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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|---|---|---|---|---|---|---|---|---|---|---|
segment_translate_preimage (a b c : E) : (λ x, a + x) ⁻¹' [a + b -[𝕜] a + c] = [b -[𝕜] c] | set.ext $ λ x, mem_segment_translate 𝕜 a | lemma | segment_translate_preimage | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"mem_segment_translate",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_translate_preimage (a b c : E) :
(λ x, a + x) ⁻¹' open_segment 𝕜 (a + b) (a + c) = open_segment 𝕜 b c | set.ext $ λ x, mem_open_segment_translate 𝕜 a | lemma | open_segment_translate_preimage | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"mem_open_segment_translate",
"open_segment",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_translate_image (a b c : E) : (λ x, a + x) '' [b -[𝕜] c] = [a + b -[𝕜] a + c] | segment_translate_preimage 𝕜 a b c ▸ image_preimage_eq _ $ add_left_surjective a | lemma | segment_translate_image | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment_translate_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_translate_image (a b c : E) :
(λ x, a + x) '' open_segment 𝕜 b c = open_segment 𝕜 (a + b) (a + c) | open_segment_translate_preimage 𝕜 a b c ▸ image_preimage_eq _ $ add_left_surjective a | lemma | open_segment_translate_image | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment",
"open_segment_translate_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray_of_mem_segment [strict_ordered_comm_ring 𝕜] [add_comm_group E] [module 𝕜 E]
{x y z : E} (h : x ∈ [y -[𝕜] z]) : same_ray 𝕜 (x - y) (z - x) | begin
rw segment_eq_image' at h,
rcases h with ⟨θ, ⟨hθ₀, hθ₁⟩, rfl⟩,
simpa only [add_sub_cancel', ←sub_sub, sub_smul, one_smul]
using (same_ray_nonneg_smul_left (z - y) hθ₀).nonneg_smul_right (sub_nonneg.2 hθ₁)
end | lemma | same_ray_of_mem_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"add_comm_group",
"module",
"one_smul",
"same_ray",
"same_ray_nonneg_smul_left",
"segment_eq_image'",
"strict_ordered_comm_ring",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
midpoint_mem_segment [invertible (2 : 𝕜)] (x y : E) : midpoint 𝕜 x y ∈ [x -[𝕜] y] | begin
rw segment_eq_image_line_map,
exact ⟨⅟2, ⟨inv_of_nonneg.mpr zero_le_two, inv_of_le_one one_le_two⟩, rfl⟩,
end | lemma | midpoint_mem_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"inv_of_le_one",
"invertible",
"midpoint",
"segment_eq_image_line_map",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_segment_sub_add [invertible (2 : 𝕜)] (x y : E) : x ∈ [x - y -[𝕜] x + y] | by { convert @midpoint_mem_segment 𝕜 _ _ _ _ _ _ _, rw midpoint_sub_add } | lemma | mem_segment_sub_add | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"invertible",
"midpoint_mem_segment",
"midpoint_sub_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_segment_add_sub [invertible (2 : 𝕜)] (x y : E) : x ∈ [x + y -[𝕜] x - y] | by { convert @midpoint_mem_segment 𝕜 _ _ _ _ _ _ _, rw midpoint_add_sub } | lemma | mem_segment_add_sub | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"invertible",
"midpoint_add_sub",
"midpoint_mem_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_mem_open_segment_iff [densely_ordered 𝕜] [no_zero_smul_divisors 𝕜 E] :
x ∈ open_segment 𝕜 x y ↔ x = y | begin
split,
{ rintro ⟨a, b, ha, hb, hab, hx⟩,
refine smul_right_injective _ hb.ne' ((add_right_inj (a • x)).1 _),
rw [hx, ←add_smul, hab, one_smul] },
{ rintro rfl,
rw open_segment_same,
exact mem_singleton _ }
end | lemma | left_mem_open_segment_iff | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"densely_ordered",
"no_zero_smul_divisors",
"one_smul",
"open_segment",
"open_segment_same",
"smul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_mem_open_segment_iff [densely_ordered 𝕜] [no_zero_smul_divisors 𝕜 E] :
y ∈ open_segment 𝕜 x y ↔ x = y | by rw [open_segment_symm, left_mem_open_segment_iff, eq_comm] | lemma | right_mem_open_segment_iff | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"densely_ordered",
"left_mem_open_segment_iff",
"no_zero_smul_divisors",
"open_segment",
"open_segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_segment_iff_div : x ∈ [y -[𝕜] z] ↔
∃ a b : 𝕜, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x | begin
split,
{ rintro ⟨a, b, ha, hb, hab, rfl⟩,
use [a, b, ha, hb],
simp * },
{ rintro ⟨a, b, ha, hb, hab, rfl⟩,
refine ⟨a / (a + b), b / (a + b), div_nonneg ha hab.le, div_nonneg hb hab.le, _, rfl⟩,
rw [←add_div, div_self hab.ne'] }
end | lemma | mem_segment_iff_div | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"div_nonneg",
