statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
concave_on_id {s : set β} (hs : convex 𝕜 s) : concave_on 𝕜 s id | ⟨hs, by { intros, refl }⟩ | lemma | concave_on_id | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.subset {t : set E} (hf : convex_on 𝕜 t f) (hst : s ⊆ t) (hs : convex 𝕜 s) :
convex_on 𝕜 s f | ⟨hs, λ x hx y hy, hf.2 (hst hx) (hst hy)⟩ | lemma | convex_on.subset | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.subset {t : set E} (hf : concave_on 𝕜 t f) (hst : s ⊆ t) (hs : convex 𝕜 s) :
concave_on 𝕜 s f | ⟨hs, λ x hx y hy, hf.2 (hst hx) (hst hy)⟩ | lemma | concave_on.subset | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.subset {t : set E} (hf : strict_convex_on 𝕜 t f) (hst : s ⊆ t)
(hs : convex 𝕜 s) :
strict_convex_on 𝕜 s f | ⟨hs, λ x hx y hy, hf.2 (hst hx) (hst hy)⟩ | lemma | strict_convex_on.subset | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.subset {t : set E} (hf : strict_concave_on 𝕜 t f) (hst : s ⊆ t)
(hs : convex 𝕜 s) :
strict_concave_on 𝕜 s f | ⟨hs, λ x hx y hy, hf.2 (hst hx) (hst hy)⟩ | lemma | strict_concave_on.subset | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.comp (hg : convex_on 𝕜 (f '' s) g) (hf : convex_on 𝕜 s f)
(hg' : monotone_on g (f '' s)) : convex_on 𝕜 s (g ∘ f) | ⟨hf.1, λ x hx y hy a b ha hb hab, (hg' (mem_image_of_mem f $ hf.1 hx hy ha hb hab)
(hg.1 (mem_image_of_mem f hx) (mem_image_of_mem f hy) ha hb hab) $ hf.2 hx hy ha hb hab).trans $
hg.2 (mem_image_of_mem f hx) (mem_image_of_mem f hy) ha hb hab⟩ | lemma | convex_on.comp | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.comp (hg : concave_on 𝕜 (f '' s) g) (hf : concave_on 𝕜 s f)
(hg' : monotone_on g (f '' s)) : concave_on 𝕜 s (g ∘ f) | ⟨hf.1, λ x hx y hy a b ha hb hab,
(hg.2 (mem_image_of_mem f hx) (mem_image_of_mem f hy) ha hb hab).trans $
hg' (hg.1 (mem_image_of_mem f hx) (mem_image_of_mem f hy) ha hb hab)
(mem_image_of_mem f $ hf.1 hx hy ha hb hab) $ hf.2 hx hy ha hb hab⟩ | lemma | concave_on.comp | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.comp_concave_on (hg : convex_on 𝕜 (f '' s) g) (hf : concave_on 𝕜 s f)
(hg' : antitone_on g (f '' s)) : convex_on 𝕜 s (g ∘ f) | hg.dual.comp hf hg' | lemma | convex_on.comp_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"antitone_on",
"concave_on",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.comp_convex_on (hg : concave_on 𝕜 (f '' s) g) (hf : convex_on 𝕜 s f)
(hg' : antitone_on g (f '' s)) : concave_on 𝕜 s (g ∘ f) | hg.dual.comp hf hg' | lemma | concave_on.comp_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"antitone_on",
"concave_on",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.comp (hg : strict_convex_on 𝕜 (f '' s) g) (hf : strict_convex_on 𝕜 s f)
(hg' : strict_mono_on g (f '' s)) (hf' : s.inj_on f) : strict_convex_on 𝕜 s (g ∘ f) | ⟨hf.1, λ x hx y hy hxy a b ha hb hab, (hg' (mem_image_of_mem f $ hf.1 hx hy ha.le hb.le hab)
(hg.1 (mem_image_of_mem f hx) (mem_image_of_mem f hy) ha.le hb.le hab) $
hf.2 hx hy hxy ha hb hab).trans $
hg.2 (mem_image_of_mem f hx) (mem_image_of_mem f hy) (mt (hf' hx hy) hxy) ha hb hab⟩ | lemma | strict_convex_on.comp | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_convex_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.comp (hg : strict_concave_on 𝕜 (f '' s) g) (hf : strict_concave_on 𝕜 s f)
(hg' : strict_mono_on g (f '' s)) (hf' : s.inj_on f) : strict_concave_on 𝕜 s (g ∘ f) | ⟨hf.1, λ x hx y hy hxy a b ha hb hab,
(hg.2 (mem_image_of_mem f hx) (mem_image_of_mem f hy) (mt (hf' hx hy) hxy) ha hb hab).trans $
hg' (hg.1 (mem_image_of_mem f hx) (mem_image_of_mem f hy) ha.le hb.le hab)
(mem_image_of_mem f $ hf.1 hx hy ha.le hb.le hab) $ hf.2 hx hy hxy ha hb hab⟩ | lemma | strict_concave_on.comp | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.comp_strict_concave_on (hg : strict_convex_on 𝕜 (f '' s) g)
(hf : strict_concave_on 𝕜 s f) (hg' : strict_anti_on g (f '' s)) (hf' : s.inj_on f) :
strict_convex_on 𝕜 s (g ∘ f) | hg.dual.comp hf hg' hf' | lemma | strict_convex_on.comp_strict_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_anti_on",
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.comp_strict_convex_on (hg : strict_concave_on 𝕜 (f '' s) g)
(hf : strict_convex_on 𝕜 s f) (hg' : strict_anti_on g (f '' s)) (hf' : s.inj_on f) :
strict_concave_on 𝕜 s (g ∘ f) | hg.dual.comp hf hg' hf' | lemma | strict_concave_on.comp_strict_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_anti_on",
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.add (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
convex_on 𝕜 s (f + g) | ⟨hf.1, λ x hx y hy a b ha hb hab,
calc
f (a • x + b • y) + g (a • x + b • y) ≤ (a • f x + b • f y) + (a • g x + b • g y)
: add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)
... = a • (f x + g x) + b • (f y + g y) : by rw [smul_add, smul_add, add_add_add_comm]⟩ | lemma | convex_on.add | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.add (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
