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
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|---|---|---|---|---|---|---|---|---|---|---|
delta (x : E) : 𝓢(E, F) →L[𝕜] F | (bounded_continuous_function.eval_clm 𝕜 x).comp (to_bounded_continuous_function_clm 𝕜 E F) | def | schwartz_map.delta | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"bounded_continuous_function.eval_clm"
] | The Dirac delta distribution | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
delta_apply (x₀ : E) (f : 𝓢(E, F)) : delta 𝕜 F x₀ f = f x₀ | rfl | lemma | schwartz_map.delta_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm (𝕜 : Type*) (E : Type*) [semi_normed_ring 𝕜] [add_group E] [has_smul 𝕜 E]
extends add_group_seminorm E | (smul' : ∀ (a : 𝕜) (x : E), to_fun (a • x) = ‖a‖ * to_fun x) | structure | seminorm | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"add_group",
"add_group_seminorm",
"has_smul",
"semi_normed_ring"
] | A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_class (F : Type*) (𝕜 E : out_param $ Type*) [semi_normed_ring 𝕜] [add_group E]
[has_smul 𝕜 E] extends add_group_seminorm_class F E ℝ | (map_smul_eq_mul (f : F) (a : 𝕜) (x : E) : f (a • x) = ‖a‖ * f x) | class | seminorm_class | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"add_group",
"add_group_seminorm_class",
"has_smul",
"semi_normed_ring"
] | `seminorm_class F 𝕜 E` states that `F` is a type of seminorms on the `𝕜`-module E.
You should extend this class when you extend `seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm.of [semi_normed_ring 𝕜] [add_comm_group E] [module 𝕜 E] (f : E → ℝ)
(add_le : ∀ (x y : E), f (x + y) ≤ f x + f y)
(smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) : seminorm 𝕜 E | { to_fun := f,
map_zero' := by rw [←zero_smul 𝕜 (0 : E), smul, norm_zero, zero_mul],
add_le' := add_le,
smul' := smul,
neg' := λ x, by rw [←neg_one_smul 𝕜, smul, norm_neg, ← smul, one_smul] } | def | seminorm.of | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"add_comm_group",
"add_le",
"module",
"one_smul",
"semi_normed_ring",
"seminorm",
"zero_mul"
] | Alternative constructor for a `seminorm` on an `add_comm_group E` that is a module over a
`semi_norm_ring 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm.of_smul_le [normed_field 𝕜] [add_comm_group E] [module 𝕜 E] (f : E → ℝ)
(map_zero : f 0 = 0) (add_le : ∀ x y, f (x + y) ≤ f x + f y)
(smul_le : ∀ (r : 𝕜) x, f (r • x) ≤ ‖r‖ * f x) : seminorm 𝕜 E | seminorm.of f add_le
(λ r x, begin
refine le_antisymm (smul_le r x) _,
by_cases r = 0,
{ simp [h, map_zero] },
rw ←mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h)),
rw inv_mul_cancel_left₀ (norm_ne_zero_iff.mpr h),
specialize smul_le r⁻¹ (r • x),
rw norm_inv at smul_le,
convert smul_... | def | seminorm.of_smul_le | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"add_comm_group",
"add_le",
"inv_mul_cancel_left₀",
"module",
"norm_inv",
"normed_field",
"seminorm",
"seminorm.of"
] | Alternative constructor for a `seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_class : seminorm_class (seminorm 𝕜 E) 𝕜 E | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero',
map_add_le_add := λ f, f.add_le',
map_neg_eq_map := λ f, f.neg',
map_smul_eq_mul := λ f, f.smul' } | instance | seminorm.seminorm_class | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm",
"seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {p q : seminorm 𝕜 E} (h : ∀ x, (p : E → ℝ) x = q x) : p = q | fun_like.ext p q h | lemma | seminorm.ext | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"fun_like.ext",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ⇑(0 : seminorm 𝕜 E) = 0 | rfl | lemma | seminorm.coe_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : E) : (0 : seminorm 𝕜 E) x = 0 | rfl | lemma | seminorm.zero_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p : seminorm 𝕜 E) : ⇑(r • p) = r • p | rfl | lemma | seminorm.coe_smul | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"has_smul",
