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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|---|---|---|---|---|---|---|---|---|---|---|
wbtw_swap_right_iff [no_zero_smul_divisors R V] (x : P) {y z : P} :
(wbtw R x y z ∧ wbtw R x z y) ↔ y = z | begin
nth_rewrite 0 wbtw_comm,
nth_rewrite 1 wbtw_comm,
rw eq_comm,
exact wbtw_swap_left_iff R x
end | lemma | wbtw_swap_right_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",
"wbtw",
"wbtw_comm",
"wbtw_swap_left_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_rotate_iff [no_zero_smul_divisors R V] (x : P) {y z : P} :
(wbtw R x y z ∧ wbtw R z x y) ↔ x = y | by rw [wbtw_comm, wbtw_swap_right_iff, eq_comm] | lemma | wbtw_rotate_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",
"wbtw",
"wbtw_comm",
"wbtw_swap_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.swap_left_iff [no_zero_smul_divisors R V] {x y z : P} (h : wbtw R x y z) :
wbtw R y x z ↔ x = y | by rw [←wbtw_swap_left_iff R z, and_iff_right 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",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.swap_right_iff [no_zero_smul_divisors R V] {x y z : P} (h : wbtw R x y z) :
wbtw R x z y ↔ y = z | by rw [←wbtw_swap_right_iff R x, and_iff_right h] | lemma | wbtw.swap_right_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",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.rotate_iff [no_zero_smul_divisors R V] {x y z : P} (h : wbtw R x y z) :
wbtw R z x y ↔ x = y | by rw [←wbtw_rotate_iff R x, and_iff_right h] | lemma | wbtw.rotate_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",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.not_swap_left [no_zero_smul_divisors R V] {x y z : P} (h : sbtw R x y z) :
¬ wbtw R y x z | λ hs, h.left_ne (h.wbtw.swap_left_iff.1 hs) | lemma | sbtw.not_swap_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.not_swap_right [no_zero_smul_divisors R V] {x y z : P} (h : sbtw R x y z) :
¬ wbtw R x z y | λ hs, h.ne_right (h.wbtw.swap_right_iff.1 hs) | lemma | sbtw.not_swap_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.not_rotate [no_zero_smul_divisors R V] {x y z : P} (h : sbtw R x y z) :
¬ wbtw R z x y | λ hs, h.left_ne (h.wbtw.rotate_iff.1 hs) | lemma | sbtw.not_rotate | 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",
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_line_map_iff [no_zero_smul_divisors R V] {x y : P} {r : R} :
wbtw R x (line_map x y r) y ↔ x = y ∨ r ∈ set.Icc (0 : R) 1 | begin
by_cases hxy : x = y, { simp [hxy] },
rw [or_iff_right hxy, wbtw, affine_segment, (line_map_injective R hxy).mem_set_image]
end | lemma | wbtw_line_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"
] | [
"affine_segment",
"no_zero_smul_divisors",
"or_iff_right",
"set.Icc",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_line_map_iff [no_zero_smul_divisors R V] {x y : P} {r : R} :
sbtw R x (line_map x y r) y ↔ x ≠ y ∧ r ∈ set.Ioo (0 : R) 1 | begin
rw [sbtw_iff_mem_image_Ioo_and_ne, and_comm, and_congr_right],
intro hxy,
rw (line_map_injective R hxy).mem_set_image
end | lemma | sbtw_line_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"sbtw_iff_mem_image_Ioo_and_ne",
"set.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_mul_sub_add_iff [no_zero_divisors R] {x y r : R} :
wbtw R x (r * (y - x) + x) y ↔ x = y ∨ r ∈ set.Icc (0 : R) 1 | wbtw_line_map_iff | lemma | wbtw_mul_sub_add_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_divisors",
"set.Icc",
"wbtw",
"wbtw_line_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_mul_sub_add_iff [no_zero_divisors R] {x y r : R} :
sbtw R x (r * (y - x) + x) y ↔ x ≠ y ∧ r ∈ set.Ioo (0 : R) 1 | sbtw_line_map_iff | lemma | sbtw_mul_sub_add_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_divisors",
"sbtw",
"sbtw_line_map_iff",
"set.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_zero_one_iff {x : R} : wbtw R 0 x 1 ↔ x ∈ set.Icc (0 : R) 1 | begin
simp_rw [wbtw, affine_segment, set.mem_image, line_map_apply_ring],
simp
end | lemma | wbtw_zero_one_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",
"set.Icc",
"set.mem_image",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_one_zero_iff {x : R} : wbtw R 1 x 0 ↔ x ∈ set.Icc (0 : R) 1 | by rw [wbtw_comm, wbtw_zero_one_iff] | lemma | wbtw_one_zero_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"
] | [
"set.Icc",
"wbtw",
"wbtw_comm",
"wbtw_zero_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_zero_one_iff {x : R} : sbtw R 0 x 1 ↔ x ∈ set.Ioo (0 : R) 1 | begin
rw [sbtw, wbtw_zero_one_iff, set.mem_Icc, set.mem_Ioo],
exact ⟨λ h, ⟨h.1.1.lt_of_ne (ne.symm h.2.1), h.1.2.lt_of_ne h.2.2⟩,
λ h, ⟨⟨h.1.le, h.2.le⟩, h.1.ne', h.2.ne⟩⟩
end | lemma | sbtw_zero_one_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",
"set.Ioo",
"set.mem_Icc",
"set.mem_Ioo",
"wbtw_zero_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_one_zero_iff {x : R} : sbtw R 1 x 0 ↔ x ∈ set.Ioo (0 : R) 1 | by rw [sbtw_comm, sbtw_zero_one_iff] | lemma | sbtw_one_zero_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",
"sbtw_comm",
"sbtw_zero_one_iff",
"set.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_left {w x y z : P} (h₁ : wbtw R w y z) (h₂ : wbtw R w x y) : wbtw R w x z | begin
rcases h₁ with ⟨t₁, ht₁, rfl⟩,
rcases h₂ with ⟨t₂, ht₂, rfl⟩,
refine ⟨t₂ * t₁, ⟨mul_nonneg ht₂.1 ht₁.1, mul_le_one ht₂.2 ht₁.1 ht₁.2⟩, _⟩,
simp [line_map_apply, smul_smul]
end | lemma | wbtw.trans_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"
] | [
"mul_le_one",
"smul_smul",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_right {w x y z : P} (h₁ : wbtw R w x z) (h₂ : wbtw R x y z) : wbtw R w y z | begin
