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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to_convex_cone_bot : (⊥ : submodule 𝕜 E).to_convex_cone = 0 | rfl | lemma | submodule.to_convex_cone_bot | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_convex_cone_top : (⊤ : submodule 𝕜 E).to_convex_cone = ⊤ | rfl | lemma | submodule.to_convex_cone_top | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_convex_cone_inf (S T : submodule 𝕜 E) :
(S ⊓ T).to_convex_cone = S.to_convex_cone ⊓ T.to_convex_cone | rfl | lemma | submodule.to_convex_cone_inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointed_to_convex_cone (S : submodule 𝕜 E) : S.to_convex_cone.pointed | S.zero_mem | lemma | submodule.pointed_to_convex_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positive : convex_cone 𝕜 E | { carrier := set.Ici 0,
smul_mem' := λ c hc x (hx : _ ≤ _), smul_nonneg hc.le hx,
add_mem' := λ x (hx : _ ≤ _) y (hy : _ ≤ _), add_nonneg hx hy } | def | convex_cone.positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"set.Ici",
"smul_nonneg"
] | The positive cone is the convex cone formed by the set of nonnegative elements in an ordered
module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_positive {x : E} : x ∈ positive 𝕜 E ↔ 0 ≤ x | iff.rfl | lemma | convex_cone.mem_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_positive : ↑(positive 𝕜 E) = set.Ici (0 : E) | rfl | lemma | convex_cone.coe_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
salient_positive : salient (positive 𝕜 E) | λ x xs hx hx', lt_irrefl (0 : E)
(calc
0 < x : lt_of_le_of_ne xs hx.symm
... ≤ x + (-x) : le_add_of_nonneg_right hx'
... = 0 : add_neg_self x) | lemma | convex_cone.salient_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | The positive cone of an ordered module is always salient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pointed_positive : pointed (positive 𝕜 E) | le_refl 0 | lemma | convex_cone.pointed_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | The positive cone of an ordered module is always pointed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strictly_positive : convex_cone 𝕜 E | { carrier := set.Ioi 0,
smul_mem' := λ c hc x (hx : _ < _), smul_pos hc hx,
add_mem' := λ x hx y hy, add_pos hx hy } | def | convex_cone.strictly_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"set.Ioi"
] | The cone of strictly positive elements.
Note that this naming diverges from the mathlib convention of `pos` and `nonneg` due to "positive
cone" (`convex_cone.positive`) being established terminology for the non-negative elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_strictly_positive {x : E} : x ∈ strictly_positive 𝕜 E ↔ 0 < x | iff.rfl | lemma | convex_cone.mem_strictly_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_strictly_positive : ↑(strictly_positive 𝕜 E) = set.Ioi (0 : E) | rfl | lemma | convex_cone.coe_strictly_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"set.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positive_le_strictly_positive : strictly_positive 𝕜 E ≤ positive 𝕜 E | λ x, le_of_lt | lemma | convex_cone.positive_le_strictly_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
salient_strictly_positive : salient (strictly_positive 𝕜 E) | (salient_positive 𝕜 E).anti $ positive_le_strictly_positive 𝕜 E | lemma | convex_cone.salient_strictly_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | The strictly positive cone of an ordered module is always salient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
blunt_strictly_positive : blunt (strictly_positive 𝕜 E) | lt_irrefl 0 | lemma | convex_cone.blunt_strictly_positive | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | The strictly positive cone of an ordered module is always blunt. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cone (s : set E) (hs : convex 𝕜 s) : convex_cone 𝕜 E | begin
apply convex_cone.mk (⋃ (c : 𝕜) (H : 0 < c), c • s);
simp only [mem_Union, mem_smul_set],
{ rintros c c_pos _ ⟨c', c'_pos, x, hx, rfl⟩,
exact ⟨c * c', mul_pos c_pos c'_pos, x, hx, (smul_smul _ _ _).symm⟩ },
{ rintros _ ⟨cx, cx_pos, x, hx, rfl⟩ _ ⟨cy, cy_pos, y, hy, rfl⟩,
have : 0 < cx + cy, fro... | def | convex.to_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex",
"convex_cone",
"mul_div_assoc'",
"mul_div_cancel_left",
"smul_add",
"smul_smul"
] | The set of vectors proportional to those in a convex set forms a convex cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_cone : x ∈ hs.to_cone s ↔ ∃ (c : 𝕜), 0 < c ∧ ∃ y ∈ s, c • y = x | by simp only [to_cone, convex_cone.mem_mk, mem_Union, mem_smul_set, eq_comm, exists_prop] | lemma | convex.mem_to_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone.mem_mk",
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_cone' : x ∈ hs.to_cone s ↔ ∃ (c : 𝕜), 0 < c ∧ c • x ∈ s | begin
refine hs.mem_to_cone.trans ⟨_, _⟩,
{ rintros ⟨c, hc, y, hy, rfl⟩,
exact ⟨c⁻¹, inv_pos.2 hc, by rwa [smul_smul, inv_mul_cancel hc.ne', one_smul]⟩ },
{ rintros ⟨c, hc, hcx⟩,
