statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
convex_halfspace_ge {f : E → β} (h : is_linear_map 𝕜 f) (r : β) :
convex 𝕜 {w | r ≤ f w} | (convex_Ici r).is_linear_preimage h | lemma | convex_halfspace_ge | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Ici",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hyperplane {f : E → β} (h : is_linear_map 𝕜 f) (r : β) :
convex 𝕜 {w | f w = r} | begin
simp_rw le_antisymm_iff,
exact (convex_halfspace_le h r).inter (convex_halfspace_ge h r),
end | lemma | convex_hyperplane | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_halfspace_ge",
"convex_halfspace_le",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Iio (r : β) : convex 𝕜 (Iio r) | begin
intros x hx y hy a b ha hb hab,
obtain rfl | ha' := ha.eq_or_lt,
{ rw zero_add at hab,
rwa [zero_smul, zero_add, hab, one_smul] },
rw mem_Iio at hx hy,
calc
a • x + b • y
< a • r + b • r
: add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hx ha') (smul_le_smul_of_nonneg hy.le hb)
.... | lemma | convex_Iio | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex.combo_self",
"one_smul",
"smul_le_smul_of_nonneg",
"smul_lt_smul_of_pos",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Ioi (r : β) : convex 𝕜 (Ioi r) | @convex_Iio 𝕜 βᵒᵈ _ _ _ _ r | lemma | convex_Ioi | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Ioo (r s : β) : convex 𝕜 (Ioo r s) | Ioi_inter_Iio.subst ((convex_Ioi r).inter $ convex_Iio s) | lemma | convex_Ioo | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Iio",
"convex_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Ico (r s : β) : convex 𝕜 (Ico r s) | Ici_inter_Iio.subst ((convex_Ici r).inter $ convex_Iio s) | lemma | convex_Ico | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Ici",
"convex_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Ioc (r s : β) : convex 𝕜 (Ioc r s) | Ioi_inter_Iic.subst ((convex_Ioi r).inter $ convex_Iic s) | lemma | convex_Ioc | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Iic",
"convex_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_lt {f : E → β} (h : is_linear_map 𝕜 f) (r : β) :
convex 𝕜 {w | f w < r} | (convex_Iio r).is_linear_preimage h | lemma | convex_halfspace_lt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Iio",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_gt {f : E → β} (h : is_linear_map 𝕜 f) (r : β) :
convex 𝕜 {w | r < f w} | (convex_Ioi r).is_linear_preimage h | lemma | convex_halfspace_gt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Ioi",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_uIcc (r s : β) : convex 𝕜 (uIcc r s) | convex_Icc _ _ | lemma | convex_uIcc | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.convex_le (hf : monotone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | f x ≤ r} | λ x hx y hy a b ha hb hab, ⟨hs hx.1 hy.1 ha hb hab,
(hf (hs hx.1 hy.1 ha hb hab) (max_rec' s hx.1 hy.1) (convex.combo_le_max x y ha hb hab)).trans
(max_rec' _ hx.2 hy.2)⟩ | lemma | monotone_on.convex_le | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex.combo_le_max",
"max_rec'",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.convex_lt (hf : monotone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | f x < r} | λ x hx y hy a b ha hb hab, ⟨hs hx.1 hy.1 ha hb hab,
(hf (hs hx.1 hy.1 ha hb hab) (max_rec' s hx.1 hy.1) (convex.combo_le_max x y ha hb hab)).trans_lt
(max_rec' _ hx.2 hy.2)⟩ | lemma | monotone_on.convex_lt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex.combo_le_max",
"max_rec'",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.convex_ge (hf : monotone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | r ≤ f x} | @monotone_on.convex_le 𝕜 Eᵒᵈ βᵒᵈ _ _ _ _ _ _ _ hf.dual hs r | lemma | monotone_on.convex_ge | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"monotone_on",
"monotone_on.convex_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.convex_gt (hf : monotone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | r < f x} | @monotone_on.convex_lt 𝕜 Eᵒᵈ βᵒᵈ _ _ _ _ _ _ _ hf.dual hs r | lemma | monotone_on.convex_gt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"monotone_on",