"div_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_open_segment_iff_div : x ∈ open_segment 𝕜 y z ↔
∃ a b : 𝕜, 0 < a ∧ 0 < b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x | begin
split,
{ rintro ⟨a, b, ha, hb, hab, rfl⟩,
use [a, b, ha, hb],
rw [hab, div_one, div_one] },
{ rintro ⟨a, b, ha, hb, rfl⟩,
have hab : 0 < a + b := by positivity,
refine ⟨a / (a + b), b / (a + b), by positivity, by positivity, _, rfl⟩,
rw [←add_div, div_self hab.ne'] }
end | lemma | mem_open_segment_iff_div | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"div_one",
"div_self",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_segment_iff_same_ray : x ∈ [y -[𝕜] z] ↔ same_ray 𝕜 (x - y) (z - x) | begin
refine ⟨same_ray_of_mem_segment, λ h, _⟩,
rcases h.exists_eq_smul_add with ⟨a, b, ha, hb, hab, hxy, hzx⟩,
rw [add_comm, sub_add_sub_cancel] at hxy hzx,
rw [←mem_segment_translate _ (-x), neg_add_self],
refine ⟨b, a, hb, ha, add_comm a b ▸ hab, _⟩,
rw [←sub_eq_neg_add, ←neg_sub, hxy, ←sub_eq_neg_add, h... | lemma | mem_segment_iff_same_ray | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"same_ray",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset_union (x y : E) {z : E} (hz : z ∈ range (line_map x y : 𝕜 → E)) :
open_segment 𝕜 x y ⊆ insert z (open_segment 𝕜 x z ∪ open_segment 𝕜 z y) | begin
rcases hz with ⟨c, rfl⟩,
simp only [open_segment_eq_image_line_map, ← maps_to'],
rintro a ⟨h₀, h₁⟩,
rcases lt_trichotomy a c with hac|rfl|hca,
{ right, left,
have hc : 0 < c, from h₀.trans hac,
refine ⟨a / c, ⟨div_pos h₀ hc, (div_lt_one hc).2 hac⟩, _⟩,
simp only [← homothety_eq_line_map, ← h... | lemma | open_segment_subset_union | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"div_lt_one",
"div_mul_cancel",
"one_mul",
"open_segment",
"open_segment_eq_image_line_map"
] | If `z = line_map x y c` is a point on the line passing through `x` and `y`, then the open
segment `open_segment 𝕜 x y` is included in the union of the open segments `open_segment 𝕜 x z`,
`open_segment 𝕜 z y`, and the point `z`. Informally, `(x, y) ⊆ {z} ∪ (x, z) ∪ (z, y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
segment_subset_Icc (h : x ≤ y) : [x -[𝕜] y] ⊆ Icc x y | begin
rintro z ⟨a, b, ha, hb, hab, rfl⟩,
split,
calc
x = a • x + b • x :(convex.combo_self hab _).symm
... ≤ a • x + b • y : add_le_add_left (smul_le_smul_of_nonneg h hb) _,
calc
a • x + b • y
≤ a • y + b • y : add_le_add_right (smul_le_smul_of_nonneg h ha) _
... = y : convex.combo_sel... | lemma | segment_subset_Icc | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset_Ioo (h : x < y) : open_segment 𝕜 x y ⊆ Ioo x y | begin
rintro z ⟨a, b, ha, hb, hab, rfl⟩,
split,
calc
x = a • x + b • x : (convex.combo_self hab _).symm
... < a • x + b • y : add_lt_add_left (smul_lt_smul_of_pos h hb) _,
calc
a • x + b • y
< a • y + b • y : add_lt_add_right (smul_lt_smul_of_pos h ha) _
... = y : convex.combo_self hab... | lemma | open_segment_subset_Ioo | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"open_segment",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_subset_uIcc (x y : E) : [x -[𝕜] y] ⊆ uIcc x y | begin
cases le_total x y,
{ rw uIcc_of_le h,
exact segment_subset_Icc h },
{ rw [uIcc_of_ge h, segment_symm],
exact segment_subset_Icc h }
end | lemma | segment_subset_uIcc | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment_subset_Icc",
"segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.min_le_combo (x y : E) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
min x y ≤ a • x + b • y | (segment_subset_uIcc x y ⟨_, _, ha, hb, hab, rfl⟩).1 | lemma | convex.min_le_combo | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment_subset_uIcc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.combo_le_max (x y : E) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
a • x + b • y ≤ max x y | (segment_subset_uIcc x y ⟨_, _, ha, hb, hab, rfl⟩).2 | lemma | convex.combo_le_max | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment_subset_uIcc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_subset_segment : Icc x y ⊆ [x -[𝕜] y] | begin
rintro z ⟨hxz, hyz⟩,
obtain rfl | h := (hxz.trans hyz).eq_or_lt,
{ rw segment_same,
exact hyz.antisymm hxz },
rw ←sub_nonneg at hxz hyz,
rw ←sub_pos at h,
refine ⟨(y - z) / (y - x), (z - x) / (y - x), div_nonneg hyz h.le, div_nonneg hxz h.le, _, _⟩,
{ rw [←add_div, sub_add_sub_cancel, div_self h... | lemma | Icc_subset_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"div_eq_iff",
"div_nonneg",
"div_self",
"mul_comm",
"segment_same",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_eq_Icc (h : x ≤ y) : [x -[𝕜] y] = Icc x y | (segment_subset_Icc h).antisymm Icc_subset_segment | lemma | segment_eq_Icc | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"Icc_subset_segment",
