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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star_convex.linear_image (hs : star_convex 𝕜 x s) (f : E →ₗ[𝕜] F) :
star_convex 𝕜 (f x) (s.image f) | begin
intros y hy a b ha hb hab,
obtain ⟨y', hy', rfl⟩ := hy,
exact ⟨a • x + b • y', hs hy' ha hb hab, by rw [f.map_add, f.map_smul, f.map_smul]⟩,
end | lemma | star_convex.linear_image | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.is_linear_image (hs : star_convex 𝕜 x s) {f : E → F} (hf : is_linear_map 𝕜 f) :
star_convex 𝕜 (f x) (f '' s) | hs.linear_image $ hf.mk' f | lemma | star_convex.is_linear_image | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"is_linear_map",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.linear_preimage {s : set F} (f : E →ₗ[𝕜] F) (hs : star_convex 𝕜 (f x) s) :
star_convex 𝕜 x (s.preimage f) | begin
intros y hy a b ha hb hab,
rw [mem_preimage, f.map_add, f.map_smul, f.map_smul],
exact hs hy ha hb hab,
end | lemma | star_convex.linear_preimage | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.is_linear_preimage {s : set F} {f : E → F} (hs : star_convex 𝕜 (f x) s)
(hf : is_linear_map 𝕜 f) :
star_convex 𝕜 x (preimage f s) | hs.linear_preimage $ hf.mk' f | lemma | star_convex.is_linear_preimage | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"is_linear_map",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.add {t : set E} (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 y t) :
star_convex 𝕜 (x + y) (s + t) | by { rw ←add_image_prod, exact (hs.prod ht).is_linear_image is_linear_map.is_linear_map_add } | lemma | star_convex.add | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"is_linear_map.is_linear_map_add",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.add_left (hs : star_convex 𝕜 x s) (z : E) :
star_convex 𝕜 (z + x) ((λ x, z + x) '' s) | begin
intros y hy a b ha hb hab,
obtain ⟨y', hy', rfl⟩ := hy,
refine ⟨a • x + b • y', hs hy' ha hb hab, _⟩,
rw [smul_add, smul_add, add_add_add_comm, ←add_smul, hab, one_smul],
end | lemma | star_convex.add_left | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"one_smul",
"smul_add",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.add_right (hs : star_convex 𝕜 x s) (z : E) :
star_convex 𝕜 (x + z) ((λ x, x + z) '' s) | begin
intros y hy a b ha hb hab,
obtain ⟨y', hy', rfl⟩ := hy,
refine ⟨a • x + b • y', hs hy' ha hb hab, _⟩,
rw [smul_add, smul_add, add_add_add_comm, ←add_smul, hab, one_smul],
end | lemma | star_convex.add_right | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"one_smul",
"smul_add",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.preimage_add_right (hs : star_convex 𝕜 (z + x) s) :
star_convex 𝕜 x ((λ x, z + x) ⁻¹' s) | begin
intros y hy a b ha hb hab,
have h := hs hy ha hb hab,
rwa [smul_add, smul_add, add_add_add_comm, ←add_smul, hab, one_smul] at h,
end | lemma | star_convex.preimage_add_right | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"one_smul",
"smul_add",
"star_convex"
] | The translation of a star-convex set is also star-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex.preimage_add_left (hs : star_convex 𝕜 (x + z) s) :
star_convex 𝕜 x ((λ x, x + z) ⁻¹' s) | begin
rw add_comm at hs,
simpa only [add_comm] using hs.preimage_add_right,
end | lemma | star_convex.preimage_add_left | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | The translation of a star-convex set is also star-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex.sub' {s : set (E × E)} (hs : star_convex 𝕜 (x, y) s) :
star_convex 𝕜 (x - y) ((λ x : E × E, x.1 - x.2) '' s) | hs.is_linear_image is_linear_map.is_linear_map_sub | lemma | star_convex.sub' | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"is_linear_map.is_linear_map_sub",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.smul (hs : star_convex 𝕜 x s) (c : 𝕜) : star_convex 𝕜 (c • x) (c • s) | hs.linear_image $ linear_map.lsmul _ _ c | lemma | star_convex.smul | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"linear_map.lsmul",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.preimage_smul {c : 𝕜} (hs : star_convex 𝕜 (c • x) s) :
star_convex 𝕜 x ((λ z, c • z) ⁻¹' s) | hs.linear_preimage (linear_map.lsmul _ _ c) | lemma | star_convex.preimage_smul | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"linear_map.lsmul",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.affinity (hs : star_convex 𝕜 x s) (z : E) (c : 𝕜) :
star_convex 𝕜 (z + c • x) ((λ x, z + c • x) '' s) | begin
have h := (hs.smul c).add_left z,
rwa [←image_smul, image_image] at h,
end | lemma | star_convex.affinity | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_zero_iff :
star_convex 𝕜 0 s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s | begin
refine forall_congr (λ x, forall_congr $ λ hx, ⟨λ h a ha₀ ha₁, _, λ h a b ha hb hab, _⟩),
{ simpa only [sub_add_cancel, eq_self_iff_true, forall_true_left, zero_add, smul_zero] using
h (sub_nonneg_of_le ha₁) ha₀ },
{ rw [smul_zero, zero_add],
exact h hb (by { rw ←hab, exact le_add_of_nonneg_left h... | lemma | star_convex_zero_iff | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"forall_true_left",
"smul_zero",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.add_smul_mem (hs : star_convex 𝕜 x s) (hy : x + y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
(ht₁ : t ≤ 1) :
x + t • y ∈ s | begin
have h : x + t • y = (1 - t) • x + t • (x + y),
{ rw [smul_add, ←add_assoc, ←add_smul, sub_add_cancel, one_smul] },
rw h,
exact hs hy (sub_nonneg_of_le ht₁) ht₀ (sub_add_cancel _ _),
end | lemma | star_convex.add_smul_mem | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"one_smul",
