statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
convex_iff_sum_mem :
convex R s ↔
(∀ (t : finset E) (w : E → R),
(∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s ) | begin
refine ⟨λ hs t w hw₀ hw₁ hts, hs.sum_mem hw₀ hw₁ hts, _⟩,
intros h x hx y hy a b ha hb hab,
by_cases h_cases: x = y,
{ rw [h_cases, ←add_smul, hab, one_smul], exact hy },
{ convert h {x, y} (λ z, if z = y then b else a) _ _ _,
{ simp only [sum_pair h_cases, if_neg h_cases, if_pos rfl] },
{ simp_... | lemma | convex_iff_sum_mem | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex",
"finset",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_mem_convex_hull (t : finset ι) {w : ι → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
(hws : 0 < ∑ i in t, w i) {z : ι → E} (hz : ∀ i ∈ t, z i ∈ s) :
t.center_mass w z ∈ convex_hull R s | (convex_convex_hull R s).center_mass_mem hw₀ hws (λ i hi, subset_convex_hull R s $ hz i hi) | lemma | finset.center_mass_mem_convex_hull | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_convex_hull",
"convex_hull",
"finset",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_id_mem_convex_hull (t : finset E) {w : E → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
(hws : 0 < ∑ i in t, w i) :
t.center_mass w id ∈ convex_hull R (t : set E) | t.center_mass_mem_convex_hull hw₀ hws (λ i, mem_coe.2) | lemma | finset.center_mass_id_mem_convex_hull | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"finset"
] | A refinement of `finset.center_mass_mem_convex_hull` when the indexed family is a `finset` of
the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_combination_eq_center_mass {ι : Type*} {t : finset ι} {p : ι → E} {w : ι → R}
(hw₂ : ∑ i in t, w i = 1) :
t.affine_combination R p w = center_mass t w p | begin
rw [affine_combination_eq_weighted_vsub_of_point_vadd_of_sum_eq_one _ w _ hw₂ (0 : E),
finset.weighted_vsub_of_point_apply, vadd_eq_add, add_zero, t.center_mass_eq_of_sum_1 _ hw₂],
simp_rw [vsub_eq_sub, sub_zero],
end | lemma | affine_combination_eq_center_mass | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset",
"finset.weighted_vsub_of_point_apply",
"vsub_eq_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_combination_mem_convex_hull
{s : finset ι} {v : ι → E} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1) :
s.affine_combination R v w ∈ convex_hull R (range v) | begin
rw affine_combination_eq_center_mass hw₁,
apply s.center_mass_mem_convex_hull hw₀,
{ simp [hw₁], },
{ simp, },
end | lemma | affine_combination_mem_convex_hull | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"affine_combination_eq_center_mass",
"convex_hull",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.centroid_eq_center_mass (s : finset ι) (hs : s.nonempty) (p : ι → E) :
s.centroid R p = s.center_mass (s.centroid_weights R) p | affine_combination_eq_center_mass (s.sum_centroid_weights_eq_one_of_nonempty R hs) | lemma | finset.centroid_eq_center_mass | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"affine_combination_eq_center_mass",
"finset"
] | The centroid can be regarded as a center of mass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.centroid_mem_convex_hull (s : finset E) (hs : s.nonempty) :
s.centroid R id ∈ convex_hull R (s : set E) | begin
rw s.centroid_eq_center_mass hs,
apply s.center_mass_id_mem_convex_hull,
{ simp only [inv_nonneg, implies_true_iff, nat.cast_nonneg, finset.centroid_weights_apply], },
{ have hs_card : (s.card : R) ≠ 0, { simp [finset.nonempty_iff_ne_empty.mp hs] },
simp only [hs_card, finset.sum_const, nsmul_eq_mul, ... | lemma | finset.centroid_mem_convex_hull | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"finset",
"finset.centroid_weights_apply",
"inv_nonneg",
"mul_inv_cancel",
"nat.cast_nonneg",
"nsmul_eq_mul",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_range_eq_exists_affine_combination (v : ι → E) :
convex_hull R (range v) = { x | ∃ (s : finset ι) (w : ι → R)
(hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1), s.affine_combination R v w = x } | begin
refine subset.antisymm (convex_hull_min _ _) _,
{ intros x hx,
obtain ⟨i, hi⟩ := set.mem_range.mp hx,
refine ⟨{i}, function.const ι (1 : R), by simp, by simp, by simp [hi]⟩, },
{ rintro x ⟨s, w, hw₀, hw₁, rfl⟩ y ⟨s', w', hw₀', hw₁', rfl⟩ a b ha hb hab,
let W : ι → R := λ i, (if i ∈ s then a * w ... | lemma | convex_hull_range_eq_exists_affine_combination | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"add_smul",
"affine_combination_mem_convex_hull",
"convex_hull",
"convex_hull_min",
"finset",
"ite_smul",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_eq (s : set E) :
convex_hull R s = {x : E | ∃ (ι : Type u') (t : finset ι) (w : ι → R) (z : ι → E)
(hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hz : ∀ i ∈ t, z i ∈ s),
t.center_mass w z = x} | begin
refine subset.antisymm (convex_hull_min _ _) _,
{ intros x hx,
use [punit, {punit.star}, λ _, 1, λ _, x, λ _ _, zero_le_one,
finset.sum_singleton, λ _ _, hx],
simp only [finset.center_mass, finset.sum_singleton, inv_one, one_smul] },
{ rintros x ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ y ⟨ι', sy,... | lemma | convex_hull_eq | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_min",
"finset",
"finset.center_mass",
"finset.center_mass_segment'",
"finset.mem_disj_sum",
"inv_one",
"one_smul",
"sum.elim_inl",
"sum.elim_inr",
"zero_le_one",
"zero_lt_one"
] | Convex hull of `s` is equal to the set of all centers of masses of `finset`s `t`, `z '' t ⊆ s`.
