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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|---|---|---|---|---|---|---|---|---|---|---|
summable.mul_of_nonneg {f : ι → ℝ} {g : ι' → ℝ}
(hf : summable f) (hg : summable g) (hf' : 0 ≤ f) (hg' : 0 ≤ g) :
summable (λ (x : ι × ι'), f x.1 * g x.2) | let ⟨s, hf⟩ := hf in
let ⟨t, hg⟩ := hg in
suffices this : ∀ u : finset (ι × ι'), ∑ x in u, f x.1 * g x.2 ≤ s*t,
from summable_of_sum_le (λ x, mul_nonneg (hf' _) (hg' _)) this,
assume u,
calc ∑ x in u, f x.1 * g x.2
≤ ∑ x in u.image prod.fst ×ˢ u.image prod.snd, f x.1 * g x.2 :
sum_mono_set_of_nonneg (λ x, ... | lemma | summable.mul_of_nonneg | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"finset",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"sum_le_has_sum",
"summable",
"summable_of_sum_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.mul_norm {f : ι → α} {g : ι' → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
summable (λ (x : ι × ι'), ‖f x.1 * g x.2‖) | summable_of_nonneg_of_le (λ x, norm_nonneg (f x.1 * g x.2)) (λ x, norm_mul_le (f x.1) (g x.2))
(hf.mul_of_nonneg hg (λ x, norm_nonneg $ f x) (λ x, norm_nonneg $ g x) : _) | lemma | summable.mul_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"norm_mul_le",
"summable",
"summable_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_mul_of_summable_norm [complete_space α] {f : ι → α} {g : ι' → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
summable (λ (x : ι × ι'), f x.1 * g x.2) | summable_of_summable_norm (hf.mul_norm hg) | lemma | summable_mul_of_summable_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"complete_space",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_tsum_of_summable_norm [complete_space α] {f : ι → α} {g : ι' → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
(∑' x, f x) * (∑' y, g y) = (∑' z : ι × ι', f z.1 * g z.2) | tsum_mul_tsum (summable_of_summable_norm hf) (summable_of_summable_norm hg)
(summable_mul_of_summable_norm hf hg) | lemma | tsum_mul_tsum_of_summable_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"complete_space",
"summable",
"summable_mul_of_summable_norm",
"summable_of_summable_norm",
"tsum_mul_tsum"
] | Product of two infinites sums indexed by arbitrary types.
See also `tsum_mul_tsum` if `f` and `g` are *not* absolutely summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_norm_sum_mul_antidiagonal_of_summable_norm {f g : ℕ → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
summable (λ n, ‖∑ kl in antidiagonal n, f kl.1 * g kl.2‖) | begin
have := summable_sum_mul_antidiagonal_of_summable_mul
(summable.mul_of_nonneg hf hg (λ _, norm_nonneg _) (λ _, norm_nonneg _)),
refine summable_of_nonneg_of_le (λ _, norm_nonneg _) _ this,
intros n,
calc ‖∑ kl in antidiagonal n, f kl.1 * g kl.2‖
≤ ∑ kl in antidiagonal n, ‖f kl.1 * g kl.2‖ : nor... | lemma | summable_norm_sum_mul_antidiagonal_of_summable_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"norm_mul_le",
"norm_sum_le",
"summable",
"summable.mul_of_nonneg",
"summable_of_nonneg_of_le",
"summable_sum_mul_antidiagonal_of_summable_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm [complete_space α] {f g : ℕ → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
(∑' n, f n) * (∑' n, g n) = ∑' n, ∑ kl in antidiagonal n, f kl.1 * g kl.2 | tsum_mul_tsum_eq_tsum_sum_antidiagonal (summable_of_summable_norm hf) (summable_of_summable_norm hg)
(summable_mul_of_summable_norm hf hg) | lemma | tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"complete_space",
"summable",
"summable_mul_of_summable_norm",
"summable_of_summable_norm",
"tsum_mul_tsum_eq_tsum_sum_antidiagonal"
] | The Cauchy product formula for the product of two infinite sums indexed by `ℕ`,
expressed by summing on `finset.nat.antidiagonal`.
See also `tsum_mul_tsum_eq_tsum_sum_antidiagonal` if `f` and `g` are
*not* absolutely summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_norm_sum_mul_range_of_summable_norm {f g : ℕ → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
summable (λ n, ‖∑ k in range (n+1), f k * g (n - k)‖) | begin
simp_rw ← sum_antidiagonal_eq_sum_range_succ (λ k l, f k * g l),
exact summable_norm_sum_mul_antidiagonal_of_summable_norm hf hg
end | lemma | summable_norm_sum_mul_range_of_summable_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"summable",
"summable_norm_sum_mul_antidiagonal_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm [complete_space α] {f g : ℕ → α}
(hf : summable (λ x, ‖f x‖)) (hg : summable (λ x, ‖g x‖)) :
(∑' n, f n) * (∑' n, g n) = ∑' n, ∑ k in range (n+1), f k * g (n - k) | begin
simp_rw ← sum_antidiagonal_eq_sum_range_succ (λ k l, f k * g l),
exact tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm hf hg
end | lemma | tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm | analysis.normed.field | src/analysis/normed/field/infinite_sum.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.infinite_sum"
] | [
"complete_space",
"summable",
"tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm"
] | The Cauchy product formula for the product of two infinite sums indexed by `ℕ`,
expressed by summing on `finset.range`.