"smul_add",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.smul_mem (hs : star_convex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
(ht₁ : t ≤ 1) :
t • x ∈ s | by simpa using hs.add_smul_mem (by simpa using hx) ht₀ ht₁ | lemma | star_convex.smul_mem | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.add_smul_sub_mem (hs : star_convex 𝕜 x s) (hy : y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
(ht₁ : t ≤ 1) :
x + t • (y - x) ∈ s | begin
apply hs.segment_subset hy,
rw segment_eq_image',
exact mem_image_of_mem _ ⟨ht₀, ht₁⟩,
end | lemma | star_convex.add_smul_sub_mem | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"segment_eq_image'",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.affine_preimage (f : E →ᵃ[𝕜] F) {s : set F} (hs : star_convex 𝕜 (f x) s) :
star_convex 𝕜 x (f ⁻¹' s) | begin
intros y hy a b ha hb hab,
rw [mem_preimage, convex.combo_affine_apply hab],
exact hs hy ha hb hab,
end | lemma | star_convex.affine_preimage | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"convex.combo_affine_apply",
"star_convex"
] | The preimage of a star-convex set under an affine map is star-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex.affine_image (f : E →ᵃ[𝕜] F) {s : set E} (hs : star_convex 𝕜 x s) :
star_convex 𝕜 (f x) (f '' s) | begin
rintro y ⟨y', ⟨hy', hy'f⟩⟩ a b ha hb hab,
refine ⟨a • x + b • y', ⟨hs hy' ha hb hab, _⟩⟩,
rw [convex.combo_affine_apply hab, hy'f],
end | lemma | star_convex.affine_image | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"convex.combo_affine_apply",
"star_convex"
] | The image of a star-convex set under an affine map is star-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex.neg (hs : star_convex 𝕜 x s) : star_convex 𝕜 (-x) (-s) | by { rw ←image_neg, exact hs.is_linear_image is_linear_map.is_linear_map_neg } | lemma | star_convex.neg | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"is_linear_map.is_linear_map_neg",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.sub (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 y t) :
star_convex 𝕜 (x - y) (s - t) | by { simp_rw sub_eq_add_neg, exact hs.add ht.neg } | lemma | star_convex.sub | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_iff_div :
star_convex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b →
(a / (a + b)) • x + (b / (a + b)) • y ∈ s | ⟨λ h y hy a b ha hb hab, begin
apply h hy,
{ have ha', from mul_le_mul_of_nonneg_left ha (inv_pos.2 hab).le,
rwa [mul_zero, ←div_eq_inv_mul] at ha' },
{ have hb', from mul_le_mul_of_nonneg_left hb (inv_pos.2 hab).le,
rwa [mul_zero, ←div_eq_inv_mul] at hb' },
{ rw ←add_div,
exact div_self hab.ne' }
e... | lemma | star_convex_iff_div | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"div_one",
"div_self",
"mul_le_mul_of_nonneg_left",
"mul_zero",
"star_convex",
"zero_lt_one"
] | Alternative definition of star-convexity, using division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex.mem_smul (hs : star_convex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t) :
x ∈ t • s | begin
rw mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne',
exact hs.smul_mem hx (inv_nonneg.2 $ zero_le_one.trans ht) (inv_le_one ht),
end | lemma | star_convex.mem_smul | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"inv_le_one",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.star_convex [ordered_semiring 𝕜] [ordered_add_comm_monoid E]
[module 𝕜 E] [ordered_smul 𝕜 E] {x : E} {s : set E} (hs : s.ord_connected) (hx : x ∈ s)
(h : ∀ y ∈ s, x ≤ y ∨ y ≤ x) :
star_convex 𝕜 x s | begin
intros y hy a b ha hb hab,
obtain hxy | hyx := h _ hy,
{ refine hs.out hx hy (mem_Icc.2 ⟨_, _⟩),
calc
x = a • x + b • x : (convex.combo_self hab _).symm
... ≤ a • x + b • y : add_le_add_left (smul_le_smul_of_nonneg hxy hb) _,
calc
a • x + b • y
≤ a • y + b • y : add_le_... | lemma | set.ord_connected.star_convex | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"convex.combo_self",
"module",
"ordered_add_comm_monoid",
"ordered_semiring",
"ordered_smul",
"smul_le_smul_of_nonneg",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_iff_ord_connected [linear_ordered_field 𝕜] {x : 𝕜} {s : set 𝕜} (hx : x ∈ s) :
star_convex 𝕜 x s ↔ s.ord_connected | by simp_rw [ord_connected_iff_uIcc_subset_left hx, star_convex_iff_segment_subset, segment_eq_uIcc] | lemma | star_convex_iff_ord_connected | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"linear_ordered_field",
"segment_eq_uIcc",
"star_convex",
"star_convex_iff_segment_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_disjoint_segment_convex_hull_triple {p q u v x y z : E}
(hz : z ∈ segment 𝕜 x y) (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬ disjoint (segment 𝕜 u v) (convex_hull 𝕜 {p, q, z}) | begin
rw not_disjoint_iff,
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz,
obtain rfl | haz' := haz.eq_or_lt,
{ rw zero_add at habz,
rw [zero_smul, zero_add, habz, one_smul],
refine ⟨v, right_mem_segment _ _ _, segment_subset_convex_hull _ _ hv⟩; simp },
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv,
obtai... | lemma | not_disjoint_segment_convex_hull_triple | analysis.convex | src/analysis/convex/stone_separation.lean | [
"analysis.convex.join"
] | [
"convex_hull",
"disjoint",
"div_eq_inv_mul",
"div_nonneg",
"div_self",
"fin.mk_bit0",
"fin.mk_one",
"finset",
"finset.center_mass",
"finset.center_mass_mem_convex_hull",
"finset.univ",
"list.foldr_cons",
"list.foldr_nil",
"list.pmap",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_right_... | In a tetrahedron with vertices `x`, `y`, `p`, `q`, any segment `[u, v]` joining the opposite
edges `[x, p]` and `[y, q]` passes through any triangle of vertices `p`, `q`, `z` where