This version allows finsets in any type in any universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.convex_hull_eq (s : finset E) :
convex_hull R ↑s = {x : E | ∃ (w : E → R) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : ∑ y in s, w y = 1),
s.center_mass w id = x} | begin
refine subset.antisymm (convex_hull_min _ _) _,
{ intros x hx,
rw [finset.mem_coe] at hx,
refine ⟨_, _, _, finset.center_mass_ite_eq _ _ _ hx⟩,
{ intros, split_ifs, exacts [zero_le_one, le_refl 0] },
{ rw [finset.sum_ite_eq, if_pos hx] } },
{ rintro x ⟨wx, hwx₀, hwx₁, rfl⟩ y ⟨wy, hwy₀, hwy₁,... | lemma | finset.convex_hull_eq | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_min",
"finset",
"finset.center_mass_ite_eq",
"finset.center_mass_segment",
"finset.mem_coe",
"mul_one",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.mem_convex_hull {s : finset E} {x : E} :
x ∈ convex_hull R (s : set E) ↔
∃ (w : E → R) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : ∑ y in s, w y = 1), s.center_mass w id = x | by rw [finset.convex_hull_eq, set.mem_set_of_eq] | lemma | finset.mem_convex_hull | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"finset",
"finset.convex_hull_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.convex_hull_eq {s : set E} (hs : s.finite) :
convex_hull R s = {x : E | ∃ (w : E → R) (hw₀ : ∀ y ∈ s, 0 ≤ w y)
(hw₁ : ∑ y in hs.to_finset, w y = 1), hs.to_finset.center_mass w id = x} | by simpa only [set.finite.coe_to_finset, set.finite.mem_to_finset, exists_prop]
using hs.to_finset.convex_hull_eq | lemma | set.finite.convex_hull_eq | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"exists_prop",
"set.finite.coe_to_finset",
"set.finite.mem_to_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_eq_union_convex_hull_finite_subsets (s : set E) :
convex_hull R s = ⋃ (t : finset E) (w : ↑t ⊆ s), convex_hull R ↑t | begin
refine subset.antisymm _ _,
{ rw convex_hull_eq,
rintros x ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩,
simp only [mem_Union],
refine ⟨t.image z, _, _⟩,
{ rw [coe_image, set.image_subset_iff],
exact hz },
{ apply t.center_mass_mem_convex_hull hw₀,
{ simp only [hw₁, zero_lt_one] },
{ ... | lemma | convex_hull_eq_union_convex_hull_finite_subsets | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_eq",
"convex_hull_mono",
"finset",
"finset.mem_image_of_mem",
"set.image_subset_iff",
"zero_lt_one"
] | A weak version of Carathéodory's theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_mem_convex_hull_prod {t : set F} {x : E} {y : F} (hx : x ∈ convex_hull R s)
(hy : y ∈ convex_hull R t) :
(x, y) ∈ convex_hull R (s ×ˢ t) | begin
rw convex_hull_eq at ⊢ hx hy,
obtain ⟨ι, a, w, S, hw, hw', hS, hSp⟩ := hx,
obtain ⟨κ, b, v, T, hv, hv', hT, hTp⟩ := hy,
have h_sum : ∑ (i : ι × κ) in a ×ˢ b, w i.fst * v i.snd = 1,
{ rw [finset.sum_product, ← hw'],
congr,
ext i,
have : ∑ (y : κ) in b, w i * v y = ∑ (y : κ) in b, v y * w i,
... | lemma | mk_mem_convex_hull_prod | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_eq",
"finset.center_mass_eq_of_sum_1",
"finset.sum_mul",
"mul_comm",
"one_smul",
"prod.smul_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_prod (s : set E) (t : set F) :
convex_hull R (s ×ˢ t) = convex_hull R s ×ˢ convex_hull R t | subset.antisymm (convex_hull_min (prod_mono (subset_convex_hull _ _) $ subset_convex_hull _ _) $
(convex_convex_hull _ _).prod $ convex_convex_hull _ _) $
prod_subset_iff.2 $ λ x hx y, mk_mem_convex_hull_prod hx | lemma | convex_hull_prod | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_convex_hull",
"convex_hull",
"convex_hull_min",
"mk_mem_convex_hull_prod",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_add (s t : set E) : convex_hull R (s + t) = convex_hull R s + convex_hull R t | by simp_rw [←image2_add, ←image_prod, is_linear_map.is_linear_map_add.convex_hull_image,
convex_hull_prod] | lemma | convex_hull_add | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_add_monoid_hom : set E →+ set E | { to_fun := convex_hull R,
map_add' := convex_hull_add,
map_zero' := convex_hull_zero } | def | convex_hull_add_monoid_hom | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_add",
"convex_hull_zero"
] | `convex_hull` is an additive monoid morphism under pointwise addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_hull_sub (s t : set E) : convex_hull R (s - t) = convex_hull R s - convex_hull R t | by simp_rw [sub_eq_add_neg, convex_hull_add, convex_hull_neg] | lemma | convex_hull_sub | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_add",
"convex_hull_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_list_sum (l : list (set E)) : convex_hull R l.sum = (l.map $ convex_hull R).sum | map_list_sum (convex_hull_add_monoid_hom R E) l | lemma | convex_hull_list_sum | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_add_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_multiset_sum (s : multiset (set E)) :
convex_hull R s.sum = (s.map $ convex_hull R).sum | map_multiset_sum (convex_hull_add_monoid_hom R E) s | lemma | convex_hull_multiset_sum | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_add_monoid_hom",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_sum {ι} (s : finset ι) (t : ι → set E) :
convex_hull R (∑ i in s, t i) = ∑ i in s, convex_hull R (t i) | map_sum (convex_hull_add_monoid_hom R E) _ _ | lemma | convex_hull_sum | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_add_monoid_hom",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_basis_eq_std_simplex :
convex_hull R (range $ λ(i j:ι), if i = j then (1:R) else 0) = std_simplex R ι | begin
refine subset.antisymm (convex_hull_min _ (convex_std_simplex R ι)) _,
{ rintros _ ⟨i, rfl⟩,
exact ite_eq_mem_std_simplex R i },
{ rintros w ⟨hw₀, hw₁⟩,
rw [pi_eq_sum_univ w, ← finset.univ.center_mass_eq_of_sum_1 _ hw₁],
exact finset.univ.center_mass_mem_convex_hull (λ i hi, hw₀ i)
(hw₁.sy... | lemma | convex_hull_basis_eq_std_simplex | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_min",
"convex_std_simplex",
"ite_eq_mem_std_simplex",
"pi_eq_sum_univ",
"std_simplex",
"zero_lt_one"
] | `std_simplex 𝕜 ι` is the convex hull of the canonical basis in `ι → 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.finite.convex_hull_eq_image {s : set E} (hs : s.finite) :
convex_hull R s = by haveI | hs.fintype; exact
(⇑(∑ x : s, (@linear_map.proj R s _ (λ i, R) _ _ x).smul_right x.1)) '' (std_simplex R s) :=
begin
rw [← convex_hull_basis_eq_std_simplex, ← linear_map.convex_hull_image, ← set.range_comp, (∘)],
apply congr_arg,
convert subtype.range_coe.symm,
ext x,
simp [linear_map.sum_apply, ite_smul,... | lemma | set.finite.convex_hull_eq_image | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex_hull",
"convex_hull_basis_eq_std_simplex",
"finset.filter_eq",
"ite_smul",
"linear_map.convex_hull_image",
"linear_map.proj",
"linear_map.sum_apply",
"set.range_comp",
"std_simplex"
] | The convex hull of a finite set is the image of the standard simplex in `s → ℝ`
under the linear map sending each function `w` to `∑ x in s, w x • x`.
