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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convex_join_singleton_segment (a b c : E) :
convex_join 𝕜 {a} (segment 𝕜 b c) = convex_hull 𝕜 {a, b, c} | by rw [←segment_same 𝕜, convex_join_segments, insert_idem] | lemma | convex_join_singleton_segment | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_hull",
"convex_join",
"convex_join_segments",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.convex_join (hs : convex 𝕜 s) (ht : convex 𝕜 t) :
convex 𝕜 (convex_join 𝕜 s t) | begin
rw convex_iff_segment_subset at ⊢ ht hs,
simp_rw mem_convex_join,
rintro x ⟨xa, hxa, xb, hxb, hx⟩ y ⟨ya, hya, yb, hyb, hy⟩,
refine (segment_subset_convex_join hx hy).trans _,
have triv : ({xa, xb, ya, yb} : set E) = {xa, ya, xb, yb} := by simp only [set.insert_comm],
rw [convex_join_segments, triv, ←c... | lemma | convex.convex_join | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex",
"convex_iff_segment_subset",
"convex_join",
"convex_join_mono",
"convex_join_segments",
"mem_convex_join",
"segment_subset_convex_join",
"set.insert_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.convex_hull_union (hs : convex 𝕜 s) (ht : convex 𝕜 t) (hs₀ : s.nonempty)
(ht₀ : t.nonempty) :
convex_hull 𝕜 (s ∪ t) = convex_join 𝕜 s t | (convex_hull_min (union_subset (subset_convex_join_left ht₀) $ subset_convex_join_right hs₀) $
hs.convex_join ht).antisymm $ convex_join_subset_convex_hull _ _ | lemma | convex.convex_hull_union | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex",
"convex_hull",
"convex_hull_min",
"convex_join",
"convex_join_subset_convex_hull",
"subset_convex_join_left",
"subset_convex_join_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_union (hs : s.nonempty) (ht : t.nonempty) :
convex_hull 𝕜 (s ∪ t) = convex_join 𝕜 (convex_hull 𝕜 s) (convex_hull 𝕜 t) | begin
rw [←convex_hull_convex_hull_union_left, ←convex_hull_convex_hull_union_right],
exact (convex_convex_hull 𝕜 s).convex_hull_union (convex_convex_hull 𝕜 t)
hs.convex_hull ht.convex_hull,
end | lemma | convex_hull_union | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_convex_hull",
"convex_hull",
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.has_extreme_point (hscomp : is_compact s) (hsnemp : s.nonempty) :
(s.extreme_points ℝ).nonempty | begin
let S : set (set E) := {t | t.nonempty ∧ is_closed t ∧ is_extreme ℝ s t},
rsuffices ⟨t, ⟨⟨x, hxt⟩, htclos, hst⟩, hBmin⟩ : ∃ t ∈ S, ∀ u ∈ S, u ⊆ t → u = t,
{ refine ⟨x, mem_extreme_points_iff_extreme_singleton.2 _⟩,
rwa ←eq_singleton_iff_unique_mem.2 ⟨hxt, λ y hyB, _⟩,
by_contra hyx,
obtain ⟨l, h... | lemma | is_compact.has_extreme_point | analysis.convex | src/analysis/convex/krein_milman.lean | [
"analysis.convex.exposed",
"analysis.normed_space.hahn_banach.separation"
] | [
"by_contra",
"geometric_hahn_banach_point_point",
"is_closed",
"is_closed_sInter",
"is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed",
"is_compact_of_is_closed_subset",
"is_exposed",
"is_extreme",
"is_extreme_sInter",
"zorn_superset"
] | **Krein-Milman lemma**: In a LCTVS, any nonempty compact set has an extreme point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_convex_hull_extreme_points (hscomp : is_compact s) (hAconv : convex ℝ s) :
closure (convex_hull ℝ $ s.extreme_points ℝ) = s | begin
apply (closure_minimal (convex_hull_min extreme_points_subset hAconv) hscomp.is_closed).antisymm,
by_contra hs,
obtain ⟨x, hxA, hxt⟩ := not_subset.1 hs,
obtain ⟨l, r, hlr, hrx⟩ := geometric_hahn_banach_closed_point (convex_convex_hull _ _).closure
is_closed_closure hxt,
have h : is_exposed ℝ s {y ∈ ... | lemma | closure_convex_hull_extreme_points | analysis.convex | src/analysis/convex/krein_milman.lean | [
"analysis.convex.exposed",
"analysis.normed_space.hahn_banach.separation"
] | [
"by_contra",
"closure",
"closure_minimal",
"convex",
"convex_convex_hull",
"convex_hull",
"convex_hull_min",
"extreme_points_subset",
"geometric_hahn_banach_closed_point",
"is_closed_closure",
"is_compact",
"is_exposed",
"subset_closure",
"subset_convex_hull"
] | **Krein-Milman theorem**: In a LCTVS, any compact convex set is the closure of the convex hull
of its extreme points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_haar_frontier (hs : convex ℝ s) : μ (frontier s) = 0 | begin
/- If `s` is included in a hyperplane, then `frontier s ⊆ closure s` is included in the same
hyperplane, hence it has measure zero. -/
cases ne_or_eq (affine_span ℝ s) ⊤ with hspan hspan,
{ refine measure_mono_null _ (add_haar_affine_subspace _ _ hspan),
exact frontier_subset_closure.trans (closure_mi... | lemma | convex.add_haar_frontier | analysis.convex | src/analysis/convex/measure.lean | [
"analysis.convex.topology",
"analysis.normed_space.add_torsor_bases",
"measure_theory.measure.lebesgue.eq_haar"
] | [
"affine_span",
"closure",
"closure_minimal",
"continuous_pow",
"convex",
"convex_ball",
"ennreal.continuous_mul_const",
"ennreal.of_real_coe_nnreal",
"finite_dimensional.finrank",
"frontier",
"frontier_inter_open_inter",
"ge_of_tendsto",
"interior",
"interior_inter",
"interior_subset",
... | Haar measure of the frontier of a convex set is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_measurable_set (hs : convex ℝ s) : null_measurable_set s μ | null_measurable_set_of_null_frontier (hs.add_haar_frontier μ) | lemma | convex.null_measurable_set | analysis.convex | src/analysis/convex/measure.lean | [
