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 |
|---|---|---|---|---|---|---|---|---|---|---|
strict_concave_on.translate_left (hf : strict_concave_on 𝕜 s f) (c : E) :
strict_concave_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, z + c)) | by simpa only [add_comm] using hf.translate_right _ | lemma | strict_concave_on.translate_left | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on"
] | Left translation preserves strict concavity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.smul {c : 𝕜} (hc : 0 ≤ c) (hf : convex_on 𝕜 s f) : convex_on 𝕜 s (λ x, c • f x) | ⟨hf.1, λ x hx y hy a b ha hb hab,
calc
c • f (a • x + b • y) ≤ c • (a • f x + b • f y)
: smul_le_smul_of_nonneg (hf.2 hx hy ha hb hab) hc
... = a • (c • f x) + b • (c • f y)
: by rw [smul_add, smul_comm c, smul_comm c]; apply_instance⟩ | lemma | convex_on.smul | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"smul_add",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.smul {c : 𝕜} (hc : 0 ≤ c) (hf : concave_on 𝕜 s f) :
concave_on 𝕜 s (λ x, c • f x) | hf.dual.smul hc | lemma | concave_on.smul | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F} (hf : convex_on 𝕜 s f) :
convex_on 𝕜 (g ⁻¹' s) (f ∘ g) | ⟨hf.1.affine_preimage _, λ x hx y hy a b ha hb hab,
calc
(f ∘ g) (a • x + b • y) = f (g (a • x + b • y)) : rfl
... = f (a • (g x) + b • (g y)) : by rw [convex.combo_affine_apply hab]
... ≤ a • f (g x) + b • f (g y) : hf.2 hx hy ha hb hab⟩ | lemma | convex_on.comp_affine_map | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_affine_apply",
"convex_on"
] | If a function is convex on `s`, it remains convex when precomposed by an affine map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F} (hf : concave_on 𝕜 s f) :
concave_on 𝕜 (g ⁻¹' s) (f ∘ g) | hf.dual.comp_affine_map g | lemma | concave_on.comp_affine_map | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | If a function is concave on `s`, it remains concave when precomposed by an affine map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_iff_div {f : E → β} :
convex_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b →
0 < a + b → f ((a/(a+b)) • x + (b/(a+b)) • y) ≤ (a/(a+b)) • f x + (b/(a+b)) • f y | and_congr iff.rfl
⟨begin
intros h x hx y hy a b ha hb hab,
apply h hx hy (div_nonneg ha hab.le) (div_nonneg hb hab.le),
rw [←add_div, div_self hab.ne'],
end,
begin
intros h x hx y hy a b ha hb hab,
simpa [hab, zero_lt_one] using h hx hy ha hb,
end⟩ | lemma | convex_on_iff_div | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on",
"div_nonneg",
"div_self",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on_iff_div {f : E → β} :
concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b
→ 0 < a + b → (a/(a+b)) • f x + (b/(a+b)) • f y ≤ f ((a/(a+b)) • x + (b/(a+b)) • y) | @convex_on_iff_div _ _ βᵒᵈ _ _ _ _ _ _ _ | lemma | concave_on_iff_div | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"convex_on_iff_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on_iff_div {f : E → β} :
strict_convex_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a
→ 0 < b → f ((a/(a+b)) • x + (b/(a+b)) • y) < (a/(a+b)) • f x + (b/(a+b)) • f y | and_congr iff.rfl
⟨begin
intros h x hx y hy hxy a b ha hb,
have hab := add_pos ha hb,
apply h hx hy hxy (div_pos ha hab) (div_pos hb hab),
rw [←add_div, div_self hab.ne'],
end,
begin
intros h x hx y hy hxy a b ha hb hab,
simpa [hab, zero_lt_one] using h hx hy hxy ha hb,
end⟩ | lemma | strict_convex_on_iff_div | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"div_pos",
"div_self",
"strict_convex_on",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on_iff_div {f : E → β} :
strict_concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a
→ 0 < b → (a/(a+b)) • f x + (b/(a+b)) • f y < f ((a/(a+b)) • x + (b/(a+b)) • y) | @strict_convex_on_iff_div _ _ βᵒᵈ _ _ _ _ _ _ _ | lemma | strict_concave_on_iff_div | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"strict_concave_on",
"strict_convex_on_iff_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.le_right_of_left_le'' (hf : convex_on 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y ≤ z) (h : f x ≤ f y) : f y ≤ f z | hyz.eq_or_lt.elim (λ hyz, (congr_arg f hyz).le)
(λ hyz, hf.le_right_of_left_le hx hz (Ioo_subset_open_segment ⟨hxy, hyz⟩) h) | lemma | convex_on.le_right_of_left_le'' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"Ioo_subset_open_segment",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.le_left_of_right_le'' (hf : convex_on 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s)
(hxy : x ≤ y) (hyz : y < z) (h : f z ≤ f y) : f y ≤ f x | hxy.eq_or_lt.elim (λ hxy, (congr_arg f hxy).ge)
(λ hxy, hf.le_left_of_right_le hx hz (Ioo_subset_open_segment ⟨hxy, hyz⟩) h) | lemma | convex_on.le_left_of_right_le'' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"Ioo_subset_open_segment",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.right_le_of_le_left'' (hf : concave_on 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y ≤ z) (h : f y ≤ f x) : f z ≤ f y | hf.dual.le_right_of_left_le'' hx hz hxy hyz h | lemma | concave_on.right_le_of_le_left'' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.left_le_of_le_right'' (hf : concave_on 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s)
(hxy : x ≤ y) (hyz : y < z) (h : f y ≤ f z) : f x ≤ f y | hf.dual.le_left_of_right_le'' hx hz hxy hyz h | lemma | concave_on.left_le_of_le_right'' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge (s : set E) (x : E) : ℝ | Inf {r : ℝ | 0 < r ∧ x ∈ r • s} | def | gauge | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [] | The Minkowski functional. Given a set `s` in a real vector space, `gauge s` is the functional
