statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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w_opp_side_self_iff {s : affine_subspace R P} {x : P} : s.w_opp_side x x ↔ x ∈ s | begin
split,
{ rintro ⟨p₁, hp₁, p₂, hp₂, h⟩,
obtain ⟨a, -, -, -, -, h₁, -⟩ := h.exists_eq_smul_add,
rw [add_comm, vsub_add_vsub_cancel, ←eq_vadd_iff_vsub_eq] at h₁,
rw h₁,
exact s.smul_vsub_vadd_mem a hp₂ hp₁ hp₁ },
{ exact λ h, ⟨x, h, x, h, same_ray.rfl⟩ }
end | lemma | affine_subspace.w_opp_side_self_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"vsub_add_vsub_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_s_opp_side_self (s : affine_subspace R P) (x : P) : ¬s.s_opp_side x x | by simp [s_opp_side] | lemma | affine_subspace.not_s_opp_side_self | 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_iff_exists_left {s : affine_subspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.w_same_side x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, same_ray R (x -ᵥ p₁) (y -ᵥ p₂) | begin
split,
{ rintro ⟨p₁', hp₁', p₂', hp₂', h0 | h0 | ⟨r₁, r₂, hr₁, hr₂, hr⟩⟩,
{ rw vsub_eq_zero_iff_eq at h0,
rw h0,
exact or.inl hp₁' },
{ refine or.inr ⟨p₂', hp₂', _⟩,
rw h0,
exact same_ray.zero_right _ },
{ refine or.inr ⟨(r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂', s.smul_vsub_vadd_mem ... | lemma | affine_subspace.w_same_side_iff_exists_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"mul_div_cancel'",
"same_ray",
"same_ray.zero_right",
"smul_smul",
"smul_sub",
"vsub_eq_zero_iff_eq",
"vsub_sub_vsub_cancel_right",
"vsub_vadd_eq_vsub_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side_iff_exists_right {s : affine_subspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
s.w_same_side x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, same_ray R (x -ᵥ p₁) (y -ᵥ p₂) | begin
rw [w_same_side_comm, w_same_side_iff_exists_left h],
simp_rw same_ray_comm
end | lemma | affine_subspace.w_same_side_iff_exists_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"same_ray",
"same_ray_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_iff_exists_left {s : affine_subspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.s_same_side x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, same_ray R (x -ᵥ p₁) (y -ᵥ p₂) | begin
rw [s_same_side, and_comm, w_same_side_iff_exists_left h, and_assoc, and.congr_right_iff],
intro hx,
rw or_iff_right hx
end | lemma | affine_subspace.s_same_side_iff_exists_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"and.congr_right_iff",
"or_iff_right",
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_iff_exists_right {s : affine_subspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
s.s_same_side x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, same_ray R (x -ᵥ p₁) (y -ᵥ p₂) | begin
rw [s_same_side_comm, s_same_side_iff_exists_left h, ←and_assoc, and_comm (y ∉ s), and_assoc],
simp_rw same_ray_comm
end | lemma | affine_subspace.s_same_side_iff_exists_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"same_ray",
"same_ray_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_iff_exists_left {s : affine_subspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.w_opp_side x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, same_ray R (x -ᵥ p₁) (p₂ -ᵥ y) | begin
split,
{ rintro ⟨p₁', hp₁', p₂', hp₂', h0 | h0 | ⟨r₁, r₂, hr₁, hr₂, hr⟩⟩,
{ rw vsub_eq_zero_iff_eq at h0,
rw h0,
exact or.inl hp₁' },
{ refine or.inr ⟨p₂', hp₂', _⟩,
rw h0,
exact same_ray.zero_right _ },
{ refine or.inr ⟨(-r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂', s.smul_vsub_vadd_mem... | lemma | affine_subspace.w_opp_side_iff_exists_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"mul_div_cancel'",
"mul_neg",
"neg_div",
"neg_smul",
"same_ray",
"same_ray.zero_right",
"smul_add",
"smul_smul",
"vadd_vsub_assoc",
"vsub_eq_zero_iff_eq",
"vsub_sub_vsub_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_iff_exists_right {s : affine_subspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
s.w_opp_side x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, same_ray R (x -ᵥ p₁) (p₂ -ᵥ y) | begin
rw [w_opp_side_comm, w_opp_side_iff_exists_left h],
split,
{ rintro (hy | ⟨p, hp, hr⟩), { exact or.inl hy },
refine or.inr ⟨p, hp, _⟩,
rwa [same_ray_comm, ←same_ray_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] },
{ rintro (hy | ⟨p, hp, hr⟩), { exact or.inl hy },
refine or.inr ⟨p, hp, _... | lemma | affine_subspace.w_opp_side_iff_exists_right | 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",
"same_ray_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_iff_exists_left {s : affine_subspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.s_opp_side x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, same_ray R (x -ᵥ p₁) (p₂ -ᵥ y) | begin
rw [s_opp_side, and_comm, w_opp_side_iff_exists_left h, and_assoc, and.congr_right_iff],
intro hx,
rw or_iff_right hx
end | lemma | affine_subspace.s_opp_side_iff_exists_left | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"and.congr_right_iff",
"or_iff_right",
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_iff_exists_right {s : affine_subspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
s.s_opp_side x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, same_ray R (x -ᵥ p₁) (p₂ -ᵥ y) | begin
rw [s_opp_side, and_comm, w_opp_side_iff_exists_right h, and_assoc, and.congr_right_iff,
and.congr_right_iff],
rintro hx hy,
rw or_iff_right hy
end | lemma | affine_subspace.s_opp_side_iff_exists_right | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"and.congr_right_iff",
"or_iff_right",
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side.trans {s : affine_subspace R P} {x y z : P} (hxy : s.w_same_side x y)
(hyz : s.w_same_side y z) (hy : y ∉ s) : s.w_same_side x z | begin
rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩,
rw [w_same_side_iff_exists_left hp₂, or_iff_right hy] at hyz,
rcases hyz with ⟨p₃, hp₃, hyz⟩,
refine ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz _⟩,