"analysis.convex.topology",
"analysis.normed_space.add_torsor_bases",
"measure_theory.measure.lebesgue.eq_haar"
] | [
"convex",
"null_measurable_set_of_null_frontier"
] | A convex set in a finite dimensional real vector space is null measurable with respect to an
additive Haar measure on this space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_norm (hs : convex ℝ s) : convex_on ℝ s norm | ⟨hs, λ x hx y hy a b ha hb hab,
calc ‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ : norm_add_le _ _
... = a * ‖x‖ + b * ‖y‖
: by rw [norm_smul, norm_smul, real.norm_of_nonneg ha, real.norm_of_nonneg hb]⟩ | lemma | convex_on_norm | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex",
"convex_on",
"norm_smul",
"real.norm_of_nonneg"
] | The norm on a real normed space is convex on any convex set. See also `seminorm.convex_on`
and `convex_on_univ_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_univ_norm : convex_on ℝ univ (norm : E → ℝ) | convex_on_norm convex_univ | lemma | convex_on_univ_norm | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_on",
"convex_on_norm",
"convex_univ"
] | The norm on a real normed space is convex on the whole space. See also `seminorm.convex_on`
and `convex_on_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_dist (z : E) (hs : convex ℝ s) : convex_on ℝ s (λ z', dist z' z) | by simpa [dist_eq_norm, preimage_preimage]
using (convex_on_norm (hs.translate (-z))).comp_affine_map
(affine_map.id ℝ E - affine_map.const ℝ E z) | lemma | convex_on_dist | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"affine_map.const",
"affine_map.id",
"convex",
"convex_on",
"convex_on_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_univ_dist (z : E) : convex_on ℝ univ (λz', dist z' z) | convex_on_dist z convex_univ | lemma | convex_on_univ_dist | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_on",
"convex_on_dist",
"convex_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_ball (a : E) (r : ℝ) : convex ℝ (metric.ball a r) | by simpa only [metric.ball, sep_univ] using (convex_on_univ_dist a).convex_lt r | lemma | convex_ball | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex",
"convex_on_univ_dist",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_closed_ball (a : E) (r : ℝ) : convex ℝ (metric.closed_ball a r) | by simpa only [metric.closed_ball, sep_univ] using (convex_on_univ_dist a).convex_le r | lemma | convex_closed_ball | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex",
"convex_on_univ_dist",
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.thickening (hs : convex ℝ s) (δ : ℝ) : convex ℝ (thickening δ s) | by { rw ←add_ball_zero, exact hs.add (convex_ball 0 _) } | lemma | convex.thickening | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex",
"convex_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.cthickening (hs : convex ℝ s) (δ : ℝ) : convex ℝ (cthickening δ s) | begin
obtain hδ | hδ := le_total 0 δ,
{ rw cthickening_eq_Inter_thickening hδ,
exact convex_Inter₂ (λ _ _, hs.thickening _) },
{ rw cthickening_of_nonpos hδ,
exact hs.closure }
end | lemma | convex.cthickening | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex",
"convex_Inter₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_exists_dist_ge {s : set E} {x : E} (hx : x ∈ convex_hull ℝ s) (y : E) :
∃ x' ∈ s, dist x y ≤ dist x' y | (convex_on_dist y (convex_convex_hull ℝ _)).exists_ge_of_mem_convex_hull hx | lemma | convex_hull_exists_dist_ge | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_convex_hull",
"convex_hull",
"convex_on_dist"
] | Given a point `x` in the convex hull of `s` and a point `y`, there exists a point
of `s` at distance at least `dist x y` from `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_hull_exists_dist_ge2 {s t : set E} {x y : E}
(hx : x ∈ convex_hull ℝ s) (hy : y ∈ convex_hull ℝ t) :
∃ (x' ∈ s) (y' ∈ t), dist x y ≤ dist x' y' | begin
rcases convex_hull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩,
rcases convex_hull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩,
use [x', hx', y', hy'],
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
end | lemma | convex_hull_exists_dist_ge2 | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_hull",
"convex_hull_exists_dist_ge",
"dist_comm"
] | Given a point `x` in the convex hull of `s` and a point `y` in the convex hull of `t`,
there exist points `x' ∈ s` and `y' ∈ t` at distance at least `dist x y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_hull_ediam (s : set E) :
emetric.diam (convex_hull ℝ s) = emetric.diam s | begin
refine (emetric.diam_le $ λ x hx y hy, _).antisymm (emetric.diam_mono $ subset_convex_hull ℝ s),
rcases convex_hull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩,
rw edist_dist,
apply le_trans (ennreal.of_real_le_of_real H),
rw ← edist_dist,
exact emetric.edist_le_diam_of_mem hx' hy'
end | lemma | convex_hull_ediam | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_hull",
"convex_hull_exists_dist_ge2",
"edist_dist",
"emetric.diam",
"emetric.diam_le",
"emetric.diam_mono",
"emetric.edist_le_diam_of_mem",
"ennreal.of_real_le_of_real",
"subset_convex_hull"
] | Emetric diameter of the convex hull of a set `s` equals the emetric diameter of `s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_hull_diam (s : set E) :