See also `tsum_mul_tsum_eq_tsum_sum_range` if `f` and `g` are
*not* absolutely summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsemigroup.unit_ball (𝕜 : Type*) [non_unital_semi_normed_ring 𝕜] :
subsemigroup 𝕜 | { carrier := ball (0 : 𝕜) 1,
mul_mem' := λ x y hx hy,
begin
rw [mem_ball_zero_iff] at *,
exact (norm_mul_le x y).trans_lt (mul_lt_one_of_nonneg_of_lt_one_left (norm_nonneg _)
hx hy.le)
end } | def | subsemigroup.unit_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"mul_lt_one_of_nonneg_of_lt_one_left",
"non_unital_semi_normed_ring",
"norm_mul_le",
"subsemigroup"
] | Unit ball in a non unital semi normed ring as a bundled `subsemigroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_unit_ball [non_unital_semi_normed_ring 𝕜] (x y : ball (0 : 𝕜) 1) :
↑(x * y) = (x * y : 𝕜) | rfl | lemma | coe_mul_unit_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"non_unital_semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemigroup.unit_closed_ball (𝕜 : Type*) [non_unital_semi_normed_ring 𝕜] :
subsemigroup 𝕜 | { carrier := closed_ball 0 1,
mul_mem' := λ x y hx hy,
begin
rw [mem_closed_ball_zero_iff] at *,
exact (norm_mul_le x y).trans (mul_le_one hx (norm_nonneg _) hy)
end } | def | subsemigroup.unit_closed_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"mul_le_one",
"non_unital_semi_normed_ring",
"norm_mul_le",
"subsemigroup"
] | Closed unit ball in a non unital semi normed ring as a bundled `subsemigroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_unit_closed_ball [non_unital_semi_normed_ring 𝕜] (x y : closed_ball (0 : 𝕜) 1) :
↑(x * y) = (x * y : 𝕜) | rfl | lemma | coe_mul_unit_closed_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"non_unital_semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.unit_closed_ball (𝕜 : Type*) [semi_normed_ring 𝕜] [norm_one_class 𝕜] :
submonoid 𝕜 | { carrier := closed_ball 0 1,
one_mem' := mem_closed_ball_zero_iff.2 norm_one.le,
.. subsemigroup.unit_closed_ball 𝕜 } | def | submonoid.unit_closed_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"norm_one_class",
"semi_normed_ring",
"submonoid",
"subsemigroup.unit_closed_ball"
] | Closed unit ball in a semi normed ring as a bundled `submonoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one_unit_closed_ball [semi_normed_ring 𝕜] [norm_one_class 𝕜] :
((1 : closed_ball (0 : 𝕜) 1) : 𝕜) = 1 | rfl | lemma | coe_one_unit_closed_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"norm_one_class",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow_unit_closed_ball [semi_normed_ring 𝕜] [norm_one_class 𝕜]
(x : closed_ball (0 : 𝕜) 1) (n : ℕ) :
↑(x ^ n) = (x ^ n : 𝕜) | rfl | lemma | coe_pow_unit_closed_ball | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"norm_one_class",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.unit_sphere (𝕜 : Type*) [normed_division_ring 𝕜] : submonoid 𝕜 | { carrier := sphere (0 : 𝕜) 1,
mul_mem' := λ x y hx hy, by { rw [mem_sphere_zero_iff_norm] at *, simp * },
one_mem' := mem_sphere_zero_iff_norm.2 norm_one } | def | submonoid.unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring",
"submonoid"
] | Unit sphere in a normed division ring as a bundled `submonoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inv_unit_sphere [normed_division_ring 𝕜] (x : sphere (0 : 𝕜) 1) : ↑x⁻¹ = (x⁻¹ : 𝕜) | rfl | lemma | coe_inv_unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div_unit_sphere [normed_division_ring 𝕜] (x y : sphere (0 : 𝕜) 1) :
↑(x / y) = (x / y : 𝕜) | rfl | lemma | coe_div_unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow_unit_sphere [normed_division_ring 𝕜] (x : sphere (0 : 𝕜) 1) (n : ℤ) :
↑(x ^ n) = (x ^ n : 𝕜) | rfl | lemma | coe_zpow_unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one_unit_sphere [normed_division_ring 𝕜] : ((1 : sphere (0 : 𝕜) 1) : 𝕜) = 1 | rfl | lemma | coe_one_unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_unit_sphere [normed_division_ring 𝕜] (x y : sphere (0 : 𝕜) 1) :
↑(x * y) = (x * y : 𝕜) | rfl | lemma | coe_mul_unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow_unit_sphere [normed_division_ring 𝕜] (x : sphere (0 : 𝕜) 1) (n : ℕ) :
↑(x ^ n) = (x ^ n : 𝕜) | rfl | lemma | coe_pow_unit_sphere | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_sphere_to_units (𝕜 : Type*) [normed_division_ring 𝕜] : sphere (0 : 𝕜) 1 →* units 𝕜 | units.lift_right (submonoid.unit_sphere 𝕜).subtype (λ x, units.mk0 x $ ne_zero_of_mem_unit_sphere _)
(λ x, rfl) | def | unit_sphere_to_units | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring",
"submonoid.unit_sphere",
"units",
"units.lift_right",
"units.mk0"
] | Monoid homomorphism from the unit sphere to the group of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_sphere_to_units_apply_coe [normed_division_ring 𝕜] (x : sphere (0 : 𝕜) 1) :
(unit_sphere_to_units 𝕜 x : 𝕜) = x | rfl | lemma | unit_sphere_to_units_apply_coe | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring",
"unit_sphere_to_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_sphere_to_units_injective [normed_division_ring 𝕜] :
