/-- The monoid hom underlying a `localization_map`. -/
add_decl_doc localization_map.to_monoid_hom
end submonoid
namespace localization
run_cmd to_additive.map_namespace `localization `add_localization
/-- The congruence relation on `M × S`, `M` a `comm_monoid` and `S` a submonoid of `M`, whose
quotient is the localization of `M` at `S`, defined as the unique congruence relation on
`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,
`(1, 1) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies
`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/
@[to_additive "The congruence relation on `M × S`, `M` an `add_comm_monoid` and `S`
an `add_submonoid` of `M`, whose quotient is the localization of `M` at `S`, defined as the unique
congruence relation on `M × S` such that for any other congruence relation `s` on `M × S` where
for all `y ∈ S`, `(0, 0) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies
`(x₁, y₁) ∼ (x₂, y₂)` by `s`."]
def r (S : submonoid M) : con (M × S) :=
Inf {c | ∀ y : S, c 1 (y, y)}
/-- An alternate form of the congruence relation on `M × S`, `M` a `comm_monoid` and `S` a
submonoid of `M`, whose quotient is the localization of `M` at `S`. -/
@[to_additive "An alternate form of the congruence relation on `M × S`, `M` a `comm_monoid` and
`S` a submonoid of `M`, whose quotient is the localization of `M` at `S`."]
def r' : con (M × S) :=
begin
refine { r := λ a b : M × S, ∃ c : S, a.1 * b.2 * c = b.1 * a.2 * c,
iseqv := ⟨λ a, ⟨1, rfl⟩, λ a b ⟨c, hc⟩, ⟨c, hc.symm⟩, _⟩,
.. },
{ rintros a b c ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩,
use b.2 * t₁ * t₂,
simp only [submonoid.coe_mul],
calc a.1 * c.2 * (b.2 * t₁ * t₂) = a.1 * b.2 * t₁ * c.2 * t₂ : by ac_refl
... = b.1 * c.2 * t₂ * a.2 * t₁ : by { rw ht₁, ac_refl }
... = c.1 * a.2 * (b.2 * t₁ * t₂) : by { rw ht₂, ac_refl } },
{ rintros a b c d ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩,
use t₁ * t₂,
calc (a.1 * c.1) * (b.2 * d.2) * (t₁ * t₂) = (a.1 * b.2 * t₁) * (c.1 * d.2 * t₂) :
by ac_refl
... = (b.1 * d.1) * (a.2 * c.2) * (t₁ * t₂) : by { rw [ht₁, ht₂], ac_refl } }
end
/-- The congruence relation used to localize a `comm_monoid` at a submonoid can be expressed
equivalently as an infimum (see `localization.r`) or explicitly
(see `localization.r'`). -/
@[to_additive "The additive congruence relation used to localize an `add_comm_monoid` at a
submonoid can be expressed equivalently as an infimum (see `add_localization.r`) or
explicitly (see `add_localization.r'`)."]
theorem r_eq_r' : r S = r' S :=
le_antisymm (Inf_le $ λ _, ⟨1, by simp⟩) $
le_Inf $ λ b H ⟨p, q⟩ y ⟨t, ht⟩,
begin
rw [← mul_one (p, q), ← mul_one y],
refine b.trans (b.mul (b.refl _) (H (y.2 * t))) _,
convert b.symm (b.mul (b.refl y) (H (q * t))) using 1,
rw [prod.mk_mul_mk, submonoid.coe_mul, ← mul_assoc, ht, mul_left_comm, mul_assoc],
refl
end
variables {S}
@[to_additive]
lemma r_iff_exists {x y : M × S} : r S x y ↔ ∃ c : S, x.1 * y.2 * c = y.1 * x.2 * c :=
by rw r_eq_r' S; refl
end localization
/-- The localization of a `comm_monoid` at one of its submonoids (as a quotient type). -/
@[to_additive add_localization "The localization of an `add_comm_monoid` at one
of its submonoids (as a quotient type)."]
def localization := (localization.r S).quotient
namespace localization
@[to_additive] instance inhabited :
inhabited (localization S) :=
con.quotient.inhabited
@[to_additive] instance : comm_monoid (localization S) :=
(r S).comm_monoid
variables {S}
/-- Given a `comm_monoid` `M` and submonoid `S`, `mk` sends `x : M`, `y ∈ S` to the equivalence
class of `(x, y)` in the localization of `M` at `S`. -/
@[to_additive "Given an `add_comm_monoid` `M` and submonoid `S`, `mk` sends `x : M`, `y ∈ S` to
the equivalence class of `(x, y)` in the localization of `M` at `S`."]
def mk (x : M) (y : S) : localization S := (r S).mk' (x, y)
@[elab_as_eliminator, to_additive]
theorem ind {p : localization S → Prop}
(H : ∀ (y : M × S), p (mk y.1 y.2)) (x) : p x :=
by rcases x; convert H x; exact prod.mk.eta.symm
@[elab_as_eliminator, to_additive]
theorem induction_on {p : localization S → Prop} (x)
(H : ∀ (y : M × S), p (mk y.1 y.2)) : p x := ind H x
@[elab_as_eliminator, to_additive]
theorem induction_on₂ {p : localization S → localization S → Prop} (x y)
(H : ∀ (x y : M × S), p (mk x.1 x.2) (mk y.1 y.2)) : p x y :=
induction_on x $ λ x, induction_on y $ H x
@[elab_as_eliminator, to_additive]
theorem induction_on₃
{p : localization S → localization S → localization S → Prop} (x y z)
(H : ∀ (x y z : M × S), p (mk x.1 x.2) (mk y.1 y.2) (mk z.1 z.2)) : p x y z :=
induction_on₂ x y $ λ x y, induction_on z $ H x y
@[to_additive] lemma one_rel (y : S) : r S 1 (y, y) := λ b hb, hb y
@[to_additive] theorem r_of_eq {x y : M × S} (h : y.1 * x.2 = x.1 * y.2) : r S x y :=
r_iff_exists.2 ⟨1, by rw h⟩
end localization
variables {S N}
namespace monoid_hom
/-- Makes a localization map from a `comm_monoid` hom satisfying the characteristic predicate. -/
@[to_additive "Makes a localization map from an `add_comm_monoid` hom satisfying the characteristic
predicate."]
def to_localization_map (f : M →* N) (H1 : ∀ y : S, is_unit (f y))
(H2 : ∀ z, ∃ x : M × S, z * f x.2 = f x.1) (H3 : ∀ x y, f x = f y ↔ ∃ c : S, x * c = y * c) :
submonoid.localization_map S N :=
{ map_units' := H1,
surj' := H2,
eq_iff_exists' := H3,
.. f }
end monoid_hom
namespace submonoid
namespace localization_map
/-- Short for `to_monoid_hom`; used to apply a localization map as a function. -/
@[to_additive "Short for `to_add_monoid_hom`; used to apply a localization map as a function."]
abbreviation to_map (f : localization_map S N) := f.to_monoid_hom
@[to_additive, ext] lemma ext {f g : localization_map S N} (h : ∀ x, f.to_map x = g.to_map x) :
f = g :=
by cases f; cases g; simp only; exact funext h
attribute [ext] add_submonoid.localization_map.ext
@[to_additive] lemma ext_iff {f g : localization_map S N} :
f = g ↔ ∀ x, f.to_map x = g.to_map x :=
⟨λ h x, h ▸ rfl, ext⟩
@[to_additive] lemma to_map_injective :
function.injective (@localization_map.to_map _ _ S N _) :=
λ _ _ h, ext $ monoid_hom.ext_iff.1 h
@[to_additive] lemma map_units (f : localization_map S N) (y : S) :
is_unit (f.to_map y) := f.4 y
@[to_additive] lemma surj (f : localization_map S N) (z : N) :
∃ x : M × S, z * f.to_map x.2 = f.to_map x.1 := f.5 z
@[to_additive] lemma eq_iff_exists (f : localization_map S N) {x y} :
f.to_map x = f.to_map y ↔ ∃ c : S, x * c = y * c := f.6 x y
/-- Given a localization map `f : M →* N`, a section function sending `z : N` to some
`(x, y) : M × S` such that `f x * (f y)⁻¹ = z`. -/
@[to_additive "Given a localization map `f : M →+ N`, a section function sending `z : N`
to some `(x, y) : M × S` such that `f x - f y = z`."]
noncomputable def sec (f : localization_map S N) (z : N) : M × S :=
classical.some $ f.surj z
@[to_additive] lemma sec_spec {f : localization_map S N} (z : N) :
z * f.to_map (f.sec z).2 = f.to_map (f.sec z).1 :=
classical.some_spec $ f.surj z
@[to_additive] lemma sec_spec' {f : localization_map S N} (z : N) :
f.to_map (f.sec z).1 = f.to_map (f.sec z).2 * z :=
by rw [mul_comm, sec_spec]
/-- Given a monoid hom `f : M →* N` and submonoid `S ⊆ M` such that `f(S) ⊆ units N`, for all
`w : M, z : N` and `y ∈ S`, we have `w * (f y)⁻¹ = z ↔ w = f y * z`. -/
@[to_additive "Given an add_monoid hom `f : M →+ N` and submonoid `S ⊆ M` such that
`f(S) ⊆ add_units N`, for all `w : M, z : N` and `y ∈ S`, we have `w - f y = z ↔ w = f y + z`."]
lemma mul_inv_left {f : M →* N} (h : ∀ y : S, is_unit (f y))
(y : S) (w z) : w * ↑(is_unit.lift_right (f.mrestrict S) h y)⁻¹ = z ↔ w = f y * z :=
by rw mul_comm; convert units.inv_mul_eq_iff_eq_mul _;
exact (is_unit.coe_lift_right (f.mrestrict S) h _).symm
/-- Given a monoid hom `f : M →* N` and submonoid `S ⊆ M` such that `f(S) ⊆ units N`, for all
`w : M, z : N` and `y ∈ S`, we have `z = w * (f y)⁻¹ ↔ z * f y = w`. -/
@[to_additive "Given an add_monoid hom `f : M →+ N` and submonoid `S ⊆ M` such that
`f(S) ⊆ add_units N`, for all `w : M, z : N` and `y ∈ S`, we have `z = w - f y ↔ z + f y = w`."]
lemma mul_inv_right {f : M →* N} (h : ∀ y : S, is_unit (f y))
(y : S) (w z) : z = w * ↑(is_unit.lift_right (f.mrestrict S) h y)⁻¹ ↔ z * f y = w :=
by rw [eq_comm, mul_inv_left h, mul_comm, eq_comm]
/-- Given a monoid hom `f : M →* N` and submonoid `S ⊆ M` such that
`f(S) ⊆ units N`, for all `x₁ x₂ : M` and `y₁, y₂ ∈ S`, we have
`f x₁ * (f y₁)⁻¹ = f x₂ * (f y₂)⁻¹ ↔ f (x₁ * y₂) = f (x₂ * y₁)`. -/
@[simp, to_additive "Given an add_monoid hom `f : M →+ N` and submonoid `S ⊆ M` such that
`f(S) ⊆ add_units N`, for all `x₁ x₂ : M` and `y₁, y₂ ∈ S`, we have
`f x₁ - f y₁ = f x₂ - f y₂ ↔ f (x₁ + y₂) = f (x₂ + y₁)`."]
lemma mul_inv {f : M →* N} (h : ∀ y : S, is_unit (f y)) {x₁ x₂} {y₁ y₂ : S} :
f x₁ * ↑(is_unit.lift_right (f.mrestrict S) h y₁)⁻¹ =
f x₂ * ↑(is_unit.lift_right (f.mrestrict S) h y₂)⁻¹ ↔ f (x₁ * y₂) = f (x₂ * y₁) :=
by rw [mul_inv_right h, mul_assoc, mul_comm _ (f y₂), ←mul_assoc, mul_inv_left h, mul_comm x₂,
f.map_mul, f.map_mul]
/-- Given a monoid hom `f : M →* N` and submonoid `S ⊆ M` such that `f(S) ⊆ units N`, for all
`y, z ∈ S`, we have `(f y)⁻¹ = (f z)⁻¹ → f y = f z`. -/
@[to_additive "Given an add_monoid hom `f : M →+ N` and submonoid `S ⊆ M` such that
`f(S) ⊆ add_units N`, for all `y, z ∈ S`, we have `- (f y) = - (f z) → f y = f z`."]
lemma inv_inj {f : M →* N} (hf : ∀ y : S, is_unit (f y)) {y z}
(h : (is_unit.lift_right (f.mrestrict S) hf y)⁻¹ = (is_unit.lift_right (f.mrestrict S) hf z)⁻¹) :
f y = f z :=
by rw [←mul_one (f y), eq_comm, ←mul_inv_left hf y (f z) 1, h];
convert units.inv_mul _; exact (is_unit.coe_lift_right (f.mrestrict S) hf _).symm
/-- Given a monoid hom `f : M →* N` and submonoid `S ⊆ M` such that `f(S) ⊆ units N`, for all
`y ∈ S`, `(f y)⁻¹` is unique. -/
@[to_additive "Given an add_monoid hom `f : M →+ N` and submonoid `S ⊆ M` such that
`f(S) ⊆ add_units N`, for all `y ∈ S`, `- (f y)` is unique."]
lemma inv_unique {f : M →* N} (h : ∀ y : S, is_unit (f y)) {y : S}
{z} (H : f y * z = 1) : ↑(is_unit.lift_right (f.mrestrict S) h y)⁻¹ = z :=
by rw [←one_mul ↑(_)⁻¹, mul_inv_left, ←H]
variables (f : localization_map S N)
@[to_additive] lemma map_right_cancel {x y} {c : S} (h : f.to_map (c * x) = f.to_map (c * y)) :
f.to_map x = f.to_map y :=
begin
rw [f.to_map.map_mul, f.to_map.map_mul] at h,
cases f.map_units c with u hu,
rw ←hu at h,
exact (units.mul_right_inj u).1 h,
end
@[to_additive] lemma map_left_cancel {x y} {c : S} (h : f.to_map (x * c) = f.to_map (y * c)) :
f.to_map x = f.to_map y :=
f.map_right_cancel $ by rw [mul_comm _ x, mul_comm _ y, h]
/-- Given a localization map `f : M →* N`, the surjection sending `(x, y) : M × S` to
`f x * (f y)⁻¹`. -/
@[to_additive "Given a localization map `f : M →+ N`, the surjection sending `(x, y) : M × S`
to `f x - f y`."]
noncomputable def mk' (f : localization_map S N) (x : M) (y : S) : N :=
f.to_map x * ↑(is_unit.lift_right (f.to_map.mrestrict S) f.map_units y)⁻¹
@[to_additive] lemma mk'_mul (x₁ x₂ : M) (y₁ y₂ : S) :
f.mk' (x₁ * x₂) (y₁ * y₂) = f.mk' x₁ y₁ * f.mk' x₂ y₂ :=
(mul_inv_left f.map_units _ _ _).2 $
show _ = _ * (_ * _ * (_ * _)), by
rw [←mul_assoc, ←mul_assoc, mul_inv_right f.map_units, mul_assoc, mul_assoc,
mul_comm _ (f.to_map x₂), ←mul_assoc, ←mul_assoc, mul_inv_right f.map_units,
submonoid.coe_mul, f.to_map.map_mul, f.to_map.map_mul];
ac_refl
@[to_additive] lemma mk'_one (x) : f.mk' x (1 : S) = f.to_map x :=
by rw [mk', monoid_hom.map_one]; exact mul_one _
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M`, for all `z : N` we have that if
`x : M, y ∈ S` are such that `z * f y = f x`, then `f x * (f y)⁻¹ = z`. -/
@[simp, to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M`, for all `z : N`
we have that if `x : M, y ∈ S` are such that `z + f y = f x`, then `f x - f y = z`."]
lemma mk'_sec (z : N) : f.mk' (f.sec z).1 (f.sec z).2 = z :=
show _ * _ = _, by rw [←sec_spec, mul_inv_left, mul_comm]
@[to_additive] lemma mk'_surjective (z : N) : ∃ x (y : S), f.mk' x y = z :=
⟨(f.sec z).1, (f.sec z).2, f.mk'_sec z⟩
@[to_additive] lemma mk'_spec (x) (y : S) :
f.mk' x y * f.to_map y = f.to_map x :=
show _ * _ * _ = _, by rw [mul_assoc, mul_comm _ (f.to_map y), ←mul_assoc, mul_inv_left, mul_comm]
@[to_additive] lemma mk'_spec' (x) (y : S) :
f.to_map y * f.mk' x y = f.to_map x :=
by rw [mul_comm, mk'_spec]
@[to_additive] theorem eq_mk'_iff_mul_eq {x} {y : S} {z} :
z = f.mk' x y ↔ z * f.to_map y = f.to_map x :=
⟨λ H, by rw [H, mk'_spec], λ H, by erw [mul_inv_right, H]; refl⟩
@[to_additive] theorem mk'_eq_iff_eq_mul {x} {y : S} {z} :
f.mk' x y = z ↔ f.to_map x = z * f.to_map y :=
by rw [eq_comm, eq_mk'_iff_mul_eq, eq_comm]
@[to_additive] lemma mk'_eq_iff_eq {x₁ x₂} {y₁ y₂ : S} :
f.mk' x₁ y₁ = f.mk' x₂ y₂ ↔ f.to_map (x₁ * y₂) = f.to_map (x₂ * y₁) :=
⟨λ H, by rw [f.to_map.map_mul, f.mk'_eq_iff_eq_mul.1 H, mul_assoc,
mul_comm (f.to_map _), ←mul_assoc, mk'_spec, f.to_map.map_mul],
λ H, by rw [mk'_eq_iff_eq_mul, mk', mul_assoc, mul_comm _ (f.to_map y₁), ←mul_assoc,
←f.to_map.map_mul, ←H, f.to_map.map_mul, mul_inv_right f.map_units]⟩
@[to_additive] protected lemma eq {a₁ b₁} {a₂ b₂ : S} :
f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ ∃ c : S, a₁ * b₂ * c = b₁ * a₂ * c :=
f.mk'_eq_iff_eq.trans $ f.eq_iff_exists
@[to_additive] protected lemma eq' {a₁ b₁} {a₂ b₂ : S} :
f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ localization.r S (a₁, a₂) (b₁, b₂) :=
by rw [f.eq, localization.r_iff_exists]
@[to_additive] lemma eq_iff_eq (g : localization_map S P) {x y} :
f.to_map x = f.to_map y ↔ g.to_map x = g.to_map y :=
f.eq_iff_exists.trans g.eq_iff_exists.symm
@[to_additive] lemma mk'_eq_iff_mk'_eq (g : localization_map S P) {x₁ x₂}
{y₁ y₂ : S} : f.mk' x₁ y₁ = f.mk' x₂ y₂ ↔ g.mk' x₁ y₁ = g.mk' x₂ y₂ :=
f.eq'.trans g.eq'.symm
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M`, for all `x₁ : M` and `y₁ ∈ S`,
if `x₂ : M, y₂ ∈ S` are such that `f x₁ * (f y₁)⁻¹ * f y₂ = f x₂`, then there exists `c ∈ S`
such that `x₁ * y₂ * c = x₂ * y₁ * c`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M`, for all `x₁ : M`
and `y₁ ∈ S`, if `x₂ : M, y₂ ∈ S` are such that `(f x₁ - f y₁) + f y₂ = f x₂`, then there exists
`c ∈ S` such that `x₁ + y₂ + c = x₂ + y₁ + c`."]
lemma exists_of_sec_mk' (x) (y : S) :
∃ c : S, x * (f.sec $ f.mk' x y).2 * c = (f.sec $ f.mk' x y).1 * y * c :=
f.eq_iff_exists.1 $ f.mk'_eq_iff_eq.1 $ (mk'_sec _ _).symm
@[to_additive] lemma mk'_eq_of_eq {a₁ b₁ : M} {a₂ b₂ : S} (H : b₁ * a₂ = a₁ * b₂) :
f.mk' a₁ a₂ = f.mk' b₁ b₂ :=
f.mk'_eq_iff_eq.2 $ H ▸ rfl
@[simp, to_additive] lemma mk'_self (y : S) :
f.mk' (y : M) y = 1 :=
show _ * _ = _, by rw [mul_inv_left, mul_one]
@[simp, to_additive] lemma mk'_self' (x) (H : x ∈ S) :
f.mk' x ⟨x, H⟩ = 1 :=
by convert mk'_self _ _; refl
@[to_additive] lemma mul_mk'_eq_mk'_of_mul (x₁ x₂) (y : S) :
f.to_map x₁ * f.mk' x₂ y = f.mk' (x₁ * x₂) y :=
by rw [←mk'_one, ←mk'_mul, one_mul]
@[to_additive] lemma mk'_mul_eq_mk'_of_mul (x₁ x₂) (y : S) :
f.mk' x₂ y * f.to_map x₁ = f.mk' (x₁ * x₂) y :=
by rw [mul_comm, mul_mk'_eq_mk'_of_mul]
@[to_additive] lemma mul_mk'_one_eq_mk' (x) (y : S) :
f.to_map x * f.mk' 1 y = f.mk' x y :=
by rw [mul_mk'_eq_mk'_of_mul, mul_one]
@[simp, to_additive] lemma mk'_mul_cancel_right (x : M) (y : S) :
f.mk' (x * y) y = f.to_map x :=
by rw [←mul_mk'_one_eq_mk', f.to_map.map_mul, mul_assoc, mul_mk'_one_eq_mk', mk'_self, mul_one]
@[to_additive] lemma mk'_mul_cancel_left (x) (y : S) :
f.mk' ((y : M) * x) y = f.to_map x :=
by rw [mul_comm, mk'_mul_cancel_right]
@[to_additive] lemma is_unit_comp (j : N →* P) (y : S) :
is_unit (j.comp f.to_map y) :=
⟨units.map j $ is_unit.lift_right (f.to_map.mrestrict S) f.map_units y,
show j _ = j _, from congr_arg j $
(is_unit.coe_lift_right (f.to_map.mrestrict S) f.map_units _)⟩
variables {g : M →* P}
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M` and a map of `comm_monoid`s
`g : M →* P` such that `g(S) ⊆ units P`, `f x = f y → g x = g y` for all `x y : M`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M` and a map
of `add_comm_monoid`s `g : M →+ P` such that `g(S) ⊆ add_units P`, `f x = f y → g x = g y`
for all `x y : M`."]
lemma eq_of_eq (hg : ∀ y : S, is_unit (g y)) {x y} (h : f.to_map x = f.to_map y) :
g x = g y :=
begin
obtain ⟨c, hc⟩ := f.eq_iff_exists.1 h,
rw [←mul_one (g x), ←is_unit.mul_lift_right_inv (g.mrestrict S) hg c],
show _ * (g c * _) = _,
rw [←mul_assoc, ←g.map_mul, hc, mul_inv_left hg, g.map_mul, mul_comm],
end
/-- Given `comm_monoid`s `M, P`, localization maps `f : M →* N, k : P →* Q` for submonoids
`S, T` respectively, and `g : M →* P` such that `g(S) ⊆ T`, `f x = f y` implies
`k (g x) = k (g y)`. -/
@[to_additive "Given `add_comm_monoid`s `M, P`, localization maps `f : M →+ N, k : P →+ Q` for
submonoids `S, T` respectively, and `g : M →+ P` such that `g(S) ⊆ T`, `f x = f y`
implies `k (g x) = k (g y)`."]
lemma comp_eq_of_eq {T : submonoid P} {Q : Type*} [comm_monoid Q]
(hg : ∀ y : S, g y ∈ T) (k : localization_map T Q)
{x y} (h : f.to_map x = f.to_map y) : k.to_map (g x) = k.to_map (g y) :=
f.eq_of_eq (λ y : S, show is_unit (k.to_map.comp g y), from k.map_units ⟨g y, hg y⟩) h
variables (hg : ∀ y : S, is_unit (g y))
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M` and a map of `comm_monoid`s
`g : M →* P` such that `g y` is invertible for all `y : S`, the homomorphism induced from
`N` to `P` sending `z : N` to `g x * (g y)⁻¹`, where `(x, y) : M × S` are such that
`z = f x * (f y)⁻¹`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M` and a map
of `add_comm_monoid`s `g : M →+ P` such that `g y` is invertible for all `y : S`, the homomorphism
induced from `N` to `P` sending `z : N` to `g x - g y`, where `(x, y) : M × S` are such that
`z = f x - f y`."]
noncomputable def lift : N →* P :=
{ to_fun := λ z, g (f.sec z).1 * ↑(is_unit.lift_right (g.mrestrict S) hg (f.sec z).2)⁻¹,
map_one' := by rw [mul_inv_left, mul_one]; exact f.eq_of_eq hg
(by rw [←sec_spec, one_mul]),
map_mul' := λ x y,
begin
rw [mul_inv_left hg, ←mul_assoc, ←mul_assoc, mul_inv_right hg,
mul_comm _ (g (f.sec y).1), ←mul_assoc, ←mul_assoc, mul_inv_right hg],
repeat { rw ←g.map_mul },
exact f.eq_of_eq hg (by repeat { rw f.to_map.map_mul <|> rw sec_spec' }; ac_refl)
end }
variables {S g}
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M` and a map of `comm_monoid`s
`g : M →* P` such that `g y` is invertible for all `y : S`, the homomorphism induced from
`N` to `P` maps `f x * (f y)⁻¹` to `g x * (g y)⁻¹` for all `x : M, y ∈ S`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M` and a map
of `add_comm_monoid`s `g : M →+ P` such that `g y` is invertible for all `y : S`, the homomorphism
induced from `N` to `P` maps `f x - f y` to `g x - g y` for all `x : M, y ∈ S`."]
lemma lift_mk' (x y) :
f.lift hg (f.mk' x y) = g x * ↑(is_unit.lift_right (g.mrestrict S) hg y)⁻¹ :=
(mul_inv hg).2 $ f.eq_of_eq hg $ by
rw [f.to_map.map_mul, f.to_map.map_mul, sec_spec', mul_assoc, f.mk'_spec, mul_comm]
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M`, if a `comm_monoid` map
`g : M →* P` induces a map `f.lift hg : N →* P` then for all `z : N, v : P`, we have
`f.lift hg z = v ↔ g x = g y * v`, where `x : M, y ∈ S` are such that `z * f y = f x`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M`, if
an `add_comm_monoid` map `g : M →+ P` induces a map `f.lift hg : N →+ P` then for all
`z : N, v : P`, we have `f.lift hg z = v ↔ g x = g y + v`, where `x : M, y ∈ S` are such that
`z + f y = f x`."]
lemma lift_spec (z v) :
f.lift hg z = v ↔ g (f.sec z).1 = g (f.sec z).2 * v :=
mul_inv_left hg _ _ v
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M`, if a `comm_monoid` map
`g : M →* P` induces a map `f.lift hg : N →* P` then for all `z : N, v w : P`, we have
`f.lift hg z * w = v ↔ g x * w = g y * v`, where `x : M, y ∈ S` are such that
`z * f y = f x`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M`, if
an `add_comm_monoid` map `g : M →+ P` induces a map `f.lift hg : N →+ P` then for all
`z : N, v w : P`, we have `f.lift hg z + w = v ↔ g x + w = g y + v`, where `x : M, y ∈ S` are such
that `z + f y = f x`."]
lemma lift_spec_mul (z w v) :
f.lift hg z * w = v ↔ g (f.sec z).1 * w = g (f.sec z).2 * v :=
begin
rw mul_comm,
show _ * (_ * _) = _ ↔ _,
rw [←mul_assoc, mul_inv_left hg, mul_comm],
end
@[to_additive] lemma lift_mk'_spec (x v) (y : S) :
f.lift hg (f.mk' x y) = v ↔ g x = g y * v :=
by rw f.lift_mk' hg; exact mul_inv_left hg _ _ _
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M`, if a `comm_monoid` map
`g : M →* P` induces a map `f.lift hg : N →* P` then for all `z : N`, we have
`f.lift hg z * g y = g x`, where `x : M, y ∈ S` are such that `z * f y = f x`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M`, if
an `add_comm_monoid` map `g : M →+ P` induces a map `f.lift hg : N →+ P` then for all `z : N`, we
have `f.lift hg z + g y = g x`, where `x : M, y ∈ S` are such that `z + f y = f x`."]
lemma lift_mul_right (z) :
f.lift hg z * g (f.sec z).2 = g (f.sec z).1 :=
show _ * _ * _ = _, by erw [mul_assoc, is_unit.lift_right_inv_mul, mul_one]
/-- Given a localization map `f : M →* N` for a submonoid `S ⊆ M`, if a `comm_monoid` map
`g : M →* P` induces a map `f.lift hg : N →* P` then for all `z : N`, we have
`g y * f.lift hg z = g x`, where `x : M, y ∈ S` are such that `z * f y = f x`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S ⊆ M`, if
an `add_comm_monoid` map `g : M →+ P` induces a map `f.lift hg : N →+ P` then for all `z : N`, we
have `g y + f.lift hg z = g x`, where `x : M, y ∈ S` are such that `z + f y = f x`."]
lemma lift_mul_left (z) :
g (f.sec z).2 * f.lift hg z = g (f.sec z).1 :=
by rw [mul_comm, lift_mul_right]
@[simp, to_additive] lemma lift_eq (x : M) :
f.lift hg (f.to_map x) = g x :=
by rw [lift_spec, ←g.map_mul]; exact f.eq_of_eq hg (by rw [sec_spec', f.to_map.map_mul])
@[to_additive] lemma lift_eq_iff {x y : M × S} :
f.lift hg (f.mk' x.1 x.2) = f.lift hg (f.mk' y.1 y.2) ↔ g (x.1 * y.2) = g (y.1 * x.2) :=
by rw [lift_mk', lift_mk', mul_inv hg]
@[simp, to_additive] lemma lift_comp : (f.lift hg).comp f.to_map = g :=
by ext; exact f.lift_eq hg _
@[simp, to_additive] lemma lift_of_comp (j : N →* P) :
f.lift (f.is_unit_comp j) = j :=
begin
ext,
rw lift_spec,
show j _ = j _ * _,
erw [←j.map_mul, sec_spec'],
end
@[to_additive] lemma epic_of_localization_map {j k : N →* P}
(h : ∀ a, j.comp f.to_map a = k.comp f.to_map a) : j = k :=
begin
rw [←f.lift_of_comp j, ←f.lift_of_comp k],
congr' 1,
ext,
exact h x,
end
@[to_additive] lemma lift_unique {j : N →* P}
(hj : ∀ x, j (f.to_map x) = g x) : f.lift hg = j :=
begin
ext,
rw [lift_spec, ←hj, ←hj, ←j.map_mul],
apply congr_arg,
rw ←sec_spec',
end
@[simp, to_additive] lemma lift_id (x) : f.lift f.map_units x = x :=
monoid_hom.ext_iff.1 (f.lift_of_comp $ monoid_hom.id N) x
/-- Given two localization maps `f : M →* N, k : M →* P` for a submonoid `S ⊆ M`,
the hom from `P` to `N` induced by `f` is left inverse to the hom from `N` to `P`
induced by `k`. -/
@[simp, to_additive] lemma lift_left_inverse {k : localization_map S P} (z : N) :
k.lift f.map_units (f.lift k.map_units z) = z :=
begin
rw lift_spec,
cases f.surj z with x hx,
conv_rhs {congr, skip, rw f.eq_mk'_iff_mul_eq.2 hx},
rw [mk', ←mul_assoc, mul_inv_right f.map_units, ←f.to_map.map_mul, ←f.to_map.map_mul],
apply k.eq_of_eq f.map_units,
rw [k.to_map.map_mul, k.to_map.map_mul, ←sec_spec, mul_assoc, lift_spec_mul],
repeat { rw ←k.to_map.map_mul },
apply f.eq_of_eq k.map_units,
repeat { rw f.to_map.map_mul },
rw [sec_spec', ←hx],
ac_refl,
end
@[to_additive] lemma lift_surjective_iff :
function.surjective (f.lift hg) ↔ ∀ v : P, ∃ x : M × S, v * g x.2 = g x.1 :=
begin
split,
{ intros H v,
obtain ⟨z, hz⟩ := H v,
obtain ⟨x, hx⟩ := f.surj z,
use x,
rw [←hz, f.eq_mk'_iff_mul_eq.2 hx, lift_mk', mul_assoc, mul_comm _ (g ↑x.2)],
erw [is_unit.mul_lift_right_inv (g.mrestrict S) hg, mul_one] },
{ intros H v,
obtain ⟨x, hx⟩ := H v,
use f.mk' x.1 x.2,
rw [lift_mk', mul_inv_left hg, mul_comm, ←hx] }
end
@[to_additive] lemma lift_injective_iff :
function.injective (f.lift hg) ↔ ∀ x y, f.to_map x = f.to_map y ↔ g x = g y :=
begin
split,
{ intros H x y,
split,
{ exact f.eq_of_eq hg },
{ intro h,
rw [←f.lift_eq hg, ←f.lift_eq hg] at h,
exact H h }},
{ intros H z w h,
obtain ⟨x, hx⟩ := f.surj z,
obtain ⟨y, hy⟩ := f.surj w,
rw [←f.mk'_sec z, ←f.mk'_sec w],
exact (mul_inv f.map_units).2 ((H _ _).2 $ (mul_inv hg).1 h) }
end
variables {T : submonoid P} (hy : ∀ y : S, g y ∈ T) {Q : Type*} [comm_monoid Q]
(k : localization_map T Q)
/-- Given a `comm_monoid` homomorphism `g : M →* P` where for submonoids `S ⊆ M, T ⊆ P` we have
`g(S) ⊆ T`, the induced monoid homomorphism from the localization of `M` at `S` to the
localization of `P` at `T`: if `f : M →* N` and `k : P →* Q` are localization maps for `S` and
`T` respectively, we send `z : N` to `k (g x) * (k (g y))⁻¹`, where `(x, y) : M × S` are such
that `z = f x * (f y)⁻¹`. -/
@[to_additive "Given a `add_comm_monoid` homomorphism `g : M →+ P` where for submonoids
`S ⊆ M, T ⊆ P` we have `g(S) ⊆ T`, the induced add_monoid homomorphism from the localization of `M`
at `S` to the localization of `P` at `T`: if `f : M →+ N` and `k : P →+ Q` are localization maps
for `S` and `T` respectively, we send `z : N` to `k (g x) - k (g y)`, where `(x, y) : M × S` are
such that `z = f x - f y`."]