metric.diam (convex_hull ℝ s) = metric.diam s | by simp only [metric.diam, convex_hull_ediam] | lemma | convex_hull_diam | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_hull",
"convex_hull_ediam",
"metric.diam"
] | Diameter of the convex hull of a set `s` equals the emetric diameter of `s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_convex_hull {s : set E} :
metric.bounded (convex_hull ℝ s) ↔ metric.bounded s | by simp only [metric.bounded_iff_ediam_ne_top, convex_hull_ediam] | lemma | bounded_convex_hull | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_hull",
"convex_hull_ediam",
"metric.bounded",
"metric.bounded_iff_ediam_ne_top"
] | Convex hull of `s` is bounded if and only if `s` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.path_connected : path_connected_space E | topological_add_group.path_connected | instance | normed_space.path_connected | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"path_connected_space",
"topological_add_group.path_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space.loc_path_connected : loc_path_connected_space E | loc_path_connected_of_bases (λ x, metric.nhds_basis_ball)
(λ x r r_pos, (convex_ball x r).is_path_connected $ by simp [r_pos]) | instance | normed_space.loc_path_connected | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"convex_ball",
"is_path_connected",
"loc_path_connected_of_bases",
"loc_path_connected_space",
"metric.nhds_basis_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_add_dist_of_mem_segment {x y z : E} (h : y ∈ [x -[ℝ] z]) :
dist x y + dist y z = dist x z | begin
simp only [dist_eq_norm, mem_segment_iff_same_ray] at *,
simpa only [sub_add_sub_cancel', norm_sub_rev] using h.norm_add.symm
end | lemma | dist_add_dist_of_mem_segment | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"mem_segment_iff_same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_set_of_same_ray (x : E) : is_connected {y | same_ray ℝ x y} | begin
by_cases hx : x = 0, { simpa [hx] using is_connected_univ },
simp_rw ←exists_nonneg_left_iff_same_ray hx,
exact is_connected_Ici.image _ ((continuous_id.smul continuous_const).continuous_on)
end | lemma | is_connected_set_of_same_ray | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"continuous_const",
"continuous_on",
"is_connected",
"is_connected_univ",
"same_ray"
] | The set of vectors in the same ray as `x` is connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected_set_of_same_ray_and_ne_zero {x : E} (hx : x ≠ 0) :
is_connected {y | same_ray ℝ x y ∧ y ≠ 0} | begin
simp_rw ←exists_pos_left_iff_same_ray_and_ne_zero hx,
exact is_connected_Ioi.image _ ((continuous_id.smul continuous_const).continuous_on)
end | lemma | is_connected_set_of_same_ray_and_ne_zero | analysis.convex | src/analysis/convex/normed.lean | [
"analysis.convex.jensen",
"analysis.convex.topology",
"analysis.normed.group.pointwise",
"analysis.normed_space.ray"
] | [
"continuous_const",
"continuous_on",
"is_connected",
"same_ray"
] | The set of nonzero vectors in the same ray as the nonzero vector `x` is connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partition_of_unity.finsum_smul_mem_convex {s : set X} (f : partition_of_unity ι X s)
{g : ι → X → E} {t : set E} {x : X} (hx : x ∈ s) (hg : ∀ i, f i x ≠ 0 → g i x ∈ t)
(ht : convex ℝ t) :
∑ᶠ i, f i x • g i x ∈ t | ht.finsum_mem (λ i, f.nonneg _ _) (f.sum_eq_one hx) hg | lemma | partition_of_unity.finsum_smul_mem_convex | analysis.convex | src/analysis/convex/partition_of_unity.lean | [
"topology.partition_of_unity",
"analysis.convex.combination"
] | [
"convex",
"partition_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_continuous_forall_mem_convex_of_local (ht : ∀ x, convex ℝ (t x))
(H : ∀ x : X, ∃ (U ∈ 𝓝 x) (g : X → E), continuous_on g U ∧ ∀ y ∈ U, g y ∈ t y) : ∃
g : C(X, E), ∀ x, g x ∈ t x | begin
choose U hU g hgc hgt using H,
obtain ⟨f, hf⟩ := partition_of_unity.exists_is_subordinate is_closed_univ (λ x, interior (U x))
(λ x, is_open_interior) (λ x hx, mem_Union.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩),
refine ⟨⟨λ x, ∑ᶠ i, f i x • g i x,
hf.continuous_finsum_smul (λ i, is_open_interior) $... | lemma | exists_continuous_forall_mem_convex_of_local | analysis.convex | src/analysis/convex/partition_of_unity.lean | [
"topology.partition_of_unity",
"analysis.convex.combination"
] | [
"continuous_on",
"convex",
"interior",
"interior_subset",
"is_closed_univ",
"is_open_interior",
"partition_of_unity.exists_is_subordinate",
"subset_closure"
] | Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
each point `x : X`, there exists a neighborhood `U ∈ 𝓝 X` and a function `g : X → E` that is
continuous on `U` and sends each... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_continuous_forall_mem_convex_of_local_const (ht : ∀ x, convex ℝ (t x))
(H : ∀ x : X, ∃ c : E, ∀ᶠ y in 𝓝 x, c ∈ t y) :
∃ g : C(X, E), ∀ x, g x ∈ t x | exists_continuous_forall_mem_convex_of_local ht $ λ x,
let ⟨c, hc⟩ := H x in ⟨_, hc, λ _, c, continuous_on_const, λ y, id⟩ | lemma | exists_continuous_forall_mem_convex_of_local_const | analysis.convex | src/analysis/convex/partition_of_unity.lean | [
"topology.partition_of_unity",
"analysis.convex.combination"
] | [
"continuous_on_const",
"convex",
"exists_continuous_forall_mem_convex_of_local"
] | Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