function.injective (unit_sphere_to_units 𝕜) | λ x y h, subtype.eq $ by convert congr_arg units.val h | lemma | unit_sphere_to_units_injective | analysis.normed.field | src/analysis/normed/field/unit_ball.lean | [
"analysis.normed.field.basic",
"analysis.normed.group.ball_sphere"
] | [
"normed_division_ring",
"unit_sphere_to_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_mul (x : ℝ) (t : ℝ) :
‖(↑(t * x) : add_circle (t * p))‖ = |t| * ‖(x : add_circle p)‖ | begin
have aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b) := λ a b c h, by
{ simp only [mem_zmultiples_iff] at ⊢ h,
obtain ⟨n, rfl⟩ := h,
exact ⟨n, (mul_smul_comm n c b).symm⟩, },
rcases eq_or_ne t 0 with rfl | ht, { simp, },
have ht' : |t| ≠ 0 := (not_congr abs_eq_zero).mpr ht,
si... | lemma | add_circle.norm_coe_mul | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_eq_zero",
"abs_inv",
"abs_mul",
"abs_nonneg",
"add_circle",
"aux",
"eq_inv_mul_iff_mul_eq₀",
"eq_or_ne",
"inv_mul_cancel_left₀",
"mul_smul_comm",
"real.Inf_smul_of_nonneg",
"real.norm_eq_abs",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_neg_period (x : ℝ) :
‖(x : add_circle (-p))‖ = ‖(x : add_circle p)‖ | begin
suffices : ‖(↑(-1 * x) : add_circle (-1 * p))‖ = ‖(x : add_circle p)‖,
{ rw [← this, neg_one_mul], simp, },
simp only [norm_coe_mul, abs_neg, abs_one, one_mul],
end | lemma | add_circle.norm_neg_period | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_neg",
"abs_one",
"add_circle",
"neg_one_mul",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_of_zero {x : ℝ} : ‖(x : add_circle (0 : ℝ))‖ = |x| | begin
suffices : {y : ℝ | (y : add_circle (0 : ℝ)) = (x : add_circle (0 : ℝ)) } = { x },
{ rw [quotient_norm_eq, this, image_singleton, real.norm_eq_abs, cInf_singleton], },
ext y,
simp [quotient_add_group.eq_iff_sub_mem, mem_zmultiples_iff, sub_eq_zero],
end | lemma | add_circle.norm_eq_of_zero | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"add_circle",
"cInf_singleton",
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq {x : ℝ} : ‖(x : add_circle p)‖ = |x - round (p⁻¹ * x) * p| | begin
suffices : ∀ (x : ℝ), ‖(x : add_circle (1 : ℝ))‖ = |x - round x|,
{ rcases eq_or_ne p 0 with rfl | hp, { simp, },
intros,
have hx := norm_coe_mul p x p⁻¹,
rw [abs_inv, eq_inv_mul_iff_mul_eq₀ ((not_congr abs_eq_zero).mpr hp)] at hx,
rw [← hx, inv_mul_cancel hp, this, ← abs_mul, mul_sub, mul_inv... | lemma | add_circle.norm_eq | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_eq_zero",
"abs_inv",
"abs_mul",
"abs_sub_comm",
"abs_sub_round_eq_min",
"add_circle",
"bdd_below",
"cInf_le_iff",
"eq_inv_mul_iff_mul_eq₀",
"eq_or_ne",
"inv_mul_cancel",
"le_cInf_iff",
"le_min_iff",
"mem_lower_bounds",
"mul_comm",
"mul_inv_cancel_left₀",
"real.norm_eq_abs",
"r... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq' (hp : 0 < p) {x : ℝ} :
‖(x : add_circle p)‖ = p * |(p⁻¹ * x) - round (p⁻¹ * x)| | begin
conv_rhs { congr, rw ← abs_eq_self.mpr hp.le, },
rw [← abs_mul, mul_sub, mul_inv_cancel_left₀ hp.ne.symm, norm_eq, mul_comm p],
end | lemma | add_circle.norm_eq' | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_mul",
"add_circle",
"mul_comm",
"mul_inv_cancel_left₀",
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_half_period {x : add_circle p} (hp : p ≠ 0) : ‖x‖ ≤ |p|/2 | begin
obtain ⟨x⟩ := x,
change ‖(x : add_circle p)‖ ≤ |p|/2,
rw [norm_eq, ← mul_le_mul_left (abs_pos.mpr (inv_ne_zero hp)), ← abs_mul, mul_sub, mul_left_comm,
← mul_div_assoc, ← abs_mul, inv_mul_cancel hp, mul_one, abs_one],
exact abs_sub_round (p⁻¹ * x),
end | lemma | add_circle.norm_le_half_period | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_mul",
"abs_one",
"abs_sub_round",
"add_circle",
"inv_mul_cancel",
"inv_ne_zero",
"mul_div_assoc",
"mul_le_mul_left",
"mul_left_comm",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_half_period_eq : ‖(↑(p/2) : add_circle p)‖ = |p|/2 | begin
rcases eq_or_ne p 0 with rfl | hp, { simp, },
rw [norm_eq, ← mul_div_assoc, inv_mul_cancel hp, one_div, round_two_inv, algebra_map.coe_one,
one_mul, (by linarith : p / 2 - p = -(p / 2)), abs_neg, abs_div, abs_two],
end | lemma | add_circle.norm_half_period_eq | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_div",
"abs_neg",
"abs_two",
"add_circle",
"algebra_map.coe_one",
"eq_or_ne",
"inv_mul_cancel",
"mul_div_assoc",
"one_div",
"one_mul",
"round_two_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_eq_abs_iff {x : ℝ} (hp : p ≠ 0) : ‖(x : add_circle p)‖ = |x| ↔ |x| ≤ |p|/2 | begin
refine ⟨λ hx, hx ▸ norm_le_half_period p hp, λ hx, _⟩,
suffices : ∀ (p : ℝ), 0 < p → |x| ≤ p/2 → ‖(x : add_circle p)‖ = |x|,
{ rcases lt_trichotomy 0 p with hp | rfl | hp,
{ rw abs_eq_self.mpr hp.le at hx,
exact this p hp hx, },
{ contradiction, },
{ rw ← norm_neg_period,
rw abs_eq_n... | lemma | add_circle.norm_coe_eq_abs_iff | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_div",
"abs_two",
"add_circle",
"eq_or_ne",
"mul_assoc",
"mul_div_assoc",
"mul_inv_cancel",
"mul_le_mul_left",
"mul_lt_mul_left",
"mul_neg",
"mul_one",
"ne.lt_of_le",
"one_mul",
"round",
"round_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_univ_of_half_period_le