noncomputable def map : N →* Q :=
@lift _ _ _ _ _ _ _ f (k.to_map.comp g) $ λ y, k.map_units ⟨g y, hy y⟩
variables {k}
@[to_additive] lemma map_eq (x) :
f.map hy k (f.to_map x) = k.to_map (g x) := f.lift_eq (λ y, k.map_units ⟨g y, hy y⟩) x
@[simp, to_additive] lemma map_comp :
(f.map hy k).comp f.to_map = k.to_map.comp g := f.lift_comp $ λ y, k.map_units ⟨g y, hy y⟩
@[to_additive] lemma map_mk' (x) (y : S) :
f.map hy k (f.mk' x y) = k.mk' (g x) ⟨g y, hy y⟩ :=
begin
rw [map, lift_mk', mul_inv_left],
{ show k.to_map (g x) = k.to_map (g y) * _,
rw mul_mk'_eq_mk'_of_mul,
exact (k.mk'_mul_cancel_left (g x) ⟨(g y), hy y⟩).symm },
end
/-- Given localization maps `f : M →* N, k : P →* Q` for submonoids `S, T` respectively, if a
`comm_monoid` homomorphism `g : M →* P` induces a `f.map hy k : N →* Q`, then for all `z : N`,
`u : Q`, we have `f.map hy k z = u ↔ k (g x) = k (g y) * u` where `x : M, y ∈ S` are such that
`z * f y = f x`. -/
@[to_additive "Given localization maps `f : M →+ N, k : P →+ Q` for submonoids `S, T` respectively,
if an `add_comm_monoid` homomorphism `g : M →+ P` induces a `f.map hy k : N →+ Q`, then for all
`z : N`, `u : Q`, we have `f.map hy k z = u ↔ k (g x) = k (g y) + u` where `x : M, y ∈ S` are such
that `z + f y = f x`."]
lemma map_spec (z u) :
f.map hy k z = u ↔ k.to_map (g (f.sec z).1) = k.to_map (g (f.sec z).2) * u :=
f.lift_spec (λ y, k.map_units ⟨g y, hy y⟩) _ _
/-- Given localization maps `f : M →* N, k : P →* Q` for submonoids `S, T` respectively, if a
`comm_monoid` homomorphism `g : M →* P` induces a `f.map hy k : N →* Q`, then for all `z : N`,
we have `f.map hy k z * k (g y) = k (g x)` where `x : M, y ∈ S` are such that
`z * f y = f x`. -/
@[to_additive "Given localization maps `f : M →+ N, k : P →+ Q` for submonoids `S, T` respectively,
if an `add_comm_monoid` homomorphism `g : M →+ P` induces a `f.map hy k : N →+ Q`, then
for all `z : N`, we have `f.map hy k z + k (g y) = k (g x)` where `x : M, y ∈ S` are such that
`z + f y = f x`."]
lemma map_mul_right (z) :
f.map hy k z * (k.to_map (g (f.sec z).2)) = k.to_map (g (f.sec z).1) :=
f.lift_mul_right (λ y, k.map_units ⟨g y, hy y⟩) _
/-- Given localization maps `f : M →* N, k : P →* Q` for submonoids `S, T` respectively, if a
`comm_monoid` homomorphism `g : M →* P` induces a `f.map hy k : N →* Q`, then for all `z : N`,
we have `k (g y) * f.map hy k z = k (g x)` where `x : M, y ∈ S` are such that
`z * f y = f x`. -/
@[to_additive "Given localization maps `f : M →+ N, k : P →+ Q` for submonoids `S, T` respectively,
if an `add_comm_monoid` homomorphism `g : M →+ P` induces a `f.map hy k : N →+ Q`, then for all
`z : N`, we have `k (g y) + f.map hy k z = k (g x)` where `x : M, y ∈ S` are such that
`z + f y = f x`."]
lemma map_mul_left (z) :
k.to_map (g (f.sec z).2) * f.map hy k z = k.to_map (g (f.sec z).1) :=
by rw [mul_comm, f.map_mul_right]
@[simp, to_additive] lemma map_id (z : N) :
f.map (λ y, show monoid_hom.id M y ∈ S, from y.2) f z = z :=
f.lift_id z
/-- If `comm_monoid` homs `g : M →* P, l : P →* A` induce maps of localizations, the composition
of the induced maps equals the map of localizations induced by `l ∘ g`. -/
@[to_additive "If `add_comm_monoid` homs `g : M →+ P, l : P →+ A` induce maps of localizations,
the composition of the induced maps equals the map of localizations induced by `l ∘ g`."]
lemma map_comp_map {A : Type*} [comm_monoid A] {U : submonoid A} {R} [comm_monoid R]
(j : localization_map U R) {l : P →* A} (hl : ∀ w : T, l w ∈ U) :
(k.map hl j).comp (f.map hy k) = f.map (λ x, show l.comp g x ∈ U, from hl ⟨g x, hy x⟩) j :=
begin
ext z,
show j.to_map _ * _ = j.to_map (l _) * _,
{ rw [mul_inv_left, ←mul_assoc, mul_inv_right],
show j.to_map _ * j.to_map (l (g _)) = j.to_map (l _) * _,
rw [←j.to_map.map_mul, ←j.to_map.map_mul, ←l.map_mul, ←l.map_mul],
exact k.comp_eq_of_eq hl j
(by rw [k.to_map.map_mul, k.to_map.map_mul, sec_spec', mul_assoc, map_mul_right]) },
end
/-- If `comm_monoid` homs `g : M →* P, l : P →* A` induce maps of localizations, the composition
of the induced maps equals the map of localizations induced by `l ∘ g`. -/
@[to_additive "If `add_comm_monoid` homs `g : M →+ P, l : P →+ A` induce maps of localizations,
the composition of the induced maps equals the map of localizations induced by `l ∘ g`."]
lemma map_map {A : Type*} [comm_monoid A] {U : submonoid A} {R} [comm_monoid R]
(j : localization_map U R) {l : P →* A} (hl : ∀ w : T, l w ∈ U) (x) :
k.map hl j (f.map hy k x) = f.map (λ x, show l.comp g x ∈ U, from hl ⟨g x, hy x⟩) j x :=
by rw ←f.map_comp_map hy j hl; refl
variables {g}
/-- If `f : M →* N` and `k : M →* P` are localization maps for a submonoid `S`, we get an
isomorphism of `N` and `P`. -/
@[to_additive "If `f : M →+ N` and `k : M →+ R` are localization maps for a submonoid `S`,
we get an isomorphism of `N` and `R`."]
noncomputable def mul_equiv_of_localizations
(k : localization_map S P) : N ≃* P :=
⟨f.lift k.map_units, k.lift f.map_units, f.lift_left_inverse,
k.lift_left_inverse, monoid_hom.map_mul _⟩
@[to_additive, simp] lemma mul_equiv_of_localizations_apply
{k : localization_map S P} {x} :
f.mul_equiv_of_localizations k x = f.lift k.map_units x := rfl
@[to_additive, simp] lemma mul_equiv_of_localizations_symm_apply
{k : localization_map S P} {x} :
(f.mul_equiv_of_localizations k).symm x = k.lift f.map_units x := rfl
@[to_additive] lemma mul_equiv_of_localizations_symm_eq_mul_equiv_of_localizations
{k : localization_map S P} :
(k.mul_equiv_of_localizations f).symm = f.mul_equiv_of_localizations k := rfl
/-- If `f : M →* N` is a localization map for a submonoid `S` and `k : N ≃* P` is an isomorphism
of `comm_monoid`s, `k ∘ f` is a localization map for `M` at `S`. -/
@[to_additive "If `f : M →+ N` is a localization map for a submonoid `S` and `k : N ≃+ P` is an
isomorphism of `add_comm_monoid`s, `k ∘ f` is a localization map for `M` at `S`."]
def of_mul_equiv_of_localizations (k : N ≃* P) : localization_map S P :=
(k.to_monoid_hom.comp f.to_map).to_localization_map (λ y, is_unit_comp f k.to_monoid_hom y)
(λ v, let ⟨z, hz⟩ := k.to_equiv.surjective v in
let ⟨x, hx⟩ := f.surj z in ⟨x, show v * k _ = k _, by rw [←hx, k.map_mul, ←hz]; refl⟩)
(λ x y, (k.to_equiv.apply_eq_iff_eq _ _).trans $ f.eq_iff_exists)
@[to_additive, simp] lemma of_mul_equiv_of_localizations_apply {k : N ≃* P} (x) :
(f.of_mul_equiv_of_localizations k).to_map x = k (f.to_map x) := rfl
@[to_additive] lemma of_mul_equiv_of_localizations_eq {k : N ≃* P} :
(f.of_mul_equiv_of_localizations k).to_map = k.to_monoid_hom.comp f.to_map := rfl
@[to_additive] lemma symm_comp_of_mul_equiv_of_localizations_apply {k : N ≃* P} (x) :
k.symm ((f.of_mul_equiv_of_localizations k).to_map x) = f.to_map x :=
k.symm_apply_apply (f.to_map x)
@[to_additive] lemma symm_comp_of_mul_equiv_of_localizations_apply' {k : P ≃* N} (x) :
k ((f.of_mul_equiv_of_localizations k.symm).to_map x) = f.to_map x :=
k.apply_symm_apply (f.to_map x)
@[to_additive] lemma of_mul_equiv_of_localizations_eq_iff_eq {k : N ≃* P} {x y} :
(f.of_mul_equiv_of_localizations k).to_map x = y ↔ f.to_map x = k.symm y :=
k.to_equiv.eq_symm_apply.symm
@[to_additive] lemma mul_equiv_of_localizations_right_inv (k : localization_map S P) :
f.of_mul_equiv_of_localizations (f.mul_equiv_of_localizations k) = k :=
to_map_injective $ f.lift_comp k.map_units
@[to_additive, simp] lemma mul_equiv_of_localizations_right_inv_apply
{k : localization_map S P} {x} :
(f.of_mul_equiv_of_localizations (f.mul_equiv_of_localizations k)).to_map x = k.to_map x :=
ext_iff.1 (f.mul_equiv_of_localizations_right_inv k) x
@[to_additive] lemma mul_equiv_of_localizations_left_inv (k : N ≃* P) :
f.mul_equiv_of_localizations (f.of_mul_equiv_of_localizations k) = k :=
mul_equiv.ext $ monoid_hom.ext_iff.1 $ f.lift_of_comp k.to_monoid_hom
@[to_additive, simp] lemma mul_equiv_of_localizations_left_inv_apply {k : N ≃* P} (x) :
f.mul_equiv_of_localizations (f.of_mul_equiv_of_localizations k) x = k x :=
by rw mul_equiv_of_localizations_left_inv
@[simp, to_additive] lemma of_mul_equiv_of_localizations_id :
f.of_mul_equiv_of_localizations (mul_equiv.refl N) = f :=
by ext; refl
@[to_additive] lemma of_mul_equiv_of_localizations_comp {k : N ≃* P} {j : P ≃* Q} :
(f.of_mul_equiv_of_localizations (k.trans j)).to_map =
j.to_monoid_hom.comp (f.of_mul_equiv_of_localizations k).to_map :=
by ext; refl
/-- Given `comm_monoid`s `M, P` and submonoids `S ⊆ M, T ⊆ P`, if `f : M →* N` is a localization
map for `S` and `k : P ≃* M` is an isomorphism of `comm_monoid`s such that `k(T) = S`, `f ∘ k`
is a localization map for `T`. -/
@[to_additive "Given `comm_monoid`s `M, P` and submonoids `S ⊆ M, T ⊆ P`, if `f : M →* N` is
a localization map for `S` and `k : P ≃* M` is an isomorphism of `comm_monoid`s such that
`k(T) = S`, `f ∘ k` is a localization map for `T`."]
def of_mul_equiv_of_dom {k : P ≃* M} (H : T.map k.to_monoid_hom = S) :
localization_map T N :=
let H' : S.comap k.to_monoid_hom = T :=
H ▸ (submonoid.ext' $ T.1.preimage_image_eq k.to_equiv.injective) in
(f.to_map.comp k.to_monoid_hom).to_localization_map
(λ y, let ⟨z, hz⟩ := f.map_units ⟨k y, H ▸ set.mem_image_of_mem k y.2⟩ in ⟨z, hz⟩)
(λ z, let ⟨x, hx⟩ := f.surj z in let ⟨v, hv⟩ := k.to_equiv.surjective x.1 in
let ⟨w, hw⟩ := k.to_equiv.surjective x.2 in ⟨(v, ⟨w, H' ▸ show k w ∈ S, from hw.symm ▸ x.2.2⟩),
show z * f.to_map (k.to_equiv w) = f.to_map (k.to_equiv v), by erw [hv, hw, hx]; refl⟩)
(λ x y, show f.to_map _ = f.to_map _ ↔ _, by erw f.eq_iff_exists;
exact ⟨λ ⟨c, hc⟩, let ⟨d, hd⟩ := k.to_equiv.surjective c in
⟨⟨d, H' ▸ show k d ∈ S, from hd.symm ▸ c.2⟩, by erw [←hd, ←k.map_mul, ←k.map_mul] at hc;
exact k.to_equiv.injective hc⟩, λ ⟨c, hc⟩, ⟨⟨k c, H ▸ set.mem_image_of_mem k c.2⟩,
by erw ←k.map_mul; rw [hc, k.map_mul]; refl⟩⟩)
@[to_additive, simp] lemma of_mul_equiv_of_dom_apply
{k : P ≃* M} (H : T.map k.to_monoid_hom = S) (x) :
(f.of_mul_equiv_of_dom H).to_map x = f.to_map (k x) := rfl
@[to_additive] lemma of_mul_equiv_of_dom_eq
{k : P ≃* M} (H : T.map k.to_monoid_hom = S) :
(f.of_mul_equiv_of_dom H).to_map = f.to_map.comp k.to_monoid_hom := rfl
@[to_additive] lemma of_mul_equiv_of_dom_comp_symm {k : P ≃* M}
(H : T.map k.to_monoid_hom = S) (x) :
(f.of_mul_equiv_of_dom H).to_map (k.symm x) = f.to_map x :=
congr_arg f.to_map $ k.apply_symm_apply x
@[to_additive] lemma of_mul_equiv_of_dom_comp {k : M ≃* P}
(H : T.map k.symm.to_monoid_hom = S) (x) :
(f.of_mul_equiv_of_dom H).to_map (k x) = f.to_map x :=
congr_arg f.to_map $ k.symm_apply_apply x
/-- A special case of `f ∘ id = f`, `f` a localization map. -/
@[simp, to_additive "A special case of `f ∘ id = f`, `f` a localization map."]
lemma of_mul_equiv_of_dom_id :
f.of_mul_equiv_of_dom (show S.map (mul_equiv.refl M).to_monoid_hom = S, from
submonoid.ext $ λ x, ⟨λ ⟨y, hy, h⟩, h ▸ hy, λ h, ⟨x, h, rfl⟩⟩) = f :=
by ext; refl
/-- Given localization maps `f : M →* N, k : P →* U` for submonoids `S, T` respectively, an
isomorphism `j : M ≃* P` such that `j(S) = T` induces an isomorphism of localizations
`N ≃* U`. -/
@[to_additive "Given localization maps `f : M →+ N, k : P →+ U` for submonoids `S, T` respectively,
an isomorphism `j : M ≃+ P` such that `j(S) = T` induces an isomorphism of
localizations `N ≃+ U`."]
noncomputable def mul_equiv_of_mul_equiv
(k : localization_map T Q) {j : M ≃* P} (H : S.map j.to_monoid_hom = T) :
N ≃* Q :=
f.mul_equiv_of_localizations $ k.of_mul_equiv_of_dom H
@[simp, to_additive] lemma mul_equiv_of_mul_equiv_eq_map_apply
{k : localization_map T Q} {j : M ≃* P} (H : S.map j.to_monoid_hom = T) (x) :
f.mul_equiv_of_mul_equiv k H x =
f.map (λ y : S, show j.to_monoid_hom y ∈ T, from H ▸ set.mem_image_of_mem j y.2) k x := rfl
@[to_additive] lemma mul_equiv_of_mul_equiv_eq_map
{k : localization_map T Q} {j : M ≃* P} (H : S.map j.to_monoid_hom = T) :
(f.mul_equiv_of_mul_equiv k H).to_monoid_hom =
f.map (λ y : S, show j.to_monoid_hom y ∈ T, from H ▸ set.mem_image_of_mem j y.2) k := rfl
@[simp, to_additive] lemma mul_equiv_of_mul_equiv_eq {k : localization_map T Q}
{j : M ≃* P} (H : S.map j.to_monoid_hom = T) (x) :
f.mul_equiv_of_mul_equiv k H (f.to_map x) = k.to_map (j x) :=
f.map_eq (λ y : S, H ▸ set.mem_image_of_mem j y.2) _
@[simp, to_additive] lemma mul_equiv_of_mul_equiv_mk' {k : localization_map T Q}
{j : M ≃* P} (H : S.map j.to_monoid_hom = T) (x y) :
f.mul_equiv_of_mul_equiv k H (f.mk' x y) = k.mk' (j x) ⟨j y, H ▸ set.mem_image_of_mem j y.2⟩ :=
f.map_mk' (λ y : S, H ▸ set.mem_image_of_mem j y.2) _ _
@[to_additive, simp] lemma of_mul_equiv_of_mul_equiv_apply
{k : localization_map T Q} {j : M ≃* P} (H : S.map j.to_monoid_hom = T) (x) :
(f.of_mul_equiv_of_localizations (f.mul_equiv_of_mul_equiv k H)).to_map x = k.to_map (j x) :=
ext_iff.1 (f.mul_equiv_of_localizations_right_inv (k.of_mul_equiv_of_dom H)) x
@[to_additive] lemma of_mul_equiv_of_mul_equiv
{k : localization_map T Q} {j : M ≃* P} (H : S.map j.to_monoid_hom = T) :
(f.of_mul_equiv_of_localizations (f.mul_equiv_of_mul_equiv k H)).to_map =
k.to_map.comp j.to_monoid_hom :=
monoid_hom.ext $ f.of_mul_equiv_of_mul_equiv_apply H
end localization_map
end submonoid
namespace localization
variables (S)
/-- Natural hom sending `x : M`, `M` a `comm_monoid`, to the equivalence class of
`(x, 1)` in the localization of `M` at a submonoid. -/
@[to_additive "Natural homomorphism sending `x : M`, `M` an `add_comm_monoid`, to the equivalence
class of `(x, 0)` in the localization of `M` at a submonoid."]
def monoid_of : submonoid.localization_map S (localization S) :=
{ map_units' := λ y, is_unit_iff_exists_inv.2 ⟨mk 1 y, (r S).eq.2 $
show r S (_, 1 * y) 1, by simpa using (r S).symm (one_rel y)⟩,
surj' := λ z, induction_on z $ λ x, ⟨x, (r S).eq.2 $
show r S (x.1 * x.2, x.2 * 1) (x.1, 1), by
rw [mul_comm x.2, ←mul_one (x.1, (1 : S))];
exact (r S).mul ((r S).refl (x.1, 1)) ((r S).symm $ one_rel x.2)⟩,
eq_iff_exists' := λ x y, (r S).eq.trans $ r_iff_exists.trans $
show (∃ (c : S), x * 1 * c = y * 1 * c) ↔ _, by rw [mul_one, mul_one],
..(r S).mk'.comp $ monoid_hom.inl M S }
variables {S}
@[to_additive] lemma mk_one_eq_monoid_of_mk (x) : mk x 1 = (monoid_of S).to_map x := rfl
@[to_additive] lemma mk_eq_monoid_of_mk'_apply (x y) : mk x y = (monoid_of S).mk' x y :=
show _ = _ * _, from (submonoid.localization_map.mul_inv_right (monoid_of S).map_units _ _ _).2 $
begin
rw [←mk_one_eq_monoid_of_mk, ←mk_one_eq_monoid_of_mk,
show mk x y * mk y 1 = mk (x * y) (1 * y), by rw mul_comm 1 y; refl,
show mk x 1 = mk (x * 1) ((1 : S) * 1), by rw [mul_one, mul_one]],
exact (con.eq _).2 (con.symm _ $ (localization.r S).mul
(con.refl _ (x, 1)) $ one_rel _),
end
@[simp, to_additive] lemma mk_eq_monoid_of_mk' : mk = (monoid_of S).mk' :=
funext $ λ _, funext $ λ _, mk_eq_monoid_of_mk'_apply _ _
variables (f : submonoid.localization_map S N)
/-- Given a localization map `f : M →* N` for a submonoid `S`, we get an isomorphism between
the localization of `M` at `S` as a quotient type and `N`. -/
@[to_additive "Given a localization map `f : M →+ N` for a submonoid `S`, we get an isomorphism
between the localization of `M` at `S` as a quotient type and `N`."]
noncomputable def mul_equiv_of_quotient (f : submonoid.localization_map S N) :
localization S ≃* N :=
(monoid_of S).mul_equiv_of_localizations f
variables {f}
@[simp, to_additive] lemma mul_equiv_of_quotient_apply (x) :
mul_equiv_of_quotient f x = (monoid_of S).lift f.map_units x := rfl
@[simp, to_additive] lemma mul_equiv_of_quotient_mk' (x y) :
mul_equiv_of_quotient f ((monoid_of S).mk' x y) = f.mk' x y :=
(monoid_of S).lift_mk' _ _ _
@[to_additive] lemma mul_equiv_of_quotient_mk (x y) :
mul_equiv_of_quotient f (mk x y) = f.mk' x y :=
by rw mk_eq_monoid_of_mk'_apply; exact mul_equiv_of_quotient_mk' _ _
@[simp, to_additive] lemma mul_equiv_of_quotient_monoid_of (x) :
mul_equiv_of_quotient f ((monoid_of S).to_map x) = f.to_map x :=
(monoid_of S).lift_eq _ _
@[simp, to_additive] lemma mul_equiv_of_quotient_symm_mk' (x y) :
(mul_equiv_of_quotient f).symm (f.mk' x y) = (monoid_of S).mk' x y :=
f.lift_mk' _ _ _
@[to_additive] lemma mul_equiv_of_quotient_symm_mk (x y) :
(mul_equiv_of_quotient f).symm (f.mk' x y) = mk x y :=
by rw mk_eq_monoid_of_mk'_apply; exact mul_equiv_of_quotient_symm_mk' _ _
@[simp, to_additive] lemma mul_equiv_of_quotient_symm_monoid_of (x) :
(mul_equiv_of_quotient f).symm (f.to_map x) = (monoid_of S).to_map x :=
f.lift_eq _ _
end localization
|
4fba764c277a673d2ac6e2326b63238ec03b75c1 | 0c1546a496eccfb56620165cad015f88d56190c5 | /tests/lean/run/using_smt3.lean | d2cfae2aba7c54eda9aa79cb1e893c4831a83718 | [
"Apache-2.0"
] | permissive | Solertis/lean | 491e0939957486f664498fbfb02546e042699958 | 84188c5aa1673fdf37a082b2de8562dddf53df3f | refs/heads/master | 1,610,174,257,606 | 1,486,263,620,000 | 1,486,263,620,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 595 | lean | open tactic
lemma ex1 : let x := 1 in ∀ y, x = y → y = 1 :=
by using_smt $ smt_tactic.intros >> get_local `x >>= (fun a, trace a)
open tactic_result
meta def fail_if_success {α : Type} (t : smt_tactic α) : smt_tactic unit :=
λ ss ts, match t ss ts with
| success _ _ := failed ts
| exception .(α × smt_state) _ _ _ := success ((), ss) ts
end
def my_smt_config : smt_config :=
{ pre_cfg := { zeta := tt } }
lemma ex2 : let x := 1 in ∀ y, x = y → y = 1 :=
by using_smt_core my_smt_config $ smt_tactic.intros >> fail_if_success (get_local `x) >> return ()
|
266724804d81a07a5c7c9ca216138d679c774053 | bcdb337a071a6642a282053e6c285336b9716772 | /Tensor/Enumerable.lean | 26937d329e1396d5416a215159da0ea6fed6fac6 | [] | no_license | kovach/lean4-sparse-matrix-test | 87a8b3a0b7785976eccfaf69b0d289940549f446 | be9f6029883c88bf1f7588a0e6d3cde033368809 | refs/heads/master | 1,678,513,001,774 | 1,614,027,522,000 | 1,614,027,522,000 | 341,329,194 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 8,249 | lean | -- Kyle Miller
import Init.Data.Nat.Basic
import Init.Core
import Tensor.Has
macro "exfalso" : tactic => `(apply False.elim)
def Function.leftInverse (g : β → α) (f : α → β) : Prop :=
∀ x, g (f x) = x
def Function.rightInverse (g : β → α) (f : α → β) : Prop :=
Function.leftInverse f g
structure Equiv (α : Sort u) (β : Sort v) where
toFun : α → β
invFun : β → α
leftInv : Function.leftInverse invFun toFun
rightInv : Function.rightInverse invFun toFun
infix:25 " ≃ " => Equiv
def Equiv.symm (f : α ≃ β) : β ≃ α where
toFun := f.invFun
invFun := f.toFun
leftInv := f.rightInv
rightInv := f.leftInv
/-- An equivalence "is" a function. -/
instance (α : Type u) (β : Type v) : CoeFun (Equiv α β) (λ _ => α → β) where
coe := Equiv.toFun
class Enumerable (α : Type u) where
card : Nat
enum : α ≃ Fin card
theorem cartEncodeProp {i j m n : Nat} (hi : i < m) (hj : j < n) : i * n + j < m * n := by
cases m with
| zero => exfalso; exact Nat.notLtZero _ hi
| succ m => {
rw [Nat.succ_mul];
exact Nat.ltOfLeOfLt (Nat.addLeAddRight (Nat.mulLeMulRight _ (Nat.leOfLtSucc hi)) _) (Nat.addLtAddLeft hj _)
}
def cartDecode {n m : Nat} (k : Fin (n * m)) : Fin n × Fin m :=
let ⟨k, h⟩ := k
(
⟨k / m, sorry⟩,
⟨k % m, Nat.modLt _ (by { cases m; exfalso; rw Nat.mul_zero at h; exact Nat.notLtZero _ h; apply Nat.succPos})⟩
)
instance [Enumerable α] [Enumerable β] : Enumerable (α × β) where
card := Enumerable.card α * Enumerable.card β
enum := {
toFun := λ (a, b) =>
let ⟨i, hi⟩ := Enumerable.enum a
let ⟨j, hj⟩ := Enumerable.enum b
⟨i * Enumerable.card β + j, cartEncodeProp hi hj⟩
invFun := λ n =>
let (i, j) := cartDecode n
(Enumerable.enum.symm i, Enumerable.enum.symm j)
leftInv := sorry
rightInv := sorry
}
instance : Enumerable (Fin n) where
card := n
enum := {
toFun := id
invFun := id
leftInv := λ _ => rfl
rightInv := λ _ => rfl
}
instance : Enumerable Bool where
card := 2
enum := {
toFun := fun
| false => 0
| true => 1
invFun := fun
| ⟨0, _⟩ => false
| ⟨1, _⟩ => true
| ⟨n+2, h⟩ => False.elim (Nat.notSuccLeZero _ (Nat.leOfSuccLeSucc (Nat.leOfSuccLeSucc h)))
leftInv := fun
| true => rfl
| false => rfl
rightInv := fun
| ⟨0, _⟩ => rfl
| ⟨1, _⟩ => rfl
| ⟨n+2, h⟩ => False.elim (Nat.notSuccLeZero _ (Nat.leOfSuccLeSucc (Nat.leOfSuccLeSucc h)))
}
instance : Enumerable Empty where
card := 0
enum := {
toFun := fun t => nomatch t
invFun := fun
| ⟨n, h⟩ => False.elim (Nat.notSuccLeZero _ h)
leftInv := fun t => nomatch t
rightInv := fun t => nomatch t
}
def Enumerable.listOf.aux (α : Type u) [Enumerable α] : Nat -> Nat -> List α
| lo, 0 => []
| lo, (left+1) =>
if h : lo < Enumerable.card α then
Enumerable.enum.symm ⟨lo, h⟩ :: aux α (lo + 1) left
else [] -- Shouldn't happen, but makes the definition easy.
