statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_open.lower_semicontinuous_on_indicator (hs : is_open s) (hy : 0 ≤ y) :
lower_semicontinuous_on (indicator s (λ x, y)) t | (hs.lower_semicontinuous_indicator hy).lower_semicontinuous_on t | lemma | is_open.lower_semicontinuous_on_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"lower_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.lower_semicontinuous_at_indicator (hs : is_open s) (hy : 0 ≤ y) :
lower_semicontinuous_at (indicator s (λ x, y)) x | (hs.lower_semicontinuous_indicator hy).lower_semicontinuous_at x | lemma | is_open.lower_semicontinuous_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"lower_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.lower_semicontinuous_within_at_indicator (hs : is_open s) (hy : 0 ≤ y) :
lower_semicontinuous_within_at (indicator s (λ x, y)) t x | (hs.lower_semicontinuous_indicator hy).lower_semicontinuous_within_at t x | lemma | is_open.lower_semicontinuous_within_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.lower_semicontinuous_indicator (hs : is_closed s) (hy : y ≤ 0) :
lower_semicontinuous (indicator s (λ x, y)) | begin
assume x z hz,
by_cases h : x ∈ s; simp [h] at hz,
{ apply filter.eventually_of_forall (λ x', _),
by_cases h' : x' ∈ s;
simp [h', hz, hz.trans_le hy], },
{ filter_upwards [hs.is_open_compl.mem_nhds h],
simp [hz] { contextual := tt } }
end | lemma | is_closed.lower_semicontinuous_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually_of_forall",
"is_closed",
"lower_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.lower_semicontinuous_on_indicator (hs : is_closed s) (hy : y ≤ 0) :
lower_semicontinuous_on (indicator s (λ x, y)) t | (hs.lower_semicontinuous_indicator hy).lower_semicontinuous_on t | lemma | is_closed.lower_semicontinuous_on_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"lower_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.lower_semicontinuous_at_indicator (hs : is_closed s) (hy : y ≤ 0) :
lower_semicontinuous_at (indicator s (λ x, y)) x | (hs.lower_semicontinuous_indicator hy).lower_semicontinuous_at x | lemma | is_closed.lower_semicontinuous_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"lower_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.lower_semicontinuous_within_at_indicator (hs : is_closed s) (hy : y ≤ 0) :
lower_semicontinuous_within_at (indicator s (λ x, y)) t x | (hs.lower_semicontinuous_indicator hy).lower_semicontinuous_within_at t x | lemma | is_closed.lower_semicontinuous_within_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_iff_is_open_preimage :
lower_semicontinuous f ↔ ∀ y, is_open (f ⁻¹' (Ioi y)) | ⟨λ H y, is_open_iff_mem_nhds.2 (λ x hx, H x y hx), λ H x y y_lt, is_open.mem_nhds (H y) y_lt⟩ | theorem | lower_semicontinuous_iff_is_open_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"is_open.mem_nhds",
"lower_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous.is_open_preimage (hf : lower_semicontinuous f) (y : β) :
is_open (f ⁻¹' (Ioi y)) | lower_semicontinuous_iff_is_open_preimage.1 hf y | lemma | lower_semicontinuous.is_open_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"lower_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_iff_is_closed_preimage {f : α → γ} :
lower_semicontinuous f ↔ ∀ y, is_closed (f ⁻¹' (Iic y)) | begin
rw lower_semicontinuous_iff_is_open_preimage,
congrm (∀ y, (_ : Prop)),
rw [← is_open_compl_iff, ← preimage_compl, compl_Iic]
end | theorem | lower_semicontinuous_iff_is_closed_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"is_open_compl_iff",
"lower_semicontinuous",
"lower_semicontinuous_iff_is_open_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous.is_closed_preimage {f : α → γ} (hf : lower_semicontinuous f) (y : γ) :
is_closed (f ⁻¹' (Iic y)) | lower_semicontinuous_iff_is_closed_preimage.1 hf y | lemma | lower_semicontinuous.is_closed_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"lower_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.lower_semicontinuous_within_at {f : α → γ}
(h : continuous_within_at f s x) : lower_semicontinuous_within_at f s x | λ y hy, h (Ioi_mem_nhds hy) | lemma | continuous_within_at.lower_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"Ioi_mem_nhds",
"continuous_within_at",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.lower_semicontinuous_at {f : α → γ}
(h : continuous_at f x) : lower_semicontinuous_at f x | λ y hy, h (Ioi_mem_nhds hy) | lemma | continuous_at.lower_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"Ioi_mem_nhds",
"continuous_at",
"lower_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.lower_semicontinuous_on {f : α → γ}
(h : continuous_on f s) : lower_semicontinuous_on f s | λ x hx, (h x hx).lower_semicontinuous_within_at | lemma | continuous_on.lower_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_on",