each point `x : X`, there exists a vector `c : E` that belongs to `t y` for all `y` in a
neighborhood of `x`. Then there exist... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.Ici_extend (hf : convex 𝕜 s) :
convex 𝕜 {x | Ici_extend (restrict (Ici z) (∈ s)) x} | by { rw convex_iff_ord_connected at ⊢ hf, exact hf.restrict.Ici_extend } | lemma | convex.Ici_extend | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"convex",
"convex_iff_ord_connected"
] | A convex set extended towards minus infinity is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.Iic_extend (hf : convex 𝕜 s) :
convex 𝕜 {x | Iic_extend (restrict (Iic z) (∈ s)) x} | by { rw convex_iff_ord_connected at ⊢ hf, exact hf.restrict.Iic_extend } | lemma | convex.Iic_extend | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"convex",
"convex_iff_ord_connected"
] | A convex set extended towards infinity is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.Ici_extend (hf : convex_on 𝕜 s f) (hf' : monotone_on f s) :
convex_on 𝕜 {x | Ici_extend (restrict (Ici z) (∈ s)) x} (Ici_extend $ restrict (Ici z) f) | begin
refine ⟨hf.1.Ici_extend, λ x hx y hy a b ha hb hab, _⟩,
dsimp [Ici_extend_apply] at ⊢ hx hy,
refine (hf' (hf.1.ord_connected.uIcc_subset hx hy $ monotone.image_uIcc_subset (λ _ _, max_le_max
le_rfl) $ mem_image_of_mem _ $ convex_uIcc _ _ left_mem_uIcc right_mem_uIcc ha hb hab)
(hf.1 hx hy ha hb hab)... | lemma | convex_on.Ici_extend | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"convex.combo_self",
"convex_on",
"convex_uIcc",
"le_rfl",
"max_le_max",
"monotone.image_uIcc_subset",
"monotone_on",
"smul_eq_mul",
"smul_max"
] | A convex monotone function extended constantly towards minus infinity is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.Iic_extend (hf : convex_on 𝕜 s f) (hf' : antitone_on f s) :
convex_on 𝕜 {x | Iic_extend (restrict (Iic z) (∈ s)) x} (Iic_extend $ restrict (Iic z) f) | begin
refine ⟨hf.1.Iic_extend, λ x hx y hy a b ha hb hab, _⟩,
dsimp [Iic_extend_apply] at ⊢ hx hy,
refine (hf' (hf.1 hx hy ha hb hab) (hf.1.ord_connected.uIcc_subset hx hy $
monotone.image_uIcc_subset (λ _ _, min_le_min le_rfl) $ mem_image_of_mem _ $
convex_uIcc _ _ left_mem_uIcc right_mem_uIcc ha hb ha... | lemma | convex_on.Iic_extend | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"antitone_on",
"convex.combo_self",
"convex_on",
"convex_uIcc",
"le_rfl",
"min_le_min",
"monotone.image_uIcc_subset",
"smul_eq_mul",
"smul_min"
] | A convex antitone function extended constantly towards infinity is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.Ici_extend (hf : concave_on 𝕜 s f) (hf' : antitone_on f s) :
concave_on 𝕜 {x | Ici_extend (restrict (Ici z) (∈ s)) x} (Ici_extend $ restrict (Ici z) f) | hf.dual.Ici_extend hf'.dual_right | lemma | concave_on.Ici_extend | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"antitone_on",
"concave_on"
] | A concave antitone function extended constantly minus towards infinity is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.Iic_extend (hf : concave_on 𝕜 s f) (hf' : monotone_on f s) :
concave_on 𝕜 {x | Iic_extend (restrict (Iic z) (∈ s)) x} (Iic_extend $ restrict (Iic z) f) | hf.dual.Iic_extend hf'.dual_right | lemma | concave_on.Iic_extend | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"concave_on",
"monotone_on"
] | A concave monotone function extended constantly towards infinity is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.Ici_extend_of_monotone (hf : convex_on 𝕜 univ f) (hf' : monotone f) :
convex_on 𝕜 univ (Ici_extend $ restrict (Ici z) f) | hf.Ici_extend $ hf'.monotone_on _ | lemma | convex_on.Ici_extend_of_monotone | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"convex_on",
"monotone"
] | A convex monotone function extended constantly towards minus infinity is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.Iic_extend_of_antitone (hf : convex_on 𝕜 univ f) (hf' : antitone f) :
convex_on 𝕜 univ (Iic_extend $ restrict (Iic z) f) | hf.Iic_extend $ hf'.antitone_on _ | lemma | convex_on.Iic_extend_of_antitone | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"antitone",
"convex_on"
] | A convex antitone function extended constantly towards infinity is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.Ici_extend_of_antitone (hf : concave_on 𝕜 univ f) (hf' : antitone f) :
concave_on 𝕜 univ (Ici_extend $ restrict (Ici z) f) | hf.Ici_extend $ hf'.antitone_on _ | lemma | concave_on.Ici_extend_of_antitone | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"antitone",
"concave_on"
] | A concave antitone function extended constantly minus towards infinity is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.Iic_extend_of_monotone (hf : concave_on 𝕜 univ f) (hf' : monotone f) :
concave_on 𝕜 univ (Iic_extend $ restrict (Iic z) f) | hf.Iic_extend $ hf'.monotone_on _ | lemma | concave_on.Iic_extend_of_monotone | analysis.convex | src/analysis/convex/proj_Icc.lean | [
"analysis.convex.function",
"data.set.intervals.proj_Icc"
] | [
"concave_on",
"monotone"
] | A concave monotone function extended constantly towards infinity is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasiconvex_on : Prop | ∀ r, convex 𝕜 {x ∈ s | f x ≤ r} | def | quasiconvex_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex"
] | A function is quasiconvex if all its sublevels are convex.
