statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
right_angle_rotation_aux₁ : E →ₗ[ℝ] E | let to_dual : E ≃ₗ[ℝ] (E →ₗ[ℝ] ℝ) :=
(inner_product_space.to_dual ℝ E).to_linear_equiv ≪≫ₗ linear_map.to_continuous_linear_map.symm in
↑to_dual.symm ∘ₗ ω | def | orientation.right_angle_rotation_aux₁ | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"inner_product_space.to_dual"
] | Auxiliary construction for `orientation.right_angle_rotation`, rotation by 90 degrees in an
oriented real inner product space of dimension 2. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_right_angle_rotation_aux₁_left (x y : E) :
⟪o.right_angle_rotation_aux₁ x, y⟫ = ω x y | by simp only [right_angle_rotation_aux₁, linear_equiv.trans_symm, linear_equiv.coe_trans,
linear_equiv.coe_coe, inner_product_space.to_dual_symm_apply, eq_self_iff_true,
linear_map.coe_to_continuous_linear_map', linear_isometry_equiv.coe_to_linear_equiv,
linear_map.comp_apply, ... | lemma | orientation.inner_right_angle_rotation_aux₁_left | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"inner_product_space.to_dual_symm_apply",
"linear_equiv.coe_coe",
"linear_equiv.coe_trans",
"linear_equiv.symm_symm",
"linear_equiv.trans_symm",
"linear_isometry_equiv.coe_to_linear_equiv",
"linear_isometry_equiv.to_linear_equiv_symm",
"linear_map.coe_to_continuous_linear_map'",
"linear_map.comp_app... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_right_angle_rotation_aux₁_right (x y : E) :
⟪x, o.right_angle_rotation_aux₁ y⟫ = - ω x y | begin
rw real_inner_comm,
simp [o.area_form_swap y x],
end | lemma | orientation.inner_right_angle_rotation_aux₁_right | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"real_inner_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_aux₂ : E →ₗᵢ[ℝ] E | { norm_map' := λ x, begin
dsimp,
refine le_antisymm _ _,
{ cases eq_or_lt_of_le (norm_nonneg (o.right_angle_rotation_aux₁ x)) with h h,
{ rw ← h,
positivity },
refine le_of_mul_le_mul_right _ h,
rw [← real_inner_self_eq_norm_mul_norm, o.inner_right_angle_rotation_aux₁_left],
... | def | orientation.right_angle_rotation_aux₂ | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"abs_real_inner_le_norm",
"eq_or_lt_of_le",
"exists_ne",
"finite_dimensional.nontrivial_of_finrank_pos",
"finrank_span_le_card",
"finset.card",
"finset.card_singleton",
"le_of_mul_le_mul_right",
"nontrivial",
"real_inner_self_eq_norm_mul_norm",
"set.to_finset_singleton",
"submodule"
] | Auxiliary construction for `orientation.right_angle_rotation`, rotation by 90 degrees in an
oriented real inner product space of dimension 2. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_angle_rotation_aux₁_right_angle_rotation_aux₁ (x : E) :
o.right_angle_rotation_aux₁ (o.right_angle_rotation_aux₁ x) = - x | begin
apply ext_inner_left ℝ,
intros y,
have : ⟪o.right_angle_rotation_aux₁ y, o.right_angle_rotation_aux₁ x⟫ = ⟪y, x⟫ :=
linear_isometry.inner_map_map o.right_angle_rotation_aux₂ y x,
rw [o.inner_right_angle_rotation_aux₁_right, ← o.inner_right_angle_rotation_aux₁_left, this,
inner_neg_right],
end | lemma | orientation.right_angle_rotation_aux₁_right_angle_rotation_aux₁ | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"ext_inner_left",
"inner_neg_right",
"linear_isometry.inner_map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation : E ≃ₗᵢ[ℝ] E | linear_isometry_equiv.of_linear_isometry
o.right_angle_rotation_aux₂
(-o.right_angle_rotation_aux₁)
(by ext; simp [right_angle_rotation_aux₂])
(by ext; simp [right_angle_rotation_aux₂]) | def | orientation.right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_isometry_equiv.of_linear_isometry"
] | An isometric automorphism of an oriented real inner product space of dimension 2 (usual notation
`J`). This automorphism squares to -1. We will define rotations in such a way that this
automorphism is equal to rotation by 90 degrees. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_right_angle_rotation_left (x y : E) : ⟪J x, y⟫ = ω x y | begin
rw right_angle_rotation,
exact o.inner_right_angle_rotation_aux₁_left x y
end | lemma | orientation.inner_right_angle_rotation_left | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_right_angle_rotation_right (x y : E) : ⟪x, J y⟫ = - ω x y | begin
rw right_angle_rotation,
exact o.inner_right_angle_rotation_aux₁_right x y
end | lemma | orientation.inner_right_angle_rotation_right | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_right_angle_rotation (x : E) : J (J x) = - x | begin
rw right_angle_rotation,
exact o.right_angle_rotation_aux₁_right_angle_rotation_aux₁ x
end | lemma | orientation.right_angle_rotation_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_symm :
linear_isometry_equiv.symm J = linear_isometry_equiv.trans J (linear_isometry_equiv.neg ℝ) | begin
rw right_angle_rotation,
exact linear_isometry_equiv.to_linear_isometry_injective rfl
end | lemma | orientation.right_angle_rotation_symm | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_isometry_equiv.neg",
"linear_isometry_equiv.symm",
"linear_isometry_equiv.to_linear_isometry_injective",