exact ⟨c⁻¹, inv_pos.2 hc, _, hcx, by rw [smul_smul, inv_mul_cancel hc.ne', one_smul]⟩ }
end | lemma | convex.mem_to_cone' | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"inv_mul_cancel",
"one_smul",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_to_cone : s ⊆ hs.to_cone s | λ x hx, hs.mem_to_cone'.2 ⟨1, zero_lt_one, by rwa one_smul⟩ | lemma | convex.subset_to_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cone_is_least : is_least { t : convex_cone 𝕜 E | s ⊆ t } (hs.to_cone s) | begin
refine ⟨hs.subset_to_cone, λ t ht x hx, _⟩,
rcases hs.mem_to_cone.1 hx with ⟨c, hc, y, hy, rfl⟩,
exact t.smul_mem hc (ht hy)
end | lemma | convex.to_cone_is_least | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"is_least"
] | `hs.to_cone s` is the least cone that includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cone_eq_Inf : hs.to_cone s = Inf { t : convex_cone 𝕜 E | s ⊆ t } | hs.to_cone_is_least.is_glb.Inf_eq.symm | lemma | convex.to_cone_eq_Inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_to_cone_is_least (s : set E) :
is_least {t : convex_cone 𝕜 E | s ⊆ t} ((convex_convex_hull 𝕜 s).to_cone _) | begin
convert (convex_convex_hull 𝕜 s).to_cone_is_least,
ext t,
exact ⟨λ h, convex_hull_min h t.convex, (subset_convex_hull 𝕜 s).trans⟩,
end | lemma | convex_hull_to_cone_is_least | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"convex_convex_hull",
"convex_hull_min",
"is_least",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_to_cone_eq_Inf (s : set E) :
(convex_convex_hull 𝕜 s).to_cone _ = Inf {t : convex_cone 𝕜 E | s ⊆ t} | eq.symm $ is_glb.Inf_eq $ is_least.is_glb $ convex_hull_to_cone_is_least s | lemma | convex_hull_to_cone_eq_Inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"convex_convex_hull",
"convex_hull_to_cone_is_least",
"is_glb.Inf_eq",
"is_least.is_glb"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x | begin
obtain ⟨y, -, hy⟩ : ∃ (y : E) (h : y ∈ ⊤), y ∉ f.domain,
{ exact @set_like.exists_of_lt (submodule ℝ E) _ _ _ _ (lt_top_iff_ne_top.2 hdom) },
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x:E) - y ∈ s → f x ≤ c) ∧ (∀ x : f.domain, (x:E) + y ∈ s → c ≤ f x),
{ set Sp := f '' {x : f.domain | (x:E) +... | lemma | riesz_extension.step | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"dense",
"exists_between_of_forall_le",
"linear_pmap.domain_sup_span_singleton",
"linear_pmap.sup_span_singleton_apply_mk",
"lower_bounds",
"mul_assoc",
"mul_inv_cancel",
"mul_le_mul_left",
"neg_mul",
"neg_smul",
"one_mul",
"one_smul",
"set.nonempty",
"set_like.exists_of_lt",
"smul_add",... | Induction step in M. Riesz extension theorem. Given a convex cone `s` in a vector space `E`,
a partially defined linear map `f : f.domain → ℝ`, assume that `f` is nonnegative on `f.domain ∩ p`
and `p + s = E`. If `f` is not defined on the whole `E`, then we can extend it to a larger
submodule without breaking the non-n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_top (p : E →ₗ.[ℝ] ℝ)
(hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
(hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x | begin
replace hp_nonneg : p ∈ { p | _ }, by { rw mem_set_of_eq, exact hp_nonneg },
obtain ⟨q, hqs, hpq, hq⟩ := zorn_nonempty_partial_order₀ _ _ _ hp_nonneg,
{ refine ⟨q, hpq, _, hqs⟩,
contrapose! hq,
rcases step s q hqs _ hq with ⟨r, hqr, hr⟩,
{ exact ⟨r, hr, hqr.le, hqr.ne'⟩ },
{ exact λ y, let ⟨... | theorem | riesz_extension.exists_top | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"directed_on",
"linear_pmap.Sup",
"linear_pmap.le_Sup",
"zorn_nonempty_partial_order₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
riesz_extension (s : convex_cone ℝ E) (f : E →ₗ.[ℝ] ℝ)
(nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x) (dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) :
∃ g : E →ₗ[ℝ] ℝ, (∀ x : f.domain, g x = f x) ∧ (∀ x ∈ s, 0 ≤ g x) | begin
rcases riesz_extension.exists_top s f nonneg dense with ⟨⟨g_dom, g⟩, ⟨hpg, hfg⟩, htop, hgs⟩,
clear hpg,
refine ⟨g ∘ₗ ↑(linear_equiv.of_top _ htop).symm, _, _⟩;
simp only [comp_apply, linear_equiv.coe_coe, linear_equiv.of_top_symm_apply],
{ exact λ x, (hfg (submodule.coe_mk _ _).symm).symm },
{ exact... | theorem | riesz_extension | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"dense",
"linear_equiv.coe_coe",
"linear_equiv.of_top",
"linear_equiv.of_top_symm_apply",
"riesz_extension.exists_top",
"submodule.coe_mk"
] | M. **Riesz extension theorem**: given a convex cone `s` in a vector space `E`, a submodule `p`,
and a linear `f : p → ℝ`, assume that `f` is nonnegative on `p ∩ s` and `p + s = E`. Then
there exists a globally defined linear function `g : E → ℝ` that agrees with `f` on `p`,
and is nonnegative on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_of_le_sublinear (f : E →ₗ.[ℝ] ℝ) (N : E → ℝ)
(N_hom : ∀ (c : ℝ), 0 < c → ∀ x, N (c • x) = c * N x)
(N_add : ∀ x y, N (x + y) ≤ N x + N y)
(hf : ∀ x : f.domain, f x ≤ N x) :
∃ g : E →ₗ[ℝ] ℝ, (∀ x : f.domain, g x = f x) ∧ (∀ x, g x ≤ N x) | begin
let s : convex_cone ℝ (E × ℝ) :=