concave_on 𝕜 s (f + g) | hf.dual.add hg | lemma | concave_on.add | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_const (c : β) (hs : convex 𝕜 s) : convex_on 𝕜 s (λ x:E, c) | ⟨hs, λ x y _ _ a b _ _ hab, (convex.combo_self hab c).ge⟩ | lemma | convex_on_const | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex.combo_self",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on_const (c : β) (hs : convex 𝕜 s) : concave_on 𝕜 s (λ x:E, c) | @convex_on_const _ _ βᵒᵈ _ _ _ _ _ _ c hs | lemma | concave_on_const | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"convex_on_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_of_convex_epigraph (h : convex 𝕜 {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2}) :
convex_on 𝕜 s f | ⟨λ x hx y hy a b ha hb hab, (@h (x, f x) ⟨hx, le_rfl⟩ (y, f y) ⟨hy, le_rfl⟩ a b ha hb hab).1,
λ x hx y hy a b ha hb hab, (@h (x, f x) ⟨hx, le_rfl⟩ (y, f y) ⟨hy, le_rfl⟩ a b ha hb hab).2⟩ | lemma | convex_on_of_convex_epigraph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on_of_convex_hypograph (h : convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1}) :
concave_on 𝕜 s f | @convex_on_of_convex_epigraph 𝕜 E βᵒᵈ _ _ _ _ _ _ _ h | lemma | concave_on_of_convex_hypograph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"convex_on_of_convex_epigraph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.convex_le (hf : convex_on 𝕜 s f) (r : β) :
convex 𝕜 {x ∈ s | f x ≤ r} | λ x hx y hy a b ha hb hab, ⟨hf.1 hx.1 hy.1 ha hb hab,
calc
f (a • x + b • y) ≤ a • f x + b • f y : hf.2 hx.1 hy.1 ha hb hab
... ≤ a • r + b • r : add_le_add (smul_le_smul_of_nonneg hx.2 ha)
(smul_le_smul_of_nonneg hy.2 hb)
... =... | lemma | convex_on.convex_le | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex.combo_self",
"convex_on",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.convex_ge (hf : concave_on 𝕜 s f) (r : β) :
convex 𝕜 {x ∈ s | r ≤ f x} | hf.dual.convex_le r | lemma | concave_on.convex_ge | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.convex_epigraph (hf : convex_on 𝕜 s f) :
convex 𝕜 {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2} | begin
rintro ⟨x, r⟩ ⟨hx, hr⟩ ⟨y, t⟩ ⟨hy, ht⟩ a b ha hb hab,
refine ⟨hf.1 hx hy ha hb hab, _⟩,
calc f (a • x + b • y) ≤ a • f x + b • f y : hf.2 hx hy ha hb hab
... ≤ a • r + b • t : add_le_add (smul_le_smul_of_nonneg hr ha)
(smul_le_smul_of_nonneg ht hb)
end | lemma | convex_on.convex_epigraph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.convex_hypograph (hf : concave_on 𝕜 s f) :
convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1} | hf.dual.convex_epigraph | lemma | concave_on.convex_hypograph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_iff_convex_epigraph :
convex_on 𝕜 s f ↔ convex 𝕜 {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2} | ⟨convex_on.convex_epigraph, convex_on_of_convex_epigraph⟩ | lemma | convex_on_iff_convex_epigraph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on_iff_convex_hypograph :
concave_on 𝕜 s f ↔ convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1} | @convex_on_iff_convex_epigraph 𝕜 E βᵒᵈ _ _ _ _ _ _ _ f | lemma | concave_on_iff_convex_hypograph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"convex_on_iff_convex_epigraph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.translate_right (hf : convex_on 𝕜 s f) (c : E) :
convex_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, c + z)) | ⟨hf.1.translate_preimage_right _, λ x hx y hy a b ha hb hab,
calc
f (c + (a • x + b • y)) = f (a • (c + x) + b • (c + y))
: by rw [smul_add, smul_add, add_add_add_comm, convex.combo_self hab]
... ≤ a • f (c + x) + b • f (c + y) : hf.2 hx hy ha hb hab⟩ | lemma | convex_on.translate_right | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_self",
"convex_on",
"smul_add"
] | Right translation preserves convexity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.translate_right (hf : concave_on 𝕜 s f) (c : E) :
concave_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, c + z)) | hf.dual.translate_right _ | lemma | concave_on.translate_right | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | Right translation preserves concavity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.translate_left (hf : convex_on 𝕜 s f) (c : E) :
convex_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, z + c)) | by simpa only [add_comm] using hf.translate_right _ | lemma | convex_on.translate_left | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on"
] | Left translation preserves convexity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.translate_left (hf : concave_on 𝕜 s f) (c : E) :
concave_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, z + c)) | hf.dual.translate_left _ | lemma | concave_on.translate_left | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | Left translation preserves concavity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_iff_forall_pos {s : set E} {f : E → β} :