which sends `x : E` to the smallest `r : ℝ` such that `x` is in `s` scaled by `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_def : gauge s x = Inf {r ∈ set.Ioi 0 | x ∈ r • s} | rfl | lemma | gauge_def | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"set.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_def' : gauge s x = Inf {r ∈ set.Ioi 0 | r⁻¹ • x ∈ s} | begin
congrm Inf (λ r, _),
exact and_congr_right (λ hr, mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _),
end | lemma | gauge_def' | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"set.Ioi"
] | An alternative definition of the gauge using scalar multiplication on the element rather than on
the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_set_bdd_below : bdd_below {r : ℝ | 0 < r ∧ x ∈ r • s} | ⟨0, λ r hr, hr.1.le⟩ | lemma | gauge_set_bdd_below | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent.gauge_set_nonempty (absorbs : absorbent ℝ s) :
{r : ℝ | 0 < r ∧ x ∈ r • s}.nonempty | let ⟨r, hr₁, hr₂⟩ := absorbs x in ⟨r, hr₁, hr₂ r (real.norm_of_nonneg hr₁.le).ge⟩ | lemma | absorbent.gauge_set_nonempty | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"real.norm_of_nonneg"
] | If the given subset is `absorbent` then the set we take an infimum over in `gauge` is nonempty,
which is useful for proving many properties about the gauge. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_mono (hs : absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s | λ x, cInf_le_cInf gauge_set_bdd_below hs.gauge_set_nonempty $ λ r hr, ⟨hr.1, smul_set_mono h hr.2⟩ | lemma | gauge_mono | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"cInf_le_cInf",
"gauge",
"gauge_set_bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_of_gauge_lt (absorbs : absorbent ℝ s) (h : gauge s x < a) :
∃ b, 0 < b ∧ b < a ∧ x ∈ b • s | begin
obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_cInf_lt absorbs.gauge_set_nonempty h,
exact ⟨b, hb, hba, hx⟩,
end | lemma | exists_lt_of_gauge_lt | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"exists_lt_of_cInf_lt",
"gauge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_zero : gauge s 0 = 0 | begin
rw gauge_def',
by_cases (0 : E) ∈ s,
{ simp only [smul_zero, sep_true, h, cInf_Ioi] },
{ simp only [smul_zero, sep_false, h, real.Inf_empty] }
end | lemma | gauge_zero | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"cInf_Ioi",
"gauge",
"gauge_def'",
"real.Inf_empty",
"smul_zero"
] | The gauge evaluated at `0` is always zero (mathematically this requires `0` to be in the set `s`
but, the real infimum of the empty set in Lean being defined as `0`, it holds unconditionally). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_zero' : gauge (0 : set E) = 0 | begin
ext,
rw gauge_def',
obtain rfl | hx := eq_or_ne x 0,
{ simp only [cInf_Ioi, mem_zero, pi.zero_apply, eq_self_iff_true, sep_true, smul_zero] },
{ simp only [mem_zero, pi.zero_apply, inv_eq_zero, smul_eq_zero],
convert real.Inf_empty,
exact eq_empty_iff_forall_not_mem.2 (λ r hr, hr.2.elim (ne_of_g... | lemma | gauge_zero' | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"cInf_Ioi",
"eq_or_ne",
"gauge",
"gauge_def'",
"inv_eq_zero",
"real.Inf_empty",
"smul_eq_zero",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_empty : gauge (∅ : set E) = 0 | by { ext, simp only [gauge_def', real.Inf_empty, mem_empty_iff_false, pi.zero_apply, sep_false] } | lemma | gauge_empty | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_def'",
"real.Inf_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 | by { obtain rfl | rfl := subset_singleton_iff_eq.1 h, exacts [gauge_empty, gauge_zero'] } | lemma | gauge_of_subset_zero | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_empty",
"gauge_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_nonneg (x : E) : 0 ≤ gauge s x | real.Inf_nonneg _ $ λ x hx, hx.1.le | lemma | gauge_nonneg | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"real.Inf_nonneg"
] | The gauge is always nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x | begin
have : ∀ x, -x ∈ s ↔ x ∈ s := λ x, ⟨λ h, by simpa using symmetric _ h, symmetric x⟩,
simp_rw [gauge_def', smul_neg, this]
end | lemma | gauge_neg | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_def'",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_neg_set_neg (x : E) : gauge (-s) (-x) = gauge s x | by simp_rw [gauge_def', smul_neg, neg_mem_neg] | lemma | gauge_neg_set_neg | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_def'",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) | by rw [← gauge_neg_set_neg, neg_neg] | lemma | gauge_neg_set_eq_gauge_neg | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_neg_set_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a | begin
obtain rfl | ha' := ha.eq_or_lt,
{ rw [mem_singleton_iff.1 (zero_smul_set_subset _ hx), gauge_zero] },
{ exact cInf_le gauge_set_bdd_below ⟨ha', hx⟩ }
end | lemma | gauge_le_of_mem | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"cInf_le",
"gauge",
"gauge_set_bdd_below",
"gauge_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_le_eq (hs₁ : convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : absorbent ℝ s) (ha : 0 ≤ a) :
{x | gauge s x ≤ a} = ⋂ (r : ℝ) (H : a < r), r • s | begin
ext,
simp_rw [set.mem_Inter, set.mem_set_of_eq],
refine ⟨λ h r hr, _, λ h, le_of_forall_pos_lt_add (λ ε hε, _)⟩,