{ carrier := {p : E × ℝ | N p.1 ≤ p.2 },
smul_mem' := λ c hc p hp,
calc N (c • p.1) = c * N p.1 : N_hom c hc p.1
... ≤ c * p.2 : mul_le_mul_of_nonneg_left hp hc.le,
add_mem' := λ x hx y hy, (N_add _ _).trans (add_le_add hx hy) },
obtain ⟨g, g_eq, g_nonneg... | theorem | exists_extension_of_le_sublinear | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"convex_cone.mem_mk",
"linear_pmap.coprod_apply",
"linear_pmap.neg_apply",
"mul_le_mul_of_nonneg_left",
"riesz_extension",
"submodule.coe_zero",
"subtype.coe_eta",
"subtype.coe_mk"
] | **Hahn-Banach theorem**: if `N : E → ℝ` is a sublinear map, `f` is a linear map
defined on a subspace of `E`, and `f x ≤ N x` for all `x` in the domain of `f`,
then `f` can be extended to the whole space to a linear map `g` such that `g x ≤ N x`
for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.inner_dual_cone (s : set H) : convex_cone ℝ H | { carrier := { y | ∀ x ∈ s, 0 ≤ ⟪ x, y ⟫ },
smul_mem' := λ c hc y hy x hx,
begin
rw real_inner_smul_right,
exact mul_nonneg hc.le (hy x hx)
end,
add_mem' := λ u hu v hv x hx,
begin
rw inner_add_right,
exact add_nonneg (hu x hx) (hv x hx)
end } | def | set.inner_dual_cone | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"convex_cone",
"inner_add_right",
"real_inner_smul_right"
] | The dual cone is the cone consisting of all points `y` such that for
all points `x` in a given set `0 ≤ ⟪ x, y ⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_inner_dual_cone (y : H) (s : set H) :
y ∈ s.inner_dual_cone ↔ ∀ x ∈ s, 0 ≤ ⟪ x, y ⟫ | iff.rfl | lemma | mem_inner_dual_cone | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_empty : (∅ : set H).inner_dual_cone = ⊤ | eq_top_iff.mpr $ λ x hy y, false.elim | lemma | inner_dual_cone_empty | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_zero : (0 : set H).inner_dual_cone = ⊤ | eq_top_iff.mpr $ λ x hy y (hy : y = 0), hy.symm ▸ (inner_zero_left _).ge | lemma | inner_dual_cone_zero | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"inner_zero_left"
] | Dual cone of the convex cone {0} is the total space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_dual_cone_univ : (univ : set H).inner_dual_cone = 0 | begin
suffices : ∀ x : H, x ∈ (univ : set H).inner_dual_cone → x = 0,
{ apply set_like.coe_injective,
exact eq_singleton_iff_unique_mem.mpr ⟨λ x hx, (inner_zero_right _).ge, this⟩ },
exact λ x hx, by simpa [←real_inner_self_nonpos] using hx (-x) (mem_univ _),
end | lemma | inner_dual_cone_univ | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"inner_zero_right",
"set_like.coe_injective"
] | Dual cone of the total space is the convex cone {0}. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_dual_cone_le_inner_dual_cone (h : t ⊆ s) :
s.inner_dual_cone ≤ t.inner_dual_cone | λ y hy x hx, hy x (h hx) | lemma | inner_dual_cone_le_inner_dual_cone | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointed_inner_dual_cone : s.inner_dual_cone.pointed | λ x hx, by rw inner_zero_right | lemma | pointed_inner_dual_cone | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"inner_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_singleton (x : H) :
({x} : set H).inner_dual_cone = (convex_cone.positive ℝ ℝ).comap (innerₛₗ ℝ x) | convex_cone.ext $ λ i, forall_eq | lemma | inner_dual_cone_singleton | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"convex_cone.ext",
"convex_cone.positive",
"forall_eq",
"innerₛₗ"
] | The inner dual cone of a singleton is given by the preimage of the positive cone under the
linear map `λ y, ⟪x, y⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_dual_cone_union (s t : set H) :
(s ∪ t).inner_dual_cone = s.inner_dual_cone ⊓ t.inner_dual_cone | le_antisymm
(le_inf (λ x hx y hy, hx _ $ or.inl hy) (λ x hx y hy, hx _ $ or.inr hy))
(λ x hx y, or.rec (hx.1 _) (hx.2 _)) | lemma | inner_dual_cone_union | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_insert (x : H) (s : set H) :
(insert x s).inner_dual_cone = set.inner_dual_cone {x} ⊓ s.inner_dual_cone | by rw [insert_eq, inner_dual_cone_union] | lemma | inner_dual_cone_insert | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"inner_dual_cone_union",
"set.inner_dual_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_Union {ι : Sort*} (f : ι → set H) :
(⋃ i, f i).inner_dual_cone = ⨅ i, (f i).inner_dual_cone | begin
refine le_antisymm (le_infi $ λ i x hx y hy, hx _ $ mem_Union_of_mem _ hy) _,
intros x hx y hy,
rw [convex_cone.mem_infi] at hx,
obtain ⟨j, hj⟩ := mem_Union.mp hy,
exact hx _ _ hj,
end | lemma | inner_dual_cone_Union | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"convex_cone.mem_infi",
"le_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_sUnion (S : set (set H)) :
(⋃₀ S).inner_dual_cone = Inf (set.inner_dual_cone '' S) | by simp_rw [Inf_image, sUnion_eq_bUnion, inner_dual_cone_Union] | lemma | inner_dual_cone_sUnion | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"Inf_image",
"inner_dual_cone_Union",
"set.inner_dual_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_dual_cone_eq_Inter_inner_dual_cone_singleton :