/-- Create a list of every term in the Enumerable type in order. -/
def Enumerable.listOf (α : Type u) [Enumerable α] : List α :=
Enumerable.listOf.aux α 0 (Enumerable.card α)
/-- A function-backed vector -/
structure Vec (ι : Type u) (α : Type v) where
toFun : ι → α
macro "vec" xs:Lean.explicitBinders " => " b:term : term => Lean.expandExplicitBinders `Vec.mk xs b
/-- support `v[i]` notation. -/
@[inline] def Vec.getOp (self : Vec ι α) (idx : ι) : α := self.toFun idx
/-- A vector as a function mapping indices to values. -/
class HasVec (β : Type u) (ι : outParam $ Type v) (α : outParam $ Type w) where
toVec : β → Vec ι α
export HasVec (toVec)
instance : HasVec (Vec ι α) ι α := ⟨ id ⟩
instance [Add α] : Add (Vec ι α) where
add v w := vec i => v[i] + w[i]
instance [Sub α] : Sub (Vec ι α) where
sub v w := vec i => v[i] - w[i]
instance [Mul α] : Mul (Vec ι α) where
mul v w := vec i => v[i] * w[i]
instance [Div α] : Div (Vec ι α) where
div v w := vec i => v[i] / w[i]
namespace Vec
def push (v : Vec ι (Vec κ α)) : Vec (ι × κ) α := vec p => v[p.1][p.2]
def pop (v : Vec (ι × κ) α) : Vec ι (Vec κ α) := vec i j => v[(i, j)]
def reindex (v : Vec ι α) (f : κ → ι) : Vec κ α := vec i => v[f i]
instance [OfNat α n] : OfNat (Vec ι α) n where
ofNat := vec i => OfNat.ofNat n
def transpose (v : Vec ι (Vec κ α)) : Vec κ (Vec ι α) := vec j i => v[i][j]
end Vec
def Vec.sum [Enumerable ι] [Add α] [OfNat α Nat.zero] (v : Vec ι α) : α := do
let mut s : α := 0
for i in Enumerable.listOf ι do
s := s + v[i]
return s
structure DenseVec (ι : Type u) [Enumerable ι] (α : Type v) where
array : Array α
hasSize : array.size = Enumerable.card ι
namespace DenseVec
variable {ι : Type u} [Enumerable ι] {α : Type v}
instance [Repr α] : Repr (DenseVec ι α) where
reprPrec d _ := repr d.array
theorem sizeMapEq (a : Array α) (f : α → β) : (a.map f).size = a.size := sorry
instance [HMul α α₁ α₁] : HMul α (DenseVec ι α₁) (DenseVec ι α₁) where
hMul r u := ⟨ u.array.map (λ x => r * x) , (sizeMapEq _ _).symm ▸ u.hasSize⟩
def fill (a : α) : DenseVec ι α where
array := Array.mkArray (Enumerable.card ι) a
hasSize := Array.size_mkArray ..
def empty [Inhabited α] : DenseVec ι α :=
fill Inhabited.default
def translateIdx (v : DenseVec ι α) (i : ι) : Fin v.array.size :=
let ⟨n, h⟩ := Enumerable.enum i
⟨n, by rw v.hasSize; exact h⟩
/-- Get the value associated to a particular index. -/
def get (v : DenseVec ι α) (i : ι) : α :=
v.array.get (v.translateIdx i)
/-- support `v[i]` notation. -/
@[inline] def getOp (self : DenseVec ι α) (idx : ι) : α := self.get idx
instance : HasVec (DenseVec ι α) ι α where
toVec v := vec i => v[i]
def of [HasVec β ι α] (v : β) : DenseVec ι α where
array := do
let v' := toVec v
let mut a := Array.empty
for i in Enumerable.listOf ι do
a := a.push $ v'[i]
return a
hasSize := sorry -- need to define differently to be able to easily prove this
/-- Set the value associated to a particular index. -/
def set (v : DenseVec ι α) (i : ι) (a : α) : DenseVec ι α where
array := v.array.set (v.translateIdx i) a
hasSize := by rw [Array.size_set, v.hasSize]
def forIn {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : DenseVec ι α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
as.array.forIn b f
instance [HasZero α] [Enumerable ι] : HasZero (DenseVec ι α) where
zero := ⟨ Array.mkArray (Enumerable.card ι) HasZero.zero, Array.size_mkArray .. ⟩
instance [HasZero α] [Enumerable ι] : OfNat (DenseVec ι α) 0 where
ofNat := HasZero.zero
def pure (x : α) : DenseVec ι α := fill x
def dot_ (v w : DenseVec ι Float) : Float := do
let mut s : Float := 0
for i in Enumerable.listOf ι do
s := s + v[i] * w[i]
return s
def add_ [Add α] (v w : DenseVec ι α) : DenseVec ι α := do
let mut v := v
for i in Enumerable.listOf ι do
v := v.set i (v.get i + w.get i)
return v
-- dsk: alternate add
def add [Add α] [Inhabited α] ( u v : DenseVec ι α) : DenseVec ι α := do
let mut values := u.array
for i in [0:values.size] do
values := values.set! i (values[i] + v.array[i])
return ⟨ values , sorry ⟩
instance [Add α] [Inhabited α]: Add (DenseVec ι α) where
add := add
end DenseVec
theorem List.toArraySizeEq (x : List α) : x.toArray.size = x.length := sorry
def List.toDenseVec (x : List α) : DenseVec (Fin x.length) α where
array := x.toArray
hasSize := List.toArraySizeEq ..
syntax "![" sepBy(term, ", ") "]" : term
macro_rules
| `(![ $elems,* ]) => `(List.toDenseVec [ $elems,* ])
example : DenseVec (Fin 3) Nat := ![2,22,222] |
d2d3b40a1631489145b3290b40f52e7126c1b735 | d7189ea2ef694124821b033e533f18905b5e87ef | /galois/vector/repeat_lemmas.lean | 7c737aba09edc1d54d31555f7e9e4017cf83bb62 | [
"Apache-2.0"
] | permissive | digama0/lean-protocol-support | eaa7e6f8b8e0d5bbfff1f7f52bfb79a3b11b0f59 | cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda | refs/heads/master | 1,625,421,450,627 | 1,506,035,462,000 | 1,506,035,462,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 624 | lean | import data.vector
universe variables u
namespace vector
variable {α : Type u}
variable {n : ℕ}
local infix `++`:65 := vector.append
@[simp]
theorem to_list_repeat (x : α) : to_list (repeat x n) = list.repeat x n :=
begin
induction n,
all_goals { simp [repeat] }
end
theorem repeat_succ_to_append {α : Type} (x : α) (n : ℕ)
: repeat x (nat.succ n) = repeat x n ++ cons x nil :=
begin
induction n with n ind,
{ apply vector.eq,
simp [repeat]
},
{
have h := congr_arg to_list ind,
simp [vector.repeat] at h,
apply vector.eq,
simp [vector.repeat],
rw h,
},
end
end vector
|
cea3657d2578208a9cf3e5812a631054884a988d | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/control/equiv_functor/instances_auto.lean | 0ea3f6dca0151b3787a1858a94791154f339f2e5 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,371 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Scott Morrison
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.fintype.basic
import Mathlib.control.equiv_functor
import Mathlib.PostPort
universes u_1
namespace Mathlib
/-!
# `equiv_functor` instances
We derive some `equiv_functor` instances, to enable `equiv_rw` to rewrite under these functions.
-/
protected instance equiv_functor_unique : equiv_functor unique :=
equiv_functor.mk fun (α β : Type u_1) (e : α ≃ β) => ⇑(equiv.unique_congr e)
protected instance equiv_functor_perm : equiv_functor equiv.perm :=
equiv_functor.mk
fun (α β : Type u_1) (e : α ≃ β) (p : equiv.perm α) =>
equiv.trans (equiv.trans (equiv.symm e) p) e
-- There is a classical instance of `is_lawful_functor finset` available,
-- but we provide this computable alternative separately.
protected instance equiv_functor_finset : equiv_functor finset :=
equiv_functor.mk
fun (α β : Type u_1) (e : α ≃ β) (s : finset α) => finset.map (equiv.to_embedding e) s
protected instance equiv_functor_fintype : equiv_functor fintype :=
equiv_functor.mk
fun (α β : Type u_1) (e : α ≃ β) (s : fintype α) =>
fintype.of_bijective (⇑e) (equiv.bijective e)
end Mathlib |
39df89c6147864bfa8c71d109dd1ad570abbbbe2 | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/algebra/group/freiman.lean | 9a04d8f0c74db5a6a9103370ead0aa1c4a84190e | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 16,478 | lean | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import algebra.big_operators.multiset
import data.fun_like.basic
/-!
# Freiman homomorphisms
In this file, we define Freiman homomorphisms. A `n`-Freiman homomorphism on `A` is a function
`f : α → β` such that `f (x₁) * ... * f (xₙ) = f (y₁) * ... * f (yₙ)` for all
`x₁, ..., xₙ, y₁, ..., yₙ ∈ A` such that `x₁ * ... * xₙ = y₁ * ... * yₙ`. In particular, any
`mul_hom` is a Freiman homomorphism.
They are of interest in additive combinatorics.
## Main declaration
* `freiman_hom`: Freiman homomorphism.
* `add_freiman_hom`: Additive Freiman homomorphism.
## Notation
* `A →*[n] β`: Multiplicative `n`-Freiman homomorphism on `A`
* `A →+[n] β`: Additive `n`-Freiman homomorphism on `A`
## Implementation notes
In the context of combinatorics, we are interested in Freiman homomorphisms over sets which are not
necessarily closed under addition/multiplication. This means we must parametrize them with a set in
an `add_monoid`/`monoid` instead of the `add_monoid`/`monoid` itself.
## References
[Yufei Zhao, *18.225: Graph Theory and Additive Combinatorics][https://yufeizhao.com/gtac/]
## TODO
`monoid_hom.to_freiman_hom` could be relaxed to `mul_hom.to_freiman_hom` by proving
`(s.map f).prod = (t.map f).prod` directly by induction instead of going through `f s.prod`.
Define `n`-Freiman isomorphisms.
Affine maps induce Freiman homs. Concretely, provide the `add_freiman_hom_class (α →ₐ[𝕜] β) A β n`
instance.
-/
open multiset
variables {F α β γ δ G : Type*}
/-- An additive `n`-Freiman homomorphism is a map which preserves sums of `n` elements. -/
structure add_freiman_hom (A : set α) (β : Type*) [add_comm_monoid α] [add_comm_monoid β] (n : ℕ) :=
(to_fun : α → β)
(map_sum_eq_map_sum' {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : s.card = n) (ht : t.card = n) (h : s.sum = t.sum) :
(s.map to_fun).sum = (t.map to_fun).sum)
/-- A `n`-Freiman homomorphism on a set `A` is a map which preserves products of `n` elements. -/
@[to_additive add_freiman_hom]
structure freiman_hom (A : set α) (β : Type*) [comm_monoid α] [comm_monoid β] (n : ℕ) :=
(to_fun : α → β)
(map_prod_eq_map_prod' {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : s.card = n) (ht : t.card = n) (h : s.prod = t.prod) :
(s.map to_fun).prod = (t.map to_fun).prod)
notation A ` →+[`:25 n:25 `] `:0 β:0 := add_freiman_hom A β n
notation A ` →*[`:25 n:25 `] `:0 β:0 := freiman_hom A β n
/-- `add_freiman_hom_class F s β n` states that `F` is a type of `n`-ary sums-preserving morphisms.
You should extend this class when you extend `add_freiman_hom`. -/
class add_freiman_hom_class (F : Type*) (A : out_param $ set α) (β : out_param $ Type*)
[add_comm_monoid α] [add_comm_monoid β] (n : ℕ) [fun_like F α (λ _, β)] :=
(map_sum_eq_map_sum' (f : F) {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A)
(htA : ∀ ⦃x⦄, x ∈ t → x ∈ A) (hs : s.card = n) (ht : t.card = n) (h : s.sum = t.sum) :
(s.map f).sum = (t.map f).sum)
/-- `freiman_hom_class F A β n` states that `F` is a type of `n`-ary products-preserving morphisms.
You should extend this class when you extend `freiman_hom`. -/
@[to_additive add_freiman_hom_class]
class freiman_hom_class (F : Type*) (A : out_param $ set α) (β : out_param $ Type*) [comm_monoid α]
[comm_monoid β] (n : ℕ) [fun_like F α (λ _, β)] :=
(map_prod_eq_map_prod' (f : F) {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A)
(htA : ∀ ⦃x⦄, x ∈ t → x ∈ A) (hs : s.card = n) (ht : t.card = n) (h : s.prod = t.prod) :
(s.map f).prod = (t.map f).prod)
variables [fun_like F α (λ _, β)]
section comm_monoid
variables [comm_monoid α] [comm_monoid β] [comm_monoid γ] [comm_monoid δ] [comm_group G] {A : set α}
{B : set β} {C : set γ} {n : ℕ}
@[to_additive]
lemma map_prod_eq_map_prod [freiman_hom_class F A β n] (f : F) {s t : multiset α}
(hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A) (hs : s.card = n) (ht : t.card = n)
(h : s.prod = t.prod) :
(s.map f).prod = (t.map f).prod :=
freiman_hom_class.map_prod_eq_map_prod' f hsA htA hs ht h
namespace freiman_hom
@[to_additive]
instance fun_like : fun_like (A →*[n] β) α (λ _, β) :=
{ coe := to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr' }
@[to_additive]
instance freiman_hom_class : freiman_hom_class (A →*[n] β) A β n :=
{ map_prod_eq_map_prod' := map_prod_eq_map_prod' }
/-- Helper instance for when there's too many metavariables to apply
`fun_like.has_coe_to_fun` directly. -/
@[to_additive]
instance : has_coe_to_fun (A →*[n] β) (λ _, α → β) := ⟨to_fun⟩
initialize_simps_projections freiman_hom (to_fun → apply)
@[simp, to_additive]
lemma to_fun_eq_coe (f : A →*[n] β) : f.to_fun = f := rfl
@[ext, to_additive]
lemma ext ⦃f g : A →*[n] β⦄ (h : ∀ x, f x = g x) : f = g := fun_like.ext f g h
@[simp, to_additive]
lemma coe_mk (f : α → β) (h : ∀ s t : multiset α, (∀ ⦃x⦄, x ∈ s → x ∈ A) → (∀ ⦃x⦄, x ∈ t → x ∈ A) →
s.card = n → t.card = n → s.prod = t.prod → (s.map f).prod = (t.map f).prod) :
⇑(mk f h) = f := rfl
@[simp, to_additive] lemma mk_coe (f : A →*[n] β) (h) : mk f h = f := ext $ λ _, rfl
/-- The identity map from a commutative monoid to itself. -/
@[to_additive "The identity map from an additive commutative monoid to itself.", simps]
protected def id (A : set α) (n : ℕ) : A →*[n] α :=
{ to_fun := λ x, x, map_prod_eq_map_prod' := λ s t _ _ _ _ h, by rw [map_id', map_id', h] }
/-- Composition of Freiman homomorphisms as a Freiman homomorphism. -/
@[to_additive "Composition of additive Freiman homomorphisms as an additive Freiman homomorphism."]
protected def comp (f : B →*[n] γ) (g : A →*[n] β) (hAB : A.maps_to g B) : A →*[n] γ :=
{ to_fun := f ∘ g,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h, begin
rw [←map_map,
map_prod_eq_map_prod f _ _ ((s.card_map _).trans hs) ((t.card_map _).trans ht)
(map_prod_eq_map_prod g hsA htA hs ht h), map_map],
{ simpa using (λ a h, hAB (hsA h)) },
{ simpa using (λ a h, hAB (htA h)) }
end }
@[simp, to_additive]
lemma coe_comp (f : B →*[n] γ) (g : A →*[n] β) {hfg} : ⇑(f.comp g hfg) = f ∘ g := rfl
@[to_additive]
lemma comp_apply (f : B →*[n] γ) (g : A →*[n] β) {hfg} (x : α) : f.comp g hfg x = f (g x) := rfl
@[to_additive]
lemma comp_assoc (f : A →*[n] β) (g : B →*[n] γ) (h : C →*[n] δ) {hf hhg hgf}
{hh : A.maps_to (g.comp f hgf) C} :
(h.comp g hhg).comp f hf = h.comp (g.comp f hgf) hh := rfl
@[to_additive]
lemma cancel_right {g₁ g₂ : B →*[n] γ} {f : A →*[n] β} (hf : function.surjective f) {hg₁ hg₂} :
g₁.comp f hg₁ = g₂.comp f hg₂ ↔ g₁ = g₂ :=
⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, λ h, h ▸ rfl⟩
@[to_additive]
lemma cancel_right_on {g₁ g₂ : B →*[n] γ} {f : A →*[n] β} (hf : A.surj_on f B) {hf'} :
A.eq_on (g₁.comp f hf') (g₂.comp f hf') ↔ B.eq_on g₁ g₂ :=
hf.cancel_right hf'
@[to_additive]
lemma cancel_left_on {g : B →*[n] γ} {f₁ f₂ : A →*[n] β} (hg : B.inj_on g) {hf₁ hf₂} :
A.eq_on (g.comp f₁ hf₁) (g.comp f₂ hf₂) ↔ A.eq_on f₁ f₂ :=
hg.cancel_left hf₁ hf₂
@[simp, to_additive] lemma comp_id (f : A →*[n] β) {hf} : f.comp (freiman_hom.id A n) hf = f :=
ext $ λ x, rfl
@[simp, to_additive] lemma id_comp (f : A →*[n] β) {hf} : (freiman_hom.id B n).comp f hf = f :=
ext $ λ x, rfl
/-- `freiman_hom.const A n b` is the Freiman homomorphism sending everything to `b`. -/
@[to_additive "`add_freiman_hom.const n b` is the Freiman homomorphism sending everything to `b`."]
def const (A : set α) (n : ℕ) (b : β) : A →*[n] β :=
{ to_fun := λ _, b,
map_prod_eq_map_prod' := λ s t _ _ hs ht _,
by rw [multiset.map_const, multiset.map_const, prod_repeat, prod_repeat, hs, ht] }
@[simp, to_additive] lemma const_apply (n : ℕ) (b : β) (x : α) : const A n b x = b := rfl
@[simp, to_additive]
lemma const_comp (n : ℕ) (c : γ) (f : A →*[n] β) {hf} : (const B n c).comp f hf = const A n c := rfl
/-- `1` is the Freiman homomorphism sending everything to `1`. -/
@[to_additive "`0` is the Freiman homomorphism sending everything to `0`."]
instance : has_one (A →*[n] β) := ⟨const A n 1⟩
@[simp, to_additive] lemma one_apply (x : α) : (1 : A →*[n] β) x = 1 := rfl
@[simp, to_additive] lemma one_comp (f : A →*[n] β) {hf} : (1 : B →*[n] γ).comp f hf = 1 := rfl
@[to_additive] instance : inhabited (A →*[n] β) := ⟨1⟩
/-- `f * g` is the Freiman homomorphism sends `x` to `f x * g x`. -/
@[to_additive "`f + g` is the Freiman homomorphism sending `x` to `f x + g x`."]
instance : has_mul (A →*[n] β) :=
⟨λ f g, { to_fun := λ x, f x * g x,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h,
by rw [prod_map_mul, prod_map_mul, map_prod_eq_map_prod f hsA htA hs ht h,
map_prod_eq_map_prod g hsA htA hs ht h] }⟩
@[simp, to_additive] lemma mul_apply (f g : A →*[n] β) (x : α) : (f * g) x = f x * g x := rfl
@[to_additive] lemma mul_comp (g₁ g₂ : B →*[n] γ) (f : A →*[n] β) {hg hg₁ hg₂} :
(g₁ * g₂).comp f hg = g₁.comp f hg₁ * g₂.comp f hg₂ := rfl
/-- If `f` is a Freiman homomorphism to a commutative group, then `f⁻¹` is the Freiman homomorphism
sending `x` to `(f x)⁻¹`. -/
@[to_additive]
instance : has_inv (A →*[n] G) :=
⟨λ f, { to_fun := λ x, (f x)⁻¹,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h,
by rw [prod_map_inv', prod_map_inv', map_prod_eq_map_prod f hsA htA hs ht h] }⟩
@[simp, to_additive] lemma inv_apply (f : A →*[n] G) (x : α) : f⁻¹ x = (f x)⁻¹ := rfl
@[simp, to_additive] lemma inv_comp (f : B →*[n] G) (g : A →*[n] β) {hf hf'} :
f⁻¹.comp g hf = (f.comp g hf')⁻¹ :=
ext $ λ x, rfl
/-- If `f` and `g` are Freiman homomorphisms to a commutative group, then `f / g` is the Freiman
homomorphism sending `x` to `f x / g x`. -/
@[to_additive "If `f` and `g` are additive Freiman homomorphisms to an additive commutative group,
then `f - g` is the additive Freiman homomorphism sending `x` to `f x - g x`"]
instance : has_div (A →*[n] G) :=
⟨λ f g, { to_fun := λ x, f x / g x,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h,
by rw [prod_map_div, prod_map_div, map_prod_eq_map_prod f hsA htA hs ht h,
map_prod_eq_map_prod g hsA htA hs ht h] }⟩
@[simp, to_additive] lemma div_apply (f g : A →*[n] G) (x : α) : (f / g) x = f x / g x := rfl
@[simp, to_additive] lemma div_comp (f₁ f₂ : B →*[n] G) (g : A →*[n] β) {hf hf₁ hf₂} :
(f₁ / f₂).comp g hf = f₁.comp g hf₁ / f₂.comp g hf₂ :=
ext $ λ x, rfl
/-! ### Instances -/
/-- `A →*[n] β` is a `comm_monoid`. -/
@[to_additive "`α →+[n] β` is an `add_comm_monoid`."]
instance : comm_monoid (A →*[n] β) :=
{ mul := (*),
mul_assoc := λ a b c, by { ext, apply mul_assoc },
one := 1,
one_mul := λ a, by { ext, apply one_mul },
mul_one := λ a, by { ext, apply mul_one },
mul_comm := λ a b, by { ext, apply mul_comm },
npow := λ m f,
{ to_fun := λ x, f x ^ m,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h,
by rw [prod_map_pow, prod_map_pow, map_prod_eq_map_prod f hsA htA hs ht h] },
npow_zero' := λ f, by { ext x, exact pow_zero _ },
npow_succ' := λ n f, by { ext x, exact pow_succ _ _ } }
/-- If `β` is a commutative group, then `A →*[n] β` is a commutative group too. -/
@[to_additive "If `β` is an additive commutative group, then `A →*[n] β` is an additive commutative
group too."]
instance {β} [comm_group β] : comm_group (A →*[n] β) :=
{ inv := has_inv.inv,
div := has_div.div,
div_eq_mul_inv := by { intros, ext, apply div_eq_mul_inv },
mul_left_inv := by { intros, ext, apply mul_left_inv },
zpow := λ n f, { to_fun := λ x, (f x) ^ n,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h,
by rw [prod_map_zpow, prod_map_zpow, map_prod_eq_map_prod f hsA htA hs ht h] },
zpow_zero' := λ f, by { ext x, exact zpow_zero _ },
zpow_succ' := λ n f, by { ext x, simp_rw [zpow_of_nat, pow_succ, mul_apply, coe_mk] },
zpow_neg' := λ n f, by { ext x, simp_rw [zpow_neg_succ_of_nat, zpow_coe_nat], refl },
..freiman_hom.comm_monoid }
end freiman_hom
/-! ### Hom hierarchy -/
--TODO: change to `monoid_hom_class F A β → freiman_hom_class F A β n` once `map_multiset_prod` is
-- generalized
/-- A monoid homomorphism is naturally a `freiman_hom` on its entire domain.
We can't leave the domain `A : set α` of the `freiman_hom` a free variable, since it wouldn't be
inferrable. -/
@[to_additive]
instance monoid_hom.freiman_hom_class : freiman_hom_class (α →* β) set.univ β n :=
{ map_prod_eq_map_prod' := λ f s t _ _ _ _ h, by rw [←f.map_multiset_prod, h, f.map_multiset_prod] }
/-- A `monoid_hom` is naturally a `freiman_hom`. -/
@[to_additive add_monoid_hom.to_add_freiman_hom "An `add_monoid_hom` is naturally an
`add_freiman_hom`"]
def monoid_hom.to_freiman_hom (A : set α) (n : ℕ) (f : α →* β) : A →*[n] β :=
{ to_fun := f,
map_prod_eq_map_prod' := λ s t hsA htA, map_prod_eq_map_prod f
(λ _ _, set.mem_univ _) (λ _ _, set.mem_univ _) }
@[simp, to_additive]
lemma monoid_hom.to_freiman_hom_coe (f : α →* β) : (f.to_freiman_hom A n : α → β) = f := rfl
@[to_additive]
lemma monoid_hom.to_freiman_hom_injective :
function.injective (monoid_hom.to_freiman_hom A n : (α →* β) → A →*[n] β) :=
λ f g h, monoid_hom.ext $ show _, from fun_like.ext_iff.mp h
end comm_monoid
section cancel_comm_monoid
variables [comm_monoid α] [cancel_comm_monoid β] {A : set α} {m n : ℕ}
@[to_additive]
lemma map_prod_eq_map_prod_of_le [freiman_hom_class F A β n] (f : F) {s t : multiset α}
(hsA : ∀ x ∈ s, x ∈ A) (htA : ∀ x ∈ t, x ∈ A) (hs : s.card = m)
(ht : t.card = m) (hst : s.prod = t.prod) (h : m ≤ n) :
(s.map f).prod = (t.map f).prod :=
begin
obtain rfl | hm := m.eq_zero_or_pos,
{ rw card_eq_zero at hs ht,
rw [hs, ht] },
rw [←hs, card_pos_iff_exists_mem] at hm,
obtain ⟨a, ha⟩ := hm,
suffices : ((s + repeat a (n - m)).map f).prod = ((t + repeat a (n - m)).map f).prod,
{ simp_rw [multiset.map_add, prod_add] at this,
exact mul_right_cancel this },
replace ha := hsA _ ha,
refine map_prod_eq_map_prod f (λ x hx, _) (λ x hx, _) _ _ _,
rotate 2, assumption, -- Can't infer `A` and `n` from the context, so do it manually.
{ rw mem_add at hx,
refine hx.elim (hsA _) (λ h, _),
rwa eq_of_mem_repeat h },
{ rw mem_add at hx,
refine hx.elim (htA _) (λ h, _),
rwa eq_of_mem_repeat h },
{ rw [card_add, hs, card_repeat, add_tsub_cancel_of_le h] },
{ rw [card_add, ht, card_repeat, add_tsub_cancel_of_le h] },
{ rw [prod_add, prod_add, hst] }
end
/-- `α →*[n] β` is naturally included in `A →*[m] β` for any `m ≤ n`. -/
@[to_additive add_freiman_hom.to_add_freiman_hom "`α →+[n] β` is naturally included in `α →+[m] β`
for any `m ≤ n`"]
def freiman_hom.to_freiman_hom (h : m ≤ n) (f : A →*[n] β) : A →*[m] β :=
{ to_fun := f,
map_prod_eq_map_prod' := λ s t hsA htA hs ht hst,
map_prod_eq_map_prod_of_le f hsA htA hs ht hst h }
/-- A `n`-Freiman homomorphism is also a `m`-Freiman homomorphism for any `m ≤ n`. -/
@[to_additive add_freiman_hom.add_freiman_hom_class_of_le "An additive `n`-Freiman homomorphism is
also an additive `m`-Freiman homomorphism for any `m ≤ n`."]
def freiman_hom.freiman_hom_class_of_le [freiman_hom_class F A β n] (h : m ≤ n) :
freiman_hom_class F A β m :=
{ map_prod_eq_map_prod' := λ f s t hsA htA hs ht hst,
map_prod_eq_map_prod_of_le f hsA htA hs ht hst h }
@[simp, to_additive add_freiman_hom.to_add_freiman_hom_coe]
lemma freiman_hom.to_freiman_hom_coe (h : m ≤ n) (f : A →*[n] β) :
(f.to_freiman_hom h : α → β) = f := rfl
@[to_additive]
lemma freiman_hom.to_freiman_hom_injective (h : m ≤ n) :
function.injective (freiman_hom.to_freiman_hom h : (A →*[n] β) → A →*[m] β) :=
λ f g hfg, freiman_hom.ext $ by convert fun_like.ext_iff.1 hfg
end cancel_comm_monoid
|
a3d1b49cfc96e6bb37d4316659e89437bd4e8e7b | 4b846d8dabdc64e7ea03552bad8f7fa74763fc67 | /library/tools/super/cdcl.lean | a891281ad3017258ad2ca2e32eb8670312426e54 | [
"Apache-2.0"
] | permissive | pacchiano/lean | 9324b33f3ac3b5c5647285160f9f6ea8d0d767dc | fdadada3a970377a6df8afcd629a6f2eab6e84e8 | refs/heads/master | 1,611,357,380,399 | 1,489,870,101,000 | 1,489,870,101,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 863 | lean | /-
Copyright (c) 2016 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner
-/
import .clause .clausifier .cdcl_solver
open tactic expr monad super
meta def cdcl_t (th_solver : tactic unit) : tactic unit := do
as_refutation, local_false ← target,
clauses ← clauses_of_context, clauses ← get_clauses_classical clauses,
for clauses (λc, do c_pp ← pp c, clause.validate c <|> fail c_pp),
res ← cdcl.solve (cdcl.theory_solver_of_tactic th_solver) local_false clauses,
match res with
| (cdcl.result.unsat proof) := exact proof
| (cdcl.result.sat interp) :=
let interp' := do e ← interp^.to_list, [if e.2 = tt then e.1 else ```(not %%e.1)] in
do pp_interp ← pp interp',
fail (to_fmt "satisfying assignment: " ++ pp_interp)
end
meta def cdcl : tactic unit := cdcl_t skip
|
47a2b9a9ef9a2e19d74a14ca57d739acf6fe52f2 | d0c6b2ba2af981e9ab0a98f6e169262caad4b9b9 | /src/Init/Data/Array/Basic.lean | 8ec1eac84e5015804c498c55fe965ca832d9ba0a | [
"Apache-2.0"
] | permissive | fizruk/lean4 | 953b7dcd76e78c17a0743a2c1a918394ab64bbc0 | 545ed50f83c570f772ade4edbe7d38a078cbd761 | refs/heads/master | 1,677,655,987,815 | 1,612,393,885,000 | 1,612,393,885,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 27,330 | lean | /-
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Data.Nat.Basic
import Init.Data.Fin.Basic
import Init.Data.UInt
import Init.Data.Repr
import Init.Data.ToString.Basic
import Init.Util
universes u v w
namespace Array
variable {α : Type u}
@[extern "lean_mk_array"]
def mkArray {α : Type u} (n : Nat) (v : α) : Array α := {
data := List.replicate n v
}
theorem sizeMkArrayEq (n : Nat) (v : α) : (mkArray n v).size = n :=
List.lengthReplicateEq ..
instance : EmptyCollection (Array α) := ⟨Array.empty⟩
instance : Inhabited (Array α) where
default := Array.empty
def isEmpty (a : Array α) : Bool :=
a.size = 0
def singleton (v : α) : Array α :=
mkArray 1 v
/- Low-level version of `fget` which is as fast as a C array read.
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
`fget` may be slightly slower than `uget`. -/
@[extern "lean_array_uget"]
def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
a.get ⟨i.toNat, h⟩
def back [Inhabited α] (a : Array α) : α :=
a.get! (a.size - 1)
def get? (a : Array α) (i : Nat) : Option α :=
if h : i < a.size then some (a.get ⟨i, h⟩) else none
def back? (a : Array α) : Option α :=
a.get? (a.size - 1)
-- auxiliary declaration used in the equation compiler when pattern matching array literals.
abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
a.get ⟨i, h₁.symm ▸ h₂⟩
theorem sizeSetEq (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
List.lengthSetEq ..
theorem sizePushEq (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
List.lengthConcatEq ..
/- Low-level version of `fset` which is as fast as a C array fset.
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
`fset` may be slightly slower than `uset`. -/
@[extern "lean_array_uset"]
def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
a.set ⟨i.toNat, h⟩ v
@[extern "lean_array_fswap"]
def swap (a : Array α) (i j : @& Fin a.size) : Array α :=
let v₁ := a.get i
let v₂ := a.get j
let a' := a.set i v₂
a'.set (sizeSetEq a i v₂ ▸ j) v₁
@[extern "lean_array_swap"]
def swap! (a : Array α) (i j : @& Nat) : Array α :=
if h₁ : i < a.size then
if h₂ : j < a.size then swap a ⟨i, h₁⟩ ⟨j, h₂⟩
else panic! "index out of bounds"
else panic! "index out of bounds"
@[inline] def swapAt (a : Array α) (i : Fin a.size) (v : α) : α × Array α :=
let e := a.get i
let a := a.set i v
(e, a)
@[inline]
def swapAt! (a : Array α) (i : Nat) (v : α) : α × Array α :=
if h : i < a.size then
swapAt a ⟨i, h⟩ v
else
have Inhabited α from ⟨v⟩
panic! ("index " ++ toString i ++ " out of bounds")
@[extern "lean_array_pop"]
def pop (a : Array α) : Array α := {
data := a.data.dropLast
}
def shrink (a : Array α) (n : Nat) : Array α :=
let rec loop
| 0, a => a
| n+1, a => loop n a.pop
loop (a.size - n) a
@[inline]
def modifyM [Monad m] [Inhabited α] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩
let v := a.get idx
let a' := a.set idx arbitrary
let v ← f v
pure <| a'.set (sizeSetEq a .. ▸ idx) v
else
pure a
@[inline]
def modify [Inhabited α] (a : Array α) (i : Nat) (f : α → α) : Array α :=
Id.run <| a.modifyM i f
@[inline]
def modifyOp [Inhabited α] (self : Array α) (idx : Nat) (f : α → α) : Array α :=
self.modify idx f
/-
We claim this unsafe implementation is correct because an array cannot have more than `usizeSz` elements in our runtime.