"segment_subset_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_subset_open_segment : Ioo x y ⊆ open_segment 𝕜 x y | λ z hz, mem_open_segment_of_ne_left_right hz.1.ne hz.2.ne' $ Icc_subset_segment $
Ioo_subset_Icc_self hz | lemma | Ioo_subset_open_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"Icc_subset_segment",
"mem_open_segment_of_ne_left_right",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_eq_Ioo (h : x < y) : open_segment 𝕜 x y = Ioo x y | (open_segment_subset_Ioo h).antisymm Ioo_subset_open_segment | lemma | open_segment_eq_Ioo | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"Ioo_subset_open_segment",
"open_segment",
"open_segment_subset_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_eq_Icc' (x y : 𝕜) : [x -[𝕜] y] = Icc (min x y) (max x y) | begin
cases le_total x y,
{ rw [segment_eq_Icc h, max_eq_right h, min_eq_left h] },
{ rw [segment_symm, segment_eq_Icc h, max_eq_left h, min_eq_right h] }
end | lemma | segment_eq_Icc' | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment_eq_Icc",
"segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_eq_Ioo' (hxy : x ≠ y) : open_segment 𝕜 x y = Ioo (min x y) (max x y) | begin
cases hxy.lt_or_lt,
{ rw [open_segment_eq_Ioo h, max_eq_right h.le, min_eq_left h.le] },
{ rw [open_segment_symm, open_segment_eq_Ioo h, max_eq_left h.le, min_eq_right h.le] }
end | lemma | open_segment_eq_Ioo' | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment",
"open_segment_eq_Ioo",
"open_segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_eq_uIcc (x y : 𝕜) : [x -[𝕜] y] = uIcc x y | segment_eq_Icc' _ _ | lemma | segment_eq_uIcc | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment_eq_Icc'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.mem_Icc (h : x ≤ y) :
z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z | by { rw ←segment_eq_Icc h, simp_rw [←exists_prop], refl } | lemma | convex.mem_Icc | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [] | A point is in an `Icc` iff it can be expressed as a convex combination of the endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mem_Ioo (h : x < y) :
z ∈ Ioo x y ↔ ∃ a b, 0 < a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z | by { rw ←open_segment_eq_Ioo h, simp_rw [←exists_prop], refl } | lemma | convex.mem_Ioo | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [] | A point is in an `Ioo` iff it can be expressed as a strict convex combination of the endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mem_Ioc (h : x < y) :
z ∈ Ioc x y ↔ ∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z | begin
refine ⟨λ hz, _, _⟩,
{ obtain ⟨a, b, ha, hb, hab, rfl⟩ := (convex.mem_Icc h.le).1 (Ioc_subset_Icc_self hz),
obtain rfl | hb' := hb.eq_or_lt,
{ rw add_zero at hab,
rw [hab, one_mul, zero_mul, add_zero] at hz,
exact (hz.1.ne rfl).elim },
{ exact ⟨a, b, ha, hb', hab, rfl⟩ } },
{ rintro ... | lemma | convex.mem_Ioc | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.mem_Icc",
"convex.mem_Ioo",
"one_mul",
"zero_mul"
] | A point is in an `Ioc` iff it can be expressed as a semistrict convex combination of the
endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mem_Ico (h : x < y) :
z ∈ Ico x y ↔ ∃ a b, 0 < a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z | begin
refine ⟨λ hz, _, _⟩,
{ obtain ⟨a, b, ha, hb, hab, rfl⟩ := (convex.mem_Icc h.le).1 (Ico_subset_Icc_self hz),
obtain rfl | ha' := ha.eq_or_lt,
{ rw zero_add at hab,
rw [hab, one_mul, zero_mul, zero_add] at hz,
exact (hz.2.ne rfl).elim },
{ exact ⟨a, b, ha', hb, hab, rfl⟩ } },
{ rintro ... | lemma | convex.mem_Ico | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.mem_Icc",
"convex.mem_Ioo",
"one_mul",
"zero_mul"
] | A point is in an `Ico` iff it can be expressed as a semistrict convex combination of the
endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
segment_subset (x y : E × F) : segment 𝕜 x y ⊆ segment 𝕜 x.1 y.1 ×ˢ segment 𝕜 x.2 y.2 | begin
rintro z ⟨a, b, ha, hb, hab, hz⟩,
exact ⟨⟨a, b, ha, hb, hab, congr_arg prod.fst hz⟩, a, b, ha, hb, hab, congr_arg prod.snd hz⟩,
end | lemma | prod.segment_subset | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset (x y : E × F) :
open_segment 𝕜 x y ⊆ open_segment 𝕜 x.1 y.1 ×ˢ open_segment 𝕜 x.2 y.2 | begin
rintro z ⟨a, b, ha, hb, hab, hz⟩,
exact ⟨⟨a, b, ha, hb, hab, congr_arg prod.fst hz⟩, a, b, ha, hb, hab, congr_arg prod.snd hz⟩,
end | lemma | prod.open_segment_subset | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mk_segment_left (x₁ x₂ : E) (y : F) :