"lower_semicontinuous_on",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.lower_semicontinuous {f : α → γ}
(h : continuous f) : lower_semicontinuous f | λ x, h.continuous_at.lower_semicontinuous_at | lemma | continuous.lower_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"lower_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_lower_semicontinuous_within_at
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_within_at f s x)
(gmon : monotone g) : lower_semicontinuous_within_at (g ∘ f) s x | begin
assume y hy,
by_cases h : ∃ l, l < f x,
{ obtain ⟨z, zlt, hz⟩ : ∃ z < f x, Ioc z (f x) ⊆ g ⁻¹' (Ioi y) :=
exists_Ioc_subset_of_mem_nhds (hg (Ioi_mem_nhds hy)) h,
filter_upwards [hf z zlt] with a ha,
calc y < g (min (f x) (f a)) : hz (by simp [zlt, ha, le_refl])
... ≤ g (f a) : gmon (min_le... | lemma | continuous_at.comp_lower_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"Ioi_mem_nhds",
"continuous_at",
"exists_Ioc_subset_of_mem_nhds",
"filter.eventually_of_forall",
"lower_semicontinuous_within_at",
"monotone",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_lower_semicontinuous_at
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_at f x)
(gmon : monotone g) : lower_semicontinuous_at (g ∘ f) x | begin
simp only [← lower_semicontinuous_within_at_univ_iff] at hf ⊢,
exact hg.comp_lower_semicontinuous_within_at hf gmon
end | lemma | continuous_at.comp_lower_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"lower_semicontinuous_at",
"lower_semicontinuous_within_at_univ_iff",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_lower_semicontinuous_on
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous_on f s)
(gmon : monotone g) : lower_semicontinuous_on (g ∘ f) s | λ x hx, (hg.continuous_at).comp_lower_semicontinuous_within_at (hf x hx) gmon | lemma | continuous.comp_lower_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"lower_semicontinuous_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_lower_semicontinuous
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous f)
(gmon : monotone g) : lower_semicontinuous (g ∘ f) | λ x, (hg.continuous_at).comp_lower_semicontinuous_at (hf x) gmon | lemma | continuous.comp_lower_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"lower_semicontinuous",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_lower_semicontinuous_within_at_antitone
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_within_at f s x)
(gmon : antitone g) : upper_semicontinuous_within_at (g ∘ f) s x | @continuous_at.comp_lower_semicontinuous_within_at α _ x s γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | lemma | continuous_at.comp_lower_semicontinuous_within_at_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous_at",
"continuous_at.comp_lower_semicontinuous_within_at",
"lower_semicontinuous_within_at",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_lower_semicontinuous_at_antitone
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_at f x)
(gmon : antitone g) : upper_semicontinuous_at (g ∘ f) x | @continuous_at.comp_lower_semicontinuous_at α _ x γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | lemma | continuous_at.comp_lower_semicontinuous_at_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous_at",
"continuous_at.comp_lower_semicontinuous_at",
"lower_semicontinuous_at",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_lower_semicontinuous_on_antitone
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous_on f s)
(gmon : antitone g) : upper_semicontinuous_on (g ∘ f) s | λ x hx, (hg.continuous_at).comp_lower_semicontinuous_within_at_antitone (hf x hx) gmon | lemma | continuous.comp_lower_semicontinuous_on_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous",
"lower_semicontinuous_on",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_lower_semicontinuous_antitone
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous f)
(gmon : antitone g) : upper_semicontinuous (g ∘ f) | λ x, (hg.continuous_at).comp_lower_semicontinuous_at_antitone (hf x) gmon | lemma | continuous.comp_lower_semicontinuous_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous",
"lower_semicontinuous",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at.add' {f g : α → γ}
(hf : lower_semicontinuous_within_at f s x) (hg : lower_semicontinuous_within_at g s x)
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
lower_semicontinuous_within_at (λ z, f z + g z) s x | begin
assume y hy,
obtain ⟨u, v, u_open, xu, v_open, xv, h⟩ : ∃ (u v : set γ), is_open u ∧ f x ∈ u ∧ is_open v ∧
g x ∈ v ∧ u ×ˢ v ⊆ {p : γ × γ | y < p.fst + p.snd} :=
mem_nhds_prod_iff'.1 (hcont (is_open_Ioi.mem_nhds hy)),
by_cases hx₁ : ∃ l, l < f x,