"linear_isometry_equiv.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_right_angle_rotation_self (x : E) : ⟪J x, x⟫ = 0 | by simp | lemma | orientation.inner_right_angle_rotation_self | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_right_angle_rotation_swap (x y : E) : ⟪x, J y⟫ = - ⟪J x, y⟫ | by simp | lemma | orientation.inner_right_angle_rotation_swap | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_right_angle_rotation_swap' (x y : E) : ⟪J x, y⟫ = - ⟪x, J y⟫ | by simp [o.inner_right_angle_rotation_swap x y] | lemma | orientation.inner_right_angle_rotation_swap' | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_comp_right_angle_rotation (x y : E) : ⟪J x, J y⟫ = ⟪x, y⟫ | linear_isometry_equiv.inner_map_map J x y | lemma | orientation.inner_comp_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_isometry_equiv.inner_map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_right_angle_rotation_left (x y : E) : ω (J x) y = - ⟪x, y⟫ | by rw [← o.inner_comp_right_angle_rotation, o.inner_right_angle_rotation_right, neg_neg] | lemma | orientation.area_form_right_angle_rotation_left | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_right_angle_rotation_right (x y : E) : ω x (J y) = ⟪x, y⟫ | by rw [← o.inner_right_angle_rotation_left, o.inner_comp_right_angle_rotation] | lemma | orientation.area_form_right_angle_rotation_right | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_comp_right_angle_rotation (x y : E) : ω (J x) (J y) = ω x y | by simp | lemma | orientation.area_form_comp_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_trans_right_angle_rotation :
linear_isometry_equiv.trans J J = linear_isometry_equiv.neg ℝ | by ext; simp | lemma | orientation.right_angle_rotation_trans_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_isometry_equiv.neg",
"linear_isometry_equiv.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_neg_orientation (x : E) :
(-o).right_angle_rotation x = - o.right_angle_rotation x | begin
apply ext_inner_right ℝ,
intros y,
rw inner_right_angle_rotation_left,
simp
end | lemma | orientation.right_angle_rotation_neg_orientation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"ext_inner_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_trans_neg_orientation :
(-o).right_angle_rotation = o.right_angle_rotation.trans (linear_isometry_equiv.neg ℝ) | linear_isometry_equiv.ext $ o.right_angle_rotation_neg_orientation | lemma | orientation.right_angle_rotation_trans_neg_orientation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_isometry_equiv.ext",
"linear_isometry_equiv.neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation_map {F : Type*}
[normed_add_comm_group F] [inner_product_space ℝ F] [fact (finrank ℝ F = 2)]
(φ : E ≃ₗᵢ[ℝ] F) (x : F) :
(orientation.map (fin 2) φ.to_linear_equiv o).right_angle_rotation x
= φ (o.right_angle_rotation (φ.symm x)) | begin
apply ext_inner_right ℝ,
intros y,
rw inner_right_angle_rotation_left,
transitivity ⟪J (φ.symm x), φ.symm y⟫,
{ simp [o.area_form_map] },
transitivity ⟪φ (J (φ.symm x)), φ (φ.symm y)⟫,
{ rw φ.inner_map_map },
{ simp },
end | lemma | orientation.right_angle_rotation_map | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"ext_inner_right",
"fact",
"inner_product_space",
"normed_add_comm_group",
"orientation.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_equiv_comp_right_angle_rotation (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < (φ.to_linear_equiv : E →ₗ[ℝ] E).det) (x : E) :
φ (J x) = J (φ x) | begin
convert (o.right_angle_rotation_map φ (φ x)).symm,
{ simp },
{ symmetry,
rwa ← o.map_eq_iff_det_pos φ.to_linear_equiv at hφ,
rw [fact.out (finrank ℝ E = 2), fintype.card_fin] },
end | lemma | orientation.linear_isometry_equiv_comp_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fintype.card_fin"
] | `J` commutes with any positively-oriented isometric automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_angle_rotation_map' {F : Type*}
[normed_add_comm_group F] [inner_product_space ℝ F] [fact (finrank ℝ F = 2)]
(φ : E ≃ₗᵢ[ℝ] F) :
(orientation.map (fin 2) φ.to_linear_equiv o).right_angle_rotation
= (φ.symm.trans o.right_angle_rotation).trans φ | linear_isometry_equiv.ext $ o.right_angle_rotation_map φ | lemma | orientation.right_angle_rotation_map' | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fact",
"inner_product_space",
"linear_isometry_equiv.ext",
"normed_add_comm_group",
"orientation.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_equiv_comp_right_angle_rotation' (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < (φ.to_linear_equiv : E →ₗ[ℝ] E).det) :
linear_isometry_equiv.trans J φ = φ.trans J | linear_isometry_equiv.ext $ o.linear_isometry_equiv_comp_right_angle_rotation φ hφ | lemma | orientation.linear_isometry_equiv_comp_right_angle_rotation' | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_isometry_equiv.ext",
"linear_isometry_equiv.trans"
] | `J` commutes with any positively-oriented isometric automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_right_angle_rotation (x : E) (hx : x ≠ 0) : basis (fin 2) ℝ E | @basis_of_linear_independent_of_card_eq_finrank ℝ _ _ _ _ _ _ _ ![x, J x]
(linear_independent_of_ne_zero_of_inner_eq_zero (λ i, by { fin_cases i; simp [hx] })
begin
intros i j hij,
fin_cases i; fin_cases j,
{ simpa },
{ simp },
{ simp },
{ simpa }
end)