(hp : p ≠ 0) (x : add_circle p) {ε : ℝ} (hε : |p|/2 ≤ ε) :
closed_ball x ε = univ | eq_univ_iff_forall.mpr $
λ x, by simpa only [mem_closed_ball, dist_eq_norm] using (norm_le_half_period p hp).trans hε | lemma | add_circle.closed_ball_eq_univ_of_half_period_le | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_real_preimage_closed_ball_period_zero (x ε : ℝ) :
coe⁻¹' closed_ball (x : add_circle (0 : ℝ)) ε = closed_ball x ε | by { ext y; simp [dist_eq_norm, ← quotient_add_group.coe_sub], } | lemma | add_circle.coe_real_preimage_closed_ball_period_zero | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_real_preimage_closed_ball_eq_Union (x ε : ℝ) :
coe⁻¹' closed_ball (x : add_circle p) ε = ⋃ (z : ℤ), closed_ball (x + z • p) ε | begin
rcases eq_or_ne p 0 with rfl | hp, { simp [Union_const], },
ext y,
simp only [dist_eq_norm, mem_preimage, mem_closed_ball, zsmul_eq_mul, mem_Union, real.norm_eq_abs,
← quotient_add_group.coe_sub, norm_eq, ← sub_sub],
refine ⟨λ h, ⟨round (p⁻¹ * (y - x)), h⟩, _⟩,
rintros ⟨n, hn⟩,
rw [← mul_le_mul_le... | lemma | add_circle.coe_real_preimage_closed_ball_eq_Union | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_mul",
"add_circle",
"eq_or_ne",
"inv_mul_cancel_left₀",
"inv_ne_zero",
"mul_comm",
"mul_le_mul_left",
"real.norm_eq_abs",
"round_le",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_real_preimage_closed_ball_inter_eq
{x ε : ℝ} (s : set ℝ) (hs : s ⊆ closed_ball x (|p|/2)) :
coe⁻¹' closed_ball (x : add_circle p) ε ∩ s = if ε < |p|/2 then (closed_ball x ε) ∩ s else s | begin
cases le_or_lt (|p|/2) ε with hε hε,
{ rcases eq_or_ne p 0 with rfl | hp,
{ simp only [abs_zero, zero_div] at hε,
simp only [not_lt.mpr hε, coe_real_preimage_closed_ball_period_zero, abs_zero, zero_div,
if_false, inter_eq_right_iff_subset],
exact hs.trans (closed_ball_subset_closed_bal... | lemma | add_circle.coe_real_preimage_closed_ball_inter_eq | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_zero",
"add_circle",
"eq_or_ne",
"int.cast_le_neg_one_or_one_le_cast_of_ne_zero",
"mul_zero",
"real.closed_ball_eq_Icc",
"zero_div",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_nat_cast {m n : ℕ} :
‖(↑((↑m / ↑n) * p) : add_circle p)‖ = p * (↑(min (m % n) (n - m % n)) / n) | begin
have : p⁻¹ * (↑m / ↑n * p) = ↑m / ↑n, { rw [mul_comm _ p, inv_mul_cancel_left₀ hp.out.ne.symm], },
rw [norm_eq' p hp.out, this, abs_sub_round_div_nat_cast_eq],
end | lemma | add_circle.norm_div_nat_cast | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"abs_sub_round_div_nat_cast_eq",
"add_circle",
"inv_mul_cancel_left₀",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_norm_eq_of_fin_add_order {u : add_circle p} (hu : is_of_fin_add_order u) :
∃ (k : ℕ), ‖u‖ = p * (k / add_order_of u) | begin
let n := add_order_of u,
change ∃ (k : ℕ), ‖u‖ = p * (k / n),
obtain ⟨m, -, -, hm⟩ := exists_gcd_eq_one_of_is_of_fin_add_order hu,
refine ⟨min (m % n) (n - m % n), _⟩,
rw [← hm, norm_div_nat_cast],
end | lemma | add_circle.exists_norm_eq_of_fin_add_order | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"add_circle",
"is_of_fin_add_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_order_smul_norm_of_is_of_fin_add_order
{u : add_circle p} (hu : is_of_fin_add_order u) (hu' : u ≠ 0) :
p ≤ add_order_of u • ‖u‖ | begin
obtain ⟨n, hn⟩ := exists_norm_eq_of_fin_add_order hu,
replace hu : (add_order_of u : ℝ) ≠ 0, { norm_cast, exact (add_order_of_pos_iff.mpr hu).ne.symm },
conv_lhs { rw ← mul_one p, },
rw [hn, nsmul_eq_mul, ← mul_assoc, mul_comm _ p, mul_assoc, mul_div_cancel' _ hu,
mul_le_mul_left hp.out, nat.one_le_ca... | lemma | add_circle.le_add_order_smul_norm_of_is_of_fin_add_order | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"add_circle",
"algebra_map.coe_zero",
"is_of_fin_add_order",
"mul_assoc",
"mul_comm",
"mul_div_cancel'",
"mul_le_mul_left",
"mul_one",
"mul_zero",
"nat.one_le_cast",
"nat.one_le_iff_ne_zero",
"norm_eq_zero",
"nsmul_eq_mul",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq {x : ℝ} : ‖(x : unit_add_circle)‖ = |x - round x| | by simp [add_circle.norm_eq] | lemma | unit_add_circle.norm_eq | analysis.normed.group | src/analysis/normed/group/add_circle.lean | [
"analysis.normed.group.quotient",
"topology.instances.add_circle"
] | [
"add_circle.norm_eq",
"round",
"unit_add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_torsor (V : out_param $ Type*) (P : Type*)
[out_param $ seminormed_add_comm_group V] [pseudo_metric_space P]
extends add_torsor V P | (dist_eq_norm' : ∀ (x y : P), dist x y = ‖(x -ᵥ y : V)‖) | class | normed_add_torsor | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"add_torsor",
"pseudo_metric_space",
"seminormed_add_comm_group"
] | A `normed_add_torsor V P` is a torsor of an additive seminormed group
action by a `seminormed_add_comm_group V` on points `P`. We bundle the pseudometric space
structure and require the distance to be the same as results from the
norm (which in fact implies the distance yields a pseudometric space, but
bundling just th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_torsor.to_add_torsor' {V P : Type*} [normed_add_comm_group V] [metric_space P]