{ obtain ⟨z₁, z₁lt, h₁⟩ : ∃ z₁ < f x, Ioc z₁ (f x) ⊆ u... | lemma | lower_semicontinuous_within_at.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"exists_Ioc_subset_of_mem_nhds",
"filter.eventually_of_forall",
"is_open",
"lower_semicontinuous_within_at",
"not_exists"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_at.add' {f g : α → γ}
(hf : lower_semicontinuous_at f x) (hg : lower_semicontinuous_at g x)
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
lower_semicontinuous_at (λ z, f z + g z) x | by { simp_rw [← lower_semicontinuous_within_at_univ_iff] at *, exact hf.add' hg hcont } | lemma | lower_semicontinuous_at.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"lower_semicontinuous_at",
"lower_semicontinuous_within_at_univ_iff"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_on.add' {f g : α → γ}
(hf : lower_semicontinuous_on f s) (hg : lower_semicontinuous_on g s)
(hcont : ∀ x ∈ s, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
lower_semicontinuous_on (λ z, f z + g z) s | λ x hx, (hf x hx).add' (hg x hx) (hcont x hx) | lemma | lower_semicontinuous_on.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"lower_semicontinuous_on"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous.add' {f g : α → γ}
(hf : lower_semicontinuous f) (hg : lower_semicontinuous g)
(hcont : ∀ x, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
lower_semicontinuous (λ z, f z + g z) | λ x, (hf x).add' (hg x) (hcont x) | lemma | lower_semicontinuous.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"lower_semicontinuous"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_within_at.add {f g : α → γ}
(hf : lower_semicontinuous_within_at f s x) (hg : lower_semicontinuous_within_at g s x) :
lower_semicontinuous_within_at (λ z, f z + g z) s x | hf.add' hg continuous_add.continuous_at | lemma | lower_semicontinuous_within_at.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_within_at"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_at.add {f g : α → γ}
(hf : lower_semicontinuous_at f x) (hg : lower_semicontinuous_at g x) :
lower_semicontinuous_at (λ z, f z + g z) x | hf.add' hg continuous_add.continuous_at | lemma | lower_semicontinuous_at.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_on.add {f g : α → γ}
(hf : lower_semicontinuous_on f s) (hg : lower_semicontinuous_on g s) :
lower_semicontinuous_on (λ z, f z + g z) s | hf.add' hg (λ x hx, continuous_add.continuous_at) | lemma | lower_semicontinuous_on.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous.add {f g : α → γ}
(hf : lower_semicontinuous f) (hg : lower_semicontinuous g) :
lower_semicontinuous (λ z, f z + g z) | hf.add' hg (λ x, continuous_add.continuous_at) | lemma | lower_semicontinuous.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_within_at_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, lower_semicontinuous_within_at (f i) s x) :
lower_semicontinuous_within_at (λ z, (∑ i in a, f i z)) s x | begin
classical,
induction a using finset.induction_on with i a ia IH generalizing ha,
{ exact lower_semicontinuous_within_at_const },
{ simp only [ia, finset.sum_insert, not_false_iff],
exact lower_semicontinuous_within_at.add (ha _ (finset.mem_insert_self i a))
(IH (λ j ja, ha j (finset.mem_insert_o... | lemma | lower_semicontinuous_within_at_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"lower_semicontinuous_within_at",
"lower_semicontinuous_within_at.add",
"lower_semicontinuous_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, lower_semicontinuous_at (f i) x) :
lower_semicontinuous_at (λ z, (∑ i in a, f i z)) x | begin
simp_rw [← lower_semicontinuous_within_at_univ_iff] at *,
exact lower_semicontinuous_within_at_sum ha
end | lemma | lower_semicontinuous_at_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"lower_semicontinuous_at",
"lower_semicontinuous_within_at_sum",
"lower_semicontinuous_within_at_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, lower_semicontinuous_on (f i) s) :
lower_semicontinuous_on (λ z, (∑ i in a, f i z)) s | λ x hx, lower_semicontinuous_within_at_sum (λ i hi, ha i hi x hx) | lemma | lower_semicontinuous_on_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"lower_semicontinuous_on",
"lower_semicontinuous_within_at_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, lower_semicontinuous (f i)) :
lower_semicontinuous (λ z, (∑ i in a, f i z)) | λ x, lower_semicontinuous_at_sum (λ i hi, ha i hi x) | lemma | lower_semicontinuous_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"lower_semicontinuous",
"lower_semicontinuous_at_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at_csupr {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝[s] x, bdd_above (range $ λ i, f i y))
(h : ∀ i, lower_semicontinuous_within_at (f i) s x) :