"monotone_on.convex_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.convex_le (hf : antitone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | f x ≤ r} | @monotone_on.convex_ge 𝕜 E βᵒᵈ _ _ _ _ _ _ _ hf hs r | lemma | antitone_on.convex_le | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone_on",
"convex",
"monotone_on.convex_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.convex_lt (hf : antitone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | f x < r} | @monotone_on.convex_gt 𝕜 E βᵒᵈ _ _ _ _ _ _ _ hf hs r | lemma | antitone_on.convex_lt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone_on",
"convex",
"monotone_on.convex_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.convex_ge (hf : antitone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | r ≤ f x} | @monotone_on.convex_le 𝕜 E βᵒᵈ _ _ _ _ _ _ _ hf hs r | lemma | antitone_on.convex_ge | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone_on",
"convex",
"monotone_on.convex_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.convex_gt (hf : antitone_on f s) (hs : convex 𝕜 s) (r : β) :
convex 𝕜 {x ∈ s | r < f x} | @monotone_on.convex_lt 𝕜 E βᵒᵈ _ _ _ _ _ _ _ hf hs r | lemma | antitone_on.convex_gt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone_on",
"convex",
"monotone_on.convex_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.convex_le (hf : monotone f) (r : β) :
convex 𝕜 {x | f x ≤ r} | set.sep_univ.subst ((hf.monotone_on univ).convex_le convex_univ r) | lemma | monotone.convex_le | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_univ",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.convex_lt (hf : monotone f) (r : β) :
convex 𝕜 {x | f x ≤ r} | set.sep_univ.subst ((hf.monotone_on univ).convex_le convex_univ r) | lemma | monotone.convex_lt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_univ",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.convex_ge (hf : monotone f ) (r : β) :
convex 𝕜 {x | r ≤ f x} | set.sep_univ.subst ((hf.monotone_on univ).convex_ge convex_univ r) | lemma | monotone.convex_ge | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_univ",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.convex_gt (hf : monotone f) (r : β) :
convex 𝕜 {x | f x ≤ r} | set.sep_univ.subst ((hf.monotone_on univ).convex_le convex_univ r) | lemma | monotone.convex_gt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_univ",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.convex_le (hf : antitone f) (r : β) :
convex 𝕜 {x | f x ≤ r} | set.sep_univ.subst ((hf.antitone_on univ).convex_le convex_univ r) | lemma | antitone.convex_le | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone",
"convex",
"convex_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.convex_lt (hf : antitone f) (r : β) :
convex 𝕜 {x | f x < r} | set.sep_univ.subst ((hf.antitone_on univ).convex_lt convex_univ r) | lemma | antitone.convex_lt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone",
"convex",
"convex_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.convex_ge (hf : antitone f) (r : β) :
convex 𝕜 {x | r ≤ f x} | set.sep_univ.subst ((hf.antitone_on univ).convex_ge convex_univ r) | lemma | antitone.convex_ge | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone",
"convex",
"convex_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.convex_gt (hf : antitone f) (r : β) :
convex 𝕜 {x | r < f x} | set.sep_univ.subst ((hf.antitone_on univ).convex_gt convex_univ r) | lemma | antitone.convex_gt | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"antitone",
"convex",