(fact.out (finrank ℝ E = 2)).symm | def | orientation.basis_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"basis",
"basis_of_linear_independent_of_card_eq_finrank",
"linear_independent_of_ne_zero_of_inner_eq_zero"
] | For a nonzero vector `x` in an oriented two-dimensional real inner product space `E`,
`![x, J x]` forms an (orthogonal) basis for `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_basis_right_angle_rotation (x : E) (hx : x ≠ 0) :
⇑(o.basis_right_angle_rotation x hx) = ![x, J x] | coe_basis_of_linear_independent_of_card_eq_finrank _ _ | lemma | orientation.coe_basis_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"coe_basis_of_linear_independent_of_card_eq_finrank"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_mul_inner_add_area_form_mul_area_form' (a x : E) :
⟪a, x⟫ • innerₛₗ ℝ a + ω a x • ω a = ‖a‖ ^ 2 • innerₛₗ ℝ x | begin
by_cases ha : a = 0,
{ simp [ha] },
apply (o.basis_right_angle_rotation a ha).ext,
intros i,
fin_cases i,
{ simp only [real_inner_self_eq_norm_sq, algebra.id.smul_eq_mul, innerₛₗ_apply,
linear_map.smul_apply, linear_map.add_apply, matrix.cons_val_zero,
o.coe_basis_right_angle_rotation, o.a... | lemma | orientation.inner_mul_inner_add_area_form_mul_area_form' | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"algebra.id.smul_eq_mul",
"innerₛₗ",
"innerₛₗ_apply",
"linear_map.add_apply",
"linear_map.smul_apply",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"real_inner_comm",
"real_inner_self_eq_norm_sq",
"ring"
] | For vectors `a x y : E`, the identity `⟪a, x⟫ * ⟪a, y⟫ + ω a x * ω a y = ‖a‖ ^ 2 * ⟪x, y⟫`. (See
`orientation.inner_mul_inner_add_area_form_mul_area_form` for the "applied" form.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_mul_inner_add_area_form_mul_area_form (a x y : E) :
⟪a, x⟫ * ⟪a, y⟫ + ω a x * ω a y = ‖a‖ ^ 2 * ⟪x, y⟫ | congr_arg (λ f : E →ₗ[ℝ] ℝ, f y) (o.inner_mul_inner_add_area_form_mul_area_form' a x) | lemma | orientation.inner_mul_inner_add_area_form_mul_area_form | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | For vectors `a x y : E`, the identity `⟪a, x⟫ * ⟪a, y⟫ + ω a x * ω a y = ‖a‖ ^ 2 * ⟪x, y⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_sq_add_area_form_sq (a b : E) : ⟪a, b⟫ ^ 2 + ω a b ^ 2 = ‖a‖ ^ 2 * ‖b‖ ^ 2 | by simpa [sq, real_inner_self_eq_norm_sq] using o.inner_mul_inner_add_area_form_mul_area_form a b b | lemma | orientation.inner_sq_add_area_form_sq | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"real_inner_self_eq_norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_mul_area_form_sub' (a x : E) :
⟪a, x⟫ • ω a - ω a x • innerₛₗ ℝ a = ‖a‖ ^ 2 • ω x | begin
by_cases ha : a = 0,
{ simp [ha] },
apply (o.basis_right_angle_rotation a ha).ext,
intros i,
fin_cases i,
{ simp only [o.coe_basis_right_angle_rotation, o.area_form_apply_self, o.area_form_swap a x,
real_inner_self_eq_norm_sq, algebra.id.smul_eq_mul, innerₛₗ_apply, linear_map.sub_apply,
li... | lemma | orientation.inner_mul_area_form_sub' | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"algebra.id.smul_eq_mul",
"innerₛₗ",
"innerₛₗ_apply",
"linear_map.smul_apply",
"linear_map.sub_apply",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"real_inner_comm",
"real_inner_self_eq_norm_sq",
"ring"
] | For vectors `a x y : E`, the identity `⟪a, x⟫ * ω a y - ω a x * ⟪a, y⟫ = ‖a‖ ^ 2 * ω x y`. (See
`orientation.inner_mul_area_form_sub` for the "applied" form.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_mul_area_form_sub (a x y : E) : ⟪a, x⟫ * ω a y - ω a x * ⟪a, y⟫ = ‖a‖ ^ 2 * ω x y | congr_arg (λ f : E →ₗ[ℝ] ℝ, f y) (o.inner_mul_area_form_sub' a x) | lemma | orientation.inner_mul_area_form_sub | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | For vectors `a x y : E`, the identity `⟪a, x⟫ * ω a y - ω a x * ⟪a, y⟫ = ‖a‖ ^ 2 * ω x y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonneg_inner_and_area_form_eq_zero_iff_same_ray (x y : E) :
0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ same_ray ℝ x y | begin
by_cases hx : x = 0,
{ simp [hx] },
split,
{ let a : ℝ := (o.basis_right_angle_rotation x hx).repr y 0,
let b : ℝ := (o.basis_right_angle_rotation x hx).repr y 1,
suffices : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → same_ray ℝ x (a • x + b • J x),
{ rw ← (o.basis_right_angle_rotation x hx).sum_repr ... | lemma | orientation.nonneg_inner_and_area_form_eq_zero_iff_same_ray | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"algebra.id.smul_eq_mul",
"eq_zero_of_ne_zero_of_mul_right_eq_zero",
"fin.succ_zero_eq_one'",
"fintype.univ_of_is_empty",
"inner_add_left",
"inner_add_right",
"inner_smul_left",
"inner_smul_right",
"inner_zero_right",
"linear_map.add_apply",
"linear_map.map_smulₛₗ",
"matrix.cons_val_one",
"m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler : E →ₗ[ℝ] E →ₗ[ℝ] ℂ | (linear_map.llcomp ℝ E ℝ ℂ complex.of_real_clm) ∘ₗ innerₛₗ ℝ
+ (linear_map.llcomp ℝ E ℝ ℂ ((linear_map.lsmul ℝ ℂ).flip complex.I)) ∘ₗ ω | def | orientation.kahler | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.I",
"complex.of_real_clm",
"innerₛₗ",
"linear_map.llcomp",
"linear_map.lsmul"
] | A complex-valued real-bilinear map on an oriented real inner product space of dimension 2. Its
real part is the inner product and its imaginary part is `orientation.area_form`.
