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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is_bounded_at_im_infty {α : Type*} [has_norm α] (f : ℍ → α) : Prop | bounded_at_filter at_im_infty f | def | upper_half_plane.is_bounded_at_im_infty | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"has_norm"
] | A function ` f : ℍ → α` is bounded at infinity if it is bounded along `at_im_infty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_zero_at_im_infty {α : Type*} [has_zero α] [topological_space α] (f : ℍ → α) : Prop | zero_at_filter at_im_infty f | def | upper_half_plane.is_zero_at_im_infty | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"topological_space"
] | A function ` f : ℍ → α` is zero at infinity it is zero along `at_im_infty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_form_is_bounded_at_im_infty {α : Type*} [normed_field α] :
is_bounded_at_im_infty (0 : ℍ → α) | const_bounded_at_filter at_im_infty (0:α) | lemma | upper_half_plane.zero_form_is_bounded_at_im_infty | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"normed_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_im_infty_submodule (α : Type*) [normed_field α] : submodule α (ℍ → α) | zero_at_filter_submodule at_im_infty | def | upper_half_plane.zero_at_im_infty_submodule | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"normed_field",
"submodule"
] | Module of functions that are zero at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_at_im_infty_subalgebra (α : Type*) [normed_field α] : subalgebra α (ℍ → α) | bounded_filter_subalgebra at_im_infty | def | upper_half_plane.bounded_at_im_infty_subalgebra | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"normed_field",
"subalgebra"
] | ubalgebra of functions that are bounded at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_at_im_infty.mul {f g : ℍ → ℂ} (hf : is_bounded_at_im_infty f)
(hg : is_bounded_at_im_infty g) : is_bounded_at_im_infty (f * g) | by simpa only [pi.one_apply, mul_one, norm_eq_abs] using hf.mul hg | lemma | upper_half_plane.is_bounded_at_im_infty.mul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"mul_one",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_mem (f : ℍ → ℂ) :
is_bounded_at_im_infty f ↔ ∃ (M A : ℝ), ∀ z : ℍ, A ≤ im z → abs (f z) ≤ M | by simp [is_bounded_at_im_infty, bounded_at_filter, asymptotics.is_O_iff, filter.eventually,
at_im_infty_mem] | lemma | upper_half_plane.bounded_mem | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"asymptotics.is_O_iff",
"filter.eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_im_infty (f : ℍ → ℂ) :
is_zero_at_im_infty f ↔ ∀ ε : ℝ, 0 < ε → ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ ε | begin
rw [is_zero_at_im_infty, zero_at_filter, tendsto_iff_forall_eventually_mem],
split,
{ simp_rw [filter.eventually, at_im_infty_mem],
intros h ε hε,
simpa using (h (metric.closed_ball (0 : ℂ) ε) (metric.closed_ball_mem_nhds (0 : ℂ) hε))},
{ simp_rw metric.mem_nhds_iff,
intros h s hs,
simp_r... | lemma | upper_half_plane.zero_at_im_infty | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"filter.eventually",
"metric.closed_ball",
"metric.closed_ball_mem_nhds",
"metric.mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smooth_coe : smooth 𝓘(ℂ) 𝓘(ℂ) (coe : ℍ → ℂ) | λ x, cont_mdiff_at_ext_chart_at | lemma | upper_half_plane.smooth_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/manifold.lean | [
"analysis.complex.upper_half_plane.topology",
"geometry.manifold.cont_mdiff_mfderiv"
] | [
"cont_mdiff_at_ext_chart_at",
"smooth"
] | The inclusion map `ℍ → ℂ` is a smooth map of manifolds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mdifferentiable_coe : mdifferentiable 𝓘(ℂ) 𝓘(ℂ) (coe : ℍ → ℂ) | smooth_coe.mdifferentiable | lemma | upper_half_plane.mdifferentiable_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/manifold.lean | [
"analysis.complex.upper_half_plane.topology",
"geometry.manifold.cont_mdiff_mfderiv"
] | [
"mdifferentiable"
] | The inclusion map `ℍ → ℂ` is a differentiable map of manifolds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im))) | rfl | lemma | upper_half_plane.dist_eq | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_half_dist (z w : ℍ) :
sinh (dist z w / 2) = dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) | by rw [dist_eq, mul_div_cancel_left (arsinh _) two_ne_zero, sinh_arsinh] | lemma | upper_half_plane.sinh_half_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"mul_div_cancel_left",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosh_half_dist (z w : ℍ) :
cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * sqrt (z.im * w.im)) | begin