rw wbtw_comm at *,
exact h₁.trans_left h₂
end | lemma | wbtw.trans_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_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_sbtw_left [no_zero_smul_divisors R V] {w x y z : P} (h₁ : wbtw R w y z)
(h₂ : sbtw R w x y) : sbtw R w x z | begin
refine ⟨h₁.trans_left h₂.wbtw, h₂.ne_left, _⟩,
rintro rfl,
exact h₂.right_ne ((wbtw_swap_right_iff R w).1 ⟨h₁, h₂.wbtw⟩)
end | lemma | wbtw.trans_sbtw_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"wbtw",
"wbtw_swap_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_sbtw_right [no_zero_smul_divisors R V] {w x y z : P} (h₁ : wbtw R w x z)
(h₂ : sbtw R x y z) : sbtw R w y z | begin
rw wbtw_comm at *,
rw sbtw_comm at *,
exact h₁.trans_sbtw_left h₂
end | lemma | wbtw.trans_sbtw_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"sbtw_comm",
"wbtw",
"wbtw_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.trans_left [no_zero_smul_divisors R V] {w x y z : P} (h₁ : sbtw R w y z)
(h₂ : sbtw R w x y) : sbtw R w x z | h₁.wbtw.trans_sbtw_left h₂ | lemma | sbtw.trans_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"
] | [
"no_zero_smul_divisors",
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.trans_right [no_zero_smul_divisors R V] {w x y z : P} (h₁ : sbtw R w x z)
(h₂ : sbtw R x y z) : sbtw R w y z | h₁.wbtw.trans_sbtw_right h₂ | lemma | sbtw.trans_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"
] | [
"no_zero_smul_divisors",
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_left_ne [no_zero_smul_divisors R V] {w x y z : P} (h₁ : wbtw R w y z)
(h₂ : wbtw R w x y) (h : y ≠ z) : x ≠ z | begin
rintro rfl,
exact h (h₁.swap_right_iff.1 h₂)
end | lemma | wbtw.trans_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"
] | [
"no_zero_smul_divisors",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_right_ne [no_zero_smul_divisors R V] {w x y z : P} (h₁ : wbtw R w x z)
(h₂ : wbtw R x y z) (h : w ≠ x) : w ≠ y | begin
rintro rfl,
exact h (h₁.swap_left_iff.1 h₂)
end | lemma | wbtw.trans_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"
] | [
"no_zero_smul_divisors",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.trans_wbtw_left_ne [no_zero_smul_divisors R V] {w x y z : P} (h₁ : sbtw R w y z)
(h₂ : wbtw R w x y) : x ≠ z | h₁.wbtw.trans_left_ne h₂ h₁.ne_right | lemma | sbtw.trans_wbtw_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.trans_wbtw_right_ne [no_zero_smul_divisors R V] {w x y z : P} (h₁ : sbtw R w x z)
(h₂ : wbtw R x y z) : w ≠ y | h₁.wbtw.trans_right_ne h₂ h₁.left_ne | lemma | sbtw.trans_wbtw_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"
] | [
"no_zero_smul_divisors",
"sbtw",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.affine_combination_of_mem_affine_span_pair [no_zero_divisors R]
[no_zero_smul_divisors R V] {ι : Type*} {p : ι → P} (ha : affine_independent R p)
{w w₁ w₂ : ι → R} {s : finset ι} (hw : ∑ i in s, w i = 1) (hw₁ : ∑ i in s, w₁ i = 1)
(hw₂ : ∑ i in s, w₂ i = 1)
(h : s.affine_combination R p w ∈
line[R, s.a... | begin
rw affine_combination_mem_affine_span_pair ha hw hw₁ hw₂ at h,
rcases h with ⟨r, hr⟩,
dsimp only at hr,
rw [hr i his, sbtw_mul_sub_add_iff] at hs,
change ∀ i ∈ s, w i = ((r • (w₂ - w₁) + w₁) i) at hr,
rw s.affine_combination_congr hr (λ _ _, rfl),
dsimp only,
rw [←s.weighted_vsub_vadd_affine_combi... | lemma | sbtw.affine_combination_of_mem_affine_span_pair | 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_combination_mem_affine_span_pair",
"affine_independent",
"finset",
"no_zero_divisors",
"no_zero_smul_divisors",
"sbtw",
"sbtw_line_map_iff",
"sbtw_mul_sub_add_iff",
"vsub_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.same_ray_vsub {x y z : P} (h : wbtw R x y z) : same_ray R (y -ᵥ x) (z -ᵥ y) | begin
rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩,
simp_rw line_map_apply,
rcases ht0.lt_or_eq with ht0' | rfl, swap, { simp },
rcases ht1.lt_or_eq with ht1' | rfl, swap, { simp },
refine or.inr (or.inr ⟨1 - t, t, sub_pos.2 ht1', ht0', _⟩),
simp [vsub_vadd_eq_vsub_sub, smul_sub, smul_smul, ←sub_smul],
ring_nf
end | lemma | wbtw.same_ray_vsub | 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"
] | [
"same_ray",
"smul_smul",
"smul_sub",
"vsub_vadd_eq_vsub_sub",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.same_ray_vsub_left {x y z : P} (h : wbtw R x y z) : same_ray R (y -ᵥ x) (z -ᵥ x) | begin
rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩,
simpa [line_map_apply] using same_ray_nonneg_smul_left (z -ᵥ x) ht0
end | lemma | wbtw.same_ray_vsub_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"
] | [
"same_ray",
"same_ray_nonneg_smul_left",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.same_ray_vsub_right {x y z : P} (h : wbtw R x y z) : same_ray R (z -ᵥ x) (z -ᵥ y) | begin
rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩,
simpa [line_map_apply, vsub_vadd_eq_vsub_sub, sub_smul] using
same_ray_nonneg_smul_right (z -ᵥ x) (sub_nonneg.2 ht1)
end | lemma | wbtw.same_ray_vsub_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"
] | [
"same_ray",
"same_ray_nonneg_smul_right",
"sub_smul",
"vsub_vadd_eq_vsub_sub",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_of_sbtw_of_sbtw_of_mem_affine_span_pair [no_zero_smul_divisors R V]
{t : affine.triangle R P} {i₁ i₂ i₃ : fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
(h₁ : sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : sbtw R (t.points i₁) p₂ (t.points i₃))
(h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
... | begin
-- Should not be needed; see comments on local instances in `data.sign`.