This means that, for all `r`, `{x ∈ s | f x ≤ r}` is `𝕜`-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasiconcave_on : Prop | ∀ r, convex 𝕜 {x ∈ s | r ≤ f x} | def | quasiconcave_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex"
] | A function is quasiconcave if all its superlevels are convex.
This means that, for all `r`, `{x ∈ s | r ≤ f x}` is `𝕜`-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasilinear_on : Prop | quasiconvex_on 𝕜 s f ∧ quasiconcave_on 𝕜 s f | def | quasilinear_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"quasiconcave_on",
"quasiconvex_on"
] | A function is quasilinear if it is both quasiconvex and quasiconcave.
This means that, for all `r`,
the sets `{x ∈ s | f x ≤ r}` and `{x ∈ s | r ≤ f x}` are `𝕜`-convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasiconvex_on.dual : quasiconvex_on 𝕜 s f → quasiconcave_on 𝕜 s (to_dual ∘ f) | id | lemma | quasiconvex_on.dual | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"quasiconcave_on",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconcave_on.dual : quasiconcave_on 𝕜 s f → quasiconvex_on 𝕜 s (to_dual ∘ f) | id | lemma | quasiconcave_on.dual | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"quasiconcave_on",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasilinear_on.dual : quasilinear_on 𝕜 s f → quasilinear_on 𝕜 s (to_dual ∘ f) | and.swap | lemma | quasilinear_on.dual | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.quasiconvex_on_of_convex_le (hs : convex 𝕜 s) (h : ∀ r, convex 𝕜 {x | f x ≤ r}) :
quasiconvex_on 𝕜 s f | λ r, hs.inter (h r) | lemma | convex.quasiconvex_on_of_convex_le | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.quasiconcave_on_of_convex_ge (hs : convex 𝕜 s) (h : ∀ r, convex 𝕜 {x | r ≤ f x}) :
quasiconcave_on 𝕜 s f | @convex.quasiconvex_on_of_convex_le 𝕜 E βᵒᵈ _ _ _ _ _ _ hs h | lemma | convex.quasiconcave_on_of_convex_ge | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"convex.quasiconvex_on_of_convex_le",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconvex_on.convex [is_directed β (≤)] (hf : quasiconvex_on 𝕜 s f) : convex 𝕜 s | λ x hx y hy a b ha hb hab,
let ⟨z, hxz, hyz⟩ := exists_ge_ge (f x) (f y) in (hf _ ⟨hx, hxz⟩ ⟨hy, hyz⟩ ha hb hab).1 | lemma | quasiconvex_on.convex | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"exists_ge_ge",
"is_directed",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconcave_on.convex [is_directed β (≥)] (hf : quasiconcave_on 𝕜 s f) : convex 𝕜 s | hf.dual.convex | lemma | quasiconcave_on.convex | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"is_directed",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconvex_on.sup (hf : quasiconvex_on 𝕜 s f) (hg : quasiconvex_on 𝕜 s g) :
quasiconvex_on 𝕜 s (f ⊔ g) | begin
intro r,
simp_rw [pi.sup_def, sup_le_iff, set.sep_and],
exact (hf r).inter (hg r),
end | lemma | quasiconvex_on.sup | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"pi.sup_def",
"quasiconvex_on",
"set.sep_and",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconcave_on.inf (hf : quasiconcave_on 𝕜 s f) (hg : quasiconcave_on 𝕜 s g) :
quasiconcave_on 𝕜 s (f ⊓ g) | hf.dual.sup hg | lemma | quasiconcave_on.inf | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconvex_on_iff_le_max :
quasiconvex_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
f (a • x + b • y) ≤ max (f x) (f y) | ⟨λ hf, ⟨hf.convex, λ x hx y hy a b ha hb hab,
(hf _ ⟨hx, le_max_left _ _⟩ ⟨hy, le_max_right _ _⟩ ha hb hab).2⟩,
λ hf r x hx y hy a b ha hb hab,
⟨hf.1 hx.1 hy.1 ha hb hab, (hf.2 hx.1 hy.1 ha hb hab).trans $ max_le hx.2 hy.2⟩⟩ | lemma | quasiconvex_on_iff_le_max | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconcave_on_iff_min_le :
quasiconcave_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
min (f x) (f y) ≤ f (a • x + b • y) | @quasiconvex_on_iff_le_max 𝕜 E βᵒᵈ _ _ _ _ _ _ | lemma | quasiconcave_on_iff_min_le | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"quasiconcave_on",
"quasiconvex_on_iff_le_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasilinear_on_iff_mem_uIcc :
quasilinear_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
f (a • x + b • y) ∈ uIcc (f x) (f y) | begin
rw [quasilinear_on, quasiconvex_on_iff_le_max, quasiconcave_on_iff_min_le, and_and_and_comm,
and_self],
apply and_congr_right',
simp_rw [←forall_and_distrib, ←Icc_min_max, mem_Icc, and_comm],
end | lemma | quasilinear_on_iff_mem_uIcc | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"and_and_and_comm",
"and_congr_right'",
"convex",
"quasiconcave_on_iff_min_le",
"quasiconvex_on_iff_le_max",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconvex_on.convex_lt (hf : quasiconvex_on 𝕜 s f) (r : β) : convex 𝕜 {x ∈ s | f x < r} | begin
refine λ x hx y hy a b ha hb hab, _,
have h := hf _ ⟨hx.1, le_max_left _ _⟩ ⟨hy.1, le_max_right _ _⟩ ha hb hab,
exact ⟨h.1, h.2.trans_lt $ max_lt hx.2 hy.2⟩,