have H₁ : (2 ^ 2 : ℝ) = 4, by norm_num1,
have H₂ : 0 < z.im * w.im, from mul_pos z.im_pos w.im_pos,
have H₃ : 0 < 2 * sqrt (z.im * w.im), from mul_pos two_pos (sqrt_pos.2 H₂),
rw [← sq_eq_sq (cosh_pos _).le (div_nonneg dist_nonneg H₃.le), cosh_sq', sinh_half_dist, div_pow,
div_pow, one_add_div (pow_ne... | lemma | upper_half_plane.cosh_half_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"complex.conj_im",
"complex.conj_re",
"complex.dist_eq",
"complex.mul_re",
"complex.norm_sq_conj",
"complex.norm_sq_sub",
"complex.sq_abs",
"dist_nonneg",
"div_nonneg",
"div_pow",
"mul_pow",
"one_add_div",
"pow_ne_zero",
"ring",
"sq_eq_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tanh_half_dist (z w : ℍ) :
tanh (dist z w / 2) = dist (z : ℂ) w / dist (z : ℂ) (conj ↑w) | begin
rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one],
exact (mul_pos (zero_lt_two' ℝ) (sqrt_pos.2 $ mul_pos z.im_pos w.im_pos)).ne'
end | lemma | upper_half_plane.tanh_half_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"div_div_div_comm",
"div_one",
"div_self",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_half_dist (z w : ℍ) :
exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * sqrt (z.im * w.im)) | by rw [← sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div] | lemma | upper_half_plane.exp_half_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"add_div",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2 * z.im * w.im) | by rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
sq_sqrt (mul_pos z.im_pos w.im_pos).le, sq (2 : ℝ), mul_assoc, ← mul_div_assoc,
mul_assoc, mul_div_mul_left _ _ (two_ne_zero' ℝ)] | lemma | upper_half_plane.cosh_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"div_pow",
"mul_assoc",
"mul_div_assoc",
"mul_div_mul_left",
"mul_pow",
"two_mul",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_half_dist_add_dist (a b c : ℍ) :
sinh ((dist a b + dist b c) / 2) =
(dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
(2 * sqrt (a.im * c.im) * dist (b : ℂ) (conj ↑b)) | begin
simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm],
rw [← add_div, complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist ↑b _),
dist_comm (b : ℂ), complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im],
congr' 2,
rw [sqrt_mul, sqrt... | lemma | upper_half_plane.sinh_half_dist_add_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"abs_of_pos",
"add_div",
"complex.dist_conj_comm",
"complex.dist_self_conj",
"dist_comm",
"div_mul_div_comm",
"mul_comm",
"mul_mul_mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_comm (z w : ℍ) : dist z w = dist w z | by simp only [dist_eq, dist_comm (z : ℂ), mul_comm] | lemma | upper_half_plane.dist_comm | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_comm",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_iff_le_sinh :
dist z w ≤ r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) ≤ sinh (r / 2) | by rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist] | lemma | upper_half_plane.dist_le_iff_le_sinh | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"div_le_div_right",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_iff_eq_sinh :
dist z w = r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) = sinh (r / 2) | by rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist] | lemma | upper_half_plane.dist_eq_iff_eq_sinh | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"div_left_inj'",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 | begin
rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc],
{ norm_num },
{ exact (mul_pos z.im_pos w.im_pos).le },
{ exact div_nonneg dist_nonneg (mul_nonneg zero_le_two $ sqrt_nonneg _) },
{ exact sinh_nonneg_iff.2 (div_nonneg hr zero_le_two) }
end | lemma | upper_half_plane.dist_eq_iff_eq_sq_sinh | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_nonneg",
"div_nonneg",
"div_pow",
"mul_assoc",
"mul_pow",
"sq_eq_sq",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c | begin
rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist,
div_mul_eq_div_div _ _ (dist _ _), le_div_iff, div_mul_eq_mul_div],
{ exact div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
(euclidean_geometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj ↑b)) },
{ rw [dist_comm, dist_pos, ne.def,... | lemma | upper_half_plane.dist_triangle | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"complex.conj_eq_iff_im",
"dist_comm",
"dist_pos",
"dist_triangle",
"div_le_div_of_le",
"div_mul_eq_div_div",
"div_mul_eq_mul_div",
"euclidean_geometry.mul_dist_le_mul_dist_add_mul_dist",