{ have hr' := ha.trans_lt hr,
rw mem_smul_set_iff_inv_smul_mem₀ hr'.ne',
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr),
suffices : (r⁻¹ * δ) • δ⁻¹ • x ∈ s,... | lemma | gauge_le_eq | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"convex",
"exists_lt_of_gauge_lt",
"gauge",
"gauge_le_of_mem",
"half_pos",
"inv_mul_le_iff",
"mul_inv_cancel_right₀",
"mul_one",
"set.mem_Inter",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_lt_eq' (absorbs : absorbent ℝ s) (a : ℝ) :
{x | gauge s x < a} = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s | begin
ext,
simp_rw [mem_set_of_eq, mem_Union, exists_prop],
exact ⟨exists_lt_of_gauge_lt absorbs,
λ ⟨r, hr₀, hr₁, hx⟩, (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩,
end | lemma | gauge_lt_eq' | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"exists_prop",
"gauge",
"gauge_le_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_lt_eq (absorbs : absorbent ℝ s) (a : ℝ) :
{x | gauge s x < a} = ⋃ (r ∈ set.Ioo 0 (a : ℝ)), r • s | begin
ext,
simp_rw [mem_set_of_eq, mem_Union, exists_prop, mem_Ioo, and_assoc],
exact ⟨exists_lt_of_gauge_lt absorbs,
λ ⟨r, hr₀, hr₁, hx⟩, (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩,
end | lemma | gauge_lt_eq | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"exists_prop",
"gauge",
"gauge_le_of_mem",
"set.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_lt_one_subset_self (hs : convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : absorbent ℝ s) :
{x | gauge s x < 1} ⊆ s | begin
rw gauge_lt_eq absorbs,
refine set.Union₂_subset (λ r hr _, _),
rintro ⟨y, hy, rfl⟩,
exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr),
end | lemma | gauge_lt_one_subset_self | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"convex",
"gauge",
"gauge_lt_eq",
"set.Union₂_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 | gauge_le_of_mem zero_le_one $ by rwa one_smul | lemma | gauge_le_one_of_mem | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_le_of_mem",
"one_smul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_subset_gauge_le_one : s ⊆ {x | gauge s x ≤ 1} | λ x, gauge_le_one_of_mem | lemma | self_subset_gauge_le_one | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_le_one_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.gauge_le (hs : convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : absorbent ℝ s) (a : ℝ) :
convex ℝ {x | gauge s x ≤ a} | begin
by_cases ha : 0 ≤ a,
{ rw gauge_le_eq hs h₀ absorbs ha,
exact convex_Inter (λ i, convex_Inter (λ hi, hs.smul _)) },
{ convert convex_empty,
exact eq_empty_iff_forall_not_mem.2 (λ x hx, ha $ (gauge_nonneg _).trans hx) }
end | lemma | convex.gauge_le | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"convex",
"convex_Inter",
"convex_empty",
"gauge",
"gauge_le_eq",
"gauge_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.star_convex (hs : balanced ℝ s) : star_convex ℝ 0 s | star_convex_zero_iff.2 $ λ x hx a ha₀ ha₁,
hs _ (by rwa real.norm_of_nonneg ha₀) (smul_mem_smul_set hx) | lemma | balanced.star_convex | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"balanced",
"real.norm_of_nonneg",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_gauge_of_not_mem (hs₀ : star_convex ℝ 0 s) (hs₂ : absorbs ℝ s {x}) (hx : x ∉ a • s) :
a ≤ gauge s x | begin
rw star_convex_zero_iff at hs₀,
obtain ⟨r, hr, h⟩ := hs₂,
refine le_cInf ⟨r, hr, singleton_subset_iff.1 $ h _ (real.norm_of_nonneg hr.le).ge⟩ _,
rintro b ⟨hb, x, hx', rfl⟩,
refine not_lt.1 (λ hba, hx _),
have ha := hb.trans hba,
refine ⟨(a⁻¹ * b) • x, hs₀ hx' (by positivity) _, _⟩,
{ rw ←div_eq_in... | lemma | le_gauge_of_not_mem | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbs",
"div_le_one_of_le",
"gauge",
"le_cInf",
"mul_inv_cancel_left₀",
"real.norm_of_nonneg",
"star_convex",
"star_convex_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_gauge_of_not_mem (hs₁ : star_convex ℝ 0 s) (hs₂ : absorbs ℝ s {x}) (hx : x ∉ s) :
1 ≤ gauge s x | le_gauge_of_not_mem hs₁ hs₂ $ by rwa one_smul | lemma | one_le_gauge_of_not_mem | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbs",
"gauge",
"le_gauge_of_not_mem",
"one_smul",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_smul_of_nonneg [mul_action_with_zero α E] [is_scalar_tower α ℝ (set E)] {s : set E}
{a : α} (ha : 0 ≤ a) (x : E) :
gauge s (a • x) = a • gauge s x | begin
obtain rfl | ha' := ha.eq_or_lt,
{ rw [zero_smul, gauge_zero, zero_smul] },
rw [gauge_def', gauge_def', ←real.Inf_smul_of_nonneg ha],
congr' 1,
ext r,
simp_rw [set.mem_smul_set, set.mem_sep_iff],
split,
{ rintro ⟨hr, hx⟩,
simp_rw mem_Ioi at ⊢ hr,
rw ←mem_smul_set_iff_inv_smul_mem₀ hr.ne' a... | lemma | gauge_smul_of_nonneg | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_def'",
"gauge_zero",
"inv_inv",
"inv_ne_zero",
"is_scalar_tower",
"mul_action_with_zero",
"set.mem_sep_iff",
"set.mem_smul_set",
"smul_assoc",
"smul_inv_smul₀",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_smul_left_of_nonneg [mul_action_with_zero α E] [smul_comm_class α ℝ ℝ]
[is_scalar_tower α ℝ ℝ] [is_scalar_tower α ℝ E] {s : set E} {a : α} (ha : 0 ≤ a) :
gauge (a • s) = a⁻¹ • gauge s | begin
obtain rfl | ha' := ha.eq_or_lt,
{ rw [inv_zero, zero_smul, gauge_of_subset_zero (zero_smul_set_subset _)] },
ext,
rw [gauge_def', pi.smul_apply, gauge_def', ←real.Inf_smul_of_nonneg (inv_nonneg.2 ha)],
congr' 1,
ext r,
simp_rw [set.mem_smul_set, set.mem_sep_iff],
split,
{ rintro ⟨hr, y, hy, h⟩,... | lemma | gauge_smul_left_of_nonneg | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge",