"convex_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.smul (hs : convex 𝕜 s) (c : 𝕜) : convex 𝕜 (c • s) | hs.linear_image (linear_map.lsmul _ _ c) | lemma | convex.smul | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"linear_map.lsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.smul_preimage (hs : convex 𝕜 s) (c : 𝕜) : convex 𝕜 ((λ z, c • z) ⁻¹' s) | hs.linear_preimage (linear_map.lsmul _ _ c) | lemma | convex.smul_preimage | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"linear_map.lsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.affinity (hs : convex 𝕜 s) (z : E) (c : 𝕜) : convex 𝕜 ((λ x, z + c • x) '' s) | by simpa only [←image_smul, ←image_vadd, image_image] using (hs.smul c).vadd z | lemma | convex.affinity | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_open_segment (a b : E) : convex 𝕜 (open_segment 𝕜 a b) | begin
rw convex_iff_open_segment_subset,
rintro p ⟨ap, bp, hap, hbp, habp, rfl⟩ q ⟨aq, bq, haq, hbq, habq, rfl⟩ z ⟨a, b, ha, hb, hab, rfl⟩,
refine ⟨a * ap + b * aq, a * bp + b * bq, by positivity, by positivity, _, _⟩,
{ rw [add_add_add_comm, ←mul_add, ←mul_add, habp, habq, mul_one, mul_one, hab] },
{ simp_rw... | lemma | convex_open_segment | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"add_smul",
"convex",
"convex_iff_open_segment_subset",
"mul_one",
"open_segment",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.add_smul_mem (hs : convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ s)
{t : 𝕜} (ht : t ∈ Icc (0 : 𝕜) 1) : x + t • y ∈ s | begin
have h : x + t • y = (1 - t) • x + t • (x + y),
{ rw [smul_add, ←add_assoc, ←add_smul, sub_add_cancel, one_smul] },
rw h,
exact hs hx hy (sub_nonneg_of_le ht.2) ht.1 (sub_add_cancel _ _),
end | lemma | convex.add_smul_mem | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"one_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.smul_mem_of_zero_mem (hs : convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
{t : 𝕜} (ht : t ∈ Icc (0 : 𝕜) 1) : t • x ∈ s | by simpa using hs.add_smul_mem zero_mem (by simpa using hx) ht | lemma | convex.smul_mem_of_zero_mem | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.add_smul_sub_mem (h : convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
{t : 𝕜} (ht : t ∈ Icc (0 : 𝕜) 1) : x + t • (y - x) ∈ s | begin
apply h.segment_subset hx hy,
rw segment_eq_image',
exact mem_image_of_mem _ ht,
end | lemma | convex.add_smul_sub_mem | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"segment_eq_image'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_subspace.convex (Q : affine_subspace 𝕜 E) : convex 𝕜 (Q : set E) | begin
intros x hx y hy a b ha hb hab,
rw [eq_sub_of_add_eq hab, ← affine_map.line_map_apply_module],
exact affine_map.line_map_mem b hx hy,
end | lemma | affine_subspace.convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"affine_map.line_map_apply_module",
"affine_map.line_map_mem",
"affine_subspace",
"convex"
] | Affine subspaces are convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.affine_preimage (f : E →ᵃ[𝕜] F) {s : set F} (hs : convex 𝕜 s) :
convex 𝕜 (f ⁻¹' s) | λ x hx, (hs hx).affine_preimage _ | lemma | convex.affine_preimage | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | The preimage of a convex set under an affine map is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.affine_image (f : E →ᵃ[𝕜] F) (hs : convex 𝕜 s) : convex 𝕜 (f '' s) | by { rintro _ ⟨x, hx, rfl⟩, exact (hs hx).affine_image _ } | lemma | convex.affine_image | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | The image of a convex set under an affine map is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.neg (hs : convex 𝕜 s) : convex 𝕜 (-s) | hs.is_linear_preimage is_linear_map.is_linear_map_neg | lemma | convex.neg | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"is_linear_map.is_linear_map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.sub (hs : convex 𝕜 s) (ht : convex 𝕜 t) : convex 𝕜 (s - t) | by { rw sub_eq_add_neg, exact hs.add ht.neg } | lemma | convex.sub | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_div :
convex 𝕜 s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄,
0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s | forall₂_congr $ λ x hx, star_convex_iff_div | lemma | convex_iff_div | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"forall₂_congr",