`z ∈ [x, y]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_convex_convex_compl_subset (hs : convex 𝕜 s) (ht : convex 𝕜 t) (hst : disjoint s t) :
∃ C : set E, convex 𝕜 C ∧ convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ | begin
let S : set (set E) := {C | convex 𝕜 C ∧ disjoint C t},
obtain ⟨C, hC, hsC, hCmax⟩ := zorn_subset_nonempty S
(λ c hcS hc ⟨t, ht⟩, ⟨⋃₀ c, ⟨hc.directed_on.convex_sUnion (λ s hs, (hcS hs).1),
disjoint_sUnion_left.2 $ λ c hc, (hcS hc).2⟩, λ s, subset_sUnion_of_mem⟩) s ⟨hs, hst⟩,
refine ⟨C, hC.1, conve... | lemma | exists_convex_convex_compl_subset | analysis.convex | src/analysis/convex/stone_separation.lean | [
"analysis.convex.join"
] | [
"convex",
"convex_hull",
"convex_hull_insert",
"convex_hull_min",
"convex_join_singleton_left",
"disjoint",
"not_disjoint_segment_convex_hull_triple",
"segment",
"subset_convex_hull",
"zorn_subset_nonempty"
] | **Stone's Separation Theorem** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex : Prop | s.pairwise $ λ x y, ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s | def | strict_convex | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior"
] | A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_iff_open_segment_subset :
strict_convex 𝕜 s ↔ s.pairwise (λ x y, open_segment 𝕜 x y ⊆ interior s) | forall₅_congr $ λ x hx y hy hxy, (open_segment_subset_iff 𝕜).symm | lemma | strict_convex_iff_open_segment_subset | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"forall₅_congr",
"interior",
"open_segment",
"open_segment_subset_iff",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.open_segment_subset (hs : strict_convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) :
open_segment 𝕜 x y ⊆ interior s | strict_convex_iff_open_segment_subset.1 hs hx hy h | lemma | strict_convex.open_segment_subset | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior",
"open_segment",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_empty : strict_convex 𝕜 (∅ : set E) | pairwise_empty _ | lemma | strict_convex_empty | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_univ : strict_convex 𝕜 (univ : set E) | begin
intros x hx y hy hxy a b ha hb hab,
rw interior_univ,
exact mem_univ _,
end | lemma | strict_convex_univ | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior_univ",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.eq (hs : strict_convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a)
(hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y | hs.eq hx hy $ λ H, h $ H ha hb hab | lemma | strict_convex.eq | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.inter {t : set E} (hs : strict_convex 𝕜 s) (ht : strict_convex 𝕜 t) :
strict_convex 𝕜 (s ∩ t) | begin
intros x hx y hy hxy a b ha hb hab,
rw interior_inter,
exact ⟨hs hx.1 hy.1 hxy ha hb hab, ht hx.2 hy.2 hxy ha hb hab⟩,
end | lemma | strict_convex.inter | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior_inter",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
directed.strict_convex_Union {ι : Sort*} {s : ι → set E} (hdir : directed (⊆) s)
(hs : ∀ ⦃i : ι⦄, strict_convex 𝕜 (s i)) :
strict_convex 𝕜 (⋃ i, s i) | begin
rintro x hx y hy hxy a b ha hb hab,
rw mem_Union at hx hy,
obtain ⟨i, hx⟩ := hx,
obtain ⟨j, hy⟩ := hy,
obtain ⟨k, hik, hjk⟩ := hdir i j,
exact interior_mono (subset_Union s k) (hs (hik hx) (hjk hy) hxy ha hb hab),
end | lemma | directed.strict_convex_Union | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"directed",
"interior_mono",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
directed_on.strict_convex_sUnion {S : set (set E)} (hdir : directed_on (⊆) S)
(hS : ∀ s ∈ S, strict_convex 𝕜 s) :
strict_convex 𝕜 (⋃₀ S) | begin
rw sUnion_eq_Union,
exact (directed_on_iff_directed.1 hdir).strict_convex_Union (λ s, hS _ s.2),
end | lemma | directed_on.strict_convex_sUnion | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"directed_on",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.convex (hs : strict_convex 𝕜 s) : convex 𝕜 s | convex_iff_pairwise_pos.2 $ λ x hx y hy hxy a b ha hb hab, interior_subset $ hs hx hy hxy ha hb hab | lemma | strict_convex.convex | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex",
"interior_subset",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.strict_convex_of_open (h : is_open s) (hs : convex 𝕜 s) :
strict_convex 𝕜 s | λ x hx y hy _ a b ha hb hab, h.interior_eq.symm ▸ hs hx hy ha.le hb.le hab | lemma | convex.strict_convex_of_open | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex",
"is_open",
"strict_convex"
] | An open convex set is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open.strict_convex_iff (h : is_open s) : strict_convex 𝕜 s ↔ convex 𝕜 s | ⟨strict_convex.convex, convex.strict_convex_of_open h⟩ | lemma | is_open.strict_convex_iff | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex",
"convex.strict_convex_of_open",
"is_open",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_singleton (c : E) : strict_convex 𝕜 ({c} : set E) | pairwise_singleton _ _ | lemma | strict_convex_singleton | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.strict_convex (hs : s.subsingleton) : strict_convex 𝕜 s | hs.pairwise _ | lemma | set.subsingleton.strict_convex | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.linear_image [semiring 𝕝] [module 𝕝 E] [module 𝕝 F]
[linear_map.compatible_smul E F 𝕜 𝕝] (hs : strict_convex 𝕜 s) (f : E →ₗ[𝕝] F)
(hf : is_open_map f) :
strict_convex 𝕜 (f '' s) | begin
rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ hxy a b ha hb hab,
refine hf.image_interior_subset _ ⟨a • x + b • y, hs hx hy (ne_of_apply_ne _ hxy) ha hb hab, _⟩,
rw [map_add, f.map_smul_of_tower a, f.map_smul_of_tower b]
end | lemma | strict_convex.linear_image | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"is_open_map",