(λ x, (x, y)) '' [x₁ -[𝕜] x₂] = [(x₁, y) -[𝕜] (x₂, y)] | begin
ext ⟨x', y'⟩,
simp_rw [set.mem_image, segment, set.mem_set_of, prod.smul_mk, prod.mk_add_mk,
prod.mk.inj_iff, ←exists_and_distrib_right, @exists_comm E, exists_eq_left'],
refine exists₅_congr (λ a b ha hb hab, _),
rw convex.combo_self hab,
end | lemma | prod.image_mk_segment_left | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"exists_comm",
"exists_eq_left'",
"exists₅_congr",
"prod.mk.inj_iff",
"prod.smul_mk",
"segment",
"set.mem_image",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mk_segment_right (x : E) (y₁ y₂ : F) :
(λ y, (x, y)) '' [y₁ -[𝕜] y₂] = [(x, y₁) -[𝕜] (x, y₂)] | begin
ext ⟨x', y'⟩,
simp_rw [set.mem_image, segment, set.mem_set_of, prod.smul_mk, prod.mk_add_mk,
prod.mk.inj_iff, ←exists_and_distrib_right, @exists_comm F, exists_eq_left'],
refine exists₅_congr (λ a b ha hb hab, _),
rw convex.combo_self hab,
end | lemma | prod.image_mk_segment_right | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"exists_comm",
"exists_eq_left'",
"exists₅_congr",
"prod.mk.inj_iff",
"prod.smul_mk",
"segment",
"set.mem_image",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mk_open_segment_left (x₁ x₂ : E) (y : F) :
(λ x, (x, y)) '' open_segment 𝕜 x₁ x₂ = open_segment 𝕜 (x₁, y) (x₂, y) | begin
ext ⟨x', y'⟩,
simp_rw [set.mem_image, open_segment, set.mem_set_of, prod.smul_mk, prod.mk_add_mk,
prod.mk.inj_iff, ←exists_and_distrib_right, @exists_comm E, exists_eq_left'],
refine exists₅_congr (λ a b ha hb hab, _),
rw convex.combo_self hab,
end | lemma | prod.image_mk_open_segment_left | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"exists_comm",
"exists_eq_left'",
"exists₅_congr",
"open_segment",
"prod.mk.inj_iff",
"prod.smul_mk",
"set.mem_image",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mk_open_segment_right (x : E) (y₁ y₂ : F) :
(λ y, (x, y)) '' open_segment 𝕜 y₁ y₂ = open_segment 𝕜 (x, y₁) (x, y₂) | begin
ext ⟨x', y'⟩,
simp_rw [set.mem_image, open_segment, set.mem_set_of, prod.smul_mk, prod.mk_add_mk,
prod.mk.inj_iff, ←exists_and_distrib_right, @exists_comm F, exists_eq_left'],
refine exists₅_congr (λ a b ha hb hab, _),
rw convex.combo_self hab,
end | lemma | prod.image_mk_open_segment_right | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"exists_comm",
"exists_eq_left'",
"exists₅_congr",
"open_segment",
"prod.mk.inj_iff",
"prod.smul_mk",
"set.mem_image",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_subset (x y : Π i, π i) : segment 𝕜 x y ⊆ s.pi (λ i, segment 𝕜 (x i) (y i)) | by { rintro z ⟨a, b, ha, hb, hab, hz⟩ i -, exact ⟨a, b, ha, hb, hab, congr_fun hz i⟩ } | lemma | pi.segment_subset | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_segment_subset (x y : Π i, π i) :
open_segment 𝕜 x y ⊆ s.pi (λ i, open_segment 𝕜 (x i) (y i)) | by { rintro z ⟨a, b, ha, hb, hab, hz⟩ i -, exact ⟨a, b, ha, hb, hab, congr_fun hz i⟩ } | lemma | pi.open_segment_subset | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_update_segment (i : ι) (x₁ x₂ : π i) (y : Π i, π i) :
update y i '' [x₁ -[𝕜] x₂] = [update y i x₁ -[𝕜] update y i x₂] | begin
ext z,
simp_rw [set.mem_image, segment, set.mem_set_of, ←update_smul, ←update_add, update_eq_iff,
←exists_and_distrib_right, @exists_comm (π i), exists_eq_left'],
refine exists₅_congr (λ a b ha hb hab, _),
rw convex.combo_self hab,
end | lemma | pi.image_update_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"exists_comm",
"exists_eq_left'",
"exists₅_congr",
"segment",
"set.mem_image",
"set.mem_set_of",
"update",
"update_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_update_open_segment (i : ι) (x₁ x₂ : π i) (y : Π i, π i) :
update y i '' open_segment 𝕜 x₁ x₂ = open_segment 𝕜 (update y i x₁) (update y i x₂) | begin
ext z,
simp_rw [set.mem_image, open_segment, set.mem_set_of, ←update_smul, ←update_add, update_eq_iff,
←exists_and_distrib_right, @exists_comm (π i), exists_eq_left'],
refine exists₅_congr (λ a b ha hb hab, _),
rw convex.combo_self hab,
end | lemma | pi.image_update_open_segment | analysis.convex | src/analysis/convex/segment.lean | [
"algebra.order.invertible",
"algebra.order.smul",
"linear_algebra.affine_space.midpoint",
"linear_algebra.ray",
"tactic.positivity"
] | [
"convex.combo_self",
"exists_comm",
"exists_eq_left'",
"exists₅_congr",
"open_segment",
"set.mem_image",
"set.mem_set_of",
"update",
"update_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side (s : affine_subspace R P) (x y : P) : Prop | ∃ p₁ p₂ ∈ s, same_ray R (x -ᵥ p₁) (y -ᵥ p₂) | def | affine_subspace.w_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"same_ray"