"star_convex_iff_div"
] | Alternative definition of set convexity, using division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mem_smul_of_zero_mem (h : convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s)
(hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t) :
x ∈ t • s | begin
rw mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne',
exact h.smul_mem_of_zero_mem zero_mem hx ⟨inv_nonneg.2 (zero_le_one.trans ht), inv_le_one ht⟩,
end | lemma | convex.mem_smul_of_zero_mem | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"inv_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.add_smul (h_conv : convex 𝕜 s) {p q : 𝕜} (hp : 0 ≤ p) (hq : 0 ≤ q) :
(p + q) • s = p • s + q • s | begin
obtain rfl | hs := s.eq_empty_or_nonempty,
{ simp_rw [smul_set_empty, add_empty] },
obtain rfl | hp' := hp.eq_or_lt,
{ rw [zero_add, zero_smul_set hs, zero_add] },
obtain rfl | hq' := hq.eq_or_lt,
{ rw [add_zero, zero_smul_set hs, add_zero] },
ext,
split,
{ rintro ⟨v, hv, rfl⟩,
exact ⟨p • v,... | lemma | convex.add_smul | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"add_div",
"add_smul",
"convex",
"mul_div_cancel'",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.convex_of_chain [ordered_semiring 𝕜] [ordered_add_comm_monoid E]
[module 𝕜 E] [ordered_smul 𝕜 E] {s : set E} (hs : s.ord_connected) (h : is_chain (≤) s) :
convex 𝕜 s | begin
refine convex_iff_segment_subset.mpr (λ x hx y hy, _),
obtain hxy | hyx := h.total hx hy,
{ exact (segment_subset_Icc hxy).trans (hs.out hx hy) },
{ rw segment_symm,
exact (segment_subset_Icc hyx).trans (hs.out hy hx) }
end | lemma | set.ord_connected.convex_of_chain | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"is_chain",
"module",
"ordered_add_comm_monoid",
"ordered_semiring",
"ordered_smul",
"segment_subset_Icc",
"segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.convex [ordered_semiring 𝕜] [linear_ordered_add_comm_monoid E] [module 𝕜 E]
[ordered_smul 𝕜 E] {s : set E} (hs : s.ord_connected) :
convex 𝕜 s | hs.convex_of_chain $ is_chain_of_trichotomous s | lemma | set.ord_connected.convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"is_chain_of_trichotomous",
"linear_ordered_add_comm_monoid",
"module",
"ordered_semiring",
"ordered_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_ord_connected [linear_ordered_field 𝕜] {s : set 𝕜} :
convex 𝕜 s ↔ s.ord_connected | by simp_rw [convex_iff_segment_subset, segment_eq_uIcc, ord_connected_iff_uIcc_subset] | lemma | convex_iff_ord_connected | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_iff_segment_subset",
"linear_ordered_field",
"segment_eq_uIcc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex (K : submodule 𝕜 E) : convex 𝕜 (↑K : set E) | by { repeat {intro}, refine add_mem (smul_mem _ _ _) (smul_mem _ _ _); assumption } | lemma | submodule.convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex (K : submodule 𝕜 E) : star_convex 𝕜 (0 : E) K | K.convex K.zero_mem | lemma | submodule.star_convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"star_convex",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
std_simplex : set (ι → 𝕜) | {f | (∀ x, 0 ≤ f x) ∧ ∑ x, f x = 1} | def | std_simplex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [] | The standard simplex in the space of functions `ι → 𝕜` is the set of vectors with non-negative
coordinates with total sum `1`. This is the free object in the category of convex spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
std_simplex_eq_inter :
std_simplex 𝕜 ι = (⋂ x, {f | 0 ≤ f x}) ∩ {f | ∑ x, f x = 1} | by { ext f, simp only [std_simplex, set.mem_inter_iff, set.mem_Inter, set.mem_set_of_eq] } | lemma | std_simplex_eq_inter | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"set.mem_Inter",
"set.mem_inter_iff",
"std_simplex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_std_simplex : convex 𝕜 (std_simplex 𝕜 ι) | begin
refine λ f hf g hg a b ha hb hab, ⟨λ x, _, _⟩,