"is_scalar_tower",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p : seminorm 𝕜 E) (x : E) : (r • p) x = r • p x | rfl | lemma | seminorm.smul_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"has_smul",
"is_scalar_tower",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (p q : seminorm 𝕜 E) : ⇑(p + q) = p + q | rfl | lemma | seminorm.coe_add | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (p q : seminorm 𝕜 E) (x : E) : (p + q) x = p x + q x | rfl | lemma | seminorm.add_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_add_monoid_hom : add_monoid_hom (seminorm 𝕜 E) (E → ℝ) | ⟨coe_fn, coe_zero, coe_add⟩ | def | seminorm.coe_fn_add_monoid_hom | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"add_monoid_hom",
"seminorm"
] | `coe_fn` as an `add_monoid_hom`. Helper definition for showing that `seminorm 𝕜 E` is
a module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_add_monoid_hom_injective : function.injective (coe_fn_add_monoid_hom 𝕜 E) | show @function.injective (seminorm 𝕜 E) (E → ℝ) coe_fn, from fun_like.coe_injective | lemma | seminorm.coe_fn_add_monoid_hom_injective | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"fun_like.coe_injective",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (p q : seminorm 𝕜 E) : ⇑(p ⊔ q) = p ⊔ q | rfl | lemma | seminorm.coe_sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_apply (p q : seminorm 𝕜 E) (x : E) : (p ⊔ q) x = p x ⊔ q x | rfl | lemma | seminorm.sup_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p q : seminorm 𝕜 E) :
r • (p ⊔ q) = r • p ⊔ r • q | have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y),
from λ x y, by simpa only [←smul_eq_mul, ←nnreal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)]
using mul_max_of_nonneg x y (r • 1 : ℝ≥0).coe_nonneg,
ext $ λ x, real.smul_max _ _ | lemma | seminorm.smul_sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"has_smul",
"is_scalar_tower",
"mul_max_of_nonneg",
"seminorm",
"smul_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe {p q : seminorm 𝕜 E} : (p : E → ℝ) ≤ q ↔ p ≤ q | iff.rfl | lemma | seminorm.coe_le_coe | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_coe {p q : seminorm 𝕜 E} : (p : E → ℝ) < q ↔ p < q | iff.rfl | lemma | seminorm.coe_lt_coe | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def {p q : seminorm 𝕜 E} : p ≤ q ↔ ∀ x, p x ≤ q x | iff.rfl | lemma | seminorm.le_def | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_def {p q : seminorm 𝕜 E} : p < q ↔ p ≤ q ∧ ∃ x, p x < q x | pi.lt_def | lemma | seminorm.lt_def | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"pi.lt_def",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : seminorm 𝕜 E | { to_fun := λ x, p (f x),
smul' := λ _ _, by rw [map_smulₛₗ, map_smul_eq_mul, ring_hom_isometric.is_iso],
..(p.to_add_group_seminorm.comp f.to_add_monoid_hom) } | def | seminorm.comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | Composition of a seminorm with a linear map is a seminorm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : ⇑(p.comp f) = p ∘ f | rfl | lemma | seminorm.coe_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) :
(p.comp f) x = p (f x) | rfl | lemma | seminorm.comp_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (p : seminorm 𝕜 E) : p.comp linear_map.id = p | ext $ λ _, rfl | lemma | seminorm.comp_id | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"linear_map.id",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_zero (p : seminorm 𝕜₂ E₂) : p.comp (0 : E →ₛₗ[σ₁₂] E₂) = 0 | ext $ λ _, map_zero p | lemma | seminorm.comp_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp (f : E →ₛₗ[σ₁₂] E₂) : (0 : seminorm 𝕜₂ E₂).comp f = 0 | ext $ λ _, rfl | lemma | seminorm.zero_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_comp [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] (p : seminorm 𝕜₃ E₃)