"gauge_def'",
"gauge_of_subset_zero",
"inv_inv",
"inv_smul_smul₀",
"inv_zero",
"is_scalar_tower",
"mul_action_with_zero",
"pi.smul_apply",
"set.mem_sep_iff",
"set.mem_smul_set",
"smul_assoc",
"smul_comm_class",
"smul_inv₀",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_smul_left [module α E] [smul_comm_class α ℝ ℝ] [is_scalar_tower α ℝ ℝ]
[is_scalar_tower α ℝ E] {s : set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
gauge (a • s) = |a|⁻¹ • gauge s | begin
rw ←gauge_smul_left_of_nonneg (abs_nonneg a),
obtain h | h := abs_choice a,
{ rw h },
{ rw [h, set.neg_smul_set, ←set.smul_set_neg],
congr,
ext y,
refine ⟨symmetric _, λ hy, _⟩,
rw ←neg_neg y,
exact symmetric _ hy },
{ apply_instance }
end | lemma | gauge_smul_left | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"abs_choice",
"abs_nonneg",
"gauge",
"is_scalar_tower",
"module",
"set.neg_smul_set",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_norm_smul (hs : balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) | begin
unfold gauge,
congr' with θ,
rw @is_R_or_C.real_smul_eq_coe_smul 𝕜,
refine and_congr_right (λ hθ, (hs.smul _).mem_smul_iff _),
rw [is_R_or_C.norm_of_real, abs_norm],
end | lemma | gauge_norm_smul | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"abs_norm",
"balanced",
"gauge",
"is_R_or_C.norm_of_real",
"is_R_or_C.real_smul_eq_coe_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_smul (hs : balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x | by { rw [←smul_eq_mul, ←gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs], apply_instance } | lemma | gauge_smul | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"balanced",
"gauge",
"gauge_norm_smul"
] | If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_subset_gauge_lt_one (s : set E) : interior s ⊆ {x | gauge s x < 1} | begin
intros x hx,
let f : ℝ → E := λ t, t • x,
have hf : continuous f,
{ continuity },
let s' := f ⁻¹' (interior s),
have hs' : is_open s' := hf.is_open_preimage _ is_open_interior,
have one_mem : (1 : ℝ) ∈ s',
{ simpa only [s', f, set.mem_preimage, one_smul] },
obtain ⟨ε, hε₀, hε⟩ := (metric.nhds_ba... | lemma | interior_subset_gauge_lt_one | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"continuity",
"continuous",
"gauge",
"gauge_le_of_mem",
"interior",
"interior_subset",
"inv_lt_one_iff",
"is_open",
"is_open_interior",
"lt_one_add",
"one_smul",
"real.closed_ball_eq_Icc",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_lt_one_eq_self_of_open (hs₁ : convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : is_open s) :
{x | gauge s x < 1} = s | begin
refine (gauge_lt_one_subset_self hs₁ ‹_› $ absorbent_nhds_zero $ hs₂.mem_nhds hs₀).antisymm _,
convert interior_subset_gauge_lt_one s,
exact hs₂.interior_eq.symm,
end | lemma | gauge_lt_one_eq_self_of_open | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent_nhds_zero",
"convex",
"gauge",
"gauge_lt_one_subset_self",
"interior_subset_gauge_lt_one",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_lt_one_of_mem_of_open (hs₁ : convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : is_open s)
{x : E} (hx : x ∈ s) :
gauge s x < 1 | by rwa ←gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂ at hx | lemma | gauge_lt_one_of_mem_of_open | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"convex",
"gauge",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s)
(hs₁ : convex ℝ s) (hs₂ : is_open s) (hx : x ∈ ε • s) :
gauge s x < ε | begin
have : ε⁻¹ • x ∈ s,
{ rwa ←mem_smul_set_iff_inv_smul_mem₀ hε.ne' },
have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₁ hs₀ hs₂ this,
rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one]
at h_gauge_lt,
apply_instance
end | lemma | gauge_lt_of_mem_smul | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"convex",
"gauge",
"gauge_lt_one_of_mem_of_open",
"gauge_smul_of_nonneg",
"inv_mul_lt_iff",
"is_open",
"mul_one",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_add_le (hs : convex ℝ s) (absorbs : absorbent ℝ s) (x y : E) :
gauge s (x + y) ≤ gauge s x + gauge s y | begin
refine le_of_forall_pos_lt_add (λ ε hε, _),
obtain ⟨a, ha, ha', hx⟩ := exists_lt_of_gauge_lt absorbs
(lt_add_of_pos_right (gauge s x) (half_pos hε)),
obtain ⟨b, hb, hb', hy⟩ := exists_lt_of_gauge_lt absorbs
(lt_add_of_pos_right (gauge s y) (half_pos hε)),
rw mem_smul_set_iff_inv_smul_mem₀ ha.ne' a... | lemma | gauge_add_le | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbs",
"convex",
"exists_lt_of_gauge_lt",
"gauge",
"gauge_le_of_mem",
"half_pos",
"mul_inv_cancel",
"one_div",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_seminorm (hs₀ : balanced 𝕜 s) (hs₁ : convex ℝ s) (hs₂ : absorbent ℝ s) :
seminorm 𝕜 E | seminorm.of (gauge s) (gauge_add_le hs₁ hs₂) (gauge_smul hs₀) | def | gauge_seminorm | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"balanced",
"convex",
"gauge",
"gauge_add_le",
"gauge_smul",
"seminorm",
"seminorm.of"
] | `gauge s` as a seminorm when `s` is balanced, convex and absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gauge_seminorm_lt_one_of_open (hs : is_open s) {x : E} (hx : x ∈ s) :
gauge_seminorm hs₀ hs₁ hs₂ x < 1 | gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx | lemma | gauge_seminorm_lt_one_of_open | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge_lt_one_of_mem_of_open",
"gauge_seminorm",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_seminorm_ball_one (hs : is_open s) :
(gauge_seminorm hs₀ hs₁ hs₂).ball 0 1 = s | begin
rw seminorm.ball_zero_eq,