"linear_map.compatible_smul",
"module",
"ne_of_apply_ne",
"semiring",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.is_linear_image (hs : strict_convex 𝕜 s) {f : E → F} (h : is_linear_map 𝕜 f)
(hf : is_open_map f) :
strict_convex 𝕜 (f '' s) | hs.linear_image (h.mk' f) hf | lemma | strict_convex.is_linear_image | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"is_linear_map",
"is_open_map",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.linear_preimage {s : set F} (hs : strict_convex 𝕜 s) (f : E →ₗ[𝕜] F)
(hf : continuous f) (hfinj : injective f) :
strict_convex 𝕜 (s.preimage f) | begin
intros x hx y hy hxy a b ha hb hab,
refine preimage_interior_subset_interior_preimage hf _,
rw [mem_preimage, f.map_add, f.map_smul, f.map_smul],
exact hs hx hy (hfinj.ne hxy) ha hb hab,
end | lemma | strict_convex.linear_preimage | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"continuous",
"preimage_interior_subset_interior_preimage",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.is_linear_preimage {s : set F} (hs : strict_convex 𝕜 s) {f : E → F}
(h : is_linear_map 𝕜 f) (hf : continuous f) (hfinj : injective f) :
strict_convex 𝕜 (s.preimage f) | hs.linear_preimage (h.mk' f) hf hfinj | lemma | strict_convex.is_linear_preimage | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"continuous",
"is_linear_map",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.strict_convex {s : set β} (hs : ord_connected s) :
strict_convex 𝕜 s | begin
refine strict_convex_iff_open_segment_subset.2 (λ x hx y hy hxy, _),
cases hxy.lt_or_lt with hlt hlt; [skip, rw [open_segment_symm]];
exact (open_segment_subset_Ioo hlt).trans (is_open_Ioo.subset_interior_iff.2 $
Ioo_subset_Icc_self.trans $ hs.out ‹_› ‹_›)
end | lemma | set.ord_connected.strict_convex | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"open_segment_subset_Ioo",
"open_segment_symm",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Iic (r : β) : strict_convex 𝕜 (Iic r) | ord_connected_Iic.strict_convex | lemma | strict_convex_Iic | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Ici (r : β) : strict_convex 𝕜 (Ici r) | ord_connected_Ici.strict_convex | lemma | strict_convex_Ici | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Iio (r : β) : strict_convex 𝕜 (Iio r) | ord_connected_Iio.strict_convex | lemma | strict_convex_Iio | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Ioi (r : β) : strict_convex 𝕜 (Ioi r) | ord_connected_Ioi.strict_convex | lemma | strict_convex_Ioi | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Icc (r s : β) : strict_convex 𝕜 (Icc r s) | ord_connected_Icc.strict_convex | lemma | strict_convex_Icc | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Ioo (r s : β) : strict_convex 𝕜 (Ioo r s) | ord_connected_Ioo.strict_convex | lemma | strict_convex_Ioo | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Ico (r s : β) : strict_convex 𝕜 (Ico r s) | ord_connected_Ico.strict_convex | lemma | strict_convex_Ico | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_Ioc (r s : β) : strict_convex 𝕜 (Ioc r s) | ord_connected_Ioc.strict_convex | lemma | strict_convex_Ioc | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_uIcc (r s : β) : strict_convex 𝕜 (uIcc r s) | strict_convex_Icc _ _ | lemma | strict_convex_uIcc | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex",
"strict_convex_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_uIoc (r s : β) : strict_convex 𝕜 (uIoc r s) | strict_convex_Ioc _ _ | lemma | strict_convex_uIoc | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex",
"strict_convex_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.preimage_add_right (hs : strict_convex 𝕜 s) (z : E) :
strict_convex 𝕜 ((λ x, z + x) ⁻¹' s) | begin
intros x hx y hy hxy a b ha hb hab,
refine preimage_interior_subset_interior_preimage (continuous_add_left _) _,
have h := hs hx hy ((add_right_injective _).ne hxy) ha hb hab,
rwa [smul_add, smul_add, add_add_add_comm, ←add_smul, hab, one_smul] at h,
end | lemma | strict_convex.preimage_add_right | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"one_smul",
"preimage_interior_subset_interior_preimage",
"smul_add",
"strict_convex"
] | The translation of a strictly convex set is also strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.preimage_add_left (hs : strict_convex 𝕜 s) (z : E) :
strict_convex 𝕜 ((λ x, x + z) ⁻¹' s) | by simpa only [add_comm] using hs.preimage_add_right z | lemma | strict_convex.preimage_add_left | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | The translation of a strictly convex set is also strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.add (hs : strict_convex 𝕜 s) (ht : strict_convex 𝕜 t) :
strict_convex 𝕜 (s + t) | begin
rintro _ ⟨v, w, hv, hw, rfl⟩ _ ⟨x, y, hx, hy, rfl⟩ h a b ha hb hab,
rw [smul_add, smul_add, add_add_add_comm],
obtain rfl | hvx := eq_or_ne v x,
{ refine interior_mono (add_subset_add (singleton_subset_iff.2 hv) subset.rfl) _,
rw [convex.combo_self hab, singleton_add],
exact (is_open_map_add_left ... | lemma | strict_convex.add | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex.combo_self",
"eq_or_ne",
"interior_mono",
"ne_of_apply_ne",
"smul_add",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.add_left (hs : strict_convex 𝕜 s) (z : E) :
strict_convex 𝕜 ((λ x, z + x) '' s) | by simpa only [singleton_add] using (strict_convex_singleton z).add hs | lemma | strict_convex.add_left | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex",
"strict_convex_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.add_right (hs : strict_convex 𝕜 s) (z : E) :