[normed_add_torsor V P] : add_torsor V P | normed_add_torsor.to_add_torsor | instance | normed_add_torsor.to_add_torsor' | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"add_torsor",
"metric_space",
"normed_add_comm_group",
"normed_add_torsor"
] | Shortcut instance to help typeclass inference out. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_torsor.to_has_isometric_vadd : has_isometric_vadd V P | ⟨λ c, isometry.of_dist_eq $ λ x y, by simp [normed_add_torsor.dist_eq_norm']⟩ | instance | normed_add_torsor.to_has_isometric_vadd | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"has_isometric_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_add_comm_group.to_normed_add_torsor : normed_add_torsor V V | { dist_eq_norm' := dist_eq_norm } | instance | seminormed_add_comm_group.to_normed_add_torsor | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"normed_add_torsor"
] | A `seminormed_add_comm_group` is a `normed_add_torsor` over itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_subspace.to_normed_add_torsor {R : Type*} [ring R] [module R V]
(s : affine_subspace R P) [nonempty s] : normed_add_torsor s.direction s | { dist_eq_norm' := λ x y, normed_add_torsor.dist_eq_norm' ↑x ↑y,
..affine_subspace.to_add_torsor s } | instance | affine_subspace.to_normed_add_torsor | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"affine_subspace",
"affine_subspace.to_add_torsor",
"module",
"normed_add_torsor",
"ring"
] | A nonempty affine subspace of a `normed_add_torsor` is itself a `normed_add_torsor`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq_norm_vsub (x y : P) : dist x y = ‖x -ᵥ y‖ | normed_add_torsor.dist_eq_norm' x y | lemma | dist_eq_norm_vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [] | The distance equals the norm of subtracting two points. In this
lemma, it is necessary to have `V` as an explicit argument; otherwise
`rw dist_eq_norm_vsub` sometimes doesn't work. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_eq_nnnorm_vsub (x y : P) : nndist x y = ‖x -ᵥ y‖₊ | nnreal.eq $ dist_eq_norm_vsub V x y | lemma | nndist_eq_nnnorm_vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_eq_norm_vsub",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_norm_vsub' (x y : P) : dist x y = ‖y -ᵥ x‖ | (dist_comm _ _).trans (dist_eq_norm_vsub _ _ _) | lemma | dist_eq_norm_vsub' | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_comm",
"dist_eq_norm_vsub"
] | The distance equals the norm of subtracting two points. In this
lemma, it is necessary to have `V` as an explicit argument; otherwise
`rw dist_eq_norm_vsub'` sometimes doesn't work. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_eq_nnnorm_vsub' (x y : P) : nndist x y = ‖y -ᵥ x‖₊ | nnreal.eq $ dist_eq_norm_vsub' V x y | lemma | nndist_eq_nnnorm_vsub' | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_eq_norm_vsub'",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_vadd_cancel_left (v : V) (x y : P) :
dist (v +ᵥ x) (v +ᵥ y) = dist x y | dist_vadd _ _ _ | lemma | dist_vadd_cancel_left | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_vadd_cancel_right (v₁ v₂ : V) (x : P) :
dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂ | by rw [dist_eq_norm_vsub V, dist_eq_norm, vadd_vsub_vadd_cancel_right] | lemma | dist_vadd_cancel_right | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_eq_norm_vsub",
"vadd_vsub_vadd_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_vadd_cancel_right (v₁ v₂ : V) (x : P) :
nndist (v₁ +ᵥ x) (v₂ +ᵥ x) = nndist v₁ v₂ | nnreal.eq $ dist_vadd_cancel_right _ _ _ | lemma | nndist_vadd_cancel_right | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vadd_cancel_right",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_vadd_left (v : V) (x : P) : dist (v +ᵥ x) x = ‖v‖ | by simp [dist_eq_norm_vsub V _ x] | lemma | dist_vadd_left | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_eq_norm_vsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_vadd_left (v : V) (x : P) : nndist (v +ᵥ x) x = ‖v‖₊ | nnreal.eq $ dist_vadd_left _ _ | lemma | nndist_vadd_left | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vadd_left",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_vadd_right (v : V) (x : P) : dist x (v +ᵥ x) = ‖v‖ | by rw [dist_comm, dist_vadd_left] | lemma | dist_vadd_right | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_comm",
"dist_vadd_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_vadd_right (v : V) (x : P) : nndist x (v +ᵥ x) = ‖v‖₊ | nnreal.eq $ dist_vadd_right _ _ | lemma | nndist_vadd_right | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vadd_right",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_equiv.vadd_const (x : P) : V ≃ᵢ P | { to_equiv := equiv.vadd_const x,
isometry_to_fun := isometry.of_dist_eq $ λ _ _, dist_vadd_cancel_right _ _ _ } | def | isometry_equiv.vadd_const | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vadd_cancel_right",
"equiv.vadd_const"
] | Isometry between the tangent space `V` of a (semi)normed add torsor `P` and `P` given by
