statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
dist_add_dist_eq_iff : dist x y + dist y z = dist x z ↔ y ∈ [x -[ℝ] z] | by simp only [mem_segment_iff_same_ray, same_ray_iff_norm_add, dist_eq_norm',
sub_add_sub_cancel', eq_comm] | lemma | dist_add_dist_eq_iff | 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"
] | [
"mem_segment_iff_same_ray",
"same_ray_iff_norm_add"
] | In a strictly convex space, the triangle inequality turns into an equality if and only if the
middle point belongs to the segment joining two other points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_midpoint_lt_iff (h : ‖x‖ = ‖y‖) : ‖(1/2 : ℝ) • (x + y)‖ < ‖x‖ ↔ x ≠ y | by rw [norm_smul, real.norm_of_nonneg (one_div_nonneg.2 zero_le_two), ←inv_eq_one_div,
←div_eq_inv_mul, div_lt_iff (zero_lt_two' ℝ), mul_two, ←not_same_ray_iff_of_norm_eq h,
not_same_ray_iff_norm_add_lt, h] | lemma | norm_midpoint_lt_iff | 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"
] | [
"div_lt_iff",
"mul_two",
"norm_smul",
"not_same_ray_iff_norm_add_lt",
"real.norm_of_nonneg",
"zero_le_two",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_line_map_of_dist_eq_mul_of_dist_eq_mul {x y z : PE} (hxy : dist x y = r * dist x z)
(hyz : dist y z = (1 - r) * dist x z) :
y = affine_map.line_map x z r | begin
have : y -ᵥ x ∈ [(0 : E) -[ℝ] z -ᵥ x],
{ rw [← dist_add_dist_eq_iff, dist_zero_left, dist_vsub_cancel_right, ← dist_eq_norm_vsub',
← dist_eq_norm_vsub', hxy, hyz, ← add_mul, add_sub_cancel'_right, one_mul] },
rcases eq_or_ne x z with rfl|hne,
{ obtain rfl : y = x, by simpa,
simp },
{ rw [← dis... | lemma | eq_line_map_of_dist_eq_mul_of_dist_eq_mul | 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",
"affine_map.line_map_apply",
"dist_add_dist_eq_iff",
"dist_eq_norm_vsub'",
"dist_ne_zero",
"dist_vsub_cancel_right",
"eq_or_ne",
"mul_left_inj'",
"norm_smul",
"one_mul",
"real.norm_of_nonneg",
"smul_zero",
"vsub_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_midpoint_of_dist_eq_half {x y z : PE} (hx : dist x y = dist x z / 2)
(hy : dist y z = dist x z / 2) : y = midpoint ℝ x z | begin
apply eq_line_map_of_dist_eq_mul_of_dist_eq_mul,
{ rwa [inv_of_eq_inv, ← div_eq_inv_mul] },
{ rwa [inv_of_eq_inv, ← one_div, sub_half, one_div, ← div_eq_inv_mul] }
end | lemma | eq_midpoint_of_dist_eq_half | 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"
] | [
"div_eq_inv_mul",
"eq_line_map_of_dist_eq_mul_of_dist_eq_mul",
"inv_of_eq_inv",
"midpoint",
"one_div",
"sub_half"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_isometry_of_strict_convex_space {f : PF → PE} (hi : isometry f) :
PF →ᵃⁱ[ℝ] PE | { norm_map := λ x, by simp [affine_map.of_map_midpoint, ←dist_eq_norm_vsub E, hi.dist_eq],
..affine_map.of_map_midpoint f (λ x y, begin
apply eq_midpoint_of_dist_eq_half,
{ rw [hi.dist_eq, hi.dist_eq, dist_left_midpoint, real.norm_of_nonneg zero_le_two,
div_eq_inv_mul] },
{ rw [hi.dist_eq, hi.dist... | def | isometry.affine_isometry_of_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"
] | [
"affine_map.of_map_midpoint",
"dist_left_midpoint",
"dist_midpoint_right",
"div_eq_inv_mul",
"eq_midpoint_of_dist_eq_half",
"isometry",
"real.norm_of_nonneg",
"zero_le_two"
] | An isometry of `normed_add_torsor`s for real normed spaces, strictly convex in the case of
the codomain, is an affine isometry. Unlike Mazur-Ulam, this does not require the isometry to
be surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_affine_isometry_of_strict_convex_space {f : PF → PE} (hi : isometry f) :
⇑(hi.affine_isometry_of_strict_convex_space) = f | rfl | lemma | isometry.coe_affine_isometry_of_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"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_isometry_of_strict_convex_space_apply {f : PF → PE} (hi : isometry f)
(p : PF) :
hi.affine_isometry_of_strict_convex_space p = f p | rfl | lemma | isometry.affine_isometry_of_strict_convex_space_apply | 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"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.convex_iff_is_preconnected {s : set ℝ} : convex ℝ s ↔ is_preconnected s | convex_iff_ord_connected.trans is_preconnected_iff_ord_connected.symm | lemma | real.convex_iff_is_preconnected | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
std_simplex_subset_closed_ball :
std_simplex ℝ ι ⊆ metric.closed_ball 0 1 | begin
assume f hf,
rw [metric.mem_closed_ball, dist_pi_le_iff zero_le_one],
intros x,
rw [pi.zero_apply, real.dist_0_eq_abs, abs_of_nonneg $ hf.1 x],
exact (mem_Icc_of_mem_std_simplex hf x).2,
end | lemma | std_simplex_subset_closed_ball | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"abs_of_nonneg",
"dist_pi_le_iff",
"mem_Icc_of_mem_std_simplex",
"metric.closed_ball",
"metric.mem_closed_ball",
"real.dist_0_eq_abs",
"std_simplex",
"zero_le_one"