(s.inner_dual_cone : set H) = ⋂ i : s, (({i} : set H).inner_dual_cone : set H) | by rw [←convex_cone.coe_infi, ←inner_dual_cone_Union, Union_of_singleton_coe] | lemma | inner_dual_cone_eq_Inter_inner_dual_cone_singleton | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [] | The dual cone of `s` equals the intersection of dual cones of the points in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_inner_dual_cone : is_closed (s.inner_dual_cone : set H) | begin
-- reduce the problem to showing that dual cone of a singleton `{x}` is closed
rw inner_dual_cone_eq_Inter_inner_dual_cone_singleton,
apply is_closed_Inter,
intros x,
-- the dual cone of a singleton `{x}` is the preimage of `[0, ∞)` under `inner x`
have h : ↑({x} : set H).inner_dual_cone = (inner x :... | lemma | is_closed_inner_dual_cone | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"continuity",
"convex_cone.coe_comap",
"convex_cone.coe_positive",
"inner_dual_cone_eq_Inter_inner_dual_cone_singleton",
"inner_dual_cone_singleton",
"innerₛₗ_apply_coe",
"is_closed",
"is_closed_Inter",
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_cone.pointed_of_nonempty_of_is_closed (K : convex_cone ℝ H)
(ne : (K : set H).nonempty) (hc : is_closed (K : set H)) : K.pointed | begin
obtain ⟨x, hx⟩ := ne,
let f : ℝ → H := (• x),
-- f (0, ∞) is a subset of K
have fI : f '' set.Ioi 0 ⊆ (K : set H),
{ rintro _ ⟨_, h, rfl⟩,
exact K.smul_mem (set.mem_Ioi.1 h) hx },
-- closure of f (0, ∞) is a subset of K
have clf : closure (f '' set.Ioi 0) ⊆ (K : set H) := hc.closure_subset_iff... | lemma | convex_cone.pointed_of_nonempty_of_is_closed | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"closure",
"closure_Ioi",
"continuous_const",
"continuous_within_at",
"convex_cone",
"convex_cone.pointed",
"is_closed",
"set.Ioi",
"set_like.mem_coe",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_cone.hyperplane_separation_of_nonempty_of_is_closed_of_nmem (K : convex_cone ℝ H)
(ne : (K : set H).nonempty) (hc : is_closed (K : set H)) {b : H} (disj : b ∉ K) :
∃ (y : H), (∀ x : H, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ) ∧ ⟪y, b⟫_ℝ < 0 | begin
-- let `z` be the point in `K` closest to `b`
obtain ⟨z, hzK, infi⟩ := exists_norm_eq_infi_of_complete_convex ne hc.is_complete K.convex b,
-- for any `w` in `K`, we have `⟪b - z, w - z⟫_ℝ ≤ 0`
have hinner := (norm_eq_infi_iff_real_inner_le_zero K.convex hzK).1 infi,
-- set `y := z - b`
use z - b,
... | theorem | convex_cone.hyperplane_separation_of_nonempty_of_is_closed_of_nmem | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"convex_cone",
"exists_norm_eq_infi_of_complete_convex",
"iff.not",
"infi",
"inner_add_right",
"inner_neg_right",
"is_closed",
"lt_of_not_le",
"neg_eq_neg_one_mul",
"neg_one_mul",
"neg_smul",
"norm_eq_infi_iff_real_inner_le_zero",
"one_smul",
"real_inner_comm",
"real_inner_self_nonpos",
... | This is a stronger version of the Hahn-Banach separation theorem for closed convex cones. This
is also the geometric interpretation of Farkas' lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_cone.inner_dual_cone_of_inner_dual_cone_eq_self (K : convex_cone ℝ H)
(ne : (K : set H).nonempty) (hc : is_closed (K : set H)) :
((K : set H).inner_dual_cone : set H).inner_dual_cone = K | begin
ext x,
split,
{ rw [mem_inner_dual_cone, ← set_like.mem_coe],
contrapose!,
exact K.hyperplane_separation_of_nonempty_of_is_closed_of_nmem ne hc },
{ rintro hxK y h,
specialize h x hxK,
rwa real_inner_comm },
end | theorem | convex_cone.inner_dual_cone_of_inner_dual_cone_eq_self | analysis.convex.cone | src/analysis/convex/cone/dual.lean | [
"analysis.convex.cone.basic",
"analysis.inner_product_space.projection"
] | [
"convex_cone",
"is_closed",
"mem_inner_dual_cone",
"real_inner_comm",
"set_like.mem_coe"
] | The inner dual of inner dual of a non-empty, closed convex cone is itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure (K : convex_cone 𝕜 E) : convex_cone 𝕜 E | { carrier := closure ↑K,
smul_mem' :=
λ c hc _ h₁, map_mem_closure (continuous_id'.const_smul c) h₁ (λ _ h₂, K.smul_mem hc h₂),
add_mem' := λ _ h₁ _ h₂, map_mem_closure₂ continuous_add h₁ h₂ K.add_mem } | def | convex_cone.closure | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"closure",
"convex_cone",
"map_mem_closure",
"map_mem_closure₂"
] | The closure of a convex cone inside a topological space as a convex cone. This
construction is mainly used for defining maps between proper cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_closure (K : convex_cone 𝕜 E) : (K.closure : set E) = closure K | rfl | lemma | convex_cone.coe_closure | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"closure",
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure {K : convex_cone 𝕜 E} {a : E} :
a ∈ K.closure ↔ a ∈ closure (K : set E) | iff.rfl | lemma | convex_cone.mem_closure | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"closure",
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq {K L : convex_cone 𝕜 E} : K.closure = L ↔ closure (K : set E) = L | set_like.ext'_iff | lemma | convex_cone.closure_eq | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"closure",