{ apply_rules [add_nonneg, mul_nonneg, hf.1, hg.1] },
{ erw [finset.sum_add_distrib, ← finset.smul_sum, ← finset.smul_sum, hf.2, hg.2,
smul_eq_mul, smul_eq_mul, mul_one, mul_one],
exact hab }
end | lemma | convex_std_simplex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"finset.smul_sum",
"mul_one",
"smul_eq_mul",
"std_simplex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_eq_mem_std_simplex (i : ι) : (λ j, ite (i = j) (1:𝕜) 0) ∈ std_simplex 𝕜 ι | ⟨λ j, by simp only; split_ifs; norm_num, by rw [finset.sum_ite_eq, if_pos (finset.mem_univ _)]⟩ | lemma | ite_eq_mem_std_simplex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"finset.mem_univ",
"std_simplex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment (x y : P) | line_map x y '' (set.Icc (0 : R) 1) | def | affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"set.Icc"
] | The segment of points weakly between `x` and `y`. When convexity is refactored to support
abstract affine combination spaces, this will no longer need to be a separate definition from
`segment`. However, lemmas involving `+ᵥ` or `-ᵥ` will still be relevant after such a
refactoring, as distinct from versions involving `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_segment_eq_segment (x y : V) : affine_segment R x y = segment R x y | by rw [segment_eq_image_line_map, affine_segment] | lemma | affine_segment_eq_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"segment",
"segment_eq_image_line_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_comm (x y : P) : affine_segment R x y = affine_segment R y x | begin
refine set.ext (λ z, _),
split;
{ rintro ⟨t, ht, hxy⟩,
refine ⟨1 - t, _, _⟩,
{ rwa [set.sub_mem_Icc_iff_right, sub_self, sub_zero] },
{ rwa [line_map_apply_one_sub] } },
end | lemma | affine_segment_comm | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"set.ext",
"set.sub_mem_Icc_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_mem_affine_segment (x y : P) : x ∈ affine_segment R x y | ⟨0, set.left_mem_Icc.2 zero_le_one, line_map_apply_zero _ _⟩ | lemma | left_mem_affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_mem_affine_segment (x y : P) : y ∈ affine_segment R x y | ⟨1, set.right_mem_Icc.2 zero_le_one, line_map_apply_one _ _⟩ | lemma | right_mem_affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_same (x : P) : affine_segment R x x = {x} | by simp_rw [affine_segment, line_map_same, affine_map.coe_const,
(set.nonempty_Icc.mpr zero_le_one).image_const] | lemma | affine_segment_same | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_map.coe_const",
"affine_segment",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_image (f : P →ᵃ[R] P') (x y : P) :
f '' affine_segment R x y = affine_segment R (f x) (f y) | begin
rw [affine_segment, affine_segment, set.image_image, ←comp_line_map],
refl
end | lemma | affine_segment_image | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"set.image_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_const_vadd_image (x y : P) (v : V) :
((+ᵥ) v) '' affine_segment R x y = affine_segment R (v +ᵥ x) (v +ᵥ y) | affine_segment_image (affine_equiv.const_vadd R P v : P →ᵃ[R] P) x y | lemma | affine_segment_const_vadd_image | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_equiv.const_vadd",
"affine_segment",
"affine_segment_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_vadd_const_image (x y : V) (p : P) :
(+ᵥ p) '' affine_segment R x y = affine_segment R (x +ᵥ p) (y +ᵥ p) | affine_segment_image (affine_equiv.vadd_const R p : V →ᵃ[R] P) x y | lemma | affine_segment_vadd_const_image | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_equiv.vadd_const",
"affine_segment",
"affine_segment_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_const_vsub_image (x y p : P) :
((-ᵥ) p) '' affine_segment R x y = affine_segment R (p -ᵥ x) (p -ᵥ y) | affine_segment_image (affine_equiv.const_vsub R p : P →ᵃ[R] V) x y | lemma | affine_segment_const_vsub_image | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_equiv.const_vsub",