On `ℂ` with the standard orientation, `kahler w z = conj w * z`; see `complex.kahler`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kahler_apply_apply (x y : E) : o.kahler x y = ⟪x, y⟫ + ω x y • complex.I | rfl | lemma | orientation.kahler_apply_apply | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.I"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_swap (x y : E) : o.kahler x y = conj (o.kahler y x) | begin
simp only [kahler_apply_apply],
rw [real_inner_comm, area_form_swap],
simp,
end | lemma | orientation.kahler_swap | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"real_inner_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_apply_self (x : E) : o.kahler x x = ‖x‖ ^ 2 | by simp [kahler_apply_apply, real_inner_self_eq_norm_sq] | lemma | orientation.kahler_apply_self | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"real_inner_self_eq_norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_right_angle_rotation_left (x y : E) :
o.kahler (J x) y = - complex.I * o.kahler x y | begin
simp only [o.area_form_right_angle_rotation_left, o.inner_right_angle_rotation_left,
o.kahler_apply_apply, complex.of_real_neg, complex.real_smul],
linear_combination ω x y * complex.I_sq,
end | lemma | orientation.kahler_right_angle_rotation_left | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.I",
"complex.I_sq",
"complex.of_real_neg",
"complex.real_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_right_angle_rotation_right (x y : E) :
o.kahler x (J y) = complex.I * o.kahler x y | begin
simp only [o.area_form_right_angle_rotation_right, o.inner_right_angle_rotation_right,
o.kahler_apply_apply, complex.of_real_neg, complex.real_smul],
linear_combination - ω x y * complex.I_sq,
end | lemma | orientation.kahler_right_angle_rotation_right | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.I",
"complex.I_sq",
"complex.of_real_neg",
"complex.real_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_comp_right_angle_rotation (x y : E) : o.kahler (J x) (J y) = o.kahler x y | begin
simp only [kahler_right_angle_rotation_left, kahler_right_angle_rotation_right],
linear_combination - o.kahler x y * complex.I_sq,
end | lemma | orientation.kahler_comp_right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.I_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_neg_orientation (x y : E) : (-o).kahler x y = conj (o.kahler x y) | by simp [kahler_apply_apply] | lemma | orientation.kahler_neg_orientation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_mul (a x y : E) : o.kahler x a * o.kahler a y = ‖a‖ ^ 2 * o.kahler x y | begin
transitivity (↑(‖a‖ ^ 2) : ℂ) * o.kahler x y,
{ ext,
{ simp only [o.kahler_apply_apply, complex.add_im, complex.add_re, complex.I_im, complex.I_re,
complex.mul_im, complex.mul_re, complex.of_real_im, complex.of_real_re, complex.real_smul],
rw [real_inner_comm a x, o.area_form_swap x a],
... | lemma | orientation.kahler_mul | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.I_im",
"complex.I_re",
"complex.add_im",
"complex.add_re",
"complex.mul_im",
"complex.mul_re",
"complex.of_real_im",
"complex.of_real_re",
"complex.real_smul",
"real_inner_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_kahler (x y : E) : complex.norm_sq (o.kahler x y) = ‖x‖ ^ 2 * ‖y‖ ^ 2 | by simpa [kahler_apply_apply, complex.norm_sq, sq] using o.inner_sq_add_area_form_sq x y | lemma | orientation.norm_sq_kahler | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_kahler (x y : E) : complex.abs (o.kahler x y) = ‖x‖ * ‖y‖ | begin
rw [← sq_eq_sq, complex.sq_abs],
{ linear_combination o.norm_sq_kahler x y },
{ positivity },
{ positivity }
end | lemma | orientation.abs_kahler | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.abs",
"complex.sq_abs",
"sq_eq_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_kahler (x y : E) : ‖o.kahler x y‖ = ‖x‖ * ‖y‖ | by simpa using o.abs_kahler x y | lemma | orientation.norm_kahler | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_eq_zero_of_kahler_eq_zero {x y : E} (hx : o.kahler x y = 0) : x = 0 ∨ y = 0 | begin
have : ‖x‖ * ‖y‖ = 0 := by simpa [hx] using (o.norm_kahler x y).symm,
cases eq_zero_or_eq_zero_of_mul_eq_zero this with h h,
{ left,
simpa using h },
{ right,
simpa using h },
end | lemma | orientation.eq_zero_or_eq_zero_of_kahler_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_eq_zero_iff (x y : E) : o.kahler x y = 0 ↔ x = 0 ∨ y = 0 | begin
refine ⟨o.eq_zero_or_eq_zero_of_kahler_eq_zero, _⟩,
rintros (rfl | rfl);
simp,
end | lemma | orientation.kahler_eq_zero_iff | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_ne_zero {x y : E} (hx : x ≠ 0) (hy : y ≠ 0) : o.kahler x y ≠ 0 | begin
apply mt o.eq_zero_or_eq_zero_of_kahler_eq_zero,
tauto,
end | lemma | orientation.kahler_ne_zero | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_ne_zero_iff (x y : E) : o.kahler x y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0 | begin
refine ⟨_, λ h, o.kahler_ne_zero h.1 h.2⟩,
contrapose,
simp only [not_and_distrib, not_not, kahler_apply_apply, complex.real_smul],
rintros (rfl | rfl);
simp,
end | lemma | orientation.kahler_ne_zero_iff | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.real_smul",