Since we have no sums over finite sets, we use sum over `@finset.univ _ hs.fintype`.
The map is defined in terms of operations on `(s → ℝ) →ₗ[ℝ] ℝ` so that later we wil... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_Icc_of_mem_std_simplex (hf : f ∈ std_simplex R ι) (x) :
f x ∈ Icc (0 : R) 1 | ⟨hf.1 x, hf.2 ▸ finset.single_le_sum (λ y hy, hf.1 y) (finset.mem_univ x)⟩ | lemma | mem_Icc_of_mem_std_simplex | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset.mem_univ",
"std_simplex"
] | All values of a function `f ∈ std_simplex 𝕜 ι` belong to `[0, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_basis.convex_hull_eq_nonneg_coord {ι : Type*} (b : affine_basis ι R E) :
convex_hull R (range b) = {x | ∀ i, 0 ≤ b.coord i x} | begin
rw convex_hull_range_eq_exists_affine_combination,
ext x,
refine ⟨_, λ hx, _⟩,
{ rintros ⟨s, w, hw₀, hw₁, rfl⟩ i,
by_cases hi : i ∈ s,
{ rw b.coord_apply_combination_of_mem hi hw₁,
exact hw₀ i hi, },
{ rw b.coord_apply_combination_of_not_mem hi hw₁, }, },
{ have hx' : x ∈ affine_span R... | lemma | affine_basis.convex_hull_eq_nonneg_coord | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"affine_basis",
"affine_span",
"affine_subspace.mem_top",
"convex_hull",
"convex_hull_range_eq_exists_affine_combination",
"mem_affine_span_iff_eq_affine_combination"
] | The convex hull of an affine basis is the intersection of the half-spaces defined by the
corresponding barycentric coordinates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_halfspace_re_lt (r : ℝ) : convex ℝ {c : ℂ | c.re < r} | convex_halfspace_lt (is_linear_map.mk complex.add_re complex.smul_re) _ | lemma | convex_halfspace_re_lt | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_re",
"complex.smul_re",
"convex",
"convex_halfspace_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_re_le (r : ℝ) : convex ℝ {c : ℂ | c.re ≤ r} | convex_halfspace_le (is_linear_map.mk complex.add_re complex.smul_re) _ | lemma | convex_halfspace_re_le | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_re",
"complex.smul_re",
"convex",
"convex_halfspace_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_re_gt (r : ℝ) : convex ℝ {c : ℂ | r < c.re } | convex_halfspace_gt (is_linear_map.mk complex.add_re complex.smul_re) _ | lemma | convex_halfspace_re_gt | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_re",
"complex.smul_re",
"convex",
"convex_halfspace_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_re_ge (r : ℝ) : convex ℝ {c : ℂ | r ≤ c.re} | convex_halfspace_ge (is_linear_map.mk complex.add_re complex.smul_re) _ | lemma | convex_halfspace_re_ge | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_re",
"complex.smul_re",
"convex",
"convex_halfspace_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_im_lt (r : ℝ) : convex ℝ {c : ℂ | c.im < r} | convex_halfspace_lt (is_linear_map.mk complex.add_im complex.smul_im) _ | lemma | convex_halfspace_im_lt | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_im",
"complex.smul_im",
"convex",
"convex_halfspace_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_im_le (r : ℝ) : convex ℝ {c : ℂ | c.im ≤ r} | convex_halfspace_le (is_linear_map.mk complex.add_im complex.smul_im) _ | lemma | convex_halfspace_im_le | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_im",
"complex.smul_im",
"convex",
"convex_halfspace_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_im_gt (r : ℝ) : convex ℝ {c : ℂ | r < c.im} | convex_halfspace_gt (is_linear_map.mk complex.add_im complex.smul_im) _ | lemma | convex_halfspace_im_gt | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_im",
"complex.smul_im",
"convex",
"convex_halfspace_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_im_ge (r : ℝ) : convex ℝ {c : ℂ | r ≤ c.im} | convex_halfspace_ge (is_linear_map.mk complex.add_im complex.smul_im) _ | lemma | convex_halfspace_im_ge | analysis.convex | src/analysis/convex/complex.lean | [
"analysis.convex.basic",
"data.complex.module"
] | [
"complex.add_im",
"complex.smul_im",
"convex",
"convex_halfspace_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.contractible_space (h : star_convex ℝ x s) (hne : s.nonempty) :
contractible_space s | begin
refine (contractible_iff_id_nullhomotopic _).2 ⟨⟨x, h.mem hne⟩,
⟨⟨⟨λ p, ⟨p.1.1 • x + (1 - p.1.1) • p.2, _⟩, _⟩, λ x, _, λ x, _⟩⟩⟩,
{ exact h p.2.2 p.1.2.1 (sub_nonneg.2 p.1.2.2) (add_sub_cancel'_right _ _) },
{ exact ((continuous_subtype_val.fst'.smul continuous_const).add
((continuous_const.sub c... | lemma | star_convex.contractible_space | analysis.convex | src/analysis/convex/contractible.lean | [
"analysis.convex.star",
"topology.homotopy.contractible"
] | [
"continuous_const",
"contractible_iff_id_nullhomotopic",
"contractible_space",
"star_convex"
] | A non-empty star convex set is a contractible space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.contractible_space (hs : convex ℝ s) (hne : s.nonempty) :
contractible_space s | let ⟨x, hx⟩ := hne in (hs.star_convex hx).contractible_space hne | lemma | convex.contractible_space | analysis.convex | src/analysis/convex/contractible.lean | [
"analysis.convex.star",
"topology.homotopy.contractible"
] | [
"contractible_space",
"convex"