] | Every vector in `std_simplex 𝕜 ι` has `max`-norm at most `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_std_simplex : metric.bounded (std_simplex ℝ ι) | (metric.bounded_iff_subset_ball 0).2 ⟨1, std_simplex_subset_closed_ball⟩ | lemma | bounded_std_simplex | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"metric.bounded",
"metric.bounded_iff_subset_ball",
"std_simplex"
] | `std_simplex ℝ ι` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_std_simplex : is_closed (std_simplex ℝ ι) | (std_simplex_eq_inter ℝ ι).symm ▸ is_closed.inter
(is_closed_Inter $ λ i, is_closed_le continuous_const (continuous_apply i))
(is_closed_eq (continuous_finset_sum _ $ λ x _, continuous_apply x) continuous_const) | lemma | is_closed_std_simplex | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"continuous_apply",
"continuous_const",
"is_closed",
"is_closed.inter",
"is_closed_Inter",
"is_closed_eq",
"is_closed_le",
"std_simplex",
"std_simplex_eq_inter"
] | `std_simplex ℝ ι` is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_std_simplex : is_compact (std_simplex ℝ ι) | metric.is_compact_iff_is_closed_bounded.2 ⟨is_closed_std_simplex ι, bounded_std_simplex ι⟩ | lemma | is_compact_std_simplex | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"bounded_std_simplex",
"is_compact",
"std_simplex"
] | `std_simplex ℝ ι` is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
segment_subset_closure_open_segment : [x -[𝕜] y] ⊆ closure (open_segment 𝕜 x y) | begin
rw [segment_eq_image, open_segment_eq_image, ←closure_Ioo (zero_ne_one' 𝕜)],
exact image_closure_subset_closure_image (by continuity),
end | lemma | segment_subset_closure_open_segment | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"continuity",
"image_closure_subset_closure_image",
"open_segment",
"open_segment_eq_image",
"segment_eq_image",
"zero_ne_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_open_segment (x y : E) : closure (open_segment 𝕜 x y) = [x -[𝕜] y] | begin
rw [segment_eq_image, open_segment_eq_image, ←closure_Ioo (zero_ne_one' 𝕜)],
exact (image_closure_of_is_compact (bounded_Ioo _ _).is_compact_closure $
continuous.continuous_on $ by continuity).symm,
end | lemma | closure_open_segment | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"continuity",
"continuous.continuous_on",
"image_closure_of_is_compact",
"open_segment",
"open_segment_eq_image",
"segment_eq_image",
"zero_ne_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.combo_interior_closure_subset_interior {s : set E} (hs : convex 𝕜 s) {a b : 𝕜}
(ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :
a • interior s + b • closure s ⊆ interior s | interior_smul₀ ha.ne' s ▸
calc interior (a • s) + b • closure s ⊆ interior (a • s) + closure (b • s) :
add_subset_add subset.rfl (smul_closure_subset b s)
... = interior (a • s) + b • s : by rw is_open_interior.add_closure (b • s)
... ⊆ interior (a • s + b • s) : subset_interior_add_left
... ⊆ interior s : ... | lemma | convex.combo_interior_closure_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior",
"interior_mono",
"interior_smul₀",
"smul_closure_subset"
] | If `s` is a convex set, then `a • interior s + b • closure s ⊆ interior s` for all `0 < a`,
`0 ≤ b`, `a + b = 1`. See also `convex.combo_interior_self_subset_interior` for a weaker version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.combo_interior_self_subset_interior {s : set E} (hs : convex 𝕜 s) {a b : 𝕜}
(ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :
a • interior s + b • s ⊆ interior s | calc a • interior s + b • s ⊆ a • interior s + b • closure s :
add_subset_add subset.rfl $ image_subset _ subset_closure
... ⊆ interior s : hs.combo_interior_closure_subset_interior ha hb hab | lemma | convex.combo_interior_self_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior",
"subset_closure"
] | If `s` is a convex set, then `a • interior s + b • s ⊆ interior s` for all `0 < a`, `0 ≤ b`,
`a + b = 1`. See also `convex.combo_interior_closure_subset_interior` for a stronger version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.combo_closure_interior_subset_interior {s : set E} (hs : convex 𝕜 s) {a b : 𝕜}
(ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :
a • closure s + b • interior s ⊆ interior s | by { rw add_comm, exact hs.combo_interior_closure_subset_interior hb ha (add_comm a b ▸ hab) } | lemma | convex.combo_closure_interior_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior"
] | If `s` is a convex set, then `a • closure s + b • interior s ⊆ interior s` for all `0 ≤ a`,
`0 < b`, `a + b = 1`. See also `convex.combo_self_interior_subset_interior` for a weaker version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.combo_self_interior_subset_interior {s : set E} (hs : convex 𝕜 s) {a b : 𝕜}
(ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :
a • s + b • interior s ⊆ interior s | by { rw add_comm, exact hs.combo_interior_self_subset_interior hb ha (add_comm a b ▸ hab) } | lemma | convex.combo_self_interior_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior"