refine λ h, false.elim _,
rw vsub_eq_zero_iff_eq at h,
exact hy (h.symm ▸ hp₂)
end | lemma | affine_subspace.w_same_side.trans | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"or_iff_right",
"vsub_eq_zero_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side.trans_s_same_side {s : affine_subspace R P} {x y z : P}
(hxy : s.w_same_side x y) (hyz : s.s_same_side y z) : s.w_same_side x z | hxy.trans hyz.1 hyz.2.1 | lemma | affine_subspace.w_same_side.trans_s_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 | |
w_same_side.trans_w_opp_side {s : affine_subspace R P} {x y z : P} (hxy : s.w_same_side x y)
(hyz : s.w_opp_side y z) (hy : y ∉ s) : s.w_opp_side x z | begin
rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩,
rw [w_opp_side_iff_exists_left hp₂, or_iff_right hy] at hyz,
rcases hyz with ⟨p₃, hp₃, hyz⟩,
refine ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz _⟩,
refine λ h, false.elim _,
rw vsub_eq_zero_iff_eq at h,
exact hy (h.symm ▸ hp₂)
end | lemma | affine_subspace.w_same_side.trans_w_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"or_iff_right",
"vsub_eq_zero_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side.trans_s_opp_side {s : affine_subspace R P} {x y z : P} (hxy : s.w_same_side x y)
(hyz : s.s_opp_side y z) : s.w_opp_side x z | hxy.trans_w_opp_side hyz.1 hyz.2.1 | lemma | affine_subspace.w_same_side.trans_s_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_same_side.trans_w_same_side {s : affine_subspace R P} {x y z : P}
(hxy : s.s_same_side x y) (hyz : s.w_same_side y z) : s.w_same_side x z | (hyz.symm.trans_s_same_side hxy.symm).symm | lemma | affine_subspace.s_same_side.trans_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.trans {s : affine_subspace R P} {x y z : P} (hxy : s.s_same_side x y)
(hyz : s.s_same_side y z) : s.s_same_side x z | ⟨hxy.w_same_side.trans_s_same_side hyz, hxy.2.1, hyz.2.2⟩ | lemma | affine_subspace.s_same_side.trans | 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.trans_w_opp_side {s : affine_subspace R P} {x y z : P} (hxy : s.s_same_side x y)
(hyz : s.w_opp_side y z) : s.w_opp_side x z | hxy.w_same_side.trans_w_opp_side hyz hxy.2.2 | lemma | affine_subspace.s_same_side.trans_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_same_side.trans_s_opp_side {s : affine_subspace R P} {x y z : P} (hxy : s.s_same_side x y)
(hyz : s.s_opp_side y z) : s.s_opp_side x z | ⟨hxy.trans_w_opp_side hyz.1, hxy.2.1, hyz.2.2⟩ | lemma | affine_subspace.s_same_side.trans_s_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 | |
w_opp_side.trans_w_same_side {s : affine_subspace R P} {x y z : P} (hxy : s.w_opp_side x y)
(hyz : s.w_same_side y z) (hy : y ∉ s) : s.w_opp_side x z | (hyz.symm.trans_w_opp_side hxy.symm hy).symm | lemma | affine_subspace.w_opp_side.trans_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 | |
w_opp_side.trans_s_same_side {s : affine_subspace R P} {x y z : P} (hxy : s.w_opp_side x y)
(hyz : s.s_same_side y z) : s.w_opp_side x z | hxy.trans_w_same_side hyz.1 hyz.2.1 | lemma | affine_subspace.w_opp_side.trans_s_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 | |
w_opp_side.trans {s : affine_subspace R P} {x y z : P} (hxy : s.w_opp_side x y)
(hyz : s.w_opp_side y z) (hy : y ∉ s) : s.w_same_side x z | begin
rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩,
rw [w_opp_side_iff_exists_left hp₂, or_iff_right hy] at hyz,
rcases hyz with ⟨p₃, hp₃, hyz⟩,
rw [←same_ray_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] at hyz,
refine ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz _⟩,
refine λ h, false.elim _,
rw vsub_eq_zero_iff_e... | lemma | affine_subspace.w_opp_side.trans | 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",
"or_iff_right",
"vsub_eq_zero_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side.trans_s_opp_side {s : affine_subspace R P} {x y z : P} (hxy : s.w_opp_side x y)
(hyz : s.s_opp_side y z) : s.w_same_side x z | hxy.trans hyz.1 hyz.2.1 | lemma | affine_subspace.w_opp_side.trans_s_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.trans_w_same_side {s : affine_subspace R P} {x y z : P} (hxy : s.s_opp_side x y)
(hyz : s.w_same_side y z) : s.w_opp_side x z | (hyz.symm.trans_s_opp_side hxy.symm).symm | lemma | affine_subspace.s_opp_side.trans_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_opp_side.trans_s_same_side {s : affine_subspace R P} {x y z : P} (hxy : s.s_opp_side x y)
(hyz : s.s_same_side y z) : s.s_opp_side x z | (hyz.symm.trans_s_opp_side hxy.symm).symm | lemma | affine_subspace.s_opp_side.trans_s_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_opp_side.trans_w_opp_side {s : affine_subspace R P} {x y z : P} (hxy : s.s_opp_side x y)
(hyz : s.w_opp_side y z) : s.w_same_side x z | (hyz.symm.trans_s_opp_side hxy.symm).symm | lemma | affine_subspace.s_opp_side.trans_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.trans {s : affine_subspace R P} {x y z : P} (hxy : s.s_opp_side x y)
(hyz : s.s_opp_side y z) : s.s_same_side x z | ⟨hxy.trans_w_opp_side hyz.1, hxy.2.1, hyz.2.2⟩ | lemma | affine_subspace.s_opp_side.trans | 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_and_w_opp_side_iff {s : affine_subspace R P} {x y : P} :
(s.w_same_side x y ∧ s.w_opp_side x y) ↔ (x ∈ s ∨ y ∈ s) | begin
split,
{ rintro ⟨hs, ho⟩,
rw w_opp_side_comm at ho,
by_contra h,
rw not_or_distrib at h,
exact h.1 (w_opp_side_self_iff.1 (hs.trans_w_opp_side ho h.2)) },
{ rintro (h | h),
{ exact ⟨w_same_side_of_left_mem y h, w_opp_side_of_left_mem y h⟩ },
{ exact ⟨w_same_side_of_right_mem x h, w_o... | lemma | affine_subspace.w_same_side_and_w_opp_side_iff | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"by_contra",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_same_side.not_s_opp_side {s : affine_subspace R P} {x y : P} (h : s.w_same_side x y) :
¬s.s_opp_side x y | begin
intro ho,
have hxy := w_same_side_and_w_opp_side_iff.1 ⟨h, ho.1⟩,
rcases hxy with hx | hy,
{ exact ho.2.1 hx },