(g : E₂ →ₛₗ[σ₂₃] E₃) (f : E →ₛₗ[σ₁₂] E₂) :
p.comp (g.comp f) = (p.comp g).comp f | ext $ λ _, rfl | lemma | seminorm.comp_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"ring_hom_comp_triple",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comp (p q : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : (p + q).comp f = p.comp f + q.comp f | ext $ λ _, rfl | lemma | seminorm.add_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_add_le (p : seminorm 𝕜₂ E₂) (f g : E →ₛₗ[σ₁₂] E₂) :
p.comp (f + g) ≤ p.comp f + p.comp g | λ _, map_add_le_add p _ _ | lemma | seminorm.comp_add_le | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comp (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : R) :
(c • p).comp f = c • (p.comp f) | ext $ λ _, rfl | lemma | seminorm.smul_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mono {p q : seminorm 𝕜₂ E₂} (f : E →ₛₗ[σ₁₂] E₂) (hp : p ≤ q) :
p.comp f ≤ q.comp f | λ _, hp _ | lemma | seminorm.comp_mono | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback (f : E →ₛₗ[σ₁₂] E₂) : seminorm 𝕜₂ E₂ →+ seminorm 𝕜 E | ⟨λ p, p.comp f, zero_comp f, λ p q, add_comp p q f⟩ | def | seminorm.pullback | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | The composition as an `add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_bot : ⇑(⊥ : seminorm 𝕜 E) = 0 | rfl | lemma | seminorm.coe_bot | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq_zero : (⊥ : seminorm 𝕜 E) = 0 | rfl | lemma | seminorm.bot_eq_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul {p q : seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q | begin
simp_rw [le_def, coe_smul],
intros x,
simp_rw [pi.smul_apply, nnreal.smul_def, smul_eq_mul],
exact mul_le_mul hab (hpq x) (map_nonneg p x) (nnreal.coe_nonneg b),
end | lemma | seminorm.smul_le_smul | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"mul_le_mul",
"nnreal.coe_nonneg",
"nnreal.smul_def",
"pi.smul_apply",
"seminorm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sup_apply (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) :
s.sup p x = ↑(s.sup (λ i, ⟨p i x, map_nonneg (p i) x⟩) : ℝ≥0) | begin
induction s using finset.cons_induction_on with a s ha ih,
{ rw [finset.sup_empty, finset.sup_empty, coe_bot, _root_.bot_eq_zero, pi.zero_apply,
nonneg.coe_zero] },
{ rw [finset.sup_cons, finset.sup_cons, coe_sup, sup_eq_max, pi.sup_apply, sup_eq_max,
nnreal.coe_max, subtype.coe_mk, ih] }
en... | lemma | seminorm.finset_sup_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.cons_induction_on",
"finset.sup_cons",
"finset.sup_empty",
"ih",
"map_nonneg",
"nnreal.coe_max",
"nonneg.coe_zero",
"pi.sup_apply",
"seminorm",
"subtype.coe_mk",
"sup_eq_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sup_le_sum (p : ι → seminorm 𝕜 E) (s : finset ι) : s.sup p ≤ ∑ i in s, p i | begin
classical,
refine finset.sup_le_iff.mpr _,
intros i hi,
rw [finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left],
exact bot_le,
end | lemma | seminorm.finset_sup_le_sum | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"bot_le",
"finset",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sup_apply_le {p : ι → seminorm 𝕜 E} {s : finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a | begin
lift a to ℝ≥0 using ha,
rw [finset_sup_apply, nnreal.coe_le_coe],
exact finset.sup_le h,
end | lemma | seminorm.finset_sup_apply_le | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"lift",
"nnreal.coe_le_coe",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sup_apply_lt {p : ι → seminorm 𝕜 E} {s : finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a | begin