addition/subtraction of `x : P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_vsub_cancel_left (x y z : P) : dist (x -ᵥ y) (x -ᵥ z) = dist y z | by rw [dist_eq_norm, vsub_sub_vsub_cancel_left, dist_comm, dist_eq_norm_vsub V] | lemma | dist_vsub_cancel_left | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_comm",
"dist_eq_norm_vsub",
"vsub_sub_vsub_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_equiv.const_vsub (x : P) : P ≃ᵢ V | { to_equiv := equiv.const_vsub x,
isometry_to_fun := isometry.of_dist_eq $ λ y z, dist_vsub_cancel_left _ _ _ } | def | isometry_equiv.const_vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vsub_cancel_left",
"equiv.const_vsub"
] | Isometry between the tangent space `V` of a (semi)normed add torsor `P` and `P` given by
subtraction from `x : P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_vsub_cancel_right (x y z : P) : dist (x -ᵥ z) (y -ᵥ z) = dist x y | (isometry_equiv.vadd_const z).symm.dist_eq x y | lemma | dist_vsub_cancel_right | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"isometry_equiv.vadd_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_vsub_cancel_right (x y z : P) : nndist (x -ᵥ z) (y -ᵥ z) = nndist x y | nnreal.eq $ dist_vsub_cancel_right _ _ _ | lemma | nndist_vsub_cancel_right | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vsub_cancel_right",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_vadd_vadd_le (v v' : V) (p p' : P) :
dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p' | by simpa using dist_triangle (v +ᵥ p) (v' +ᵥ p) (v' +ᵥ p') | lemma | dist_vadd_vadd_le | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_triangle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_vadd_vadd_le (v v' : V) (p p' : P) :
nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p' | dist_vadd_vadd_le _ _ _ _ | lemma | nndist_vadd_vadd_le | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_vadd_vadd_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄ | by { rw [dist_eq_norm, vsub_sub_vsub_comm, dist_eq_norm_vsub V, dist_eq_norm_vsub V],
exact norm_sub_le _ _ } | lemma | dist_vsub_vsub_le | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_eq_norm_vsub",
"vsub_sub_vsub_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄ | by simp only [← nnreal.coe_le_coe, nnreal.coe_add, ← dist_nndist, dist_vsub_vsub_le] | lemma | nndist_vsub_vsub_le | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"dist_nndist",
"dist_vsub_vsub_le",
"nnreal.coe_add",
"nnreal.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_vadd_vadd_le (v v' : V) (p p' : P) :
edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p' | by { simp only [edist_nndist], apply_mod_cast nndist_vadd_vadd_le } | lemma | edist_vadd_vadd_le | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"edist_nndist",
"nndist_vadd_vadd_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄ | by { simp only [edist_nndist], apply_mod_cast nndist_vsub_vsub_le } | lemma | edist_vsub_vsub_le | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"edist_nndist",
"nndist_vsub_vsub_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_metric_space_of_normed_add_comm_group_of_add_torsor (V P : Type*)
[seminormed_add_comm_group V] [add_torsor V P] : pseudo_metric_space P | { dist := λ x y, ‖(x -ᵥ y : V)‖,
dist_self := λ x, by simp,
dist_comm := λ x y, by simp only [←neg_vsub_eq_vsub_rev y x, norm_neg],
dist_triangle := begin
intros x y z,
change ‖x -ᵥ z‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖,
rw ←vsub_add_vsub_cancel,
apply norm_add_le
end } | def | pseudo_metric_space_of_normed_add_comm_group_of_add_torsor | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"add_torsor",
"dist_comm",
"dist_self",
"dist_triangle",
"pseudo_metric_space",
"seminormed_add_comm_group"
] | The pseudodistance defines a pseudometric space structure on the torsor. This
is not an instance because it depends on `V` to define a `metric_space
P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space_of_normed_add_comm_group_of_add_torsor (V P : Type*)
[normed_add_comm_group V] [add_torsor V P] :
metric_space P | { dist := λ x y, ‖(x -ᵥ y : V)‖,
dist_self := λ x, by simp,
eq_of_dist_eq_zero := λ x y h, by simpa using h,
dist_comm := λ x y, by simp only [←neg_vsub_eq_vsub_rev y x, norm_neg],
dist_triangle := begin
intros x y z,
change ‖x -ᵥ z‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖,
rw ←vsub_add_vsub_cancel,
apply norm_ad... | def | metric_space_of_normed_add_comm_group_of_add_torsor | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"add_torsor",
"dist_comm",
"dist_self",
"dist_triangle",
"eq_of_dist_eq_zero",
"metric_space",
"normed_add_comm_group"
] | The distance defines a metric space structure on the torsor. This
is not an instance because it depends on `V` to define a `metric_space
P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_with.vadd [pseudo_emetric_space α] {f : α → V} {g : α → P} {Kf Kg : ℝ≥0}
(hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :
lipschitz_with (Kf + Kg) (f +ᵥ g) | λ x y,
calc edist (f x +ᵥ g x) (f y +ᵥ g y) ≤ edist (f x) (f y) + edist (g x) (g y) :
edist_vadd_vadd_le _ _ _ _