{ exact ho.2.2 hy }
end | lemma | affine_subspace.w_same_side.not_s_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_same_side.not_w_opp_side {s : affine_subspace R P} {x y : P} (h : s.s_same_side x y) :
¬s.w_opp_side x y | begin
intro ho,
have hxy := w_same_side_and_w_opp_side_iff.1 ⟨h.1, ho⟩,
rcases hxy with hx | hy,
{ exact h.2.1 hx },
{ exact h.2.2 hy }
end | lemma | affine_subspace.s_same_side.not_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_same_side.not_s_opp_side {s : affine_subspace R P} {x y : P} (h : s.s_same_side x y) :
¬s.s_opp_side x y | λ ho, h.not_w_opp_side ho.1 | lemma | affine_subspace.s_same_side.not_s_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 | |
w_opp_side.not_s_same_side {s : affine_subspace R P} {x y : P} (h : s.w_opp_side x y) :
¬s.s_same_side x y | λ hs, hs.not_w_opp_side h | lemma | affine_subspace.w_opp_side.not_s_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_opp_side.not_w_same_side {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
¬s.w_same_side x y | λ hs, hs.not_s_opp_side h | lemma | affine_subspace.s_opp_side.not_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_opp_side.not_s_same_side {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
¬s.s_same_side x y | λ hs, h.not_w_same_side hs.1 | lemma | affine_subspace.s_opp_side.not_s_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 | |
w_opp_side_iff_exists_wbtw {s : affine_subspace R P} {x y : P} :
s.w_opp_side x y ↔ ∃ p ∈ s, wbtw R x p y | begin
refine ⟨λ h, _, λ ⟨p, hp, h⟩, h.w_opp_side₁₃ hp⟩,
rcases h with ⟨p₁, hp₁, p₂, hp₂, (h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩)⟩,
{ rw vsub_eq_zero_iff_eq at h,
rw h,
exact ⟨p₁, hp₁, wbtw_self_left _ _ _⟩ },
{ rw vsub_eq_zero_iff_eq at h,
rw ←h,
exact ⟨p₂, hp₂, wbtw_self_right _ _ _⟩ },
{ refine ⟨lin... | lemma | affine_subspace.w_opp_side_iff_exists_wbtw | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"div_eq_inv_mul",
"div_le_one_of_le",
"div_self",
"neg_div",
"neg_one_smul",
"set.mem_image_of_mem",
"smul_add",
"smul_neg",
"smul_smul",
"smul_sub",
"vadd_vsub_assoc",
"vsub_eq_zero_iff_eq",
"vsub_vadd_eq_vsub_sub",
"wbtw",
"wbtw_self_left",
"wbtw_self_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side.exists_sbtw {s : affine_subspace R P} {x y : P} (h : s.s_opp_side x y) :
∃ p ∈ s, sbtw R x p y | begin
obtain ⟨p, hp, hw⟩ := w_opp_side_iff_exists_wbtw.1 h.w_opp_side,
refine ⟨p, hp, hw, _, _⟩,
{ rintro rfl,
exact h.2.1 hp },
{ rintro rfl,
exact h.2.2 hp },
end | lemma | affine_subspace.s_opp_side.exists_sbtw | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"sbtw"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.sbtw.s_opp_side_of_not_mem_of_mem {s : affine_subspace R P} {x y z : P}
(h : sbtw R x y z) (hx : x ∉ s) (hy : y ∈ s) : s.s_opp_side x z | begin
refine ⟨h.wbtw.w_opp_side₁₃ hy, hx, λ hz, hx _⟩,
rcases h with ⟨⟨t, ⟨ht0, ht1⟩, rfl⟩, hyx, hyz⟩,
rw line_map_apply at hy,
have ht : t ≠ 1, { rintro rfl, simpa [line_map_apply] using hyz },
have hy' := vsub_mem_direction hy hz,
rw [vadd_vsub_assoc, ←neg_vsub_eq_vsub_rev z, ←neg_one_smul R (z -ᵥ x), ←ad... | lemma | sbtw.s_opp_side_of_not_mem_of_mem | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"sbtw",
"submodule.smul_mem",
"vadd_vsub_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_smul_vsub_vadd_left {s : affine_subspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : s.s_same_side (t • (x -ᵥ p₁) +ᵥ p₂) x | begin
refine ⟨w_same_side_smul_vsub_vadd_left x hp₁ hp₂ ht.le, λ h, hx _, hx⟩,
rwa [vadd_mem_iff_mem_direction _ hp₂, s.direction.smul_mem_iff ht.ne.symm,
vsub_right_mem_direction_iff_mem hp₁] at h
end | lemma | affine_subspace.s_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_same_side_smul_vsub_vadd_right {s : affine_subspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : s.s_same_side x (t • (x -ᵥ p₁) +ᵥ p₂) | (s_same_side_smul_vsub_vadd_left hx hp₁ hp₂ ht).symm | lemma | affine_subspace.s_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 | |
s_same_side_line_map_left {s : affine_subspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s)
{t : R} (ht : 0 < t) : s.s_same_side (line_map x y t) y | s_same_side_smul_vsub_vadd_left hy hx hx ht | lemma | affine_subspace.s_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 | |
s_same_side_line_map_right {s : affine_subspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s)
{t : R} (ht : 0 < t) : s.s_same_side y (line_map x y t) | (s_same_side_line_map_left hx hy ht).symm | lemma | affine_subspace.s_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 | |
s_opp_side_smul_vsub_vadd_left {s : affine_subspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : s.s_opp_side (t • (x -ᵥ p₁) +ᵥ p₂) x | begin
refine ⟨w_opp_side_smul_vsub_vadd_left x hp₁ hp₂ ht.le, λ h, hx _, hx⟩,
rwa [vadd_mem_iff_mem_direction _ hp₂, s.direction.smul_mem_iff ht.ne,
vsub_right_mem_direction_iff_mem hp₁] at h
end | lemma | affine_subspace.s_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_smul_vsub_vadd_right {s : affine_subspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : s.s_opp_side x (t • (x -ᵥ p₁) +ᵥ p₂) | (s_opp_side_smul_vsub_vadd_left hx hp₁ hp₂ ht).symm | lemma | affine_subspace.s_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 | |
s_opp_side_line_map_left {s : affine_subspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s)
{t : R} (ht : t < 0) : s.s_opp_side (line_map x y t) y | s_opp_side_smul_vsub_vadd_left hy hx hx ht | lemma | affine_subspace.s_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 | |
s_opp_side_line_map_right {s : affine_subspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s)
{t : R} (ht : t < 0) : s.s_opp_side y (line_map x y t) | (s_opp_side_line_map_left hx hy ht).symm | lemma | affine_subspace.s_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 | |
set_of_w_same_side_eq_image2 {s : affine_subspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{y | s.w_same_side x y} = set.image2 (λ (t : R) q, t • (x -ᵥ p) +ᵥ q) (set.Ici 0) s | begin