letI : decidable_rel ((<) : R → R → Prop) := linear_ordered_ring.decidable_lt,
have h₁₃ : i₁ ≠ i₃, { rintro rfl, simpa using h₂ },
have h₂₃ : i₂ ≠ i₃, { rintro rfl, simpa using h₁ },
have h3 : ∀ i : fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃,... | lemma | sbtw_of_sbtw_of_sbtw_of_mem_affine_span_pair | 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.triangle",
"affine_span",
"affine_span_mono",
"affine_span_pair_le_of_mem_of_mem",
"affine_subspace.le_def'",
"by_contra",
"eq_affine_combination_of_mem_affine_span_of_fintype",
"finset",
"finset.affine_combination_line_map_weights_apply_of_ne",
"finset.affine_combination_single_weights_ap... | Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wbtw_iff_left_eq_or_right_mem_image_Ici {x y z : P} :
wbtw R x y z ↔ x = y ∨ z ∈ line_map x y '' (set.Ici (1 : R)) | begin
refine ⟨λ h, _, λ h, _⟩,
{ rcases h with ⟨r, ⟨hr0, hr1⟩, rfl⟩,
rcases hr0.lt_or_eq with hr0' | rfl,
{ rw set.mem_image,
refine or.inr ⟨r⁻¹, one_le_inv hr0' hr1, _⟩,
simp only [line_map_apply, smul_smul, vadd_vsub],
rw [inv_mul_cancel hr0'.ne', one_smul, vsub_vadd] },
{ simp } },
... | lemma | wbtw_iff_left_eq_or_right_mem_image_Ici | 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"
] | [
"inv_le_one",
"inv_mul_cancel",
"one_le_inv",
"one_smul",
"set.Ici",
"set.mem_Ici",
"set.mem_image",
"smul_smul",
"vadd_vsub",
"vsub_vadd",
"wbtw",
"wbtw_self_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.right_mem_image_Ici_of_left_ne {x y z : P} (h : wbtw R x y z) (hne : x ≠ y) :
z ∈ line_map x y '' (set.Ici (1 : R)) | (wbtw_iff_left_eq_or_right_mem_image_Ici.1 h).resolve_left hne | lemma | wbtw.right_mem_image_Ici_of_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"
] | [
"set.Ici",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.right_mem_affine_span_of_left_ne {x y z : P} (h : wbtw R x y z) (hne : x ≠ y) :
z ∈ line[R, x, y] | begin
rcases h.right_mem_image_Ici_of_left_ne hne with ⟨r, ⟨-, rfl⟩⟩,
exact line_map_mem_affine_span_pair _ _ _
end | lemma | wbtw.right_mem_affine_span_of_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"
] | [
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_iff_left_ne_and_right_mem_image_IoI {x y z : P} :
sbtw R x y z ↔ x ≠ y ∧ z ∈ line_map x y '' (set.Ioi (1 : R)) | begin
refine ⟨λ h, ⟨h.left_ne, _⟩, λ h, _⟩,
{ obtain ⟨r, ⟨hr, rfl⟩⟩ := h.wbtw.right_mem_image_Ici_of_left_ne h.left_ne,
rw [set.mem_Ici] at hr,
rcases hr.lt_or_eq with hrlt | rfl,
{ exact set.mem_image_of_mem _ hrlt },
{ exfalso, simpa using h } },
{ rcases h with ⟨hne, r, hr, rfl⟩,
rw set.mem... | lemma | sbtw_iff_left_ne_and_right_mem_image_IoI | 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.Ioi",
"set.Ioi_subset_Ici_self",
"set.mem_Ici",
"set.mem_Ioi",
"set.mem_image_of_mem",
"set.mem_of_mem_of_subset",
"smul_ne_zero_iff",
"vsub_ne_zero",
"vsub_vadd_eq_vsub_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.right_mem_image_Ioi {x y z : P} (h : sbtw R x y z) :
z ∈ line_map x y '' (set.Ioi (1 : R)) | (sbtw_iff_left_ne_and_right_mem_image_IoI.1 h).2 | lemma | sbtw.right_mem_image_Ioi | 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.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.right_mem_affine_span {x y z : P} (h : sbtw R x y z) : z ∈ line[R, x, y] | h.wbtw.right_mem_affine_span_of_left_ne h.left_ne | lemma | sbtw.right_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"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_iff_right_eq_or_left_mem_image_Ici {x y z : P} :
wbtw R x y z ↔ z = y ∨ x ∈ line_map z y '' (set.Ici (1 : R)) | by rw [wbtw_comm, wbtw_iff_left_eq_or_right_mem_image_Ici] | lemma | wbtw_iff_right_eq_or_left_mem_image_Ici | 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.Ici",
"wbtw",
"wbtw_comm",
"wbtw_iff_left_eq_or_right_mem_image_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.left_mem_image_Ici_of_right_ne {x y z : P} (h : wbtw R x y z) (hne : z ≠ y) :
x ∈ line_map z y '' (set.Ici (1 : R)) | h.symm.right_mem_image_Ici_of_left_ne hne | lemma | wbtw.left_mem_image_Ici_of_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"
] | [
"set.Ici",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.left_mem_affine_span_of_right_ne {x y z : P} (h : wbtw R x y z) (hne : z ≠ y) :
x ∈ line[R, z, y] | h.symm.right_mem_affine_span_of_left_ne hne | lemma | wbtw.left_mem_affine_span_of_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"
] | [
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_iff_right_ne_and_left_mem_image_IoI {x y z : P} :
sbtw R x y z ↔ z ≠ y ∧ x ∈ line_map z y '' (set.Ioi (1 : R)) | by rw [sbtw_comm, sbtw_iff_left_ne_and_right_mem_image_IoI] | lemma | sbtw_iff_right_ne_and_left_mem_image_IoI | 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",
"sbtw_comm",
"sbtw_iff_left_ne_and_right_mem_image_IoI",
"set.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.left_mem_image_Ioi {x y z : P} (h : sbtw R x y z) :
x ∈ line_map z y '' (set.Ioi (1 : R)) | h.symm.right_mem_image_Ioi | lemma | sbtw.left_mem_image_Ioi | 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.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.left_mem_affine_span {x y z : P} (h : sbtw R x y z) : x ∈ line[R, z, y] | h.symm.right_mem_affine_span | lemma | sbtw.left_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"