convex_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1
→ f (a • x + b • y) ≤ a • f x + b • f y | begin
refine and_congr_right' ⟨λ h x hx y hy a b ha hb hab, h hx hy ha.le hb.le hab,
λ h x hx y hy a b ha hb hab, _⟩,
obtain rfl | ha' := ha.eq_or_lt,
{ rw [zero_add] at hab, subst b, simp_rw [zero_smul, zero_add, one_smul] },
obtain rfl | hb' := hb.eq_or_lt,
{ rw [add_zero] at hab, subst a, simp_rw [zero... | lemma | convex_on_iff_forall_pos | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"and_congr_right'",
"convex",
"convex_on",
"one_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on_iff_forall_pos {s : set E} {f : E → β} :
concave_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1
→ a • f x + b • f y ≤ f (a • x + b • y) | @convex_on_iff_forall_pos 𝕜 E βᵒᵈ _ _ _ _ _ _ _ | lemma | concave_on_iff_forall_pos | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"convex_on_iff_forall_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_iff_pairwise_pos {s : set E} {f : E → β} :
convex_on 𝕜 s f ↔ convex 𝕜 s ∧
s.pairwise (λ x y, ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1
→ f (a • x + b • y) ≤ a • f x + b • f y) | begin
rw convex_on_iff_forall_pos,
refine and_congr_right' ⟨λ h x hx y hy _ a b ha hb hab, h hx hy ha hb hab,
λ h x hx y hy a b ha hb hab, _⟩,
obtain rfl | hxy := eq_or_ne x y,
{ rw [convex.combo_self hab, convex.combo_self hab] },
exact h hx hy hxy ha hb hab,
end | lemma | convex_on_iff_pairwise_pos | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"and_congr_right'",
"convex",
"convex.combo_self",
"convex_on",
"convex_on_iff_forall_pos",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on_iff_pairwise_pos {s : set E} {f : E → β} :
concave_on 𝕜 s f ↔ convex 𝕜 s ∧
s.pairwise (λ x y, ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1
→ a • f x + b • f y ≤ f (a • x + b • y)) | @convex_on_iff_pairwise_pos 𝕜 E βᵒᵈ _ _ _ _ _ _ _ | lemma | concave_on_iff_pairwise_pos | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"convex_on_iff_pairwise_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.convex_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : convex_on 𝕜 s f | ⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩ | lemma | linear_map.convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | A linear map is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.concave_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : concave_on 𝕜 s f | ⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩ | lemma | linear_map.concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | A linear map is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.convex_on {s : set E} {f : E → β} (hf : strict_convex_on 𝕜 s f) :
convex_on 𝕜 s f | convex_on_iff_pairwise_pos.mpr ⟨hf.1, λ x hx y hy hxy a b ha hb hab, (hf.2 hx hy hxy ha hb hab).le⟩ | lemma | strict_convex_on.convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.concave_on {s : set E} {f : E → β} (hf : strict_concave_on 𝕜 s f) :
concave_on 𝕜 s f | hf.dual.convex_on | lemma | strict_concave_on.concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.convex_lt (hf : strict_convex_on 𝕜 s f) (r : β) :
convex 𝕜 {x ∈ s | f x < r} | convex_iff_pairwise_pos.2 $ λ x hx y hy hxy a b ha hb hab, ⟨hf.1 hx.1 hy.1 ha.le hb.le hab,
calc
f (a • x + b • y) < a • f x + b • f y : hf.2 hx.1 hy.1 hxy ha hb hab
... ≤ a • r + b • r : add_le_add (smul_lt_smul_of_pos hx.2 ha).le
(smul_lt_smul_... | lemma | strict_convex_on.convex_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex.combo_self",
"smul_lt_smul_of_pos",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.convex_gt (hf : strict_concave_on 𝕜 s f) (r : β) :
convex 𝕜 {x ∈ s | r < f x} | hf.dual.convex_lt r | lemma | strict_concave_on.convex_gt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_order.convex_on_of_lt (hs : convex 𝕜 s)
(hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
f (a • x + b • y) ≤ a • f x + b • f y) : convex_on 𝕜 s f | begin
refine convex_on_iff_pairwise_pos.2 ⟨hs, λ x hx y hy hxy a b ha hb hab, _⟩,
wlog h : x < y,
{ rw [add_comm (a • x), add_comm (a • f x)], rw add_comm at hab,
refine this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_lt.resolve_left h), },
exact hf hx hy h ha hb hab,
end | lemma | linear_order.convex_on_of_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | For a function on a convex set in a linearly ordered space (where the order and the algebraic
structures aren't necessarily compatible), in order to prove that it is convex, it suffices to
verify the inequality `f (a • x + b • y) ≤ a • f x + b • f y` only for `x < y` and positive `a`,
`b`. The main use case is `E = 𝕜`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_order.concave_on_of_lt (hs : convex 𝕜 s)
(hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
a • f x + b • f y ≤ f (a • x + b • y)) : concave_on 𝕜 s f | @linear_order.convex_on_of_lt _ _ βᵒᵈ _ _ _ _ _ _ s f hs hf | lemma | linear_order.concave_on_of_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex",