ext y,
simp_rw [set.mem_set_of, set.mem_image2, set.mem_Ici, mem_coe],
split,
{ rw [w_same_side_iff_exists_left hp, or_iff_right hx],
rintro ⟨p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩,
{ rw vsub_eq_zero_iff_eq at h,
exact false.elim (hx (h.symm ▸ hp)) },
{ rw vsub_eq_zero_iff_eq at h,
... | lemma | affine_subspace.set_of_w_same_side_eq_image2 | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"div_eq_inv_mul",
"div_pos",
"inv_mul_cancel",
"one_smul",
"or_iff_right",
"set.Ici",
"set.image2",
"set.mem_Ici",
"set.mem_image2",
"set.mem_set_of",
"smul_smul",
"vsub_eq_zero_iff_eq",
"vsub_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_s_same_side_eq_image2 {s : affine_subspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{y | s.s_same_side x y} = set.image2 (λ (t : R) q, t • (x -ᵥ p) +ᵥ q) (set.Ioi 0) s | begin
ext y,
simp_rw [set.mem_set_of, set.mem_image2, set.mem_Ioi, mem_coe],
split,
{ rw s_same_side_iff_exists_left hp,
rintro ⟨-, hy, p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩,
{ rw vsub_eq_zero_iff_eq at h,
exact false.elim (hx (h.symm ▸ hp)) },
{ rw vsub_eq_zero_iff_eq at h,
exact fals... | lemma | affine_subspace.set_of_s_same_side_eq_image2 | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"div_eq_inv_mul",
"div_pos",
"inv_mul_cancel",
"one_smul",
"set.Ioi",
"set.image2",
"set.mem_Ioi",
"set.mem_image2",
"set.mem_set_of",
"smul_smul",
"vsub_eq_zero_iff_eq",
"vsub_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_w_opp_side_eq_image2 {s : affine_subspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{y | s.w_opp_side x y} = set.image2 (λ (t : R) q, t • (x -ᵥ p) +ᵥ q) (set.Iic 0) s | begin
ext y,
simp_rw [set.mem_set_of, set.mem_image2, set.mem_Iic, mem_coe],
split,
{ rw [w_opp_side_iff_exists_left hp, or_iff_right hx],
rintro ⟨p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩,
{ rw vsub_eq_zero_iff_eq at h,
exact false.elim (hx (h.symm ▸ hp)) },
{ rw vsub_eq_zero_iff_eq at h,
... | lemma | affine_subspace.set_of_w_opp_side_eq_image2 | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"div_eq_inv_mul",
"div_neg_of_neg_of_pos",
"inv_mul_cancel",
"neg_smul",
"neg_vsub_eq_vsub_rev",
"one_smul",
"or_iff_right",
"set.Iic",
"set.image2",
"set.mem_Iic",
"set.mem_image2",
"set.mem_set_of",
"smul_neg",
"smul_smul",
"vsub_eq_zero_iff_eq",
"vsub_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_s_opp_side_eq_image2 {s : affine_subspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{y | s.s_opp_side x y} = set.image2 (λ (t : R) q, t • (x -ᵥ p) +ᵥ q) (set.Iio 0) s | begin
ext y,
simp_rw [set.mem_set_of, set.mem_image2, set.mem_Iio, mem_coe],
split,
{ rw s_opp_side_iff_exists_left hp,
rintro ⟨-, hy, p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩,
{ rw vsub_eq_zero_iff_eq at h,
exact false.elim (hx (h.symm ▸ hp)) },
{ rw vsub_eq_zero_iff_eq at h,
exact false... | lemma | affine_subspace.set_of_s_opp_side_eq_image2 | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"div_eq_inv_mul",
"div_neg_of_neg_of_pos",
"inv_mul_cancel",
"neg_smul",
"neg_vsub_eq_vsub_rev",
"one_smul",
"set.Iio",
"set.image2",
"set.mem_Iio",
"set.mem_image2",
"set.mem_set_of",
"smul_neg",
"smul_smul",
"vsub_eq_zero_iff_eq",
"vsub_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_opp_side_point_reflection {s : affine_subspace R P} {x : P} (y : P) (hx : x ∈ s) :
s.w_opp_side y (point_reflection R x y) | (wbtw_point_reflection R _ _).w_opp_side₁₃ hx | lemma | affine_subspace.w_opp_side_point_reflection | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"wbtw_point_reflection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
s_opp_side_point_reflection {s : affine_subspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) :
s.s_opp_side y (point_reflection R x y) | begin
refine (sbtw_point_reflection_of_ne R (λ h, hy _)).s_opp_side_of_not_mem_of_mem hy hx,
rwa ←h
end | lemma | affine_subspace.s_opp_side_point_reflection | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"sbtw_point_reflection_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_set_of_w_same_side {s : affine_subspace ℝ P} (x : P)
(h : (s : set P).nonempty) : is_connected {y | s.w_same_side x y} | begin
obtain ⟨p, hp⟩ := h,
haveI : nonempty s := ⟨⟨p, hp⟩⟩,
by_cases hx : x ∈ s,
{ convert is_connected_univ,
{ simp [w_same_side_of_left_mem, hx] },
{ exact add_torsor.connected_space V P } },
{ rw [set_of_w_same_side_eq_image2 hx hp, ←set.image_prod],
refine (is_connected_Ici.prod (is_connected_... | lemma | affine_subspace.is_connected_set_of_w_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"add_torsor.connected_space",
"affine_subspace",
"continuous_const",
"continuous_on",
"continuous_snd",
"is_connected",
"is_connected_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_set_of_w_same_side (s : affine_subspace ℝ P) (x : P) :
is_preconnected {y | s.w_same_side x y} | begin
rcases set.eq_empty_or_nonempty (s : set P) with h | h,
{ convert is_preconnected_empty,
rw coe_eq_bot_iff at h,
simp only [h, not_w_same_side_bot],
refl },
{ exact (is_connected_set_of_w_same_side x h).is_preconnected }
end | lemma | affine_subspace.is_preconnected_set_of_w_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"is_preconnected",
"is_preconnected_empty",
"set.eq_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_set_of_s_same_side {s : affine_subspace ℝ P} {x : P} (hx : x ∉ s)
(h : (s : set P).nonempty) : is_connected {y | s.s_same_side x y} | begin
obtain ⟨p, hp⟩ := h,
haveI : nonempty s := ⟨⟨p, hp⟩⟩,
rw [set_of_s_same_side_eq_image2 hx hp, ←set.image_prod],
refine (is_connected_Ioi.prod (is_connected_iff_connected_space.2 _)).image _
((continuous_fst.smul continuous_const).vadd continuous_snd).continuous_on,
convert add_torsor.connected_... | lemma | affine_subspace.is_connected_set_of_s_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"add_torsor.connected_space",
"affine_subspace",
"continuous_const",
"continuous_on",