... ≤ Kf * edist x y + Kg * edist x y :
add_le_add (hf x y) (hg x y)
... = (Kf + Kg) * edist x y :
(add_mul _ _ _).symm | lemma | lipschitz_with.vadd | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"edist_vadd_vadd_le",
"lipschitz_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.vsub [pseudo_emetric_space α] {f g : α → P} {Kf Kg : ℝ≥0}
(hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :
lipschitz_with (Kf + Kg) (f -ᵥ g) | λ x y,
calc edist (f x -ᵥ g x) (f y -ᵥ g y) ≤ edist (f x) (f y) + edist (g x) (g y) :
edist_vsub_vsub_le _ _ _ _
... ≤ Kf * edist x y + Kg * edist x y :
add_le_add (hf x y) (hg x y)
... = (Kf + Kg) * edist x y :
(add_mul _ _ _).symm | lemma | lipschitz_with.vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"edist_vsub_vsub_le",
"lipschitz_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_vadd : uniform_continuous (λ x : V × P, x.1 +ᵥ x.2) | (lipschitz_with.prod_fst.vadd lipschitz_with.prod_snd).uniform_continuous | lemma | uniform_continuous_vadd | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"lipschitz_with.prod_snd",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_vsub : uniform_continuous (λ x : P × P, x.1 -ᵥ x.2) | (lipschitz_with.prod_fst.vsub lipschitz_with.prod_snd).uniform_continuous | lemma | uniform_continuous_vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"lipschitz_with.prod_snd",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_torsor.to_has_continuous_vadd : has_continuous_vadd V P | { continuous_vadd := uniform_continuous_vadd.continuous } | instance | normed_add_torsor.to_has_continuous_vadd | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"has_continuous_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_vsub : continuous (λ x : P × P, x.1 -ᵥ x.2) | uniform_continuous_vsub.continuous | lemma | continuous_vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.vsub {l : filter α} {f g : α → P} {x y : P}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) :
tendsto (f -ᵥ g) l (𝓝 (x -ᵥ y)) | (continuous_vsub.tendsto (x, y)).comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.vsub {f g : α → P} (hf : continuous f) (hg : continuous g) :
continuous (f -ᵥ g) | continuous_vsub.comp (hf.prod_mk hg : _) | lemma | continuous.vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.vsub {f g : α → P} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (f -ᵥ g) x | hf.vsub hg | lemma | continuous_at.vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.vsub {f g : α → P} {x : α} {s : set α}
(hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :
continuous_within_at (f -ᵥ g) s x | hf.vsub hg | lemma | continuous_within_at.vsub | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.line_map {l : filter α} {f₁ f₂ : α → P} {g : α → R} {p₁ p₂ : P} {c : R}
(h₁ : tendsto f₁ l (𝓝 p₁)) (h₂ : tendsto f₂ l (𝓝 p₂)) (hg : tendsto g l (𝓝 c)) :
tendsto (λ x, affine_map.line_map (f₁ x) (f₂ x) (g x)) l (𝓝 $ affine_map.line_map p₁ p₂ c) | (hg.smul (h₂.vsub h₁)).vadd h₁ | lemma | filter.tendsto.line_map | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"affine_map.line_map",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.midpoint [invertible (2:R)] {l : filter α} {f₁ f₂ : α → P} {p₁ p₂ : P}
(h₁ : tendsto f₁ l (𝓝 p₁)) (h₂ : tendsto f₂ l (𝓝 p₂)) :
tendsto (λ x, midpoint R (f₁ x) (f₂ x)) l (𝓝 $ midpoint R p₁ p₂) | h₁.line_map h₂ tendsto_const_nhds | lemma | filter.tendsto.midpoint | analysis.normed.group | src/analysis/normed/group/add_torsor.lean | [
"analysis.normed.group.basic",
"linear_algebra.affine_space.affine_subspace",
"linear_algebra.affine_space.midpoint"
] | [
"filter",
"invertible",
"midpoint",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg_sphere {r : ℝ} (v : sphere (0 : E) r) :
↑(-v) = (-v : E) | rfl | lemma | coe_neg_sphere | analysis.normed.group | src/analysis/normed/group/ball_sphere.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg_ball {r : ℝ} (v : ball (0 : E) r) :
↑(-v) = (-v : E) | rfl | lemma | coe_neg_ball | analysis.normed.group | src/analysis/normed/group/ball_sphere.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg_closed_ball {r : ℝ} (v : closed_ball (0 : E) r) :
↑(-v) = (-v : E) | rfl | lemma | coe_neg_closed_ball | analysis.normed.group | src/analysis/normed/group/ball_sphere.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_norm (E : Type*) | (norm : E → ℝ) | class | has_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | Auxiliary class, endowing a type `E` with a function `norm : E → ℝ` with notation `‖x‖`. This
class is designed to be extended in more interesting classes specifying the properties of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_nnnorm (E : Type*) | (nnnorm : E → ℝ≥0) | class | has_nnnorm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | Auxiliary class, endowing a type `α` with a function `nnnorm : α → ℝ≥0` with notation `‖x‖₊`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_add_group (E : Type*) extends has_norm E, add_group E, pseudo_metric_space E | (dist := λ x y, ‖x - y‖)
(dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously) | class | seminormed_add_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"add_group",