"not_and_distrib",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_map {F : Type*}
[normed_add_comm_group F] [inner_product_space ℝ F] [fact (finrank ℝ F = 2)]
(φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
(orientation.map (fin 2) φ.to_linear_equiv o).kahler x y = o.kahler (φ.symm x) (φ.symm y) | by simp [kahler_apply_apply, area_form_map] | lemma | orientation.kahler_map | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fact",
"inner_product_space",
"normed_add_comm_group",
"orientation.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler_comp_linear_isometry_equiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < (φ.to_linear_equiv : E →ₗ[ℝ] E).det) (x y : E) :
o.kahler (φ x) (φ y) = o.kahler x y | by simp [kahler_apply_apply, o.area_form_comp_linear_isometry_equiv φ hφ] | lemma | orientation.kahler_comp_linear_isometry_equiv | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | The bilinear map `kahler` is invariant under pullback by a positively-oriented isometric
automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
area_form (w z : ℂ) : complex.orientation.area_form w z = (conj w * z).im | begin
let o := complex.orientation,
simp only [o.area_form_to_volume_form, o.volume_form_robust complex.orthonormal_basis_one_I rfl,
basis.det_apply, matrix.det_fin_two, basis.to_matrix_apply,to_basis_orthonormal_basis_one_I,
matrix.cons_val_zero, coe_basis_one_I_repr, matrix.cons_val_one, matrix.head_cons,... | lemma | complex.area_form | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"basis.det_apply",
"basis.to_matrix_apply",
"complex.orientation",
"complex.orthonormal_basis_one_I",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.det_fin_two",
"matrix.head_cons",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_angle_rotation (z : ℂ) :
complex.orientation.right_angle_rotation z = I * z | begin
apply ext_inner_right ℝ,
intros w,
rw orientation.inner_right_angle_rotation_left,
simp only [complex.area_form, complex.inner, mul_re, mul_im, conj_re, conj_im, map_mul, conj_I,
neg_re, neg_im, I_re, I_im],
ring,
end | lemma | complex.right_angle_rotation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.area_form",
"complex.inner",
"ext_inner_right",
"map_mul",
"orientation.inner_right_angle_rotation_left",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kahler (w z : ℂ) :
complex.orientation.kahler w z = conj w * z | begin
rw orientation.kahler_apply_apply,
ext1; simp,
end | lemma | complex.kahler | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"orientation.kahler_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_map_complex (f : E ≃ₗᵢ[ℝ] ℂ)
(hf : (orientation.map (fin 2) f.to_linear_equiv o) = complex.orientation) (x y : E) :
ω x y = (conj (f x) * f y).im | begin
rw [← complex.area_form, ← hf, o.area_form_map],
simp,
end | lemma | orientation.area_form_map_complex | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.area_form",
"complex.orientation",
"orientation.map"
] | The area form on an oriented real inner product space of dimension 2 can be evaluated in terms
of a complex-number representation of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_angle_rotation_map_complex (f : E ≃ₗᵢ[ℝ] ℂ)
(hf : (orientation.map (fin 2) f.to_linear_equiv o) = complex.orientation) (x : E) :
f (J x) = I * f x | begin
rw [← complex.right_angle_rotation, ← hf, o.right_angle_rotation_map],
simp,
end | lemma | orientation.right_angle_rotation_map_complex | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.orientation",
"complex.right_angle_rotation",
"orientation.map"
] | The rotation by 90 degrees on an oriented real inner product space of dimension 2 can be
evaluated in terms of a complex-number representation of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kahler_map_complex (f : E ≃ₗᵢ[ℝ] ℂ)
(hf : (orientation.map (fin 2) f.to_linear_equiv o) = complex.orientation) (x y : E) :
o.kahler x y = conj (f x) * f y | begin
rw [← complex.kahler, ← hf, o.kahler_map],
simp,
end | lemma | orientation.kahler_map_complex | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"complex.kahler",
"complex.orientation",
"orientation.map"
] | The Kahler form on an oriented real inner product space of dimension 2 can be evaluated in terms
of a complex-number representation of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_abs_convex : (𝓝 (0 : E)).has_basis
(λ (s : set E), s ∈ 𝓝 (0 : E) ∧ balanced 𝕜 s ∧ convex ℝ s) id | begin
refine (locally_convex_space.convex_basis_zero ℝ E).to_has_basis (λ s hs, _)
(λ s hs, ⟨s, ⟨hs.1, hs.2.2⟩, rfl.subset⟩),
refine ⟨convex_hull ℝ (balanced_core 𝕜 s), _, convex_hull_min (balanced_core_subset s) hs.2⟩,
refine ⟨filter.mem_of_superset (balanced_core_mem_nhds_zero hs.1) (subset_convex_hull ℝ _... | lemma | nhds_basis_abs_convex | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"balanced",
"balanced_core",
"balanced_core_balanced",
"balanced_core_mem_nhds_zero",
"balanced_core_subset",
"convex",
"convex_convex_hull",
"convex_hull_min",
"locally_convex_space.convex_basis_zero",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_abs_convex_open : (𝓝 (0 : E)).has_basis