] | A non-empty convex set is a contractible space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_topological_vector_space.contractible_space : contractible_space E | (homeomorph.set.univ E).contractible_space_iff.mp $ convex_univ.contractible_space set.univ_nonempty | instance | real_topological_vector_space.contractible_space | analysis.convex | src/analysis/convex/contractible.lean | [
"analysis.convex.star",
"topology.homotopy.contractible"
] | [
"contractible_space",
"homeomorph.set.univ",
"set.univ_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_exposed (A B : set E) : Prop | B.nonempty → ∃ l : E →L[𝕜] 𝕜, B = {x ∈ A | ∀ y ∈ A, l y ≤ l x} | def | is_exposed | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [] | A set `B` is exposed with respect to `A` iff it maximizes some functional over `A` (and contains
all points maximizing it). Written `is_exposed 𝕜 A B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.to_exposed (l : E →L[𝕜] 𝕜) (A : set E) : set E | {x ∈ A | ∀ y ∈ A, l y ≤ l x} | def | continuous_linear_map.to_exposed | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [] | A useful way to build exposed sets from intersecting `A` with halfspaces (modelled by an
inequality with a functional). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.to_exposed.is_exposed : is_exposed 𝕜 A (l.to_exposed A) | λ h, ⟨l, rfl⟩ | lemma | continuous_linear_map.to_exposed.is_exposed | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_exposed_empty : is_exposed 𝕜 A ∅ | λ ⟨x, hx⟩, by { exfalso, exact hx } | lemma | is_exposed_empty | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset (hAB : is_exposed 𝕜 A B) : B ⊆ A | begin
rintro x hx,
obtain ⟨_, rfl⟩ := hAB ⟨x, hx⟩,
exact hx.1,
end | lemma | is_exposed.subset | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (A : set E) : is_exposed 𝕜 A A | λ ⟨w, hw⟩, ⟨0, subset.antisymm (λ x hx, ⟨hx, λ y hy, by exact le_refl 0⟩) (λ x hx, hx.1)⟩ | lemma | is_exposed.refl | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antisymm (hB : is_exposed 𝕜 A B) (hA : is_exposed 𝕜 B A) :
A = B | hA.subset.antisymm hB.subset | lemma | is_exposed.antisymm | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (hC : is_exposed 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B) :
is_exposed 𝕜 B C | begin
rintro ⟨w, hw⟩,
obtain ⟨l, rfl⟩ := hC ⟨w, hw⟩,
exact ⟨l, subset.antisymm (λ x hx, ⟨hCB hx, λ y hy, hx.2 y (hBA hy)⟩)
(λ x hx, ⟨hBA hx.1, λ y hy, (hw.2 y hy).trans (hx.2 w (hCB hw))⟩)⟩,
end | lemma | is_exposed.mono | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inter_halfspace' {A B : set E} (hAB : is_exposed 𝕜 A B) (hB : B.nonempty) :
∃ l : E →L[𝕜] 𝕜, ∃ a, B = {x ∈ A | a ≤ l x} | begin
obtain ⟨l, rfl⟩ := hAB hB,
obtain ⟨w, hw⟩ := hB,
exact ⟨l, l w, subset.antisymm (λ x hx, ⟨hx.1, hx.2 w hw.1⟩)
(λ x hx, ⟨hx.1, λ y hy, (hw.2 y hy).trans hx.2⟩)⟩,
end | lemma | is_exposed.eq_inter_halfspace' | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | If `B` is a nonempty exposed subset of `A`, then `B` is the intersection of `A` with some closed
halfspace. The converse is *not* true. It would require that the corresponding open halfspace
doesn't intersect `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_inter_halfspace [nontrivial 𝕜] {A B : set E} (hAB : is_exposed 𝕜 A B) :
∃ l : E →L[𝕜] 𝕜, ∃ a, B = {x ∈ A | a ≤ l x} | begin
obtain rfl | hB := B.eq_empty_or_nonempty,
{ refine ⟨0, 1, _⟩,
rw [eq_comm, eq_empty_iff_forall_not_mem],
rintro x ⟨-, h⟩,
rw continuous_linear_map.zero_apply at h,
have : ¬ ((1:𝕜) ≤ 0) := not_le_of_lt zero_lt_one,
contradiction },
exact hAB.eq_inter_halfspace' hB,
end | lemma | is_exposed.eq_inter_halfspace | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"continuous_linear_map.zero_apply",
"is_exposed",
"nontrivial",
"not_le_of_lt",
"zero_lt_one"
] | For nontrivial `𝕜`, if `B` is an exposed subset of `A`, then `B` is the intersection of `A` with
some closed halfspace. The converse is *not* true. It would require that the corresponding open
halfspace doesn't intersect `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter [has_continuous_add 𝕜] {A B C : set E} (hB : is_exposed 𝕜 A B)
(hC : is_exposed 𝕜 A C) :
is_exposed 𝕜 A (B ∩ C) | begin
rintro ⟨w, hwB, hwC⟩,
obtain ⟨l₁, rfl⟩ := hB ⟨w, hwB⟩,
obtain ⟨l₂, rfl⟩ := hC ⟨w, hwC⟩,
refine ⟨l₁ + l₂, subset.antisymm _ _⟩,
{ rintro x ⟨⟨hxA, hxB⟩, ⟨-, hxC⟩⟩,
exact ⟨hxA, λ z hz, add_le_add (hxB z hz) (hxC z hz)⟩ },
rintro x ⟨hxA, hx⟩,
refine ⟨⟨hxA, λ y hy, _⟩, hxA, λ y hy, _⟩,
{ exact (add... | lemma | is_exposed.inter | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"has_continuous_add",
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter [has_continuous_add 𝕜] {F : finset (set E)} (hF : F.nonempty)
(hAF : ∀ B ∈ F, is_exposed 𝕜 A B) :
is_exposed 𝕜 A (⋂₀ F) | begin
revert hF F,
refine finset.induction _ _,
{ rintro h,
exfalso,
exact not_nonempty_empty h },
rintro C F _ hF _ hCF,
rw [finset.coe_insert, sInter_insert],
obtain rfl | hFnemp := F.eq_empty_or_nonempty,
{ rw [finset.coe_empty, sInter_empty, inter_univ],
exact hCF C (finset.mem_singleton_s... | lemma | is_exposed.sInter | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"finset",
"finset.coe_empty",
"finset.coe_insert",
"finset.induction",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_singleton_self",