] | If `s` is a convex set, then `a • s + b • interior s ⊆ interior s` for all `0 ≤ a`, `0 < b`,
`a + b = 1`. See also `convex.combo_closure_interior_subset_interior` for a stronger version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.combo_interior_closure_mem_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ interior s) (hy : y ∈ closure s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :
a • x + b • y ∈ interior s | hs.combo_interior_closure_subset_interior ha hb hab $
add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy) | lemma | convex.combo_interior_closure_mem_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.combo_interior_self_mem_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ interior s) (hy : y ∈ s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :
a • x + b • y ∈ interior s | hs.combo_interior_closure_mem_interior hx (subset_closure hy) ha hb hab | lemma | convex.combo_interior_self_mem_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.combo_closure_interior_mem_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ closure s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :
a • x + b • y ∈ interior s | hs.combo_closure_interior_subset_interior ha hb hab $
add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy) | lemma | convex.combo_closure_interior_mem_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.combo_self_interior_mem_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :
a • x + b • y ∈ interior s | hs.combo_closure_interior_mem_interior (subset_closure hx) hy ha hb hab | lemma | convex.combo_self_interior_mem_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.open_segment_interior_closure_subset_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ interior s) (hy : y ∈ closure s) : open_segment 𝕜 x y ⊆ interior s | begin
rintro _ ⟨a, b, ha, hb, hab, rfl⟩,
exact hs.combo_interior_closure_mem_interior hx hy ha hb.le hab
end | lemma | convex.open_segment_interior_closure_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.open_segment_interior_self_subset_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ interior s) (hy : y ∈ s) : open_segment 𝕜 x y ⊆ interior s | hs.open_segment_interior_closure_subset_interior hx (subset_closure hy) | lemma | convex.open_segment_interior_self_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"open_segment",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.open_segment_closure_interior_subset_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ closure s) (hy : y ∈ interior s) : open_segment 𝕜 x y ⊆ interior s | begin
rintro _ ⟨a, b, ha, hb, hab, rfl⟩,
exact hs.combo_closure_interior_mem_interior hx hy ha.le hb hab
end | lemma | convex.open_segment_closure_interior_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.open_segment_self_interior_subset_interior {s : set E} (hs : convex 𝕜 s) {x y : E}
(hx : x ∈ s) (hy : y ∈ interior s) : open_segment 𝕜 x y ⊆ interior s | hs.open_segment_closure_interior_subset_interior (subset_closure hx) hy | lemma | convex.open_segment_self_interior_subset_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"open_segment",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.add_smul_sub_mem_interior' {s : set E} (hs : convex 𝕜 s)
{x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) :
x + t • (y - x) ∈ interior s | by simpa only [sub_smul, smul_sub, one_smul, add_sub, add_comm]
using hs.combo_interior_closure_mem_interior hy hx ht.1 (sub_nonneg.mpr ht.2)
(add_sub_cancel'_right _ _) | lemma | convex.add_smul_sub_mem_interior' | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior",
"one_smul",
"smul_sub",
"sub_smul"
] | If `x ∈ closure s` and `y ∈ interior s`, then the segment `(x, y]` is included in `interior s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.add_smul_sub_mem_interior {s : set E} (hs : convex 𝕜 s)
{x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) :
x + t • (y - x) ∈ interior s | hs.add_smul_sub_mem_interior' (subset_closure hx) hy ht | lemma | convex.add_smul_sub_mem_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"subset_closure"
] | If `x ∈ s` and `y ∈ interior s`, then the segment `(x, y]` is included in `interior s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.add_smul_mem_interior' {s : set E} (hs : convex 𝕜 s)
{x y : E} (hx : x ∈ closure s) (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) :
x + t • y ∈ interior s | by simpa only [add_sub_cancel'] using hs.add_smul_sub_mem_interior' hx hy ht | lemma | convex.add_smul_mem_interior' | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior"
] | If `x ∈ closure s` and `x + y ∈ interior s`, then `x + t y ∈ interior s` for `t ∈ (0, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.add_smul_mem_interior {s : set E} (hs : convex 𝕜 s)