(λ (s : set E), (0 : E) ∈ s ∧ is_open s ∧ balanced 𝕜 s ∧ convex ℝ s) id | begin
refine (nhds_basis_abs_convex 𝕜 E).to_has_basis _ _,
{ rintros s ⟨hs_nhds, hs_balanced, hs_convex⟩,
refine ⟨interior s, _, interior_subset⟩,
exact ⟨mem_interior_iff_mem_nhds.mpr hs_nhds, is_open_interior,
hs_balanced.interior (mem_interior_iff_mem_nhds.mpr hs_nhds), hs_convex.interior⟩ },
rin... | lemma | nhds_basis_abs_convex_open | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"balanced",
"convex",
"is_open",
"is_open_interior",
"nhds_basis_abs_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_convex_open_sets | { s : set E // (0 : E) ∈ s ∧ is_open s ∧ balanced 𝕜 s ∧ convex ℝ s } | def | abs_convex_open_sets | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"balanced",
"convex",
"is_open"
] | The type of absolutely convex open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_convex_open_sets.has_coe : has_coe (abs_convex_open_sets 𝕜 E) (set E) | ⟨subtype.val⟩ | instance | abs_convex_open_sets.has_coe | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero_mem (s : abs_convex_open_sets 𝕜 E) : (0 : E) ∈ (s : set E) | s.2.1 | lemma | abs_convex_open_sets.coe_zero_mem | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_is_open (s : abs_convex_open_sets 𝕜 E) : is_open (s : set E) | s.2.2.1 | lemma | abs_convex_open_sets.coe_is_open | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nhds (s : abs_convex_open_sets 𝕜 E) : (s : set E) ∈ 𝓝 (0 : E) | s.coe_is_open.mem_nhds s.coe_zero_mem | lemma | abs_convex_open_sets.coe_nhds | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_balanced (s : abs_convex_open_sets 𝕜 E) : balanced 𝕜 (s : set E) | s.2.2.2.1 | lemma | abs_convex_open_sets.coe_balanced | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets",
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_convex (s : abs_convex_open_sets 𝕜 E) : convex ℝ (s : set E) | s.2.2.2.2 | lemma | abs_convex_open_sets.coe_convex | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_seminorm_family : seminorm_family 𝕜 E (abs_convex_open_sets 𝕜 E) | λ s, gauge_seminorm s.coe_balanced s.coe_convex (absorbent_nhds_zero s.coe_nhds) | def | gauge_seminorm_family | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets",
"absorbent_nhds_zero",
"gauge_seminorm",
"seminorm_family"
] | The family of seminorms defined by the gauges of absolute convex open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_seminorm_family_ball (s : abs_convex_open_sets 𝕜 E) :
(gauge_seminorm_family 𝕜 E s).ball 0 1 = (s : set E) | begin
dunfold gauge_seminorm_family,
rw seminorm.ball_zero_eq,
simp_rw gauge_seminorm_to_fun,
exact gauge_lt_one_eq_self_of_open s.coe_convex s.coe_zero_mem s.coe_is_open,
end | lemma | gauge_seminorm_family_ball | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_convex_open_sets",
"gauge_lt_one_eq_self_of_open",
"gauge_seminorm_family",
"seminorm.ball_zero_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_gauge_seminorm_family : with_seminorms (gauge_seminorm_family 𝕜 E) | begin
refine seminorm_family.with_seminorms_of_has_basis _ _,
refine (nhds_basis_abs_convex_open 𝕜 E).to_has_basis (λ s hs, _) (λ s hs, _),
{ refine ⟨s, ⟨_, rfl.subset⟩⟩,
convert (gauge_seminorm_family _ _).basis_sets_singleton_mem ⟨s, hs⟩ one_pos,
rw [gauge_seminorm_family_ball, subtype.coe_mk] },
ref... | lemma | with_gauge_seminorm_family | analysis.locally_convex | src/analysis/locally_convex/abs_convex.lean | [
"analysis.locally_convex.balanced_core_hull",
"analysis.locally_convex.with_seminorms",
"analysis.convex.gauge"
] | [
"abs_of_pos",
"balanced_Inter₂",
"convex_Inter₂",
"gauge_seminorm_family",
"gauge_seminorm_family_ball",
"is_open_bInter",
"nhds_basis_abs_convex_open",
"seminorm.balanced_ball_zero",
"seminorm.ball_finset_sup_eq_Inter",
"seminorm.convex_ball",
"seminorm.mem_ball_zero",
"seminorm.smul_ball_zer... | The topology of a locally convex space is induced by the gauge seminorm family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_core (s : set E) | ⋃₀ {t : set E | balanced 𝕜 t ∧ t ⊆ s} | def | balanced_core | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced"
] | The largest balanced subset of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_core_aux (s : set E) | ⋂ (r : 𝕜) (hr : 1 ≤ ‖r‖), r • s | def | balanced_core_aux | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [] | Helper definition to prove `balanced_core_eq_Inter` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_hull (s : set E) | ⋃ (r : 𝕜) (hr : ‖r‖ ≤ 1), r • s | def | balanced_hull | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [] | The smallest balanced superset of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_core_subset (s : set E) : balanced_core 𝕜 s ⊆ s | sUnion_subset $ λ t ht, ht.2 | lemma | balanced_core_subset | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_empty : balanced_core 𝕜 (∅ : set E) = ∅ | eq_empty_of_subset_empty (balanced_core_subset _) | lemma | balanced_core_empty | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_core_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_balanced_core_iff : x ∈ balanced_core 𝕜 s ↔ ∃ t, balanced 𝕜 t ∧ t ⊆ s ∧ x ∈ t | by simp_rw [balanced_core, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc] | lemma | mem_balanced_core_iff | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core",