"le_div_iff",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_dist_coe_div_sqrt (z w : ℍ) :
dist z w ≤ dist (z : ℂ) w / sqrt (z.im * w.im) | begin
rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff],
exact div_nonneg dist_nonneg (mul_nonneg zero_le_two (sqrt_nonneg _))
end | lemma | upper_half_plane.dist_le_dist_coe_div_sqrt | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_nonneg",
"div_mul_eq_div_div_swap",
"div_nonneg",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space_aux : metric_space ℍ | { dist := dist,
dist_self := λ z, by rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero],
dist_comm := upper_half_plane.dist_comm,
dist_triangle := upper_half_plane.dist_triangle,
eq_of_dist_eq_zero := λ z w h,
by simpa [dist_eq, real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, subtype.coe_inj]
... | def | upper_half_plane.metric_space_aux | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"eq_of_dist_eq_zero",
"metric_space",
"mul_zero",
"real.sqrt_eq_zero'",
"subtype.coe_inj",
"upper_half_plane.dist_comm",
"upper_half_plane.dist_triangle",
"zero_div"
] | An auxiliary `metric_space` instance on the upper half-plane. This instance has bad projection
to `topological_space`. We replace it later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cosh_dist' (z w : ℍ) :
real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) | have H : 0 < 2 * z.im * w.im, from mul_pos (mul_pos two_pos z.im_pos) w.im_pos,
by { field_simp [cosh_dist, complex.dist_eq, complex.sq_abs, norm_sq_apply, H, H.ne'], ring } | lemma | upper_half_plane.cosh_dist' | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"complex.dist_eq",
"complex.sq_abs",
"real.cosh",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center (z : ℍ) (r : ℝ) : ℍ | ⟨⟨z.re, z.im * cosh r⟩, mul_pos z.im_pos (cosh_pos _)⟩ | def | upper_half_plane.center | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [] | Euclidean center of the circle with center `z` and radius `r` in the hyperbolic metric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
center_re (z r) : (center z r).re = z.re | rfl | lemma | upper_half_plane.center_re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_im (z r) : (center z r).im = z.im * cosh r | rfl | lemma | upper_half_plane.center_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_zero (z : ℍ) : center z 0 = z | subtype.ext $ ext rfl $ by rw [coe_im, coe_im, center_im, real.cosh_zero, mul_one] | lemma | upper_half_plane.center_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"mul_one",
"real.cosh_zero",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_coe_center_sq (z w : ℍ) (r : ℝ) :
dist (z : ℂ) (w.center r) ^ 2 =
2 * z.im * w.im * (cosh (dist z w) - cosh r) + (w.im * sinh r) ^ 2 | begin
have H : 2 * z.im * w.im ≠ 0, by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero],
simp only [complex.dist_eq, complex.sq_abs, norm_sq_apply, coe_re, coe_im, center_re, center_im,
cosh_dist', mul_div_cancel' _ H, sub_sq z.im, mul_pow, real.cosh_sq, sub_re, sub_im, mul_sub,
← sq],
ring
end | lemma | upper_half_plane.dist_coe_center_sq | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"complex.dist_eq",
"complex.sq_abs",
"mul_div_cancel'",
"mul_ne_zero",
"mul_pow",
"real.cosh_sq",
"ring",
"sub_sq",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_coe_center (z w : ℍ) (r : ℝ) :
dist (z : ℂ) (w.center r) =
sqrt (2 * z.im * w.im * (cosh (dist z w) - cosh r) + (w.im * sinh r) ^ 2) | by rw [← sqrt_sq dist_nonneg, dist_coe_center_sq] | lemma | upper_half_plane.dist_coe_center | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * sinh r) | begin
letI := metric_space_aux,
cases lt_or_le r 0 with hr₀ hr₀,
{ transitivity ordering.gt,
exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le
dist_nonneg).cmp_eq_gt.symm] },
have hr₀' : 0 ≤ w.im * sinh r, from mul_nonneg w.im_pos.le ... | lemma | upper_half_plane.cmp_dist_eq_cmp_dist_coe_center | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"cmp_mul_pos_left",
"dist_nonneg",
"mul_neg_of_pos_of_neg",
"strict_mono_on_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_iff_dist_coe_center_eq : dist z w = r ↔ dist (z : ℂ) (w.center r) = w.im * sinh r | eq_iff_eq_of_cmp_eq_cmp (cmp_dist_eq_cmp_dist_coe_center z w r) | lemma | upper_half_plane.dist_eq_iff_dist_coe_center_eq | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"eq_iff_eq_of_cmp_eq_cmp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_center (z : ℍ) (r : ℝ) : dist (z : ℂ) (z.center r) = z.im * (cosh r - 1) | begin
rw [dist_of_re_eq (z.center_re r).symm, dist_comm, real.dist_eq, mul_sub, mul_one],