{x y : E} (hx : x ∈ s) (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) :
x + t • y ∈ interior s | hs.add_smul_mem_interior' (subset_closure hx) hy ht | lemma | convex.add_smul_mem_interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"subset_closure"
] | If `x ∈ s` and `x + y ∈ interior s`, then `x + t y ∈ interior s` for `t ∈ (0, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.interior {s : set E} (hs : convex 𝕜 s) : convex 𝕜 (interior s) | convex_iff_open_segment_subset.mpr $ λ x hx y hy,
hs.open_segment_closure_interior_subset_interior (interior_subset_closure hx) hy | lemma | convex.interior | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"interior_subset_closure"
] | In a topological vector space, the interior of a convex set is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.closure {s : set E} (hs : convex 𝕜 s) : convex 𝕜 (closure s) | λ x hx y hy a b ha hb hab,
let f : E → E → E := λ x' y', a • x' + b • y' in
have hf : continuous (function.uncurry f),
from (continuous_fst.const_smul _).add (continuous_snd.const_smul _),
show f x y ∈ closure s,
from map_mem_closure₂ hf hx hy (λ x' hx' y' hy', hs hx' hy' ha hb hab) | lemma | convex.closure | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"continuous",
"convex",
"map_mem_closure₂"
] | In a topological vector space, the closure of a convex set is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.strict_convex' {s : set E} (hs : convex 𝕜 s)
(h : (s \ interior s).pairwise $ λ x y, ∃ c : 𝕜, line_map x y c ∈ interior s) :
strict_convex 𝕜 s | begin
refine strict_convex_iff_open_segment_subset.2 _,
intros x hx y hy hne,
by_cases hx' : x ∈ interior s, { exact hs.open_segment_interior_self_subset_interior hx' hy },
by_cases hy' : y ∈ interior s, { exact hs.open_segment_self_interior_subset_interior hx hy' },
rcases h ⟨hx, hx'⟩ ⟨hy, hy'⟩ hne with ⟨c, ... | lemma | convex.strict_convex' | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior",
"open_segment_subset_union",
"pairwise",
"strict_convex"
] | A convex set `s` is strictly convex provided that for any two distinct points of
`s \ interior s`, the line passing through these points has nonempty intersection with
`interior s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.strict_convex {s : set E} (hs : convex 𝕜 s)
(h : (s \ interior s).pairwise $ λ x y, ([x -[𝕜] y] \ frontier s).nonempty) :
strict_convex 𝕜 s | begin
refine (hs.strict_convex' $ h.imp_on $ λ x hx y hy hne, _),
simp only [segment_eq_image_line_map, ← self_diff_frontier],
rintro ⟨_, ⟨⟨c, hc, rfl⟩, hcs⟩⟩,
refine ⟨c, hs.segment_subset hx.1 hy.1 _, hcs⟩,
exact (segment_eq_image_line_map 𝕜 x y).symm ▸ mem_image_of_mem _ hc
end | lemma | convex.strict_convex | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"frontier",
"interior",
"pairwise",
"segment_eq_image_line_map",
"self_diff_frontier",
"strict_convex"
] | A convex set `s` is strictly convex provided that for any two distinct points `x`, `y` of
`s \ interior s`, the segment with endpoints `x`, `y` has nonempty intersection with
`interior s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.finite.compact_convex_hull {s : set E} (hs : s.finite) :
is_compact (convex_hull ℝ s) | begin
rw [hs.convex_hull_eq_image],
apply (is_compact_std_simplex _).image,
haveI := hs.fintype,
apply linear_map.continuous_on_pi
end | lemma | set.finite.compact_convex_hull | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex_hull",
"is_compact",
"is_compact_std_simplex",
"linear_map.continuous_on_pi"
] | Convex hull of a finite set is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.finite.is_closed_convex_hull [t2_space E] {s : set E} (hs : s.finite) :
is_closed (convex_hull ℝ s) | hs.compact_convex_hull.is_closed | lemma | set.finite.is_closed_convex_hull | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex_hull",
"is_closed",
"t2_space"
] | Convex hull of a finite set is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.closure_subset_image_homothety_interior_of_one_lt {s : set E} (hs : convex ℝ s)
{x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) :
closure s ⊆ homothety x t '' interior s | begin
intros y hy,
have hne : t ≠ 0, from (one_pos.trans ht).ne',
refine ⟨homothety x t⁻¹ y, hs.open_segment_interior_closure_subset_interior hx hy _,
(affine_equiv.homothety_units_mul_hom x (units.mk0 t hne)).apply_symm_apply y⟩,
rw [open_segment_eq_image_line_map, ← inv_one, ← inv_Ioi (zero_lt_one' ℝ), ← ... | lemma | convex.closure_subset_image_homothety_interior_of_one_lt | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"affine_equiv.homothety_units_mul_hom",
"closure",
"convex",
"interior",
"inv_one",
"open_segment_eq_image_line_map",
"units.mk0",
"zero_lt_one'"
] | If we dilate the interior of a convex set about a point in its interior by a scale `t > 1`,
the result includes the closure of the original set.
TODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.closure_subset_interior_image_homothety_of_one_lt {s : set E} (hs : convex ℝ s)
{x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) :
closure s ⊆ interior (homothety x t '' s) | (hs.closure_subset_image_homothety_interior_of_one_lt hx t ht).trans $
(homothety_is_open_map x t (one_pos.trans ht).ne').image_interior_subset _ | lemma | convex.closure_subset_interior_image_homothety_of_one_lt | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"closure",
"convex",
"interior"
] | If we dilate a convex set about a point in its interior by a scale `t > 1`, the interior of
the result includes the closure of the original set.
TODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.subset_interior_image_homothety_of_one_lt {s : set E} (hs : convex ℝ s)
{x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) :
s ⊆ interior (homothety x t '' s) | subset_closure.trans $ hs.closure_subset_interior_image_homothety_of_one_lt hx t ht | lemma | convex.subset_interior_image_homothety_of_one_lt | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"interior"
] | If we dilate a convex set about a point in its interior by a scale `t > 1`, the interior of
the result includes the closure of the original set.
TODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.is_path_connected {s : set E} (hconv : convex ℝ s) (hne : s.nonempty) :
is_path_connected s | begin
refine is_path_connected_iff.mpr ⟨hne, _⟩,
intros x x_in y y_in,
have H := hconv.segment_subset x_in y_in,
rw segment_eq_image_line_map at H,
exact joined_in.of_line affine_map.line_map_continuous.continuous_on (line_map_apply_zero _ _)
(line_map_apply_one _ _) H
end | lemma | convex.is_path_connected | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"is_path_connected",
"joined_in.of_line",
"segment_eq_image_line_map"
] | A nonempty convex set is path connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.is_connected {s : set E} (h : convex ℝ s) (hne : s.nonempty) :
is_connected s | (h.is_path_connected hne).is_connected | lemma | convex.is_connected | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"is_connected"
] | A nonempty convex set is connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.is_preconnected {s : set E} (h : convex ℝ s) : is_preconnected s | s.eq_empty_or_nonempty.elim (λ h, h.symm ▸ is_preconnected_empty)
(λ hne, (h.is_connected hne).is_preconnected) | lemma | convex.is_preconnected | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"convex",
"is_preconnected",
"is_preconnected_empty"
] | A convex set is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_add_group.path_connected : path_connected_space E | path_connected_space_iff_univ.mpr $ convex_univ.is_path_connected ⟨(0 : E), trivial⟩ | lemma | topological_add_group.path_connected | analysis.convex | src/analysis/convex/topology.lean | [
"analysis.convex.combination",
"analysis.convex.strict",
"topology.path_connected",
"topology.algebra.affine",
"topology.algebra.module.basic"
] | [
"path_connected_space"
] | Every topological vector space over ℝ is path connected.
Not an instance, because it creates enormous TC subproblems (turn on `pp.all`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_convex_space (E : Type*) [seminormed_add_comm_group E] : Prop | (uniform_convex : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ δ, 0 < δ ∧
∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ) | class | uniform_convex_space | analysis.convex | src/analysis/convex/uniform.lean | [
"analysis.convex.strict_convex_space"
] | [
"seminormed_add_comm_group"
] | A *uniformly convex space* is a real normed space where the triangle inequality is strict with a
uniform bound. Namely, over the `x` and `y` of norm `1`, `‖x + y‖` is uniformly bounded above
by a constant `< 2` when `‖x - y‖` is uniformly bounded below by a positive constant.