end | lemma | quasiconvex_on.convex_lt | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasiconcave_on.convex_gt (hf : quasiconcave_on 𝕜 s f) (r : β) : convex 𝕜 {x ∈ s | r < f x} | hf.dual.convex_lt r | lemma | quasiconcave_on.convex_gt | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.quasiconvex_on (hf : convex_on 𝕜 s f) : quasiconvex_on 𝕜 s f | hf.convex_le | lemma | convex_on.quasiconvex_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex_on",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.quasiconcave_on (hf : concave_on 𝕜 s f) : quasiconcave_on 𝕜 s f | hf.convex_ge | lemma | concave_on.quasiconcave_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"concave_on",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.quasiconvex_on (hf : monotone_on f s) (hs : convex 𝕜 s) : quasiconvex_on 𝕜 s f | hf.convex_le hs | lemma | monotone_on.quasiconvex_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"monotone_on",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.quasiconcave_on (hf : monotone_on f s) (hs : convex 𝕜 s) :
quasiconcave_on 𝕜 s f | hf.convex_ge hs | lemma | monotone_on.quasiconcave_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"monotone_on",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.quasilinear_on (hf : monotone_on f s) (hs : convex 𝕜 s) : quasilinear_on 𝕜 s f | ⟨hf.quasiconvex_on hs, hf.quasiconcave_on hs⟩ | lemma | monotone_on.quasilinear_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex",
"monotone_on",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.quasiconvex_on (hf : antitone_on f s) (hs : convex 𝕜 s) : quasiconvex_on 𝕜 s f | hf.convex_le hs | lemma | antitone_on.quasiconvex_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone_on",
"convex",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.quasiconcave_on (hf : antitone_on f s) (hs : convex 𝕜 s) :
quasiconcave_on 𝕜 s f | hf.convex_ge hs | lemma | antitone_on.quasiconcave_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone_on",
"convex",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.quasilinear_on (hf : antitone_on f s) (hs : convex 𝕜 s) : quasilinear_on 𝕜 s f | ⟨hf.quasiconvex_on hs, hf.quasiconcave_on hs⟩ | lemma | antitone_on.quasilinear_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone_on",
"convex",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.quasiconvex_on (hf : monotone f) : quasiconvex_on 𝕜 univ f | (hf.monotone_on _).quasiconvex_on convex_univ | lemma | monotone.quasiconvex_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex_univ",
"monotone",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.quasiconcave_on (hf : monotone f) : quasiconcave_on 𝕜 univ f | (hf.monotone_on _).quasiconcave_on convex_univ | lemma | monotone.quasiconcave_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"convex_univ",
"monotone",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.quasilinear_on (hf : monotone f) : quasilinear_on 𝕜 univ f | ⟨hf.quasiconvex_on, hf.quasiconcave_on⟩ | lemma | monotone.quasilinear_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"monotone",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.quasiconvex_on (hf : antitone f) : quasiconvex_on 𝕜 univ f | (hf.antitone_on _).quasiconvex_on convex_univ | lemma | antitone.quasiconvex_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone",
"convex_univ",
"quasiconvex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.quasiconcave_on (hf : antitone f) : quasiconcave_on 𝕜 univ f | (hf.antitone_on _).quasiconcave_on convex_univ | lemma | antitone.quasiconcave_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone",
"convex_univ",
"quasiconcave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.quasilinear_on (hf : antitone f) : quasilinear_on 𝕜 univ f | ⟨hf.quasiconvex_on, hf.quasiconcave_on⟩ | lemma | antitone.quasilinear_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasilinear_on.monotone_on_or_antitone_on (hf : quasilinear_on 𝕜 s f) :
monotone_on f s ∨ antitone_on f s | begin
simp_rw [monotone_on_or_antitone_on_iff_uIcc, ←segment_eq_uIcc],
rintro a ha b hb c hc h,
refine ⟨((hf.2 _).segment_subset _ _ h).2, ((hf.1 _).segment_subset _ _ h).2⟩; simp [*],
end | lemma | quasilinear_on.monotone_on_or_antitone_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone_on",
"monotone_on",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasilinear_on_iff_monotone_on_or_antitone_on (hs : convex 𝕜 s) :
quasilinear_on 𝕜 s f ↔ monotone_on f s ∨ antitone_on f s | ⟨λ h, h.monotone_on_or_antitone_on,
λ h, h.elim (λ h, h.quasilinear_on hs) (λ h, h.quasilinear_on hs)⟩ | lemma | quasilinear_on_iff_monotone_on_or_antitone_on | analysis.convex | src/analysis/convex/quasiconvex.lean | [
"analysis.convex.function"
] | [
"antitone_on",
"convex",
"monotone_on",