lower_semicontinuous_within_at (λ x', ⨆ i, f i x') s x | begin
casesI is_empty_or_nonempty ι,
{ simpa only [supr_of_empty'] using lower_semicontinuous_within_at_const },
{ assume y hy,
rcases exists_lt_of_lt_csupr hy with ⟨i, hi⟩,
filter_upwards [h i y hi, bdd] with y hy hy' using hy.trans_le (le_csupr hy' i) }
end | lemma | lower_semicontinuous_within_at_csupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_above",
"exists_lt_of_lt_csupr",
"is_empty_or_nonempty",
"le_csupr",
"lower_semicontinuous_within_at",
"lower_semicontinuous_within_at_const",
"supr_of_empty'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at_supr {f : ι → α → δ}
(h : ∀ i, lower_semicontinuous_within_at (f i) s x) :
lower_semicontinuous_within_at (λ x', ⨆ i, f i x') s x | lower_semicontinuous_within_at_csupr (by simp) h | lemma | lower_semicontinuous_within_at_supr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_within_at",
"lower_semicontinuous_within_at_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, lower_semicontinuous_within_at (f i hi) s x) :
lower_semicontinuous_within_at (λ x', ⨆ i hi, f i hi x') s x | lower_semicontinuous_within_at_supr $ λ i, lower_semicontinuous_within_at_supr $ λ hi, h i hi | lemma | lower_semicontinuous_within_at_bsupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_within_at",
"lower_semicontinuous_within_at_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at_csupr {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝 x, bdd_above (range $ λ i, f i y))
(h : ∀ i, lower_semicontinuous_at (f i) x) :
lower_semicontinuous_at (λ x', ⨆ i, f i x') x | begin
simp_rw [← lower_semicontinuous_within_at_univ_iff] at *,
rw ← nhds_within_univ at bdd,
exact lower_semicontinuous_within_at_csupr bdd h
end | lemma | lower_semicontinuous_at_csupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_above",
"lower_semicontinuous_at",
"lower_semicontinuous_within_at_csupr",
"lower_semicontinuous_within_at_univ_iff",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at_supr {f : ι → α → δ}
(h : ∀ i, lower_semicontinuous_at (f i) x) :
lower_semicontinuous_at (λ x', ⨆ i, f i x') x | lower_semicontinuous_at_csupr (by simp) h | lemma | lower_semicontinuous_at_supr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at",
"lower_semicontinuous_at_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, lower_semicontinuous_at (f i hi) x) :
lower_semicontinuous_at (λ x', ⨆ i hi, f i hi x') x | lower_semicontinuous_at_supr $ λ i, lower_semicontinuous_at_supr $ λ hi, h i hi | lemma | lower_semicontinuous_at_bsupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at",
"lower_semicontinuous_at_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_csupr {f : ι → α → δ'}
(bdd : ∀ x ∈ s, bdd_above (range $ λ i, f i x))
(h : ∀ i, lower_semicontinuous_on (f i) s) :
lower_semicontinuous_on (λ x', ⨆ i, f i x') s | λ x hx, lower_semicontinuous_within_at_csupr (eventually_nhds_within_of_forall bdd)
(λ i, h i x hx) | lemma | lower_semicontinuous_on_csupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_above",
"eventually_nhds_within_of_forall",
"lower_semicontinuous_on",
"lower_semicontinuous_within_at_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_supr {f : ι → α → δ}
(h : ∀ i, lower_semicontinuous_on (f i) s) :
lower_semicontinuous_on (λ x', ⨆ i, f i x') s | lower_semicontinuous_on_csupr (by simp) h | lemma | lower_semicontinuous_on_supr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on",
"lower_semicontinuous_on_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, lower_semicontinuous_on (f i hi) s) :
lower_semicontinuous_on (λ x', ⨆ i hi, f i hi x') s | lower_semicontinuous_on_supr $ λ i, lower_semicontinuous_on_supr $ λ hi, h i hi | lemma | lower_semicontinuous_on_bsupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on",
"lower_semicontinuous_on_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_csupr {f : ι → α → δ'}
(bdd : ∀ x, bdd_above (range $ λ i, f i x))
(h : ∀ i, lower_semicontinuous (f i)) :
lower_semicontinuous (λ x', ⨆ i, f i x') | λ x, lower_semicontinuous_at_csupr (eventually_of_forall bdd) (λ i, h i x) | lemma | lower_semicontinuous_csupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_above",
"lower_semicontinuous",
"lower_semicontinuous_at_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_supr {f : ι → α → δ}
(h : ∀ i, lower_semicontinuous (f i)) :
lower_semicontinuous (λ x', ⨆ i, f i x') | lower_semicontinuous_csupr (by simp) h | lemma | lower_semicontinuous_supr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, lower_semicontinuous (f i hi)) :
lower_semicontinuous (λ x', ⨆ i hi, f i hi x') | lower_semicontinuous_supr $ λ i, lower_semicontinuous_supr $ λ hi, h i hi | lemma | lower_semicontinuous_bsupr | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at_tsum {f : ι → α → ℝ≥0∞}