lift a to ℝ≥0 using ha.le,
rw [finset_sup_apply, nnreal.coe_lt_coe, finset.sup_lt_iff],
{ exact h },
{ exact nnreal.coe_pos.mpr ha },
end | lemma | seminorm.finset_sup_apply_lt | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.sup_lt_iff",
"lift",
"nnreal.coe_lt_coe",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sub_map_le_sub (p : seminorm 𝕜 E) (x y : E) : ‖p x - p y‖ ≤ p (x - y) | abs_sub_map_le_sub p x y | lemma | seminorm.norm_sub_map_le_sub | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_smul (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : 𝕜₂) :
p.comp (c • f) = ‖c‖₊ • p.comp f | ext $ λ _, by rw [comp_apply, smul_apply, linear_map.smul_apply, map_smul_eq_mul, nnreal.smul_def,
coe_nnnorm, smul_eq_mul, comp_apply] | lemma | seminorm.comp_smul | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"linear_map.smul_apply",
"nnreal.smul_def",
"seminorm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_smul_apply (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : 𝕜₂) (x : E) :
p.comp (c • f) x = ‖c‖ * p (f x) | map_smul_eq_mul p _ _ | lemma | seminorm.comp_smul_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_range_add : bdd_below (range $ λ u, p u + q (x - u)) | ⟨0, by { rintro _ ⟨x, rfl⟩, dsimp, positivity }⟩ | lemma | seminorm.bdd_below_range_add | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"bdd_below"
] | Auxiliary lemma to show that the infimum of seminorms is well-defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_apply (p q : seminorm 𝕜 E) (x : E) : (p ⊓ q) x = ⨅ u : E, p u + q (x-u) | rfl | lemma | seminorm.inf_apply | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inf [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p q : seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q | begin
ext,
simp_rw [smul_apply, inf_apply, smul_apply, ←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def,
smul_eq_mul, real.mul_infi_of_nonneg (subtype.prop _), mul_add],
end | lemma | seminorm.smul_inf | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"has_smul",
"is_scalar_tower",
"nnreal.smul_def",
"real.mul_infi_of_nonneg",
"seminorm",
"smul_eq_mul",
"subtype.prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup_eq' {s : set $ seminorm 𝕜 E} (hs : bdd_above (coe_fn '' s : set (E → ℝ))) :
coe_fn (Sup s) = ⨆ p : s, p | congr_arg _ (dif_pos hs) | lemma | seminorm.coe_Sup_eq' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"bdd_above",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_iff {s : set $ seminorm 𝕜 E} :
bdd_above s ↔ bdd_above (coe_fn '' s : set (E → ℝ)) | ⟨λ ⟨q, hq⟩, ⟨q, ball_image_of_ball $ λ p hp, hq hp⟩,
λ H, ⟨Sup s, λ p hp x,
begin
rw [seminorm.coe_Sup_eq' H, supr_apply],
rcases H with ⟨q, hq⟩,
exact le_csupr ⟨q x, forall_range_iff.mpr $ λ i : s, hq (mem_image_of_mem _ i.2) x⟩ ⟨p, hp⟩
end ⟩⟩ | lemma | seminorm.bdd_above_iff | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"bdd_above",
"le_csupr",
"seminorm",
"seminorm.coe_Sup_eq'",
"supr_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup_eq {s : set $ seminorm 𝕜 E} (hs : bdd_above s) :
coe_fn (Sup s) = ⨆ p : s, p | seminorm.coe_Sup_eq' (seminorm.bdd_above_iff.mp hs) | lemma | seminorm.coe_Sup_eq | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"bdd_above",
"seminorm",
"seminorm.coe_Sup_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr_eq {ι : Type*} {p : ι → seminorm 𝕜 E} (hp : bdd_above (range p)) :
coe_fn (⨆ i, p i) = ⨆ i, p i | by rw [← Sup_range, seminorm.coe_Sup_eq hp]; exact supr_range' (coe_fn : seminorm 𝕜 E → E → ℝ) p | lemma | seminorm.coe_supr_eq | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"Sup_range",
"bdd_above",
"seminorm",
"seminorm.coe_Sup_eq",
"supr_range'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm.is_lub_Sup (s : set (seminorm 𝕜 E)) (hs₁ : bdd_above s) (hs₂ : s.nonempty) :