"quasilinear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment (x y : E) : set E | {z : E | ∃ (a b : 𝕜) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1), a • x + b • y = z} | def | segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [] | Segments in a vector space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_segment (x y : E) : set E | {z : E | ∃ (a b : 𝕜) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1), a • x + b • y = z} | def | open_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [] | Open segment in a vector space. Note that `open_segment 𝕜 x x = {x}` instead of being `∅` when
the base semiring has some element between `0` and `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
segment_eq_image₂ (x y : E) :
[x -[𝕜] y] = (λ p : 𝕜 × 𝕜, p.1 • x + p.2 • y) '' {p | 0 ≤ p.1 ∧ 0 ≤ p.2 ∧ p.1 + p.2 = 1} | by simp only [segment, image, prod.exists, mem_set_of_eq, exists_prop, and_assoc] | lemma | segment_eq_image₂ | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"exists_prop",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_eq_image₂ (x y : E) :
open_segment 𝕜 x y =
(λ p : 𝕜 × 𝕜, p.1 • x + p.2 • y) '' {p | 0 < p.1 ∧ 0 < p.2 ∧ p.1 + p.2 = 1} | by simp only [open_segment, image, prod.exists, mem_set_of_eq, exists_prop, and_assoc] | lemma | open_segment_eq_image₂ | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"exists_prop",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_symm (x y : E) : [x -[𝕜] y] = [y -[𝕜] x] | set.ext $ λ z,
⟨λ ⟨a, b, ha, hb, hab, H⟩, ⟨b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H⟩,
λ ⟨a, b, ha, hb, hab, H⟩, ⟨b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H⟩⟩ | lemma | segment_symm | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_symm (x y : E) : open_segment 𝕜 x y = open_segment 𝕜 y x | set.ext $ λ z,
⟨λ ⟨a, b, ha, hb, hab, H⟩, ⟨b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H⟩,
λ ⟨a, b, ha, hb, hab, H⟩, ⟨b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H⟩⟩ | lemma | open_segment_symm | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset_segment (x y : E) : open_segment 𝕜 x y ⊆ [x -[𝕜] y] | λ z ⟨a, b, ha, hb, hab, hz⟩, ⟨a, b, ha.le, hb.le, hab, hz⟩ | lemma | open_segment_subset_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_subset_iff :
[x -[𝕜] y] ⊆ s ↔ ∀ a b : 𝕜, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s | ⟨λ H a b ha hb hab, H ⟨a, b, ha, hb, hab, rfl⟩,
λ H z ⟨a, b, ha, hb, hab, hz⟩, hz ▸ H a b ha hb hab⟩ | lemma | segment_subset_iff | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset_iff :
open_segment 𝕜 x y ⊆ s ↔ ∀ a b : 𝕜, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s | ⟨λ H a b ha hb hab, H ⟨a, b, ha, hb, hab, rfl⟩,
λ H z ⟨a, b, ha, hb, hab, hz⟩, hz ▸ H a b ha hb hab⟩ | lemma | open_segment_subset_iff | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_mem_segment (x y : E) : x ∈ [x -[𝕜] y] | ⟨1, 0, zero_le_one, le_refl 0, add_zero 1, by rw [zero_smul, one_smul, add_zero]⟩ | lemma | left_mem_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"one_smul",
"zero_le_one",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_mem_segment (x y : E) : y ∈ [x -[𝕜] y] | segment_symm 𝕜 y x ▸ left_mem_segment 𝕜 y x | lemma | right_mem_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"left_mem_segment",
"segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_same (x : E) : [x -[𝕜] x] = {x} | set.ext $ λ z, ⟨λ ⟨a, b, ha, hb, hab, hz⟩,
by simpa only [(add_smul _ _ _).symm, mem_singleton_iff, hab, one_smul, eq_comm] using hz,
λ h, mem_singleton_iff.1 h ▸ left_mem_segment 𝕜 z z⟩ | lemma | segment_same | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"add_smul",
"left_mem_segment",
"one_smul",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
insert_endpoints_open_segment (x y : E) :
insert x (insert y (open_segment 𝕜 x y)) = [x -[𝕜] y] | begin
simp only [subset_antisymm_iff, insert_subset, left_mem_segment, right_mem_segment,
open_segment_subset_segment, true_and],
rintro z ⟨a, b, ha, hb, hab, rfl⟩,
refine hb.eq_or_gt.imp _ (λ hb', ha.eq_or_gt.imp _ $ λ ha', _),
{ rintro rfl,
rw [← add_zero a, hab, one_smul, zero_smul, add_zero] },
{ ... | lemma | insert_endpoints_open_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"left_mem_segment",
"one_smul",
"open_segment",
"open_segment_subset_segment",
"right_mem_segment",
"subset_antisymm_iff",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_open_segment_of_ne_left_right (hx : x ≠ z) (hy : y ≠ z) (hz : z ∈ [x -[𝕜] y]) :
z ∈ open_segment 𝕜 x y | begin
rw [←insert_endpoints_open_segment] at hz,
exact ((hz.resolve_left hx.symm).resolve_left hy.symm)
end | lemma | mem_open_segment_of_ne_left_right | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset_iff_segment_subset (hx : x ∈ s) (hy : y ∈ s) :
open_segment 𝕜 x y ⊆ s ↔ [x -[𝕜] y] ⊆ s | by simp only [←insert_endpoints_open_segment, insert_subset, *, true_and] | lemma | open_segment_subset_iff_segment_subset | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_same (x : E) : open_segment 𝕜 x x = {x} | set.ext $ λ z, ⟨λ ⟨a, b, ha, hb, hab, hz⟩,