"convex_cone",
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proper_cone (𝕜 : Type*) (E : Type*)
[ordered_semiring 𝕜] [add_comm_monoid E] [topological_space E] [has_smul 𝕜 E]
extends convex_cone 𝕜 E | (nonempty' : (carrier : set E).nonempty)
(is_closed' : is_closed (carrier : set E)) | structure | proper_cone | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"add_comm_monoid",
"convex_cone",
"has_smul",
"is_closed",
"ordered_semiring",
"topological_space"
] | A proper cone is a convex cone `K` that is nonempty and closed. Proper cones have the nice
property that the dual of the dual of a proper cone is itself. This makes them useful for defining
cone programs and proving duality theorems. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_convex_cone_eq_coe (K : proper_cone 𝕜 E) : K.to_convex_cone = K | rfl | lemma | proper_cone.to_convex_cone_eq_coe | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext' : function.injective (coe : proper_cone 𝕜 E → convex_cone 𝕜 E) | λ S T h, by cases S; cases T; congr' | lemma | proper_cone.ext' | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {S T : proper_cone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | lemma | proper_cone.ext | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone",
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe {x : E} {K : proper_cone 𝕜 E} : x ∈ (K : convex_cone 𝕜 E) ↔ x ∈ K | iff.rfl | lemma | proper_cone.mem_coe | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty (K : proper_cone 𝕜 E) : (K : set E).nonempty | K.nonempty' | lemma | proper_cone.nonempty | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed (K : proper_cone 𝕜 E) : is_closed (K : set E) | K.is_closed' | lemma | proper_cone.is_closed | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"is_closed",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_zero (x : E) : x ∈ (0 : proper_cone 𝕜 E) ↔ x = 0 | iff.rfl | lemma | proper_cone.mem_zero | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ↑(0 : proper_cone 𝕜 E) = (0 : convex_cone 𝕜 E) | rfl | lemma | proper_cone.coe_zero | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointed_zero : (0 : proper_cone 𝕜 E).pointed | by simp [convex_cone.pointed_zero] | lemma | proper_cone.pointed_zero | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone.pointed_zero",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointed (K : proper_cone ℝ E) : (K : convex_cone ℝ E).pointed | (K : convex_cone ℝ E).pointed_of_nonempty_of_is_closed K.nonempty K.is_closed | lemma | proper_cone.pointed | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : E →L[ℝ] F) (K : proper_cone ℝ E) : proper_cone ℝ F | { to_convex_cone := convex_cone.closure (convex_cone.map (f : E →ₗ[ℝ] F) ↑K),
nonempty' := ⟨ 0, subset_closure $ set_like.mem_coe.2 $ convex_cone.mem_map.2
⟨0, K.pointed, map_zero _⟩ ⟩,
is_closed' := is_closed_closure } | def | proper_cone.map | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone.closure",
"convex_cone.map",
"is_closed_closure",
"proper_cone",
"subset_closure"
] | The closure of image of a proper cone under a continuous `ℝ`-linear map is a proper cone. We
use continuous maps here so that the comap of f is also a map between proper cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (f : E →L[ℝ] F) (K : proper_cone ℝ E) :
↑(K.map f) = (convex_cone.map (f : E →ₗ[ℝ] F) ↑K).closure | rfl | lemma | proper_cone.coe_map | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"closure",
"convex_cone.map",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {f : E →L[ℝ] F} {K : proper_cone ℝ E} {y : F} :
y ∈ K.map f ↔ y ∈ (convex_cone.map (f : E →ₗ[ℝ] F) ↑K).closure | iff.rfl | lemma | proper_cone.mem_map | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"closure",
"convex_cone.map",
"mem_map",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (K : proper_cone ℝ E) : K.map (continuous_linear_map.id ℝ E) = K | proper_cone.ext' $ by simpa using is_closed.closure_eq K.is_closed | lemma | proper_cone.map_id | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"continuous_linear_map.id",
"is_closed.closure_eq",
"map_id",
"proper_cone",
"proper_cone.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual (K : proper_cone ℝ E): (proper_cone ℝ E) | { to_convex_cone := (K : set E).inner_dual_cone,
nonempty' := ⟨0, pointed_inner_dual_cone _⟩,
is_closed' := is_closed_inner_dual_cone _ } | def | proper_cone.dual | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"is_closed_inner_dual_cone",
"pointed_inner_dual_cone",
"proper_cone"
] | The inner dual cone of a proper cone is a proper cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_dual (K : proper_cone ℝ E) : ↑(dual K) = (K : set E).inner_dual_cone | rfl | lemma | proper_cone.coe_dual | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_dual {K : proper_cone ℝ E} {y : E} :