See also `uniform_convex_space.of_uniform_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_sphere_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ | uniform_convex_space.uniform_convex hε | lemma | exists_forall_sphere_dist_add_le_two_sub | analysis.convex | src/analysis/convex/uniform.lean | [
"analysis.convex.strict_convex_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_forall_closed_ball_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y⦄, ‖y‖ ≤ 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ | begin
have hε' : 0 < ε / 3 := div_pos hε zero_lt_three,
obtain ⟨δ, hδ, h⟩ := exists_forall_sphere_dist_add_le_two_sub E hε',
set δ' := min (1/2) (min (ε/3) $ δ/3),
refine ⟨δ', lt_min one_half_pos $ lt_min hε' (div_pos hδ zero_lt_three), λ x hx y hy hxy, _⟩,
obtain hx' | hx' := le_or_lt (‖x‖) (1 - δ'),
{ exa... | lemma | exists_forall_closed_ball_dist_add_le_two_sub | analysis.convex | src/analysis/convex/uniform.lean | [
"analysis.convex.strict_convex_space"
] | [
"div_pos",
"exists_forall_sphere_dist_add_le_two_sub",
"inv_mul_cancel",
"min_le_of_right_le",
"min_lt_of_left_lt",
"mul_le_mul_of_nonneg_left",
"norm_smul_of_nonneg",
"one_half_lt_one",
"one_half_pos",
"one_le_inv",
"one_mul",
"ring",
"three_ne_zero",
"zero_lt_three"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_forall_closed_ball_dist_add_le_two_mul_sub (hε : 0 < ε) (r : ℝ) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ | begin
obtain hr | hr := le_or_lt r 0,
{ exact ⟨1, one_pos, λ x hx y hy h, (hε.not_le $ h.trans $ (norm_sub_le _ _).trans $
add_nonpos (hx.trans hr) (hy.trans hr)).elim⟩ },
obtain ⟨δ, hδ, h⟩ := exists_forall_closed_ball_dist_add_le_two_sub E (div_pos hε hr),
refine ⟨δ * r, mul_pos hδ hr, λ x hx y hy hxy, _⟩... | lemma | exists_forall_closed_ball_dist_add_le_two_mul_sub | analysis.convex | src/analysis/convex/uniform.lean | [
"analysis.convex.strict_convex_space"
] | [
"div_eq_inv_mul",
"div_le_div_right",
"div_le_iff",
"div_pos",
"exists_forall_closed_ball_dist_add_le_two_sub",
"norm_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_convex_space.to_strict_convex_space : strict_convex_space ℝ E | strict_convex_space.of_norm_add_ne_two $ λ x y hx hy hxy,
let ⟨δ, hδ, h⟩ := exists_forall_closed_ball_dist_add_le_two_sub E (norm_sub_pos_iff.2 hxy)
in ((h hx.le hy.le le_rfl).trans_lt $ sub_lt_self _ hδ).ne | instance | uniform_convex_space.to_strict_convex_space | analysis.convex | src/analysis/convex/uniform.lean | [
"analysis.convex.strict_convex_space"
] | [
"exists_forall_closed_ball_dist_add_le_two_sub",
"le_rfl",
"strict_convex_space",
"strict_convex_space.of_norm_add_ne_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_cone [add_comm_monoid E] [has_smul 𝕜 E] | (carrier : set E)
(smul_mem' : ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : E⦄, x ∈ carrier → c • x ∈ carrier)
(add_mem' : ∀ ⦃x⦄ (hx : x ∈ carrier) ⦃y⦄ (hy : y ∈ carrier), x + y ∈ carrier) | structure | convex_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"add_comm_monoid",
"has_smul"
] | A convex cone is a subset `s` of a `𝕜`-module such that `a • x + b • y ∈ s` whenever `a, b > 0`
and `x, y ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk {s : set E} {h₁ h₂} : ↑(@mk 𝕜 _ _ _ _ s h₁ h₂) = s | rfl | lemma | convex_cone.coe_mk | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk {s : set E} {h₁ h₂ x} : x ∈ @mk 𝕜 _ _ _ _ s h₁ h₂ ↔ x ∈ s | iff.rfl | lemma | convex_cone.mem_mk | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {S T : convex_cone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | convex_cone.ext | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"set_like.ext"
] | Two `convex_cone`s are equal if they have the same elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_mem {c : 𝕜} {x : E} (hc : 0 < c) (hx : x ∈ S) : c • x ∈ S | S.smul_mem' hc hx | lemma | convex_cone.smul_mem | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem ⦃x⦄ (hx : x ∈ S) ⦃y⦄ (hy : y ∈ S) : x + y ∈ S | S.add_mem' hx hy | lemma | convex_cone.add_mem | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf : ((S ⊓ T : convex_cone 𝕜 E) : set E) = ↑S ∩ ↑T | rfl | lemma | convex_cone.coe_inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {x} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T | iff.rfl | lemma | convex_cone.mem_inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf (S : set (convex_cone 𝕜 E)) : ↑(Inf S) = ⋂ s ∈ S, (s : set E) | rfl | lemma | convex_cone.coe_Inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {x : E} {S : set (convex_cone 𝕜 E)} : x ∈ Inf S ↔ ∀ s ∈ S, x ∈ s | mem_Inter₂ | lemma | convex_cone.mem_Inf | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi {ι : Sort*} (f : ι → convex_cone 𝕜 E) : ↑(infi f) = ⋂ i, (f i : set E) | by simp [infi] | lemma | convex_cone.coe_infi | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι : Sort*} {x : E} {f : ι → convex_cone 𝕜 E} : x ∈ infi f ↔ ∀ i, x ∈ f i | mem_Inter₂.trans $ by simp | lemma | convex_cone.mem_infi | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot (x : E) : x ∈ (⊥ : convex_cone 𝕜 E) = false | rfl | lemma | convex_cone.mem_bot | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ↑(⊥ : convex_cone 𝕜 E) = (∅ : set E) | rfl | lemma | convex_cone.coe_bot | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : E) : x ∈ (⊤ : convex_cone 𝕜 E) | mem_univ x | lemma | convex_cone.mem_top | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ↑(⊤ : convex_cone 𝕜 E) = (univ : set E) | rfl | lemma | convex_cone.coe_top | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex : convex 𝕜 (S : set E) | convex_iff_forall_pos.2 $ λ x hx y hy a b ha hb _, S.add_mem (S.smul_mem ha hx) (S.smul_mem hb hy) | lemma | convex_cone.convex | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_iff {c : 𝕜} (hc : 0 < c) {x : E} :
c • x ∈ S ↔ x ∈ S | ⟨λ h, inv_smul_smul₀ hc.ne' x ▸ S.smul_mem (inv_pos.2 hc) h, S.smul_mem hc⟩ | lemma | convex_cone.smul_mem_iff | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"inv_smul_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 E) : convex_cone 𝕜 F | { carrier := f '' S,
smul_mem' := λ c hc y ⟨x, hx, hy⟩, hy ▸ f.map_smul c x ▸ mem_image_of_mem f (S.smul_mem hc hx),
add_mem' := λ y₁ ⟨x₁, hx₁, hy₁⟩ y₂ ⟨x₂, hx₂, hy₂⟩, hy₁ ▸ hy₂ ▸ f.map_add x₁ x₂ ▸
mem_image_of_mem f (S.add_mem hx₁ hx₂) } | def | convex_cone.map | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | The image of a convex cone under a `𝕜`-linear map is a convex cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map {f : E →ₗ[𝕜] F} {S : convex_cone 𝕜 E} {y : F} :
y ∈ S.map f ↔ ∃ x ∈ S, f x = y | mem_image_iff_bex | lemma | convex_cone.mem_map | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 E) :
(S.map f).map g = S.map (g.comp f) | set_like.coe_injective $ image_image g f S | lemma | convex_cone.map_map | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (S : convex_cone 𝕜 E) : S.map linear_map.id = S | set_like.coe_injective $ image_id _ | lemma | convex_cone.map_id | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"linear_map.id",
"map_id",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : convex_cone 𝕜 E | { carrier := f ⁻¹' S,
smul_mem' := λ c hc x hx, by { rw [mem_preimage, f.map_smul c], exact S.smul_mem hc hx },
add_mem' := λ x hx y hy, by { rw [mem_preimage, f.map_add], exact S.add_mem hx hy } } | def | convex_cone.comap | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | The preimage of a convex cone under a `𝕜`-linear map is a convex cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : (S.comap f : set E) = f ⁻¹' S | rfl | lemma | convex_cone.coe_comap | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id (S : convex_cone 𝕜 E) : S.comap linear_map.id = S | set_like.coe_injective preimage_id | lemma | convex_cone.comap_id | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"linear_map.id",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 G) :
(S.comap g).comap f = S.comap (g.comp f) | set_like.coe_injective $ preimage_comp.symm | lemma | convex_cone.comap_comap | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {f : E →ₗ[𝕜] F} {S : convex_cone 𝕜 F} {x : E} : x ∈ S.comap f ↔ f x ∈ S | iff.rfl | lemma | convex_cone.mem_comap | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_smul (S : convex_cone 𝕜 E) (h : ∀ x y : E, x ≤ y ↔ y - x ∈ S) :
ordered_smul 𝕜 E | ordered_smul.mk'
begin
intros x y z xy hz,
rw [h (z • x) (z • y), ←smul_sub z y x],
exact smul_mem S hz ((h x y).mp xy.le),
end | lemma | convex_cone.to_ordered_smul | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"ordered_smul",
"ordered_smul.mk'"
] | Constructs an ordered module given an `ordered_add_comm_group`, a cone, and a proof that
the order relation is the one defined by the cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pointed (S : convex_cone 𝕜 E) : Prop | (0 : E) ∈ S | def | convex_cone.pointed | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | A convex cone is pointed if it includes `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
blunt (S : convex_cone 𝕜 E) : Prop | (0 : E) ∉ S | def | convex_cone.blunt | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | A convex cone is blunt if it doesn't include `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pointed_iff_not_blunt (S : convex_cone 𝕜 E) : S.pointed ↔ ¬S.blunt | ⟨λ h₁ h₂, h₂ h₁, not_not.mp⟩ | lemma | convex_cone.pointed_iff_not_blunt | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blunt_iff_not_pointed (S : convex_cone 𝕜 E) : S.blunt ↔ ¬S.pointed | by rw [pointed_iff_not_blunt, not_not] | lemma | convex_cone.blunt_iff_not_pointed | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointed.mono {S T : convex_cone 𝕜 E} (h : S ≤ T) : S.pointed → T.pointed | @h _ | lemma | convex_cone.pointed.mono | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blunt.anti {S T : convex_cone 𝕜 E} (h : T ≤ S) : S.blunt → T.blunt | (∘ @@h) | lemma | convex_cone.blunt.anti | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flat : Prop | ∃ x ∈ S, x ≠ (0 : E) ∧ -x ∈ S | def | convex_cone.flat | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | A convex cone is flat if it contains some nonzero vector `x` and its opposite `-x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