is_lub s (Sup s) | begin
refine ⟨λ p hp x, _, λ p hp x, _⟩;
haveI : nonempty ↥s := hs₂.coe_sort;
rw [seminorm.coe_Sup_eq hs₁, supr_apply],
{ rcases hs₁ with ⟨q, hq⟩,
exact le_csupr ⟨q x, forall_range_iff.mpr $ λ i : s, hq i.2 x⟩ ⟨p, hp⟩ },
{ exact csupr_le (λ q, hp q.2 x) }
end | lemma | seminorm.seminorm.is_lub_Sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"bdd_above",
"csupr_le",
"is_lub",
"le_csupr",
"seminorm",
"seminorm.coe_Sup_eq",
"supr_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball (x : E) (r : ℝ) | { y : E | p (y - x) < r } | def | seminorm.ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_ball (x : E) (r : ℝ) | { y : E | p (y - x) ≤ r } | def | seminorm.closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) ≤ r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ball : y ∈ ball p x r ↔ p (y - x) < r | iff.rfl | lemma | seminorm.mem_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball : y ∈ closed_ball p x r ↔ p (y - x) ≤ r | iff.rfl | lemma | seminorm.mem_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_self (hr : 0 < r) : x ∈ ball p x r | by simp [hr] | lemma | seminorm.mem_ball_self | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_self (hr : 0 ≤ r) : x ∈ closed_ball p x r | by simp [hr] | lemma | seminorm.mem_closed_ball_self | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_zero : y ∈ ball p 0 r ↔ p y < r | by rw [mem_ball, sub_zero] | lemma | seminorm.mem_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_zero : y ∈ closed_ball p 0 r ↔ p y ≤ r | by rw [mem_closed_ball, sub_zero] | lemma | seminorm.mem_closed_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_zero_eq : ball p 0 r = { y : E | p y < r } | set.ext $ λ x, p.mem_ball_zero | lemma | seminorm.ball_zero_eq | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_zero_eq : closed_ball p 0 r = { y : E | p y ≤ r } | set.ext $ λ x, p.mem_closed_ball_zero | lemma | seminorm.closed_ball_zero_eq | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_subset_closed_ball (x r) : ball p x r ⊆ closed_ball p x r | λ y (hy : _ < _), hy.le | lemma | seminorm.ball_subset_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_bInter_ball (x r) : closed_ball p x r = ⋂ ρ > r, ball p x ρ | by ext y; simp_rw [mem_closed_ball, mem_Inter₂, mem_ball, ← forall_lt_iff_le'] | lemma | seminorm.closed_ball_eq_bInter_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"forall_lt_iff_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_zero' (x : E) (hr : 0 < r) : ball (0 : seminorm 𝕜 E) x r = set.univ | begin
rw [set.eq_univ_iff_forall, ball],
simp [hr],
end | lemma | seminorm.ball_zero' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm",
"set.eq_univ_iff_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_zero' (x : E) (hr : 0 < r) :
closed_ball (0 : seminorm 𝕜 E) x r = set.univ | eq_univ_of_subset (ball_subset_closed_ball _ _ _) (ball_zero' x hr) | lemma | seminorm.closed_ball_zero' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_smul (p : seminorm 𝕜 E) {c : nnreal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) | by { ext, rw [mem_ball, mem_ball, smul_apply, nnreal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (nnreal.coe_pos.mpr hc)] } | lemma | seminorm.ball_smul | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"lt_div_iff",
"mul_comm",
"nnreal",
"nnreal.smul_def",
"seminorm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_smul (p : seminorm 𝕜 E) {c : nnreal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).closed_ball x r = p.closed_ball x (r / c) | by { ext, rw [mem_closed_ball, mem_closed_ball, smul_apply, nnreal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (nnreal.coe_pos.mpr hc)] } | lemma | seminorm.closed_ball_smul | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"le_div_iff",