"continuous_snd",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_set_of_s_same_side (s : affine_subspace ℝ P) (x : P) :
is_preconnected {y | s.s_same_side x y} | begin
rcases set.eq_empty_or_nonempty (s : set P) with h | h,
{ convert is_preconnected_empty,
rw coe_eq_bot_iff at h,
simp only [h, not_s_same_side_bot],
refl },
{ by_cases hx : x ∈ s,
{ convert is_preconnected_empty,
simp only [hx, s_same_side, not_true, false_and, and_false],
refl }... | lemma | affine_subspace.is_preconnected_set_of_s_same_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"is_preconnected",
"is_preconnected_empty",
"set.eq_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_set_of_w_opp_side {s : affine_subspace ℝ P} (x : P)
(h : (s : set P).nonempty) : is_connected {y | s.w_opp_side x y} | begin
obtain ⟨p, hp⟩ := h,
haveI : nonempty s := ⟨⟨p, hp⟩⟩,
by_cases hx : x ∈ s,
{ convert is_connected_univ,
{ simp [w_opp_side_of_left_mem, hx] },
{ exact add_torsor.connected_space V P } },
{ rw [set_of_w_opp_side_eq_image2 hx hp, ←set.image_prod],
refine (is_connected_Iic.prod (is_connected_if... | lemma | affine_subspace.is_connected_set_of_w_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"add_torsor.connected_space",
"affine_subspace",
"continuous_const",
"continuous_on",
"continuous_snd",
"is_connected",
"is_connected_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_set_of_w_opp_side (s : affine_subspace ℝ P) (x : P) :
is_preconnected {y | s.w_opp_side x y} | begin
rcases set.eq_empty_or_nonempty (s : set P) with h | h,
{ convert is_preconnected_empty,
rw coe_eq_bot_iff at h,
simp only [h, not_w_opp_side_bot],
refl },
{ exact (is_connected_set_of_w_opp_side x h).is_preconnected }
end | lemma | affine_subspace.is_preconnected_set_of_w_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"is_preconnected",
"is_preconnected_empty",
"set.eq_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_set_of_s_opp_side {s : affine_subspace ℝ P} {x : P} (hx : x ∉ s)
(h : (s : set P).nonempty) : is_connected {y | s.s_opp_side x y} | begin
obtain ⟨p, hp⟩ := h,
haveI : nonempty s := ⟨⟨p, hp⟩⟩,
rw [set_of_s_opp_side_eq_image2 hx hp, ←set.image_prod],
refine (is_connected_Iio.prod (is_connected_iff_connected_space.2 _)).image _
((continuous_fst.smul continuous_const).vadd continuous_snd).continuous_on,
convert add_torsor.connected_s... | lemma | affine_subspace.is_connected_set_of_s_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"add_torsor.connected_space",
"affine_subspace",
"continuous_const",
"continuous_on",
"continuous_snd",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_set_of_s_opp_side (s : affine_subspace ℝ P) (x : P) :
is_preconnected {y | s.s_opp_side x y} | begin
rcases set.eq_empty_or_nonempty (s : set P) with h | h,
{ convert is_preconnected_empty,
rw coe_eq_bot_iff at h,
simp only [h, not_s_opp_side_bot],
refl },
{ by_cases hx : x ∈ s,
{ convert is_preconnected_empty,
simp only [hx, s_opp_side, not_true, false_and, and_false],
refl },
... | lemma | affine_subspace.is_preconnected_set_of_s_opp_side | analysis.convex | src/analysis/convex/side.lean | [
"analysis.convex.between",
"analysis.convex.normed",
"analysis.normed.group.add_torsor"
] | [
"affine_subspace",
"is_preconnected",
"is_preconnected_empty",
"set.eq_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.slope_mono_adjacent (hf : convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | begin
have hxz := hxy.trans hyz,
rw ←sub_pos at hxy hxz hyz,
suffices : f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y),
{ ring_nf at this ⊢, linarith },
set a := (z - y) / (z - x),
set b := (y - x) / (z - x),
have hy : a • x + b • z = y, by { field_simp, rw div_eq_iff; [ring, linarith] },
... | lemma | convex_on.slope_mono_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex_on",
"div_eq_iff",
"div_le_div_right",
"div_nonneg",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"ring"
] | If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.slope_anti_adjacent (hf : concave_on 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
(hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f z - f y) / (z - y) ≤ (f y - f x) / (y - x) | begin
rw [←neg_le_neg_iff, ←neg_sub_neg (f x), ←neg_sub_neg (f y)],
simp_rw [←pi.neg_apply, ←neg_div, neg_sub],
exact convex_on.slope_mono_adjacent hf.neg hx hz hxy hyz,
end | lemma | concave_on.slope_anti_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"concave_on",
"convex_on.slope_mono_adjacent"
] | If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.slope_strict_mono_adjacent (hf : strict_convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f y - f x) / (y - x) < (f z - f y) / (z - y) | begin
have hxz := hxy.trans hyz,
have hxz' := hxz.ne,
rw ←sub_pos at hxy hxz hyz,
suffices : f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y),
{ ring_nf at this ⊢, linarith },
set a := (z - y) / (z - x),
set b := (y - x) / (z - x),
have hy : a • x + b • z = y, by { field_simp, rw div_eq_iff... | lemma | strict_convex_on.slope_strict_mono_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"div_eq_iff",
"div_lt_div_right",
"div_pos",
"mul_comm",
"mul_lt_mul_of_pos_left",
"ring",
"strict_convex_on"
] | If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on.slope_anti_adjacent (hf : strict_concave_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f z - f y) / (z - y) < (f y - f x) / (y - x) | begin
rw [←neg_lt_neg_iff, ←neg_sub_neg (f x), ←neg_sub_neg (f y)],
simp_rw [←pi.neg_apply, ←neg_div, neg_sub],
exact strict_convex_on.slope_strict_mono_adjacent hf.neg hx hz hxy hyz,
end | lemma | strict_concave_on.slope_anti_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"strict_concave_on",
"strict_convex_on.slope_strict_mono_adjacent"
] | If `f : 𝕜 → 𝕜` is strictly concave, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly greater than the slope of the secant line of `f` on
`[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_of_slope_mono_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :
convex_on 𝕜 s f | linear_order.convex_on_of_lt hs $ λ x hx z hz hxz a b ha hb hab,
begin
let y := a * x + b * z,
have hxy : x < y,
{ rw [← one_mul x, ← hab, add_mul],
exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },
have hyz : y < z,
{ rw [← one_mul z, ← hab, add_mul],
exact add_lt_add_right ((mul_lt_mul_left ha... | lemma | convex_on_of_slope_mono_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"add_div",
"convex",
"convex_on",
"div_eq_iff",
"div_le_div_iff",
"le_div_iff",
"linear_order.convex_on_of_lt",
"mul_comm",
"mul_div_assoc",
"mul_lt_mul_left",
"one_mul",
"ring"
] | If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
less than the slope of the secant line of `f` on `[x, z]`, then `f` is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on_of_slope_anti_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : concave_on 𝕜 s f | begin
rw ←neg_convex_on_iff,
refine convex_on_of_slope_mono_adjacent hs (λ x y z hx hz hxy hyz, _),
rw ←neg_le_neg_iff,
simp_rw [←neg_div, neg_sub, pi.neg_apply, neg_sub_neg],
exact hf hx hz hxy hyz,
end | lemma | concave_on_of_slope_anti_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"concave_on",
"convex",
"convex_on_of_slope_mono_adjacent"
] | If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
greater than the slope of the secant line of `f` on `[x, z]`, then `f` is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on_of_slope_strict_mono_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) < (f z - f y) / (z - y)) :
strict_convex_on 𝕜 s f | linear_order.strict_convex_on_of_lt hs $ λ x hx z hz hxz a b ha hb hab,
begin
let y := a * x + b * z,
have hxy : x < y,
{ rw [← one_mul x, ← hab, add_mul],
exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },
have hyz : y < z,
{ rw [← one_mul z, ← hab, add_mul],
exact add_lt_add_right ((mul_lt_mul_... | lemma | strict_convex_on_of_slope_strict_mono_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"add_div",
"convex",
"div_eq_iff",
"div_lt_div_iff",
"linear_order.strict_convex_on_of_lt",
"lt_div_iff",
"mul_comm",
"mul_div_assoc",
"mul_lt_mul_left",
"one_mul",
"ring",
"strict_convex_on"
] | If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
strictly less than the slope of the secant line of `f` on `[x, z]`, then `f` is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on_of_slope_strict_anti_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) < (f y - f x) / (y - x)) : strict_concave_on 𝕜 s f | begin
rw ←neg_strict_convex_on_iff,
refine strict_convex_on_of_slope_strict_mono_adjacent hs (λ x y z hx hz hxy hyz, _),
rw ←neg_lt_neg_iff,
simp_rw [←neg_div, neg_sub, pi.neg_apply, neg_sub_neg],
exact hf hx hz hxy hyz,
end | lemma | strict_concave_on_of_slope_strict_anti_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex",
"strict_concave_on",
"strict_convex_on_of_slope_strict_mono_adjacent"
] | If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
strictly greater than the slope of the secant line of `f` on `[x, z]`, then `f` is strictly concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_iff_slope_mono_adjacent :
convex_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | ⟨λ h, ⟨h.1, λ x y z, h.slope_mono_adjacent⟩, λ h, convex_on_of_slope_mono_adjacent h.1 h.2⟩ | lemma | convex_on_iff_slope_mono_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex",
"convex_on",
"convex_on_of_slope_mono_adjacent"
] | A function `f : 𝕜 → 𝕜` is convex iff for any three points `x < y < z` the slope of the secant
line of `f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on_iff_slope_anti_adjacent :
concave_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) ≤ (f y - f x) / (y - x) | ⟨λ h, ⟨h.1, λ x y z, h.slope_anti_adjacent⟩, λ h, concave_on_of_slope_anti_adjacent h.1 h.2⟩ | lemma | concave_on_iff_slope_anti_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"concave_on",
"concave_on_of_slope_anti_adjacent",
"convex"
] | A function `f : 𝕜 → 𝕜` is concave iff for any three points `x < y < z` the slope of the secant
line of `f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on_iff_slope_strict_mono_adjacent :
strict_convex_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) < (f z - f y) / (z - y) | ⟨λ h, ⟨h.1, λ x y z, h.slope_strict_mono_adjacent⟩,
λ h, strict_convex_on_of_slope_strict_mono_adjacent h.1 h.2⟩ | lemma | strict_convex_on_iff_slope_strict_mono_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex",
"strict_convex_on",
"strict_convex_on_of_slope_strict_mono_adjacent"
] | A function `f : 𝕜 → 𝕜` is strictly convex iff for any three points `x < y < z` the slope of
the secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on_iff_slope_strict_anti_adjacent :
strict_concave_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) < (f y - f x) / (y - x) | ⟨λ h, ⟨h.1, λ x y z, h.slope_anti_adjacent⟩,
λ h, strict_concave_on_of_slope_strict_anti_adjacent h.1 h.2⟩ | lemma | strict_concave_on_iff_slope_strict_anti_adjacent | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex",
"strict_concave_on",
"strict_concave_on_of_slope_strict_anti_adjacent"
] | A function `f : 𝕜 → 𝕜` is strictly concave iff for any three points `x < y < z` the slope of
the secant line of `f` on `[x, y]` is strictly greater than the slope of the secant line of `f` on
`[x, z]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.secant_mono_aux1 (hf : convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(z - x) * f y ≤ (z - y) * f x + (y - x) * f z | begin
have hxy' : 0 < y - x := by linarith,
have hyz' : 0 < z - y := by linarith,
have hxz' : 0 < z - x := by linarith,
rw ← le_div_iff' hxz',
have ha : 0 ≤ (z - y) / (z - x) := by positivity,
have hb : 0 ≤ (y - x) / (z - x) := by positivity,
calc f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) : ... | lemma | convex_on.secant_mono_aux1 | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex_on",