exact gauge_lt_one_eq_self_of_open hs₁ hs₂.zero_mem hs
end | lemma | gauge_seminorm_ball_one | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"gauge_lt_one_eq_self_of_open",
"gauge_seminorm",
"is_open",
"seminorm.ball_zero_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm.gauge_ball (p : seminorm ℝ E) : gauge (p.ball 0 1) = p | begin
ext,
obtain hp | hp := {r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1}.eq_empty_or_nonempty,
{ rw [gauge, hp, real.Inf_empty],
by_contra,
have hpx : 0 < p x := (map_nonneg _ _).lt_of_ne h,
have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx,
refine hp.subset ⟨hpx₂, (2 * p x)⁻¹ • x, _, smul_inv_smul₀ hpx... | lemma | seminorm.gauge_ball | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"abs_of_pos",
"by_contra",
"gauge",
"inv_mul_lt_iff",
"is_glb.cInf_eq",
"lt_mul_of_one_lt_left",
"map_nonneg",
"mul_le_of_le_one_right",
"mul_one",
"one_lt_two",
"real.Inf_empty",
"real.norm_eq_abs",
"seminorm",
"smul_inv_smul₀",
"zero_lt_two"
] | Any seminorm arises as the gauge of its unit ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm.gauge_seminorm_ball (p : seminorm ℝ E) :
gauge_seminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1)
(p.absorbent_ball_zero zero_lt_one) = p | fun_like.coe_injective p.gauge_ball | lemma | seminorm.gauge_seminorm_ball | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"fun_like.coe_injective",
"gauge_seminorm",
"seminorm",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_unit_ball (x : E) : gauge (metric.ball (0 : E) 1) x = ‖x‖ | by rw [← ball_norm_seminorm ℝ, seminorm.gauge_ball, coe_norm_seminorm] | lemma | gauge_unit_ball | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"ball_norm_seminorm",
"coe_norm_seminorm",
"gauge",
"metric.ball",
"seminorm.gauge_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gauge_ball (hr : 0 < r) (x : E) : gauge (metric.ball (0 : E) r) x = ‖x‖ / r | begin
rw [←smul_unit_ball_of_pos hr, gauge_smul_left, pi.smul_apply, gauge_unit_ball, smul_eq_mul,
abs_of_nonneg hr.le, div_eq_inv_mul],
simp_rw [mem_ball_zero_iff, norm_neg],
exact λ _, id,
end | lemma | gauge_ball | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"abs_of_nonneg",
"div_eq_inv_mul",
"gauge",
"gauge_smul_left",
"gauge_unit_ball",
"metric.ball",
"pi.smul_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_gauge_le_norm (hs : metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤ ‖x‖ | begin
obtain hr | hr := le_or_lt r 0,
{ exact (mul_nonpos_of_nonpos_of_nonneg hr $ gauge_nonneg _).trans (norm_nonneg _) },
rw [mul_comm, ←le_div_iff hr, ←gauge_ball hr],
exact gauge_mono (absorbent_ball_zero hr) hs x,
end | lemma | mul_gauge_le_norm | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent_ball_zero",
"gauge",
"gauge_mono",
"gauge_nonneg",
"metric.ball",
"mul_comm",
"mul_nonpos_of_nonpos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.lipschitz_with_gauge {r : ℝ≥0} (hc : convex ℝ s) (hr : 0 < r)
(hs : metric.ball (0 : E) r ⊆ s) :
lipschitz_with r⁻¹ (gauge s) | have absorbent ℝ (metric.ball (0 : E) r) := absorbent_ball_zero hr,
lipschitz_with.of_le_add_mul _ $ λ x y,
calc gauge s x = gauge s (y + (x - y)) : by simp
... ≤ gauge s y + gauge s (x - y) : gauge_add_le hc (this.subset hs) _ _
... ≤ gauge s y + ‖x - y‖ / r :
add_le_add_left ((gauge_mono this hs (x - y)).tr... | lemma | convex.lipschitz_with_gauge | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"absorbent",
"absorbent_ball_zero",
"convex",
"div_eq_inv_mul",
"gauge",
"gauge_add_le",
"gauge_ball",
"gauge_mono",
"lipschitz_with",
"lipschitz_with.of_le_add_mul",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.uniform_continuous_gauge (hc : convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
uniform_continuous (gauge s) | begin
obtain ⟨r, hr₀, hr⟩ := metric.mem_nhds_iff.1 h₀,
lift r to ℝ≥0 using le_of_lt hr₀,
exact (hc.lipschitz_with_gauge hr₀ hr).uniform_continuous
end | lemma | convex.uniform_continuous_gauge | analysis.convex | src/analysis/convex/gauge.lean | [
"analysis.convex.basic",
"analysis.normed_space.pointwise",
"analysis.seminorm",
"data.is_R_or_C.basic",
"tactic.congrm"
] | [
"convex",
"gauge",
"lift",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull : closure_operator (set E) | closure_operator.mk₃
(λ s, ⋂ (t : set E) (hst : s ⊆ t) (ht : convex 𝕜 t), t)
(convex 𝕜)
(λ s, set.subset_Inter (λ t, set.subset_Inter $ λ hst, set.subset_Inter $ λ ht, hst))
(λ s, convex_Inter $ λ t, convex_Inter $ λ ht, convex_Inter id)
(λ s t hst ht, set.Inter_subset_of_subset t $ set.Inter_subset_of_subs... | def | convex_hull | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator",
"closure_operator.mk₃",
"convex",
"convex_Inter",
"set.Inter_subset",
"set.Inter_subset_of_subset",
"set.subset_Inter"
] | The convex hull of a set `s` is the minimal convex set that includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_convex_hull : s ⊆ convex_hull 𝕜 s | (convex_hull 𝕜).le_closure s | lemma | subset_convex_hull | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_convex_hull : convex 𝕜 (convex_hull 𝕜 s) | closure_operator.closure_mem_mk₃ s | lemma | convex_convex_hull | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator.closure_mem_mk₃",
"convex",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_eq_Inter : convex_hull 𝕜 s = ⋂ (t : set E) (hst : s ⊆ t) (ht : convex 𝕜 t), t | rfl | lemma | convex_hull_eq_Inter | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_convex_hull_iff : x ∈ convex_hull 𝕜 s ↔ ∀ t, s ⊆ t → convex 𝕜 t → x ∈ t | by simp_rw [convex_hull_eq_Inter, mem_Inter] | lemma | mem_convex_hull_iff | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex",