] | The points `x` and `y` are weakly on the same side of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
s_same_side (s : affine_subspace R P) (x y : P) : Prop | s.w_same_side x y ∧ x ∉ s ∧ y ∉ s | def | affine_subspace.s_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | The points `x` and `y` are strictly on the same side of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
w_opp_side (s : affine_subspace R P) (x y : P) : Prop | ∃ p₁ p₂ ∈ s, same_ray R (x -ᵥ p₁) (p₂ -ᵥ y) | def | affine_subspace.w_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"same_ray"
] | The points `x` and `y` are weakly on opposite sides of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
s_opp_side (s : affine_subspace R P) (x y : P) : Prop | s.w_opp_side x y ∧ x ∉ s ∧ y ∉ s | def | affine_subspace.s_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | The points `x` and `y` are strictly on opposite sides of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
w_same_side.map {s : affine_subspace R P} {x y : P} (h : s.w_same_side x y)
(f : P →ᵃ[R] P') : (s.map f).w_same_side (f x) (f y) | begin
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, _⟩,
simp_rw [←linear_map_vsub],
exact h.map f.linear
end | lemma | affine_subspace.w_same_side.map | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.injective.w_same_side_map_iff {s : affine_subspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : function.injective f) :
(s.map f).w_same_side (f x) (f y) ↔ s.w_same_side x y | begin
refine ⟨λ h, _, λ h, h.map _⟩,
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩,
rw mem_map at hfp₁ hfp₂,
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩,
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩,
refine ⟨p₁, hp₁, p₂, hp₂, _⟩,
simp_rw [←linear_map_vsub, (f.linear_injective_iff.2 hf).same_ray_map_iff] at h,
exact h
end | lemma | function.injective.w_same_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"mem_map",
"same_ray_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.injective.s_same_side_map_iff {s : affine_subspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : function.injective f) :
(s.map f).s_same_side (f x) (f y) ↔ s.s_same_side x y | by simp_rw [s_same_side, hf.w_same_side_map_iff, mem_map_iff_mem_of_injective hf] | lemma | function.injective.s_same_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.affine_equiv.w_same_side_map_iff {s : affine_subspace R P} {x y : P}
(f : P ≃ᵃ[R] P') : (s.map ↑f).w_same_side (f x) (f y) ↔ s.w_same_side x y | (show function.injective f.to_affine_map, from f.injective).w_same_side_map_iff | lemma | affine_equiv.w_same_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.affine_equiv.s_same_side_map_iff {s : affine_subspace R P} {x y : P}
(f : P ≃ᵃ[R] P') : (s.map ↑f).s_same_side (f x) (f y) ↔ s.s_same_side x y | (show function.injective f.to_affine_map, from f.injective).s_same_side_map_iff | lemma | affine_equiv.s_same_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side.map {s : affine_subspace R P} {x y : P} (h : s.w_opp_side x y)
(f : P →ᵃ[R] P') : (s.map f).w_opp_side (f x) (f y) | begin
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, _⟩,
simp_rw [←linear_map_vsub],
exact h.map f.linear
end | lemma | affine_subspace.w_opp_side.map | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.injective.w_opp_side_map_iff {s : affine_subspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : function.injective f) :
(s.map f).w_opp_side (f x) (f y) ↔ s.w_opp_side x y | begin
refine ⟨λ h, _, λ h, h.map _⟩,
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩,
rw mem_map at hfp₁ hfp₂,
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩,
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩,
refine ⟨p₁, hp₁, p₂, hp₂, _⟩,
simp_rw [←linear_map_vsub, (f.linear_injective_iff.2 hf).same_ray_map_iff] at h,
exact h
end | lemma | function.injective.w_opp_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"mem_map",
"same_ray_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.injective.s_opp_side_map_iff {s : affine_subspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : function.injective f) :
(s.map f).s_opp_side (f x) (f y) ↔ s.s_opp_side x y | by simp_rw [s_opp_side, hf.w_opp_side_map_iff, mem_map_iff_mem_of_injective hf] | lemma | function.injective.s_opp_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.affine_equiv.w_opp_side_map_iff {s : affine_subspace R P} {x y : P}
(f : P ≃ᵃ[R] P') : (s.map ↑f).w_opp_side (f x) (f y) ↔ s.w_opp_side x y | (show function.injective f.to_affine_map, from f.injective).w_opp_side_map_iff | lemma | affine_equiv.w_opp_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.affine_equiv.s_opp_side_map_iff {s : affine_subspace R P} {x y : P}