"le_div_iff'",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.secant_mono_aux2 (hf : convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f y - f x) / (y - x) ≤ (f z - f x) / (z - x) | begin
have hxy' : 0 < y - x := by linarith,
have hxz' : 0 < z - x := by linarith,
rw div_le_div_iff hxy' hxz',
linarith only [hf.secant_mono_aux1 hx hz hxy hyz],
end | lemma | convex_on.secant_mono_aux2 | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex_on",
"div_le_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.secant_mono_aux3 (hf : convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f z - f x) / (z - x) ≤ (f z - f y) / (z - y) | begin
have hyz' : 0 < z - y := by linarith,
have hxz' : 0 < z - x := by linarith,
rw div_le_div_iff hxz' hyz',
linarith only [hf.secant_mono_aux1 hx hz hxy hyz],
end | lemma | convex_on.secant_mono_aux3 | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex_on",
"div_le_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.secant_mono (hf : convex_on 𝕜 s f)
{a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x ≤ y) :
(f x - f a) / (x - a) ≤ (f y - f a) / (y - a) | begin
rcases eq_or_lt_of_le hxy with rfl | hxy,
{ simp },
cases lt_or_gt_of_ne hxa with hxa hxa,
{ cases lt_or_gt_of_ne hya with hya hya,
{ convert hf.secant_mono_aux3 hx ha hxy hya using 1;
rw ← neg_div_neg_eq;
field_simp, },
{ convert hf.slope_mono_adjacent hx hy hxa hya using 1,
rw ... | lemma | convex_on.secant_mono | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex_on",
"eq_or_lt_of_le",
"neg_div_neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.secant_strict_mono_aux1 (hf : strict_convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(z - x) * f y < (z - y) * f x + (y - x) * f z | begin
have hxy' : 0 < y - x := by linarith,
have hyz' : 0 < z - y := by linarith,
have hxz' : 0 < z - x := by linarith,
rw ← lt_div_iff' hxz',
have ha : 0 < (z - y) / (z - x) := by positivity,
have hb : 0 < (y - x) / (z - x) := by positivity,
calc f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) : ... | lemma | strict_convex_on.secant_strict_mono_aux1 | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"lt_div_iff'",
"ring",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.secant_strict_mono_aux2 (hf : strict_convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f y - f x) / (y - x) < (f z - f x) / (z - x) | begin
have hxy' : 0 < y - x := by linarith,
have hxz' : 0 < z - x := by linarith,
rw div_lt_div_iff hxy' hxz',
linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz],
end | lemma | strict_convex_on.secant_strict_mono_aux2 | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"div_lt_div_iff",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.secant_strict_mono_aux3 (hf : strict_convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f z - f x) / (z - x) < (f z - f y) / (z - y) | begin
have hyz' : 0 < z - y := by linarith,
have hxz' : 0 < z - x := by linarith,
rw div_lt_div_iff hxz' hyz',
linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz],
end | lemma | strict_convex_on.secant_strict_mono_aux3 | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"div_lt_div_iff",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.secant_strict_mono (hf : strict_convex_on 𝕜 s f)
{a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y) :
(f x - f a) / (x - a) < (f y - f a) / (y - a) | begin
cases lt_or_gt_of_ne hxa with hxa hxa,
{ cases lt_or_gt_of_ne hya with hya hya,
{ convert hf.secant_strict_mono_aux3 hx ha hxy hya using 1;
rw ← neg_div_neg_eq;
field_simp, },
{ convert hf.slope_strict_mono_adjacent hx hy hxa hya using 1,
rw ← neg_div_neg_eq;
field_simp, } },
... | lemma | strict_convex_on.secant_strict_mono | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"neg_div_neg_eq",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.secant_strict_mono (hf : strict_concave_on 𝕜 s f)
{a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y) :
(f y - f a) / (y - a) < (f x - f a) / (x - a) | begin
have key := hf.neg.secant_strict_mono ha hx hy hxa hya hxy,
simp only [pi.neg_apply] at key,
rw ← neg_lt_neg_iff,
convert key using 1; field_simp,
end | lemma | strict_concave_on.secant_strict_mono | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.strict_mono_of_lt (hf : convex_on 𝕜 s f)
{x y : 𝕜} (hx : x ∈ s) (hxy : x < y) (hxy' : f x < f y) :
strict_mono_on f (s ∩ set.Ici y) | begin
intros u hu v hv huv,
have step1 : ∀ {z : 𝕜}, z ∈ s ∩ set.Ioi y → f y < f z,
{ refine λ z hz, hf.lt_right_of_left_lt hx hz.1 _ hxy',
rw open_segment_eq_Ioo (hxy.trans hz.2),
exact ⟨hxy, hz.2⟩ },
rcases eq_or_lt_of_le hu.2 with rfl | hu2,
{ exact step1 ⟨hv.1, huv⟩ },
{ refine hf.lt_right_of_le... | lemma | convex_on.strict_mono_of_lt | analysis.convex | src/analysis/convex/slope.lean | [
"analysis.convex.function",
"tactic.field_simp",
"tactic.linarith"
] | [
"convex_on",
"eq_or_lt_of_le",
"open_segment_eq_Ioo",
"segment_eq_Icc",
"set.Ici",
"set.Ioi",
"strict_mono_on"
] | If `f` is convex on a set `s` in a linearly ordered field, and `f x < f y` for two points
`x < y` in `s`, then `f` is strictly monotone on `s ∩ [y, ∞)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex : Prop | ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s | def | star_convex | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [] | Star-convexity of sets. `s` is star-convex at `x` if every segment from `x` to a point in `s` is
contained in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex_iff_segment_subset : star_convex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → [x -[𝕜] y] ⊆ s | begin
split,
{ rintro h y hy z ⟨a, b, ha, hb, hab, rfl⟩,
exact h hy ha hb hab },
{ rintro h y hy a b ha hb hab,
exact h hy ⟨a, b, ha, hb, hab, rfl⟩ }
end | lemma | star_convex_iff_segment_subset | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.segment_subset (h : star_convex 𝕜 x s) {y : E} (hy : y ∈ s) : [x -[𝕜] y] ⊆ s | star_convex_iff_segment_subset.1 h hy | lemma | star_convex.segment_subset | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.open_segment_subset (h : star_convex 𝕜 x s) {y : E} (hy : y ∈ s) :