This kind of low level trick can be removed with a little bit of compiler support. For example, if the compiler simplifies `as.size < usizeSz` to true. -/
@[inline] unsafe def forInUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
let sz := USize.ofNat as.size
let rec @[specialize] loop (i : USize) (b : β) : m β := do
if i < sz then
let a := as.uget i lcProof
match (← f a b) with
| ForInStep.done b => pure b
| ForInStep.yield b => loop (i+1) b
else
pure b
loop 0 b
-- Move?
private theorem zeroLtOfLt : {a b : Nat} → a < b → 0 < b
| 0, _, h => h
| a+1, b, h =>
have a < b from Nat.ltTrans (Nat.ltSuccSelf _) h
zeroLtOfLt this
/- Reference implementation for `forIn` -/
@[implementedBy Array.forInUnsafe]
def forIn {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
let rec loop (i : Nat) (h : i ≤ as.size) (b : β) : m β := do
match i, h with
| 0, _ => pure b
| i+1, h =>
have h' : i < as.size from Nat.ltOfLtOfLe (Nat.ltSuccSelf i) h
have as.size - 1 < as.size from Nat.subLt (zeroLtOfLt h') (decide! : 0 < 1)
have as.size - 1 - i < as.size from Nat.ltOfLeOfLt (Nat.subLe (as.size - 1) i) this
match (← f (as.get ⟨as.size - 1 - i, this⟩) b) with
| ForInStep.done b => pure b
| ForInStep.yield b => loop i (Nat.leOfLt h') b
loop as.size (Nat.leRefl _) b
/- See comment at forInUnsafe -/
@[inline]
unsafe def foldlMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Array α) (start := 0) (stop := as.size) : m β :=
let rec @[specialize] fold (i : USize) (stop : USize) (b : β) : m β := do
if i == stop then
pure b
else
fold (i+1) stop (← f b (as.uget i lcProof))
if start < stop then
if stop ≤ as.size then
fold (USize.ofNat start) (USize.ofNat stop) init
else
pure init
else
pure init
/- Reference implementation for `foldlM` -/
@[implementedBy foldlMUnsafe]
def foldlM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Array α) (start := 0) (stop := as.size) : m β :=
let fold (stop : Nat) (h : stop ≤ as.size) :=
let rec loop (i : Nat) (j : Nat) (b : β) : m β := do
if hlt : j < stop then
match i with
| 0 => pure b
| i'+1 =>
loop i' (j+1) (← f b (as.get ⟨j, Nat.ltOfLtOfLe hlt h⟩))
else
pure b
loop (stop - start) start init
if h : stop ≤ as.size then
fold stop h
else
fold as.size (Nat.leRefl _)
/- See comment at forInUnsafe -/
@[inline]
unsafe def foldrMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Array α) (start := as.size) (stop := 0) : m β :=
let rec @[specialize] fold (i : USize) (stop : USize) (b : β) : m β := do
if i == stop then
pure b
else
fold (i-1) stop (← f (as.uget (i-1) lcProof) b)
if start ≤ as.size then
if stop < start then
fold (USize.ofNat start) (USize.ofNat stop) init
else
pure init
else if stop < as.size then
fold (USize.ofNat as.size) (USize.ofNat stop) init
else
pure init
/- Reference implementation for `foldrM` -/
@[implementedBy foldrMUnsafe]
def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Array α) (start := as.size) (stop := 0) : m β :=
let rec fold (i : Nat) (h : i ≤ as.size) (b : β) : m β := do
if i == stop then
pure b
else match i, h with
| 0, _ => pure b
| i+1, h =>
have i < as.size from Nat.ltOfLtOfLe (Nat.ltSuccSelf _) h
fold i (Nat.leOfLt this) (← f (as.get ⟨i, this⟩) b)
if h : start ≤ as.size then
if stop < start then
fold start h init
else
pure init
else if stop < as.size then
fold as.size (Nat.leRefl _) init
else
pure init
/- See comment at forInUnsafe -/
@[inline]
unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
let sz := USize.ofNat as.size
let rec @[specialize] map (i : USize) (r : Array NonScalar) : m (Array PNonScalar.{v}) := do
if i < sz then
let v := r.uget i lcProof
let r := r.uset i arbitrary lcProof
let vNew ← f (unsafeCast v)
map (i+1) (r.uset i (unsafeCast vNew) lcProof)
else
pure (unsafeCast r)
unsafeCast <| map 0 (unsafeCast as)
/- Reference implementation for `mapM` -/
@[implementedBy mapMUnsafe]
def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
as.foldlM (fun bs a => do let b ← f a; pure (bs.push b)) (mkEmpty as.size)
@[inline]
def mapIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : Fin as.size → α → m β) : m (Array β) :=
let rec @[specialize] map (i : Nat) (j : Nat) (inv : i + j = as.size) (bs : Array β) : m (Array β) := do
match i, inv with
| 0, _ => pure bs
| i+1, inv =>
have j < as.size by rw [← inv, Nat.addAssoc, Nat.addComm 1 j, Nat.addLeftComm]; apply Nat.leAddRight
let idx : Fin as.size := ⟨j, this⟩
have i + (j + 1) = as.size by rw [← inv, Nat.addComm j 1, Nat.addAssoc]
map i (j+1) this (bs.push (← f idx (as.get idx)))
map as.size 0 rfl (mkEmpty as.size)
@[inline]
def findSomeM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m (Option β)) : m (Option β) := do
for a in as do
match (← f a) with
| some b => return b
| _ => pure ⟨⟩
return none
@[inline]
def findM? {α : Type} {m : Type → Type} [Monad m] (as : Array α) (p : α → m Bool) : m (Option α) := do
for a in as do
if (← p a) then
return a
return none
@[inline]
def findIdxM? [Monad m] (as : Array α) (p : α → m Bool) : m (Option Nat) := do
let mut i := 0
for a in as do
if (← p a) then
return i
i := i + 1
return none
@[inline]
unsafe def anyMUnsafe {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
let rec @[specialize] any (i : USize) (stop : USize) : m Bool := do
if i == stop then
pure false
else
if (← p (as.uget i lcProof)) then
pure true
else
any (i+1) stop
if start < stop then
if stop ≤ as.size then
any (USize.ofNat start) (USize.ofNat stop)
else
pure false
else
pure false
@[implementedBy anyMUnsafe]
def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
let any (stop : Nat) (h : stop ≤ as.size) :=
let rec loop (i : Nat) (j : Nat) : m Bool := do
if hlt : j < stop then
match i with
| 0 => pure false
| i'+1 =>
if (← p (as.get ⟨j, Nat.ltOfLtOfLe hlt h⟩)) then
pure true
else
loop i' (j+1)
else
pure false
loop (stop - start) start
if h : stop ≤ as.size then
any stop h
else
any as.size (Nat.leRefl _)
@[inline]
def allM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
return !(← as.anyM fun v => return !(← p v))
@[inline]
def findSomeRevM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m (Option β)) : m (Option β) :=
let rec @[specialize] find : (i : Nat) → i ≤ as.size → m (Option β)
| 0, h => pure none
| i+1, h => do
have i < as.size from Nat.ltOfLtOfLe (Nat.ltSuccSelf _) h
let r ← f (as.get ⟨i, this⟩)
match r with
| some v => pure r
| none =>
have i ≤ as.size from Nat.leOfLt this
find i this
find as.size (Nat.leRefl _)
@[inline]
def findRevM? {α : Type} {m : Type → Type w} [Monad m] (as : Array α) (p : α → m Bool) : m (Option α) :=
as.findSomeRevM? fun a => return if (← p a) then some a else none
@[inline]
def forM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Array α) (start := 0) (stop := as.size) : m PUnit :=
as.foldlM (fun _ => f) ⟨⟩ start stop
@[inline]
def forRevM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Array α) (start := as.size) (stop := 0) : m PUnit :=
as.foldrM (fun a _ => f a) ⟨⟩ start stop
@[inline]
def foldl {α : Type u} {β : Type v} (f : β → α → β) (init : β) (as : Array α) (start := 0) (stop := as.size) : β :=
Id.run <| as.foldlM f init start stop
@[inline]
def foldr {α : Type u} {β : Type v} (f : α → β → β) (init : β) (as : Array α) (start := as.size) (stop := 0) : β :=
Id.run <| as.foldrM f init start stop
@[inline]
def map {α : Type u} {β : Type v} (f : α → β) (as : Array α) : Array β :=
Id.run <| as.mapM f
@[inline]
def mapIdx {α : Type u} {β : Type v} (as : Array α) (f : Fin as.size → α → β) : Array β :=
Id.run <| as.mapIdxM f
@[inline]
def find? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
Id.run <| as.findM? p
@[inline]
def findSome? {α : Type u} {β : Type v} (as : Array α) (f : α → Option β) : Option β :=
Id.run <| as.findSomeM? f
@[inline]
def findSome! {α : Type u} {β : Type v} [Inhabited β] (a : Array α) (f : α → Option β) : β :=
match findSome? a f with
| some b => b
| none => panic! "failed to find element"
@[inline]
def findSomeRev? {α : Type u} {β : Type v} (as : Array α) (f : α → Option β) : Option β :=
Id.run <| as.findSomeRevM? f
@[inline]
def findRev? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
Id.run <| as.findRevM? p
@[inline]
def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
let rec loop (i : Nat) (j : Nat) (inv : i + j = as.size) : Option Nat :=
if hlt : j < as.size then
match i, inv with
| 0, inv => by
apply False.elim
rw [Nat.zeroAdd] at inv
rw [inv] at hlt
exact absurd hlt (Nat.ltIrrefl _)
| i+1, inv =>
if p (as.get ⟨j, hlt⟩) then
some j
else
have i + (j+1) = as.size by
rw [← inv, Nat.addComm j 1, Nat.addAssoc]
loop i (j+1) this
else
none
loop as.size 0 rfl
def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
a.findIdx? fun a => a == v
@[inline]
def any (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Bool :=
Id.run <| as.anyM p start stop
@[inline]
def all (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Bool :=
Id.run <| as.allM p start stop
def contains [BEq α] (as : Array α) (a : α) : Bool :=
as.any fun b => a == b
def elem [BEq α] (a : α) (as : Array α) : Bool :=
as.contains a
-- TODO(Leo): justify termination using wf-rec, and use `swap`
partial def reverse (as : Array α) : Array α :=
let n := as.size
let mid := n / 2
let rec rev (as : Array α) (i : Nat) :=
if i < mid then
rev (as.swap! i (n - i - 1)) (i+1)
else
as
rev as 0
@[inline] def getEvenElems (as : Array α) : Array α :=
(·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
if even then
(false, r.push a)
else
(true, r)
@[export lean_array_to_list]
def toList (as : Array α) : List α :=
as.foldr List.cons []
instance {α : Type u} [Repr α] : Repr (Array α) where
reprPrec a _ :=
if a.size == 0 then
"#[]"
else
Std.Format.bracketFill "#[" (@Std.Format.joinSep _ ⟨repr⟩ (toList a) ("," ++ Std.Format.line)) "]"
instance [ToString α] : ToString (Array α) where
toString a := "#" ++ toString a.toList
protected def append (as : Array α) (bs : Array α) : Array α :=
bs.foldl (init := as) fun r v => r.push v
instance : Append (Array α) := ⟨Array.append⟩
@[inline]
def concatMapM [Monad m] (f : α → m (Array β)) (as : Array α) : m (Array β) :=
as.foldlM (init := empty) fun bs a => do return bs ++ (← f a)
@[inline]
def concatMap (f : α → Array β) (as : Array α) : Array β :=
as.foldl (init := empty) fun bs a => bs ++ f a
end Array
@[inlineIfReduce]
def List.toArrayAux : List α → Array α → Array α
| [], r => r
| a::as, r => toArrayAux as (r.push a)
@[inlineIfReduce]
def List.redLength : List α → Nat
| [] => 0
| _::as => as.redLength + 1
@[inline, matchPattern, export lean_list_to_array]
def List.toArray (as : List α) : Array α :=
as.toArrayAux (Array.mkEmpty as.redLength)
export Array (mkArray)
syntax "#[" sepBy(term, ", ") "]" : term
macro_rules
| `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
namespace Array
-- TODO(Leo): cleanup
@[specialize]
partial def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
if h : i < a.size then
let aidx : Fin a.size := ⟨i, h⟩;
let bidx : Fin b.size := ⟨i, hsz ▸ h⟩;
match p (a.get aidx) (b.get bidx) with
| true => isEqvAux a b hsz p (i+1)
| false => false
else
true
@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
if h : a.size = b.size then
isEqvAux a b h p 0
else
false
instance [BEq α] : BEq (Array α) :=
⟨fun a b => isEqv a b BEq.beq⟩
@[inline]
def filter (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Array α :=
as.foldl (init := #[]) (start := start) (stop := stop) fun r a =>
if p a then r.push a else r
@[inline]
def filterM [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
as.foldlM (init := #[]) (start := start) (stop := stop) fun r a => do
if (← p a) then r.push a else r
@[specialize]
def filterMapM [Monad m] (f : α → m (Option β)) (as : Array α) (start := 0) (stop := as.size) : m (Array β) :=
as.foldlM (init := #[]) (start := start) (stop := stop) fun bs a => do
match (← f a) with
| some b => pure (bs.push b)
| none => pure bs
@[inline]
def filterMap (f : α → Option β) (as : Array α) (start := 0) (stop := as.size) : Array β :=
Id.run <| as.filterMapM f (start := start) (stop := stop)
@[specialize]
def getMax? (as : Array α) (lt : α → α → Bool) : Option α :=
if h : 0 < as.size then
let a0 := as.get ⟨0, h⟩
some <| as.foldl (init := a0) (start := 1) fun best a =>
if lt best a then a else best
else
none
@[inline]
def partition (p : α → Bool) (as : Array α) : Array α × Array α := do
let mut bs := #[]
let mut cs := #[]
for a in as do
if p a then
bs := bs.push a
else
cs := cs.push a
return (bs, cs)
theorem ext (a b : Array α)
(h₁ : a.size = b.size)
(h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
: a = b := by
let rec extAux (a b : List α)
(h₁ : a.length = b.length)
(h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get i hi₁ = b.get i hi₂)
: a = b := by
induction a generalizing b with
| nil =>
cases b with
| nil => rfl
| cons b bs => rw [List.lengthConsEq] at h₁; injection h₁
| cons a as ih =>
cases b with
| nil => rw [List.lengthConsEq] at h₁; injection h₁
| cons b bs =>
have hz₁ : 0 < (a::as).length by rw [List.lengthConsEq]; apply Nat.zeroLtSucc
have hz₂ : 0 < (b::bs).length by rw [List.lengthConsEq]; apply Nat.zeroLtSucc
have headEq : a = b from h₂ 0 hz₁ hz₂
have h₁' : as.length = bs.length by rw [List.lengthConsEq, List.lengthConsEq] at h₁; injection h₁; assumption
have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get i hi₁ = bs.get i hi₂ by
intro i hi₁ hi₂
have hi₁' : i+1 < (a::as).length by rw [List.lengthConsEq]; apply Nat.succLtSucc; assumption
have hi₂' : i+1 < (b::bs).length by rw [List.lengthConsEq]; apply Nat.succLtSucc; assumption
have (a::as).get (i+1) hi₁' = (b::bs).get (i+1) hi₂' from h₂ (i+1) hi₁' hi₂'
apply this
have tailEq : as = bs from ih bs h₁' h₂'
rw [headEq, tailEq]
cases a; cases b
apply congrArg
apply extAux
assumption
assumption
theorem extLit {n : Nat}
(a b : Array α)
(hsz₁ : a.size = n) (hsz₂ : b.size = n)
(h : (i : Nat) → (hi : i < n) → a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ hi₂ => h i (hsz₁ ▸ hi₁)
end Array
-- CLEANUP the following code
namespace Array
partial def indexOfAux [BEq α] (a : Array α) (v : α) : Nat → Option (Fin a.size)
| i =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
if a.get idx == v then some idx
else indexOfAux a v (i+1)
else none
def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
indexOfAux a v 0
partial def eraseIdxAux : Nat → Array α → Array α
| i, a =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
let idx1 : Fin a.size := ⟨i - 1, by exact Nat.ltOfLeOfLt (Nat.predLe i) h⟩;
eraseIdxAux (i+1) (a.swap idx idx1)
else
a.pop
def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
eraseIdxAux (i.val + 1) a
def eraseIdx (a : Array α) (i : Nat) : Array α :=
if i < a.size then eraseIdxAux (i+1) a else a
theorem sizeSwapEq (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size := by
show ((a.set i (a.get j)).set (sizeSetEq a i _ ▸ j) (a.get i)).size = a.size
rw [sizeSetEq, sizeSetEq]
theorem sizePopEq (a : Array α) : a.pop.size = a.size - 1 :=
List.lengthDropLast ..
section
/- Instance for justifying `partial` declaration.
We should be able to delete it as soon as we restore support for well-founded recursion. -/
instance eraseIdxSzAuxInstance (a : Array α) : Inhabited { r : Array α // r.size = a.size - 1 } where
default := ⟨a.pop, sizePopEq a⟩
partial def eraseIdxSzAux (a : Array α) : ∀ (i : Nat) (r : Array α), r.size = a.size → { r : Array α // r.size = a.size - 1 }
| i, r, heq =>
if h : i < r.size then
let idx : Fin r.size := ⟨i, h⟩;
let idx1 : Fin r.size := ⟨i - 1, by exact Nat.ltOfLeOfLt (Nat.predLe i) h⟩;
eraseIdxSzAux a (i+1) (r.swap idx idx1) ((sizeSwapEq r idx idx1).trans heq)
else
⟨r.pop, (sizePopEq r).trans (heq ▸ rfl)⟩
end
def eraseIdx' (a : Array α) (i : Fin a.size) : { r : Array α // r.size = a.size - 1 } :=
eraseIdxSzAux a (i.val + 1) a rfl
def erase [BEq α] (as : Array α) (a : α) : Array α :=
match as.indexOf? a with
| none => as
| some i => as.feraseIdx i
partial def insertAtAux (i : Nat) : Array α → Nat → Array α
| as, j =>
if i == j then as
else
let as := as.swap! (j-1) j;
insertAtAux i as (j-1)
/--
Insert element `a` at position `i`.
Pre: `i < as.size` -/
def insertAt (as : Array α) (i : Nat) (a : α) : Array α :=
if i > as.size then panic! "invalid index"
else
let as := as.push a;
as.insertAtAux i as.size
def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
| 0, hi, acc => acc
| (i+1), hi, acc => toListLitAux a n hsz i (Nat.leOfSuccLe hi) (a.getLit i hsz (Nat.ltOfLtOfEq (Nat.ltOfLtOfLe (Nat.ltSuccSelf i) hi) hsz) :: acc)
def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.leRefl _) []
theorem toArrayLitEq (a : Array α) (n : Nat) (hsz : a.size = n) : a = toArrayLit a n hsz :=
-- TODO: this is painful to prove without proper automation
sorry
/-
First, we need to prove
∀ i j acc, i ≤ a.size → (toListLitAux a n hsz (i+1) hi acc).index j = if j < i then a.getLit j hsz _ else acc.index (j - i)
by induction
Base case is trivial
(j : Nat) (acc : List α) (hi : 0 ≤ a.size)
|- (toListLitAux a n hsz 0 hi acc).index j = if j < 0 then a.getLit j hsz _ else acc.index (j - 0)
... |- acc.index j = acc.index j
Induction
(j : Nat) (acc : List α) (hi : i+1 ≤ a.size)
|- (toListLitAux a n hsz (i+1) hi acc).index j = if j < i + 1 then a.getLit j hsz _ else acc.index (j - (i + 1))
... |- (toListLitAux a n hsz i hi' (a.getLit i hsz _ :: acc)).index j = if j < i + 1 then a.getLit j hsz _ else acc.index (j - (i + 1)) * by def
... |- if j < i then a.getLit j hsz _ else (a.getLit i hsz _ :: acc).index (j-i) * by induction hypothesis
=
if j < i + 1 then a.getLit j hsz _ else acc.index (j - (i + 1))
If j < i, then both are a.getLit j hsz _
If j = i, then lhs reduces else-branch to (a.getLit i hsz _) and rhs is then-brachn (a.getLit i hsz _)
If j >= i + 1, we use
- j - i >= 1 > 0
- (a::as).index k = as.index (k-1) If k > 0
- j - (i + 1) = (j - i) - 1
Then lhs = (a.getLit i hsz _ :: acc).index (j-i) = acc.index (j-i-1) = acc.index (j-(i+1)) = rhs
With this proof, we have
∀ j, j < n → (toListLitAux a n hsz n _ []).index j = a.getLit j hsz _
We also need
- (toListLitAux a n hsz n _ []).length = n
- j < n -> (List.toArray as).getLit j _ _ = as.index j
Then using Array.extLit, we have that a = List.toArray <| toListLitAux a n hsz n _ []
-/
partial def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) : Nat → Bool
| i =>
if h : i < as.size then
let a := as.get ⟨i, h⟩;
let b := bs.get ⟨i, Nat.ltOfLtOfLe h hle⟩;
if a == b then
isPrefixOfAux as bs hle (i+1)
else
false
else
true
/- Return true iff `as` is a prefix of `bs` -/
def isPrefixOf [BEq α] (as bs : Array α) : Bool :=
if h : as.size ≤ bs.size then
isPrefixOfAux as bs h 0
else
false
private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat), i < as.size → Bool
| 0, h => true
| i+1, h =>
have i < as.size from Nat.ltTrans (Nat.ltSuccSelf _) h;
a != as.get ⟨i, this⟩ && allDiffAuxAux as a i this
private partial def allDiffAux [BEq α] (as : Array α) : Nat → Bool
| i =>
if h : i < as.size then
allDiffAuxAux as (as.get ⟨i, h⟩) i h && allDiffAux as (i+1)
else
true
def allDiff [BEq α] (as : Array α) : Bool :=
allDiffAux as 0
@[specialize] partial def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) : Nat → Array γ → Array γ
| i, cs =>
if h : i < as.size then
let a := as.get ⟨i, h⟩;
if h : i < bs.size then
let b := bs.get ⟨i, h⟩;
zipWithAux f as bs (i+1) <| cs.push <| f a b
else
cs
else
cs
@[inline] def zipWith (as : Array α) (bs : Array β) (f : α → β → γ) : Array γ :=
zipWithAux f as bs 0 #[]
def zip (as : Array α) (bs : Array β) : Array (α × β) :=
zipWith as bs Prod.mk
end Array
|
a51455059ebc86e0c2409eeb588d04deb81335ef | c777c32c8e484e195053731103c5e52af26a25d1 | /src/analysis/special_functions/complex/log_deriv.lean | 2d43df3418d50bb5b86becb401f9a63ba97c6ba6 | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 6,336 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import analysis.special_functions.complex.log
import analysis.special_functions.exp_deriv
/-!
# Differentiability of the complex `log` function
-/
noncomputable theory
namespace complex
open set filter
open_locale real topology
/-- `complex.exp` as a `local_homeomorph` with `source = {z | -π < im z < π}` and
`target = {z | 0 < re z} ∪ {z | im z ≠ 0}`. This definition is used to prove that `complex.log`
is complex differentiable at all points but the negative real semi-axis. -/
def exp_local_homeomorph : local_homeomorph ℂ ℂ :=
local_homeomorph.of_continuous_open
{ to_fun := exp,
inv_fun := log,
source := {z : ℂ | z.im ∈ Ioo (- π) π},
target := {z : ℂ | 0 < z.re} ∪ {z : ℂ | z.im ≠ 0},
map_source' :=
begin
rintro ⟨x, y⟩ ⟨h₁ : -π < y, h₂ : y < π⟩,
refine (not_or_of_imp $ λ hz, _).symm,
obtain rfl : y = 0,
{ rw exp_im at hz,
simpa [(real.exp_pos _).ne', real.sin_eq_zero_iff_of_lt_of_lt h₁ h₂] using hz },
rw [mem_set_of_eq, ← of_real_def, exp_of_real_re],
exact real.exp_pos x
end,
map_target' := λ z h,
suffices 0 ≤ z.re ∨ z.im ≠ 0,
by simpa [log_im, neg_pi_lt_arg, (arg_le_pi _).lt_iff_ne, arg_eq_pi_iff, not_and_distrib],
h.imp (λ h, le_of_lt h) id,
left_inv' := λ x hx, log_exp hx.1 (le_of_lt hx.2),
right_inv' := λ x hx, exp_log $ by { rintro rfl, simpa [lt_irrefl] using hx } }
continuous_exp.continuous_on is_open_map_exp (is_open_Ioo.preimage continuous_im)
lemma has_strict_deriv_at_log {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) :
has_strict_deriv_at log x⁻¹ x :=
have h0 : x ≠ 0, by { rintro rfl, simpa [lt_irrefl] using h },
exp_local_homeomorph.has_strict_deriv_at_symm h h0 $
by simpa [exp_log h0] using has_strict_deriv_at_exp (log x)
lemma has_strict_fderiv_at_log_real {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) :
has_strict_fderiv_at log (x⁻¹ • (1 : ℂ →L[ℝ] ℂ)) x :=
(has_strict_deriv_at_log h).complex_to_real_fderiv
lemma cont_diff_at_log {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) {n : ℕ∞} :
cont_diff_at ℂ n log x :=
exp_local_homeomorph.cont_diff_at_symm_deriv (exp_ne_zero $ log x) h
(has_deriv_at_exp _) cont_diff_exp.cont_diff_at
end complex
section log_deriv
open complex filter
open_locale topology
variables {α : Type*} [topological_space α] {E : Type*} [normed_add_comm_group E] [normed_space ℂ E]
lemma has_strict_fderiv_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E}
(h₁ : has_strict_fderiv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_fderiv_at (λ t, log (f t)) ((f x)⁻¹ • f') x :=
(has_strict_deriv_at_log h₂).comp_has_strict_fderiv_at x h₁
lemma has_strict_deriv_at.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : has_strict_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_deriv_at (λ t, log (f t)) (f' / f x) x :=
by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).comp x h₁ }
lemma has_strict_deriv_at.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : has_strict_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_deriv_at (λ t, log (f t)) (f' / f x) x :=
by simpa only [div_eq_inv_mul]
using (has_strict_fderiv_at_log_real h₂).comp_has_strict_deriv_at x h₁
lemma has_fderiv_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E}
(h₁ : has_fderiv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_fderiv_at (λ t, log (f t)) ((f x)⁻¹ • f') x :=
(has_strict_deriv_at_log h₂).has_deriv_at.comp_has_fderiv_at x h₁
lemma has_deriv_at.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : has_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_at (λ t, log (f t)) (f' / f x) x :=
by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).has_deriv_at.comp x h₁ }
lemma has_deriv_at.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : has_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_at (λ t, log (f t)) (f' / f x) x :=
by simpa only [div_eq_inv_mul]
using (has_strict_fderiv_at_log_real h₂).has_fderiv_at.comp_has_deriv_at x h₁
lemma differentiable_at.clog {f : E → ℂ} {x : E} (h₁ : differentiable_at ℂ f x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_at ℂ (λ t, log (f t)) x :=
(h₁.has_fderiv_at.clog h₂).differentiable_at
lemma has_fderiv_within_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {s : set E} {x : E}
(h₁ : has_fderiv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_fderiv_within_at (λ t, log (f t)) ((f x)⁻¹ • f') s x :=
(has_strict_deriv_at_log h₂).has_deriv_at.comp_has_fderiv_within_at x h₁
lemma has_deriv_within_at.clog {f : ℂ → ℂ} {f' x : ℂ} {s : set ℂ}
(h₁ : has_deriv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_within_at (λ t, log (f t)) (f' / f x) s x :=
by { rw div_eq_inv_mul,
exact (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_deriv_within_at x h₁ }
lemma has_deriv_within_at.clog_real {f : ℝ → ℂ} {s : set ℝ} {x : ℝ} {f' : ℂ}
(h₁ : has_deriv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_within_at (λ t, log (f t)) (f' / f x) s x :=
by simpa only [div_eq_inv_mul]
using (has_strict_fderiv_at_log_real h₂).has_fderiv_at.comp_has_deriv_within_at x h₁
lemma differentiable_within_at.clog {f : E → ℂ} {s : set E} {x : E}
(h₁ : differentiable_within_at ℂ f s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_within_at ℂ (λ t, log (f t)) s x :=
(h₁.has_fderiv_within_at.clog h₂).differentiable_within_at
lemma differentiable_on.clog {f : E → ℂ} {s : set E}
(h₁ : differentiable_on ℂ f s) (h₂ : ∀ x ∈ s, 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_on ℂ (λ t, log (f t)) s :=
λ x hx, (h₁ x hx).clog (h₂ x hx)
lemma differentiable.clog {f : E → ℂ} (h₁ : differentiable ℂ f)
(h₂ : ∀ x, 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable ℂ (λ t, log (f t)) :=
λ x, (h₁ x).clog (h₂ x)
end log_deriv
|
d9f238899496a308fcb947f4643ec44680385b4b | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/set_theory/cardinal_ordinal.lean | 0306d8e4902a4941e6bfbcf54f024d14547bf30a | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 38,202 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import set_theory.ordinal_arithmetic
import tactic.linarith
import logic.small
/-!
# Cardinals and ordinals
Relationships between cardinals and ordinals, properties of cardinals that are proved
using ordinals.
## Main definitions
* The function `cardinal.aleph'` gives the cardinals listed by their ordinal
index, and is the inverse of `cardinal.aleph_idx`.
`aleph' n = n`, `aleph' ω = cardinal.omega = ℵ₀`, `aleph' (ω + 1) = ℵ₁`, etc.
It is an order isomorphism between ordinals and cardinals.
* The function `cardinal.aleph` gives the infinite cardinals listed by their
ordinal index. `aleph 0 = cardinal.omega = ℵ₀`, `aleph 1 = ℵ₁` is the first
uncountable cardinal, and so on.
## Main Statements
* `cardinal.mul_eq_max` and `cardinal.add_eq_max` state that the product (resp. sum) of two infinite
cardinals is just their maximum. Several variations around this fact are also given.
* `cardinal.mk_list_eq_mk` : when `α` is infinite, `α` and `list α` have the same cardinality.
* simp lemmas for inequalities between `bit0 a` and `bit1 b` are registered, making `simp`
able to prove inequalities about numeral cardinals.
## Tags
cardinal arithmetic (for infinite cardinals)
-/
noncomputable theory
open function cardinal set equiv
open_locale classical cardinal
universes u v w
namespace cardinal
section using_ordinals
open ordinal
theorem ord_is_limit {c} (co : omega ≤ c) : (ord c).is_limit :=
begin
refine ⟨λ h, omega_ne_zero _, λ a, lt_imp_lt_of_le_imp_le _⟩,
{ rw [← ordinal.le_zero, ord_le] at h,
simpa only [card_zero, nonpos_iff_eq_zero] using le_trans co h },
{ intro h, rw [ord_le] at h ⊢,
rwa [← @add_one_of_omega_le (card a), ← card_succ],
rw [← ord_le, ← le_succ_of_is_limit, ord_le],
{ exact le_trans co h },
{ rw ord_omega, exact omega_is_limit } }
end
/-- The `aleph'` index function, which gives the ordinal index of a cardinal.
(The `aleph'` part is because unlike `aleph` this counts also the
finite stages. So `aleph_idx n = n`, `aleph_idx ω = ω`,
`aleph_idx ℵ₁ = ω + 1` and so on.)