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_balanced_core_subset (s : set E) {a : 𝕜} (ha : ‖a‖ ≤ 1) :
a • balanced_core 𝕜 s ⊆ balanced_core 𝕜 s | begin
rintro x ⟨y, hy, rfl⟩,
rw mem_balanced_core_iff at hy,
rcases hy with ⟨t, ht1, ht2, hy⟩,
exact ⟨t, ⟨ht1, ht2⟩, ht1 a ha (smul_mem_smul_set hy)⟩,
end | lemma | smul_balanced_core_subset | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"mem_balanced_core_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_balanced (s : set E) : balanced 𝕜 (balanced_core 𝕜 s) | λ _, smul_balanced_core_subset s | lemma | balanced_core_balanced | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core",
"smul_balanced_core_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.subset_core_of_subset (hs : balanced 𝕜 s) (h : s ⊆ t) : s ⊆ balanced_core 𝕜 t | subset_sUnion_of_mem ⟨hs, h⟩ | lemma | balanced.subset_core_of_subset | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core"
] | The balanced core of `t` is maximal in the sense that it contains any balanced subset
`s` of `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_balanced_core_aux_iff : x ∈ balanced_core_aux 𝕜 s ↔ ∀ r : 𝕜, 1 ≤ ‖r‖ → x ∈ r • s | mem_Inter₂ | lemma | mem_balanced_core_aux_iff | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_balanced_hull_iff : x ∈ balanced_hull 𝕜 s ↔ ∃ (r : 𝕜) (hr : ‖r‖ ≤ 1), x ∈ r • s | mem_Union₂ | lemma | mem_balanced_hull_iff | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.hull_subset_of_subset (ht : balanced 𝕜 t) (h : s ⊆ t) : balanced_hull 𝕜 s ⊆ t | λ x hx, by { obtain ⟨r, hr, y, hy, rfl⟩ := mem_balanced_hull_iff.1 hx, exact ht.smul_mem hr (h hy) } | lemma | balanced.hull_subset_of_subset | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_hull"
] | The balanced hull of `s` is minimal in the sense that it is contained in any balanced superset
`t` of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_core_zero_mem (hs : (0 : E) ∈ s) : (0 : E) ∈ balanced_core 𝕜 s | mem_balanced_core_iff.2 ⟨0, balanced_zero, zero_subset.2 hs, zero_mem_zero⟩ | lemma | balanced_core_zero_mem | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_nonempty_iff : (balanced_core 𝕜 s).nonempty ↔ (0 : E) ∈ s | ⟨λ h, zero_subset.1 $ (zero_smul_set h).superset.trans $ (balanced_core_balanced s (0 : 𝕜) $
norm_zero.trans_le zero_le_one).trans $ balanced_core_subset _,
λ h, ⟨0, balanced_core_zero_mem h⟩⟩ | lemma | balanced_core_nonempty_iff | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_core_balanced",
"balanced_core_subset",
"balanced_core_zero_mem",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_balanced_hull [norm_one_class 𝕜] {s : set E} : s ⊆ balanced_hull 𝕜 s | λ _ hx, mem_balanced_hull_iff.2 ⟨1, norm_one.le, _, hx, one_smul _ _⟩ | lemma | subset_balanced_hull | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_hull",
"norm_one_class",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_hull.balanced (s : set E) : balanced 𝕜 (balanced_hull 𝕜 s) | begin
intros a ha,
simp_rw [balanced_hull, smul_set_Union₂, subset_def, mem_Union₂],
rintro x ⟨r, hr, hx⟩,
rw ←smul_assoc at hx,
exact ⟨a • r, (semi_normed_ring.norm_mul _ _).trans (mul_le_one ha (norm_nonneg r) hr), hx⟩,
end | lemma | balanced_hull.balanced | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_hull",
"mul_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_aux_empty : balanced_core_aux 𝕜 (∅ : set E) = ∅ | begin
simp_rw [balanced_core_aux, Inter₂_eq_empty_iff, smul_set_empty],
exact λ _, ⟨1, norm_one.ge, not_mem_empty _⟩,
end | lemma | balanced_core_aux_empty | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_aux_subset (s : set E) : balanced_core_aux 𝕜 s ⊆ s | λ x hx, by simpa only [one_smul] using mem_balanced_core_aux_iff.1 hx 1 norm_one.ge | lemma | balanced_core_aux_subset | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core_aux",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_aux_balanced (h0 : (0 : E) ∈ balanced_core_aux 𝕜 s):
balanced 𝕜 (balanced_core_aux 𝕜 s) | begin
rintro a ha x ⟨y, hy, rfl⟩,
obtain rfl | h := eq_or_ne a 0,
{ rwa zero_smul },
rw mem_balanced_core_aux_iff at ⊢ hy,
intros r hr,
have h'' : 1 ≤ ‖a⁻¹ • r‖,
{ rw [norm_smul, norm_inv],
exact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr },
have h' := hy (a⁻¹ • r) h'',
... | lemma | balanced_core_aux_balanced | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core_aux",
"eq_or_ne",
"mem_balanced_core_aux_iff",
"norm_inv",
"norm_smul",
"one_le_inv",
"one_le_mul_of_one_le_of_one_le",
"smul_assoc",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_aux_maximal (h : t ⊆ s) (ht : balanced 𝕜 t) : t ⊆ balanced_core_aux 𝕜 s | begin
refine λ x hx, mem_balanced_core_aux_iff.2 (λ r hr, _),
rw mem_smul_set_iff_inv_smul_mem₀ (norm_pos_iff.mp $ zero_lt_one.trans_le hr),
refine h (ht.smul_mem _ hx),
rw norm_inv,
exact inv_le_one hr,
end | lemma | balanced_core_aux_maximal | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core_aux",