(h : ∀ i, lower_semicontinuous_within_at (f i) s x) :
lower_semicontinuous_within_at (λ x', ∑' i, f i x') s x | begin
simp_rw ennreal.tsum_eq_supr_sum,
apply lower_semicontinuous_within_at_supr (λ b, _),
exact lower_semicontinuous_within_at_sum (λ i hi, h i),
end | lemma | lower_semicontinuous_within_at_tsum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"ennreal.tsum_eq_supr_sum",
"lower_semicontinuous_within_at",
"lower_semicontinuous_within_at_sum",
"lower_semicontinuous_within_at_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at_tsum {f : ι → α → ℝ≥0∞}
(h : ∀ i, lower_semicontinuous_at (f i) x) :
lower_semicontinuous_at (λ x', ∑' i, f i x') x | begin
simp_rw [← lower_semicontinuous_within_at_univ_iff] at *,
exact lower_semicontinuous_within_at_tsum h
end | lemma | lower_semicontinuous_at_tsum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at",
"lower_semicontinuous_within_at_tsum",
"lower_semicontinuous_within_at_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_tsum {f : ι → α → ℝ≥0∞}
(h : ∀ i, lower_semicontinuous_on (f i) s) :
lower_semicontinuous_on (λ x', ∑' i, f i x') s | λ x hx, lower_semicontinuous_within_at_tsum (λ i, h i x hx) | lemma | lower_semicontinuous_on_tsum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on",
"lower_semicontinuous_within_at_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_tsum {f : ι → α → ℝ≥0∞}
(h : ∀ i, lower_semicontinuous (f i)) :
lower_semicontinuous (λ x', ∑' i, f i x') | λ x, lower_semicontinuous_at_tsum (λ i, h i x) | lemma | lower_semicontinuous_tsum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_at_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_within_at.mono (h : upper_semicontinuous_within_at f s x)
(hst : t ⊆ s) : upper_semicontinuous_within_at f t x | λ y hy, filter.eventually.filter_mono (nhds_within_mono _ hst) (h y hy) | lemma | upper_semicontinuous_within_at.mono | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually.filter_mono",
"nhds_within_mono",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_within_at_univ_iff :
upper_semicontinuous_within_at f univ x ↔ upper_semicontinuous_at f x | by simp [upper_semicontinuous_within_at, upper_semicontinuous_at, nhds_within_univ] | lemma | upper_semicontinuous_within_at_univ_iff | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"nhds_within_univ",
"upper_semicontinuous_at",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_at.upper_semicontinuous_within_at
(s : set α) (h : upper_semicontinuous_at f x) : upper_semicontinuous_within_at f s x | λ y hy, filter.eventually.filter_mono nhds_within_le_nhds (h y hy) | lemma | upper_semicontinuous_at.upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually.filter_mono",
"nhds_within_le_nhds",
"upper_semicontinuous_at",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on.upper_semicontinuous_within_at
(h : upper_semicontinuous_on f s) (hx : x ∈ s) :
upper_semicontinuous_within_at f s x | h x hx | lemma | upper_semicontinuous_on.upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_on",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on.mono (h : upper_semicontinuous_on f s) (hst : t ⊆ s) :
upper_semicontinuous_on f t | λ x hx, (h x (hst hx)).mono hst | lemma | upper_semicontinuous_on.mono | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on_univ_iff :
upper_semicontinuous_on f univ ↔ upper_semicontinuous f | by simp [upper_semicontinuous_on, upper_semicontinuous, upper_semicontinuous_within_at_univ_iff] | lemma | upper_semicontinuous_on_univ_iff | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_on",
"upper_semicontinuous_within_at_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous.upper_semicontinuous_at
(h : upper_semicontinuous f) (x : α) : upper_semicontinuous_at f x | h x | lemma | upper_semicontinuous.upper_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous.upper_semicontinuous_within_at
(h : upper_semicontinuous f) (s : set α) (x : α) : upper_semicontinuous_within_at f s x | (h x).upper_semicontinuous_within_at s | lemma | upper_semicontinuous.upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous.upper_semicontinuous_on
(h : upper_semicontinuous f) (s : set α) : upper_semicontinuous_on f s | λ x hx, h.upper_semicontinuous_within_at s x | lemma | upper_semicontinuous.upper_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_within_at_const :
upper_semicontinuous_within_at (λ x, z) s x | λ y hy, filter.eventually_of_forall (λ x, hy) | lemma | upper_semicontinuous_within_at_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually_of_forall",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_at_const :