(f : P ≃ᵃ[R] P') : (s.map ↑f).s_opp_side (f x) (f y) ↔ s.s_opp_side x y | (show function.injective f.to_affine_map, from f.injective).s_opp_side_map_iff | lemma | affine_equiv.s_opp_side_map_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side.nonempty {s : affine_subspace R P} {x y : P} (h : s.w_same_side x y) :
(s : set P).nonempty | ⟨h.some, h.some_spec.some⟩ | lemma | affine_subspace.w_same_side.nonempty | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side.nonempty {s : affine_subspace R P} {x y : P} (h : s.s_same_side x y) :
(s : set P).nonempty | ⟨h.1.some, h.1.some_spec.some⟩ | lemma | affine_subspace.s_same_side.nonempty | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side.nonempty {s : affine_subspace R P} {x y : P} (h : s.w_opp_side x y) :
(s : set P).nonempty | ⟨h.some, h.some_spec.some⟩ | lemma | affine_subspace.w_opp_side.nonempty | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side.nonempty {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
(s : set P).nonempty | ⟨h.1.some, h.1.some_spec.some⟩ | lemma | affine_subspace.s_opp_side.nonempty | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side.w_same_side {s : affine_subspace R P} {x y : P} (h : s.s_same_side x y) :
s.w_same_side x y | h.1 | lemma | affine_subspace.s_same_side.w_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side.left_not_mem {s : affine_subspace R P} {x y : P} (h : s.s_same_side x y) :
x ∉ s | h.2.1 | lemma | affine_subspace.s_same_side.left_not_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side.right_not_mem {s : affine_subspace R P} {x y : P} (h : s.s_same_side x y) :
y ∉ s | h.2.2 | lemma | affine_subspace.s_same_side.right_not_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side.w_opp_side {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
s.w_opp_side x y | h.1 | lemma | affine_subspace.s_opp_side.w_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side.left_not_mem {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
x ∉ s | h.2.1 | lemma | affine_subspace.s_opp_side.left_not_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side.right_not_mem {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
y ∉ s | h.2.2 | lemma | affine_subspace.s_opp_side.right_not_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_comm {s : affine_subspace R P} {x y : P} :
s.w_same_side x y ↔ s.w_same_side y x | ⟨λ ⟨p₁, hp₁, p₂, hp₂, h⟩, ⟨p₂, hp₂, p₁, hp₁, h.symm⟩,
λ ⟨p₁, hp₁, p₂, hp₂, h⟩, ⟨p₂, hp₂, p₁, hp₁, h.symm⟩⟩ | lemma | affine_subspace.w_same_side_comm | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_comm {s : affine_subspace R P} {x y : P} :
s.s_same_side x y ↔ s.s_same_side y x | by rw [s_same_side, s_same_side, w_same_side_comm, and_comm (x ∉ s)] | lemma | affine_subspace.s_same_side_comm | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_comm {s : affine_subspace R P} {x y : P} :
s.w_opp_side x y ↔ s.w_opp_side y x | begin
split,
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨p₂, hp₂, p₁, hp₁, _⟩,
rwa [same_ray_comm, ←same_ray_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] },
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨p₂, hp₂, p₁, hp₁, _⟩,
rwa [same_ray_comm, ←same_ray_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vs... | lemma | affine_subspace.w_opp_side_comm | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"neg_vsub_eq_vsub_rev",
"same_ray_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_comm {s : affine_subspace R P} {x y : P} :
s.s_opp_side x y ↔ s.s_opp_side y x | by rw [s_opp_side, s_opp_side, w_opp_side_comm, and_comm (x ∉ s)] | lemma | affine_subspace.s_opp_side_comm | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_w_same_side_bot (x y : P) : ¬ (⊥ : affine_subspace R P).w_same_side x y | by simp [w_same_side, not_mem_bot] | lemma | affine_subspace.not_w_same_side_bot | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_s_same_side_bot (x y : P) : ¬ (⊥ : affine_subspace R P).s_same_side x y | λ h, not_w_same_side_bot x y h.w_same_side | lemma | affine_subspace.not_s_same_side_bot | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_w_opp_side_bot (x y : P) : ¬ (⊥ : affine_subspace R P).w_opp_side x y | by simp [w_opp_side, not_mem_bot] | lemma | affine_subspace.not_w_opp_side_bot | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_s_opp_side_bot (x y : P) : ¬ (⊥ : affine_subspace R P).s_opp_side x y | λ h, not_w_opp_side_bot x y h.w_opp_side | lemma | affine_subspace.not_s_opp_side_bot | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_self_iff {s : affine_subspace R P} {x : P} :