"inv_le_one",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_subset_balanced_core_aux : balanced_core 𝕜 s ⊆ balanced_core_aux 𝕜 s | balanced_core_aux_maximal (balanced_core_subset s) (balanced_core_balanced s) | lemma | balanced_core_subset_balanced_core_aux | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_core_aux",
"balanced_core_aux_maximal",
"balanced_core_balanced",
"balanced_core_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_eq_Inter (hs : (0 : E) ∈ s) :
balanced_core 𝕜 s = ⋂ (r : 𝕜) (hr : 1 ≤ ‖r‖), r • s | begin
refine balanced_core_subset_balanced_core_aux.antisymm _,
refine (balanced_core_aux_balanced _).subset_core_of_subset (balanced_core_aux_subset s),
exact balanced_core_subset_balanced_core_aux (balanced_core_zero_mem hs),
end | lemma | balanced_core_eq_Inter | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_core_aux_balanced",
"balanced_core_aux_subset",
"balanced_core_subset_balanced_core_aux",
"balanced_core_zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_balanced_core (ht : (0 : E) ∈ t) (hst : ∀ (a : 𝕜) (ha : ‖a‖ ≤ 1), a • s ⊆ t) :
s ⊆ balanced_core 𝕜 t | begin
rw balanced_core_eq_Inter ht,
refine subset_Inter₂ (λ a ha, _),
rw ←smul_inv_smul₀ (norm_pos_iff.mp $ zero_lt_one.trans_le ha) s,
refine smul_set_mono (hst _ _),
rw [norm_inv],
exact inv_le_one ha,
end | lemma | subset_balanced_core | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_core_eq_Inter",
"inv_le_one",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.balanced_core (hU : is_closed U) : is_closed (balanced_core 𝕜 U) | begin
by_cases h : (0 : E) ∈ U,
{ rw balanced_core_eq_Inter h,
refine is_closed_Inter (λ a, _),
refine is_closed_Inter (λ ha, _),
have ha' := lt_of_lt_of_le zero_lt_one ha,
rw norm_pos_iff at ha',
refine is_closed_map_smul_of_ne_zero ha' U hU },
convert is_closed_empty,
contrapose! h,
exac... | lemma | is_closed.balanced_core | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced_core",
"balanced_core_eq_Inter",
"is_closed",
"is_closed_Inter",
"is_closed_empty",
"is_closed_map_smul_of_ne_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_core_mem_nhds_zero (hU : U ∈ 𝓝 (0 : E)) : balanced_core 𝕜 U ∈ 𝓝 (0 : E) | begin
-- Getting neighborhoods of the origin for `0 : 𝕜` and `0 : E`
obtain ⟨r, V, hr, hV, hrVU⟩ : ∃ (r : ℝ) (V : set E), 0 < r ∧ V ∈ 𝓝 (0 : E) ∧
∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U,
{ have h : filter.tendsto (λ (x : 𝕜 × E), x.fst • x.snd) (𝓝 (0,0)) (𝓝 0),
from continuous_smul.tendsto' ... | lemma | balanced_core_mem_nhds_zero | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"and_imp",
"balanced_core",
"exists_prop",
"filter.mem_of_superset",
"filter.tendsto",
"mem_of_mem_nhds",
"mul_lt_mul'",
"norm_mul",
"normed_field.exists_norm_lt",
"one_mul",
"prod.exists'",
"prod.forall'",
"set_smul_mem_nhds_zero_iff",
"smul_smul",
"smul_zero",
"subset_balanced_core"
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_balanced : (𝓝 (0 : E)).has_basis
(λ (s : set E), s ∈ 𝓝 (0 : E) ∧ balanced 𝕜 s) id | filter.has_basis_self.mpr
(λ s hs, ⟨balanced_core 𝕜 s, balanced_core_mem_nhds_zero hs,
balanced_core_balanced s, balanced_core_subset s⟩) | lemma | nhds_basis_balanced | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core_balanced",
"balanced_core_mem_nhds_zero",
"balanced_core_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_closed_balanced [regular_space E] : (𝓝 (0 : E)).has_basis
(λ (s : set E), s ∈ 𝓝 (0 : E) ∧ is_closed s ∧ balanced 𝕜 s) id | begin
refine (closed_nhds_basis 0).to_has_basis (λ s hs, _) (λ s hs, ⟨s, ⟨hs.1, hs.2.1⟩, rfl.subset⟩),
refine ⟨balanced_core 𝕜 s, ⟨balanced_core_mem_nhds_zero hs.1, _⟩, balanced_core_subset s⟩,
exact ⟨hs.2.balanced_core, balanced_core_balanced s⟩
end | lemma | nhds_basis_closed_balanced | analysis.locally_convex | src/analysis/locally_convex/balanced_core_hull.lean | [
"analysis.locally_convex.basic"
] | [
"balanced",
"balanced_core_balanced",
"balanced_core_subset",
"closed_nhds_basis",
"is_closed",
"regular_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs (A B : set E) | ∃ r, 0 < r ∧ ∀ a : 𝕜, r ≤ ‖a‖ → B ⊆ a • A | def | absorbs | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [] | A set `A` absorbs another set `B` if `B` is contained in all scalings of `A` by elements of
sufficiently large norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absorbs_empty {s : set E}: absorbs 𝕜 s (∅ : set E) | ⟨1, one_pos, λ a ha, set.empty_subset _⟩ | lemma | absorbs_empty | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"set.empty_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.mono (hs : absorbs 𝕜 s u) (hst : s ⊆ t) (hvu : v ⊆ u) : absorbs 𝕜 t v | let ⟨r, hr, h⟩ := hs in ⟨r, hr, λ a ha, hvu.trans $ (h _ ha).trans $ smul_set_mono hst⟩ | lemma | absorbs.mono | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.mono_left (hs : absorbs 𝕜 s u) (h : s ⊆ t) : absorbs 𝕜 t u | hs.mono h subset.rfl | lemma | absorbs.mono_left | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.mono_right (hs : absorbs 𝕜 s u) (h : v ⊆ u) : absorbs 𝕜 s v | hs.mono subset.rfl h | lemma | absorbs.mono_right | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.