] | [
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_smul_vadd_smul_vadd_of_nonneg_of_le (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁)
(hr₂ : r₁ ≤ r₂) : wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) | begin
refine ⟨r₁ / r₂, ⟨div_nonneg hr₁ (hr₁.trans hr₂), div_le_one_of_le hr₂ (hr₁.trans hr₂)⟩, _⟩,
by_cases h : r₁ = 0, { simp [h] },
simp [line_map_apply, smul_smul, ((hr₁.lt_of_ne' h).trans_le hr₂).ne.symm]
end | lemma | wbtw_smul_vadd_smul_vadd_of_nonneg_of_le | 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"
] | [
"div_le_one_of_le",
"smul_smul",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_or_wbtw_smul_vadd_of_nonneg (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂) :
wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x) | begin
rcases le_total r₁ r₂ with h|h,
{ exact or.inl (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr₁ h) },
{ exact or.inr (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr₂ h) }
end | lemma | wbtw_or_wbtw_smul_vadd_of_nonneg | 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_smul_vadd_smul_vadd_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_smul_vadd_smul_vadd_of_nonpos_of_le (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0)
(hr₂ : r₂ ≤ r₁) : wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) | begin
convert wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x (-v) (left.nonneg_neg_iff.2 hr₁)
(neg_le_neg_iff.2 hr₂) using 1;
rw neg_smul_neg
end | lemma | wbtw_smul_vadd_smul_vadd_of_nonpos_of_le | 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_smul_neg",
"wbtw",
"wbtw_smul_vadd_smul_vadd_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_or_wbtw_smul_vadd_of_nonpos (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0) (hr₂ : r₂ ≤ 0) :
wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x) | begin
rcases le_total r₁ r₂ with h|h,
{ exact or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_le x v hr₂ h) },
{ exact or.inl (wbtw_smul_vadd_smul_vadd_of_nonpos_of_le x v hr₁ h) }
end | lemma | wbtw_or_wbtw_smul_vadd_of_nonpos | 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_smul_vadd_smul_vadd_of_nonpos_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0)
(hr₂ : 0 ≤ r₂) : wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x) | begin
convert wbtw_smul_vadd_smul_vadd_of_nonneg_of_le (r₁ • v +ᵥ x) v (left.nonneg_neg_iff.2 hr₁)
(neg_le_sub_iff_le_add.2 ((le_add_iff_nonneg_left r₁).2 hr₂)) using 1;
simp [sub_smul, ←add_vadd]
end | lemma | wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg | 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"
] | [
"sub_smul",
"wbtw",
"wbtw_smul_vadd_smul_vadd_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁)
(hr₂ : r₂ ≤ 0) : wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x) | begin
rw wbtw_comm,
exact wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg x v hr₂ hr₁
end | lemma | wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos | 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_comm",
"wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_left_right {w x y z : P} (h₁ : wbtw R w y z) (h₂ : wbtw R w x y) : wbtw R x y z | begin
rcases h₁ with ⟨t₁, ht₁, rfl⟩,
rcases h₂ with ⟨t₂, ht₂, rfl⟩,
refine ⟨(t₁ - t₂ * t₁) / (1 - t₂ * t₁),
⟨div_nonneg (sub_nonneg.2 (mul_le_of_le_one_left ht₁.1 ht₂.2))
(sub_nonneg.2 (mul_le_one ht₂.2 ht₁.1 ht₁.2)),
div_le_one_of_le (sub_le_sub_right ht₁.2 _)
... | lemma | wbtw.trans_left_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"
] | [
"div_le_one_of_le",
"div_mul_eq_mul_div",
"div_self",
"div_sub_div_same",
"eq_of_le_of_not_lt",
"mul_div_assoc",
"mul_le_of_le_one_left",
"mul_le_one",
"mul_lt_one_of_nonneg_of_lt_one_right",
"smul_smul",
"smul_sub",
"vadd_right_cancel_iff",
"vadd_vsub",
"vsub_vadd_eq_vsub_sub",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw.trans_right_left {w x y z : P} (h₁ : wbtw R w x z) (h₂ : wbtw R x y z) : wbtw R w x y | begin
rw wbtw_comm at *,
exact h₁.trans_left_right h₂
end | lemma | wbtw.trans_right_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_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw.trans_left_right {w x y z : P} (h₁ : sbtw R w y z) (h₂ : sbtw R w x y) : sbtw R x y z | ⟨h₁.wbtw.trans_left_right h₂.wbtw, h₂.right_ne, h₁.ne_right⟩ | lemma | sbtw.trans_left_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.trans_right_left {w x y z : P} (h₁ : sbtw R w x z) (h₂ : sbtw R x y z) : sbtw R w x y | ⟨h₁.wbtw.trans_right_left h₂.wbtw, h₁.ne_left, h₂.left_ne⟩ | lemma | sbtw.trans_right_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 | |
wbtw.collinear {x y z : P} (h : wbtw R x y z) : collinear R ({x, y, z} : set P) | begin
rw collinear_iff_exists_forall_eq_smul_vadd,
refine ⟨x, z -ᵥ x, _⟩,
intros p hp,
simp_rw [set.mem_insert_iff, set.mem_singleton_iff] at hp,
rcases hp with rfl|rfl|rfl,
{ refine ⟨0, _⟩, simp },
{ rcases h with ⟨t, -, rfl⟩,
exact ⟨t, rfl⟩ },
{ refine ⟨1, _⟩, simp }
end | lemma | wbtw.collinear | 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"
] | [
"collinear",
"collinear_iff_exists_forall_eq_smul_vadd",
"set.mem_insert_iff",