strict_convex 𝕜 ((λ x, x + z) '' s) | by simpa only [add_comm] using hs.add_left z | lemma | strict_convex.add_right | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.vadd (hs : strict_convex 𝕜 s) (x : E) : strict_convex 𝕜 (x +ᵥ s) | hs.add_left x | lemma | strict_convex.vadd | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | The translation of a strictly convex set is also strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.smul (hs : strict_convex 𝕜 s) (c : 𝕝) : strict_convex 𝕜 (c • s) | begin
obtain rfl | hc := eq_or_ne c 0,
{ exact (subsingleton_zero_smul_set _).strict_convex },
{ exact hs.linear_image (linear_map.lsmul _ _ c) (is_open_map_smul₀ hc) }
end | lemma | strict_convex.smul | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"eq_or_ne",
"is_open_map_smul₀",
"linear_map.lsmul",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.affinity [has_continuous_add E] (hs : strict_convex 𝕜 s) (z : E) (c : 𝕝) :
strict_convex 𝕜 (z +ᵥ c • s) | (hs.smul c).vadd z | lemma | strict_convex.affinity | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"has_continuous_add",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.preimage_smul (hs : strict_convex 𝕜 s) (c : 𝕜) :
strict_convex 𝕜 ((λ z, c • z) ⁻¹' s) | begin
classical,
obtain rfl | hc := eq_or_ne c 0,
{ simp_rw [zero_smul, preimage_const],
split_ifs,
{ exact strict_convex_univ },
{ exact strict_convex_empty } },
refine hs.linear_preimage (linear_map.lsmul _ _ c) _ (smul_right_injective E hc),
unfold linear_map.lsmul linear_map.mk₂ linear_map.mk₂... | lemma | strict_convex.preimage_smul | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"eq_or_ne",
"linear_map.lsmul",
"linear_map.mk₂",
"linear_map.mk₂'",
"linear_map.mk₂'ₛₗ",
"smul_right_injective",
"strict_convex",
"strict_convex_empty",
"strict_convex_univ",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.eq_of_open_segment_subset_frontier [nontrivial 𝕜] [densely_ordered 𝕜]
(hs : strict_convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (h : open_segment 𝕜 x y ⊆ frontier s) :
x = y | begin
obtain ⟨a, ha₀, ha₁⟩ := densely_ordered.dense (0 : 𝕜) 1 zero_lt_one,
classical,
by_contra hxy,
exact (h ⟨a, 1 - a, ha₀, sub_pos_of_lt ha₁, add_sub_cancel'_right _ _, rfl⟩).2
(hs hx hy hxy ha₀ (sub_pos_of_lt ha₁) $ add_sub_cancel'_right _ _),
end | lemma | strict_convex.eq_of_open_segment_subset_frontier | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"by_contra",
"densely_ordered",
"frontier",
"nontrivial",
"open_segment",
"strict_convex",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.add_smul_mem (hs : strict_convex 𝕜 s) (hx : x ∈ s) (hxy : x + y ∈ s)
(hy : y ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) :
x + t • y ∈ interior s | begin
have h : x + t • y = (1 - t) • x + t • (x + y),
{ rw [smul_add, ←add_assoc, ←add_smul, sub_add_cancel, one_smul] },
rw h,
refine hs hx hxy (λ h, hy $ add_left_cancel _) (sub_pos_of_lt ht₁) ht₀ (sub_add_cancel _ _),
exact x,
rw [←h, add_zero],
end | lemma | strict_convex.add_smul_mem | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior",
"one_smul",
"smul_add",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.smul_mem_of_zero_mem (hs : strict_convex 𝕜 s) (zero_mem : (0 : E) ∈ s)
(hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) :
t • x ∈ interior s | by simpa using hs.add_smul_mem zero_mem (by simpa using hx) hx₀ ht₀ ht₁ | lemma | strict_convex.smul_mem_of_zero_mem | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.add_smul_sub_mem (h : strict_convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y)
{t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : x + t • (y - x) ∈ interior s | begin
apply h.open_segment_subset hx hy hxy,
rw open_segment_eq_image',
exact mem_image_of_mem _ ⟨ht₀, ht₁⟩,
end | lemma | strict_convex.add_smul_sub_mem | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior",
"open_segment_eq_image'",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.affine_preimage {s : set F} (hs : strict_convex 𝕜 s) {f : E →ᵃ[𝕜] F}
(hf : continuous f) (hfinj : injective f) :
strict_convex 𝕜 (f ⁻¹' s) | begin
intros x hx y hy hxy a b ha hb hab,
refine preimage_interior_subset_interior_preimage hf _,
rw [mem_preimage, convex.combo_affine_apply hab],
exact hs hx hy (hfinj.ne hxy) ha hb hab,
end | lemma | strict_convex.affine_preimage | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"continuous",
"convex.combo_affine_apply",
"preimage_interior_subset_interior_preimage",
"strict_convex"
] | The preimage of a strictly convex set under an affine map is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.affine_image (hs : strict_convex 𝕜 s) {f : E →ᵃ[𝕜] F} (hf : is_open_map f) :
strict_convex 𝕜 (f '' s) | begin
rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ hxy a b ha hb hab,
exact hf.image_interior_subset _ ⟨a • x + b • y, ⟨hs hx hy (ne_of_apply_ne _ hxy) ha hb hab,
convex.combo_affine_apply hab⟩⟩,
end | lemma | strict_convex.affine_image | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex.combo_affine_apply",
"is_open_map",
"ne_of_apply_ne",
"strict_convex"
] | The image of a strictly convex set under an affine map is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.neg (hs : strict_convex 𝕜 s) : strict_convex 𝕜 (-s) | hs.is_linear_preimage is_linear_map.is_linear_map_neg continuous_id.neg neg_injective | lemma | strict_convex.neg | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"is_linear_map.is_linear_map_neg",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex.sub (hs : strict_convex 𝕜 s) (ht : strict_convex 𝕜 t) :
strict_convex 𝕜 (s - t) | (sub_eq_add_neg s t).symm ▸ hs.add ht.neg | lemma | strict_convex.sub | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_iff_div :
strict_convex 𝕜 s ↔ s.pairwise