"linear_order.convex_on_of_lt"
] | For a function on a convex set in a linearly ordered space (where the order and the algebraic
structures aren't necessarily compatible), in order to prove that it is concave it suffices to
verify the inequality `a • f x + b • f y ≤ f (a • x + b • y)` for `x < y` and positive `a`, `b`. The
main use case is `E = ℝ` howev... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_order.strict_convex_on_of_lt (hs : convex 𝕜 s)
(hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
f (a • x + b • y) < a • f x + b • f y) : strict_convex_on 𝕜 s f | begin
refine ⟨hs, λ x hx y hy hxy a b ha hb hab, _⟩,
wlog h : x < y,
{ rw [add_comm (a • x), add_comm (a • f x)], rw add_comm at hab,
refine this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_lt.resolve_left h), },
exact hf hx hy h ha hb hab,
end | lemma | linear_order.strict_convex_on_of_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"strict_convex_on"
] | For a function on a convex set in a linearly ordered space (where the order and the algebraic
structures aren't necessarily compatible), in order to prove that it is strictly convex, it suffices
to verify the inequality `f (a • x + b • y) < a • f x + b • f y` for `x < y` and positive `a`, `b`.
The main use case is `E =... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_order.strict_concave_on_of_lt (hs : convex 𝕜 s)
(hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
a • f x + b • f y < f (a • x + b • y)) : strict_concave_on 𝕜 s f | @linear_order.strict_convex_on_of_lt _ _ βᵒᵈ _ _ _ _ _ _ _ _ hs hf | lemma | linear_order.strict_concave_on_of_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"linear_order.strict_convex_on_of_lt",
"strict_concave_on"
] | For a function on a convex set in a linearly ordered space (where the order and the algebraic
structures aren't necessarily compatible), in order to prove that it is strictly concave it suffices
to verify the inequality `a • f x + b • f y < f (a • x + b • y)` for `x < y` and positive `a`, `b`.
The main use case is `E =... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.comp_linear_map {f : F → β} {s : set F} (hf : convex_on 𝕜 s f) (g : E →ₗ[𝕜] F) :
convex_on 𝕜 (g ⁻¹' s) (f ∘ g) | ⟨hf.1.linear_preimage _, λ x hx y hy a b ha hb hab,
calc
f (g (a • x + b • y)) = f (a • (g x) + b • (g y)) : by rw [g.map_add, g.map_smul, g.map_smul]
... ≤ a • f (g x) + b • f (g y) : hf.2 hx hy ha hb hab⟩ | lemma | convex_on.comp_linear_map | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on"
] | If `g` is convex on `s`, so is `(f ∘ g)` on `f ⁻¹' s` for a linear `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.comp_linear_map {f : F → β} {s : set F} (hf : concave_on 𝕜 s f) (g : E →ₗ[𝕜] F) :
concave_on 𝕜 (g ⁻¹' s) (f ∘ g) | hf.dual.comp_linear_map g | lemma | concave_on.comp_linear_map | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | If `g` is concave on `s`, so is `(g ∘ f)` on `f ⁻¹' s` for a linear `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.add_convex_on (hf : strict_convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
strict_convex_on 𝕜 s (f + g) | ⟨hf.1, λ x hx y hy hxy a b ha hb hab,
calc
f (a • x + b • y) + g (a • x + b • y) < (a • f x + b • f y) + (a • g x + b • g y)
: add_lt_add_of_lt_of_le (hf.2 hx hy hxy ha hb hab) (hg.2 hx hy ha.le hb.le hab)
... = a • (f x + g x) + b • (f y + g y) : by rw [smul_add, smul_add, add_add_add_comm]⟩ | lemma | strict_convex_on.add_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"smul_add",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.add_strict_convex_on (hf : convex_on 𝕜 s f) (hg : strict_convex_on 𝕜 s g) :
strict_convex_on 𝕜 s (f + g) | (add_comm g f) ▸ hg.add_convex_on hf | lemma | convex_on.add_strict_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.add (hf : strict_convex_on 𝕜 s f) (hg : strict_convex_on 𝕜 s g) :
strict_convex_on 𝕜 s (f + g) | ⟨hf.1, λ x hx y hy hxy a b ha hb hab,
calc
f (a • x + b • y) + g (a • x + b • y) < (a • f x + b • f y) + (a • g x + b • g y)
: add_lt_add (hf.2 hx hy hxy ha hb hab) (hg.2 hx hy hxy ha hb hab)
... = a • (f x + g x) + b • (f y + g y) : by rw [smul_add, smul_add, add_add_add_comm]⟩ | lemma | strict_convex_on.add | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"smul_add",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.add_concave_on (hf : strict_concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
strict_concave_on 𝕜 s (f + g) | hf.dual.add_convex_on hg.dual | lemma | strict_concave_on.add_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.add_strict_concave_on (hf : concave_on 𝕜 s f) (hg : strict_concave_on 𝕜 s g) :
strict_concave_on 𝕜 s (f + g) | hf.dual.add_strict_convex_on hg.dual | lemma | concave_on.add_strict_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.add (hf : strict_concave_on 𝕜 s f) (hg : strict_concave_on 𝕜 s g) :