exact abs_of_nonneg (sub_nonneg.2 $ le_mul_of_one_le_right z.im_pos.le (one_le_cosh _))
end | lemma | upper_half_plane.dist_self_center | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"abs_of_nonneg",
"dist_comm",
"le_mul_of_one_le_right",
"mul_one",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_center_dist (z w : ℍ) :
dist (z : ℂ) (w.center (dist z w)) = w.im * sinh (dist z w) | dist_eq_iff_dist_coe_center_eq.1 rfl | lemma | upper_half_plane.dist_center_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_iff_dist_coe_center_lt :
dist z w < r ↔ dist (z : ℂ) (w.center r) < w.im * sinh r | lt_iff_lt_of_cmp_eq_cmp (cmp_dist_eq_cmp_dist_coe_center z w r) | lemma | upper_half_plane.dist_lt_iff_dist_coe_center_lt | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"lt_iff_lt_of_cmp_eq_cmp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_dist_iff_lt_dist_coe_center :
r < dist z w ↔ w.im * sinh r < dist (z : ℂ) (w.center r) | lt_iff_lt_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 $ cmp_dist_eq_cmp_dist_coe_center z w r) | lemma | upper_half_plane.lt_dist_iff_lt_dist_coe_center | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"lt_iff_lt_of_cmp_eq_cmp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_iff_dist_coe_center_le :
dist z w ≤ r ↔ dist (z : ℂ) (w.center r) ≤ w.im * sinh r | le_iff_le_of_cmp_eq_cmp (cmp_dist_eq_cmp_dist_coe_center z w r) | lemma | upper_half_plane.dist_le_iff_dist_coe_center_le | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"le_iff_le_of_cmp_eq_cmp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_dist_iff_le_dist_coe_center :
r < dist z w ↔ w.im * sinh r < dist (z : ℂ) (w.center r) | lt_iff_lt_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 $ cmp_dist_eq_cmp_dist_coe_center z w r) | lemma | upper_half_plane.le_dist_iff_le_dist_coe_center | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"lt_iff_lt_of_cmp_eq_cmp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) | begin
have h₀ : 0 < z.im / w.im, from div_pos z.im_pos w.im_pos,
rw [dist_eq_iff_dist_coe_center_eq, real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div];
[skip, exact h],
nth_rewrite 3 [← abs_of_pos w.im_pos]... | lemma | upper_half_plane.dist_of_re_eq | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"abs_of_pos",
"div_pos",
"inv_div",
"real.dist_eq",
"ring"
] | For two points on the same vertical line, the distance is equal to the distance between the
logarithms of their imaginary parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w | calc dist (log z.im) (log w.im) = @dist ℍ _ ⟨⟨0, z.im⟩, z.im_pos⟩ ⟨⟨0, w.im⟩, w.im_pos⟩ :
eq.symm $ @dist_of_re_eq ⟨⟨0, z.im⟩, z.im_pos⟩ ⟨⟨0, w.im⟩, w.im_pos⟩ rfl
... ≤ dist z w :
mul_le_mul_of_nonneg_left (arsinh_le_arsinh.2 $ div_le_div_of_le
(mul_nonneg zero_le_two (sqrt_nonneg _)) $
by simpa [sqrt_sq_... | lemma | upper_half_plane.dist_log_im_le | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"complex.abs_im_le_abs",
"div_le_div_of_le",
"mul_le_mul_of_nonneg_left",
"zero_le_two"
] | Hyperbolic distance between two points is greater than or equal to the distance between the
logarithms of their imaginary parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * exp (dist z w) | begin
rw [← div_le_iff' w.im_pos, ← exp_log z.im_pos, ← exp_log w.im_pos, ← real.exp_sub, exp_le_exp],
exact (le_abs_self _).trans (dist_log_im_le z w)
end | lemma | upper_half_plane.im_le_im_mul_exp_dist | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"div_le_iff'",
"exp",
"le_abs_self",
"real.exp_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_div_exp_dist_le (z w : ℍ) : z.im / exp (dist z w) ≤ w.im | (div_le_iff (exp_pos _)).2 (im_le_im_mul_exp_dist z w) | lemma | upper_half_plane.im_div_exp_dist_le | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"div_le_iff",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (exp (dist z w) - 1) | calc dist (z : ℂ) w ≤ dist (z : ℂ) (w.center (dist z w)) + dist (w : ℂ) (w.center (dist z w)) :
dist_triangle_right _ _ _
... = w.im * (exp (dist z w) - 1) :
by rw [dist_center_dist, dist_self_center, ← mul_add, ← add_sub_assoc, real.sinh_add_cosh] | lemma | upper_half_plane.dist_coe_le | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_triangle_right",
"exp",
"real.sinh_add_cosh"
] | An upper estimate on the complex distance between two points in terms of the hyperbolic distance
and the imaginary part of one of the points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_dist_coe (z w : ℍ) : w.im * (1 - exp (-dist z w)) ≤ dist (z : ℂ) w | calc w.im * (1 - exp (-dist z w))
= dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) :
by { rw [dist_center_dist, dist_self_center, ← real.cosh_sub_sinh], ring }
... ≤ dist (z : ℂ) w : sub_le_iff_le_add.2 $ dist_triangle _ _ _ | lemma | upper_half_plane.le_dist_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_triangle",
"exp",
"real.cosh_sub_sinh",
"ring"
] | An upper estimate on the complex distance between two points in terms of the hyperbolic distance
and the imaginary part of one of the points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ} (h : dist w (center z r) ≤ z.im * sinh r) :
0 < w.im | calc 0 < z.im * (cosh r - sinh r) : mul_pos z.im_pos (sub_pos.2 $ sinh_lt_cosh _)
... = (z.center r).im - z.im * sinh r : mul_sub _ _ _
... ≤ (z.center r).im - dist (z.center r : ℂ) w : sub_le_sub_left (by rwa [dist_comm]) _
... ≤ w.im : sub_le_comm.1 $ (le_abs_self _).trans (abs_im_le_abs $ z.center r - w) | lemma | upper_half_plane.im_pos_of_dist_center_le | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_comm",
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_coe_closed_ball (z : ℍ) (r : ℝ) :
(coe : ℍ → ℂ) '' closed_ball z r = closed_ball (z.center r) (z.im * sinh r) | begin
ext w, split,
{ rintro ⟨w, hw, rfl⟩,
exact dist_le_iff_dist_coe_center_le.1 hw },
{ intro hw,
lift w to ℍ using im_pos_of_dist_center_le hw,
exact mem_image_of_mem _ (dist_le_iff_dist_coe_center_le.2 hw) },
end | lemma | upper_half_plane.image_coe_closed_ball | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_coe_ball (z : ℍ) (r : ℝ) :
(coe : ℍ → ℂ) '' ball z r = ball (z.center r) (z.im * sinh r) | begin
ext w, split,
{ rintro ⟨w, hw, rfl⟩,
exact dist_lt_iff_dist_coe_center_lt.1 hw },
{ intro hw,
lift w to ℍ using im_pos_of_dist_center_le (ball_subset_closed_ball hw),
exact mem_image_of_mem _ (dist_lt_iff_dist_coe_center_lt.2 hw) },
end | lemma | upper_half_plane.image_coe_ball | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_coe_sphere (z : ℍ) (r : ℝ) :
(coe : ℍ → ℂ) '' sphere z r = sphere (z.center r) (z.im * sinh r) | begin
ext w, split,
{ rintro ⟨w, hw, rfl⟩,
exact dist_eq_iff_dist_coe_center_eq.1 hw },
{ intro hw,
lift w to ℍ using im_pos_of_dist_center_le (sphere_subset_closed_ball hw),
exact mem_image_of_mem _ (dist_eq_iff_dist_coe_center_eq.2 hw) },
end | lemma | upper_half_plane.image_coe_sphere | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_vertical_line (a : ℝ) : isometry (λ y, mk ⟨a, exp y⟩ (exp_pos y)) | begin
refine isometry.of_dist_eq (λ y₁ y₂, _),
rw [dist_of_re_eq],
exacts [congr_arg2 _ (log_exp _) (log_exp _), rfl]
end | lemma | upper_half_plane.isometry_vertical_line | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"congr_arg2",
"exp",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_real_vadd (a : ℝ) : isometry ((+ᵥ) a : ℍ → ℍ) | isometry.of_dist_eq $ λ y₁ y₂, by simp only [dist_eq, coe_vadd, vadd_im, dist_add_left] | lemma | upper_half_plane.isometry_real_vadd | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_pos_mul (a : {x : ℝ // 0 < x}) : isometry ((•) a : ℍ → ℍ) | begin
refine isometry.of_dist_eq (λ y₁ y₂, _),
simp only [dist_eq, coe_pos_real_smul, pos_real_im], congr' 2,
rw [dist_smul₀, mul_mul_mul_comm, real.sqrt_mul (mul_self_nonneg _), real.sqrt_mul_self_eq_abs,
real.norm_eq_abs, mul_left_comm],
exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne')
end | lemma | upper_half_plane.isometry_pos_mul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/metric.lean | [
"analysis.complex.upper_half_plane.topology",
"analysis.special_functions.arsinh",
"geometry.euclidean.inversion"
] | [
"dist_smul₀",
"isometry",
"mul_div_mul_left",
"mul_left_comm",
"mul_mul_mul_comm",
"mul_self_nonneg",
"real.norm_eq_abs",
"real.sqrt_mul",
"real.sqrt_mul_self_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_coe : open_embedding (coe : ℍ → ℂ) | is_open.open_embedding_subtype_coe $ is_open_lt continuous_const complex.continuous_im | lemma | upper_half_plane.open_embedding_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/topology.lean | [
"analysis.complex.upper_half_plane.basic",
"analysis.convex.contractible",
"analysis.convex.normed",
"analysis.convex.complex",
"analysis.complex.re_im_topology",
"topology.homotopy.contractible"
] | [
"complex.continuous_im",
"continuous_const",
"is_open.open_embedding_subtype_coe",
"is_open_lt",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_coe : embedding (coe : ℍ → ℂ) | embedding_subtype_coe | lemma | upper_half_plane.embedding_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/topology.lean | [
"analysis.complex.upper_half_plane.basic",
"analysis.convex.contractible",