"mul_comm",
"nnreal",
"nnreal.smul_def",
"seminorm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_sup (p : seminorm 𝕜 E) (q : seminorm 𝕜 E) (e : E) (r : ℝ) :
ball (p ⊔ q) e r = ball p e r ∩ ball q e r | by simp_rw [ball, ←set.set_of_and, coe_sup, pi.sup_apply, sup_lt_iff] | lemma | seminorm.ball_sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"pi.sup_apply",
"seminorm",
"sup_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_sup (p : seminorm 𝕜 E) (q : seminorm 𝕜 E) (e : E) (r : ℝ) :
closed_ball (p ⊔ q) e r = closed_ball p e r ∩ closed_ball q e r | by simp_rw [closed_ball, ←set.set_of_and, coe_sup, pi.sup_apply, sup_le_iff] | lemma | seminorm.closed_ball_sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"pi.sup_apply",
"seminorm",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_finset_sup' (p : ι → seminorm 𝕜 E) (s : finset ι) (H : s.nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H (λ i, ball (p i) e r) | begin
induction H using finset.nonempty.cons_induction with a a s ha hs ih,
{ classical, simp },
{ rw [finset.sup'_cons hs, finset.inf'_cons hs, ball_sup, inf_eq_inter, ih] },
end | lemma | seminorm.ball_finset_sup' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.inf'_cons",
"finset.nonempty.cons_induction",
"finset.sup'_cons",
"ih",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_finset_sup' (p : ι → seminorm 𝕜 E) (s : finset ι) (H : s.nonempty) (e : E)
(r : ℝ) : closed_ball (s.sup' H p) e r = s.inf' H (λ i, closed_ball (p i) e r) | begin
induction H using finset.nonempty.cons_induction with a a s ha hs ih,
{ classical, simp },
{ rw [finset.sup'_cons hs, finset.inf'_cons hs, closed_ball_sup, inf_eq_inter, ih] },
end | lemma | seminorm.closed_ball_finset_sup' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.inf'_cons",
"finset.nonempty.cons_induction",
"finset.sup'_cons",
"ih",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mono {p : seminorm 𝕜 E} {r₁ r₂ : ℝ} (h : r₁ ≤ r₂) : p.ball x r₁ ⊆ p.ball x r₂ | λ _ (hx : _ < _), hx.trans_le h | lemma | seminorm.ball_mono | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"ball_mono",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_mono {p : seminorm 𝕜 E} {r₁ r₂ : ℝ} (h : r₁ ≤ r₂) :
p.closed_ball x r₁ ⊆ p.closed_ball x r₂ | λ _ (hx : _ ≤ _), hx.trans h | lemma | seminorm.closed_ball_mono | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_antitone {p q : seminorm 𝕜 E} (h : q ≤ p) : p.ball x r ⊆ q.ball x r | λ _, (h _).trans_lt | lemma | seminorm.ball_antitone | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_antitone {p q : seminorm 𝕜 E} (h : q ≤ p) :
p.closed_ball x r ⊆ q.closed_ball x r | λ _, (h _).trans | lemma | seminorm.closed_ball_antitone | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_add_ball_subset (p : seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E):
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) | begin
rintros x ⟨y₁, y₂, hy₁, hy₂, rfl⟩,
rw [mem_ball, add_sub_add_comm],
exact (map_add_le_add p _ _).trans_lt (add_lt_add hy₁ hy₂),
end | lemma | seminorm.ball_add_ball_subset | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_add_closed_ball_subset (p : seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closed_ball (x₁ : E) r₁ + p.closed_ball (x₂ : E) r₂ ⊆ p.closed_ball (x₁ + x₂) (r₁ + r₂) | begin
rintros x ⟨y₁, y₂, hy₁, hy₂, rfl⟩,
rw [mem_closed_ball, add_sub_add_comm],
exact (map_add_le_add p _ _).trans (add_le_add hy₁ hy₂)
end | lemma | seminorm.closed_ball_add_closed_ball_subset | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem_ball (p : seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.ball y r ↔ x₁ ∈ p.ball (x₂ + y) r | by simp_rw [mem_ball, sub_sub] | lemma | seminorm.sub_mem_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_ball (p : seminorm 𝕜 E) :