y ∈ dual K ↔ ∀ ⦃x⦄, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ | by {rw [← mem_coe, coe_dual, mem_inner_dual_cone _ _], refl} | lemma | proper_cone.mem_dual | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"mem_inner_dual_cone",
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : E →L[ℝ] F) (S : proper_cone ℝ F) : proper_cone ℝ E | { to_convex_cone := convex_cone.comap (f : E →ₗ[ℝ] F) S,
nonempty' := ⟨ 0,
begin
simp only [convex_cone.comap, mem_preimage, map_zero, set_like.mem_coe, mem_coe],
apply proper_cone.pointed,
end ⟩,
is_closed' :=
begin
simp only [convex_cone.comap, continuous_linear_map.coe_coe],
apply is_closed... | def | proper_cone.comap | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"continuous_linear_map.coe_coe",
"convex_cone.comap",
"is_closed.preimage",
"proper_cone",
"proper_cone.pointed",
"set_like.mem_coe"
] | The preimage of a proper cone under a continuous `ℝ`-linear map is a proper cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (f : E →L[ℝ] F) (S : proper_cone ℝ F) : (S.comap f : set E) = f ⁻¹' S | rfl | lemma | proper_cone.coe_comap | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id (S : convex_cone ℝ E) : S.comap linear_map.id = S | set_like.coe_injective preimage_id | lemma | proper_cone.comap_id | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone",
"linear_map.id",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (g : F →L[ℝ] G) (f : E →L[ℝ] F) (S : proper_cone ℝ G) :
(S.comap g).comap f = S.comap (g.comp f) | set_like.coe_injective $ preimage_comp.symm | lemma | proper_cone.comap_comap | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {f : E →L[ℝ] F} {S : proper_cone ℝ F} {x : E} : x ∈ S.comap f ↔ f x ∈ S | iff.rfl | lemma | proper_cone.mem_comap | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_dual (K : proper_cone ℝ E) : K.dual.dual = K | proper_cone.ext' $
(K : convex_cone ℝ E).inner_dual_cone_of_inner_dual_cone_eq_self K.nonempty K.is_closed | theorem | proper_cone.dual_dual | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"convex_cone",
"proper_cone",
"proper_cone.ext'"
] | The dual of the dual of a proper cone is itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hyperplane_separation (K : proper_cone ℝ E) {f : E →L[ℝ] F} {b : F} :
b ∈ K.map f ↔ ∀ y : F, (adjoint f y) ∈ K.dual → 0 ≤ ⟪y, b⟫_ℝ | iff.intro
begin
-- suppose `b ∈ K.map f`
simp only [proper_cone.mem_map, proper_cone.mem_dual, adjoint_inner_right,
convex_cone.mem_closure, mem_closure_iff_seq_limit],
-- there is a sequence `seq : ℕ → F` in the image of `f` that converges to `b`
rintros ⟨seq, hmem, htends⟩ y hinner,
suffices h : ∀ n, ... | theorem | proper_cone.hyperplane_separation | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"continuous.inner",
"continuous.seq_continuous",
"continuous_const",
"continuous_id",
"convex_cone.hyperplane_separation_of_nonempty_of_is_closed_of_nmem",
"convex_cone.mem_closure",
"convex_cone.mem_map",
"ge_of_tendsto'",
"is_closed",
"mem_closure_iff_seq_limit",
"proper_cone",
"proper_cone.... | This is a relative version of
`convex_cone.hyperplane_separation_of_nonempty_of_is_closed_of_nmem`, which we recover by setting
`f` to be the identity map. This is a geometric interpretation of the Farkas' lemma
stated using proper cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hyperplane_separation_of_nmem (K : proper_cone ℝ E) {f : E →L[ℝ] F} {b : F}
(disj : b ∉ K.map f) : ∃ y : F, (adjoint f y) ∈ K.dual ∧ ⟪y, b⟫_ℝ < 0 | by { contrapose! disj, rwa K.hyperplane_separation } | theorem | proper_cone.hyperplane_separation_of_nmem | analysis.convex.cone | src/analysis/convex/cone/proper.lean | [
"analysis.convex.cone.dual",
"analysis.inner_product_space.adjoint"
] | [
"proper_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simplicial_complex | (faces : set (finset E))
(not_empty_mem : ∅ ∉ faces)
(indep : ∀ {s}, s ∈ faces → affine_independent 𝕜 (coe : (s : set E) → E))
(down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ≠ ∅ → t ∈ faces)
(inter_subset_convex_hull : ∀ {s t}, s ∈ faces → t ∈ faces →
convex_hull 𝕜 ↑s ∩ convex_hull 𝕜 ↑t ⊆ convex_hull 𝕜 (s ∩ t : se... | structure | geometry.simplicial_complex | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"affine_independent",
"convex_hull",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
space (K : simplicial_complex 𝕜 E) : set E | ⋃ s ∈ K.faces, convex_hull 𝕜 (s : set E) | def | geometry.simplicial_complex.space | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"convex_hull"
] | The underlying space of a simplicial complex is the union of its faces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_space_iff : x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ convex_hull 𝕜 (s : set E) | mem_Union₂ | lemma | geometry.simplicial_complex.mem_space_iff | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_subset_space (hs : s ∈ K.faces) : convex_hull 𝕜 ↑s ⊆ K.space | subset_bUnion_of_mem hs | lemma | geometry.simplicial_complex.convex_hull_subset_space | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_space (hs : s ∈ K.faces) : (s : set E) ⊆ K.space | (subset_convex_hull 𝕜 _).trans $ convex_hull_subset_space hs | lemma | geometry.simplicial_complex.subset_space | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_inter_convex_hull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