"analysis.convex.normed",
"analysis.convex.complex",
"analysis.complex.re_im_topology",
"topology.homotopy.contractible"
] | [
"embedding",
"embedding_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (coe : ℍ → ℂ) | embedding_coe.continuous | lemma | upper_half_plane.continuous_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/topology.lean | [
"analysis.complex.upper_half_plane.basic",
"analysis.convex.contractible",
"analysis.convex.normed",
"analysis.convex.complex",
"analysis.complex.re_im_topology",
"topology.homotopy.contractible"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_re : continuous re | complex.continuous_re.comp continuous_coe | lemma | upper_half_plane.continuous_re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/topology.lean | [
"analysis.complex.upper_half_plane.basic",
"analysis.convex.contractible",
"analysis.convex.normed",
"analysis.convex.complex",
"analysis.complex.re_im_topology",
"topology.homotopy.contractible"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_im : continuous im | complex.continuous_im.comp continuous_coe | lemma | upper_half_plane.continuous_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/topology.lean | [
"analysis.complex.upper_half_plane.basic",
"analysis.convex.contractible",
"analysis.convex.normed",
"analysis.convex.complex",
"analysis.complex.re_im_topology",
"topology.homotopy.contractible"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex : Prop | ∀ ⦃x : E⦄, x ∈ s → star_convex 𝕜 x s | def | convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"star_convex"
] | Convexity of sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.star_convex (hs : convex 𝕜 s) (hx : x ∈ s) : star_convex 𝕜 x s | hs hx | lemma | convex.star_convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_segment_subset : convex 𝕜 s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → [x -[𝕜] y] ⊆ s | forall₂_congr $ λ x hx, star_convex_iff_segment_subset | lemma | convex_iff_segment_subset | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"forall₂_congr",
"star_convex_iff_segment_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.segment_subset (h : convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) :
[x -[𝕜] y] ⊆ s | convex_iff_segment_subset.1 h hx hy | lemma | convex.segment_subset | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.open_segment_subset (h : convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) :
open_segment 𝕜 x y ⊆ s | (open_segment_subset_segment 𝕜 x y).trans (h.segment_subset hx hy) | lemma | convex.open_segment_subset | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"open_segment",
"open_segment_subset_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_pointwise_add_subset :
convex 𝕜 s ↔ ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • s + b • s ⊆ s | iff.intro
begin
rintro hA a b ha hb hab w ⟨au, bv, ⟨u, hu, rfl⟩, ⟨v, hv, rfl⟩, rfl⟩,
exact hA hu hv ha hb hab
end
(λ h x hx y hy a b ha hb hab,
(h ha hb hab) (set.add_mem_add ⟨_, hx, rfl⟩ ⟨_, hy, rfl⟩)) | lemma | convex_iff_pointwise_add_subset | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | Alternative definition of set convexity, in terms of pointwise set operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_empty : convex 𝕜 (∅ : set E) | λ x, false.elim | lemma | convex_empty | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_univ : convex 𝕜 (set.univ : set E) | λ _ _, star_convex_univ _ | lemma | convex_univ | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"star_convex_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.inter {t : set E} (hs : convex 𝕜 s) (ht : convex 𝕜 t) : convex 𝕜 (s ∩ t) | λ x hx, (hs hx.1).inter (ht hx.2) | lemma | convex.inter | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_sInter {S : set (set E)} (h : ∀ s ∈ S, convex 𝕜 s) : convex 𝕜 (⋂₀ S) | λ x hx, star_convex_sInter $ λ s hs, h _ hs $ hx _ hs | lemma | convex_sInter | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"star_convex_sInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Inter {ι : Sort*} {s : ι → set E} (h : ∀ i, convex 𝕜 (s i)) : convex 𝕜 (⋂ i, s i) | (sInter_range s) ▸ convex_sInter $ forall_range_iff.2 h | lemma | convex_Inter | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_sInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Inter₂ {ι : Sort*} {κ : ι → Sort*} {s : Π i, κ i → set E}
(h : ∀ i j, convex 𝕜 (s i j)) :
convex 𝕜 (⋂ i j, s i j) | convex_Inter $ λ i, convex_Inter $ h i | lemma | convex_Inter₂ | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.prod {s : set E} {t : set F} (hs : convex 𝕜 s) (ht : convex 𝕜 t) : convex 𝕜 (s ×ˢ t) | λ x hx, (hs hx.1).prod (ht hx.2) | lemma | convex.prod | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_pi {ι : Type*} {E : ι → Type*} [Π i, add_comm_monoid (E i)]