"convex_hull",
"convex_hull_eq_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_min (hst : s ⊆ t) (ht : convex 𝕜 t) : convex_hull 𝕜 s ⊆ t | closure_operator.closure_le_mk₃_iff (show s ≤ t, from hst) ht | lemma | convex_hull_min | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator.closure_le_mk₃_iff",
"convex",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.convex_hull_subset_iff (ht : convex 𝕜 t) : convex_hull 𝕜 s ⊆ t ↔ s ⊆ t | ⟨(subset_convex_hull _ _).trans, λ h, convex_hull_min h ht⟩ | lemma | convex.convex_hull_subset_iff | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex",
"convex_hull",
"convex_hull_min",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_mono (hst : s ⊆ t) : convex_hull 𝕜 s ⊆ convex_hull 𝕜 t | (convex_hull 𝕜).monotone hst | lemma | convex_hull_mono | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.convex_hull_eq (hs : convex 𝕜 s) : convex_hull 𝕜 s = s | closure_operator.mem_mk₃_closed hs | lemma | convex.convex_hull_eq | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator.mem_mk₃_closed",
"convex",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_univ : convex_hull 𝕜 (univ : set E) = univ | closure_operator.closure_top (convex_hull 𝕜) | lemma | convex_hull_univ | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator.closure_top",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_empty : convex_hull 𝕜 (∅ : set E) = ∅ | convex_empty.convex_hull_eq | lemma | convex_hull_empty | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_empty_iff : convex_hull 𝕜 s = ∅ ↔ s = ∅ | begin
split,
{ intro h,
rw [←set.subset_empty_iff, ←h],
exact subset_convex_hull 𝕜 _ },
{ rintro rfl,
exact convex_hull_empty }
end | lemma | convex_hull_empty_iff | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"convex_hull_empty",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_nonempty_iff : (convex_hull 𝕜 s).nonempty ↔ s.nonempty | begin
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, ne.def, ne.def],
exact not_congr convex_hull_empty_iff,
end | lemma | convex_hull_nonempty_iff | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"convex_hull_empty_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_subset_convex_hull (hx : x ∈ s) (hy : y ∈ s) : segment 𝕜 x y ⊆ convex_hull 𝕜 s | (convex_convex_hull _ _).segment_subset (subset_convex_hull _ _ hx) (subset_convex_hull _ _ hy) | lemma | segment_subset_convex_hull | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_convex_hull",
"convex_hull",
"segment",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_singleton (x : E) : convex_hull 𝕜 ({x} : set E) = {x} | (convex_singleton x).convex_hull_eq | lemma | convex_hull_singleton | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"convex_hull_eq",
"convex_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_zero : convex_hull 𝕜 (0 : set E) = 0 | convex_hull_singleton 0 | lemma | convex_hull_zero | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"convex_hull_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_pair (x y : E) : convex_hull 𝕜 {x, y} = segment 𝕜 x y | begin
refine (convex_hull_min _ $ convex_segment _ _).antisymm
(segment_subset_convex_hull (mem_insert _ _) $ mem_insert_of_mem _ $ mem_singleton _),
rw [insert_subset, singleton_subset_iff],
exact ⟨left_mem_segment _ _ _, right_mem_segment _ _ _⟩,
end | lemma | convex_hull_pair | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"convex_hull_min",
"convex_segment",
"right_mem_segment",
"segment",
"segment_subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_convex_hull_union_left (s t : set E) :
convex_hull 𝕜 (convex_hull 𝕜 s ∪ t) = convex_hull 𝕜 (s ∪ t) | closure_operator.closure_sup_closure_left _ _ _ | lemma | convex_hull_convex_hull_union_left | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator.closure_sup_closure_left",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_convex_hull_union_right (s t : set E) :
convex_hull 𝕜 (s ∪ convex_hull 𝕜 t) = convex_hull 𝕜 (s ∪ t) | closure_operator.closure_sup_closure_right _ _ _ | lemma | convex_hull_convex_hull_union_right | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"closure_operator.closure_sup_closure_right",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.convex_remove_iff_not_mem_convex_hull_remove {s : set E} (hs : convex 𝕜 s) (x : E) :
convex 𝕜 (s \ {x}) ↔ x ∉ convex_hull 𝕜 (s \ {x}) | begin
split,
{ rintro hsx hx,
rw hsx.convex_hull_eq at hx,
exact hx.2 (mem_singleton _) },
rintro hx,
suffices h : s \ {x} = convex_hull 𝕜 (s \ {x}), { convert convex_convex_hull 𝕜 _ },
exact subset.antisymm (subset_convex_hull 𝕜 _) (λ y hy, ⟨convex_hull_min (diff_subset _ _) hs hy,
by { rintro... | lemma | convex.convex_remove_iff_not_mem_convex_hull_remove | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex",
"convex_convex_hull",
"convex_hull",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_linear_map.convex_hull_image {f : E → F} (hf : is_linear_map 𝕜 f) (s : set E) :
convex_hull 𝕜 (f '' s) = f '' convex_hull 𝕜 s | set.subset.antisymm (convex_hull_min (image_subset _ (subset_convex_hull 𝕜 s)) $
(convex_convex_hull 𝕜 s).is_linear_image hf)
(image_subset_iff.2 $ convex_hull_min
(image_subset_iff.1 $ subset_convex_hull 𝕜 _)
((convex_convex_hull 𝕜 _).is_linear_preimage hf)) | lemma | is_linear_map.convex_hull_image | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_convex_hull",
"convex_hull",