"has_continuous_add",
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_left (hC : is_exposed 𝕜 A C) (hCB : C ⊆ B) :
is_exposed 𝕜 (A ∩ B) C | begin
rintro ⟨w, hw⟩,
obtain ⟨l, rfl⟩ := hC ⟨w, hw⟩,
exact ⟨l, subset.antisymm (λ x hx, ⟨⟨hx.1, hCB hx⟩, λ y hy, hx.2 y hy.1⟩)
(λ x ⟨⟨hxC, _⟩, hx⟩, ⟨hxC, λ y hy, (hw.2 y hy).trans (hx w ⟨hC.subset hw, hCB hw⟩)⟩)⟩,
end | lemma | is_exposed.inter_left | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_right (hC : is_exposed 𝕜 B C) (hCA : C ⊆ A) :
is_exposed 𝕜 (A ∩ B) C | begin
rw inter_comm,
exact hC.inter_left hCA,
end | lemma | is_exposed.inter_right | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed [order_closed_topology 𝕜] {A B : set E}
(hAB : is_exposed 𝕜 A B) (hA : is_closed A) : is_closed B | begin
obtain rfl | hB := B.eq_empty_or_nonempty,
{ simp },
obtain ⟨l, a, rfl⟩ := hAB.eq_inter_halfspace' hB,
exact hA.is_closed_le continuous_on_const l.continuous.continuous_on,
end | lemma | is_exposed.is_closed | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"continuous_on_const",
"is_closed",
"is_exposed",
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact [order_closed_topology 𝕜] [t2_space E] {A B : set E}
(hAB : is_exposed 𝕜 A B) (hA : is_compact A) : is_compact B | is_compact_of_is_closed_subset hA (hAB.is_closed hA.is_closed) hAB.subset | lemma | is_exposed.is_compact | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_compact",
"is_compact_of_is_closed_subset",
"is_exposed",
"order_closed_topology",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.exposed_points (A : set E) :
set E | {x ∈ A | ∃ l : E →L[𝕜] 𝕜, ∀ y ∈ A, l y ≤ l x ∧ (l x ≤ l y → y = x)} | def | set.exposed_points | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [] | A point is exposed with respect to `A` iff there exists an hyperplane whose intersection with
`A` is exactly that point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exposed_point_def :
x ∈ A.exposed_points 𝕜 ↔ x ∈ A ∧ ∃ l : E →L[𝕜] 𝕜, ∀ y ∈ A, l y ≤ l x ∧ (l x ≤ l y → y = x) | iff.rfl | lemma | exposed_point_def | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exposed_points_subset :
A.exposed_points 𝕜 ⊆ A | λ x hx, hx.1 | lemma | exposed_points_subset | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exposed_points_empty :
(∅ : set E).exposed_points 𝕜 = ∅ | subset_empty_iff.1 exposed_points_subset | lemma | exposed_points_empty | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"exposed_points_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_exposed_points_iff_exposed_singleton :
x ∈ A.exposed_points 𝕜 ↔ is_exposed 𝕜 A {x} | begin
use λ ⟨hxA, l, hl⟩ h, ⟨l, eq.symm $ eq_singleton_iff_unique_mem.2 ⟨⟨hxA, λ y hy, (hl y hy).1⟩,
λ z hz, (hl z hz.1).2 (hz.2 x hxA)⟩⟩,
rintro h,
obtain ⟨l, hl⟩ := h ⟨x, mem_singleton _⟩,
rw [eq_comm, eq_singleton_iff_unique_mem] at hl,
exact ⟨hl.1.1, l, λ y hy, ⟨hl.1.2 y hy, λ hxy, hl.2 y ⟨hy, λ z hz,... | lemma | mem_exposed_points_iff_exposed_singleton | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_exposed"
] | Exposed points exactly correspond to exposed singletons. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex (hAB : is_exposed 𝕜 A B) (hA : convex 𝕜 A) :
convex 𝕜 B | begin
obtain rfl | hB := B.eq_empty_or_nonempty,
{ exact convex_empty },
obtain ⟨l, rfl⟩ := hAB hB,
exact λ x₁ hx₁ x₂ hx₂ a b ha hb hab, ⟨hA hx₁.1 hx₂.1 ha hb hab, λ y hy,
((l.to_linear_map.concave_on convex_univ).convex_ge _
⟨mem_univ _, hx₁.2 y hy⟩ ⟨mem_univ _, hx₂.2 y hy⟩ ha hb hab).2⟩,
end | lemma | is_exposed.convex | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"convex",
"convex_empty",
"convex_univ",
"is_exposed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme (hAB : is_exposed 𝕜 A B) :
is_extreme 𝕜 A B | begin
refine ⟨hAB.subset, λ x₁ hx₁A x₂ hx₂A x hxB hx, _⟩,
obtain ⟨l, rfl⟩ := hAB ⟨x, hxB⟩,
have hl : convex_on 𝕜 univ l := l.to_linear_map.convex_on convex_univ,
have hlx₁ := hxB.2 x₁ hx₁A,
have hlx₂ := hxB.2 x₂ hx₂A,
refine ⟨⟨hx₁A, λ y hy, _⟩, ⟨hx₂A, λ y hy, _⟩⟩,
{ have := @convex_on.le_left_of_right_le... | lemma | is_exposed.is_extreme | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"convex_on",
"convex_on.le_left_of_right_le",
"convex_univ",
"is_exposed",
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exposed_points_subset_extreme_points :
A.exposed_points 𝕜 ⊆ A.extreme_points 𝕜 | λ x hx, mem_extreme_points_iff_extreme_singleton.2
(mem_exposed_points_iff_exposed_singleton.1 hx).is_extreme | lemma | exposed_points_subset_extreme_points | analysis.convex | src/analysis/convex/exposed.lean | [
"analysis.convex.extreme",
"analysis.convex.function",
"topology.algebra.module.basic",
"topology.order.basic"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.of_is_local_min_on_of_convex_on_Icc {f : ℝ → β} {a b : ℝ} (a_lt_b : a < b)
(h_local_min : is_local_min_on f (Icc a b) a) (h_conv : convex_on ℝ (Icc a b) f) :
is_min_on f (Icc a b) a | begin
rintro c hc, dsimp only [mem_set_of_eq],
rw [is_local_min_on, nhds_within_Icc_eq_nhds_within_Ici a_lt_b] at h_local_min,
rcases hc.1.eq_or_lt with rfl|a_lt_c, { exact le_rfl },
have H₁ : ∀ᶠ y in 𝓝[>] a, f a ≤ f y,
from h_local_min.filter_mono (nhds_within_mono _ Ioi_subset_Ici_self),