strict_concave_on 𝕜 s (f + g) | hf.dual.add hg | lemma | strict_concave_on.add | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.convex_lt (hf : convex_on 𝕜 s f) (r : β) : convex 𝕜 {x ∈ s | f x < r} | convex_iff_forall_pos.2 $ λ x hx y hy a b ha hb hab, ⟨hf.1 hx.1 hy.1 ha.le hb.le hab,
calc
f (a • x + b • y)
≤ a • f x + b • f y : hf.2 hx.1 hy.1 ha.le hb.le hab
... < a • r + b • r : add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hx.2 ha)
(smul_le_smul_of_nonneg hy.2.... | lemma | convex_on.convex_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex.combo_self",
"convex_on",
"smul_le_smul_of_nonneg",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.convex_gt (hf : concave_on 𝕜 s f) (r : β) : convex 𝕜 {x ∈ s | r < f x} | hf.dual.convex_lt r | lemma | concave_on.convex_gt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.open_segment_subset_strict_epigraph (hf : convex_on 𝕜 s f) (p q : E × β)
(hp : p.1 ∈ s ∧ f p.1 < p.2) (hq : q.1 ∈ s ∧ f q.1 ≤ q.2) :
open_segment 𝕜 p q ⊆ {p : E × β | p.1 ∈ s ∧ f p.1 < p.2} | begin
rintro _ ⟨a, b, ha, hb, hab, rfl⟩,
refine ⟨hf.1 hp.1 hq.1 ha.le hb.le hab, _⟩,
calc f (a • p.1 + b • q.1) ≤ a • f p.1 + b • f q.1 : hf.2 hp.1 hq.1 ha.le hb.le hab
... < a • p.2 + b • q.2 :
add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hp.2 ha) (smul_le_smul_of_nonneg hq.2 hb.le)
end | lemma | convex_on.open_segment_subset_strict_epigraph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"open_segment",
"smul_le_smul_of_nonneg",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.open_segment_subset_strict_hypograph (hf : concave_on 𝕜 s f) (p q : E × β)
(hp : p.1 ∈ s ∧ p.2 < f p.1) (hq : q.1 ∈ s ∧ q.2 ≤ f q.1) :
open_segment 𝕜 p q ⊆ {p : E × β | p.1 ∈ s ∧ p.2 < f p.1} | hf.dual.open_segment_subset_strict_epigraph p q hp hq | lemma | concave_on.open_segment_subset_strict_hypograph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.convex_strict_epigraph (hf : convex_on 𝕜 s f) :
convex 𝕜 {p : E × β | p.1 ∈ s ∧ f p.1 < p.2} | convex_iff_open_segment_subset.mpr $
λ p hp q hq, hf.open_segment_subset_strict_epigraph p q hp ⟨hq.1, hq.2.le⟩ | lemma | convex_on.convex_strict_epigraph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.convex_strict_hypograph (hf : concave_on 𝕜 s f) :
convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 < f p.1} | hf.dual.convex_strict_epigraph | lemma | concave_on.convex_strict_hypograph | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.sup (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
convex_on 𝕜 s (f ⊔ g) | begin
refine ⟨hf.left, λ x hx y hy a b ha hb hab, sup_le _ _⟩,
{ calc f (a • x + b • y) ≤ a • f x + b • f y : hf.right hx hy ha hb hab
... ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y) : add_le_add
(smul_le_smul_of_nonneg le_sup_left ha)
(smul_le_smul_of_nonneg le_sup_left hb) },
{ calc g (... | lemma | convex_on.sup | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"le_sup_left",
"le_sup_right",
"smul_le_smul_of_nonneg",
"sup_le"
] | The pointwise maximum of convex functions is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.inf (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
concave_on 𝕜 s (f ⊓ g) | hf.dual.sup hg | lemma | concave_on.inf | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | The pointwise minimum of concave functions is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.sup (hf : strict_convex_on 𝕜 s f) (hg : strict_convex_on 𝕜 s g) :
strict_convex_on 𝕜 s (f ⊔ g) | ⟨hf.left, λ x hx y hy hxy a b ha hb hab, max_lt
(calc f (a • x + b • y) < a • f x + b • f y : hf.2 hx hy hxy ha hb hab
... ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y) : add_le_add
(smul_le_smul_of_nonneg le_sup_left ha.le)
(smul_le_smul_of_nonneg le_sup_left hb.le))
(calc g (a • x + b • y) < a... | lemma | strict_convex_on.sup | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"le_sup_left",
"le_sup_right",
"smul_le_smul_of_nonneg",
"strict_convex_on"
] | The pointwise maximum of strictly convex functions is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on.inf (hf : strict_concave_on 𝕜 s f) (hg : strict_concave_on 𝕜 s g) :
strict_concave_on 𝕜 s (f ⊓ g) | hf.dual.sup hg | lemma | strict_concave_on.inf | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on"
] | The pointwise minimum of strictly concave functions is strictly concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.le_on_segment' (hf : convex_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
{a b : 𝕜} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
f (a • x + b • y) ≤ max (f x) (f y) | calc
f (a • x + b • y) ≤ a • f x + b • f y : hf.2 hx hy ha hb hab
... ≤ a • max (f x) (f y) + b • max (f x) (f y) :
add_le_add (smul_le_smul_of_nonneg (le_max_left _ _) ha)
(smul_le_smul_of_nonneg (le_max_right _ _) hb)
... = max (f x) (f y) : convex.combo_self hab _ | lemma | convex_on.le_on_segment' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_self",
"convex_on",
"smul_le_smul_of_nonneg"