"has_norm",
"pseudo_metric_space"
] | A seminormed group is an additive group endowed with a norm for which `dist x y = ‖x - y‖`
defines a pseudometric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_group (E : Type*) extends has_norm E, group E, pseudo_metric_space E | (dist := λ x y, ‖x / y‖)
(dist_eq : ∀ x y, dist x y = ‖x / y‖ . obviously) | class | seminormed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"group",
"has_norm",
"pseudo_metric_space"
] | A seminormed group is a group endowed with a norm for which `dist x y = ‖x / y‖` defines a
pseudometric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_group (E : Type*) extends has_norm E, add_group E, metric_space E | (dist := λ x y, ‖x - y‖)
(dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously) | class | normed_add_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"add_group",
"has_norm",
"metric_space"
] | A normed group is an additive group endowed with a norm for which `dist x y = ‖x - y‖` defines a
metric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_group (E : Type*) extends has_norm E, group E, metric_space E | (dist := λ x y, ‖x / y‖)
(dist_eq : ∀ x y, dist x y = ‖x / y‖ . obviously) | class | normed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"group",
"has_norm",
"metric_space"
] | A normed group is a group endowed with a norm for which `dist x y = ‖x / y‖` defines a metric
space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_add_comm_group (E : Type*)
extends has_norm E, add_comm_group E, pseudo_metric_space E | (dist := λ x y, ‖x - y‖)
(dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously) | class | seminormed_add_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"add_comm_group",
"has_norm",
"pseudo_metric_space"
] | A seminormed group is an additive group endowed with a norm for which `dist x y = ‖x - y‖`
defines a pseudometric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group (E : Type*)
extends has_norm E, comm_group E, pseudo_metric_space E | (dist := λ x y, ‖x / y‖)
(dist_eq : ∀ x y, dist x y = ‖x / y‖ . obviously) | class | seminormed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"has_norm",
"pseudo_metric_space"
] | A seminormed group is a group endowed with a norm for which `dist x y = ‖x / y‖`
defines a pseudometric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_comm_group (E : Type*) extends has_norm E, add_comm_group E, metric_space E | (dist := λ x y, ‖x - y‖)
(dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously) | class | normed_add_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"add_comm_group",
"has_norm",
"metric_space"
] | A normed group is an additive group endowed with a norm for which `dist x y = ‖x - y‖` defines a
metric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_group (E : Type*) extends has_norm E, comm_group E, metric_space E | (dist := λ x y, ‖x / y‖)
(dist_eq : ∀ x y, dist x y = ‖x / y‖ . obviously) | class | normed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"has_norm",
"metric_space"
] | A normed group is a group endowed with a norm for which `dist x y = ‖x / y‖` defines a metric
space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_group.to_seminormed_group [normed_group E] : seminormed_group E | { ..‹normed_group E› } | instance | normed_group.to_seminormed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_group",
"seminormed_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.to_seminormed_comm_group [normed_comm_group E] :
seminormed_comm_group E | { ..‹normed_comm_group E› } | instance | normed_comm_group.to_seminormed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group",
"seminormed_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_comm_group.to_seminormed_group [seminormed_comm_group E] : seminormed_group E | { ..‹seminormed_comm_group E› } | instance | seminormed_comm_group.to_seminormed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_comm_group",
"seminormed_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.to_normed_group [normed_comm_group E] : normed_group E | { ..‹normed_comm_group E› } | instance | normed_comm_group.to_normed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group",
"normed_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_group.of_separation [seminormed_group E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
normed_group E | { to_metric_space :=
{ eq_of_dist_eq_zero := λ x y hxy, div_eq_one.1 $ h _ $ by rwa ←‹seminormed_group E›.dist_eq },
..‹seminormed_group E› } | def | normed_group.of_separation | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"eq_of_dist_eq_zero",
"normed_group",
"seminormed_group"
] | Construct a `normed_group` from a `seminormed_group` satisfying `∀ x, ‖x‖ = 0 → x = 1`. This
avoids having to go back to the `(pseudo_)metric_space` level when declaring a `normed_group`
instance as a special case of a more general `seminormed_group` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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