In this definition, we register additionally that this function is an initial segment,
i.e., it is order preserving and its range is an initial segment of the ordinals.
For the basic function version, see `aleph_idx`.
For an upgraded version stating that the range is everything, see `aleph_idx.rel_iso`. -/
def aleph_idx.initial_seg : @initial_seg cardinal ordinal (<) (<) :=
@rel_embedding.collapse cardinal ordinal (<) (<) _ cardinal.ord.order_embedding.lt_embedding
/-- The `aleph'` index function, which gives the ordinal index of a cardinal.
(The `aleph'` part is because unlike `aleph` this counts also the
finite stages. So `aleph_idx n = n`, `aleph_idx ω = ω`,
`aleph_idx ℵ₁ = ω + 1` and so on.)
For an upgraded version stating that the range is everything, see `aleph_idx.rel_iso`. -/
def aleph_idx : cardinal → ordinal := aleph_idx.initial_seg
@[simp] theorem aleph_idx.initial_seg_coe :
(aleph_idx.initial_seg : cardinal → ordinal) = aleph_idx := rfl
@[simp] theorem aleph_idx_lt {a b} : aleph_idx a < aleph_idx b ↔ a < b :=
aleph_idx.initial_seg.to_rel_embedding.map_rel_iff
@[simp] theorem aleph_idx_le {a b} : aleph_idx a ≤ aleph_idx b ↔ a ≤ b :=
by rw [← not_lt, ← not_lt, aleph_idx_lt]
theorem aleph_idx.init {a b} : b < aleph_idx a → ∃ c, aleph_idx c = b :=
aleph_idx.initial_seg.init _ _
/-- The `aleph'` index function, which gives the ordinal index of a cardinal.
(The `aleph'` part is because unlike `aleph` this counts also the
finite stages. So `aleph_idx n = n`, `aleph_idx ω = ω`,
`aleph_idx ℵ₁ = ω + 1` and so on.)
In this version, we register additionally that this function is an order isomorphism
between cardinals and ordinals.
For the basic function version, see `aleph_idx`. -/
def aleph_idx.rel_iso : @rel_iso cardinal.{u} ordinal.{u} (<) (<) :=
@rel_iso.of_surjective cardinal.{u} ordinal.{u} (<) (<) aleph_idx.initial_seg.{u} $
(initial_seg.eq_or_principal aleph_idx.initial_seg.{u}).resolve_right $
λ ⟨o, e⟩, begin
have : ∀ c, aleph_idx c < o := λ c, (e _).2 ⟨_, rfl⟩,
refine ordinal.induction_on o _ this, introsI α r _ h,
let s := sup.{u u} (λ a:α, inv_fun aleph_idx (ordinal.typein r a)),
apply not_le_of_gt (lt_succ_self s),
have I : injective aleph_idx := aleph_idx.initial_seg.to_embedding.injective,
simpa only [typein_enum, left_inverse_inv_fun I (succ s)] using
le_sup.{u u} (λ a, inv_fun aleph_idx (ordinal.typein r a))
(ordinal.enum r _ (h (succ s))),
end
@[simp] theorem aleph_idx.rel_iso_coe :
(aleph_idx.rel_iso : cardinal → ordinal) = aleph_idx := rfl
@[simp] theorem type_cardinal : @ordinal.type cardinal (<) _ = ordinal.univ.{u (u+1)} :=
by rw ordinal.univ_id; exact quotient.sound ⟨aleph_idx.rel_iso⟩
@[simp] theorem mk_cardinal : #cardinal = univ.{u (u+1)} :=
by simpa only [card_type, card_univ] using congr_arg card type_cardinal
/-- The `aleph'` function gives the cardinals listed by their ordinal
index, and is the inverse of `aleph_idx`.
`aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = ℵ₁`, etc.
In this version, we register additionally that this function is an order isomorphism
between ordinals and cardinals.
For the basic function version, see `aleph'`. -/
def aleph'.rel_iso := cardinal.aleph_idx.rel_iso.symm
/-- The `aleph'` function gives the cardinals listed by their ordinal
index, and is the inverse of `aleph_idx`.
`aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = ℵ₁`, etc. -/
def aleph' : ordinal → cardinal := aleph'.rel_iso
@[simp] theorem aleph'.rel_iso_coe :
(aleph'.rel_iso : ordinal → cardinal) = aleph' := rfl
@[simp] theorem aleph'_lt {o₁ o₂ : ordinal.{u}} : aleph' o₁ < aleph' o₂ ↔ o₁ < o₂ :=
aleph'.rel_iso.map_rel_iff
@[simp] theorem aleph'_le {o₁ o₂ : ordinal.{u}} : aleph' o₁ ≤ aleph' o₂ ↔ o₁ ≤ o₂ :=
le_iff_le_iff_lt_iff_lt.2 aleph'_lt
@[simp] theorem aleph'_aleph_idx (c : cardinal.{u}) : aleph' c.aleph_idx = c :=
cardinal.aleph_idx.rel_iso.to_equiv.symm_apply_apply c
@[simp] theorem aleph_idx_aleph' (o : ordinal.{u}) : (aleph' o).aleph_idx = o :=
cardinal.aleph_idx.rel_iso.to_equiv.apply_symm_apply o
@[simp] theorem aleph'_zero : aleph' 0 = 0 :=
by rw [← nonpos_iff_eq_zero, ← aleph'_aleph_idx 0, aleph'_le];
apply ordinal.zero_le
@[simp] theorem aleph'_succ {o : ordinal.{u}} : aleph' o.succ = (aleph' o).succ :=
le_antisymm
(cardinal.aleph_idx_le.1 $
by rw [aleph_idx_aleph', ordinal.succ_le, ← aleph'_lt, aleph'_aleph_idx];
apply cardinal.lt_succ_self)
(cardinal.succ_le.2 $ aleph'_lt.2 $ ordinal.lt_succ_self _)
@[simp] theorem aleph'_nat : ∀ n : ℕ, aleph' n = n
| 0 := aleph'_zero
| (n+1) := show aleph' (ordinal.succ n) = n.succ,
by rw [aleph'_succ, aleph'_nat, nat_succ]
theorem aleph'_le_of_limit {o : ordinal.{u}} (l : o.is_limit) {c} :
aleph' o ≤ c ↔ ∀ o' < o, aleph' o' ≤ c :=
⟨λ h o' h', le_trans (aleph'_le.2 $ le_of_lt h') h,
λ h, begin
rw [← aleph'_aleph_idx c, aleph'_le, ordinal.limit_le l],
intros x h',
rw [← aleph'_le, aleph'_aleph_idx],
exact h _ h'
end⟩
@[simp] theorem aleph'_omega : aleph' ordinal.omega = omega :=
eq_of_forall_ge_iff $ λ c, begin
simp only [aleph'_le_of_limit omega_is_limit, ordinal.lt_omega, exists_imp_distrib, omega_le],
exact forall_swap.trans (forall_congr $ λ n, by simp only [forall_eq, aleph'_nat]),
end
/-- `aleph'` and `aleph_idx` form an equivalence between `ordinal` and `cardinal` -/
@[simp] def aleph'_equiv : ordinal ≃ cardinal :=
⟨aleph', aleph_idx, aleph_idx_aleph', aleph'_aleph_idx⟩
/-- The `aleph` function gives the infinite cardinals listed by their
ordinal index. `aleph 0 = ω`, `aleph 1 = succ ω` is the first
uncountable cardinal, and so on. -/
def aleph (o : ordinal) : cardinal := aleph' (ordinal.omega + o)
@[simp] theorem aleph_lt {o₁ o₂ : ordinal.{u}} : aleph o₁ < aleph o₂ ↔ o₁ < o₂ :=
aleph'_lt.trans (ordinal.add_lt_add_iff_left _)
@[simp] theorem aleph_le {o₁ o₂ : ordinal.{u}} : aleph o₁ ≤ aleph o₂ ↔ o₁ ≤ o₂ :=
le_iff_le_iff_lt_iff_lt.2 aleph_lt
@[simp] theorem aleph_succ {o : ordinal.{u}} : aleph o.succ = (aleph o).succ :=
by rw [aleph, ordinal.add_succ, aleph'_succ]; refl
@[simp] theorem aleph_zero : aleph 0 = omega :=
by simp only [aleph, add_zero, aleph'_omega]
theorem omega_le_aleph' {o : ordinal} : omega ≤ aleph' o ↔ ordinal.omega ≤ o :=
by rw [← aleph'_omega, aleph'_le]
theorem omega_le_aleph (o : ordinal) : omega ≤ aleph o :=
by rw [aleph, omega_le_aleph']; apply ordinal.le_add_right
theorem ord_aleph_is_limit (o : ordinal) : is_limit (aleph o).ord :=
ord_is_limit $ omega_le_aleph _
theorem exists_aleph {c : cardinal} : omega ≤ c ↔ ∃ o, c = aleph o :=
⟨λ h, ⟨aleph_idx c - ordinal.omega,
by rw [aleph, ordinal.add_sub_cancel_of_le, aleph'_aleph_idx];
rwa [← omega_le_aleph', aleph'_aleph_idx]⟩,
λ ⟨o, e⟩, e.symm ▸ omega_le_aleph _⟩
theorem aleph'_is_normal : is_normal (ord ∘ aleph') :=
⟨λ o, ord_lt_ord.2 $ aleph'_lt.2 $ ordinal.lt_succ_self _,
λ o l a, by simp only [ord_le, aleph'_le_of_limit l]⟩
theorem aleph_is_normal : is_normal (ord ∘ aleph) :=
aleph'_is_normal.trans $ add_is_normal ordinal.omega
lemma countable_iff_lt_aleph_one {α : Type*} (s : set α) : countable s ↔ #s < aleph 1 :=
begin
have : aleph 1 = (aleph 0).succ, by simp only [← aleph_succ, ordinal.succ_zero],
rw [countable_iff, ← aleph_zero, this, lt_succ],
end
/-! ### Properties of `mul` -/
/-- If `α` is an infinite type, then `α × α` and `α` have the same cardinality. -/
theorem mul_eq_self {c : cardinal} (h : omega ≤ c) : c * c = c :=
begin
refine le_antisymm _
(by simpa only [mul_one] using
mul_le_mul_left' (one_lt_omega.le.trans h) c),
-- the only nontrivial part is `c * c ≤ c`. We prove it inductively.
refine acc.rec_on (cardinal.wf.apply c) (λ c _,
quotient.induction_on c $ λ α IH ol, _) h,
-- consider the minimal well-order `r` on `α` (a type with cardinality `c`).
rcases ord_eq α with ⟨r, wo, e⟩, resetI,
letI := linear_order_of_STO' r,
haveI : is_well_order α (<) := wo,
-- Define an order `s` on `α × α` by writing `(a, b) < (c, d)` if `max a b < max c d`, or
-- the max are equal and `a < c`, or the max are equal and `a = c` and `b < d`.
let g : α × α → α := λ p, max p.1 p.2,
let f : α × α ↪ ordinal × (α × α) :=
⟨λ p:α×α, (typein (<) (g p), p), λ p q, congr_arg prod.snd⟩,
let s := f ⁻¹'o (prod.lex (<) (prod.lex (<) (<))),
-- this is a well order on `α × α`.
haveI : is_well_order _ s := (rel_embedding.preimage _ _).is_well_order,
/- it suffices to show that this well order is smaller than `r`
if it were larger, then `r` would be a strict prefix of `s`. It would be contained in
`β × β` for some `β` of cardinality `< c`. By the inductive assumption, this set has the
same cardinality as `β` (or it is finite if `β` is finite), so it is `< c`, which is a
contradiction. -/
suffices : type s ≤ type r, {exact card_le_card this},
refine le_of_forall_lt (λ o h, _),
rcases typein_surj s h with ⟨p, rfl⟩,
rw [← e, lt_ord],
refine lt_of_le_of_lt (_ : _ ≤ card (typein (<) (g p)).succ * card (typein (<) (g p)).succ) _,
{ have : {q|s q p} ⊆ (insert (g p) {x | x < (g p)}).prod (insert (g p) {x | x < (g p)}),
{ intros q h,
simp only [s, embedding.coe_fn_mk, order.preimage, typein_lt_typein, prod.lex_def, typein_inj]
at h,
exact max_le_iff.1 (le_iff_lt_or_eq.2 $ h.imp_right and.left) },
suffices H : (insert (g p) {x | r x (g p)} : set α) ≃ ({x | r x (g p)} ⊕ punit),
{ exact ⟨(set.embedding_of_subset _ _ this).trans
((equiv.set.prod _ _).trans (H.prod_congr H)).to_embedding⟩ },
refine (equiv.set.insert _).trans
((equiv.refl _).sum_congr punit_equiv_punit),
apply @irrefl _ r },
cases lt_or_ge (card (typein (<) (g p)).succ) omega with qo qo,
{ exact lt_of_lt_of_le (mul_lt_omega qo qo) ol },
{ suffices, {exact lt_of_le_of_lt (IH _ this qo) this},
rw ← lt_ord, apply (ord_is_limit ol).2,
rw [mk_def, e], apply typein_lt_type }
end
end using_ordinals
/-- If `α` and `β` are infinite types, then the cardinality of `α × β` is the maximum
of the cardinalities of `α` and `β`. -/
theorem mul_eq_max {a b : cardinal} (ha : omega ≤ a) (hb : omega ≤ b) : a * b = max a b :=
le_antisymm
(mul_eq_self (le_trans ha (le_max_left a b)) ▸
mul_le_mul' (le_max_left _ _) (le_max_right _ _)) $
max_le
(by simpa only [mul_one] using
mul_le_mul_left' (one_lt_omega.le.trans hb) a)
(by simpa only [one_mul] using
mul_le_mul_right' (one_lt_omega.le.trans ha) b)
theorem mul_lt_of_lt {a b c : cardinal} (hc : omega ≤ c)
(h1 : a < c) (h2 : b < c) : a * b < c :=
lt_of_le_of_lt (mul_le_mul' (le_max_left a b) (le_max_right a b)) $
(lt_or_le (max a b) omega).elim
(λ h, lt_of_lt_of_le (mul_lt_omega h h) hc)
(λ h, by rw mul_eq_self h; exact max_lt h1 h2)
lemma mul_le_max_of_omega_le_left {a b : cardinal} (h : omega ≤ a) : a * b ≤ max a b :=
begin
convert mul_le_mul' (le_max_left a b) (le_max_right a b),
rw [mul_eq_self],
refine le_trans h (le_max_left a b)
end
lemma mul_eq_max_of_omega_le_left {a b : cardinal} (h : omega ≤ a) (h' : b ≠ 0) : a * b = max a b :=
begin
apply le_antisymm, apply mul_le_max_of_omega_le_left h,
cases le_or_gt omega b with hb hb, rw [mul_eq_max h hb],
have : b ≤ a, exact le_trans (le_of_lt hb) h,
rw [max_eq_left this],
convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') _, rw [mul_one],
end
theorem mul_le_max (a b : cardinal) : a * b ≤ max (max a b) ω :=
begin
by_cases ha0 : a = 0,
{ simp [ha0] },
by_cases hb0 : b = 0,
{ simp [hb0] },
by_cases ha : ω ≤ a,
{ rw [mul_eq_max_of_omega_le_left ha hb0],
exact le_max_left _ _ },
{ by_cases hb : ω ≤ b,
{ rw [mul_comm, mul_eq_max_of_omega_le_left hb ha0, max_comm],
exact le_max_left _ _ },
{ exact le_max_of_le_right (le_of_lt (mul_lt_omega (lt_of_not_ge ha) (lt_of_not_ge hb))) } }
end
lemma mul_eq_left {a b : cardinal} (ha : omega ≤ a) (hb : b ≤ a) (hb' : b ≠ 0) : a * b = a :=
by { rw [mul_eq_max_of_omega_le_left ha hb', max_eq_left hb] }
lemma mul_eq_right {a b : cardinal} (hb : omega ≤ b) (ha : a ≤ b) (ha' : a ≠ 0) : a * b = b :=
by { rw [mul_comm, mul_eq_left hb ha ha'] }
lemma le_mul_left {a b : cardinal} (h : b ≠ 0) : a ≤ b * a :=
by { convert mul_le_mul_right' (one_le_iff_ne_zero.mpr h) _,
rw [one_mul] }
lemma le_mul_right {a b : cardinal} (h : b ≠ 0) : a ≤ a * b :=
by { rw [mul_comm], exact le_mul_left h }
lemma mul_eq_left_iff {a b : cardinal} : a * b = a ↔ ((max omega b ≤ a ∧ b ≠ 0) ∨ b = 1 ∨ a = 0) :=
begin
rw [max_le_iff], split,
{ intro h,
cases (le_or_lt omega a) with ha ha,
{ have : a ≠ 0, { rintro rfl, exact not_lt_of_le ha omega_pos },
left, use ha,
{ rw [← not_lt], intro hb, apply ne_of_gt _ h, refine lt_of_lt_of_le hb (le_mul_left this) },
{ rintro rfl, apply this, rw [_root_.mul_zero] at h, subst h }},
right, by_cases h2a : a = 0, { right, exact h2a },
have hb : b ≠ 0, { rintro rfl, apply h2a, rw [mul_zero] at h, subst h },
left, rw [← h, mul_lt_omega_iff, lt_omega, lt_omega] at ha,
rcases ha with rfl|rfl|⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩, contradiction, contradiction,
rw [← ne] at h2a, rw [← one_le_iff_ne_zero] at h2a hb, norm_cast at h2a hb h ⊢,
apply le_antisymm _ hb, rw [← not_lt], intro h2b,
apply ne_of_gt _ h, conv_lhs { rw [← mul_one n] },
rwa [mul_lt_mul_left], apply nat.lt_of_succ_le h2a },
{ rintro (⟨⟨ha, hab⟩, hb⟩|rfl|rfl),
{ rw [mul_eq_max_of_omega_le_left ha hb, max_eq_left hab] },
all_goals {simp}}
end
/-! ### Properties of `add` -/
/-- If `α` is an infinite type, then `α ⊕ α` and `α` have the same cardinality. -/
theorem add_eq_self {c : cardinal} (h : omega ≤ c) : c + c = c :=
le_antisymm
(by simpa only [nat.cast_bit0, nat.cast_one, mul_eq_self h, two_mul] using
mul_le_mul_right' ((nat_lt_omega 2).le.trans h) c)
(self_le_add_left c c)
/-- If `α` is an infinite type, then the cardinality of `α ⊕ β` is the maximum
of the cardinalities of `α` and `β`. -/
theorem add_eq_max {a b : cardinal} (ha : omega ≤ a) : a + b = max a b :=
le_antisymm
(add_eq_self (le_trans ha (le_max_left a b)) ▸
add_le_add (le_max_left _ _) (le_max_right _ _)) $
max_le (self_le_add_right _ _) (self_le_add_left _ _)
theorem add_le_max (a b : cardinal) : a + b ≤ max (max a b) ω :=
begin
by_cases ha : ω ≤ a,
{ rw [add_eq_max ha],
exact le_max_left _ _ },
{ by_cases hb : ω ≤ b,
{ rw [add_comm, add_eq_max hb, max_comm],
exact le_max_left _ _ },
{ exact le_max_of_le_right (le_of_lt (add_lt_omega (lt_of_not_ge ha) (lt_of_not_ge hb))) } }
end
theorem add_lt_of_lt {a b c : cardinal} (hc : omega ≤ c)
(h1 : a < c) (h2 : b < c) : a + b < c :=
lt_of_le_of_lt (add_le_add (le_max_left a b) (le_max_right a b)) $
(lt_or_le (max a b) omega).elim
(λ h, lt_of_lt_of_le (add_lt_omega h h) hc)
(λ h, by rw add_eq_self h; exact max_lt h1 h2)
lemma eq_of_add_eq_of_omega_le {a b c : cardinal} (h : a + b = c) (ha : a < c) (hc : omega ≤ c) :
b = c :=
begin
apply le_antisymm,
{ rw [← h], apply self_le_add_left },
rw[← not_lt], intro hb,
have : a + b < c := add_lt_of_lt hc ha hb,
simpa [h, lt_irrefl] using this
end
lemma add_eq_left {a b : cardinal} (ha : omega ≤ a) (hb : b ≤ a) : a + b = a :=
by { rw [add_eq_max ha, max_eq_left hb] }
lemma add_eq_right {a b : cardinal} (hb : omega ≤ b) (ha : a ≤ b) : a + b = b :=
by { rw [add_comm, add_eq_left hb ha] }
lemma add_eq_left_iff {a b : cardinal} : a + b = a ↔ (max omega b ≤ a ∨ b = 0) :=
begin
rw [max_le_iff], split,
{ intro h, cases (le_or_lt omega a) with ha ha,
{ left, use ha, rw [← not_lt], intro hb, apply ne_of_gt _ h,
exact lt_of_lt_of_le hb (self_le_add_left b a) },
right, rw [← h, add_lt_omega_iff, lt_omega, lt_omega] at ha,
rcases ha with ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩, norm_cast at h ⊢,
rw [← add_right_inj, h, add_zero] },
{ rintro (⟨h1, h2⟩|h3), rw [add_eq_max h1, max_eq_left h2], rw [h3, add_zero] }
end
lemma add_eq_right_iff {a b : cardinal} : a + b = b ↔ (max omega a ≤ b ∨ a = 0) :=
by { rw [add_comm, add_eq_left_iff] }
lemma add_one_eq {a : cardinal} (ha : omega ≤ a) : a + 1 = a :=
have 1 ≤ a, from le_trans (le_of_lt one_lt_omega) ha,
add_eq_left ha this
protected lemma eq_of_add_eq_add_left {a b c : cardinal} (h : a + b = a + c) (ha : a < omega) :
b = c :=
begin
cases le_or_lt omega b with hb hb,
{ have : a < b := lt_of_lt_of_le ha hb,
rw [add_eq_right hb (le_of_lt this), eq_comm] at h,
rw [eq_of_add_eq_of_omega_le h this hb] },
{ have hc : c < omega,
{ rw [← not_le], intro hc,
apply lt_irrefl omega, apply lt_of_le_of_lt (le_trans hc (self_le_add_left _ a)),
rw [← h], apply add_lt_omega ha hb },
rw [lt_omega] at *,
rcases ha with ⟨n, rfl⟩, rcases hb with ⟨m, rfl⟩, rcases hc with ⟨k, rfl⟩,
norm_cast at h ⊢, apply add_left_cancel h }
end
protected lemma eq_of_add_eq_add_right {a b c : cardinal} (h : a + b = c + b) (hb : b < omega) :
a = c :=
by { rw [add_comm a b, add_comm c b] at h, exact cardinal.eq_of_add_eq_add_left h hb }
/-! ### Properties about power -/
theorem pow_le {κ μ : cardinal.{u}} (H1 : omega ≤ κ) (H2 : μ < omega) : κ ^ μ ≤ κ :=
let ⟨n, H3⟩ := lt_omega.1 H2 in
H3.symm ▸ (quotient.induction_on κ (λ α H1, nat.rec_on n
(le_of_lt $ lt_of_lt_of_le (by rw [nat.cast_zero, power_zero];
from one_lt_omega) H1)
(λ n ih, trans_rel_left _
(by { rw [nat.cast_succ, power_add, power_one];
exact mul_le_mul_right' ih _ })
(mul_eq_self H1))) H1)
lemma power_self_eq {c : cardinal} (h : omega ≤ c) : c ^ c = 2 ^ c :=
begin
apply le_antisymm,
{ apply le_trans (power_le_power_right $ le_of_lt $ cantor c), rw [← power_mul, mul_eq_self h] },
{ convert power_le_power_right (le_trans (le_of_lt $ nat_lt_omega 2) h), apply nat.cast_two.symm }
end
lemma nat_power_eq {c : cardinal.{u}} (h : omega ≤ c) {n : ℕ} (hn : 2 ≤ n) :
(n : cardinal.{u}) ^ c = 2 ^ c :=
le_antisymm (power_self_eq h ▸ power_le_power_right ((nat_lt_omega n).le.trans h))
(power_le_power_right $ by assumption_mod_cast)
lemma power_nat_le {c : cardinal.{u}} {n : ℕ} (h : omega ≤ c) : c ^ (n : cardinal.{u}) ≤ c :=
pow_le h (nat_lt_omega n)
lemma power_nat_le_max {c : cardinal.{u}} {n : ℕ} : c ^ (n : cardinal.{u}) ≤ max c ω :=
begin
by_cases hc : ω ≤ c,
{ exact le_max_of_le_left (power_nat_le hc) },
{ exact le_max_of_le_right (le_of_lt (power_lt_omega (lt_of_not_ge hc) (nat_lt_omega _))) }
end
lemma powerlt_omega {c : cardinal} (h : omega ≤ c) : c ^< omega = c :=
begin
apply le_antisymm,
{ rw [powerlt_le], intro c', rw [lt_omega], rintro ⟨n, rfl⟩, apply power_nat_le h },
convert le_powerlt one_lt_omega, rw [power_one]
end
lemma powerlt_omega_le (c : cardinal) : c ^< omega ≤ max c omega :=
begin
cases le_or_gt omega c,
{ rw [powerlt_omega h], apply le_max_left },
rw [powerlt_le], intros c' hc',
refine le_trans (le_of_lt $ power_lt_omega h hc') (le_max_right _ _)
end
/-! ### Computing cardinality of various types -/
theorem mk_list_eq_mk {α : Type u} (H1 : ω ≤ #α) : #(list α) = #α :=
eq.symm $ le_antisymm ⟨⟨λ x, [x], λ x y H, (list.cons.inj H).1⟩⟩ $
calc #(list α)
= sum (λ n : ℕ, #α ^ (n : cardinal.{u})) : mk_list_eq_sum_pow α
... ≤ sum (λ n : ℕ, #α) : sum_le_sum _ _ $ λ n, pow_le H1 $ nat_lt_omega n
... = sum (λ n : ulift.{u} ℕ, #α) : quotient.sound
⟨@sigma_congr_left _ _ (λ _, quotient.out (#α)) equiv.ulift.symm⟩
... = omega * #α : sum_const _ _
... = max ω (#α) : mul_eq_max (le_refl _) H1
... = #α : max_eq_right H1
theorem mk_finset_eq_mk {α : Type u} (h : omega ≤ #α) : #(finset α) = #α :=
eq.symm $ le_antisymm (mk_le_of_injective (λ x y, finset.singleton_inj.1)) $
calc #(finset α) ≤ #(list α) : mk_le_of_surjective list.to_finset_surjective
... = #α : mk_list_eq_mk h
lemma mk_bounded_set_le_of_omega_le (α : Type u) (c : cardinal) (hα : omega ≤ #α) :
#{t : set α // mk t ≤ c} ≤ #α ^ c :=
begin
refine le_trans _ (by rw [←add_one_eq hα]), refine quotient.induction_on c _, clear c, intro β,
fapply mk_le_of_surjective,
{ intro f, use sum.inl ⁻¹' range f,
refine le_trans (mk_preimage_of_injective _ _ (λ x y, sum.inl.inj)) _,
apply mk_range_le },
rintro ⟨s, ⟨g⟩⟩,
use λ y, if h : ∃(x : s), g x = y then sum.inl (classical.some h).val else sum.inr ⟨⟩,
apply subtype.eq, ext,
split,
{ rintro ⟨y, h⟩, dsimp only at h, by_cases h' : ∃ (z : s), g z = y,
{ rw [dif_pos h'] at h, cases sum.inl.inj h, exact (classical.some h').2 },
{ rw [dif_neg h'] at h, cases h }},
{ intro h, have : ∃(z : s), g z = g ⟨x, h⟩, exact ⟨⟨x, h⟩, rfl⟩,
use g ⟨x, h⟩, dsimp only, rw [dif_pos this], congr',
suffices : classical.some this = ⟨x, h⟩, exact congr_arg subtype.val this,
apply g.2, exact classical.some_spec this }
end
lemma mk_bounded_set_le (α : Type u) (c : cardinal) :
#{t : set α // #t ≤ c} ≤ max (#α) omega ^ c :=
begin
transitivity #{t : set (ulift.{u} nat ⊕ α) // #t ≤ c},
{ refine ⟨embedding.subtype_map _ _⟩, apply embedding.image,
use sum.inr, apply sum.inr.inj, intros s hs, exact le_trans mk_image_le hs },
refine le_trans
(mk_bounded_set_le_of_omega_le (ulift.{u} nat ⊕ α) c (self_le_add_right omega (#α))) _,
rw [max_comm, ←add_eq_max]; refl
end
lemma mk_bounded_subset_le {α : Type u} (s : set α) (c : cardinal.{u}) :
#{t : set α // t ⊆ s ∧ #t ≤ c} ≤ max (#s) omega ^ c :=
begin
refine le_trans _ (mk_bounded_set_le s c),
refine ⟨embedding.cod_restrict _ _ _⟩,
use λ t, coe ⁻¹' t.1,
{ rintros ⟨t, ht1, ht2⟩ ⟨t', h1t', h2t'⟩ h, apply subtype.eq, dsimp only at h ⊢,
refine (preimage_eq_preimage' _ _).1 h; rw [subtype.range_coe]; assumption },
rintro ⟨t, h1t, h2t⟩, exact le_trans (mk_preimage_of_injective _ _ subtype.val_injective) h2t
end
/-! ### Properties of `compl` -/
lemma mk_compl_of_omega_le {α : Type*} (s : set α) (h : omega ≤ #α) (h2 : #s < #α) :
#(sᶜ : set α) = #α :=
by { refine eq_of_add_eq_of_omega_le _ h2 h, exact mk_sum_compl s }
lemma mk_compl_finset_of_omega_le {α : Type*} (s : finset α) (h : omega ≤ #α) :
#((↑s)ᶜ : set α) = #α :=
by { apply mk_compl_of_omega_le _ h, exact lt_of_lt_of_le (finset_card_lt_omega s) h }
lemma mk_compl_eq_mk_compl_infinite {α : Type*} {s t : set α} (h : omega ≤ #α) (hs : #s < #α)
(ht : #t < #α) : #(sᶜ : set α) = #(tᶜ : set α) :=
by { rw [mk_compl_of_omega_le s h hs, mk_compl_of_omega_le t h ht] }
lemma mk_compl_eq_mk_compl_finite_lift {α : Type u} {β : Type v} {s : set α} {t : set β}
(hα : #α < omega) (h1 : lift.{(max v w)} (#α) = lift.{(max u w)} (#β))
(h2 : lift.{(max v w)} (#s) = lift.{(max u w)} (#t)) :
lift.{(max v w)} (#(sᶜ : set α)) = lift.{(max u w)} (#(tᶜ : set β)) :=
begin
have hα' := hα, have h1' := h1,
rw [← mk_sum_compl s, ← mk_sum_compl t] at h1,
rw [← mk_sum_compl s, add_lt_omega_iff] at hα,
lift #s to ℕ using hα.1 with n hn,
lift #(sᶜ : set α) to ℕ using hα.2 with m hm,
have : #(tᶜ : set β) < omega,
{ refine lt_of_le_of_lt (mk_subtype_le _) _,
rw [← lift_lt, lift_omega, ← h1', ← lift_omega.{u (max v w)}, lift_lt], exact hα' },
lift #(tᶜ : set β) to ℕ using this with k hk,
simp [nat_eq_lift_eq_iff] at h2, rw [nat_eq_lift_eq_iff.{v (max u w)}] at h2,
simp [h2.symm] at h1 ⊢, norm_cast at h1, simp at h1, exact h1
end
lemma mk_compl_eq_mk_compl_finite {α β : Type u} {s : set α} {t : set β}
(hα : #α < omega) (h1 : #α = #β) (h : #s = #t) : #(sᶜ : set α) = #(tᶜ : set β) :=
by { rw [← lift_inj], apply mk_compl_eq_mk_compl_finite_lift hα; rw [lift_inj]; assumption }
lemma mk_compl_eq_mk_compl_finite_same {α : Type*} {s t : set α} (hα : #α < omega)
(h : #s = #t) : #(sᶜ : set α) = #(tᶜ : set α) :=
mk_compl_eq_mk_compl_finite hα rfl h
/-! ### Extending an injection to an equiv -/
theorem extend_function {α β : Type*} {s : set α} (f : s ↪ β)
(h : nonempty ((sᶜ : set α) ≃ ((range f)ᶜ : set β))) :
∃ (g : α ≃ β), ∀ x : s, g x = f x :=
begin
intros, have := h, cases this with g,
let h : α ≃ β := (set.sum_compl (s : set α)).symm.trans
((sum_congr (equiv.of_injective f f.2) g).trans
(set.sum_compl (range f))),
refine ⟨h, _⟩, rintro ⟨x, hx⟩, simp [set.sum_compl_symm_apply_of_mem, hx]
end
theorem extend_function_finite {α β : Type*} {s : set α} (f : s ↪ β)
(hs : #α < omega) (h : nonempty (α ≃ β)) : ∃ (g : α ≃ β), ∀ x : s, g x = f x :=
begin
apply extend_function f,
have := h, cases this with g,
rw [← lift_mk_eq] at h,
rw [←lift_mk_eq, mk_compl_eq_mk_compl_finite_lift hs h],
rw [mk_range_eq_lift], exact f.2
end
theorem extend_function_of_lt {α β : Type*} {s : set α} (f : s ↪ β) (hs : #s < #α)
(h : nonempty (α ≃ β)) : ∃ (g : α ≃ β), ∀ x : s, g x = f x :=
begin
cases (le_or_lt omega (#α)) with hα hα,
{ apply extend_function f, have := h, cases this with g, rw [← lift_mk_eq] at h,
cases cardinal.eq.mp (mk_compl_of_omega_le s hα hs) with g2,
cases cardinal.eq.mp (mk_compl_of_omega_le (range f) _ _) with g3,
{ constructor, exact g2.trans (g.trans g3.symm) },
{ rw [← lift_le, ← h], refine le_trans _ (lift_le.mpr hα), simp },
rwa [← lift_lt, ← h, mk_range_eq_lift, lift_lt], exact f.2 },
{ exact extend_function_finite f hα h }
end
section bit
/-!