by simpa only [←add_smul, mem_singleton_iff, hab, one_smul, eq_comm] using hz,
λ (h : z = x), begin
obtain ⟨a, ha₀, ha₁⟩ := densely_ordered.dense (0 : 𝕜) 1 zero_lt_one,
refine ⟨a, 1 - a, ha₀, sub_pos_of_lt ha₁, add_sub_cancel'_right _ _, _⟩,
rw [←add_smul, add... | lemma | open_segment_same | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"one_smul",
"open_segment",
"set.ext",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_eq_image (x y : E) : [x -[𝕜] y] = (λ θ : 𝕜, (1 - θ) • x + θ • y) '' Icc (0 : 𝕜) 1 | set.ext $ λ z,
⟨λ ⟨a, b, ha, hb, hab, hz⟩,
⟨b, ⟨hb, hab ▸ le_add_of_nonneg_left ha⟩, hab ▸ hz ▸ by simp only [add_sub_cancel]⟩,
λ ⟨θ, ⟨hθ₀, hθ₁⟩, hz⟩, ⟨1-θ, θ, sub_nonneg.2 hθ₁, hθ₀, sub_add_cancel _ _, hz⟩⟩ | lemma | segment_eq_image | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_eq_image (x y : E) :
open_segment 𝕜 x y = (λ (θ : 𝕜), (1 - θ) • x + θ • y) '' Ioo (0 : 𝕜) 1 | set.ext $ λ z,
⟨λ ⟨a, b, ha, hb, hab, hz⟩,
⟨b, ⟨hb, hab ▸ lt_add_of_pos_left _ ha⟩, hab ▸ hz ▸ by simp only [add_sub_cancel]⟩,
λ ⟨θ, ⟨hθ₀, hθ₁⟩, hz⟩, ⟨1 - θ, θ, sub_pos.2 hθ₁, hθ₀, sub_add_cancel _ _, hz⟩⟩ | lemma | open_segment_eq_image | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_eq_image' (x y : E) : [x -[𝕜] y] = (λ (θ : 𝕜), x + θ • (y - x)) '' Icc (0 : 𝕜) 1 | by { convert segment_eq_image 𝕜 x y, ext θ, simp only [smul_sub, sub_smul, one_smul], abel } | lemma | segment_eq_image' | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"one_smul",
"segment_eq_image",
"smul_sub",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_eq_image' (x y : E) :
open_segment 𝕜 x y = (λ (θ : 𝕜), x + θ • (y - x)) '' Ioo (0 : 𝕜) 1 | by { convert open_segment_eq_image 𝕜 x y, ext θ, simp only [smul_sub, sub_smul, one_smul], abel } | lemma | open_segment_eq_image' | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"one_smul",
"open_segment",
"open_segment_eq_image",
"smul_sub",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_eq_image_line_map (x y : E) : [x -[𝕜] y] = affine_map.line_map x y '' Icc (0 : 𝕜) 1 | by { convert segment_eq_image 𝕜 x y, ext, exact affine_map.line_map_apply_module _ _ _ } | lemma | segment_eq_image_line_map | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"affine_map.line_map",
"affine_map.line_map_apply_module",
"segment_eq_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_eq_image_line_map (x y : E) :
open_segment 𝕜 x y = affine_map.line_map x y '' Ioo (0 : 𝕜) 1 | by { convert open_segment_eq_image 𝕜 x y, ext, exact affine_map.line_map_apply_module _ _ _ } | lemma | open_segment_eq_image_line_map | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"affine_map.line_map",
"affine_map.line_map_apply_module",
"open_segment",
"open_segment_eq_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_segment (f : E →ᵃ[𝕜] F) (a b : E) : f '' [a -[𝕜] b] = [f a -[𝕜] f b] | set.ext $ λ x, by simp_rw [segment_eq_image_line_map, mem_image, exists_exists_and_eq_and,
affine_map.apply_line_map] | lemma | image_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"affine_map.apply_line_map",
"exists_exists_and_eq_and",
"segment_eq_image_line_map",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_open_segment (f : E →ᵃ[𝕜] F) (a b : E) :
f '' open_segment 𝕜 a b = open_segment 𝕜 (f a) (f b) | set.ext $ λ x, by simp_rw [open_segment_eq_image_line_map, mem_image, exists_exists_and_eq_and,
affine_map.apply_line_map] | lemma | image_open_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"affine_map.apply_line_map",
"exists_exists_and_eq_and",
"open_segment",
"open_segment_eq_image_line_map",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_segment [add_torsor G E] [vadd_comm_class G E E] (a : G) (b c : E) :
a +ᵥ [b -[𝕜] c] = [a +ᵥ b -[𝕜] a +ᵥ c] | image_segment 𝕜 ⟨_, linear_map.id, λ _ _, vadd_comm _ _ _⟩ b c | lemma | vadd_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"add_torsor",
"image_segment",
"linear_map.id",
"vadd_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_open_segment [add_torsor G E] [vadd_comm_class G E E] (a : G) (b c : E) :
a +ᵥ open_segment 𝕜 b c = open_segment 𝕜 (a +ᵥ b) (a +ᵥ c) | image_open_segment 𝕜 ⟨_, linear_map.id, λ _ _, vadd_comm _ _ _⟩ b c | lemma | vadd_open_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"add_torsor",
"image_open_segment",
"linear_map.id",
"open_segment",
"vadd_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_segment_translate (a : E) {x b c} : a + x ∈ [a + b -[𝕜] a + c] ↔ x ∈ [b -[𝕜] c] | by simp_rw [←vadd_eq_add, ←vadd_segment, vadd_mem_vadd_set_iff] | lemma | mem_segment_translate | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_open_segment_translate (a : E) {x b c : E} :
a + x ∈ open_segment 𝕜 (a + b) (a + c) ↔ x ∈ open_segment 𝕜 b c | by simp_rw [←vadd_eq_add, ←vadd_open_segment, vadd_mem_vadd_set_iff] | lemma | mem_open_segment_translate | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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