s.w_same_side x x ↔ (s : set P).nonempty | ⟨λ h, h.nonempty, λ ⟨p, hp⟩, ⟨p, hp, p, hp, same_ray.rfl⟩⟩ | lemma | affine_subspace.w_same_side_self_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_self_iff {s : affine_subspace R P} {x : P} :
s.s_same_side x x ↔ (s : set P).nonempty ∧ x ∉ s | ⟨λ ⟨h, hx, _⟩, ⟨w_same_side_self_iff.1 h, hx⟩, λ ⟨h, hx⟩, ⟨w_same_side_self_iff.2 h, hx, hx⟩⟩ | lemma | affine_subspace.s_same_side_self_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_of_left_mem {s : affine_subspace R P} {x : P} (y : P) (hx : x ∈ s) :
s.w_same_side x y | begin
refine ⟨x, hx, x, hx, _⟩,
simp
end | lemma | affine_subspace.w_same_side_of_left_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_of_right_mem {s : affine_subspace R P} (x : P) {y : P} (hy : y ∈ s) :
s.w_same_side x y | (w_same_side_of_left_mem x hy).symm | lemma | affine_subspace.w_same_side_of_right_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_of_left_mem {s : affine_subspace R P} {x : P} (y : P) (hx : x ∈ s) :
s.w_opp_side x y | begin
refine ⟨x, hx, x, hx, _⟩,
simp
end | lemma | affine_subspace.w_opp_side_of_left_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_of_right_mem {s : affine_subspace R P} (x : P) {y : P} (hy : y ∈ s) :
s.w_opp_side x y | (w_opp_side_of_left_mem x hy).symm | lemma | affine_subspace.w_opp_side_of_right_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_vadd_left_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.w_same_side (v +ᵥ x) y ↔ s.w_same_side x y | begin
split,
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨-v +ᵥ p₁,
affine_subspace.vadd_mem_of_mem_direction (submodule.neg_mem _ hv) hp₁, p₂, hp₂, _⟩,
rwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, ←vadd_vsub_assoc] },
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨v +ᵥ p₁,
affine... | lemma | affine_subspace.w_same_side_vadd_left_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"affine_subspace.vadd_mem_of_mem_direction",
"submodule.neg_mem",
"vadd_vsub_vadd_cancel_left",
"vsub_vadd_eq_vsub_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_vadd_right_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.w_same_side x (v +ᵥ y) ↔ s.w_same_side x y | by rw [w_same_side_comm, w_same_side_vadd_left_iff hv, w_same_side_comm] | lemma | affine_subspace.w_same_side_vadd_right_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_vadd_left_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.s_same_side (v +ᵥ x) y ↔ s.s_same_side x y | by rw [s_same_side, s_same_side, w_same_side_vadd_left_iff hv,
vadd_mem_iff_mem_of_mem_direction hv] | lemma | affine_subspace.s_same_side_vadd_left_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_vadd_right_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.s_same_side x (v +ᵥ y) ↔ s.s_same_side x y | by rw [s_same_side_comm, s_same_side_vadd_left_iff hv, s_same_side_comm] | lemma | affine_subspace.s_same_side_vadd_right_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_vadd_left_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.w_opp_side (v +ᵥ x) y ↔ s.w_opp_side x y | begin
split,
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨-v +ᵥ p₁,
affine_subspace.vadd_mem_of_mem_direction (submodule.neg_mem _ hv) hp₁, p₂, hp₂, _⟩,
rwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, ←vadd_vsub_assoc] },
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
refine ⟨v +ᵥ p₁,
affine... | lemma | affine_subspace.w_opp_side_vadd_left_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"affine_subspace.vadd_mem_of_mem_direction",
"submodule.neg_mem",
"vadd_vsub_vadd_cancel_left",
"vsub_vadd_eq_vsub_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_vadd_right_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.w_opp_side x (v +ᵥ y) ↔ s.w_opp_side x y | by rw [w_opp_side_comm, w_opp_side_vadd_left_iff hv, w_opp_side_comm] | lemma | affine_subspace.w_opp_side_vadd_right_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_vadd_left_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.s_opp_side (v +ᵥ x) y ↔ s.s_opp_side x y | by rw [s_opp_side, s_opp_side, w_opp_side_vadd_left_iff hv,
vadd_mem_iff_mem_of_mem_direction hv] | lemma | affine_subspace.s_opp_side_vadd_left_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_vadd_right_iff {s : affine_subspace R P} {x y : P} {v : V}
(hv : v ∈ s.direction) : s.s_opp_side x (v +ᵥ y) ↔ s.s_opp_side x y | by rw [s_opp_side_comm, s_opp_side_vadd_left_iff hv, s_opp_side_comm] | lemma | affine_subspace.s_opp_side_vadd_right_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_smul_vsub_vadd_left {s : affine_subspace R P} {p₁ p₂ : P} (x : P)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t) : s.w_same_side (t • (x -ᵥ p₁) +ᵥ p₂) x | begin