This section proves inequalities for `bit0` and `bit1`, enabling `simp` to solve inequalities
for numeral cardinals. The complexity of the resulting algorithm is not good, as in some cases
`simp` reduces an inequality to a disjunction of two situations, depending on whether a cardinal
is finite or infinite. Since the evaluation of the branches is not lazy, this is bad. It is good
enough for practical situations, though.
For specific numbers, these inequalities could also be deduced from the corresponding
inequalities of natural numbers using `norm_cast`:
```
example : (37 : cardinal) < 42 :=
by { norm_cast, norm_num }
```
-/
lemma bit0_ne_zero (a : cardinal) : ¬bit0 a = 0 ↔ ¬a = 0 :=
by simp [bit0]
@[simp] lemma bit1_ne_zero (a : cardinal) : ¬bit1 a = 0 :=
by simp [bit1]
@[simp] lemma zero_lt_bit0 (a : cardinal) : 0 < bit0 a ↔ 0 < a :=
by { rw ←not_iff_not, simp [bit0], }
@[simp] lemma zero_lt_bit1 (a : cardinal) : 0 < bit1 a :=
lt_of_lt_of_le zero_lt_one (self_le_add_left _ _)
@[simp] lemma one_le_bit0 (a : cardinal) : 1 ≤ bit0 a ↔ 0 < a :=
⟨λ h, (zero_lt_bit0 a).mp (lt_of_lt_of_le zero_lt_one h),
λ h, le_trans (one_le_iff_pos.mpr h) (self_le_add_left a a)⟩
@[simp] lemma one_le_bit1 (a : cardinal) : 1 ≤ bit1 a :=
self_le_add_left _ _
theorem bit0_eq_self {c : cardinal} (h : omega ≤ c) : bit0 c = c :=
add_eq_self h
@[simp] theorem bit0_lt_omega {c : cardinal} : bit0 c < omega ↔ c < omega :=
by simp [bit0, add_lt_omega_iff]
@[simp] theorem omega_le_bit0 {c : cardinal} : omega ≤ bit0 c ↔ omega ≤ c :=
by { rw ← not_iff_not, simp }
@[simp] theorem bit1_eq_self_iff {c : cardinal} : bit1 c = c ↔ omega ≤ c :=
begin
by_cases h : omega ≤ c,
{ simp only [bit1, bit0_eq_self h, h, eq_self_iff_true, add_one_of_omega_le] },
{ simp only [h, iff_false],
apply ne_of_gt,
rcases lt_omega.1 (not_le.1 h) with ⟨n, rfl⟩,
norm_cast,
dsimp [bit1, bit0],
linarith }
end
@[simp] theorem bit1_lt_omega {c : cardinal} : bit1 c < omega ↔ c < omega :=
by simp [bit1, bit0, add_lt_omega_iff, one_lt_omega]
@[simp] theorem omega_le_bit1 {c : cardinal} : omega ≤ bit1 c ↔ omega ≤ c :=
by { rw ← not_iff_not, simp }
@[simp] lemma bit0_le_bit0 {a b : cardinal} : bit0 a ≤ bit0 b ↔ a ≤ b :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ rw [bit0_eq_self ha, bit0_eq_self hb] },
{ rw bit0_eq_self ha,
have I1 : ¬ (a ≤ b),
{ assume h, apply hb, exact le_trans ha h },
have I2 : ¬ (a ≤ bit0 b),
{ assume h,
have A : bit0 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_of_lt_of_le (lt_of_lt_of_le A ha) h) },
simp [I1, I2] },
{ rw [bit0_eq_self hb],
simp only [not_le] at ha,
have I1 : a ≤ b := le_of_lt (lt_of_lt_of_le ha hb),
have I2 : bit0 a ≤ b := le_trans (le_of_lt (bit0_lt_omega.2 ha)) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp }
end
@[simp] lemma bit0_le_bit1 {a b : cardinal} : bit0 a ≤ bit1 b ↔ a ≤ b :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ rw [bit0_eq_self ha, bit1_eq_self_iff.2 hb], },
{ rw bit0_eq_self ha,
have I1 : ¬ (a ≤ b),
{ assume h, apply hb, exact le_trans ha h },
have I2 : ¬ (a ≤ bit1 b),
{ assume h,
have A : bit1 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_of_lt_of_le (lt_of_lt_of_le A ha) h) },
simp [I1, I2] },
{ rw [bit1_eq_self_iff.2 hb],
simp only [not_le] at ha,
have I1 : a ≤ b := le_of_lt (lt_of_lt_of_le ha hb),
have I2 : bit0 a ≤ b := le_trans (le_of_lt (bit0_lt_omega.2 ha)) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp }
end
@[simp] lemma bit1_le_bit1 {a b : cardinal} : bit1 a ≤ bit1 b ↔ a ≤ b :=
begin
split,
{ assume h,
apply bit0_le_bit1.1 (le_trans (self_le_add_right (bit0 a) 1) h) },
{ assume h,
calc a + a + 1 ≤ a + b + 1 : add_le_add_right (add_le_add_left h a) 1
... ≤ b + b + 1 : add_le_add_right (add_le_add_right h b) 1 }
end
@[simp] lemma bit1_le_bit0 {a b : cardinal} : bit1 a ≤ bit0 b ↔ (a < b ∨ (a ≤ b ∧ omega ≤ a)) :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ simp only [bit1_eq_self_iff.mpr ha, bit0_eq_self hb, ha, and_true],
refine ⟨λ h, or.inr h, λ h, _⟩,
cases h,
{ exact le_of_lt h },
{ exact h } },
{ rw bit1_eq_self_iff.2 ha,
have I1 : ¬ (a ≤ b),
{ assume h, apply hb, exact le_trans ha h },
have I2 : ¬ (a ≤ bit0 b),
{ assume h,
have A : bit0 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_of_lt_of_le (lt_of_lt_of_le A ha) h) },
simp [I1, I2, le_of_not_ge I1] },
{ rw [bit0_eq_self hb],
simp only [not_le] at ha,
have I1 : a < b := lt_of_lt_of_le ha hb,
have I2 : bit1 a ≤ b := le_trans (le_of_lt (bit1_lt_omega.2 ha)) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp [not_le.mpr ha], }
end
@[simp] lemma bit0_lt_bit0 {a b : cardinal} : bit0 a < bit0 b ↔ a < b :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ rw [bit0_eq_self ha, bit0_eq_self hb] },
{ rw bit0_eq_self ha,
have I1 : ¬ (a < b),
{ assume h, apply hb, exact le_trans ha (le_of_lt h) },
have I2 : ¬ (a < bit0 b),
{ assume h,
have A : bit0 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_trans (lt_of_lt_of_le A ha) h) },
simp [I1, I2] },
{ rw [bit0_eq_self hb],
simp only [not_le] at ha,
have I1 : a < b := lt_of_lt_of_le ha hb,
have I2 : bit0 a < b := lt_of_lt_of_le (bit0_lt_omega.2 ha) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp }
end
@[simp] lemma bit1_lt_bit0 {a b : cardinal} : bit1 a < bit0 b ↔ a < b :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ rw [bit1_eq_self_iff.2 ha, bit0_eq_self hb], },
{ rw bit1_eq_self_iff.2 ha,
have I1 : ¬ (a < b),
{ assume h, apply hb, exact le_of_lt (lt_of_le_of_lt ha h) },
have I2 : ¬ (a < bit0 b),
{ assume h,
have A : bit0 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_trans (lt_of_lt_of_le A ha) h) },
simp [I1, I2] },
{ rw [bit0_eq_self hb],
simp only [not_le] at ha,
have I1 : a < b := (lt_of_lt_of_le ha hb),
have I2 : bit1 a < b := lt_of_lt_of_le (bit1_lt_omega.2 ha) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp }
end
@[simp] lemma bit1_lt_bit1 {a b : cardinal} : bit1 a < bit1 b ↔ a < b :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ rw [bit1_eq_self_iff.2 ha, bit1_eq_self_iff.2 hb], },
{ rw bit1_eq_self_iff.2 ha,
have I1 : ¬ (a < b),
{ assume h, apply hb, exact le_of_lt (lt_of_le_of_lt ha h) },
have I2 : ¬ (a < bit1 b),
{ assume h,
have A : bit1 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_trans (lt_of_lt_of_le A ha) h) },
simp [I1, I2] },
{ rw [bit1_eq_self_iff.2 hb],
simp only [not_le] at ha,
have I1 : a < b := (lt_of_lt_of_le ha hb),
have I2 : bit1 a < b := lt_of_lt_of_le (bit1_lt_omega.2 ha) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp }
end
@[simp] lemma bit0_lt_bit1 {a b : cardinal} : bit0 a < bit1 b ↔ (a < b ∨ (a ≤ b ∧ a < omega)) :=
begin
by_cases ha : omega ≤ a; by_cases hb : omega ≤ b,
{ simp [bit0_eq_self ha, bit1_eq_self_iff.2 hb, not_lt.mpr ha] },
{ rw bit0_eq_self ha,
have I1 : ¬ (a < b),
{ assume h, apply hb, exact le_of_lt (lt_of_le_of_lt ha h) },
have I2 : ¬ (a < bit1 b),
{ assume h,
have A : bit1 b < omega, by simpa using hb,
exact lt_irrefl _ (lt_trans (lt_of_lt_of_le A ha) h) },
simp [I1, I2, not_lt.mpr ha] },
{ rw [bit1_eq_self_iff.2 hb],
simp only [not_le] at ha,
have I1 : a < b := (lt_of_lt_of_le ha hb),
have I2 : bit0 a < b := lt_of_lt_of_le (bit0_lt_omega.2 ha) hb,
simp [I1, I2] },
{ simp at ha hb,
rcases lt_omega.1 ha with ⟨m, rfl⟩,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
norm_cast,
simp only [ha, and_true, nat.bit0_lt_bit1_iff],
refine ⟨λ h, or.inr h, λ h, _⟩,
cases h,
{ exact le_of_lt h },
{ exact h } }
end
lemma one_lt_two : (1 : cardinal) < 2 :=
-- This strategy works generally to prove inequalities between numerals in `cardinality`.
by { norm_cast, norm_num }
@[simp] lemma one_lt_bit0 {a : cardinal} : 1 < bit0 a ↔ 0 < a :=
by simp [← bit1_zero]
@[simp] lemma one_lt_bit1 (a : cardinal) : 1 < bit1 a ↔ 0 < a :=
by simp [← bit1_zero]
@[simp] lemma one_le_one : (1 : cardinal) ≤ 1 :=
le_refl _
end bit
end cardinal
lemma not_injective_of_ordinal {α : Type u} (f : ordinal.{u} → α) :
¬ function.injective f :=
begin
let g : ordinal.{u} → ulift.{u+1} α := λ o, ulift.up (f o),
suffices : ¬ function.injective g,
{ intro hf, exact this (equiv.ulift.symm.injective.comp hf) },
intro hg,
replace hg := cardinal.mk_le_of_injective hg,
rw ← cardinal.lift_mk at hg,
have := hg.trans_lt (cardinal.lift_lt_univ _),
rw cardinal.univ_id at this,
exact lt_irrefl _ this
end
lemma not_injective_of_ordinal_of_small {α : Type v} [small.{u} α] (f : ordinal.{u} → α) :
¬ function.injective f :=
begin
intro hf,
apply not_injective_of_ordinal (equiv_shrink α ∘ f),
exact (equiv_shrink _).injective.comp hf,
end
|
f19d198a766feea461f9a9606091c653ec5867b9 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/tactic/show_term_auto.lean | 6ea89dd1b925a2dc87171e942eaca66e62b7a0c0 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,127 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.tactic.core
import Mathlib.PostPort
namespace Mathlib
namespace tactic.interactive
/--
`show_term { tac }` runs the tactic `tac`,
and then prints the term that was constructed.
This is useful for
* constructing term mode proofs from tactic mode proofs, and
* understanding what tactics are doing, and how metavariables are handled.
As an example, in
```
example {P Q R : Prop} (h₁ : Q → P) (h₂ : R) (h₃ : R → Q) : P ∧ R :=
by show_term { tauto }
```
the term mode proof `⟨h₁ (h₃ h₂), eq.mpr rfl h₂⟩` produced by `tauto` will be printed.
As another example, if the goal is `ℕ × ℕ`, `show_term { split, exact 0 }` will
print `refine (0, _)`, and afterwards there will be one remaining goal (of type `ℕ`).
This indicates that `split, exact 0` partially filled in the original metavariable,
but created a new metavariable for the resulting sub-goal.
-/
end Mathlib |
bcef69b6e80970d67cda24e9e3e5ba5b7aa4375f | 618003631150032a5676f229d13a079ac875ff77 | /src/linear_algebra/finsupp.lean | 0deea29944d3c39e96585ae6b131217fb7e50b2d | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 17,115 | lean | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Johannes Hölzl
Linear structures on function with finite support `α →₀ M`.
-/
import data.monoid_algebra
noncomputable theory
open set linear_map submodule
open_locale classical
namespace finsupp
variables {α : Type*} {M : Type*} {R : Type*}
variables [ring R] [add_comm_group M] [module R M]
def lsingle (a : α) : M →ₗ[R] (α →₀ M) :=
⟨single a, assume a b, single_add, assume c b, (smul_single _ _ _).symm⟩
def lapply (a : α) : (α →₀ M) →ₗ[R] M := ⟨λg, g a, assume a b, rfl, assume a b, rfl⟩
section lsubtype_domain
variables (s : set α)
def lsubtype_domain : (α →₀ M) →ₗ[R] (s →₀ M) :=
⟨subtype_domain (λx, x ∈ s), assume a b, subtype_domain_add, assume c a, ext $ assume a, rfl⟩
lemma lsubtype_domain_apply (f : α →₀ M) :
(lsubtype_domain s : (α →₀ M) →ₗ[R] (s →₀ M)) f = subtype_domain (λx, x ∈ s) f := rfl
end lsubtype_domain
@[simp] lemma lsingle_apply (a : α) (b : M) : (lsingle a : M →ₗ[R] (α →₀ M)) b = single a b :=
rfl
@[simp] lemma lapply_apply (a : α) (f : α →₀ M) : (lapply a : (α →₀ M) →ₗ[R] M) f = f a :=
rfl
@[simp] lemma ker_lsingle (a : α) : (lsingle a : M →ₗ[R] (α →₀ M)).ker = ⊥ :=
ker_eq_bot.2 (injective_single a)
lemma lsingle_range_le_ker_lapply (s t : set α) (h : disjoint s t) :
(⨆a∈s, (lsingle a : M →ₗ[R] (α →₀ M)).range) ≤ (⨅a∈t, ker (lapply a)) :=
begin
refine supr_le (assume a₁, supr_le $ assume h₁, range_le_iff_comap.2 _),
simp only [(ker_comp _ _).symm, eq_top_iff, le_def', mem_ker, comap_infi, mem_infi],
assume b hb a₂ h₂,
have : a₁ ≠ a₂ := assume eq, h ⟨h₁, eq.symm ▸ h₂⟩,
exact single_eq_of_ne this
end
lemma infi_ker_lapply_le_bot : (⨅a, ker (lapply a : (α →₀ M) →ₗ[R] M)) ≤ ⊥ :=
begin
simp only [le_def', mem_infi, mem_ker, mem_bot, lapply_apply],
exact assume a h, finsupp.ext h
end
lemma supr_lsingle_range : (⨆a, (lsingle a : M →ₗ[R] (α →₀ M)).range) = ⊤ :=
begin
refine (eq_top_iff.2 $ le_def'.2 $ assume f _, _),
rw [← sum_single f],
refine sum_mem _ (assume a ha, submodule.mem_supr_of_mem _ a $ set.mem_image_of_mem _ trivial)
end
lemma disjoint_lsingle_lsingle (s t : set α) (hs : disjoint s t) :
disjoint (⨆a∈s, (lsingle a : M →ₗ[R] (α →₀ M)).range) (⨆a∈t, (lsingle a).range) :=
begin
refine disjoint.mono
(lsingle_range_le_ker_lapply _ _ $ disjoint_compl s)
(lsingle_range_le_ker_lapply _ _ $ disjoint_compl t)
(le_trans (le_infi $ assume i, _) infi_ker_lapply_le_bot),
classical,
by_cases his : i ∈ s,
{ by_cases hit : i ∈ t,
{ exact (hs ⟨his, hit⟩).elim },
exact inf_le_right_of_le (infi_le_of_le i $ infi_le _ hit) },
exact inf_le_left_of_le (infi_le_of_le i $ infi_le _ his)
end
lemma span_single_image (s : set M) (a : α) :
submodule.span R (single a '' s) = (submodule.span R s).map (lsingle a) :=
by rw ← span_image; refl
variables (M R)
def supported (s : set α) : submodule R (α →₀ M) :=
begin
refine ⟨ {p | ↑p.support ⊆ s }, _, _, _ ⟩,
{ simp only [subset_def, finset.mem_coe, set.mem_set_of_eq, mem_support_iff, zero_apply],
assume h ha, exact (ha rfl).elim },
{ assume p q hp hq,
refine subset.trans
(subset.trans (finset.coe_subset.2 support_add) _) (union_subset hp hq),
rw [finset.coe_union] },
{ assume a p hp,
refine subset.trans (finset.coe_subset.2 support_smul) hp }
end
variables {M}
lemma mem_supported {s : set α} (p : α →₀ M) : p ∈ (supported M R s) ↔ ↑p.support ⊆ s :=
iff.rfl
lemma mem_supported' {s : set α} (p : α →₀ M) :
p ∈ supported M R s ↔ ∀ x ∉ s, p x = 0 :=
by haveI := classical.dec_pred (λ (x : α), x ∈ s);
simp [mem_supported, set.subset_def, not_imp_comm]
lemma single_mem_supported {s : set α} {a : α} (b : M) (h : a ∈ s) :
single a b ∈ supported M R s :=
set.subset.trans support_single_subset (finset.singleton_subset_set_iff.2 h)
lemma supported_eq_span_single (s : set α) :
supported R R s = span R ((λ i, single i 1) '' s) :=
begin
refine (span_eq_of_le _ _ (le_def'.2 $ λ l hl, _)).symm,
{ rintro _ ⟨_, hp, rfl ⟩ , exact single_mem_supported R 1 hp },
{ rw ← l.sum_single,
refine sum_mem _ (λ i il, _),
convert @smul_mem R (α →₀ R) _ _ _ _ (single i 1) (l i) _,
{ simp },
apply subset_span,
apply set.mem_image_of_mem _ (hl il) }
end
variables (M R)
def restrict_dom (s : set α) : (α →₀ M) →ₗ supported M R s :=
linear_map.cod_restrict _
{ to_fun := filter (∈ s),
add := λ l₁ l₂, filter_add,
smul := λ a l, filter_smul }
(λ l, (mem_supported' _ _).2 $ λ x, filter_apply_neg (∈ s) l)
variables {M R}
section
@[simp] theorem restrict_dom_apply (s : set α) (l : α →₀ M) :
((restrict_dom M R s : (α →₀ M) →ₗ supported M R s) l : α →₀ M) = finsupp.filter (∈ s) l := rfl
end
theorem restrict_dom_comp_subtype (s : set α) :
(restrict_dom M R s).comp (submodule.subtype _) = linear_map.id :=
begin
ext l,
apply subtype.coe_ext.2,
simp,
ext a,
by_cases a ∈ s,
{ simp [h] },
{ rw [filter_apply_neg (λ x, x ∈ s) _ h],
exact ((mem_supported' R l.1).1 l.2 a h).symm }
end
theorem range_restrict_dom (s : set α) :
(restrict_dom M R s).range = ⊤ :=
begin
have := linear_map.range_comp (submodule.subtype _) (restrict_dom M R s),
rw [restrict_dom_comp_subtype, linear_map.range_id] at this,
exact eq_top_mono (submodule.map_mono le_top) this.symm
end
theorem supported_mono {s t : set α} (st : s ⊆ t) :
supported M R s ≤ supported M R t :=
λ l h, set.subset.trans h st
@[simp] theorem supported_empty : supported M R (∅ : set α) = ⊥ :=
eq_bot_iff.2 $ λ l h, (submodule.mem_bot R).2 $
by ext; simp [*, mem_supported'] at *
@[simp] theorem supported_univ : supported M R (set.univ : set α) = ⊤ :=
eq_top_iff.2 $ λ l _, set.subset_univ _
theorem supported_Union {δ : Type*} (s : δ → set α) :
supported M R (⋃ i, s i) = ⨆ i, supported M R (s i) :=
begin
refine le_antisymm _ (supr_le $ λ i, supported_mono $ set.subset_Union _ _),
haveI := classical.dec_pred (λ x, x ∈ (⋃ i, s i)),
suffices : ((submodule.subtype _).comp (restrict_dom M R (⋃ i, s i))).range ≤ ⨆ i, supported M R (s i),
{ rwa [linear_map.range_comp, range_restrict_dom, map_top, range_subtype] at this },
rw [range_le_iff_comap, eq_top_iff],
rintro l ⟨⟩,
apply finsupp.induction l, {exact zero_mem _},
refine λ x a l hl a0, add_mem _ _,
haveI := classical.dec_pred (λ x, ∃ i, x ∈ s i),
by_cases (∃ i, x ∈ s i); simp [h],
{ cases h with i hi,
exact le_supr (λ i, supported M R (s i)) i (single_mem_supported R _ hi) },
{ rw filter_single_of_neg,
{ simp },
{ exact h } }
end
theorem supported_union (s t : set α) :
supported M R (s ∪ t) = supported M R s ⊔ supported M R t :=
by erw [set.union_eq_Union, supported_Union, supr_bool_eq]; refl
theorem supported_Inter {ι : Type*} (s : ι → set α) :
supported M R (⋂ i, s i) = ⨅ i, supported M R (s i) :=
begin
refine le_antisymm (le_infi $ λ i, supported_mono $ set.Inter_subset _ _) _,
simp [le_def, infi_coe, set.subset_def],
exact λ l, set.subset_Inter
end
def supported_equiv_finsupp (s : set α) : (supported M R s) ≃ₗ[R] (s →₀ M) :=
begin
let F : (supported M R s) ≃ (s →₀ M) := restrict_support_equiv s M,
refine F.to_linear_equiv _,
have : (F : (supported M R s) → (↥s →₀ M)) = ((lsubtype_domain s : (α →₀ M) →ₗ[R] (s →₀ M)).comp
(submodule.subtype (supported M R s))) := rfl,
rw this,
exact linear_map.is_linear _
end
def lsum (f : α → R →ₗ[R] M) : (α →₀ R) →ₗ[R] M :=
⟨λ d, d.sum (λ i, f i),
assume d₁ d₂, by simp [sum_add_index],
assume a d, by simp [sum_smul_index, smul_sum, -smul_eq_mul, smul_eq_mul.symm]⟩
@[simp] theorem lsum_apply (f : α → R →ₗ[R] M) (l : α →₀ R) :
(finsupp.lsum f : (α →₀ R) →ₗ M) l = l.sum (λ b, f b) := rfl
section lmap_domain
variables {α' : Type*} {α'' : Type*} (M R)
def lmap_domain (f : α → α') : (α →₀ M) →ₗ[R] (α' →₀ M) :=
⟨map_domain f, assume a b, map_domain_add, map_domain_smul⟩
@[simp] theorem lmap_domain_apply (f : α → α') (l : α →₀ M) :
(lmap_domain M R f : (α →₀ M) →ₗ[R] (α' →₀ M)) l = map_domain f l := rfl
@[simp] theorem lmap_domain_id : (lmap_domain M R id : (α →₀ M) →ₗ[R] α →₀ M) = linear_map.id :=
linear_map.ext $ λ l, map_domain_id
theorem lmap_domain_comp (f : α → α') (g : α' → α'') :
lmap_domain M R (g ∘ f) = (lmap_domain M R g).comp (lmap_domain M R f) :=
linear_map.ext $ λ l, map_domain_comp
theorem supported_comap_lmap_domain (f : α → α') (s : set α') :
supported M R (f ⁻¹' s) ≤ (supported M R s).comap (lmap_domain M R f) :=
λ l (hl : ↑l.support ⊆ f ⁻¹' s),
show ↑(map_domain f l).support ⊆ s, begin
rw [← set.image_subset_iff, ← finset.coe_image] at hl,
exact set.subset.trans map_domain_support hl
end
theorem lmap_domain_supported [nonempty α] (f : α → α') (s : set α) :
(supported M R s).map (lmap_domain M R f) = supported M R (f '' s) :=
begin
inhabit α,
refine le_antisymm (map_le_iff_le_comap.2 $
le_trans (supported_mono $ set.subset_preimage_image _ _)
(supported_comap_lmap_domain _ _ _ _)) _,
intros l hl,
refine ⟨(lmap_domain M R (function.inv_fun_on f s) : (α' →₀ M) →ₗ α →₀ M) l, λ x hx, _, _⟩,
{ rcases finset.mem_image.1 (map_domain_support hx) with ⟨c, hc, rfl⟩,
exact function.inv_fun_on_mem (by simpa using hl hc) },
{ rw [← linear_map.comp_apply, ← lmap_domain_comp],
refine (map_domain_congr $ λ c hc, _).trans map_domain_id,
exact function.inv_fun_on_eq (by simpa using hl hc) }
end
theorem lmap_domain_disjoint_ker (f : α → α') {s : set α}
(H : ∀ a b ∈ s, f a = f b → a = b) :
disjoint (supported M R s) (lmap_domain M R f).ker :=
begin
rintro l ⟨h₁, h₂⟩,
rw [mem_coe, mem_ker, lmap_domain_apply, map_domain] at h₂,
simp, ext x,
haveI := classical.dec_pred (λ x, x ∈ s),
by_cases xs : x ∈ s,
{ have : finsupp.sum l (λ a, finsupp.single (f a)) (f x) = 0, {rw h₂, refl},
rw [finsupp.sum_apply, finsupp.sum, finset.sum_eq_single x] at this,
{ simpa [finsupp.single_apply] },
{ intros y hy xy, simp [mt (H _ _ (h₁ hy) xs) xy] },
{ simp {contextual := tt} } },
{ by_contra h, exact xs (h₁ $ finsupp.mem_support_iff.2 h) }
end
end lmap_domain
section total
variables (α) {α' : Type*} (M) {M' : Type*} (R)
[add_comm_group M'] [module R M']
(v : α → M) {v' : α' → M'}
/-- Interprets (l : α →₀ R) as linear combination of the elements in the family (v : α → M) and
evaluates this linear combination. -/
protected def total : (α →₀ R) →ₗ M := finsupp.lsum (λ i, linear_map.id.smul_right (v i))
variables {α M v}
theorem total_apply (l : α →₀ R) :
finsupp.total α M R v l = l.sum (λ i a, a • v i) := rfl
@[simp] theorem total_single (c : R) (a : α) :
finsupp.total α M R v (single a c) = c • (v a) :=
by simp [total_apply, sum_single_index]
theorem total_range (h : function.surjective v) : (finsupp.total α M R v).range = ⊤ :=
begin
apply range_eq_top.2,
intros x,
apply exists.elim (h x),
exact λ i hi, ⟨single i 1, by simp [hi]⟩
end
lemma range_total : (finsupp.total α M R v).range = span R (range v) :=
begin
ext x,
split,
{ intros hx,
rw [linear_map.mem_range] at hx,
rcases hx with ⟨l, hl⟩,
rw ← hl,
rw finsupp.total_apply,
unfold finsupp.sum,
apply sum_mem (span R (range v)),
exact λ i hi, submodule.smul _ _ (subset_span (mem_range_self i)) },
{ apply span_le.2,
intros x hx,
rcases hx with ⟨i, hi⟩,
rw [mem_coe, linear_map.mem_range],
use finsupp.single i 1,
simp [hi] }
end
theorem lmap_domain_total (f : α → α') (g : M →ₗ[R] M') (h : ∀ i, g (v i) = v' (f i)) :
(finsupp.total α' M' R v').comp (lmap_domain R R f) = g.comp (finsupp.total α M R v) :=
by ext l; simp [total_apply, finsupp.sum_map_domain_index, add_smul, h]
theorem total_emb_domain (f : α ↪ α') (l : α →₀ R) :
(finsupp.total α' M' R v') (emb_domain f l) = (finsupp.total α M' R (v' ∘ f)) l :=
by simp [total_apply, finsupp.sum, support_emb_domain, emb_domain_apply]
theorem total_map_domain (f : α → α') (hf : function.injective f) (l : α →₀ R) :
(finsupp.total α' M' R v') (map_domain f l) = (finsupp.total α M' R (v' ∘ f)) l :=
begin
have : map_domain f l = emb_domain ⟨f, hf⟩ l,
{ rw emb_domain_eq_map_domain ⟨f, hf⟩,
refl },
rw this,
apply total_emb_domain R ⟨f, hf⟩ l
end
theorem span_eq_map_total (s : set α):
span R (v '' s) = submodule.map (finsupp.total α M R v) (supported R R s) :=
begin
apply span_eq_of_le,
{ intros x hx,
rw set.mem_image at hx,
apply exists.elim hx,
intros i hi,
exact ⟨_, finsupp.single_mem_supported R 1 hi.1, by simp [hi.2]⟩ },
{ refine map_le_iff_le_comap.2 (λ z hz, _),
have : ∀i, z i • v i ∈ span R (v '' s),
{ intro c,
haveI := classical.dec_pred (λ x, x ∈ s),
by_cases c ∈ s,
{ exact smul_mem _ _ (subset_span (set.mem_image_of_mem _ h)) },
{ simp [(finsupp.mem_supported' R _).1 hz _ h] } },
refine sum_mem _ _, simp [this] }
end
theorem mem_span_iff_total {s : set α} {x : M} :
x ∈ span R (v '' s) ↔ ∃ l ∈ supported R R s, finsupp.total α M R v l = x :=
by rw span_eq_map_total; simp
variables (α) (M) (v)
protected def total_on (s : set α) : supported R R s →ₗ[R] span R (v '' s) :=
linear_map.cod_restrict _ ((finsupp.total _ _ _ v).comp (submodule.subtype (supported R R s))) $
λ ⟨l, hl⟩, (mem_span_iff_total _).2 ⟨l, hl, rfl⟩
variables {α} {M} {v}
theorem total_on_range (s : set α) : (finsupp.total_on α M R v s).range = ⊤ :=
by rw [finsupp.total_on, linear_map.range, linear_map.map_cod_restrict, ← linear_map.range_le_iff_comap,
range_subtype, map_top, linear_map.range_comp, range_subtype]; exact le_of_eq (span_eq_map_total _ _)
theorem total_comp (f : α' → α) :
(finsupp.total α' M R (v ∘ f)) = (finsupp.total α M R v).comp (lmap_domain R R f) :=
begin
ext l,
simp [total_apply],
rw sum_map_domain_index; simp [add_smul],
end
lemma total_comap_domain
(f : α → α') (l : α' →₀ R) (hf : set.inj_on f (f ⁻¹' ↑l.support)) :
finsupp.total α M R v (finsupp.comap_domain f l hf) = (l.support.preimage hf).sum (λ i, (l (f i)) • (v i)) :=
by rw finsupp.total_apply; refl
end total
protected def dom_lcongr
{α₁ : Type*} {α₂ : Type*} (e : α₁ ≃ α₂) :
(α₁ →₀ M) ≃ₗ[R] (α₂ →₀ M) :=
(finsupp.dom_congr e).to_linear_equiv
begin
change is_linear_map R (lmap_domain M R e : (α₁ →₀ M) →ₗ[R] (α₂ →₀ M)),
exact linear_map.is_linear _
end
noncomputable def congr {α' : Type*} (s : set α) (t : set α') (e : s ≃ t) :
supported M R s ≃ₗ[R] supported M R t :=
begin
haveI := classical.dec_pred (λ x, x ∈ s),
haveI := classical.dec_pred (λ x, x ∈ t),
refine linear_equiv.trans (finsupp.supported_equiv_finsupp s)
(linear_equiv.trans _ (finsupp.supported_equiv_finsupp t).symm),
exact finsupp.dom_lcongr e
end
end finsupp
variables {R : Type*} {M : Type*} {N : Type*}
variables [ring R] [add_comm_group M] [module R M] [add_comm_group N] [module R N]
lemma linear_map.map_finsupp_total
(f : M →ₗ[R] N) {ι : Type*} {g : ι → M} (l : ι →₀ R) :
f (finsupp.total ι M R g l) = finsupp.total ι N R (f ∘ g) l :=
by simp only [finsupp.total_apply, finsupp.total_apply, finsupp.sum, f.map_sum, f.map_smul]
lemma submodule.exists_finset_of_mem_supr
{ι : Sort*} (p : ι → submodule R M) {m : M} (hm : m ∈ ⨆ i, p i) :
∃ s : finset ι, m ∈ ⨆ i ∈ s, p i :=
begin
obtain ⟨f, hf, rfl⟩ : ∃ f ∈ finsupp.supported R R (⋃ i, ↑(p i)), finsupp.total M M R id f = m,
{ have aux : (id : M → M) '' (⋃ (i : ι), ↑(p i)) = (⋃ (i : ι), ↑(p i)) := set.image_id _,
rwa [supr_eq_span, ← aux, finsupp.mem_span_iff_total R] at hm },
let t : finset M := f.support,
have ht : ∀ x : {x // x ∈ t}, ∃ i, ↑x ∈ p i,
{ intros x,
rw finsupp.mem_supported at hf,
specialize hf x.2,
rwa set.mem_Union at hf },
choose g hg using ht,
let s : finset ι := finset.univ.image g,
use s,
simp only [mem_supr, supr_le_iff],
assume N hN,
rw [finsupp.total_apply, finsupp.sum, ← submodule.mem_coe],
apply is_add_submonoid.finset_sum_mem,
assume x hx,
apply submodule.smul_mem,
let i : ι := g ⟨x, hx⟩,
have hi : i ∈ s, { rw finset.mem_image, exact ⟨⟨x, hx⟩, finset.mem_univ _, rfl⟩ },
exact hN i hi (hg _),
end
|
87bdc47f89f4ddbf2b54eed90400fa28bcd908f4 | b70031c8e2c5337b91d7e70f1e0c5f528f7b0e77 | /test/ring_exp.lean | 3d3b04537e08f1a56b05c327219bdb63b5a8d888 | [
"Apache-2.0"
] | permissive | molodiuc/mathlib | cae2ba3ef1601c1f42ca0b625c79b061b63fef5b | 98ebe5a6739fbe254f9ee9d401882d4388f91035 | refs/heads/master | 1,674,237,127,059 | 1,606,353,533,000 | 1,606,353,533,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,367 | lean | import tactic.ring_exp
import algebra.group_with_zero.power
universes u
section addition
/-!