(λ x y, ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s) | ⟨λ h x hx y hy hxy a b ha hb, begin
apply h hx hy hxy (div_pos ha $ add_pos ha hb) (div_pos hb $ add_pos ha hb),
rw ←add_div,
exact div_self (add_pos ha hb).ne',
end, λ h x hx y hy hxy a b ha hb hab, by convert h hx hy hxy ha hb; rw [hab, div_one] ⟩ | lemma | strict_convex_iff_div | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"div_one",
"div_pos",
"div_self",
"interior",
"strict_convex"
] | Alternative definition of set strict convexity, using division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.mem_smul_of_zero_mem (hs : strict_convex 𝕜 s) (zero_mem : (0 : E) ∈ s)
(hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht : 1 < t) :
x ∈ t • interior s | begin
rw mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans ht).ne',
exact hs.smul_mem_of_zero_mem zero_mem hx hx₀ (inv_pos.2 $ zero_lt_one.trans ht) (inv_lt_one ht),
end | lemma | strict_convex.mem_smul_of_zero_mem | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"interior",
"inv_lt_one",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_iff_convex : strict_convex 𝕜 s ↔ convex 𝕜 s | ⟨strict_convex.convex, λ hs, hs.ord_connected.strict_convex⟩ | lemma | strict_convex_iff_convex | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex",
"strict_convex"
] | A set in a linear ordered field is strictly convex if and only if it is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_iff_ord_connected : strict_convex 𝕜 s ↔ s.ord_connected | strict_convex_iff_convex.trans convex_iff_ord_connected | lemma | strict_convex_iff_ord_connected | analysis.convex | src/analysis/convex/strict.lean | [
"analysis.convex.basic",
"topology.algebra.order.group"
] | [
"convex_iff_ord_connected",
"strict_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.dist_lt_max_dist (p : P) {p₁ p₂ p₃ : P} (h : sbtw ℝ p₁ p₂ p₃) :
dist p₂ p < max (dist p₁ p) (dist p₃ p) | begin
have hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p, { by simpa using h.left_ne_right },
rw [sbtw, ←wbtw_vsub_const_iff p, wbtw, affine_segment_eq_segment,
←insert_endpoints_open_segment, set.mem_insert_iff, set.mem_insert_iff] at h,
rcases h with ⟨h | h | h, hp₂p₁, hp₂p₃⟩,
{ rw vsub_left_cancel_iff at h, exact false.el... | lemma | sbtw.dist_lt_max_dist | analysis.convex | src/analysis/convex/strict_convex_between.lean | [
"analysis.convex.between",
"analysis.convex.strict_convex_space"
] | [
"affine_segment_eq_segment",
"dist_eq_norm_vsub",
"norm_combo_lt_of_ne",
"open_segment_eq_image",
"sbtw",
"set.mem_image",
"set.mem_insert_iff",
"vsub_left_cancel_iff",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.dist_le_max_dist (p : P) {p₁ p₂ p₃ : P} (h : wbtw ℝ p₁ p₂ p₃) :
dist p₂ p ≤ max (dist p₁ p) (dist p₃ p) | begin
by_cases hp₁ : p₂ = p₁, { simp [hp₁] },
by_cases hp₃ : p₂ = p₃, { simp [hp₃] },
have hs : sbtw ℝ p₁ p₂ p₃ := ⟨h, hp₁, hp₃⟩,
exact (hs.dist_lt_max_dist _).le
end | lemma | wbtw.dist_le_max_dist | analysis.convex | src/analysis/convex/strict_convex_between.lean | [
"analysis.convex.between",
"analysis.convex.strict_convex_space"
] | [
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
collinear.wbtw_of_dist_eq_of_dist_le {p p₁ p₂ p₃ : P} {r : ℝ}
(h : collinear ℝ ({p₁, p₂, p₃} : set P)) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p ≤ r)
(hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃) : wbtw ℝ p₁ p₂ p₃ | begin
rcases h.wbtw_or_wbtw_or_wbtw with hw | hw | hw,
{ exact hw },
{ by_cases hp₃p₂ : p₃ = p₂, { simp [hp₃p₂] },
have hs : sbtw ℝ p₂ p₃ p₁ := ⟨hw, hp₃p₂, hp₁p₃.symm⟩,
have hs' := hs.dist_lt_max_dist p,
rw [hp₁, hp₃, lt_max_iff, lt_self_iff_false, or_false] at hs',
exact false.elim (hp₂.not_lt hs... | lemma | collinear.wbtw_of_dist_eq_of_dist_le | analysis.convex | src/analysis/convex/strict_convex_between.lean | [
"analysis.convex.between",
"analysis.convex.strict_convex_space"
] | [
"collinear",
"lt_max_iff",
"lt_self_iff_false",
"sbtw",
"wbtw"
] | Given three collinear points, two (not equal) with distance `r` from `p` and one with
distance at most `r` from `p`, the third point is weakly between the other two points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
collinear.sbtw_of_dist_eq_of_dist_lt {p p₁ p₂ p₃ : P} {r : ℝ}
(h : collinear ℝ ({p₁, p₂, p₃} : set P)) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p < r)
(hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃) : sbtw ℝ p₁ p₂ p₃ | begin
refine ⟨h.wbtw_of_dist_eq_of_dist_le hp₁ hp₂.le hp₃ hp₁p₃, _, _⟩,
{ rintro rfl, exact hp₂.ne hp₁ },
{ rintro rfl, exact hp₂.ne hp₃ }
end | lemma | collinear.sbtw_of_dist_eq_of_dist_lt | analysis.convex | src/analysis/convex/strict_convex_between.lean | [
"analysis.convex.between",
"analysis.convex.strict_convex_space"
] | [
"collinear",
"sbtw"
] | Given three collinear points, two (not equal) with distance `r` from `p` and one with
distance less than `r` from `p`, the third point is strictly between the other two points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_space (𝕜 E : Type*) [normed_linear_ordered_field 𝕜] [normed_add_comm_group E]
[normed_space 𝕜 E] : Prop | (strict_convex_closed_ball : ∀ r : ℝ, 0 < r → strict_convex 𝕜 (closed_ball (0 : E) r)) | class | strict_convex_space | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"normed_add_comm_group",
"normed_linear_ordered_field",
"normed_space",
"strict_convex",
"strict_convex_closed_ball"
] | A *strictly convex space* is a normed space where the closed balls are strictly convex. We only
require balls of positive radius with center at the origin to be strictly convex in the definition,
then prove that any closed ball is strictly convex in `strict_convex_closed_ball` below.