upper_semicontinuous_at (λ x, z) x | λ y hy, filter.eventually_of_forall (λ x, hy) | lemma | upper_semicontinuous_at_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually_of_forall",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on_const :
upper_semicontinuous_on (λ x, z) s | λ x hx, upper_semicontinuous_within_at_const | lemma | upper_semicontinuous_on_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_on",
"upper_semicontinuous_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_const :
upper_semicontinuous (λ (x : α), z) | λ x, upper_semicontinuous_at_const | lemma | upper_semicontinuous_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.upper_semicontinuous_indicator (hs : is_open s) (hy : y ≤ 0) :
upper_semicontinuous (indicator s (λ x, y)) | @is_open.lower_semicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy | lemma | is_open.upper_semicontinuous_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"is_open.lower_semicontinuous_indicator",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.upper_semicontinuous_on_indicator (hs : is_open s) (hy : y ≤ 0) :
upper_semicontinuous_on (indicator s (λ x, y)) t | (hs.upper_semicontinuous_indicator hy).upper_semicontinuous_on t | lemma | is_open.upper_semicontinuous_on_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.upper_semicontinuous_at_indicator (hs : is_open s) (hy : y ≤ 0) :
upper_semicontinuous_at (indicator s (λ x, y)) x | (hs.upper_semicontinuous_indicator hy).upper_semicontinuous_at x | lemma | is_open.upper_semicontinuous_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.upper_semicontinuous_within_at_indicator (hs : is_open s) (hy : y ≤ 0) :
upper_semicontinuous_within_at (indicator s (λ x, y)) t x | (hs.upper_semicontinuous_indicator hy).upper_semicontinuous_within_at t x | lemma | is_open.upper_semicontinuous_within_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.upper_semicontinuous_indicator (hs : is_closed s) (hy : 0 ≤ y) :
upper_semicontinuous (indicator s (λ x, y)) | @is_closed.lower_semicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy | lemma | is_closed.upper_semicontinuous_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"is_closed.lower_semicontinuous_indicator",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.upper_semicontinuous_on_indicator (hs : is_closed s) (hy : 0 ≤ y) :
upper_semicontinuous_on (indicator s (λ x, y)) t | (hs.upper_semicontinuous_indicator hy).upper_semicontinuous_on t | lemma | is_closed.upper_semicontinuous_on_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.upper_semicontinuous_at_indicator (hs : is_closed s) (hy : 0 ≤ y) :
upper_semicontinuous_at (indicator s (λ x, y)) x | (hs.upper_semicontinuous_indicator hy).upper_semicontinuous_at x | lemma | is_closed.upper_semicontinuous_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.upper_semicontinuous_within_at_indicator (hs : is_closed s) (hy : 0 ≤ y) :
upper_semicontinuous_within_at (indicator s (λ x, y)) t x | (hs.upper_semicontinuous_indicator hy).upper_semicontinuous_within_at t x | lemma | is_closed.upper_semicontinuous_within_at_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_iff_is_open_preimage :
upper_semicontinuous f ↔ ∀ y, is_open (f ⁻¹' (Iio y)) | ⟨λ H y, is_open_iff_mem_nhds.2 (λ x hx, H x y hx), λ H x y y_lt, is_open.mem_nhds (H y) y_lt⟩ | theorem | upper_semicontinuous_iff_is_open_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"is_open.mem_nhds",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous.is_open_preimage (hf : upper_semicontinuous f) (y : β) :
is_open (f ⁻¹' (Iio y)) | upper_semicontinuous_iff_is_open_preimage.1 hf y | lemma | upper_semicontinuous.is_open_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_open",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_iff_is_closed_preimage {f : α → γ} :
upper_semicontinuous f ↔ ∀ y, is_closed (f ⁻¹' (Ici y)) | begin
rw upper_semicontinuous_iff_is_open_preimage,
congrm (∀ y, (_ : Prop)),
rw [← is_open_compl_iff, ← preimage_compl, compl_Ici]
end | theorem | upper_semicontinuous_iff_is_closed_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"is_open_compl_iff",
"upper_semicontinuous",
"upper_semicontinuous_iff_is_open_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous.is_closed_preimage {f : α → γ} (hf : upper_semicontinuous f) (y : γ) :
is_closed (f ⁻¹' (Ici y)) | upper_semicontinuous_iff_is_closed_preimage.1 hf y | lemma | upper_semicontinuous.is_closed_preimage | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"is_closed",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.upper_semicontinuous_within_at {f : α → γ}
(h : continuous_within_at f s x) : upper_semicontinuous_within_at f s x | λ y hy, h (Iio_mem_nhds hy) | lemma | continuous_within_at.upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"Iio_mem_nhds",