"set.mem_singleton_iff",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
collinear.wbtw_or_wbtw_or_wbtw {x y z : P} (h : collinear R ({x, y, z} : set P)) :
wbtw R x y z ∨ wbtw R y z x ∨ wbtw R z x y | begin
rw collinear_iff_of_mem (set.mem_insert _ _) at h,
rcases h with ⟨v, h⟩,
simp_rw [set.mem_insert_iff, set.mem_singleton_iff] at h,
have hy := h y (or.inr (or.inl rfl)),
have hz := h z (or.inr (or.inr rfl)),
rcases hy with ⟨ty, rfl⟩,
rcases hz with ⟨tz, rfl⟩,
rcases lt_trichotomy ty 0 with hy0|rfl|... | lemma | collinear.wbtw_or_wbtw_or_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"
] | [
"collinear",
"collinear_iff_of_mem",
"set.mem_insert",
"set.mem_insert_iff",
"set.mem_singleton_iff",
"wbtw",
"wbtw_comm",
"wbtw_or_wbtw_smul_vadd_of_nonneg",
"wbtw_or_wbtw_smul_vadd_of_nonpos",
"wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos",
"wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_iff_same_ray_vsub {x y z : P} : wbtw R x y z ↔ same_ray R (y -ᵥ x) (z -ᵥ y) | begin
refine ⟨wbtw.same_ray_vsub, λ h, _⟩,
rcases h with h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩,
{ rw vsub_eq_zero_iff_eq at h, simp [h] },
{ rw vsub_eq_zero_iff_eq at h, simp [h] },
{ refine ⟨r₂ / (r₁ + r₂),
⟨div_nonneg hr₂.le (add_nonneg hr₁.le hr₂.le),
div_le_one_of_le (le_add_of_nonneg_lef... | lemma | wbtw_iff_same_ray_vsub | 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"
] | [
"div_le_one_of_le",
"eq_vadd_iff_vsub_eq",
"one_smul",
"ring",
"same_ray",
"smul_add",
"smul_smul",
"vadd_vsub_assoc",
"vsub_eq_zero_iff_eq",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wbtw_point_reflection (x y : P) : wbtw R y x (point_reflection R x y) | begin
refine ⟨2⁻¹, ⟨by norm_num, by norm_num⟩, _⟩,
rw [line_map_apply, point_reflection_apply, vadd_vsub_assoc, ←two_smul R (x -ᵥ y)],
simp
end | lemma | wbtw_point_reflection | 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"
] | [
"vadd_vsub_assoc",
"wbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_point_reflection_of_ne {x y : P} (h : x ≠ y) : sbtw R y x (point_reflection R x y) | begin
refine ⟨wbtw_point_reflection _ _ _, h, _⟩,
nth_rewrite 0 [←point_reflection_self R x],
exact (point_reflection_involutive R x).injective.ne h
end | lemma | sbtw_point_reflection_of_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 | |
wbtw_midpoint (x y : P) : wbtw R x (midpoint R x y) y | by { convert wbtw_point_reflection R (midpoint R x y) x, simp } | lemma | wbtw_midpoint | 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"
] | [
"midpoint",
"wbtw",
"wbtw_point_reflection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sbtw_midpoint_of_ne {x y : P} (h : x ≠ y) : sbtw R x (midpoint R x y) y | begin
have h : midpoint R x y ≠ x, { simp [h] },
convert sbtw_point_reflection_of_ne R h,
simp
end | lemma | sbtw_midpoint_of_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"
] | [
"midpoint",
"sbtw",
"sbtw_point_reflection_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_body (V : Type*) [topological_space V] [add_comm_monoid V] [has_smul ℝ V] | (carrier : set V)
(convex' : convex ℝ carrier)
(is_compact' : is_compact carrier)
(nonempty' : carrier.nonempty) | structure | convex_body | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"add_comm_monoid",
"convex",
"has_smul",
"is_compact",
"topological_space"
] | Let `V` be a real topological vector space. A subset of `V` is a convex body if and only if
it is convex, compact, and nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex (K : convex_body V) : convex ℝ (K : set V) | K.convex' | lemma | convex_body.convex | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex",
"convex_body"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact (K : convex_body V) : is_compact (K : set V) | K.is_compact' | lemma | convex_body.is_compact | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty (K : convex_body V) : (K : set V).nonempty | K.nonempty' | lemma | convex_body.nonempty | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {K L : convex_body V} (h : (K : set V) = L) : K = L | set_like.ext' h | lemma | convex_body.ext | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (s : set V) (h₁ h₂ h₃) : (mk s h₁ h₂ h₃ : set V) = s | rfl | lemma | convex_body.coe_mk | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (K L : convex_body V) : (↑(K + L) : set V) = (K : set V) + L | rfl | lemma | convex_body.coe_add | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : (↑(0 : convex_body V) : set V) = 0 | rfl | lemma | convex_body.coe_zero | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (c : ℝ) (K : convex_body V) : (↑(c • K) : set V) = c • (K : set V) | rfl | lemma | convex_body.coe_smul | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul' (c : ℝ≥0) (K : convex_body V) : (↑(c • K) : set V) = c • (K : set V) | rfl | lemma | convex_body.coe_smul' | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded : metric.bounded (K : set V) | K.is_compact.bounded | lemma | convex_body.bounded | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"metric.bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Hausdorff_edist_ne_top {K L : convex_body V} : emetric.Hausdorff_edist (K : set V) L ≠ ⊤ | by apply_rules [metric.Hausdorff_edist_ne_top_of_nonempty_of_bounded, convex_body.nonempty,