convex_hull 𝕜 ↑s ∩ convex_hull 𝕜 ↑t = convex_hull 𝕜 (s ∩ t : set E) | (K.inter_subset_convex_hull hs ht).antisymm $ subset_inter
(convex_hull_mono $ set.inter_subset_left _ _) $ convex_hull_mono $ set.inter_subset_right _ _ | lemma | geometry.simplicial_complex.convex_hull_inter_convex_hull | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"convex_hull",
"convex_hull_mono",
"set.inter_subset_left",
"set.inter_subset_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_or_exists_inter_eq_convex_hull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
disjoint (convex_hull 𝕜 (s : set E)) (convex_hull 𝕜 ↑t) ∨
∃ u ∈ K.faces, convex_hull 𝕜 (s : set E) ∩ convex_hull 𝕜 ↑t = convex_hull 𝕜 ↑u | begin
classical,
by_contra' h,
refine h.2 (s ∩ t) (K.down_closed hs (inter_subset_left _ _) $ λ hst, h.1 $
disjoint_iff_inf_le.mpr $ (K.inter_subset_convex_hull hs ht).trans _) _,
{ rw [←coe_inter, hst, coe_empty, convex_hull_empty],
refl },
{ rw [coe_inter, convex_hull_inter_convex_hull hs ht] }
end | lemma | geometry.simplicial_complex.disjoint_or_exists_inter_eq_convex_hull | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"convex_hull",
"convex_hull_empty",
"disjoint"
] | The conclusion is the usual meaning of "glue nicely" in textbooks. It turns out to be quite
unusable, as it's about faces as sets in space rather than simplices. Further, additional structure
on `𝕜` means the only choice of `u` is `s ∩ t` (but it's hard to prove). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_erase
(faces : set (finset E))
(indep : ∀ s ∈ faces, affine_independent 𝕜 (coe : (s : set E) → E))
(down_closed : ∀ s ∈ faces, ∀ t ⊆ s, t ∈ faces)
(inter_subset_convex_hull : ∀ s t ∈ faces,
convex_hull 𝕜 ↑s ∩ convex_hull 𝕜 ↑t ⊆ convex_hull 𝕜 (s ∩ t : set E)) :
simplicial_complex 𝕜 E | { faces := faces \ {∅},
not_empty_mem := λ h, h.2 (mem_singleton _),
indep := λ s hs, indep _ hs.1,
down_closed := λ s t hs hts ht, ⟨down_closed _ hs.1 _ hts, ht⟩,
inter_subset_convex_hull := λ s t hs ht, inter_subset_convex_hull _ hs.1 _ ht.1 } | def | geometry.simplicial_complex.of_erase | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"affine_independent",
"convex_hull",
"finset"
] | Construct a simplicial complex by removing the empty face for you. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_subcomplex (K : simplicial_complex 𝕜 E)
(faces : set (finset E))
(subset : faces ⊆ K.faces)
(down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ∈ faces) :
simplicial_complex 𝕜 E | { faces := faces,
not_empty_mem := λ h, K.not_empty_mem (subset h),
indep := λ s hs, K.indep (subset hs),
down_closed := λ s t hs hts _, down_closed hs hts,
inter_subset_convex_hull := λ s t hs ht, K.inter_subset_convex_hull (subset hs) (subset ht) } | def | geometry.simplicial_complex.of_subcomplex | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"finset"
] | Construct a simplicial complex as a subset of a given simplicial complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vertices (K : simplicial_complex 𝕜 E) : set E | {x | {x} ∈ K.faces} | def | geometry.simplicial_complex.vertices | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | The vertices of a simplicial complex are its zero dimensional faces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_vertices : x ∈ K.vertices ↔ {x} ∈ K.faces | iff.rfl | lemma | geometry.simplicial_complex.mem_vertices | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vertices_eq : K.vertices = ⋃ k ∈ K.faces, (k : set E) | begin
ext x,
refine ⟨λ h, mem_bUnion h $ mem_coe.2 $ mem_singleton_self x, λ h, _⟩,
obtain ⟨s, hs, hx⟩ := mem_Union₂.1 h,
exact K.down_closed hs (finset.singleton_subset_iff.2 $ mem_coe.1 hx) (singleton_ne_empty _),
end | lemma | geometry.simplicial_complex.vertices_eq | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vertices_subset_space : K.vertices ⊆ K.space | vertices_eq.subset.trans $ Union₂_mono $ λ x hx, subset_convex_hull 𝕜 x | lemma | geometry.simplicial_complex.vertices_subset_space | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vertex_mem_convex_hull_iff (hx : x ∈ K.vertices) (hs : s ∈ K.faces) :
x ∈ convex_hull 𝕜 (s : set E) ↔ x ∈ s | begin
refine ⟨λ h, _, λ h, subset_convex_hull _ _ h⟩,
classical,
have h := K.inter_subset_convex_hull hx hs ⟨by simp, h⟩,
by_contra H,
rwa [←coe_inter, finset.disjoint_iff_inter_eq_empty.1
(finset.disjoint_singleton_right.2 H).symm, coe_empty, convex_hull_empty] at h,
end | lemma | geometry.simplicial_complex.vertex_mem_convex_hull_iff | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"by_contra",
"convex_hull",
"convex_hull_empty",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
face_subset_face_iff (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