### `addition` section
Test associativity and commutativity of `(+)`.
-/
example (a b : ℚ) : a = a := by ring_exp
example (a b : ℚ) : a + b = a + b := by ring_exp
example (a b : ℚ) : b + a = a + b := by ring_exp
example (a b : ℤ) : a + b + b = b + (a + b) := by ring_exp
example (a b c : ℕ) : a + b + b + (c + c) = c + (b + c) + (a + b) := by ring_exp
end addition
section numerals
/-!
### `numerals` section
Test that numerals behave like rational numbers.
-/
example (a : ℕ) : a + 5 + 5 = 0 + a + 10 := by ring_exp
example (a : ℤ) : a + 5 + 5 = 0 + a + 10 := by ring_exp
example (a : ℚ) : (1/2) * a + (1/2) * a = a := by ring_exp
end numerals
section multiplication
/-!
### `multiplication section`
Test that multiplication is associative and commutative.
Also test distributivity of `(+)` and `(*)`.
-/
example (a : ℕ) : 0 = a * 0 := by ring_exp_eq
example (a : ℕ) : a = a * 1 := by ring_exp
example (a : ℕ) : a + a = a * 2 := by ring_exp
example (a b : ℤ) : a * b = b * a := by ring_exp
example (a b : ℕ) : a * 4 * b + a = a * (4 * b + 1) := by ring_exp
end multiplication
section exponentiation
/-!
### `exponentiation` section
Test that exponentiation has the correct distributivity properties.
-/
example : 0 ^ 1 = 0 := by ring_exp
example : 0 ^ 2 = 0 := by ring_exp
example (a : ℕ) : a ^ 0 = 1 := by ring_exp
example (a : ℕ) : a ^ 1 = a := by ring_exp
example (a : ℕ) : a ^ 2 = a * a := by ring_exp
example (a b : ℕ) : a ^ b = a ^ b := by ring_exp
example (a b : ℕ) : a ^ (b + 1) = a * a ^ b := by ring_exp
example (n : ℕ) (a m : ℕ) : a * a^n * m = a^(n+1) * m := by ring_exp
example (n : ℕ) (a m : ℕ) : m * a^n * a = a^(n+1) * m := by ring_exp
example (n : ℕ) (a m : ℤ) : a * a^n * m = a^(n+1) * m := by ring_exp
example (n : ℕ) (m : ℤ) : 2 * 2^n * m = 2^(n+1) * m := by ring_exp
example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring_exp
example (n m : ℕ) (a : ℤ) : (a ^ n)^m = a^(n * m) := by ring_exp
example (n m : ℕ) (a : ℤ) : a^(n^0) = a^1 := by ring_exp
example (n : ℕ) : 0^(n + 1) = 0 := by ring_exp
example {α} [comm_ring α] (x : α) (k : ℕ) : x ^ (k + 2) = x * x * x^k := by ring_exp
example {α} [comm_ring α] (k : ℕ) (x y z : α) :
x * (z * (x - y)) + (x * (y * y ^ k) - y * (y * y ^ k)) = (z * x + y * y ^ k) * (x - y)
:= by ring_exp
-- We can represent a large exponent `n` more efficiently than just `n` multiplications:
example (a b : ℚ) : (a * b) ^ 1000000 = (b * a) ^ 1000000 := by ring_exp
example (n : ℕ) : 2 ^ (n + 1 + 1) = 2 * 2 ^ (n + 1) :=
by ring_exp_eq
end exponentiation
section power_of_sum
/-!
### `power_of_sum` section
Test that raising a sum to a power behaves like repeated multiplication,
if needed.
-/
example (a b : ℤ) : (a + b)^2 = a^2 + b^2 + a * b + b * a := by ring_exp
example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring_exp
end power_of_sum
section negation
/-!
### `negation` section
Test that negation and subtraction satisfy the expected properties,
also in semirings such as `ℕ`.
-/
example {α} [comm_ring α] (a : α) : a - a = 0 := by ring_exp_eq
example (a : ℤ) : a - a = 0 := by ring_exp
example (a : ℤ) : a + - a = 0 := by ring_exp
example (a : ℤ) : - a = (-1) * a := by ring_exp
-- Here, (a - b) is treated as an atom.
example (a b : ℕ) : a - b + a + a = a - b + 2 * a := by ring_exp
example (n : ℕ) : n + 1 - 1 = n := by ring_exp! -- But we can force a bit of evaluation anyway.
end negation
constant f {α} : α → α
section complicated
/-!
### `complicated` section
Test that complicated, real-life expressions also get normalized.
-/
example {α : Type} [linear_ordered_field α] (x : α) :
2 * x + 1 * 1 - (2 * f (x + 1 / 2) + 2 * 1) + (1 * 1 - (2 * x - 2 * f (x + 1 / 2))) = 0
:= by ring_exp_eq
example {α : Type u} [linear_ordered_field α] (x : α) :
f (x + 1 / 2) ^ 1 * -2 + (f (x + 1 / 2) ^ 1 * 2 + 0) = 0
:= by ring_exp_eq
example (x y : ℕ) : x + id y = y + id x := by ring_exp!
-- Here, we check that `n - s` is not treated as `n + additive_inverse s`,
-- if `s` doesn't have an additive inverse.
example (B s n : ℕ) : B * (f s * ((n - s) * f (n - s - 1))) = B * (n - s) * (f s * f (n - s - 1)) :=
by ring_exp
-- This is a somewhat subtle case: `-c/b` is parsed as `(-c)/b`,
-- so we can't simply treat both sides of the division as atoms.
-- Instead, we follow the `ring` tactic in interpreting `-c / b` as `-c * b⁻¹`,
-- with only `b⁻¹` an atom.
example {α} [linear_ordered_field α] (a b c : α) : a*(-c/b)*(-c/b) = a*((c/b)*(c/b)) := by ring_exp
-- test that `field_simp` works fine with powers and `ring_exp`.
example (x y : ℚ) (n : ℕ) (hx : x ≠ 0) (hy : y ≠ 0) :
1/ (2/(x / y))^(2 * n) + y / y^(n+1) - (x/y)^n * (x/(2 * y))^n / 2 ^n = 1/y^n :=
begin
simp [sub_eq_add_neg],
field_simp [hx, hy],
ring_exp
end
end complicated
section conv
/-!
### `conv` section
Test that `ring_exp` works inside of `conv`, both with and without `!`.
-/
example (n : ℕ) : (2^n * 2 + 1)^10 = (2^(n+1) + 1)^10 :=
begin
conv_rhs
{ congr,
ring_exp, },
conv_lhs
{ congr,
ring_exp, },
end
example (x y : ℤ) : x + id y - y + id x = x * 2 := begin
conv_lhs { ring_exp!, },
end
end conv
section benchmark
/-!
### `benchmark` section
The `ring_exp` tactic shouldn't be too slow.
-/
-- This last example was copied from `data/polynomial.lean`, because it timed out.
-- After some optimization, it doesn't.
variables {α : Type} [comm_ring α]
def pow_sub_pow_factor (x y : α) : Π {i : ℕ},{z // x^i - y^i = z*(x - y)}
| 0 := ⟨0, by simp⟩
| 1 := ⟨1, by simp⟩
| (k+2) :=
begin
cases @pow_sub_pow_factor (k+1) with z hz,
existsi z*x + y^(k+1),
rw [pow_succ x, pow_succ y, ←sub_add_sub_cancel (x*x^(k+1)) (x*y^(k+1)),
←mul_sub x, hz],
ring_exp_eq
end
-- Another benchmark: bound the growth of the complexity somewhat.
example {α} [comm_semiring α] (x : α) : (x + 1) ^ 4 = (1 + x) ^ 4 := by try_for 5000 {ring_exp}
example {α} [comm_semiring α] (x : α) : (x + 1) ^ 6 = (1 + x) ^ 6 := by try_for 10000 {ring_exp}
example {α} [comm_semiring α] (x : α) : (x + 1) ^ 8 = (1 + x) ^ 8 := by try_for 15000 {ring_exp}
end benchmark
|
15e8d2bcc752fd6b666f15e5380cbaa7e8739332 | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/algebra/big_operators/finprod.lean | 619f1f4b5c014c1d6ccc698f2ad44b41fed32aca | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 38,328 | lean | /-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import algebra.big_operators.order
import algebra.indicator_function
import data.set.pairwise
/-!
# Finite products and sums over types and sets
We define products and sums over types and subsets of types, with no finiteness hypotheses.
All infinite products and sums are defined to be junk values (i.e. one or zero).
This approach is sometimes easier to use than `finset.sum`,
when issues arise with `finset` and `fintype` being data.
## Main definitions
We use the following variables:
* `α`, `β` - types with no structure;
* `s`, `t` - sets
* `M`, `N` - additive or multiplicative commutative monoids
* `f`, `g` - functions
Definitions in this file:
* `finsum f : M` : the sum of `f x` as `x` ranges over the support of `f`, if it's finite.
Zero otherwise.
* `finprod f : M` : the product of `f x` as `x` ranges over the multiplicative support of `f`, if
it's finite. One otherwise.
## Notation
* `∑ᶠ i, f i` and `∑ᶠ i : α, f i` for `finsum f`
* `∏ᶠ i, f i` and `∏ᶠ i : α, f i` for `finprod f`
This notation works for functions `f : p → M`, where `p : Prop`, so the following works:
* `∑ᶠ i ∈ s, f i`, where `f : α → M`, `s : set α` : sum over the set `s`;
* `∑ᶠ n < 5, f n`, where `f : ℕ → M` : same as `f 0 + f 1 + f 2 + f 3 + f 4`;
* `∏ᶠ (n >= -2) (hn : n < 3), f n`, where `f : ℤ → M` : same as `f (-2) * f (-1) * f 0 * f 1 * f 2`.
## Implementation notes
`finsum` and `finprod` is "yet another way of doing finite sums and products in Lean". However
experiments in the wild (e.g. with matroids) indicate that it is a helpful approach in settings
where the user is not interested in computability and wants to do reasoning without running into
typeclass diamonds caused by the constructive finiteness used in definitions such as `finset` and
`fintype`. By sticking solely to `set.finite` we avoid these problems. We are aware that there are
other solutions but for beginner mathematicians this approach is easier in practice.
Another application is the construction of a partition of unity from a collection of “bump”
function. In this case the finite set depends on the point and it's convenient to have a definition
that does not mention the set explicitly.
The first arguments in all definitions and lemmas is the codomain of the function of the big
operator. This is necessary for the heuristic in `@[to_additive]`.
See the documentation of `to_additive.attr` for more information.
## Todo
We did not add `is_finite (X : Type) : Prop`, because it is simply `nonempty (fintype X)`.
There is work on `fincard` in the pipeline, which returns the cardinality of `X` if it
is finite, and 0 otherwise.
## Tags
finsum, finprod, finite sum, finite product
-/
open function set
/-!
### Definition and relation to `finset.sum` and `finset.prod`
-/
section sort
variables {M N : Type*} {α β ι : Sort*} [comm_monoid M] [comm_monoid N]
open_locale big_operators
section
/- Note: we use classical logic only for these definitions, to ensure that we do not write lemmas
with `classical.dec` in their statement. -/
open_locale classical
/-- Sum of `f x` as `x` ranges over the elements of the support of `f`, if it's finite. Zero
otherwise. -/
@[irreducible] noncomputable def finsum {M α} [add_comm_monoid M] (f : α → M) : M :=
if h : finite (support (f ∘ plift.down)) then ∑ i in h.to_finset, f i.down else 0
/-- Product of `f x` as `x` ranges over the elements of the multiplicative support of `f`, if it's
finite. One otherwise. -/
@[irreducible, to_additive]
noncomputable def finprod (f : α → M) : M :=
if h : finite (mul_support (f ∘ plift.down)) then ∏ i in h.to_finset, f i.down else 1
end
localized "notation `∑ᶠ` binders `, ` r:(scoped:67 f, finsum f) := r" in big_operators
localized "notation `∏ᶠ` binders `, ` r:(scoped:67 f, finprod f) := r" in big_operators
@[to_additive] lemma finprod_eq_prod_plift_of_mul_support_to_finset_subset
{f : α → M} (hf : finite (mul_support (f ∘ plift.down))) {s : finset (plift α)}
(hs : hf.to_finset ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i.down :=
begin
rw [finprod, dif_pos],
refine finset.prod_subset hs (λ x hx hxf, _),
rwa [hf.mem_to_finset, nmem_mul_support] at hxf
end
@[to_additive] lemma finprod_eq_prod_plift_of_mul_support_subset
{f : α → M} {s : finset (plift α)} (hs : mul_support (f ∘ plift.down) ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i.down :=
finprod_eq_prod_plift_of_mul_support_to_finset_subset
(s.finite_to_set.subset hs) $ λ x hx, by { rw finite.mem_to_finset at hx, exact hs hx }
@[simp, to_additive] lemma finprod_one : ∏ᶠ i : α, (1 : M) = 1 :=
begin
have : mul_support (λ x : plift α, (λ _, 1 : α → M) x.down) ⊆ (∅ : finset (plift α)),
from λ x h, h rfl,
rw [finprod_eq_prod_plift_of_mul_support_subset this, finset.prod_empty]
end
@[to_additive] lemma finprod_of_is_empty [is_empty α] (f : α → M) : ∏ᶠ i, f i = 1 :=
by { rw ← finprod_one, congr }
@[simp, to_additive] lemma finprod_false (f : false → M) : ∏ᶠ i, f i = 1 :=
finprod_of_is_empty _
@[to_additive] lemma finprod_eq_single (f : α → M) (a : α) (ha : ∀ x ≠ a, f x = 1) :
∏ᶠ x, f x = f a :=
begin
have : mul_support (f ∘ plift.down) ⊆ ({plift.up a} : finset (plift α)),
{ intro x, contrapose,
simpa [plift.eq_up_iff_down_eq] using ha x.down },
rw [finprod_eq_prod_plift_of_mul_support_subset this, finset.prod_singleton],
end
@[to_additive] lemma finprod_unique [unique α] (f : α → M) : ∏ᶠ i, f i = f (default α) :=
finprod_eq_single f (default α) $ λ x hx, (hx $ unique.eq_default _).elim
@[simp, to_additive] lemma finprod_true (f : true → M) : ∏ᶠ i, f i = f trivial :=
@finprod_unique M true _ ⟨⟨trivial⟩, λ _, rfl⟩ f
@[to_additive] lemma finprod_eq_dif {p : Prop} [decidable p] (f : p → M) :
∏ᶠ i, f i = if h : p then f h else 1 :=
begin
split_ifs,
{ haveI : unique p := ⟨⟨h⟩, λ _, rfl⟩, exact finprod_unique f },
{ haveI : is_empty p := ⟨h⟩, exact finprod_of_is_empty f }
end
@[to_additive] lemma finprod_eq_if {p : Prop} [decidable p] {x : M} :
∏ᶠ i : p, x = if p then x else 1 :=
finprod_eq_dif (λ _, x)
@[to_additive] lemma finprod_congr {f g : α → M} (h : ∀ x, f x = g x) :
finprod f = finprod g :=
congr_arg _ $ funext h
@[congr, to_additive] lemma finprod_congr_Prop {p q : Prop} {f : p → M} {g : q → M} (hpq : p = q)
(hfg : ∀ h : q, f (hpq.mpr h) = g h) :
finprod f = finprod g :=
by { subst q, exact finprod_congr hfg }
attribute [congr] finsum_congr_Prop
/-- To prove a property of a finite product, it suffices to prove that the property is
multiplicative and holds on multipliers. -/
@[to_additive] lemma finprod_induction {f : α → M} (p : M → Prop) (hp₀ : p 1)
(hp₁ : ∀ x y, p x → p y → p (x * y)) (hp₂ : ∀ i, p (f i)) :
p (∏ᶠ i, f i) :=
begin
rw finprod,
split_ifs,
exacts [finset.prod_induction _ _ hp₁ hp₀ (λ i hi, hp₂ _), hp₀]
end
/-- To prove a property of a finite sum, it suffices to prove that the property is
additive and holds on summands. -/
add_decl_doc finsum_induction
lemma finprod_nonneg {R : Type*} [ordered_comm_semiring R] {f : α → R} (hf : ∀ x, 0 ≤ f x) :
0 ≤ ∏ᶠ x, f x :=
finprod_induction (λ x, 0 ≤ x) zero_le_one (λ x y, mul_nonneg) hf
@[to_additive finsum_nonneg]
lemma one_le_finprod' {M : Type*} [ordered_comm_monoid M] {f : α → M} (hf : ∀ i, 1 ≤ f i) :
1 ≤ ∏ᶠ i, f i :=
finprod_induction _ le_rfl (λ _ _, one_le_mul) hf
@[to_additive] lemma monoid_hom.map_finprod_plift (f : M →* N) (g : α → M)
(h : finite (mul_support $ g ∘ plift.down)) :
f (∏ᶠ x, g x) = ∏ᶠ x, f (g x) :=
begin
rw [finprod_eq_prod_plift_of_mul_support_subset h.coe_to_finset.ge,
finprod_eq_prod_plift_of_mul_support_subset, f.map_prod],
rw [h.coe_to_finset],
exact mul_support_comp_subset f.map_one (g ∘ plift.down)
end
@[to_additive] lemma monoid_hom.map_finprod_Prop {p : Prop} (f : M →* N) (g : p → M) :
f (∏ᶠ x, g x) = ∏ᶠ x, f (g x) :=
f.map_finprod_plift g (finite.of_fintype _)
@[to_additive] lemma monoid_hom.map_finprod_of_preimage_one (f : M →* N)
(hf : ∀ x, f x = 1 → x = 1) (g : α → M) :
f (∏ᶠ i, g i) = ∏ᶠ i, f (g i) :=
begin
by_cases hg : (mul_support $ g ∘ plift.down).finite, { exact f.map_finprod_plift g hg },
rw [finprod, dif_neg, f.map_one, finprod, dif_neg],
exacts [infinite.mono (λ x hx, mt (hf (g x.down)) hx) hg, hg]
end
@[to_additive] lemma monoid_hom.map_finprod_of_injective (g : M →* N) (hg : injective g)
(f : α → M) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) :=
g.map_finprod_of_preimage_one (λ x, (hg.eq_iff' g.map_one).mp) f
@[to_additive] lemma mul_equiv.map_finprod (g : M ≃* N) (f : α → M) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) :=
g.to_monoid_hom.map_finprod_of_injective g.injective f
lemma finsum_smul {R M : Type*} [ring R] [add_comm_group M] [module R M]
[no_zero_smul_divisors R M] (f : ι → R) (x : M) :
(∑ᶠ i, f i) • x = (∑ᶠ i, (f i) • x) :=
begin
rcases eq_or_ne x 0 with rfl|hx, { simp },
exact ((smul_add_hom R M).flip x).map_finsum_of_injective (smul_left_injective R hx) _
end
lemma smul_finsum {R M : Type*} [ring R] [add_comm_group M] [module R M]
[no_zero_smul_divisors R M] (c : R) (f : ι → M) :
c • (∑ᶠ i, f i) = (∑ᶠ i, c • f i) :=
begin
rcases eq_or_ne c 0 with rfl|hc, { simp },
exact (smul_add_hom R M c).map_finsum_of_injective (smul_right_injective M hc) _
end
@[to_additive] lemma finprod_inv_distrib {G : Type*} [comm_group G] (f : α → G) :
∏ᶠ x, (f x)⁻¹ = (∏ᶠ x, f x)⁻¹ :=
((mul_equiv.inv G).map_finprod f).symm
end sort
section type
variables {α β ι M N : Type*} [comm_monoid M] [comm_monoid N]
open_locale big_operators
@[to_additive] lemma finprod_eq_mul_indicator_apply (s : set α)
(f : α → M) (a : α) :
∏ᶠ (h : a ∈ s), f a = mul_indicator s f a :=
by convert finprod_eq_if
@[simp, to_additive] lemma finprod_mem_mul_support (f : α → M) (a : α) :
∏ᶠ (h : f a ≠ 1), f a = f a :=
by rw [← mem_mul_support, finprod_eq_mul_indicator_apply, mul_indicator_mul_support]
@[to_additive] lemma finprod_mem_def (s : set α) (f : α → M) :
∏ᶠ a ∈ s, f a = ∏ᶠ a, mul_indicator s f a :=
finprod_congr $ finprod_eq_mul_indicator_apply s f
@[to_additive] lemma finprod_eq_prod_of_mul_support_subset (f : α → M) {s : finset α}
(h : mul_support f ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i :=
begin
have A : mul_support (f ∘ plift.down) = equiv.plift.symm '' mul_support f,
{ rw mul_support_comp_eq_preimage,
exact (equiv.plift.symm.image_eq_preimage _).symm },
have : mul_support (f ∘ plift.down) ⊆ s.map equiv.plift.symm.to_embedding,
{ rw [A, finset.coe_map], exact image_subset _ h },
rw [finprod_eq_prod_plift_of_mul_support_subset this],
simp
end
@[to_additive] lemma finprod_eq_prod_of_mul_support_to_finset_subset (f : α → M)
(hf : finite (mul_support f)) {s : finset α} (h : hf.to_finset ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i :=
finprod_eq_prod_of_mul_support_subset _ $ λ x hx, h $ hf.mem_to_finset.2 hx
@[to_additive] lemma finprod_def (f : α → M) [decidable (mul_support f).finite] :
∏ᶠ i : α, f i = if h : (mul_support f).finite then ∏ i in h.to_finset, f i else 1 :=
begin
split_ifs,
{ exact finprod_eq_prod_of_mul_support_to_finset_subset _ h (finset.subset.refl _) },
{ rw [finprod, dif_neg],
rw [mul_support_comp_eq_preimage],
exact mt (λ hf, hf.of_preimage equiv.plift.surjective) h}
end
@[to_additive] lemma finprod_of_infinite_mul_support {f : α → M} (hf : (mul_support f).infinite) :
∏ᶠ i, f i = 1 :=
by { classical, rw [finprod_def, dif_neg hf] }
@[to_additive] lemma finprod_eq_prod (f : α → M) (hf : (mul_support f).finite) :
∏ᶠ i : α, f i = ∏ i in hf.to_finset, f i :=
by { classical, rw [finprod_def, dif_pos hf] }
@[to_additive] lemma finprod_eq_prod_of_fintype [fintype α] (f : α → M) :
∏ᶠ i : α, f i = ∏ i, f i :=
finprod_eq_prod_of_mul_support_to_finset_subset _ (finite.of_fintype _) $ finset.subset_univ _
@[to_additive] lemma finprod_cond_eq_prod_of_cond_iff (f : α → M) {p : α → Prop} {t : finset α}
(h : ∀ {x}, f x ≠ 1 → (p x ↔ x ∈ t)) :
∏ᶠ i (hi : p i), f i = ∏ i in t, f i :=
begin
set s := {x | p x},
have : mul_support (s.mul_indicator f) ⊆ t,
{ rw [set.mul_support_mul_indicator], intros x hx, exact (h hx.2).1 hx.1 },
erw [finprod_mem_def, finprod_eq_prod_of_mul_support_subset _ this],
refine finset.prod_congr rfl (λ x hx, mul_indicator_apply_eq_self.2 $ λ hxs, _),
contrapose! hxs,
exact (h hxs).2 hx
end
@[to_additive] lemma finprod_mem_eq_prod_of_inter_mul_support_eq (f : α → M) {s : set α}
{t : finset α} (h : s ∩ mul_support f = t ∩ mul_support f) :
∏ᶠ i ∈ s, f i = ∏ i in t, f i :=
finprod_cond_eq_prod_of_cond_iff _ $ by simpa [set.ext_iff] using h
@[to_additive] lemma finprod_mem_eq_prod_of_subset (f : α → M) {s : set α} {t : finset α}
(h₁ : s ∩ mul_support f ⊆ t) (h₂ : ↑t ⊆ s) :
∏ᶠ i ∈ s, f i = ∏ i in t, f i :=
finprod_cond_eq_prod_of_cond_iff _ $ λ x hx, ⟨λ h, h₁ ⟨h, hx⟩, λ h, h₂ h⟩
@[to_additive] lemma finprod_mem_eq_prod (f : α → M) {s : set α}
(hf : (s ∩ mul_support f).finite) :
∏ᶠ i ∈ s, f i = ∏ i in hf.to_finset, f i :=
finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by simp [inter_assoc]
@[to_additive] lemma finprod_mem_eq_prod_filter (f : α → M) (s : set α) [decidable_pred (∈ s)]
(hf : (mul_support f).finite) :
∏ᶠ i ∈ s, f i = ∏ i in finset.filter (∈ s) hf.to_finset, f i :=
finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by simp [inter_comm, inter_left_comm]
@[to_additive] lemma finprod_mem_eq_to_finset_prod (f : α → M) (s : set α) [fintype s] :
∏ᶠ i ∈ s, f i = ∏ i in s.to_finset, f i :=
finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by rw [coe_to_finset]
@[to_additive] lemma finprod_mem_eq_finite_to_finset_prod (f : α → M) {s : set α} (hs : s.finite) :
∏ᶠ i ∈ s, f i = ∏ i in hs.to_finset, f i :=
finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by rw [hs.coe_to_finset]
@[to_additive] lemma finprod_mem_finset_eq_prod (f : α → M) (s : finset α) :
∏ᶠ i ∈ s, f i = ∏ i in s, f i :=
finprod_mem_eq_prod_of_inter_mul_support_eq _ rfl
@[to_additive] lemma finprod_mem_coe_finset (f : α → M) (s : finset α) :
∏ᶠ i ∈ (s : set α), f i = ∏ i in s, f i :=
finprod_mem_eq_prod_of_inter_mul_support_eq _ rfl
@[to_additive] lemma finprod_mem_eq_one_of_infinite {f : α → M} {s : set α}
(hs : (s ∩ mul_support f).infinite) : ∏ᶠ i ∈ s, f i = 1 :=
begin
rw finprod_mem_def,
apply finprod_of_infinite_mul_support,
rwa [← mul_support_mul_indicator] at hs
end
@[to_additive] lemma finprod_mem_inter_mul_support (f : α → M) (s : set α) :
∏ᶠ i ∈ (s ∩ mul_support f), f i = ∏ᶠ i ∈ s, f i :=
by rw [finprod_mem_def, finprod_mem_def, mul_indicator_inter_mul_support]
@[to_additive] lemma finprod_mem_inter_mul_support_eq (f : α → M) (s t : set α)
(h : s ∩ mul_support f = t ∩ mul_support f) :
∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, f i :=
by rw [← finprod_mem_inter_mul_support, h, finprod_mem_inter_mul_support]
@[to_additive] lemma finprod_mem_inter_mul_support_eq' (f : α → M) (s t : set α)
(h : ∀ x ∈ mul_support f, x ∈ s ↔ x ∈ t) :
∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, f i :=
begin
apply finprod_mem_inter_mul_support_eq,
ext x,
exact and_congr_left (h x)
end
@[to_additive] lemma finprod_mem_univ (f : α → M) : ∏ᶠ i ∈ @set.univ α, f i = ∏ᶠ i : α, f i :=
finprod_congr $ λ i, finprod_true _
variables {f g : α → M} {a b : α} {s t : set α}
@[to_additive] lemma finprod_mem_congr (h₀ : s = t) (h₁ : ∀ x ∈ t, f x = g x) :
∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, g i :=
h₀.symm ▸ (finprod_congr $ λ i, finprod_congr_Prop rfl (h₁ i))
/-!