convex_body.bounded] | lemma | convex_body.Hausdorff_edist_ne_top | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"convex_body",
"convex_body.bounded",
"convex_body.nonempty",
"emetric.Hausdorff_edist",
"metric.Hausdorff_edist_ne_top_of_nonempty_of_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Hausdorff_dist_coe : metric.Hausdorff_dist (K : set V) L = dist K L | rfl | lemma | convex_body.Hausdorff_dist_coe | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"metric.Hausdorff_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Hausdorff_edist_coe : emetric.Hausdorff_edist (K : set V) L = edist K L | by { rw edist_dist, exact (ennreal.of_real_to_real Hausdorff_edist_ne_top).symm } | lemma | convex_body.Hausdorff_edist_coe | analysis.convex | src/analysis/convex/body.lean | [
"analysis.convex.basic",
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"edist_dist",
"emetric.Hausdorff_edist",
"ennreal.of_real_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_convex_hull_erase [decidable_eq E] {t : finset E}
(h : ¬ affine_independent 𝕜 (coe : t → E)) {x : E} (m : x ∈ convex_hull 𝕜 (↑t : set E)) :
∃ (y : (↑t : set E)), x ∈ convex_hull 𝕜 (↑(t.erase y) : set E) | begin
simp only [finset.convex_hull_eq, mem_set_of_eq] at m ⊢,
obtain ⟨f, fpos, fsum, rfl⟩ := m,
obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h,
replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos,
clear h,
let s := @finset.filter _ (λ z, 0 < g z) ... | lemma | caratheodory.mem_convex_hull_erase | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"affine_independent",
"and_imp",
"convex_hull",
"div_nonneg",
"exists_nontrivial_relation_sum_zero_of_not_affine_ind",
"finset",
"finset.convex_hull_eq",
"finset.filter",
"inv_one",
"le_div_iff",
"mul_nonpos_of_nonneg_of_nonpos",
"mul_zero",
"one_smul",
"smul_zero",
"sub_smul",
"subtyp... | If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
min_card_finset_of_mem_convex_hull : finset E | function.argmin_on finset.card nat.lt_wf { t | ↑t ⊆ s ∧ x ∈ convex_hull 𝕜 (t : set E) }
(by simpa only [convex_hull_eq_union_convex_hull_finite_subsets s, exists_prop, mem_Union] using hx) | def | caratheodory.min_card_finset_of_mem_convex_hull | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"convex_hull",
"convex_hull_eq_union_convex_hull_finite_subsets",
"exists_prop",
"finset",
"finset.card",
"function.argmin_on"
] | Given a point `x` in the convex hull of a set `s`, this is a finite subset of `s` of minimum
cardinality, whose convex hull contains `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
min_card_finset_of_mem_convex_hull_subseteq : ↑(min_card_finset_of_mem_convex_hull hx) ⊆ s | (function.argmin_on_mem _ _ { t : finset E | ↑t ⊆ s ∧ x ∈ convex_hull 𝕜 (t : set E) } _).1 | lemma | caratheodory.min_card_finset_of_mem_convex_hull_subseteq | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"convex_hull",
"finset",
"function.argmin_on_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_min_card_finset_of_mem_convex_hull :
x ∈ convex_hull 𝕜 (min_card_finset_of_mem_convex_hull hx : set E) | (function.argmin_on_mem _ _ { t : finset E | ↑t ⊆ s ∧ x ∈ convex_hull 𝕜 (t : set E) } _).2 | lemma | caratheodory.mem_min_card_finset_of_mem_convex_hull | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"convex_hull",
"finset",
"function.argmin_on_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_card_finset_of_mem_convex_hull_nonempty :
(min_card_finset_of_mem_convex_hull hx).nonempty | begin
rw [← finset.coe_nonempty, ← @convex_hull_nonempty_iff 𝕜],
exact ⟨x, mem_min_card_finset_of_mem_convex_hull hx⟩,
end | lemma | caratheodory.min_card_finset_of_mem_convex_hull_nonempty | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"convex_hull_nonempty_iff",
"finset.coe_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_card_finset_of_mem_convex_hull_card_le_card
{t : finset E} (ht₁ : ↑t ⊆ s) (ht₂ : x ∈ convex_hull 𝕜 (t : set E)) :
(min_card_finset_of_mem_convex_hull hx).card ≤ t.card | function.argmin_on_le _ _ _ ⟨ht₁, ht₂⟩ | lemma | caratheodory.min_card_finset_of_mem_convex_hull_card_le_card | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"convex_hull",
"finset",
"function.argmin_on_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_independent_min_card_finset_of_mem_convex_hull :
affine_independent 𝕜 (coe : min_card_finset_of_mem_convex_hull hx → E) | begin
let k := (min_card_finset_of_mem_convex_hull hx).card - 1,
have hk : (min_card_finset_of_mem_convex_hull hx).card = k + 1,
{ exact (nat.succ_pred_eq_of_pos
(finset.card_pos.mpr (min_card_finset_of_mem_convex_hull_nonempty hx))).symm },
classical,
by_contra,
obtain ⟨p, hp⟩ := mem_convex_hull_eras... | lemma | caratheodory.affine_independent_min_card_finset_of_mem_convex_hull | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"affine_independent",
"by_contra",
"finset.erase_subset",
"lt_add_one",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_eq_union :
convex_hull 𝕜 s =
⋃ (t : finset E) (hss : ↑t ⊆ s) (hai : affine_independent 𝕜 (coe : t → E)), convex_hull 𝕜 ↑t | begin
apply set.subset.antisymm,
{ intros x hx,