] | A convex function on a segment is upper-bounded by the max of its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.ge_on_segment' (hf : concave_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
{a b : 𝕜} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
min (f x) (f y) ≤ f (a • x + b • y) | hf.dual.le_on_segment' hx hy ha hb hab | lemma | concave_on.ge_on_segment' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | A concave function on a segment is lower-bounded by the min of its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.le_on_segment (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s) (hy : y ∈ s)
(hz : z ∈ [x -[𝕜] y]) :
f z ≤ max (f x) (f y) | let ⟨a, b, ha, hb, hab, hz⟩ := hz in hz ▸ hf.le_on_segment' hx hy ha hb hab | lemma | convex_on.le_on_segment | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on"
] | A convex function on a segment is upper-bounded by the max of its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.ge_on_segment (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s) (hy : y ∈ s)
(hz : z ∈ [x -[𝕜] y]) :
min (f x) (f y) ≤ f z | hf.dual.le_on_segment hx hy hz | lemma | concave_on.ge_on_segment | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | A concave function on a segment is lower-bounded by the min of its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.lt_on_open_segment' (hf : strict_convex_on 𝕜 s f) {x y : E} (hx : x ∈ s)
(hy : y ∈ s) (hxy : x ≠ y) {a b : 𝕜} (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
f (a • x + b • y) < max (f x) (f y) | calc
f (a • x + b • y) < a • f x + b • f y : hf.2 hx hy hxy ha hb hab
... ≤ a • max (f x) (f y) + b • max (f x) (f y) :
add_le_add (smul_le_smul_of_nonneg (le_max_left _ _) ha.le)
(smul_le_smul_of_nonneg (le_max_right _ _) hb.le)
... = max (f x) (f y) : convex.combo_self hab _ | lemma | strict_convex_on.lt_on_open_segment' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_self",
"smul_le_smul_of_nonneg",
"strict_convex_on"
] | A strictly convex function on an open segment is strictly upper-bounded by the max of its
endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on.lt_on_open_segment' (hf : strict_concave_on 𝕜 s f) {x y : E} (hx : x ∈ s)
(hy : y ∈ s) (hxy : x ≠ y) {a b : 𝕜} (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
min (f x) (f y) < f (a • x + b • y) | hf.dual.lt_on_open_segment' hx hy hxy ha hb hab | lemma | strict_concave_on.lt_on_open_segment' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on"
] | A strictly concave function on an open segment is strictly lower-bounded by the min of its
endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.lt_on_open_segment (hf : strict_convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hxy : x ≠ y) (hz : z ∈ open_segment 𝕜 x y) :
f z < max (f x) (f y) | let ⟨a, b, ha, hb, hab, hz⟩ := hz in hz ▸ hf.lt_on_open_segment' hx hy hxy ha hb hab | lemma | strict_convex_on.lt_on_open_segment | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"open_segment",
"strict_convex_on"
] | A strictly convex function on an open segment is strictly upper-bounded by the max of its
endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on.lt_on_open_segment (hf : strict_concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hxy : x ≠ y) (hz : z ∈ open_segment 𝕜 x y) :
min (f x) (f y) < f z | hf.dual.lt_on_open_segment hx hy hxy hz | lemma | strict_concave_on.lt_on_open_segment | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"open_segment",
"strict_concave_on"
] | A strictly concave function on an open segment is strictly lower-bounded by the min of its
endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.le_left_of_right_le' (hf : convex_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
{a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) (hfy : f y ≤ f (a • x + b • y)) :
f (a • x + b • y) ≤ f x | le_of_not_lt $ λ h, lt_irrefl (f (a • x + b • y)) $
calc
f (a • x + b • y)
≤ a • f x + b • f y : hf.2 hx hy ha.le hb hab
... < a • f (a • x + b • y) + b • f (a • x + b • y)
: add_lt_add_of_lt_of_le (smul_lt_smul_of_pos h ha) (smul_le_smul_of_nonneg hfy hb)
... = f (a • x + b • y) : convex.... | lemma | convex_on.le_left_of_right_le' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_self",
"convex_on",
"smul_le_smul_of_nonneg",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.left_le_of_le_right' (hf : concave_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
{a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) (hfy : f (a • x + b • y) ≤ f y) :
f x ≤ f (a • x + b • y) | hf.dual.le_left_of_right_le' hx hy ha hb hab hfy | lemma | concave_on.left_le_of_le_right' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.le_right_of_left_le' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
(hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
(hfx : f x ≤ f (a • x + b • y)) :
f (a • x + b • y) ≤ f y | begin
rw add_comm at ⊢ hab hfx,
exact hf.le_left_of_right_le' hy hx hb ha hab hfx,