have H₂ : ∀ᶠ y... | lemma | is_min_on.of_is_local_min_on_of_convex_on_Icc | analysis.convex | src/analysis/convex/extrema.lean | [
"analysis.convex.function",
"topology.algebra.affine",
"topology.local_extr",
"topology.metric_space.basic"
] | [
"Ioc_mem_nhds_within_Ioi",
"add_smul",
"convex.mem_Ioc",
"convex_on",
"is_local_min_on",
"is_min_on",
"le_rfl",
"nhds_within_Icc_eq_nhds_within_Ici",
"nhds_within_mono",
"one_smul",
"smul_le_smul_iff_of_pos"
] | Helper lemma for the more general case: `is_min_on.of_is_local_min_on_of_convex_on`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_min_on.of_is_local_min_on_of_convex_on {f : E → β} {a : E}
(a_in_s : a ∈ s) (h_localmin : is_local_min_on f s a) (h_conv : convex_on ℝ s f) :
is_min_on f s a | begin
intros x x_in_s,
let g : ℝ →ᵃ[ℝ] E := affine_map.line_map a x,
have hg0 : g 0 = a := affine_map.line_map_apply_zero a x,
have hg1 : g 1 = x := affine_map.line_map_apply_one a x,
have hgc : continuous g, from affine_map.line_map_continuous,
have h_maps : maps_to g (Icc 0 1) s,
{ simpa only [maps_to',... | lemma | is_min_on.of_is_local_min_on_of_convex_on | analysis.convex | src/analysis/convex/extrema.lean | [
"analysis.convex.function",
"topology.algebra.affine",
"topology.local_extr",
"topology.metric_space.basic"
] | [
"affine_map.line_map",
"affine_map.line_map_apply_one",
"affine_map.line_map_apply_zero",
"affine_map.line_map_continuous",
"continuous",
"convex_Icc",
"convex_on",
"is_local_min_on",
"is_min_on",
"is_min_on.of_is_local_min_on_of_convex_on_Icc",
"segment_eq_image_line_map",
"zero_le_one"
] | A local minimum of a convex function is a global minimum, restricted to a set `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_max_on.of_is_local_max_on_of_concave_on {f : E → β} {a : E}
(a_in_s : a ∈ s) (h_localmax: is_local_max_on f s a) (h_conc : concave_on ℝ s f) :
is_max_on f s a | @is_min_on.of_is_local_min_on_of_convex_on _ βᵒᵈ _ _ _ _ _ _ _ _ s f a a_in_s h_localmax h_conc | lemma | is_max_on.of_is_local_max_on_of_concave_on | analysis.convex | src/analysis/convex/extrema.lean | [
"analysis.convex.function",
"topology.algebra.affine",
"topology.local_extr",
"topology.metric_space.basic"
] | [
"concave_on",
"is_local_max_on",
"is_max_on",
"is_min_on.of_is_local_min_on_of_convex_on"
] | A local maximum of a concave function is a global maximum, restricted to a set `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_min_on.of_is_local_min_of_convex_univ {f : E → β} {a : E}
(h_local_min : is_local_min f a) (h_conv : convex_on ℝ univ f) : ∀ x, f a ≤ f x | λ x, (is_min_on.of_is_local_min_on_of_convex_on (mem_univ a)
(h_local_min.on univ) h_conv) (mem_univ x) | lemma | is_min_on.of_is_local_min_of_convex_univ | analysis.convex | src/analysis/convex/extrema.lean | [
"analysis.convex.function",
"topology.algebra.affine",
"topology.local_extr",
"topology.metric_space.basic"
] | [
"convex_on",
"is_local_min",
"is_min_on.of_is_local_min_on_of_convex_on"
] | A local minimum of a convex function is a global minimum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_max_on.of_is_local_max_of_convex_univ {f : E → β} {a : E}
(h_local_max : is_local_max f a) (h_conc : concave_on ℝ univ f) : ∀ x, f x ≤ f a | @is_min_on.of_is_local_min_of_convex_univ _ βᵒᵈ _ _ _ _ _ _ _ _ f a h_local_max h_conc | lemma | is_max_on.of_is_local_max_of_convex_univ | analysis.convex | src/analysis/convex/extrema.lean | [
"analysis.convex.function",
"topology.algebra.affine",
"topology.local_extr",
"topology.metric_space.basic"
] | [
"concave_on",
"is_local_max",
"is_min_on.of_is_local_min_of_convex_univ"
] | A local maximum of a concave function is a global maximum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_extreme (A B : set E) : Prop | B ⊆ A ∧ ∀ ⦃x₁⦄, x₁ ∈ A → ∀ ⦃x₂⦄, x₂ ∈ A → ∀ ⦃x⦄, x ∈ B → x ∈ open_segment 𝕜 x₁ x₂ → x₁ ∈ B ∧ x₂ ∈ B | def | is_extreme | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"open_segment"
] | A set `B` is an extreme subset of `A` if `B ⊆ A` and all points of `B` only belong to open
segments whose ends are in `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.extreme_points (A : set E) : set E | {x ∈ A | ∀ ⦃x₁⦄, x₁ ∈ A → ∀ ⦃x₂⦄, x₂ ∈ A → x ∈ open_segment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x} | def | set.extreme_points | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"open_segment"
] | A point `x` is an extreme point of a set `A` if `x` belongs to no open segment with ends in
`A`, except for the obvious `open_segment x x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_extreme.refl (A : set E) :
is_extreme 𝕜 A A | ⟨subset.rfl, λ x₁ hx₁A x₂ hx₂A x hxA hx, ⟨hx₁A, hx₂A⟩⟩ | lemma | is_extreme.refl | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.rfl :
is_extreme 𝕜 A A | is_extreme.refl 𝕜 A | lemma | is_extreme.rfl | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme",
"is_extreme.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.trans (hAB : is_extreme 𝕜 A B) (hBC : is_extreme 𝕜 B C) :
is_extreme 𝕜 A C | begin
refine ⟨subset.trans hBC.1 hAB.1, λ x₁ hx₁A x₂ hx₂A x hxC hx, _⟩,
obtain ⟨hx₁B, hx₂B⟩ := hAB.2 hx₁A hx₂A (hBC.1 hxC) hx,
exact hBC.2 hx₁B hx₂B hxC hx,
end | lemma | is_extreme.trans | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.antisymm :