[Π i, has_smul 𝕜 (E i)] {s : set ι} {t : Π i, set (E i)} (ht : ∀ ⦃i⦄, i ∈ s → convex 𝕜 (t i)) :
convex 𝕜 (s.pi t) | λ x hx, star_convex_pi $ λ i hi, ht hi $ hx _ hi | lemma | convex_pi | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"add_comm_monoid",
"convex",
"has_smul",
"star_convex_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
directed.convex_Union {ι : Sort*} {s : ι → set E} (hdir : directed (⊆) s)
(hc : ∀ ⦃i : ι⦄, convex 𝕜 (s i)) :
convex 𝕜 (⋃ i, s i) | begin
rintro x hx y hy a b ha hb hab,
rw mem_Union at ⊢ hx hy,
obtain ⟨i, hx⟩ := hx,
obtain ⟨j, hy⟩ := hy,
obtain ⟨k, hik, hjk⟩ := hdir i j,
exact ⟨k, hc (hik hx) (hjk hy) ha hb hab⟩,
end | lemma | directed.convex_Union | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
directed_on.convex_sUnion {c : set (set E)} (hdir : directed_on (⊆) c)
(hc : ∀ ⦃A : set E⦄, A ∈ c → convex 𝕜 A) :
convex 𝕜 (⋃₀c) | begin
rw sUnion_eq_Union,
exact (directed_on_iff_directed.1 hdir).convex_Union (λ A, hc A.2),
end | lemma | directed_on.convex_sUnion | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"directed_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_open_segment_subset :
convex 𝕜 s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → open_segment 𝕜 x y ⊆ s | forall₂_congr $ λ x, star_convex_iff_open_segment_subset | lemma | convex_iff_open_segment_subset | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"forall₂_congr",
"open_segment",
"star_convex_iff_open_segment_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_forall_pos :
convex 𝕜 s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1
→ a • x + b • y ∈ s | forall₂_congr $ λ x, star_convex_iff_forall_pos | lemma | convex_iff_forall_pos | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"forall₂_congr",
"star_convex_iff_forall_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_iff_pairwise_pos :
convex 𝕜 s ↔ s.pairwise (λ x y, ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s) | begin
refine convex_iff_forall_pos.trans ⟨λ h x hx y hy _, h hx hy, _⟩,
intros h x hx y hy a b ha hb hab,
obtain rfl | hxy := eq_or_ne x y,
{ rwa convex.combo_self hab },
{ exact h hx hy hxy ha hb hab },
end | lemma | convex_iff_pairwise_pos | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex.combo_self",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.star_convex_iff (hs : convex 𝕜 s) (h : s.nonempty) : star_convex 𝕜 x s ↔ x ∈ s | ⟨λ hxs, hxs.mem h, hs.star_convex⟩ | lemma | convex.star_convex_iff | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"star_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.convex {s : set E} (h : s.subsingleton) : convex 𝕜 s | convex_iff_pairwise_pos.mpr (h.pairwise _) | lemma | set.subsingleton.convex | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_singleton (c : E) : convex 𝕜 ({c} : set E) | subsingleton_singleton.convex | lemma | convex_singleton | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_zero : convex 𝕜 (0 : set E) | convex_singleton _ | lemma | convex_zero | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_segment (x y : E) : convex 𝕜 [x -[𝕜] y] | begin
rintro p ⟨ap, bp, hap, hbp, habp, rfl⟩ q ⟨aq, bq, haq, hbq, habq, rfl⟩ a b ha hb hab,
refine ⟨a * ap + b * aq, a * bp + b * bq,
add_nonneg (mul_nonneg ha hap) (mul_nonneg hb haq),
add_nonneg (mul_nonneg ha hbp) (mul_nonneg hb hbq), _, _⟩,
{ rw [add_add_add_comm, ←mul_add, ←mul_add, habp, habq, mul_o... | lemma | convex_segment | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"add_smul",
"convex",
"mul_one",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.linear_image (hs : convex 𝕜 s) (f : E →ₗ[𝕜] F) : convex 𝕜 (f '' s) | begin
intros x hx y hy a b ha hb hab,
obtain ⟨x', hx', rfl⟩ := mem_image_iff_bex.1 hx,
obtain ⟨y', hy', rfl⟩ := mem_image_iff_bex.1 hy,
exact ⟨a • x' + b • y', hs hx' hy' ha hb hab, by rw [f.map_add, f.map_smul, f.map_smul]⟩,
end | lemma | convex.linear_image | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.is_linear_image (hs : convex 𝕜 s) {f : E → F} (hf : is_linear_map 𝕜 f) :
convex 𝕜 (f '' s) | hs.linear_image $ hf.mk' f | lemma | convex.is_linear_image | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.linear_preimage {s : set F} (hs : convex 𝕜 s) (f : E →ₗ[𝕜] F) :
convex 𝕜 (f ⁻¹' s) | begin
intros x hx y hy a b ha hb hab,
rw [mem_preimage, f.map_add, f.map_smul, f.map_smul],
exact hs hx hy ha hb hab,