end | lemma | convex_on.le_right_of_left_le' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.right_le_of_le_left' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
(hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
(hfx : f (a • x + b • y) ≤ f x) :
f y ≤ f (a • x + b • y) | hf.dual.le_right_of_left_le' hx hy ha hb hab hfx | lemma | concave_on.right_le_of_le_left' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.le_left_of_right_le (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f y ≤ f z) :
f z ≤ f x | begin
obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz,
exact hf.le_left_of_right_le' hx hy ha hb.le hab hyz,
end | lemma | convex_on.le_left_of_right_le | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.left_le_of_le_right (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f z ≤ f y) :
f x ≤ f z | hf.dual.le_left_of_right_le hx hy hz hyz | lemma | concave_on.left_le_of_le_right | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.le_right_of_left_le (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f x ≤ f z) :
f z ≤ f y | begin
obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz,
exact hf.le_right_of_left_le' hx hy ha.le hb hab hxz,
end | lemma | convex_on.le_right_of_left_le | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.right_le_of_le_left (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f z ≤ f x) :
f y ≤ f z | hf.dual.le_right_of_left_le hx hy hz hxz | lemma | concave_on.right_le_of_le_left | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.lt_left_of_right_lt' (hf : convex_on 𝕜 s f) {x y : E} (hx : x ∈ s)
(hy : y ∈ s) {a b : 𝕜} (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
(hfy : f y < f (a • x + b • y)) :
f (a • x + b • y) < f x | not_le.1 $ λ h, lt_irrefl (f (a • x + b • y)) $
calc
f (a • x + b • y)
≤ a • f x + b • f y : hf.2 hx hy ha.le hb.le hab
... < a • f (a • x + b • y) + b • f (a • x + b • y)
: add_lt_add_of_le_of_lt (smul_le_smul_of_nonneg h ha.le) (smul_lt_smul_of_pos hfy hb)
... = f (a • x + b • y) : conve... | lemma | convex_on.lt_left_of_right_lt' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_self",
"convex_on",
"smul_le_smul_of_nonneg",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.left_lt_of_lt_right' (hf : concave_on 𝕜 s f) {x y : E} (hx : x ∈ s)
(hy : y ∈ s) {a b : 𝕜} (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
(hfy : f (a • x + b • y) < f y) :
f x < f (a • x + b • y) | hf.dual.lt_left_of_right_lt' hx hy ha hb hab hfy | lemma | concave_on.left_lt_of_lt_right' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.lt_right_of_left_lt' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
(hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
(hfx : f x < f (a • x + b • y)) :
f (a • x + b • y) < f y | begin
rw add_comm at ⊢ hab hfx,
exact hf.lt_left_of_right_lt' hy hx hb ha hab hfx,
end | lemma | convex_on.lt_right_of_left_lt' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.lt_right_of_left_lt' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
(hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
(hfx : f (a • x + b • y) < f x) :
f y < f (a • x + b • y) | hf.dual.lt_right_of_left_lt' hx hy ha hb hab hfx | lemma | concave_on.lt_right_of_left_lt' | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.lt_left_of_right_lt (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f y < f z) :
f z < f x | begin
obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz,
exact hf.lt_left_of_right_lt' hx hy ha hb hab hyz,
end | lemma | convex_on.lt_left_of_right_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.left_lt_of_lt_right (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f z < f y) :
f x < f z | hf.dual.lt_left_of_right_lt hx hy hz hyz | lemma | concave_on.left_lt_of_lt_right | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.lt_right_of_left_lt (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f x < f z) :
f z < f y | begin
obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz,
exact hf.lt_right_of_left_lt' hx hy ha hb hab hxz,
end | lemma | convex_on.lt_right_of_left_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.lt_right_of_left_lt (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
(hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f z < f x) :
f y < f z | hf.dual.lt_right_of_left_lt hx hy hz hxz | lemma | concave_on.lt_right_of_left_lt | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"open_segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_convex_on_iff : convex_on 𝕜 s (-f) ↔ concave_on 𝕜 s f | begin
split,
{ rintro ⟨hconv, h⟩,
refine ⟨hconv, λ x hx y hy a b ha hb hab, _⟩,
simp [neg_apply, neg_le, add_comm] at h,
exact h hx hy ha hb hab },
{ rintro ⟨hconv, h⟩,
refine ⟨hconv, λ x hx y hy a b ha hb hab, _⟩,
rw ←neg_le_neg_iff,
simp_rw [neg_add, pi.neg_apply, smul_neg, neg_neg],
... | lemma | neg_convex_on_iff | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex_on",
"smul_neg"