"affine_segment",
"affine_segment_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_segment_vsub_const_image (x y p : P) :
(-ᵥ p) '' affine_segment R x y = affine_segment R (x -ᵥ p) (y -ᵥ p) | affine_segment_image ((affine_equiv.vadd_const R p).symm : P →ᵃ[R] V) x y | lemma | affine_segment_vsub_const_image | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_equiv.vadd_const",
"affine_segment",
"affine_segment_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_const_vadd_affine_segment {x y z : P} (v : V) :
v +ᵥ z ∈ affine_segment R (v +ᵥ x) (v +ᵥ y) ↔ z ∈ affine_segment R x y | by rw [←affine_segment_const_vadd_image, (add_action.injective v).mem_set_image] | lemma | mem_const_vadd_affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_vadd_const_affine_segment {x y z : V} (p : P) :
z +ᵥ p ∈ affine_segment R (x +ᵥ p) (y +ᵥ p) ↔ z ∈ affine_segment R x y | by rw [←affine_segment_vadd_const_image, (vadd_right_injective p).mem_set_image] | lemma | mem_vadd_const_affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"vadd_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_const_vsub_affine_segment {x y z : P} (p : P) :
p -ᵥ z ∈ affine_segment R (p -ᵥ x) (p -ᵥ y) ↔ z ∈ affine_segment R x y | by rw [←affine_segment_const_vsub_image, (vsub_right_injective p).mem_set_image] | lemma | mem_const_vsub_affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"vsub_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_vsub_const_affine_segment {x y z : P} (p : P) :
z -ᵥ p ∈ affine_segment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affine_segment R x y | by rw [←affine_segment_vsub_const_image, (vsub_left_injective p).mem_set_image] | lemma | mem_vsub_const_affine_segment | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"vsub_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw (x y z : P) : Prop | y ∈ affine_segment R x z | def | wbtw | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment"
] | The point `y` is weakly between `x` and `z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sbtw (x y z : P) : Prop | wbtw R x y z ∧ y ≠ x ∧ y ≠ z | def | sbtw | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"wbtw"
] | The point `y` is strictly between `x` and `z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wbtw.map {x y z : P} (h : wbtw R x y z) (f : P →ᵃ[R] P') : wbtw R (f x) (f y) (f z) | begin
rw [wbtw, ←affine_segment_image],
exact set.mem_image_of_mem _ h
end | lemma | wbtw.map | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"set.mem_image_of_mem",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.wbtw_map_iff {x y z : P} {f : P →ᵃ[R] P'} (hf : function.injective f) :
wbtw R (f x) (f y) (f z) ↔ wbtw R x y z | begin
refine ⟨λ h, _, λ h, h.map _⟩,
rwa [wbtw, ←affine_segment_image, hf.mem_set_image] at h
end | lemma | function.injective.wbtw_map_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.sbtw_map_iff {x y z : P} {f : P →ᵃ[R] P'} (hf : function.injective f) :
sbtw R (f x) (f y) (f z) ↔ sbtw R x y z | by simp_rw [sbtw, hf.wbtw_map_iff, hf.ne_iff] | lemma | function.injective.sbtw_map_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_equiv.wbtw_map_iff {x y z : P} (f : P ≃ᵃ[R] P') :
wbtw R (f x) (f y) (f z) ↔ wbtw R x y z | begin
refine function.injective.wbtw_map_iff (_ : function.injective f.to_affine_map),
exact f.injective
end | lemma | affine_equiv.wbtw_map_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"function.injective.wbtw_map_iff",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_equiv.sbtw_map_iff {x y z : P} (f : P ≃ᵃ[R] P') :
sbtw R (f x) (f y) (f z) ↔ sbtw R x y z | begin
refine function.injective.sbtw_map_iff (_ : function.injective f.to_affine_map),
exact f.injective
end | lemma | affine_equiv.sbtw_map_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"function.injective.sbtw_map_iff",
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_const_vadd_iff {x y z : P} (v : V) :
wbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ wbtw R x y z | mem_const_vadd_affine_segment _ | lemma | wbtw_const_vadd_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"mem_const_vadd_affine_segment",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_vadd_const_iff {x y z : V} (p : P) :
wbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ wbtw R x y z | mem_vadd_const_affine_segment _ | lemma | wbtw_vadd_const_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"mem_vadd_const_affine_segment",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_const_vsub_iff {x y z : P} (p : P) :
wbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ wbtw R x y z | mem_const_vsub_affine_segment _ | lemma | wbtw_const_vsub_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"mem_const_vsub_affine_segment",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_vsub_const_iff {x y z : P} (p : P) :
wbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ wbtw R x y z | mem_vsub_const_affine_segment _ | lemma | wbtw_vsub_const_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"mem_vsub_const_affine_segment",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_const_vadd_iff {x y z : P} (v : V) :
sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ sbtw R x y z | by simp_rw [sbtw, wbtw_const_vadd_iff, (add_action.injective v).ne_iff] | lemma | sbtw_const_vadd_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw",
"wbtw_const_vadd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_vadd_const_iff {x y z : V} (p : P) :
sbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ sbtw R x y z | by simp_rw [sbtw, wbtw_vadd_const_iff, (vadd_right_injective p).ne_iff] | lemma | sbtw_vadd_const_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw",
"vadd_right_injective",
"wbtw_vadd_const_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_const_vsub_iff {x y z : P} (p : P) :
sbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ sbtw R x y z | by simp_rw [sbtw, wbtw_const_vsub_iff, (vsub_right_injective p).ne_iff] | lemma | sbtw_const_vsub_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw",
"vsub_right_injective",
"wbtw_const_vsub_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_vsub_const_iff {x y z : P} (p : P) :
sbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ sbtw R x y z | by simp_rw [sbtw, wbtw_vsub_const_iff, (vsub_left_injective p).ne_iff] | lemma | sbtw_vsub_const_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw",
"vsub_left_injective",
"wbtw_vsub_const_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.wbtw {x y z : P} (h : sbtw R x y z) : wbtw R x y z | h.1 | lemma | sbtw.wbtw | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.ne_left {x y z : P} (h : sbtw R x y z) : y ≠ x | h.2.1 | lemma | sbtw.ne_left | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.left_ne {x y z : P} (h : sbtw R x y z) : x ≠ y | h.2.1.symm | lemma | sbtw.left_ne | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.ne_right {x y z : P} (h : sbtw R x y z) : y ≠ z | h.2.2 | lemma | sbtw.ne_right | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.right_ne {x y z : P} (h : sbtw R x y z) : z ≠ y | h.2.2.symm | lemma | sbtw.right_ne | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.mem_image_Ioo {x y z : P} (h : sbtw R x y z) :
y ∈ line_map x z '' (set.Ioo (0 : R) 1) | begin
rcases h with ⟨⟨t, ht, rfl⟩, hyx, hyz⟩,
rcases set.eq_endpoints_or_mem_Ioo_of_mem_Icc ht with rfl|rfl|ho,
{ exfalso, simpa using hyx },
{ exfalso, simpa using hyz },
{ exact ⟨t, ho, rfl⟩ }
end | lemma | sbtw.mem_image_Ioo | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw",
"set.Ioo",
"set.eq_endpoints_or_mem_Ioo_of_mem_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.mem_affine_span {x y z : P} (h : wbtw R x y z) : y ∈ line[R, x, z] | begin
rcases h with ⟨r, ⟨-, rfl⟩⟩,
exact line_map_mem_affine_span_pair _ _ _