anti_symmetric (is_extreme 𝕜 : set E → set E → Prop) | λ A B hAB hBA, subset.antisymm hBA.1 hAB.1 | lemma | is_extreme.antisymm | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.inter (hAB : is_extreme 𝕜 A B) (hAC : is_extreme 𝕜 A C) :
is_extreme 𝕜 A (B ∩ C) | begin
use subset.trans (inter_subset_left _ _) hAB.1,
rintro x₁ hx₁A x₂ hx₂A x ⟨hxB, hxC⟩ hx,
obtain ⟨hx₁B, hx₂B⟩ := hAB.2 hx₁A hx₂A hxB hx,
obtain ⟨hx₁C, hx₂C⟩ := hAC.2 hx₁A hx₂A hxC hx,
exact ⟨⟨hx₁B, hx₁C⟩, hx₂B, hx₂C⟩,
end | lemma | is_extreme.inter | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.mono (hAC : is_extreme 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B) :
is_extreme 𝕜 B C | ⟨hCB, λ x₁ hx₁B x₂ hx₂B x hxC hx, hAC.2 (hBA hx₁B) (hBA hx₂B) hxC hx⟩ | lemma | is_extreme.mono | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme_Inter {ι : Sort*} [nonempty ι] {F : ι → set E}
(hAF : ∀ i : ι, is_extreme 𝕜 A (F i)) :
is_extreme 𝕜 A (⋂ i : ι, F i) | begin
obtain i := classical.arbitrary ι,
refine ⟨Inter_subset_of_subset i (hAF i).1, λ x₁ hx₁A x₂ hx₂A x hxF hx, _⟩,
simp_rw mem_Inter at ⊢ hxF,
have h := λ i, (hAF i).2 hx₁A hx₂A (hxF i) hx,
exact ⟨λ i, (h i).1, λ i, (h i).2⟩,
end | lemma | is_extreme_Inter | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"classical.arbitrary",
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme_bInter {F : set (set E)} (hF : F.nonempty) (hA : ∀ B ∈ F, is_extreme 𝕜 A B) :
is_extreme 𝕜 A (⋂ B ∈ F, B) | by { haveI := hF.to_subtype, simpa only [Inter_subtype] using is_extreme_Inter (λ i : F, hA _ i.2) } | lemma | is_extreme_bInter | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme",
"is_extreme_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme_sInter {F : set (set E)} (hF : F.nonempty)
(hAF : ∀ B ∈ F, is_extreme 𝕜 A B) :
is_extreme 𝕜 A (⋂₀ F) | begin
obtain ⟨B, hB⟩ := hF,
refine ⟨(sInter_subset_of_mem hB).trans (hAF B hB).1, λ x₁ hx₁A x₂ hx₂A x hxF hx, _⟩,
simp_rw mem_sInter at ⊢ hxF,
have h := λ B hB, (hAF B hB).2 hx₁A hx₂A (hxF B hB) hx,
exact ⟨λ B hB, (h B hB).1, λ B hB, (h B hB).2⟩,
end | lemma | is_extreme_sInter | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_extreme_points :
x ∈ A.extreme_points 𝕜 ↔ x ∈ A ∧ ∀ (x₁ x₂ ∈ A), x ∈ open_segment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x | iff.rfl | lemma | mem_extreme_points | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_extreme_points_iff_extreme_singleton :
x ∈ A.extreme_points 𝕜 ↔ is_extreme 𝕜 A {x} | begin
refine ⟨_, λ hx, ⟨singleton_subset_iff.1 hx.1, λ x₁ hx₁ x₂ hx₂, hx.2 hx₁ hx₂ rfl⟩⟩,
rintro ⟨hxA, hAx⟩,
use singleton_subset_iff.2 hxA,
rintro x₁ hx₁A x₂ hx₂A y (rfl : y = x),
exact hAx hx₁A hx₂A,
end | lemma | mem_extreme_points_iff_extreme_singleton | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | x is an extreme point to A iff {x} is an extreme set of A. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extreme_points_subset : A.extreme_points 𝕜 ⊆ A | λ x hx, hx.1 | lemma | extreme_points_subset | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extreme_points_empty :
(∅ : set E).extreme_points 𝕜 = ∅ | subset_empty_iff.1 extreme_points_subset | lemma | extreme_points_empty | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"extreme_points_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extreme_points_singleton :
({x} : set E).extreme_points 𝕜 = {x} | extreme_points_subset.antisymm $ singleton_subset_iff.2
⟨mem_singleton x, λ x₁ hx₁ x₂ hx₂ _, ⟨hx₁, hx₂⟩⟩ | lemma | extreme_points_singleton | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_extreme_points_subset_extreme_points_of_subset (hBA : B ⊆ A) :
B ∩ A.extreme_points 𝕜 ⊆ B.extreme_points 𝕜 | λ x ⟨hxB, hxA⟩, ⟨hxB, λ x₁ hx₁ x₂ hx₂ hx, hxA.2 (hBA hx₁) (hBA hx₂) hx⟩ | lemma | inter_extreme_points_subset_extreme_points_of_subset | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.extreme_points_subset_extreme_points (hAB : is_extreme 𝕜 A B) :
B.extreme_points 𝕜 ⊆ A.extreme_points 𝕜 | λ x hx, mem_extreme_points_iff_extreme_singleton.2 (hAB.trans
(mem_extreme_points_iff_extreme_singleton.1 hx)) | lemma | is_extreme.extreme_points_subset_extreme_points | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.extreme_points_eq (hAB : is_extreme 𝕜 A B) :
B.extreme_points 𝕜 = B ∩ A.extreme_points 𝕜 | subset.antisymm (λ x hx, ⟨hx.1, hAB.extreme_points_subset_extreme_points hx⟩)
(inter_extreme_points_subset_extreme_points_of_subset hAB.1) | lemma | is_extreme.extreme_points_eq | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"inter_extreme_points_subset_extreme_points_of_subset",
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extreme.convex_diff (hA : convex 𝕜 A) (hAB : is_extreme 𝕜 A B) :
convex 𝕜 (A \ B) | convex_iff_open_segment_subset.2 (λ x₁ ⟨hx₁A, hx₁B⟩ x₂ ⟨hx₂A, hx₂B⟩ x hx,
⟨hA.open_segment_subset hx₁A hx₂A hx, λ hxB, hx₁B (hAB.2 hx₁A hx₂A hxB hx).1⟩) | lemma | is_extreme.convex_diff | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"convex",
"is_extreme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extreme_points_prod (s : set E) (t : set F) :
(s ×ˢ t).extreme_points 𝕜 = s.extreme_points 𝕜 ×ˢ t.extreme_points 𝕜 | begin
ext,
refine (and_congr_right $ λ hx, ⟨λ h, _, λ h, _⟩).trans (and_and_and_comm _ _ _ _),