] | A function `-f` is convex iff `f` is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_concave_on_iff : concave_on 𝕜 s (-f) ↔ convex_on 𝕜 s f | by rw [← neg_convex_on_iff, neg_neg f] | lemma | neg_concave_on_iff | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex_on",
"neg_convex_on_iff"
] | A function `-f` is concave iff `f` is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_strict_convex_on_iff : strict_convex_on 𝕜 s (-f) ↔ strict_concave_on 𝕜 s f | begin
split,
{ rintro ⟨hconv, h⟩,
refine ⟨hconv, λ x hx y hy hxy a b ha hb hab, _⟩,
simp [neg_apply, neg_lt, add_comm] at h,
exact h hx hy hxy ha hb hab },
{ rintro ⟨hconv, h⟩,
refine ⟨hconv, λ x hx y hy hxy a b ha hb hab, _⟩,
rw ←neg_lt_neg_iff,
simp_rw [neg_add, pi.neg_apply, smul_neg, n... | lemma | neg_strict_convex_on_iff | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"smul_neg",
"strict_concave_on",
"strict_convex_on"
] | A function `-f` is strictly convex iff `f` is strictly concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_strict_concave_on_iff : strict_concave_on 𝕜 s (-f) ↔ strict_convex_on 𝕜 s f | by rw [← neg_strict_convex_on_iff, neg_neg f] | lemma | neg_strict_concave_on_iff | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"neg_strict_convex_on_iff",
"strict_concave_on",
"strict_convex_on"
] | A function `-f` is strictly concave iff `f` is strictly convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.sub (hf : convex_on 𝕜 s f) (hg : concave_on 𝕜 s g) : convex_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add hg.neg | lemma | convex_on.sub | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.sub (hf : concave_on 𝕜 s f) (hg : convex_on 𝕜 s g) : concave_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add hg.neg | lemma | concave_on.sub | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.sub (hf : strict_convex_on 𝕜 s f) (hg : strict_concave_on 𝕜 s g) :
strict_convex_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add hg.neg | lemma | strict_convex_on.sub | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.sub (hf : strict_concave_on 𝕜 s f) (hg : strict_convex_on 𝕜 s g) :
strict_concave_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add hg.neg | lemma | strict_concave_on.sub | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.sub_strict_concave_on (hf : convex_on 𝕜 s f) (hg : strict_concave_on 𝕜 s g) :
strict_convex_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add_strict_convex_on hg.neg | lemma | convex_on.sub_strict_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.sub_strict_convex_on (hf : concave_on 𝕜 s f) (hg : strict_convex_on 𝕜 s g) :
strict_concave_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add_strict_concave_on hg.neg | lemma | concave_on.sub_strict_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"strict_concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.sub_concave_on (hf : strict_convex_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
strict_convex_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add_convex_on hg.neg | lemma | strict_convex_on.sub_concave_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"concave_on",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.sub_convex_on (hf : strict_concave_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
strict_concave_on 𝕜 s (f - g) | (sub_eq_add_neg f g).symm ▸ hf.add_concave_on hg.neg | lemma | strict_concave_on.sub_convex_on | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex_on",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.translate_right (hf : strict_convex_on 𝕜 s f) (c : E) :
strict_convex_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, c + z)) | ⟨hf.1.translate_preimage_right _, λ x hx y hy hxy a b ha hb hab,
calc
f (c + (a • x + b • y)) = f (a • (c + x) + b • (c + y))
: by rw [smul_add, smul_add, add_add_add_comm, convex.combo_self hab]
... < a • f (c + x) + b • f (c + y) : hf.2 hx hy ((add_right_injective c).ne hxy) ha hb hab⟩ | lemma | strict_convex_on.translate_right | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"convex.combo_self",
"smul_add",
"strict_convex_on"
] | Right translation preserves strict convexity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on.translate_right (hf : strict_concave_on 𝕜 s f) (c : E) :
strict_concave_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, c + z)) | hf.dual.translate_right _ | lemma | strict_concave_on.translate_right | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_concave_on"
] | Right translation preserves strict concavity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.translate_left (hf : strict_convex_on 𝕜 s f) (c : E) :
strict_convex_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, z + c)) | by simpa only [add_comm] using hf.translate_right _ | lemma | strict_convex_on.translate_left | analysis.convex | src/analysis/convex/function.lean | [
"analysis.convex.basic"
] | [
"strict_convex_on"
] | Left translation preserves strict convexity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.