"continuous_within_at",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.upper_semicontinuous_at {f : α → γ}
(h : continuous_at f x) : upper_semicontinuous_at f x | λ y hy, h (Iio_mem_nhds hy) | lemma | continuous_at.upper_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"Iio_mem_nhds",
"continuous_at",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.upper_semicontinuous_on {f : α → γ}
(h : continuous_on f s) : upper_semicontinuous_on f s | λ x hx, (h x hx).upper_semicontinuous_within_at | lemma | continuous_on.upper_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_on",
"upper_semicontinuous_on",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.upper_semicontinuous {f : α → γ}
(h : continuous f) : upper_semicontinuous f | λ x, h.continuous_at.upper_semicontinuous_at | lemma | continuous.upper_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_upper_semicontinuous_within_at
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_within_at f s x)
(gmon : monotone g) : upper_semicontinuous_within_at (g ∘ f) s x | @continuous_at.comp_lower_semicontinuous_within_at α _ x s γᵒᵈ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon.dual | lemma | continuous_at.comp_upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"continuous_at.comp_lower_semicontinuous_within_at",
"monotone",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_upper_semicontinuous_at
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_at f x)
(gmon : monotone g) : upper_semicontinuous_at (g ∘ f) x | @continuous_at.comp_lower_semicontinuous_at α _ x γᵒᵈ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon.dual | lemma | continuous_at.comp_upper_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"continuous_at.comp_lower_semicontinuous_at",
"monotone",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_upper_semicontinuous_on
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous_on f s)
(gmon : monotone g) : upper_semicontinuous_on (g ∘ f) s | λ x hx, (hg.continuous_at).comp_upper_semicontinuous_within_at (hf x hx) gmon | lemma | continuous.comp_upper_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"monotone",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_upper_semicontinuous
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous f)
(gmon : monotone g) : upper_semicontinuous (g ∘ f) | λ x, (hg.continuous_at).comp_upper_semicontinuous_at (hf x) gmon | lemma | continuous.comp_upper_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"monotone",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_upper_semicontinuous_within_at_antitone
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_within_at f s x)
(gmon : antitone g) : lower_semicontinuous_within_at (g ∘ f) s x | @continuous_at.comp_upper_semicontinuous_within_at α _ x s γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | lemma | continuous_at.comp_upper_semicontinuous_within_at_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous_at",
"continuous_at.comp_upper_semicontinuous_within_at",
"lower_semicontinuous_within_at",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_upper_semicontinuous_at_antitone
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_at f x)
(gmon : antitone g) : lower_semicontinuous_at (g ∘ f) x | @continuous_at.comp_upper_semicontinuous_at α _ x γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | lemma | continuous_at.comp_upper_semicontinuous_at_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous_at",
"continuous_at.comp_upper_semicontinuous_at",
"lower_semicontinuous_at",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_upper_semicontinuous_on_antitone
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous_on f s)
(gmon : antitone g) : lower_semicontinuous_on (g ∘ f) s | λ x hx, (hg.continuous_at).comp_upper_semicontinuous_within_at_antitone (hf x hx) gmon | lemma | continuous.comp_upper_semicontinuous_on_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous",
"lower_semicontinuous_on",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_upper_semicontinuous_antitone
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous f)
(gmon : antitone g) : lower_semicontinuous (g ∘ f) | λ x, (hg.continuous_at).comp_upper_semicontinuous_at_antitone (hf x) gmon | lemma | continuous.comp_upper_semicontinuous_antitone | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"antitone",
"continuous",
"lower_semicontinuous",
"upper_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_within_at.add' {f g : α → γ}
(hf : upper_semicontinuous_within_at f s x) (hg : upper_semicontinuous_within_at g s x)
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