See also `strict_convex_space.of_s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_closed_ball [strict_convex_space 𝕜 E] (x : E) (r : ℝ) :
strict_convex 𝕜 (closed_ball x r) | begin
cases le_or_lt r 0 with hr hr,
{ exact (subsingleton_closed_ball x hr).strict_convex },
rw ← vadd_closed_ball_zero,
exact (strict_convex_space.strict_convex_closed_ball r hr).vadd _,
end | lemma | strict_convex_closed_ball | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"strict_convex",
"strict_convex_space",
"vadd_closed_ball_zero"
] | A closed ball in a strictly convex space is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_space.of_strict_convex_closed_unit_ball
[linear_map.compatible_smul E E 𝕜 ℝ] (h : strict_convex 𝕜 (closed_ball (0 : E) 1)) :
strict_convex_space 𝕜 E | ⟨λ r hr, by simpa only [smul_closed_unit_ball_of_nonneg hr.le] using h.smul r⟩ | lemma | strict_convex_space.of_strict_convex_closed_unit_ball | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.compatible_smul",
"smul_closed_unit_ball_of_nonneg",
"strict_convex",
"strict_convex_space"
] | A real normed vector space is strictly convex provided that the unit ball is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_space.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
strict_convex_space ℝ E | begin
refine strict_convex_space.of_strict_convex_closed_unit_ball ℝ
((convex_closed_ball _ _).strict_convex' $ λ x hx y hy hne, _),
rw [interior_closed_ball (0 : E) one_ne_zero, closed_ball_diff_ball, mem_sphere_zero_iff_norm]
at hx hy,
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩,
use b,
rwa [affine... | lemma | strict_convex_space.of_norm_combo_lt_one | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"affine_map.line_map_apply_module",
"convex_closed_ball",
"interior_closed_ball",
"one_ne_zero",
"strict_convex_space",
"strict_convex_space.of_strict_convex_closed_unit_ball"
] | Strict convexity is equivalent to `‖a • x + b • y‖ < 1` for all `x` and `y` of norm at most `1`
and all strictly positive `a` and `b` such that `a + b = 1`. This lemma shows that it suffices to
check this for points of norm one and some `a`, `b` such that `a + b = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_space.of_norm_combo_ne_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y →
∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
strict_convex_space ℝ E | begin
refine strict_convex_space.of_strict_convex_closed_unit_ball ℝ
((convex_closed_ball _ _).strict_convex _),
simp only [interior_closed_ball _ one_ne_zero, closed_ball_diff_ball, set.pairwise,
frontier_closed_ball _ one_ne_zero, mem_sphere_zero_iff_norm],
intros x hx y hy hne,
rcases h x y hx hy hne... | lemma | strict_convex_space.of_norm_combo_ne_one | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"convex_closed_ball",
"frontier_closed_ball",
"interior_closed_ball",
"one_ne_zero",
"set.pairwise",
"strict_convex",
"strict_convex_space",
"strict_convex_space.of_strict_convex_closed_unit_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_space.of_norm_add_ne_two
(h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) :
strict_convex_space ℝ E | begin
refine strict_convex_space.of_norm_combo_ne_one
(λ x y hx hy hne, ⟨1/2, 1/2, one_half_pos.le, one_half_pos.le, add_halves _, _⟩),
rw [← smul_add, norm_smul, real.norm_of_nonneg one_half_pos.le, one_div,
← div_eq_inv_mul, ne.def, div_eq_one_iff_eq (two_ne_zero' ℝ)],
exact h hx hy hne,
end | lemma | strict_convex_space.of_norm_add_ne_two | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"add_halves",
"div_eq_inv_mul",
"div_eq_one_iff_eq",
"norm_smul",
"one_div",
"real.norm_of_nonneg",
"smul_add",
"strict_convex_space",
"strict_convex_space.of_norm_combo_ne_one",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_space.of_pairwise_sphere_norm_ne_two
(h : (sphere (0 : E) 1).pairwise $ λ x y, ‖x + y‖ ≠ 2) :
strict_convex_space ℝ E | strict_convex_space.of_norm_add_ne_two $ λ x y hx hy,
h (mem_sphere_zero_iff_norm.2 hx) (mem_sphere_zero_iff_norm.2 hy) | lemma | strict_convex_space.of_pairwise_sphere_norm_ne_two | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"pairwise",
"strict_convex_space",
"strict_convex_space.of_norm_add_ne_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_space.of_norm_add
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → same_ray ℝ x y) :
strict_convex_space ℝ E | begin
refine strict_convex_space.of_pairwise_sphere_norm_ne_two (λ x hx y hy, mt $ λ h₂, _),
rw mem_sphere_zero_iff_norm at hx hy,
exact (same_ray_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h₂)
end | lemma | strict_convex_space.of_norm_add | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"same_ray",
"same_ray_iff_of_norm_eq",
"strict_convex_space",
"strict_convex_space.of_pairwise_sphere_norm_ne_two"
] | If `‖x + y‖ = ‖x‖ + ‖y‖` implies that `x y : E` are in the same ray, then `E` is a strictly
convex space. See also a more | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
combo_mem_ball_of_ne (hx : x ∈ closed_ball z r) (hy : y ∈ closed_ball z r) (hne : x ≠ y)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : a • x + b • y ∈ ball z r | begin
rcases eq_or_ne r 0 with rfl|hr,
{ rw [closed_ball_zero, mem_singleton_iff] at hx hy,
exact (hne (hx.trans hy.symm)).elim },
{ simp only [← interior_closed_ball _ hr] at hx hy ⊢,
exact strict_convex_closed_ball ℝ z r hx hy hne ha hb hab }