simp only [exists_prop, set.mem_Union],
exact ⟨caratheodory.min_card_finset_of_mem_convex_hull hx,
caratheodory.min_card_finset_of_mem_convex_hull_subseteq hx,
caratheodory.affine_independent_min_card_finset_of_mem_convex_hull hx,
... | lemma | convex_hull_eq_union | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"affine_independent",
"caratheodory.affine_independent_min_card_finset_of_mem_convex_hull",
"caratheodory.mem_min_card_finset_of_mem_convex_hull",
"caratheodory.min_card_finset_of_mem_convex_hull_subseteq",
"convex_hull",
"convex_hull_mono",
"exists_prop",
"finset",
"set.Union_subset",
"set.mem_Un... | **Carathéodory's convexity theorem** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_pos_convex_span_of_mem_convex_hull {x : E} (hx : x ∈ convex_hull 𝕜 s) :
∃ (ι : Sort (u+1)) (_ : fintype ι), by exactI ∃ (z : ι → E) (w : ι → 𝕜)
(hss : set.range z ⊆ s) (hai : affine_independent 𝕜 z)
(hw : ∀ i, 0 < w i), ∑ i, w i = 1 ∧ ∑ i, w i • z i = x | begin
rw convex_hull_eq_union at hx,
simp only [exists_prop, set.mem_Union] at hx,
obtain ⟨t, ht₁, ht₂, ht₃⟩ := hx,
simp only [t.convex_hull_eq, exists_prop, set.mem_set_of_eq] at ht₃,
obtain ⟨w, hw₁, hw₂, hw₃⟩ := ht₃,
let t' := t.filter (λ i, w i ≠ 0),
refine ⟨t', t'.fintype_coe_sort, (coe : t' → E), w ∘... | theorem | eq_pos_convex_span_of_mem_convex_hull | analysis.convex | src/analysis/convex/caratheodory.lean | [
"analysis.convex.combination",
"linear_algebra.affine_space.independent",
"tactic.field_simp"
] | [
"affine_independent",
"convex_hull",
"convex_hull_eq_union",
"exists_prop",
"finset.filter_subset",
"fintype",
"set.mem_Union",
"set.range",
"subtype.range_coe_subtype",
"zero_smul"
] | A more explicit version of `convex_hull_eq_union`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.center_mass (t : finset ι) (w : ι → R) (z : ι → E) : E | (∑ i in t, w i)⁻¹ • (∑ i in t, w i • z i) | def | finset.center_mass | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset"
] | Center of mass of a finite collection of points with prescribed weights.
Note that we require neither `0 ≤ w i` nor `∑ w = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.center_mass_empty : (∅ : finset ι).center_mass w z = 0 | by simp only [center_mass, sum_empty, smul_zero] | lemma | finset.center_mass_empty | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_pair (hne : i ≠ j) :
({i, j} : finset ι).center_mass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j | by simp only [center_mass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul] | lemma | finset.center_mass_pair | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"div_eq_inv_mul",
"finset",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_insert (ha : i ∉ t) (hw : ∑ j in t, w j ≠ 0) :
(insert i t).center_mass w z = (w i / (w i + ∑ j in t, w j)) • z i +
((∑ j in t, w j) / (w i + ∑ j in t, w j)) • t.center_mass w z | begin
simp only [center_mass, sum_insert ha, smul_add, (mul_smul _ _ _).symm, ← div_eq_inv_mul],
congr' 2,
rw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div]
end | lemma | finset.center_mass_insert | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"div_eq_inv_mul",
"div_mul_eq_mul_div",
"mul_inv_cancel",
"one_div",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_singleton (hw : w i ≠ 0) : ({i} : finset ι).center_mass w z = z i | by rw [center_mass, sum_singleton, sum_singleton, ← mul_smul, inv_mul_cancel hw, one_smul] | lemma | finset.center_mass_singleton | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset",
"inv_mul_cancel",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_eq_of_sum_1 (hw : ∑ i in t, w i = 1) :
t.center_mass w z = ∑ i in t, w i • z i | by simp only [finset.center_mass, hw, inv_one, one_smul] | lemma | finset.center_mass_eq_of_sum_1 | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset.center_mass",
"inv_one",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_smul : t.center_mass w (λ i, c • z i) = c • t.center_mass w z | by simp only [finset.center_mass, finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc] | lemma | finset.center_mass_smul | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset.center_mass",
"finset.smul_sum",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_segment'
(s : finset ι) (t : finset ι') (ws : ι → R) (zs : ι → E) (wt : ι' → R) (zt : ι' → E)
(hws : ∑ i in s, ws i = 1) (hwt : ∑ i in t, wt i = 1) (a b : R) (hab : a + b = 1) :
a • s.center_mass ws zs + b • t.center_mass wt zt =
(s.disj_sum t).center_mass (sum.elim (λ i, a * ws i) (λ j, b ... | begin
rw [s.center_mass_eq_of_sum_1 _ hws, t.center_mass_eq_of_sum_1 _ hwt,
smul_sum, smul_sum, ← finset.sum_sum_elim, finset.center_mass_eq_of_sum_1],
{ congr' with ⟨⟩; simp only [sum.elim_inl, sum.elim_inr, mul_smul] },
{ rw [sum_sum_elim, ← mul_sum, ← mul_sum, hws, hwt, mul_one, mul_one, hab] }
end | lemma | finset.center_mass_segment' | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset",
"finset.center_mass_eq_of_sum_1",
"mul_one",
"sum.elim",
"sum.elim_inl",
"sum.elim_inr"
] | A convex combination of two centers of mass is a center of mass as well. This version
deals with two different index types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.center_mass_segment