upper_semicontinuous_within_at (λ z, f z + g z) s x | @lower_semicontinuous_within_at.add' α _ x s γᵒᵈ _ _ _ _ _ hf hg hcont | lemma | upper_semicontinuous_within_at.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"lower_semicontinuous_within_at.add'",
"upper_semicontinuous_within_at"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_at.add' {f g : α → γ}
(hf : upper_semicontinuous_at f x) (hg : upper_semicontinuous_at g x)
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
upper_semicontinuous_at (λ z, f z + g z) x | by { simp_rw [← upper_semicontinuous_within_at_univ_iff] at *, exact hf.add' hg hcont } | lemma | upper_semicontinuous_at.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"upper_semicontinuous_at",
"upper_semicontinuous_within_at_univ_iff"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_on.add' {f g : α → γ}
(hf : upper_semicontinuous_on f s) (hg : upper_semicontinuous_on g s)
(hcont : ∀ x ∈ s, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
upper_semicontinuous_on (λ z, f z + g z) s | λ x hx, (hf x hx).add' (hg x hx) (hcont x hx) | lemma | upper_semicontinuous_on.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"upper_semicontinuous_on"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous.add' {f g : α → γ}
(hf : upper_semicontinuous f) (hg : upper_semicontinuous g)
(hcont : ∀ x, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) :
upper_semicontinuous (λ z, f z + g z) | λ x, (hf x).add' (hg x) (hcont x) | lemma | upper_semicontinuous.add' | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"upper_semicontinuous"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of
the lemma uses `[has_continuous_add]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_within_at.add {f g : α → γ}
(hf : upper_semicontinuous_within_at f s x) (hg : upper_semicontinuous_within_at g s x) :
upper_semicontinuous_within_at (λ z, f z + g z) s x | hf.add' hg continuous_add.continuous_at | lemma | upper_semicontinuous_within_at.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_within_at"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_at.add {f g : α → γ}
(hf : upper_semicontinuous_at f x) (hg : upper_semicontinuous_at g x) :
upper_semicontinuous_at (λ z, f z + g z) x | hf.add' hg continuous_add.continuous_at | lemma | upper_semicontinuous_at.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_at"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_on.add {f g : α → γ}
(hf : upper_semicontinuous_on f s) (hg : upper_semicontinuous_on g s) :
upper_semicontinuous_on (λ z, f z + g z) s | hf.add' hg (λ x hx, continuous_add.continuous_at) | lemma | upper_semicontinuous_on.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_on"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous.add {f g : α → γ}
(hf : upper_semicontinuous f) (hg : upper_semicontinuous g) :
upper_semicontinuous (λ z, f z + g z) | hf.add' hg (λ x, continuous_add.continuous_at) | lemma | upper_semicontinuous.add | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous"
] | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `ereal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_within_at_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, upper_semicontinuous_within_at (f i) s x) :
upper_semicontinuous_within_at (λ z, (∑ i in a, f i z)) s x | @lower_semicontinuous_within_at_sum α _ x s ι γᵒᵈ _ _ _ _ f a ha | lemma | upper_semicontinuous_within_at_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"lower_semicontinuous_within_at_sum",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_at_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, upper_semicontinuous_at (f i) x) :
upper_semicontinuous_at (λ z, (∑ i in a, f i z)) x | begin
simp_rw [← upper_semicontinuous_within_at_univ_iff] at *,
exact upper_semicontinuous_within_at_sum ha
end | lemma | upper_semicontinuous_at_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"upper_semicontinuous_at",
"upper_semicontinuous_within_at_sum",
"upper_semicontinuous_within_at_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, upper_semicontinuous_on (f i) s) :
upper_semicontinuous_on (λ z, (∑ i in a, f i z)) s | λ x hx, upper_semicontinuous_within_at_sum (λ i hi, ha i hi x hx) | lemma | upper_semicontinuous_on_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"upper_semicontinuous_on",
"upper_semicontinuous_within_at_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_sum {f : ι → α → γ} {a : finset ι}
(ha : ∀ i ∈ a, upper_semicontinuous (f i)) :
upper_semicontinuous (λ z, (∑ i in a, f i z)) | λ x, upper_semicontinuous_at_sum (λ i hi, ha i hi x) | lemma | upper_semicontinuous_sum | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"finset",
"upper_semicontinuous",
"upper_semicontinuous_at_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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