x +ᵥ p.ball y r = p.ball (x +ᵥ y) r | begin
letI := add_group_seminorm.to_seminormed_add_comm_group p.to_add_group_seminorm,
exact metric.vadd_ball x y r,
end | lemma | seminorm.vadd_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | The image of a ball under addition with a singleton is another ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_closed_ball (p : seminorm 𝕜 E) :
x +ᵥ p.closed_ball y r = p.closed_ball (x +ᵥ y) r | begin
letI := add_group_seminorm.to_seminormed_add_comm_group p.to_add_group_seminorm,
exact metric.vadd_closed_ball x y r,
end | lemma | seminorm.vadd_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | The image of a closed ball under addition with a singleton is another closed ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_comp (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' (p.ball (f x) r) | begin
ext,
simp_rw [ball, mem_preimage, comp_apply, set.mem_set_of_eq, map_sub],
end | lemma | seminorm.ball_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_comp (p : seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closed_ball x r = f ⁻¹' (p.closed_ball (f x) r) | begin
ext,
simp_rw [closed_ball, mem_preimage, comp_apply, set.mem_set_of_eq, map_sub],
end | lemma | seminorm.closed_ball_comp | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_metric_ball {r : ℝ} :
p ⁻¹' (metric.ball 0 r) = {x | p x < r} | begin
ext x,
simp only [mem_set_of, mem_preimage, mem_ball_zero_iff, real.norm_of_nonneg (map_nonneg p _)]
end | lemma | seminorm.preimage_metric_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"metric.ball",
"real.norm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_metric_closed_ball {r : ℝ} :
p ⁻¹' (metric.closed_ball 0 r) = {x | p x ≤ r} | begin
ext x,
simp only [mem_set_of, mem_preimage, mem_closed_ball_zero_iff,
real.norm_of_nonneg (map_nonneg p _)]
end | lemma | seminorm.preimage_metric_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"metric.closed_ball",
"real.norm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_zero_eq_preimage_ball {r : ℝ} :
p.ball 0 r = p ⁻¹' (metric.ball 0 r) | by rw [ball_zero_eq, preimage_metric_ball] | lemma | seminorm.ball_zero_eq_preimage_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_zero_eq_preimage_closed_ball {r : ℝ} :
p.closed_ball 0 r = p ⁻¹' (metric.closed_ball 0 r) | by rw [closed_ball_zero_eq, preimage_metric_closed_ball] | lemma | seminorm.closed_ball_zero_eq_preimage_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_bot {r : ℝ} (x : E) (hr : 0 < r) :
ball (⊥ : seminorm 𝕜 E) x r = set.univ | ball_zero' x hr | lemma | seminorm.ball_bot | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_bot {r : ℝ} (x : E) (hr : 0 < r) :
closed_ball (⊥ : seminorm 𝕜 E) x r = set.univ | closed_ball_zero' x hr | lemma | seminorm.closed_ball_bot | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_ball_zero (r : ℝ) : balanced 𝕜 (ball p 0 r) | begin
rintro a ha x ⟨y, hy, hx⟩,
rw [mem_ball_zero, ←hx, map_smul_eq_mul],
calc _ ≤ p y : mul_le_of_le_one_left (map_nonneg p _) ha
... < r : by rwa mem_ball_zero at hy,
end | lemma | seminorm.balanced_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"balanced",
"balanced_ball_zero",
"map_nonneg",
"mul_le_of_le_one_left"
] | Seminorm-balls at the origin are balanced. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_closed_ball_zero (r : ℝ) : balanced 𝕜 (closed_ball p 0 r) | begin
rintro a ha x ⟨y, hy, hx⟩,
rw [mem_closed_ball_zero, ←hx, map_smul_eq_mul],
calc _ ≤ p y : mul_le_of_le_one_left (map_nonneg p _) ha
... ≤ r : by rwa mem_closed_ball_zero at hy
end | lemma | seminorm.balanced_closed_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"balanced",
"map_nonneg",
"mul_le_of_le_one_left"