(s : finset ι) (w₁ w₂ : ι → R) (z : ι → E)
(hw₁ : ∑ i in s, w₁ i = 1) (hw₂ : ∑ i in s, w₂ i = 1) (a b : R) (hab : a + b = 1) :
a • s.center_mass w₁ z + b • s.center_mass w₂ z =
s.center_mass (λ i, a * w₁ i + b * w₂ i) z | have hw : ∑ i in s, (a * w₁ i + b * w₂ i) = 1,
by simp only [mul_sum.symm, sum_add_distrib, mul_one, *],
by simp only [finset.center_mass_eq_of_sum_1, smul_sum, sum_add_distrib, add_smul, mul_smul, *] | lemma | finset.center_mass_segment | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"add_smul",
"finset",
"finset.center_mass_eq_of_sum_1",
"mul_one"
] | A convex combination of two centers of mass is a center of mass as well. This version
works if two centers of mass share the set of original points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.center_mass_ite_eq (hi : i ∈ t) :
t.center_mass (λ j, if (i = j) then (1 : R) else 0) z = z i | begin
rw [finset.center_mass_eq_of_sum_1],
transitivity ∑ j in t, if (i = j) then z i else 0,
{ congr' with i, split_ifs, exacts [h ▸ one_smul _ _, zero_smul _ _] },
{ rw [sum_ite_eq, if_pos hi] },
{ rw [sum_ite_eq, if_pos hi] }
end | lemma | finset.center_mass_ite_eq | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset.center_mass_eq_of_sum_1",
"one_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_subset {t' : finset ι} (ht : t ⊆ t')
(h : ∀ i ∈ t', i ∉ t → w i = 0) :
t.center_mass w z = t'.center_mass w z | begin
rw [center_mass, sum_subset ht h, smul_sum, center_mass, smul_sum],
apply sum_subset ht,
assume i hit' hit,
rw [h i hit' hit, zero_smul, smul_zero]
end | lemma | finset.center_mass_subset | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.center_mass_filter_ne_zero :
(t.filter (λ i, w i ≠ 0)).center_mass w z = t.center_mass w z | finset.center_mass_subset z (filter_subset _ _) $ λ i hit hit',
by simpa only [hit, mem_filter, true_and, ne.def, not_not] using hit' | lemma | finset.center_mass_filter_ne_zero | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset.center_mass_subset",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_mass_le_sup {s : finset ι} {f : ι → α} {w : ι → R}
(hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : 0 < ∑ i in s, w i) :
s.center_mass w f ≤ s.sup' (nonempty_of_ne_empty $ by { rintro rfl, simpa using hw₁ }) f | begin
rw [center_mass, inv_smul_le_iff hw₁, sum_smul],
exact sum_le_sum (λ i hi, smul_le_smul_of_nonneg (le_sup' _ hi) $ hw₀ i hi),
apply_instance,
end | lemma | finset.center_mass_le_sup | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset",
"inv_smul_le_iff",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_le_center_mass {s : finset ι} {f : ι → α} {w : ι → R}
(hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : 0 < ∑ i in s, w i) :
s.inf' (nonempty_of_ne_empty $ by { rintro rfl, simpa using hw₁ }) f ≤ s.center_mass w f | @center_mass_le_sup R _ αᵒᵈ _ _ _ _ _ _ _ hw₀ hw₁ | lemma | finset.inf_le_center_mass | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.center_mass_mem (hs : convex R s) :
(∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i in t, w i) → (∀ i ∈ t, z i ∈ s) → t.center_mass w z ∈ s | begin
induction t using finset.induction with i t hi ht, { simp [lt_irrefl] },
intros h₀ hpos hmem,
have zi : z i ∈ s, from hmem _ (mem_insert_self _ _),
have hs₀ : ∀ j ∈ t, 0 ≤ w j, from λ j hj, h₀ j $ mem_insert_of_mem hj,
rw [sum_insert hi] at hpos,
by_cases hsum_t : ∑ j in t, w j = 0,
{ have ws : ∀ j ... | lemma | convex.center_mass_mem | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex",
"finset.center_mass_insert",
"finset.induction",
"inv_mul_cancel",
"one_smul"
] | The center of mass of a finite subset of a convex set belongs to the set
provided that all weights are non-negative, and the total weight is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.sum_mem (hs : convex R s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
(hz : ∀ i ∈ t, z i ∈ s) :
∑ i in t, w i • z i ∈ s | by simpa only [h₁, center_mass, inv_one, one_smul] using
hs.center_mass_mem h₀ (h₁.symm ▸ zero_lt_one) hz | lemma | convex.sum_mem | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"convex",
"inv_one",
"one_smul",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : set E}
(hs : convex R s) (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
∑ᶠ i, w i • z i ∈ s | begin
have hfin_w : (support (w ∘ plift.down)).finite,
{ by_contra H,
rw [finsum, dif_neg H] at h₁,
exact zero_ne_one h₁ },
have hsub : support ((λ i, w i • z i) ∘ plift.down) ⊆ hfin_w.to_finset,
from (support_smul_subset_left _ _).trans hfin_w.coe_to_finset.ge,
rw [finsum_eq_sum_plift_of_support_su... | lemma | convex.finsum_mem | analysis.convex | src/analysis/convex/combination.lean | [
"algebra.big_operators.order",
"analysis.convex.hull",
"linear_algebra.affine_space.basis"
] | [
"by_contra",
"convex",
"finite",
"finsum",
"zero_ne_one"
] | A version of `convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of
nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such
that `z i ∈ s` whenever `w i ≠ 0``, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also
`partition_of_unity.finsum_smul_m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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