refine ⟨p₂, hp₂, p₁, hp₁, _⟩,
rw vadd_vsub,
exact same_ray_nonneg_smul_left _ ht
end | lemma | affine_subspace.w_same_side_smul_vsub_vadd_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"same_ray_nonneg_smul_left",
"vadd_vsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_smul_vsub_vadd_right {s : affine_subspace R P} {p₁ p₂ : P} (x : P)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t) : s.w_same_side x (t • (x -ᵥ p₁) +ᵥ p₂) | (w_same_side_smul_vsub_vadd_left x hp₁ hp₂ ht).symm | lemma | affine_subspace.w_same_side_smul_vsub_vadd_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_line_map_left {s : affine_subspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
(ht : 0 ≤ t) : s.w_same_side (line_map x y t) y | w_same_side_smul_vsub_vadd_left y h h ht | lemma | affine_subspace.w_same_side_line_map_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_line_map_right {s : affine_subspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
(ht : 0 ≤ t) : s.w_same_side y (line_map x y t) | (w_same_side_line_map_left y h ht).symm | lemma | affine_subspace.w_same_side_line_map_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_smul_vsub_vadd_left {s : affine_subspace R P} {p₁ p₂ : P} (x : P)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.w_opp_side (t • (x -ᵥ p₁) +ᵥ p₂) x | begin
refine ⟨p₂, hp₂, p₁, hp₁, _⟩,
rw [vadd_vsub, ←neg_neg t, neg_smul, ←smul_neg, neg_vsub_eq_vsub_rev],
exact same_ray_nonneg_smul_left _ (neg_nonneg.2 ht)
end | lemma | affine_subspace.w_opp_side_smul_vsub_vadd_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"neg_smul",
"neg_vsub_eq_vsub_rev",
"same_ray_nonneg_smul_left",
"vadd_vsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_smul_vsub_vadd_right {s : affine_subspace R P} {p₁ p₂ : P} (x : P)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.w_opp_side x (t • (x -ᵥ p₁) +ᵥ p₂) | (w_opp_side_smul_vsub_vadd_left x hp₁ hp₂ ht).symm | lemma | affine_subspace.w_opp_side_smul_vsub_vadd_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_line_map_left {s : affine_subspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
(ht : t ≤ 0) : s.w_opp_side (line_map x y t) y | w_opp_side_smul_vsub_vadd_left y h h ht | lemma | affine_subspace.w_opp_side_line_map_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_line_map_right {s : affine_subspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
(ht : t ≤ 0) : s.w_opp_side y (line_map x y t) | (w_opp_side_line_map_left y h ht).symm | lemma | affine_subspace.w_opp_side_line_map_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wbtw.w_same_side₂₃ {s : affine_subspace R P} {x y z : P} (h : wbtw R x y z)
(hx : x ∈ s) : s.w_same_side y z | begin
rcases h with ⟨t, ⟨ht0, -⟩, rfl⟩,
exact w_same_side_line_map_left z hx ht0
end | lemma | wbtw.w_same_side₂₃ | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wbtw.w_same_side₃₂ {s : affine_subspace R P} {x y z : P} (h : wbtw R x y z)
(hx : x ∈ s) : s.w_same_side z y | (h.w_same_side₂₃ hx).symm | lemma | wbtw.w_same_side₃₂ | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wbtw.w_same_side₁₂ {s : affine_subspace R P} {x y z : P} (h : wbtw R x y z)
(hz : z ∈ s) : s.w_same_side x y | h.symm.w_same_side₃₂ hz | lemma | wbtw.w_same_side₁₂ | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wbtw.w_same_side₂₁ {s : affine_subspace R P} {x y z : P} (h : wbtw R x y z)
(hz : z ∈ s) : s.w_same_side y x | h.symm.w_same_side₂₃ hz | lemma | wbtw.w_same_side₂₁ | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wbtw.w_opp_side₁₃ {s : affine_subspace R P} {x y z : P} (h : wbtw R x y z)
(hy : y ∈ s) : s.w_opp_side x z | begin
rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩,
refine ⟨_, hy, _, hy, _⟩,
rcases ht1.lt_or_eq with ht1' | rfl, swap, { simp },
rcases ht0.lt_or_eq with ht0' | rfl, swap, { simp },
refine or.inr (or.inr ⟨1 - t, t, sub_pos.2 ht1', ht0', _⟩),
simp_rw [line_map_apply, vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, ←neg_vsub... | lemma | wbtw.w_opp_side₁₃ | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"smul_neg",
"smul_smul",
"vadd_vsub_assoc",
"vsub_self",
"vsub_vadd_eq_vsub_sub",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wbtw.w_opp_side₃₁ {s : affine_subspace R P} {x y z : P} (h : wbtw R x y z)
(hy : y ∈ s) : s.w_opp_side z x | h.symm.w_opp_side₁₃ hy | lemma | wbtw.w_opp_side₃₁ | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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