end | lemma | convex.linear_preimage | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.is_linear_preimage {s : set F} (hs : convex 𝕜 s) {f : E → F} (hf : is_linear_map 𝕜 f) :
convex 𝕜 (f ⁻¹' s) | hs.linear_preimage $ hf.mk' f | lemma | convex.is_linear_preimage | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.add {t : set E} (hs : convex 𝕜 s) (ht : convex 𝕜 t) : convex 𝕜 (s + t) | by { rw ← add_image_prod, exact (hs.prod ht).is_linear_image is_linear_map.is_linear_map_add } | lemma | convex.add | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"is_linear_map.is_linear_map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_add_submonoid : add_submonoid (set E) | { carrier := {s : set E | convex 𝕜 s},
zero_mem' := convex_zero,
add_mem' := λ s t, convex.add } | def | convex_add_submonoid | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"add_submonoid",
"convex",
"convex.add",
"convex_zero"
] | The convex sets form an additive submonoid under pointwise addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_convex_add_submonoid : ↑(convex_add_submonoid 𝕜 E) = {s : set E | convex 𝕜 s} | rfl | lemma | coe_convex_add_submonoid | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_add_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_convex_add_submonoid {s : set E} :
s ∈ convex_add_submonoid 𝕜 E ↔ convex 𝕜 s | iff.rfl | lemma | mem_convex_add_submonoid | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_add_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_list_sum {l : list (set E)} (h : ∀ i ∈ l, convex 𝕜 i) : convex 𝕜 l.sum | (convex_add_submonoid 𝕜 E).list_sum_mem h | lemma | convex_list_sum | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_add_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_multiset_sum {s : multiset (set E)} (h : ∀ i ∈ s, convex 𝕜 i) : convex 𝕜 s.sum | (convex_add_submonoid 𝕜 E).multiset_sum_mem _ h | lemma | convex_multiset_sum | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_add_submonoid",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_sum {ι} {s : finset ι} (t : ι → set E) (h : ∀ i ∈ s, convex 𝕜 (t i)) :
convex 𝕜 (∑ i in s, t i) | (convex_add_submonoid 𝕜 E).sum_mem h | lemma | convex_sum | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_add_submonoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.vadd (hs : convex 𝕜 s) (z : E) : convex 𝕜 (z +ᵥ s) | by { simp_rw [←image_vadd, vadd_eq_add, ←singleton_add], exact (convex_singleton _).add hs } | lemma | convex.vadd | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.translate (hs : convex 𝕜 s) (z : E) : convex 𝕜 ((λ x, z + x) '' s) | hs.vadd _ | lemma | convex.translate | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.translate_preimage_right (hs : convex 𝕜 s) (z : E) : convex 𝕜 ((λ x, z + x) ⁻¹' s) | begin
intros x hx y hy a b ha hb hab,
have h := hs hx hy ha hb hab,
rwa [smul_add, smul_add, add_add_add_comm, ←add_smul, hab, one_smul] at h,
end | lemma | convex.translate_preimage_right | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"one_smul",
"smul_add"
] | The translation of a convex set is also convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.translate_preimage_left (hs : convex 𝕜 s) (z : E) : convex 𝕜 ((λ x, x + z) ⁻¹' s) | by simpa only [add_comm] using hs.translate_preimage_right z | lemma | convex.translate_preimage_left | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex"
] | The translation of a convex set is also convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_Iic (r : β) : convex 𝕜 (Iic r) | λ x hx y hy a b ha hb hab,
calc
a • x + b • y
≤ a • r + b • r
: add_le_add (smul_le_smul_of_nonneg hx ha) (smul_le_smul_of_nonneg hy hb)
... = r : convex.combo_self hab _ | lemma | convex_Iic | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex.combo_self",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Ici (r : β) : convex 𝕜 (Ici r) | @convex_Iic 𝕜 βᵒᵈ _ _ _ _ r | lemma | convex_Ici | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_Icc (r s : β) : convex 𝕜 (Icc r s) | Ici_inter_Iic.subst ((convex_Ici r).inter $ convex_Iic s) | lemma | convex_Icc | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Ici",
"convex_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_halfspace_le {f : E → β} (h : is_linear_map 𝕜 f) (r : β) :
convex 𝕜 {w | f w ≤ r} | (convex_Iic r).is_linear_preimage h | lemma | convex_halfspace_le | analysis.convex | src/analysis/convex/basic.lean | [
"algebra.order.module",
"analysis.convex.star",
"linear_algebra.affine_space.affine_subspace"
] | [
"convex",
"convex_Iic",
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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