] | Closed seminorm-balls at the origin are balanced. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_finset_sup_eq_Inter (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = ⋂ (i ∈ s), ball (p i) x r | begin
lift r to nnreal using hr.le,
simp_rw [ball, Inter_set_of, finset_sup_apply, nnreal.coe_lt_coe,
finset.sup_lt_iff (show ⊥ < r, from hr), ←nnreal.coe_lt_coe, subtype.coe_mk],
end | lemma | seminorm.ball_finset_sup_eq_Inter | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.sup_lt_iff",
"lift",
"nnreal",
"nnreal.coe_lt_coe",
"seminorm",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_finset_sup_eq_Inter (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closed_ball (s.sup p) x r = ⋂ (i ∈ s), closed_ball (p i) x r | begin
lift r to nnreal using hr,
simp_rw [closed_ball, Inter_set_of, finset_sup_apply, nnreal.coe_le_coe,
finset.sup_le_iff, ←nnreal.coe_le_coe, subtype.coe_mk]
end | lemma | seminorm.closed_ball_finset_sup_eq_Inter | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.sup_le_iff",
"lift",
"nnreal",
"nnreal.coe_le_coe",
"seminorm",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_finset_sup (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf (λ i, ball (p i) x r) | begin
rw finset.inf_eq_infi,
exact ball_finset_sup_eq_Inter _ _ _ hr,
end | lemma | seminorm.ball_finset_sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.inf_eq_infi",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_finset_sup (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
closed_ball (s.sup p) x r = s.inf (λ i, closed_ball (p i) x r) | begin
rw finset.inf_eq_infi,
exact closed_ball_finset_sup_eq_Inter _ _ _ hr,
end | lemma | seminorm.closed_ball_finset_sup | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"finset",
"finset.inf_eq_infi",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_smul_ball (p : seminorm 𝕜 E) (r₁ r₂ : ℝ) :
metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | begin
rw set.subset_def,
intros x hx,
rw set.mem_smul at hx,
rcases hx with ⟨a, y, ha, hy, hx⟩,
rw [←hx, mem_ball_zero, map_smul_eq_mul],
exact mul_lt_mul'' (mem_ball_zero_iff.mp ha) (p.mem_ball_zero.mp hy) (norm_nonneg a)
(map_nonneg p y),
end | lemma | seminorm.ball_smul_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"metric.ball",
"mul_lt_mul''",
"seminorm",
"set.mem_smul",
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_smul_closed_ball (p : seminorm 𝕜 E) (r₁ r₂ : ℝ) :
metric.closed_ball (0 : 𝕜) r₁ • p.closed_ball 0 r₂ ⊆ p.closed_ball 0 (r₁ * r₂) | begin
rw set.subset_def,
intros x hx,
rw set.mem_smul at hx,
rcases hx with ⟨a, y, ha, hy, hx⟩,
rw [←hx, mem_closed_ball_zero, map_smul_eq_mul],
rw mem_closed_ball_zero_iff at ha,
exact mul_le_mul ha (p.mem_closed_ball_zero.mp hy) (map_nonneg _ y) ((norm_nonneg a).trans ha)
end | lemma | seminorm.closed_ball_smul_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"metric.closed_ball",
"mul_le_mul",
"seminorm",
"set.mem_smul",
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_eq_emptyset (p : seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ | begin
ext,
rw [seminorm.mem_ball, set.mem_empty_iff_false, iff_false, not_lt],
exact hr.trans (map_nonneg p _),
end | lemma | seminorm.ball_eq_emptyset | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"seminorm",
"seminorm.mem_ball",
"set.mem_empty_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_emptyset (p : seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closed_ball x r = ∅ | begin
ext,
rw [seminorm.mem_closed_ball, set.mem_empty_iff_false, iff_false, not_le],
exact hr.trans_le (map_nonneg _ _),
end | lemma | seminorm.closed_ball_eq_emptyset | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"map_nonneg",
"seminorm",
"seminorm.mem_closed_ball",
"set.mem_empty_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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