"convex_hull_min",
"is_linear_map",
"set.subset.antisymm",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.convex_hull_image (f : E →ₗ[𝕜] F) (s : set E) :
convex_hull 𝕜 (f '' s) = f '' convex_hull 𝕜 s | f.is_linear.convex_hull_image s | lemma | linear_map.convex_hull_image | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_smul (a : 𝕜) (s : set E) : convex_hull 𝕜 (a • s) = a • convex_hull 𝕜 s | (linear_map.lsmul _ _ a).convex_hull_image _ | lemma | convex_hull_smul | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_hull",
"linear_map.lsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_map.image_convex_hull (f : E →ᵃ[𝕜] F) :
f '' convex_hull 𝕜 s = convex_hull 𝕜 (f '' s) | begin
apply set.subset.antisymm,
{ rw set.image_subset_iff,
refine convex_hull_min _ ((convex_convex_hull 𝕜 (⇑f '' s)).affine_preimage f),
rw ← set.image_subset_iff,
exact subset_convex_hull 𝕜 (f '' s) },
{ exact convex_hull_min (set.image_subset _ (subset_convex_hull 𝕜 s))
((convex_convex_hull... | lemma | affine_map.image_convex_hull | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"convex_convex_hull",
"convex_hull",
"convex_hull_min",
"set.image_subset",
"set.image_subset_iff",
"set.subset.antisymm",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_subset_affine_span : convex_hull 𝕜 s ⊆ (affine_span 𝕜 s : set E) | convex_hull_min (subset_affine_span 𝕜 s) (affine_span 𝕜 s).convex | lemma | convex_hull_subset_affine_span | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"affine_span",
"convex",
"convex_hull",
"convex_hull_min",
"subset_affine_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_span_convex_hull : affine_span 𝕜 (convex_hull 𝕜 s) = affine_span 𝕜 s | begin
refine le_antisymm _ (affine_span_mono 𝕜 (subset_convex_hull 𝕜 s)),
rw affine_span_le,
exact convex_hull_subset_affine_span s,
end | lemma | affine_span_convex_hull | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"affine_span",
"affine_span_le",
"affine_span_mono",
"convex_hull",
"convex_hull_subset_affine_span",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_neg (s : set E) : convex_hull 𝕜 (-s) = -convex_hull 𝕜 s | by { simp_rw ←image_neg, exact (affine_map.image_convex_hull _ $ -1).symm } | lemma | convex_hull_neg | analysis.convex | src/analysis/convex/hull.lean | [
"analysis.convex.basic",
"order.closure"
] | [
"affine_map.image_convex_hull",
"convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_independent (p : ι → E) : Prop | ∀ (s : set ι) (x : ι), p x ∈ convex_hull 𝕜 (p '' s) → x ∈ s | def | convex_independent | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_hull"
] | An indexed family is said to be convex independent if every point only belongs to convex hulls
of sets containing it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton.convex_independent [subsingleton ι] (p : ι → E) :
convex_independent 𝕜 p | λ s x hx, begin
have : (convex_hull 𝕜 (p '' s)).nonempty := ⟨p x, hx⟩,
rw [convex_hull_nonempty_iff, set.nonempty_image_iff] at this,
rwa subsingleton.mem_iff_nonempty,
end | lemma | subsingleton.convex_independent | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_hull",
"convex_hull_nonempty_iff",
"convex_independent",
"set.nonempty_image_iff",
"subsingleton.mem_iff_nonempty"
] | A family with at most one point is convex independent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent.injective {p : ι → E} (hc : convex_independent 𝕜 p) :
function.injective p | begin
refine λ i j hij, hc {j} i _,
rw [hij, set.image_singleton, convex_hull_singleton],
exact set.mem_singleton _,
end | lemma | convex_independent.injective | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_hull_singleton",
"convex_independent",
"set.image_singleton",
"set.mem_singleton"
] | A convex independent family is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent.comp_embedding {ι' : Type*} (f : ι' ↪ ι) {p : ι → E}
(hc : convex_independent 𝕜 p) :
convex_independent 𝕜 (p ∘ f) | begin
intros s x hx,
rw ←f.injective.mem_set_image,
exact hc _ _ (by rwa set.image_image),
end | lemma | convex_independent.comp_embedding | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_independent",
"set.image_image"
] | If a family is convex independent, so is any subfamily given by composition of an embedding into
index type with the original family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent.subtype {p : ι → E} (hc : convex_independent 𝕜 p) (s : set ι) :
convex_independent 𝕜 (λ i : s, p i) | hc.comp_embedding (embedding.subtype _) | lemma | convex_independent.subtype | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_independent"
] | If a family is convex independent, so is any subfamily indexed by a subtype of the index type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent.range {p : ι → E} (hc : convex_independent 𝕜 p) :
convex_independent 𝕜 (λ x, x : set.range p → E) | begin
let f : set.range p → ι := λ x, x.property.some,
have hf : ∀ x, p (f x) = x := λ x, x.property.some_spec,
let fe : set.range p ↪ ι := ⟨f, λ x₁ x₂ he, subtype.ext (hf x₁ ▸ hf x₂ ▸ he ▸ rfl)⟩,
convert hc.comp_embedding fe,
ext,
rw [embedding.coe_fn_mk, comp_app, hf],
end | lemma | convex_independent.range | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_independent",
"set.range",
"subtype.ext"
] | If an indexed family of points is convex independent, so is the corresponding set of points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent.mono {s t : set E} (hc : convex_independent 𝕜 (λ x, x : t → E))
(hs : s ⊆ t) :