end | lemma | combo_mem_ball_of_ne | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"eq_or_ne",
"interior_closed_ball",
"strict_convex_closed_ball"
] | If `x ≠ y` belong to the same closed ball, then a convex combination of `x` and `y` with
positive coefficients belongs to the corresponding open ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_segment_subset_ball_of_ne (hx : x ∈ closed_ball z r) (hy : y ∈ closed_ball z r)
(hne : x ≠ y) : open_segment ℝ x y ⊆ ball z r | (open_segment_subset_iff _).2 $ λ a b, combo_mem_ball_of_ne hx hy hne | lemma | open_segment_subset_ball_of_ne | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"combo_mem_ball_of_ne",
"open_segment",
"open_segment_subset_iff"
] | If `x ≠ y` belong to the same closed ball, then the open segment with endpoints `x` and `y` is
included in the corresponding open ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_combo_lt_of_ne (hx : ‖x‖ ≤ r) (hy : ‖y‖ ≤ r) (hne : x ≠ y) (ha : 0 < a) (hb : 0 < b)
(hab : a + b = 1) : ‖a • x + b • y‖ < r | begin
simp only [← mem_ball_zero_iff, ← mem_closed_ball_zero_iff] at hx hy ⊢,
exact combo_mem_ball_of_ne hx hy hne ha hb hab
end | lemma | norm_combo_lt_of_ne | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"combo_mem_ball_of_ne"
] | If `x` and `y` are two distinct vectors of norm at most `r`, then a convex combination of `x`
and `y` with positive coefficients has norm strictly less than `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_lt_of_not_same_ray (h : ¬same_ray ℝ x y) : ‖x + y‖ < ‖x‖ + ‖y‖ | begin
simp only [same_ray_iff_inv_norm_smul_eq, not_or_distrib, ← ne.def] at h,
rcases h with ⟨hx, hy, hne⟩,
rw ← norm_pos_iff at hx hy,
have hxy : 0 < ‖x‖ + ‖y‖ := add_pos hx hy,
have := combo_mem_ball_of_ne (inv_norm_smul_mem_closed_unit_ball x)
(inv_norm_smul_mem_closed_unit_ball y) hne (div_pos hx hxy... | lemma | norm_add_lt_of_not_same_ray | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"add_div",
"combo_mem_ball_of_ne",
"div_eq_inv_mul",
"div_lt_one",
"div_pos",
"div_self",
"inv_norm_smul_mem_closed_unit_ball",
"norm_smul",
"not_or_distrib",
"real.norm_of_nonneg",
"same_ray",
"same_ray_iff_inv_norm_smul_eq",
"smul_add",
"smul_inv_smul₀"
] | In a strictly convex space, if `x` and `y` are not in the same ray, then `‖x + y‖ < ‖x‖ +
‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_norm_sub_of_not_same_ray (h : ¬same_ray ℝ x y) : ‖x‖ - ‖y‖ < ‖x - y‖ | begin
nth_rewrite 0 ←sub_add_cancel x y at ⊢ h,
exact sub_lt_iff_lt_add.2 (norm_add_lt_of_not_same_ray $ λ H', h $ H'.add_left same_ray.rfl),
end | lemma | lt_norm_sub_of_not_same_ray | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"norm_add_lt_of_not_same_ray",
"same_ray",
"same_ray.rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_lt_norm_sub_of_not_same_ray (h : ¬same_ray ℝ x y) : |‖x‖ - ‖y‖| < ‖x - y‖ | begin
refine abs_sub_lt_iff.2 ⟨lt_norm_sub_of_not_same_ray h, _⟩,
rw norm_sub_rev,
exact lt_norm_sub_of_not_same_ray (mt same_ray.symm h),
end | lemma | abs_lt_norm_sub_of_not_same_ray | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"lt_norm_sub_of_not_same_ray",
"same_ray",
"same_ray.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray_iff_norm_add : same_ray ℝ x y ↔ ‖x + y‖ = ‖x‖ + ‖y‖ | ⟨same_ray.norm_add, λ h, not_not.1 $ λ h', (norm_add_lt_of_not_same_ray h').ne h⟩ | lemma | same_ray_iff_norm_add | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"norm_add_lt_of_not_same_ray",
"same_ray"
] | In a strictly convex space, two vectors `x`, `y` are in the same ray if and only if the triangle
inequality for `x` and `y` becomes an equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_norm_eq_of_norm_add_eq (h₁ : ‖x‖ = ‖y‖) (h₂ : ‖x + y‖ = ‖x‖ + ‖y‖) : x = y | (same_ray_iff_norm_add.mpr h₂).eq_of_norm_eq h₁ | lemma | eq_of_norm_eq_of_norm_add_eq | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [] | If `x` and `y` are two vectors in a strictly convex space have the same norm and the norm of
their sum is equal to the sum of their norms, then they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_same_ray_iff_norm_add_lt : ¬ same_ray ℝ x y ↔ ‖x + y‖ < ‖x‖ + ‖y‖ | same_ray_iff_norm_add.not.trans (norm_add_le _ _).lt_iff_ne.symm | lemma | not_same_ray_iff_norm_add_lt | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"same_ray"
] | In a strictly convex space, two vectors `x`, `y` are not in the same ray if and only if the
triangle inequality for `x` and `y` is strict. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
same_ray_iff_norm_sub : same_ray ℝ x y ↔ ‖x - y‖ = |‖x‖ - ‖y‖| | ⟨same_ray.norm_sub, λ h, not_not.1 $ λ h', (abs_lt_norm_sub_of_not_same_ray h').ne' h⟩ | lemma | same_ray_iff_norm_sub | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"abs_lt_norm_sub_of_not_same_ray",
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_same_ray_iff_abs_lt_norm_sub : ¬ same_ray ℝ x y ↔ |‖x‖ - ‖y‖| < ‖x - y‖ | same_ray_iff_norm_sub.not.trans $ ne_comm.trans (abs_norm_sub_norm_le _ _).lt_iff_ne.symm | lemma | not_same_ray_iff_abs_lt_norm_sub | analysis.convex | src/analysis/convex/strict_convex_space.lean | [
"analysis.convex.normed",
"analysis.convex.strict",
"analysis.normed.order.basic",
"analysis.normed_space.add_torsor",
"analysis.normed_space.pointwise",
"analysis.normed_space.affine_isometry"
] | [
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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