end | lemma | wbtw.mem_affine_span | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_comm {x y z : P} : wbtw R x y z ↔ wbtw R z y x | by rw [wbtw, wbtw, affine_segment_comm] | lemma | wbtw_comm | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment_comm",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_comm {x y z : P} : sbtw R x y z ↔ sbtw R z y x | by rw [sbtw, sbtw, wbtw_comm, ←and_assoc, ←and_assoc, and.right_comm] | lemma | sbtw_comm | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"and.right_comm",
"sbtw",
"wbtw_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_self_left (x y : P) : wbtw R x x y | left_mem_affine_segment _ _ _ | lemma | wbtw_self_left | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"left_mem_affine_segment",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_self_right (x y : P) : wbtw R x y y | right_mem_affine_segment _ _ _ | lemma | wbtw_self_right | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"right_mem_affine_segment",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_self_iff {x y : P} : wbtw R x y x ↔ y = x | begin
refine ⟨λ h, _, λ h, _⟩,
{ simpa [wbtw, affine_segment] using h },
{ rw h,
exact wbtw_self_left R x x }
end | lemma | wbtw_self_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"affine_segment",
"wbtw",
"wbtw_self_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_sbtw_self_left (x y : P) : ¬ sbtw R x x y | λ h, h.ne_left rfl | lemma | not_sbtw_self_left | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_sbtw_self_right (x y : P) : ¬ sbtw R x y y | λ h, h.ne_right rfl | lemma | not_sbtw_self_right | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.left_ne_right_of_ne_left {x y z : P} (h : wbtw R x y z) (hne : y ≠ x) : x ≠ z | begin
rintro rfl,
rw wbtw_self_iff at h,
exact hne h
end | lemma | wbtw.left_ne_right_of_ne_left | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"wbtw",
"wbtw_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.left_ne_right_of_ne_right {x y z : P} (h : wbtw R x y z) (hne : y ≠ z) : x ≠ z | begin
rintro rfl,
rw wbtw_self_iff at h,
exact hne h
end | lemma | wbtw.left_ne_right_of_ne_right | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"wbtw",
"wbtw_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.left_ne_right {x y z : P} (h : sbtw R x y z) : x ≠ z | h.wbtw.left_ne_right_of_ne_left h.2.1 | lemma | sbtw.left_ne_right | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_iff_mem_image_Ioo_and_ne [no_zero_smul_divisors R V] {x y z : P} :
sbtw R x y z ↔ y ∈ line_map x z '' (set.Ioo (0 : R) 1) ∧ x ≠ z | begin
refine ⟨λ h, ⟨h.mem_image_Ioo, h.left_ne_right⟩, λ h, _⟩,
rcases h with ⟨⟨t, ht, rfl⟩, hxz⟩,
refine ⟨⟨t, set.mem_Icc_of_Ioo ht, rfl⟩, _⟩,
rw [line_map_apply, ←@vsub_ne_zero V, ←@vsub_ne_zero V _ _ _ _ z, vadd_vsub_assoc,
vadd_vsub_assoc, ←neg_vsub_eq_vsub_rev z x, ←@neg_one_smul R, ←add_smul,
... | lemma | sbtw_iff_mem_image_Ioo_and_ne | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"neg_one_smul",
"no_zero_smul_divisors",
"sbtw",
"set.Ioo",
"set.mem_Icc_of_Ioo",
"smul_ne_zero",
"vadd_vsub_assoc",
"vsub_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_sbtw_self (x y : P) : ¬ sbtw R x y x | λ h, h.left_ne_right rfl | lemma | not_sbtw_self | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_swap_left_iff [no_zero_smul_divisors R V] {x y : P} (z : P) :
(wbtw R x y z ∧ wbtw R y x z) ↔ x = y | begin
split,
{ rintro ⟨hxyz, hyxz⟩,
rcases hxyz with ⟨ty, hty, rfl⟩,
rcases hyxz with ⟨tx, htx, hx⟩,
simp_rw [line_map_apply, ←add_vadd] at hx,
rw [←@vsub_eq_zero_iff_eq V, vadd_vsub, vsub_vadd_eq_vsub_sub, smul_sub, smul_smul,
←sub_smul, ←add_smul, smul_eq_zero] at hx,
rcases hx with h|... | lemma | wbtw_swap_left_iff | analysis.convex | src/analysis/convex/between.lean | [
"data.set.intervals.group",
"analysis.convex.segment",
"linear_algebra.affine_space.finite_dimensional",
"tactic.field_simp",
"algebra.char_p.invertible"
] | [
"no_zero_smul_divisors",
"smul_eq_zero",
"smul_smul",
"smul_sub",
"vadd_vsub",
"vsub_eq_zero_iff_eq",
"vsub_vadd_eq_vsub_sub",
"wbtw",
"wbtw_self_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.