convex_independent 𝕜 (λ x, x : s → E) | hc.comp_embedding (s.embedding_of_subset t hs) | lemma | convex_independent.mono | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_independent"
] | A subset of a convex independent set of points is convex independent as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.convex_independent_iff_set {p : ι → E}
(hi : function.injective p) :
convex_independent 𝕜 (λ x, x : set.range p → E) ↔ convex_independent 𝕜 p | ⟨λ hc, hc.comp_embedding
(⟨λ i, ⟨p i, set.mem_range_self _⟩, λ x y h, hi (subtype.mk_eq_mk.1 h)⟩ : ι ↪ set.range p),
convex_independent.range⟩ | lemma | function.injective.convex_independent_iff_set | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_independent",
"set.mem_range_self",
"set.range"
] | The range of an injective indexed family of points is convex independent iff that family is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent.mem_convex_hull_iff {p : ι → E}
(hc : convex_independent 𝕜 p) (s : set ι) (i : ι) :
p i ∈ convex_hull 𝕜 (p '' s) ↔ i ∈ s | ⟨hc _ _, λ hi, subset_convex_hull 𝕜 _ (set.mem_image_of_mem p hi)⟩ | lemma | convex_independent.mem_convex_hull_iff | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_hull",
"convex_independent",
"set.mem_image_of_mem",
"subset_convex_hull"
] | If a family is convex independent, a point in the family is in the convex hull of some of the
points given by a subset of the index type if and only if the point's index is in this subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent_iff_not_mem_convex_hull_diff {p : ι → E} :
convex_independent 𝕜 p ↔ ∀ i s, p i ∉ convex_hull 𝕜 (p '' (s \ {i})) | begin
refine ⟨λ hc i s h, _, λ h s i hi, _⟩,
{ rw hc.mem_convex_hull_iff at h,
exact h.2 (set.mem_singleton _) },
{ by_contra H,
refine h i s _,
rw set.diff_singleton_eq_self H,
exact hi }
end | lemma | convex_independent_iff_not_mem_convex_hull_diff | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"by_contra",
"convex_hull",
"convex_independent",
"set.diff_singleton_eq_self",
"set.mem_singleton"
] | If a family is convex independent, a point in the family is not in the convex hull of the other
points. See `convex_independent_set_iff_not_mem_convex_hull_diff` for the `set` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent_set_iff_inter_convex_hull_subset {s : set E} :
convex_independent 𝕜 (λ x, x : s → E) ↔ ∀ t, t ⊆ s → s ∩ convex_hull 𝕜 t ⊆ t | begin
split,
{ rintro hc t h x ⟨hxs, hxt⟩,
refine hc {x | ↑x ∈ t} ⟨x, hxs⟩ _,
rw subtype.coe_image_of_subset h,
exact hxt },
{ intros hc t x h,
rw ←subtype.coe_injective.mem_set_image,
exact hc (t.image coe) (subtype.coe_image_subset s t) ⟨x.prop, h⟩ }
end | lemma | convex_independent_set_iff_inter_convex_hull_subset | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_hull",
"convex_independent",
"subtype.coe_image_of_subset",
"subtype.coe_image_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_independent_set_iff_not_mem_convex_hull_diff {s : set E} :
convex_independent 𝕜 (λ x, x : s → E) ↔ ∀ x ∈ s, x ∉ convex_hull 𝕜 (s \ {x}) | begin
rw convex_independent_set_iff_inter_convex_hull_subset,
split,
{ rintro hs x hxs hx,
exact (hs _ (set.diff_subset _ _) ⟨hxs, hx⟩).2 (set.mem_singleton _) },
{ rintro hs t ht x ⟨hxs, hxt⟩,
by_contra h,
exact hs _ hxs (convex_hull_mono (set.subset_diff_singleton ht h) hxt) }
end | lemma | convex_independent_set_iff_not_mem_convex_hull_diff | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"by_contra",
"convex_hull",
"convex_hull_mono",
"convex_independent",
"convex_independent_set_iff_inter_convex_hull_subset",
"set.diff_subset",
"set.mem_singleton",
"set.subset_diff_singleton"
] | If a set is convex independent, a point in the set is not in the convex hull of the other
points. See `convex_independent_iff_not_mem_convex_hull_diff` for the indexed family version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_independent_iff_finset {p : ι → E} :
convex_independent 𝕜 p
↔ ∀ (s : finset ι) (x : ι), p x ∈ convex_hull 𝕜 (s.image p : set E) → x ∈ s | begin
refine ⟨λ hc s x hx, hc s x _, λ h s x hx, _⟩,
{ rwa finset.coe_image at hx },
have hp : injective p,
{ rintro a b hab,
rw ←mem_singleton,
refine h {b} a _,
rw [hab, image_singleton, coe_singleton, convex_hull_singleton],
exact set.mem_singleton _ },
rw convex_hull_eq_union_convex_hull_f... | lemma | convex_independent_iff_finset | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex_hull",
"convex_hull_eq_union_convex_hull_finite_subsets",
"convex_hull_singleton",
"convex_independent",
"finset",
"finset.coe_image",
"set.mem_Union",
"set.mem_singleton"
] | To check convex independence, one only has to check finsets thanks to Carathéodory's theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.convex_independent_extreme_points (hs : convex 𝕜 s) :
convex_independent 𝕜 (λ p, p : s.extreme_points 𝕜 → E) | convex_independent_set_iff_not_mem_convex_hull_diff.2 $ λ x hx h,
(extreme_points_convex_hull_subset
(inter_extreme_points_subset_extreme_points_of_subset (convex_hull_min
((set.diff_subset _ _).trans extreme_points_subset) hs) ⟨h, hx⟩)).2 (set.mem_singleton _) | lemma | convex.convex_independent_extreme_points | analysis.convex | src/analysis/convex/independent.lean | [
"analysis.convex.combination",
"analysis.convex.extreme"
] | [
"convex",
"convex_hull_min",
"convex_independent",
"extreme_points_convex_hull_subset",
"extreme_points_subset",
"inter_extreme_points_subset_extreme_points_of_subset",
"set.diff_subset",
"set.mem_singleton"
] | 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.