convex_hull 𝕜 (s : set E) ⊆ convex_hull 𝕜 ↑t ↔ s ⊆ t | ⟨λ h x hxs, (vertex_mem_convex_hull_iff (K.down_closed hs (finset.singleton_subset_iff.2 hxs) $
singleton_ne_empty _) ht).1 (h (subset_convex_hull 𝕜 ↑s hxs)), convex_hull_mono⟩ | lemma | geometry.simplicial_complex.face_subset_face_iff | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"convex_hull",
"subset_convex_hull"
] | A face is a subset of another one iff its vertices are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
facets (K : simplicial_complex 𝕜 E) : set (finset E) | {s ∈ K.faces | ∀ ⦃t⦄, t ∈ K.faces → s ⊆ t → s = t} | def | geometry.simplicial_complex.facets | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"finset"
] | A facet of a simplicial complex is a maximal face. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_facets : s ∈ K.facets ↔ s ∈ K.faces ∧ ∀ t ∈ K.faces, s ⊆ t → s = t | mem_sep_iff | lemma | geometry.simplicial_complex.mem_facets | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
facets_subset : K.facets ⊆ K.faces | λ s hs, hs.1 | lemma | geometry.simplicial_complex.facets_subset | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_facet_iff_subface (hs : s ∈ K.faces) : (s ∉ K.facets ↔ ∃ t, t ∈ K.faces ∧ s ⊂ t) | begin
refine ⟨λ (hs' : ¬ (_ ∧ _)), _, _⟩,
{ push_neg at hs',
obtain ⟨t, ht⟩ := hs' hs,
exact ⟨t, ht.1, ⟨ht.2.1, (λ hts, ht.2.2 (subset.antisymm ht.2.1 hts))⟩⟩ },
{ rintro ⟨t, ht⟩ ⟨hs, hs'⟩,
have := hs' ht.1 ht.2.1,
rw this at ht,
exact ht.2.2 (subset.refl t) } -- `has_ssubset.ssubset.ne` would... | lemma | geometry.simplicial_complex.not_facet_iff_subface | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faces_bot : (⊥ : simplicial_complex 𝕜 E).faces = ∅ | rfl | lemma | geometry.simplicial_complex.faces_bot | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
space_bot : (⊥ : simplicial_complex 𝕜 E).space = ∅ | set.bUnion_empty _ | lemma | geometry.simplicial_complex.space_bot | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [
"set.bUnion_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
facets_bot : (⊥ : simplicial_complex 𝕜 E).facets = ∅ | eq_empty_of_subset_empty facets_subset | lemma | geometry.simplicial_complex.facets_bot | analysis.convex.simplicial_complex | src/analysis/convex/simplicial_complex/basic.lean | [
"analysis.convex.hull",
"linear_algebra.affine_space.independent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on_exp : strict_convex_on ℝ univ exp | begin
apply strict_convex_on_of_slope_strict_mono_adjacent convex_univ,
rintros x y z - - hxy hyz,
transitivity exp y,
{ have h1 : 0 < y - x := by linarith,
have h2 : x - y < 0 := by linarith,
rw div_lt_iff h1,
calc exp y - exp x = exp y - exp y * exp (x - y) : by rw ← exp_add; ring_nf
... = exp... | lemma | strict_convex_on_exp | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_univ",
"div_lt_iff",
"exp",
"exp_add",
"lt_div_iff",
"mul_lt_mul_of_pos_left",
"ring",
"strict_convex_on",
"strict_convex_on_of_slope_strict_mono_adjacent"
] | `exp` is strictly convex on the whole real line.
We give an elementary proof rather than using the second derivative test, since this lemma is
needed early in the analysis library. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_exp : convex_on ℝ univ exp | strict_convex_on_exp.convex_on | lemma | convex_on_exp | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_on",
"exp"
] | `exp` is convex on the whole real line. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_pow (n : ℕ) : convex_on ℝ (Ici 0) (λ x : ℝ, x^n) | begin
induction n with k IH,
{ exact convex_on_const (1:ℝ) (convex_Ici _) },
refine ⟨convex_Ici _, _⟩,
rintros a (ha : 0 ≤ a) b (hb : 0 ≤ b) μ ν hμ hν h,
have H := IH.2 ha hb hμ hν h,
have : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν,
{ cases le_or_lt a b with hab hab,
{ have : a ^ k ≤ b ^ k := pow_le_pow_... | lemma | convex_on_pow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_Ici",
"convex_on",
"convex_on_const",
"mul_le_mul_of_nonneg_left",
"pow_le_pow_of_le_left"
] | `x^n`, `n : ℕ` is convex on `[0, +∞)` for all `n`.
We give an elementary proof rather than using the second derivative test, since this lemma is
needed early in the analysis library. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
even.convex_on_pow {n : ℕ} (hn : even n) : convex_on ℝ set.univ (λ x : ℝ, x^n) | begin
refine ⟨convex_univ, _⟩,
intros a ha b hb μ ν hμ hν h,
obtain ⟨k, rfl⟩ := hn.exists_two_nsmul _,
have : 0 ≤ (a - b) ^ 2 * μ * ν := by positivity,
calc (μ * a + ν * b) ^ (2 * k) = ((μ * a + ν * b) ^ 2) ^ k : by rw pow_mul
... ≤ ((μ + ν) * (μ * a ^ 2 + ν * b ^ 2)) ^ k : pow_le_pow_of_le_left (by positiv... | lemma | even.convex_on_pow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_on",
"convex_on_pow",
"pow_le_pow_of_le_left",
"pow_mul"
] | `x^n`, `n : ℕ` is convex on the whole real line whenever `n` is even.
We give an elementary proof rather than using the second derivative test, since this lemma is
needed early in the analysis library. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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