### Distributivity w.r.t. addition, subtraction, and (scalar) multiplication
-/
/-- If the multiplicative supports of `f` and `g` are finite, then the product of `f i * g i` equals
the product of `f i` multiplied by the product over `g i`. -/
@[to_additive] lemma finprod_mul_distrib (hf : (mul_support f).finite)
(hg : (mul_support g).finite) :
∏ᶠ i, (f i * g i) = (∏ᶠ i, f i) * ∏ᶠ i, g i :=
begin
classical,
rw [finprod_eq_prod_of_mul_support_to_finset_subset _ hf (finset.subset_union_left _ _),
finprod_eq_prod_of_mul_support_to_finset_subset _ hg (finset.subset_union_right _ _),
← finset.prod_mul_distrib],
refine finprod_eq_prod_of_mul_support_subset _ _,
simp [mul_support_mul]
end
/-- A more general version of `finprod_mem_mul_distrib` that requires `s ∩ mul_support f` and
`s ∩ mul_support g` instead of `s` to be finite. -/
@[to_additive] lemma finprod_mem_mul_distrib' (hf : (s ∩ mul_support f).finite)
(hg : (s ∩ mul_support g).finite) :
∏ᶠ i ∈ s, (f i * g i) = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ s, g i :=
begin
rw [← mul_support_mul_indicator] at hf hg,
simp only [finprod_mem_def, mul_indicator_mul, finprod_mul_distrib hf hg]
end
/-- The product of constant one over any set equals one. -/
@[to_additive] lemma finprod_mem_one (s : set α) : ∏ᶠ i ∈ s, (1 : M) = 1 := by simp
/-- If a function `f` equals one on a set `s`, then the product of `f i` over `i ∈ s` equals one. -/
@[to_additive] lemma finprod_mem_of_eq_on_one (hf : eq_on f 1 s) : ∏ᶠ i ∈ s, f i = 1 :=
by { rw ← finprod_mem_one s, exact finprod_mem_congr rfl hf }
/-- If the product of `f i` over `i ∈ s` is not equal to one, then there is some `x ∈ s`
such that `f x ≠ 1`. -/
@[to_additive] lemma exists_ne_one_of_finprod_mem_ne_one (h : ∏ᶠ i ∈ s, f i ≠ 1) :
∃ x ∈ s, f x ≠ 1 :=
begin
by_contra h', push_neg at h',
exact h (finprod_mem_of_eq_on_one h')
end
/-- Given a finite set `s`, the product of `f i * g i` over `i ∈ s` equals the product of `f i`
over `i ∈ s` times the product of `g i` over `i ∈ s`. -/
@[to_additive] lemma finprod_mem_mul_distrib (hs : s.finite) :
∏ᶠ i ∈ s, (f i * g i) = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ s, g i :=
finprod_mem_mul_distrib' (hs.inter_of_left _) (hs.inter_of_left _)
@[to_additive] lemma monoid_hom.map_finprod {f : α → M} (g : M →* N) (hf : (mul_support f).finite) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) :=
g.map_finprod_plift f $ hf.preimage $ equiv.plift.injective.inj_on _
/-- A more general version of `monoid_hom.map_finprod_mem` that requires `s ∩ mul_support f` and
instead of `s` to be finite. -/
@[to_additive] lemma monoid_hom.map_finprod_mem' {f : α → M} (g : M →* N)
(h₀ : (s ∩ mul_support f).finite) :
g (∏ᶠ j ∈ s, f j) = ∏ᶠ i ∈ s, (g (f i)) :=
begin
rw [g.map_finprod],
{ simp only [g.map_finprod_Prop] },
{ simpa only [finprod_eq_mul_indicator_apply, mul_support_mul_indicator] }
end
/-- Given a monoid homomorphism `g : M →* N`, and a function `f : α → M`, the value of `g` at the
product of `f i` over `i ∈ s` equals the product of `(g ∘ f) i` over `s`. -/
@[to_additive] lemma monoid_hom.map_finprod_mem (f : α → M) (g : M →* N) (hs : s.finite) :
g (∏ᶠ j ∈ s, f j) = ∏ᶠ i ∈ s, g (f i) :=
g.map_finprod_mem' (hs.inter_of_left _)
/-!
### `∏ᶠ x ∈ s, f x` and set operations
-/
/-- The product of any function over an empty set is one. -/
@[to_additive] lemma finprod_mem_empty : ∏ᶠ i ∈ (∅ : set α), f i = 1 := by simp
/-- A set `s` is not empty if the product of some function over `s` is not equal to one. -/
@[to_additive] lemma nonempty_of_finprod_mem_ne_one (h : ∏ᶠ i ∈ s, f i ≠ 1) : s.nonempty :=
ne_empty_iff_nonempty.1 $ λ h', h $ h'.symm ▸ finprod_mem_empty
/-- Given finite sets `s` and `t`, the product of `f i` over `i ∈ s ∪ t` times the product of
`f i` over `i ∈ s ∩ t` equals the product of `f i` over `i ∈ s` times the product of `f i`
over `i ∈ t`. -/
@[to_additive] lemma finprod_mem_union_inter (hs : s.finite) (ht : t.finite) :
(∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i :=
begin
lift s to finset α using hs, lift t to finset α using ht,
classical,
rw [← finset.coe_union, ← finset.coe_inter],
simp only [finprod_mem_coe_finset, finset.prod_union_inter]
end
/-- A more general version of `finprod_mem_union_inter` that requires `s ∩ mul_support f` and
`t ∩ mul_support f` instead of `s` and `t` to be finite. -/
@[to_additive] lemma finprod_mem_union_inter'
(hs : (s ∩ mul_support f).finite) (ht : (t ∩ mul_support f).finite) :
(∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i :=
begin
rw [← finprod_mem_inter_mul_support f s, ← finprod_mem_inter_mul_support f t,
← finprod_mem_union_inter hs ht, ← union_inter_distrib_right, finprod_mem_inter_mul_support,
← finprod_mem_inter_mul_support f (s ∩ t)],
congr' 2,
rw [inter_left_comm, inter_assoc, inter_assoc, inter_self, inter_left_comm]
end
/-- A more general version of `finprod_mem_union` that requires `s ∩ mul_support f` and
`t ∩ mul_support f` instead of `s` and `t` to be finite. -/
@[to_additive] lemma finprod_mem_union' (hst : disjoint s t) (hs : (s ∩ mul_support f).finite)
(ht : (t ∩ mul_support f).finite) :
∏ᶠ i ∈ s ∪ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i :=
by rw [← finprod_mem_union_inter' hs ht, disjoint_iff_inter_eq_empty.1 hst, finprod_mem_empty,
mul_one]
/-- Given two finite disjoint sets `s` and `t`, the product of `f i` over `i ∈ s ∪ t` equals the
product of `f i` over `i ∈ s` times the product of `f i` over `i ∈ t`. -/
@[to_additive] lemma finprod_mem_union (hst : disjoint s t) (hs : s.finite) (ht : t.finite) :
∏ᶠ i ∈ s ∪ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i :=
finprod_mem_union' hst (hs.inter_of_left _) (ht.inter_of_left _)
/-- A more general version of `finprod_mem_union'` that requires `s ∩ mul_support f` and
`t ∩ mul_support f` instead of `s` and `t` to be disjoint -/
@[to_additive] lemma finprod_mem_union'' (hst : disjoint (s ∩ mul_support f) (t ∩ mul_support f))
(hs : (s ∩ mul_support f).finite) (ht : (t ∩ mul_support f).finite) :
∏ᶠ i ∈ s ∪ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i :=
by rw [← finprod_mem_inter_mul_support f s, ← finprod_mem_inter_mul_support f t,
← finprod_mem_union hst hs ht, ← union_inter_distrib_right, finprod_mem_inter_mul_support]
/-- The product of `f i` over `i ∈ {a}` equals `f a`. -/
@[to_additive] lemma finprod_mem_singleton : ∏ᶠ i ∈ ({a} : set α), f i = f a :=
by rw [← finset.coe_singleton, finprod_mem_coe_finset, finset.prod_singleton]
@[simp, to_additive] lemma finprod_cond_eq_left : ∏ᶠ i = a, f i = f a :=
finprod_mem_singleton
@[simp, to_additive] lemma finprod_cond_eq_right : ∏ᶠ i (hi : a = i), f i = f a :=
by simp [@eq_comm _ a]
/-- A more general version of `finprod_mem_insert` that requires `s ∩ mul_support f` instead of
`s` to be finite. -/
@[to_additive] lemma finprod_mem_insert' (f : α → M) (h : a ∉ s)
(hs : (s ∩ mul_support f).finite) :
∏ᶠ i ∈ insert a s, f i = f a * ∏ᶠ i ∈ s, f i :=
begin
rw [insert_eq, finprod_mem_union' _ _ hs, finprod_mem_singleton],
{ rwa disjoint_singleton_left },
{ exact (finite_singleton a).inter_of_left _ }
end
/-- Given a finite set `s` and an element `a ∉ s`, the product of `f i` over `i ∈ insert a s` equals
`f a` times the product of `f i` over `i ∈ s`. -/
@[to_additive] lemma finprod_mem_insert (f : α → M) (h : a ∉ s) (hs : s.finite) :
∏ᶠ i ∈ insert a s, f i = f a * ∏ᶠ i ∈ s, f i :=
finprod_mem_insert' f h $ hs.inter_of_left _
/-- If `f a = 1` for all `a ∉ s`, then the product of `f i` over `i ∈ insert a s` equals the
product of `f i` over `i ∈ s`. -/
@[to_additive] lemma finprod_mem_insert_of_eq_one_if_not_mem (h : a ∉ s → f a = 1) :
∏ᶠ i ∈ (insert a s), f i = ∏ᶠ i ∈ s, f i :=
begin
refine finprod_mem_inter_mul_support_eq' _ _ _ (λ x hx, ⟨_, or.inr⟩),
rintro (rfl|hxs),
exacts [not_imp_comm.1 h hx, hxs]
end
/-- If `f a = 1`, then the product of `f i` over `i ∈ insert a s` equals the product of `f i` over
`i ∈ s`. -/
@[to_additive] lemma finprod_mem_insert_one (h : f a = 1) :
∏ᶠ i ∈ (insert a s), f i = ∏ᶠ i ∈ s, f i :=
finprod_mem_insert_of_eq_one_if_not_mem (λ _, h)
/-- The product of `f i` over `i ∈ {a, b}`, `a ≠ b`, is equal to `f a * f b`. -/
@[to_additive] lemma finprod_mem_pair (h : a ≠ b) : ∏ᶠ i ∈ ({a, b} : set α), f i = f a * f b :=
by { rw [finprod_mem_insert, finprod_mem_singleton], exacts [h, finite_singleton b] }
/-- The product of `f y` over `y ∈ g '' s` equals the product of `f (g i)` over `s`
provided that `g` is injective on `s ∩ mul_support (f ∘ g)`. -/
@[to_additive] lemma finprod_mem_image' {s : set β} {g : β → α}
(hg : set.inj_on g (s ∩ mul_support (f ∘ g))) :
∏ᶠ i ∈ (g '' s), f i = ∏ᶠ j ∈ s, f (g j) :=
begin
classical,
by_cases hs : finite (s ∩ mul_support (f ∘ g)),
{ have hg : ∀ (x ∈ hs.to_finset) (y ∈ hs.to_finset), g x = g y → x = y,
by simpa only [hs.mem_to_finset],
rw [finprod_mem_eq_prod _ hs, ← finset.prod_image hg],
refine finprod_mem_eq_prod_of_inter_mul_support_eq f _,
rw [finset.coe_image, hs.coe_to_finset, ← image_inter_mul_support_eq, inter_assoc,
inter_self] },
{ rw [finprod_mem_eq_one_of_infinite hs, finprod_mem_eq_one_of_infinite],
rwa [image_inter_mul_support_eq, infinite_image_iff hg] }
end
/-- The product of `f y` over `y ∈ g '' s` equals the product of `f (g i)` over `s`
provided that `g` is injective on `s`. -/
@[to_additive] lemma finprod_mem_image {β} {s : set β} {g : β → α} (hg : set.inj_on g s) :
∏ᶠ i ∈ (g '' s), f i = ∏ᶠ j ∈ s, f (g j) :=
finprod_mem_image' $ hg.mono $ inter_subset_left _ _
/-- The product of `f y` over `y ∈ set.range g` equals the product of `f (g i)` over all `i`
provided that `g` is injective on `mul_support (f ∘ g)`. -/
@[to_additive] lemma finprod_mem_range' {g : β → α} (hg : set.inj_on g (mul_support (f ∘ g))) :
∏ᶠ i ∈ range g, f i = ∏ᶠ j, f (g j) :=
begin
rw [← image_univ, finprod_mem_image', finprod_mem_univ],
rwa univ_inter
end
/-- The product of `f y` over `y ∈ set.range g` equals the product of `f (g i)` over all `i`
provided that `g` is injective. -/
@[to_additive] lemma finprod_mem_range {g : β → α} (hg : injective g) :
∏ᶠ i ∈ range g, f i = ∏ᶠ j, f (g j) :=
finprod_mem_range' (hg.inj_on _)
/-- The product of `f i` over `s : set α` is equal to the product of `g j` over `t : set β`
if there exists a function `e : α → β` such that `e` is bijective from `s` to `t` and for all
`x` in `s` we have `f x = g (e x)`. -/
@[to_additive] lemma finprod_mem_eq_of_bij_on {s : set α} {t : set β} {f : α → M} {g : β → M}
(e : α → β) (he₀ : set.bij_on e s t) (he₁ : ∀ x ∈ s, f x = g (e x)) :
∏ᶠ i ∈ s, f i = ∏ᶠ j ∈ t, g j :=
begin
rw [← set.bij_on.image_eq he₀, finprod_mem_image he₀.2.1],
exact finprod_mem_congr rfl he₁
end
@[to_additive] lemma finprod_set_coe_eq_finprod_mem (s : set α) : ∏ᶠ j : s, f j = ∏ᶠ i ∈ s, f i :=
begin
rw [← finprod_mem_range, subtype.range_coe],
exact subtype.coe_injective
end
@[to_additive] lemma finprod_subtype_eq_finprod_cond (p : α → Prop) :
∏ᶠ j : subtype p, f j = ∏ᶠ i (hi : p i), f i :=
finprod_set_coe_eq_finprod_mem {i | p i}
@[to_additive] lemma finprod_mem_inter_mul_diff' (t : set α) (h : (s ∩ mul_support f).finite) :
(∏ᶠ i ∈ s ∩ t, f i) * ∏ᶠ i ∈ s \ t, f i = ∏ᶠ i ∈ s, f i :=
begin
rw [← finprod_mem_union', inter_union_diff],
exacts [λ x hx, hx.2.2 hx.1.2, h.subset (λ x hx, ⟨hx.1.1, hx.2⟩),
h.subset (λ x hx, ⟨hx.1.1, hx.2⟩)],
end
@[to_additive] lemma finprod_mem_inter_mul_diff (t : set α) (h : s.finite) :
(∏ᶠ i ∈ s ∩ t, f i) * ∏ᶠ i ∈ s \ t, f i = ∏ᶠ i ∈ s, f i :=
finprod_mem_inter_mul_diff' _ $ h.inter_of_left _
/-- A more general version of `finprod_mem_mul_diff` that requires `t ∩ mul_support f` instead of
`t` to be finite. -/
@[to_additive] lemma finprod_mem_mul_diff' (hst : s ⊆ t) (ht : (t ∩ mul_support f).finite) :
(∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t \ s, f i = ∏ᶠ i ∈ t, f i :=
by rw [← finprod_mem_inter_mul_diff' _ ht, inter_eq_self_of_subset_right hst]
/-- Given a finite set `t` and a subset `s` of `t`, the product of `f i` over `i ∈ s`
times the product of `f i` over `t \ s` equals the product of `f i` over `i ∈ t`. -/
@[to_additive] lemma finprod_mem_mul_diff (hst : s ⊆ t) (ht : t.finite) :
(∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t \ s, f i = ∏ᶠ i ∈ t, f i :=
finprod_mem_mul_diff' hst (ht.inter_of_left _)
/-- Given a family of pairwise disjoint finite sets `t i` indexed by a finite type,
the product of `f a` over the union `⋃ i, t i` is equal to the product over all indexes `i`
of the products of `f a` over `a ∈ t i`. -/
@[to_additive] lemma finprod_mem_Union [fintype ι] {t : ι → set α}
(h : pairwise (disjoint on t)) (ht : ∀ i, (t i).finite) :
∏ᶠ a ∈ (⋃ i : ι, t i), f a = ∏ᶠ i, (∏ᶠ a ∈ t i, f a) :=
begin
lift t to ι → finset α using ht,
classical,
rw [← bUnion_univ, ← finset.coe_univ, ← finset.coe_bUnion,
finprod_mem_coe_finset, finset.prod_bUnion],
{ simp only [finprod_mem_coe_finset, finprod_eq_prod_of_fintype] },
{ exact λ x _ y _ hxy, finset.disjoint_iff_disjoint_coe.2 (h x y hxy) }
end
/-- Given a family of sets `t : ι → set α`, a finite set `I` in the index type such that all
sets `t i`, `i ∈ I`, are finite, if all `t i`, `i ∈ I`, are pairwise disjoint, then
the product of `f a` over `a ∈ ⋃ i ∈ I, t i` is equal to the product over `i ∈ I`
of the products of `f a` over `a ∈ t i`. -/
@[to_additive] lemma finprod_mem_bUnion {I : set ι} {t : ι → set α}
(h : pairwise_on I (disjoint on t)) (hI : I.finite) (ht : ∀ i ∈ I, (t i).finite) :
∏ᶠ a ∈ ⋃ x ∈ I, t x, f a = ∏ᶠ i ∈ I, ∏ᶠ j ∈ t i, f j :=
begin
haveI := hI.fintype,
rw [← Union_subtype, finprod_mem_Union, ← finprod_set_coe_eq_finprod_mem],
exacts [λ x y hxy, h x x.2 y y.2 (subtype.coe_injective.ne hxy), λ b, ht b b.2]
end
/-- If `t` is a finite set of pairwise disjoint finite sets, then the product of `f a`
over `a ∈ ⋃₀ t` is the product over `s ∈ t` of the products of `f a` over `a ∈ s`. -/
@[to_additive] lemma finprod_mem_sUnion {t : set (set α)} (h : pairwise_on t disjoint)
(ht₀ : t.finite) (ht₁ : ∀ x ∈ t, set.finite x):
∏ᶠ a ∈ ⋃₀ t, f a = ∏ᶠ s ∈ t, ∏ᶠ a ∈ s, f a :=
by rw [set.sUnion_eq_bUnion, finprod_mem_bUnion h ht₀ ht₁]
/-- If `s : set α` and `t : set β` are finite sets, then the product over `s` commutes
with the product over `t`. -/
@[to_additive] lemma finprod_mem_comm {s : set α} {t : set β}
(f : α → β → M) (hs : s.finite) (ht : t.finite) :
∏ᶠ i ∈ s, ∏ᶠ j ∈ t, f i j = ∏ᶠ j ∈ t, ∏ᶠ i ∈ s, f i j :=
begin
lift s to finset α using hs, lift t to finset β using ht,
simp only [finprod_mem_coe_finset],
exact finset.prod_comm
end
/-- To prove a property of a finite product, it suffices to prove that the property is
multiplicative and holds on multipliers. -/
@[to_additive] lemma finprod_mem_induction (p : M → Prop) (hp₀ : p 1)
(hp₁ : ∀ x y, p x → p y → p (x * y)) (hp₂ : ∀ x ∈ s, p $ f x) :
p (∏ᶠ i ∈ s, f i) :=
finprod_induction _ hp₀ hp₁ $ λ x, finprod_induction _ hp₀ hp₁ $ hp₂ x
lemma finprod_cond_nonneg {R : Type*} [ordered_comm_semiring R] {p : α → Prop} {f : α → R}
(hf : ∀ x, p x → 0 ≤ f x) :
0 ≤ ∏ᶠ x (h : p x), f x :=
finprod_nonneg $ λ x, finprod_nonneg $ hf x
@[to_additive]
lemma single_le_finprod {M : Type*} [ordered_comm_monoid M] (i : α) {f : α → M}
(hf : finite (mul_support f)) (h : ∀ j, 1 ≤ f j) :
f i ≤ ∏ᶠ j, f j :=
by classical;
calc f i ≤ ∏ j in insert i hf.to_finset, f j :
finset.single_le_prod' (λ j hj, h j) (finset.mem_insert_self _ _)
... = ∏ᶠ j, f j :
(finprod_eq_prod_of_mul_support_to_finset_subset _ hf (finset.subset_insert _ _)).symm
lemma finprod_eq_zero {M₀ : Type*} [comm_monoid_with_zero M₀] (f : α → M₀) (x : α)
(hx : f x = 0) (hf : finite (mul_support f)) :
∏ᶠ x, f x = 0 :=
begin
nontriviality,
rw [finprod_eq_prod f hf],
refine finset.prod_eq_zero (hf.mem_to_finset.2 _) hx,
simp [hx]
end
@[to_additive] lemma finprod_prod_comm (s : finset β) (f : α → β → M)
(h : ∀ b ∈ s, (mul_support (λ a, f a b)).finite) :
∏ᶠ a : α, ∏ b in s, f a b = ∏ b in s, ∏ᶠ a : α, f a b :=
begin
have hU : mul_support (λ a, ∏ b in s, f a b) ⊆
(s.finite_to_set.bUnion (λ b hb, h b (finset.mem_coe.1 hb))).to_finset,
{ rw finite.coe_to_finset,
intros x hx,
simp only [exists_prop, mem_Union, ne.def, mem_mul_support, finset.mem_coe],
contrapose! hx,
rw [mem_mul_support, not_not, finset.prod_congr rfl hx, finset.prod_const_one] },
rw [finprod_eq_prod_of_mul_support_subset _ hU, finset.prod_comm],
refine finset.prod_congr rfl (λ b hb, (finprod_eq_prod_of_mul_support_subset _ _).symm),
intros a ha,
simp only [finite.coe_to_finset, mem_Union],
exact ⟨b, hb, ha⟩
end
@[to_additive] lemma prod_finprod_comm (s : finset α) (f : α → β → M)
(h : ∀ a ∈ s, (mul_support (f a)).finite) :
∏ a in s, ∏ᶠ b : β, f a b = ∏ᶠ b : β, ∏ a in s, f a b :=
(finprod_prod_comm s (λ b a, f a b) h).symm
lemma mul_finsum {R : Type*} [semiring R] (f : α → R) (r : R)
(h : (function.support f).finite) :
r * ∑ᶠ a : α, f a = ∑ᶠ a : α, r * f a :=
(add_monoid_hom.mul_left r).map_finsum h
lemma finsum_mul {R : Type*} [semiring R] (f : α → R) (r : R)
(h : (function.support f).finite) :
(∑ᶠ a : α, f a) * r = ∑ᶠ a : α, f a * r :=
(add_monoid_hom.mul_right r).map_finsum h
@[to_additive] lemma finset.mul_support_of_fiberwise_prod_subset_image [decidable_eq β]
(s : finset α) (f : α → M) (g : α → β) :
mul_support (λ b, (s.filter (λ a, g a = b)).prod f) ⊆ s.image g :=
begin
simp only [finset.coe_image, set.mem_image, finset.mem_coe, function.support_subset_iff],
intros b h,
suffices : (s.filter (λ (a : α), g a = b)).nonempty,
{ simpa only [s.fiber_nonempty_iff_mem_image g b, finset.mem_image, exists_prop], },
exact finset.nonempty_of_prod_ne_one h,
end
/-- Note that `b ∈ (s.filter (λ ab, prod.fst ab = a)).image prod.snd` iff `(a, b) ∈ s` so we can
simplify the right hand side of this lemma. However the form stated here is more useful for
iterating this lemma, e.g., if we have `f : α × β × γ → M`. -/
@[to_additive] lemma finprod_mem_finset_product' [decidable_eq α] [decidable_eq β]
(s : finset (α × β)) (f : α × β → M) :
∏ᶠ ab (h : ab ∈ s), f ab =
∏ᶠ a b (h : b ∈ (s.filter (λ ab, prod.fst ab = a)).image prod.snd), f (a, b) :=
begin
have : ∀ a, ∏ (i : β) in (s.filter (λ ab, prod.fst ab = a)).image prod.snd, f (a, i) =
(finset.filter (λ ab, prod.fst ab = a) s).prod f,
{ intros a, apply finset.prod_bij (λ b _, (a, b)); finish, },
rw finprod_mem_finset_eq_prod,
simp_rw [finprod_mem_finset_eq_prod, this],
rw [finprod_eq_prod_of_mul_support_subset _
(s.mul_support_of_fiberwise_prod_subset_image f prod.fst),
← finset.prod_fiberwise_of_maps_to _ f],
finish,
end
/-- See also `finprod_mem_finset_product'`. -/
@[to_additive] lemma finprod_mem_finset_product (s : finset (α × β)) (f : α × β → M) :
∏ᶠ ab (h : ab ∈ s), f ab = ∏ᶠ a b (h : (a, b) ∈ s), f (a, b) :=
by { classical, rw finprod_mem_finset_product', simp, }
@[to_additive] lemma finprod_mem_finset_product₃ {γ : Type*}
(s : finset (α × β × γ)) (f : α × β × γ → M) :
∏ᶠ abc (h : abc ∈ s), f abc = ∏ᶠ a b c (h : (a, b, c) ∈ s), f (a, b, c) :=
by { classical, rw finprod_mem_finset_product', simp_rw finprod_mem_finset_product', simp, }
@[to_additive] lemma finprod_curry (f : α × β → M) (hf : (mul_support f).finite) :
∏ᶠ ab, f ab = ∏ᶠ a b, f (a, b) :=
begin
have h₁ : ∀ a, ∏ᶠ (h : a ∈ hf.to_finset), f a = f a, { simp, },
have h₂ : ∏ᶠ a, f a = ∏ᶠ a (h : a ∈ hf.to_finset), f a, { simp, },
simp_rw [h₂, finprod_mem_finset_product, h₁],
end
@[to_additive] lemma finprod_curry₃ {γ : Type*} (f : α × β × γ → M) (h : (mul_support f).finite) :
∏ᶠ abc, f abc = ∏ᶠ a b c, f (a, b, c) :=
by { rw finprod_curry f h, congr, ext a, rw finprod_curry, simp [h], }
@[to_additive]
lemma finprod_dmem {s : set α} [decidable_pred (∈ s)] (f : (Π (a : α), a ∈ s → M)) :
∏ᶠ (a : α) (h : a ∈ s), f a h = ∏ᶠ (a : α) (h : a ∈ s), if h' : a ∈ s then f a h' else 1 :=
finprod_congr (λ a, finprod_congr (λ ha, (dif_pos ha).symm))
@[to_additive]
lemma finprod_emb_domain' {f : α → β} (hf : function.injective f)
[decidable_pred (∈ set.range f)] (g : α → M) :
∏ᶠ (b : β), (if h : b ∈ set.range f then g (classical.some h) else 1) = ∏ᶠ (a : α), g a :=
begin
simp_rw [← finprod_eq_dif],
rw [finprod_dmem, finprod_mem_range hf, finprod_congr (λ a, _)],
rw [dif_pos (set.mem_range_self a), hf (classical.some_spec (set.mem_range_self a))]
end
@[to_additive]
lemma finprod_emb_domain (f : α ↪ β) [decidable_pred (∈ set.range f)] (g : α → M) :
∏ᶠ (b : β), (if h : b ∈ set.range f then g (classical.some h) else 1) = ∏ᶠ (a : α), g a :=
finprod_emb_domain' f.injective g
end type
|
70b5d514a7c784503987fe5cae253b9af2ebad1c | cf39355caa609c0f33405126beee2739aa3cb77e | /library/system/io_interface.lean | 67faa4450c63e915c19b48dcf0ec139271450864 | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 4,200 | lean | /-
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import data.buffer system.random
inductive io.error
| other : string → io.error
| sys : nat → io.error
inductive io.mode
| read | write | read_write | append
inductive io.process.stdio
| piped
| inherit
| null
structure io.process.spawn_args :=
/- Command name. -/
(cmd : string)
/- Arguments for the process -/
(args : list string := [])
/- Configuration for the process' stdin handle. -/
(stdin := stdio.inherit)
/- Configuration for the process' stdout handle. -/
(stdout := stdio.inherit)
/- Configuration for the process' stderr handle. -/
(stderr := stdio.inherit)
/- Working directory for the process. -/
(cwd : option string := none)
/- Environment variables for the process. -/
(env : list (string × option string) := [])
class monad_io (m : Type → Type → Type) :=
[monad : Π e, monad (m e)]
-- TODO(Leo): use monad_except after it is merged
(catch : Π e₁ e₂ α, m e₁ α → (e₁ → m e₂ α) → m e₂ α)
(fail : Π e α, e → m e α)
(iterate : Π e α, α → (α → m e (option α)) → m e α)
-- Primitive Types
(handle : Type)
class monad_io_terminal (m : Type → Type → Type) :=
(put_str : string → m io.error unit)
(get_line : m io.error string)
(cmdline_args : list string)
open monad_io (handle)
class monad_io_net_system (m : Type → Type → Type) [monad_io m] :=
(socket : Type)
(listen : string → nat → m io.error socket)
(accept : socket → m io.error socket)
(connect : string → m io.error socket)
(recv : socket → nat → m io.error char_buffer)
(send : socket → char_buffer → m io.error unit)
(close : socket → m io.error unit)
class monad_io_file_system (m : Type → Type → Type) [monad_io m] :=
/- Remark: in Haskell, they also provide (Maybe TextEncoding) and NewlineMode -/
(mk_file_handle : string → io.mode → bool → m io.error (handle m))
(is_eof : (handle m) → m io.error bool)
(flush : (handle m) → m io.error unit)
(close : (handle m) → m io.error unit)
(read : (handle m) → nat → m io.error char_buffer)
(write : (handle m) → char_buffer → m io.error unit)
(get_line : (handle m) → m io.error char_buffer)
(stdin : m io.error (handle m))
(stdout : m io.error (handle m))
(stderr : m io.error (handle m))
(file_exists : string → m io.error bool)
(dir_exists : string → m io.error bool)
(remove : string → m io.error unit)
(rename : string → string → m io.error unit)
(mkdir : string → bool → m io.error bool)
(rmdir : string → m io.error bool)
meta class monad_io_serial (m : Type → Type → Type) [monad_io m] :=
(serialize : (handle m) → expr → m io.error unit)
(deserialize : (handle m) → m io.error expr)
class monad_io_environment (m : Type → Type → Type) :=
(get_env : string → m io.error (option string))
-- we don't provide set_env as it is (thread-)unsafe (at least with glibc)
(get_cwd : m io.error string)
(set_cwd : string → m io.error unit)
class monad_io_process (m : Type → Type → Type) [monad_io m] :=
(child : Type)
(stdin : child → (handle m))
(stdout : child → (handle m))
(stderr : child → (handle m))
(spawn : io.process.spawn_args → m io.error child)
(wait : child → m io.error nat)
(sleep : nat → m io.error unit)
class monad_io_random (m : Type → Type → Type) :=
(set_rand_gen : std_gen → m io.error unit)
(rand : nat → nat → m io.error nat)
instance monad_io_is_monad (m : Type → Type → Type) (e : Type) [monad_io m] : monad (m e) :=
monad_io.monad e
instance monad_io_is_monad_fail (m : Type → Type → Type) [monad_io m] : monad_fail (m io.error) :=
{ fail := λ α s, monad_io.fail _ _ (io.error.other s) }
instance monad_io_is_alternative (m : Type → Type → Type) [monad_io m] : alternative (m io.error) :=
{ orelse := λ α a b, monad_io.catch _ _ _ a (λ _, b),
failure := λ α, monad_io.fail _ _ (io.error.other "failure") }
|
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