split,
{ rintro x₁ hx₁ x₂ hx₂ hx_fst,
refine (h (mk_mem_prod hx₁ hx.2) (mk_mem_prod hx₂ hx.2) _).imp
(congr_arg prod.fst) (congr_arg prod.fst),
rw ←prod.image_mk_open_segment_left,
exact ⟨_, hx_fst, ... | lemma | extreme_points_prod | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"and_and_and_comm",
"prod.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extreme_points_pi (s : Π i, set (π i)) :
(univ.pi s).extreme_points 𝕜 = univ.pi (λ i, (s i).extreme_points 𝕜) | begin
ext,
simp only [mem_extreme_points, mem_pi, mem_univ, true_implies_iff, @forall_and_distrib ι],
refine and_congr_right (λ hx, ⟨λ h i, _, λ h, _⟩),
{ rintro x₁ hx₁ x₂ hx₂ hi,
refine (h (update x i x₁) _ (update x i x₂) _ _).imp (λ h₁, by rw [←h₁, update_same])
(λ h₂, by rw [←h₂, update_same]),
... | lemma | extreme_points_pi | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"eq_or_ne",
"forall_and_distrib",
"mem_extreme_points",
"update",
"update_eq_self",
"update_noteq",
"update_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_extreme_points_iff_forall_segment :
x ∈ A.extreme_points 𝕜 ↔ x ∈ A ∧ ∀ (x₁ x₂ ∈ A), x ∈ segment 𝕜 x₁ x₂ → x₁ = x ∨ x₂ = x | begin
refine and_congr_right (λ hxA, forall₄_congr $ λ x₁ h₁ x₂ h₂, _),
split,
{ rw ← insert_endpoints_open_segment,
rintro H (rfl|rfl|hx),
exacts [or.inl rfl, or.inr rfl, or.inl $ (H hx).1] },
{ intros H hx,
rcases H (open_segment_subset_segment _ _ _ hx) with rfl | rfl,
exacts [⟨rfl, (left_mem... | lemma | mem_extreme_points_iff_forall_segment | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"forall₄_congr",
"insert_endpoints_open_segment",
"open_segment_subset_segment",
"segment"
] | A useful restatement using `segment`: `x` is an extreme point iff the only (closed) segments
that contain it are those with `x` as one of their endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mem_extreme_points_iff_convex_diff (hA : convex 𝕜 A) :
x ∈ A.extreme_points 𝕜 ↔ x ∈ A ∧ convex 𝕜 (A \ {x}) | begin
use λ hx, ⟨hx.1, (mem_extreme_points_iff_extreme_singleton.1 hx).convex_diff hA⟩,
rintro ⟨hxA, hAx⟩,
refine mem_extreme_points_iff_forall_segment.2 ⟨hxA, λ x₁ hx₁ x₂ hx₂ hx, _⟩,
rw convex_iff_segment_subset at hAx,
by_contra' h,
exact (hAx ⟨hx₁, λ hx₁, h.1 (mem_singleton_iff.2 hx₁)⟩
⟨hx₂, λ hx₂, h... | lemma | convex.mem_extreme_points_iff_convex_diff | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"convex",
"convex_iff_segment_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.mem_extreme_points_iff_mem_diff_convex_hull_diff (hA : convex 𝕜 A) :
x ∈ A.extreme_points 𝕜 ↔ x ∈ A \ convex_hull 𝕜 (A \ {x}) | by rw [hA.mem_extreme_points_iff_convex_diff, hA.convex_remove_iff_not_mem_convex_hull_remove,
mem_diff] | lemma | convex.mem_extreme_points_iff_mem_diff_convex_hull_diff | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"convex",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extreme_points_convex_hull_subset :
(convex_hull 𝕜 A).extreme_points 𝕜 ⊆ A | begin
rintro x hx,
rw (convex_convex_hull 𝕜 _).mem_extreme_points_iff_convex_diff at hx,
by_contra,
exact (convex_hull_min (subset_diff.2 ⟨subset_convex_hull 𝕜 _, disjoint_singleton_right.2 h⟩) hx.2
hx.1).2 rfl,
apply_instance
end | lemma | extreme_points_convex_hull_subset | analysis.convex | src/analysis/convex/extreme.lean | [
"analysis.convex.hull"
] | [
"by_contra",
"convex_convex_hull",
"convex_hull",
"convex_hull_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on : Prop | convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
f (a • x + b • y) ≤ a • f x + b • f y | def | convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex"
] | Convexity of functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on : Prop | convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
a • f x + b • f y ≤ f (a • x + b • y) | def | concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex"
] | Concavity of functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on : Prop | convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
f (a • x + b • y) < a • f x + b • f y | def | strict_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex"
] | Strict convexity of functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on : Prop | convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
a • f x + b • f y < f (a • x + b • y) | def | strict_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex"
] | Strict concavity of functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.dual (hf : convex_on 𝕜 s f) : concave_on 𝕜 s (to_dual ∘ f) | hf | lemma | convex_on.dual | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.dual (hf : concave_on 𝕜 s f) : convex_on 𝕜 s (to_dual ∘ f) | hf | lemma | concave_on.dual | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.dual (hf : strict_convex_on 𝕜 s f) : strict_concave_on 𝕜 s (to_dual ∘ f) | hf | lemma | strict_convex_on.dual | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.dual (hf : strict_concave_on 𝕜 s f) : strict_convex_on 𝕜 s (to_dual ∘ f) | hf | lemma | strict_concave_on.dual | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_id {s : set β} (hs : convex 𝕜 s) : convex_on 𝕜 s id | ⟨hs, by { intros, refl }⟩ | lemma | convex_on_id | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.