open_segment 𝕜 x y ⊆ s | (open_segment_subset_segment 𝕜 x y).trans (h.segment_subset hy) | lemma | star_convex.open_segment_subset | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"open_segment",
"open_segment_subset_segment",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_iff_pointwise_add_subset :
star_convex 𝕜 x s ↔ ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • {x} + b • s ⊆ s | begin
refine ⟨_, λ h y hy a b ha hb hab,
h ha hb hab (add_mem_add (smul_mem_smul_set $ mem_singleton _) ⟨_, hy, rfl⟩)⟩,
rintro hA a b ha hb hab w ⟨au, bv, ⟨u, (rfl : u = x), rfl⟩, ⟨v, hv, rfl⟩, rfl⟩,
exact hA hv ha hb hab,
end | lemma | star_convex_iff_pointwise_add_subset | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | Alternative definition of star-convexity, in terms of pointwise set operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_convex_empty (x : E) : star_convex 𝕜 x ∅ | λ y hy, hy.elim | lemma | star_convex_empty | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_univ (x : E) : star_convex 𝕜 x univ | λ _ _ _ _ _ _ _, trivial | lemma | star_convex_univ | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.inter (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 x t) :
star_convex 𝕜 x (s ∩ t) | λ y hy a b ha hb hab, ⟨hs hy.left ha hb hab, ht hy.right ha hb hab⟩ | lemma | star_convex.inter | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_sInter {S : set (set E)} (h : ∀ s ∈ S, star_convex 𝕜 x s) :
star_convex 𝕜 x (⋂₀ S) | λ y hy a b ha hb hab s hs, h s hs (hy s hs) ha hb hab | lemma | star_convex_sInter | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_Inter {ι : Sort*} {s : ι → set E} (h : ∀ i, star_convex 𝕜 x (s i)) :
star_convex 𝕜 x (⋂ i, s i) | (sInter_range s) ▸ star_convex_sInter $ forall_range_iff.2 h | lemma | star_convex_Inter | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex",
"star_convex_sInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.union (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 x t) :
star_convex 𝕜 x (s ∪ t) | begin
rintro y (hy | hy) a b ha hb hab,
{ exact or.inl (hs hy ha hb hab) },
{ exact or.inr (ht hy ha hb hab) }
end | lemma | star_convex.union | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_Union {ι : Sort*} {s : ι → set E} (hs : ∀ i, star_convex 𝕜 x (s i)) :
star_convex 𝕜 x (⋃ i, s i) | begin
rintro y hy a b ha hb hab,
rw mem_Union at ⊢ hy,
obtain ⟨i, hy⟩ := hy,
exact ⟨i, hs i hy ha hb hab⟩,
end | lemma | star_convex_Union | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_sUnion {S : set (set E)} (hS : ∀ s ∈ S, star_convex 𝕜 x s) :
star_convex 𝕜 x (⋃₀ S) | begin
rw sUnion_eq_Union,
exact star_convex_Union (λ s, hS _ s.2),
end | lemma | star_convex_sUnion | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex",
"star_convex_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.prod {y : F} {s : set E} {t : set F} (hs : star_convex 𝕜 x s)
(ht : star_convex 𝕜 y t) :
star_convex 𝕜 (x, y) (s ×ˢ t) | λ y hy a b ha hb hab, ⟨hs hy.1 ha hb hab, ht hy.2 ha hb hab⟩ | lemma | star_convex.prod | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_pi {ι : Type*} {E : ι → Type*} [Π i, add_comm_monoid (E i)]
[Π i, has_smul 𝕜 (E i)] {x : Π i, E i} {s : set ι} {t : Π i, set (E i)}
(ht : ∀ ⦃i⦄, i ∈ s → star_convex 𝕜 (x i) (t i)) :
star_convex 𝕜 x (s.pi t) | λ y hy a b ha hb hab i hi, ht hi (hy i hi) ha hb hab | lemma | star_convex_pi | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"add_comm_monoid",
"has_smul",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex.mem (hs : star_convex 𝕜 x s) (h : s.nonempty) : x ∈ s | begin
obtain ⟨y, hy⟩ := h,
convert hs hy zero_le_one le_rfl (add_zero 1),
rw [one_smul, zero_smul, add_zero],
end | lemma | star_convex.mem | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"le_rfl",
"one_smul",
"star_convex",
"zero_le_one",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_iff_forall_pos (hx : x ∈ s) :
star_convex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s | begin
refine ⟨λ h y hy a b ha hb hab, h hy ha.le hb.le hab, _⟩,
intros h y hy a b ha hb hab,
obtain rfl | ha := ha.eq_or_lt,
{ rw zero_add at hab,
rwa [hab, one_smul, zero_smul, zero_add] },
obtain rfl | hb := hb.eq_or_lt,
{ rw add_zero at hab,
rwa [hab, one_smul, zero_smul, add_zero] },
exact h h... | lemma | star_convex_iff_forall_pos | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"one_smul",
"star_convex",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_iff_forall_ne_pos (hx : x ∈ s) :
star_convex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
a • x + b • y ∈ s | begin
refine ⟨λ h y hy _ a b ha hb hab, h hy ha.le hb.le hab, _⟩,
intros h y hy a b ha hb hab,
obtain rfl | ha' := ha.eq_or_lt,
{ rw [zero_add] at hab, rwa [hab, zero_smul, one_smul, zero_add] },
obtain rfl | hb' := hb.eq_or_lt,
{ rw [add_zero] at hab, rwa [hab, zero_smul, one_smul, add_zero] },
obtain rf... | lemma | star_convex_iff_forall_ne_pos | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"convex.combo_self",
"eq_or_ne",
"one_smul",
"star_convex",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_iff_open_segment_subset (hx : x ∈ s) :
star_convex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → open_segment 𝕜 x y ⊆ s | star_convex_iff_segment_subset.trans $ forall₂_congr $ λ y hy,
(open_segment_subset_iff_segment_subset hx hy).symm | lemma | star_convex_iff_open_segment_subset | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"forall₂_congr",
"open_segment",
"open_segment_subset_iff_segment_subset",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_convex_singleton (x : E) : star_convex 𝕜 x {x} | begin
rintro y (rfl : y = x) a b ha hb hab,
exact convex.combo_self hab _,
end | lemma | star_convex_singleton | analysis.convex | src/analysis/convex/star.lean | [
"analysis.convex.segment"
] | [
"convex.combo_self",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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