salient : Prop | ∀ x ∈ S, x ≠ (0 : E) → -x ∉ S | def | convex_cone.salient | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | A convex cone is salient if it doesn't include `x` and `-x` for any nonzero `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
salient_iff_not_flat (S : convex_cone 𝕜 E) : S.salient ↔ ¬S.flat | begin
split,
{ rintros h₁ ⟨x, xs, H₁, H₂⟩,
exact h₁ x xs H₁ H₂ },
{ intro h,
unfold flat at h,
push_neg at h,
exact h }
end | lemma | convex_cone.salient_iff_not_flat | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flat.mono {S T : convex_cone 𝕜 E} (h : S ≤ T) : S.flat → T.flat | | ⟨x, hxS, hx, hnxS⟩ := ⟨x, h hxS, hx, h hnxS⟩ | lemma | convex_cone.flat.mono | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
salient.anti {S T : convex_cone 𝕜 E} (h : T ≤ S) : S.salient → T.salient | λ hS x hxT hx hnT, hS x (h hxT) hx (h hnT) | lemma | convex_cone.salient.anti | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flat.pointed {S : convex_cone 𝕜 E} (hS : S.flat) : S.pointed | begin
obtain ⟨x, hx, _, hxneg⟩ := hS,
rw [pointed, ←add_neg_self x],
exact add_mem S hx hxneg,
end | lemma | convex_cone.flat.pointed | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | A flat cone is always pointed (contains `0`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
blunt.salient {S : convex_cone 𝕜 E} : S.blunt → S.salient | begin
rw [salient_iff_not_flat, blunt_iff_not_pointed],
exact mt flat.pointed,
end | lemma | convex_cone.blunt.salient | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | A blunt cone (one not containing `0`) is always salient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_preorder (h₁ : S.pointed) : preorder E | { le := λ x y, y - x ∈ S,
le_refl := λ x, by change x - x ∈ S; rw [sub_self x]; exact h₁,
le_trans := λ x y z xy zy, by simpa using add_mem S zy xy } | def | convex_cone.to_preorder | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [] | A pointed convex cone defines a preorder. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_partial_order (h₁ : S.pointed) (h₂ : S.salient) : partial_order E | { le_antisymm :=
begin
intros a b ab ba,
by_contradiction h,
have h' : b - a ≠ 0 := λ h'', h (eq_of_sub_eq_zero h'').symm,
have H := h₂ (b-a) ab h',
rw neg_sub b a at H,
exact H ba,
end,
..to_preorder S h₁ } | def | convex_cone.to_partial_order | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"by_contradiction"
] | A pointed and salient cone defines a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_add_comm_group (h₁ : S.pointed) (h₂ : S.salient) :
ordered_add_comm_group E | { add_le_add_left :=
begin
intros a b hab c,
change c + b - (c + a) ∈ S,
rw add_sub_add_left_eq_sub,
exact hab,
end,
..to_partial_order S h₁ h₂,
..show add_comm_group E, by apply_instance } | def | convex_cone.to_ordered_add_comm_group | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"add_comm_group",
"ordered_add_comm_group"
] | A pointed and salient cone defines an `ordered_add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_zero (x : E) : x ∈ (0 : convex_cone 𝕜 E) ↔ x = 0 | iff.rfl | lemma | convex_cone.mem_zero | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : convex_cone 𝕜 E) : set E) = 0 | rfl | lemma | convex_cone.coe_zero | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointed_zero : (0 : convex_cone 𝕜 E).pointed | by rw [pointed, mem_zero] | lemma | convex_cone.pointed_zero | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_add {K₁ K₂ : convex_cone 𝕜 E} {a : E} :
a ∈ K₁ + K₂ ↔ ∃ (x y : E), x ∈ K₁ ∧ y ∈ K₂ ∧ x + y = a | iff.rfl | lemma | convex_cone.mem_add | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_convex_cone (S : submodule 𝕜 E) : convex_cone 𝕜 E | { carrier := S,
smul_mem' := λ c hc x hx, S.smul_mem c hx,
add_mem' := λ x hx y hy, S.add_mem hx hy } | def | submodule.to_convex_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"convex_cone",
"submodule"
] | Every submodule is trivially a convex cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_convex_cone (S : submodule 𝕜 E) : ↑S.to_convex_cone = (S : set E) | rfl | lemma | submodule.coe_to_convex_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_convex_cone {x : E} {S : submodule 𝕜 E} : x ∈ S.to_convex_cone ↔ x ∈ S | iff.rfl | lemma | submodule.mem_to_convex_cone | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_convex_cone_le_iff {S T : submodule 𝕜 E} :
S.to_convex_cone ≤ T.to_convex_cone ↔ S ≤ T | iff.rfl | lemma | submodule.to_convex_cone_le_iff | analysis.convex.cone | src/analysis/convex/cone/basic.lean | [
"analysis.convex.hull",
"data.real.basic",
"linear_algebra.linear_pmap"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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