statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
value | commit stringclasses 1
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IsOpen.lowerSemicontinuousOn_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
LowerSemicontinuousOn (indicator s fun _x => y) t | (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousOn t | theorem | IsOpen.lowerSemicontinuousOn_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"LowerSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.lowerSemicontinuousAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
LowerSemicontinuousAt (indicator s fun _x => y) x | (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x | theorem | IsOpen.lowerSemicontinuousAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"LowerSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.lowerSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
LowerSemicontinuousWithinAt (indicator s fun _x => y) t x | (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousWithinAt t x | theorem | IsOpen.lowerSemicontinuousWithinAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"LowerSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.lowerSemicontinuous_indicator (hs : IsClosed s) (hy : y ≤ 0) :
LowerSemicontinuous (indicator s fun _x => y) | by
intro x z hz
by_cases h : x ∈ s <;> simp [h] at hz
· refine Filter.Eventually.of_forall fun x' => ?_
by_cases h' : x' ∈ s <;> simp [h', hz, hz.trans_le hy]
· filter_upwards [hs.isOpen_compl.mem_nhds h]
simp +contextual [hz] | theorem | IsClosed.lowerSemicontinuous_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Filter.Eventually.of_forall",
"IsClosed",
"LowerSemicontinuous",
"by_cases"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.lowerSemicontinuousOn_indicator (hs : IsClosed s) (hy : y ≤ 0) :
LowerSemicontinuousOn (indicator s fun _x => y) t | (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousOn t | theorem | IsClosed.lowerSemicontinuousOn_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.lowerSemicontinuousAt_indicator (hs : IsClosed s) (hy : y ≤ 0) :
LowerSemicontinuousAt (indicator s fun _x => y) x | (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x | theorem | IsClosed.lowerSemicontinuousAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.lowerSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : y ≤ 0) :
LowerSemicontinuousWithinAt (indicator s fun _x => y) t x | (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousWithinAt t x | theorem | IsClosed.lowerSemicontinuousWithinAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_iff_isOpen_preimage :
LowerSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Ioi y) | ⟨fun H y => isOpen_iff_mem_nhds.2 fun x hx => H x y hx, fun H _x y y_lt =>
IsOpen.mem_nhds (H y) y_lt⟩ | theorem | lowerSemicontinuous_iff_isOpen_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"IsOpen.mem_nhds",
"LowerSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.isOpen_preimage (hf : LowerSemicontinuous f) (y : β) :
IsOpen (f ⁻¹' Ioi y) | lowerSemicontinuous_iff_isOpen_preimage.1 hf y | theorem | LowerSemicontinuous.isOpen_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"LowerSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_iff_preimage_Ioi :
LowerSemicontinuousOn f s ↔ ∀ b, ∃ u, IsOpen u ∧ s ∩ f ⁻¹' Set.Ioi b = s ∩ u | by
simp only [← lowerSemicontinuous_restrict_iff, restrict_eq,
lowerSemicontinuous_iff_isOpen_preimage, preimage_comp, isOpen_induced_iff,
Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm] | theorem | lowerSemicontinuousOn_iff_preimage_Ioi | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"LowerSemicontinuousOn",
"Subtype.preimage_coe_eq_preimage_coe_iff",
"isOpen_induced_iff",
"lowerSemicontinuous_iff_isOpen_preimage",
"lowerSemicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_of_forall_isMaxOn_and_mem
(hfy : ∀ i ∈ I, LowerSemicontinuousOn (f i) s)
{M : α → ι}
(M_mem : ∀ x ∈ s, M x ∈ I)
(M_max : ∀ x ∈ s, IsMaxOn (fun y ↦ f y x) I (M x)) :
LowerSemicontinuousOn (fun x ↦ f (M x) x) s | by
intro x hx b hb
apply Filter.Eventually.mp <| hfy (M x) (M_mem x hx) x hx b hb
apply eventually_nhdsWithin_of_forall
intro z hz h
exact lt_of_lt_of_le h (M_max z hz (M_mem x hx)) | theorem | lowerSemicontinuousOn_of_forall_isMaxOn_and_mem | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Filter.Eventually.mp",
"IsMaxOn",
"LowerSemicontinuousOn",
"eventually_nhdsWithin_of_forall",
"lt_of_lt_of_le"
] | Given a family of lower semicontinuous functions `f : ι → α → β` such that
for each `x : α`, there is a choice `M x : ι` such that the maximum value of
evaluation at `x` is achieved by the function `f (M x)`, then the pointwise
maximum of the family `f` is lower semicontinuous.
In the statement we restrict to subsets `... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upperSemicontinuousOn_of_forall_isMinOn_and_mem
(hfy : ∀ i ∈ I, UpperSemicontinuousOn (f i) s)
{m : α → ι}
(m_mem : ∀ x ∈ s, m x ∈ I)
(m_min : ∀ x ∈ s, IsMinOn (fun i ↦ f i x) I (m x)) :
UpperSemicontinuousOn (fun x ↦ f (m x) x) s | lowerSemicontinuousOn_of_forall_isMaxOn_and_mem (β := βᵒᵈ) hfy m_mem m_min | theorem | upperSemicontinuousOn_of_forall_isMinOn_and_mem | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsMinOn",
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_of_forall_isMaxOn_and_mem"
] | Given a family of upper semicontinuous functions `f : ι → α → β` such that
for each `x : α`, there is a choice `m x : ι` such that the minimum value of
evaluation at `x` is achieved by the function `f (m x)`, then the pointwise
maximum of the family `f` is upper semicontinuous.
In the statement we restrict to subsets `... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerSemicontinuous_iff_isClosed_preimage {f : α → γ} :
LowerSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Iic y) | by
rw [lowerSemicontinuous_iff_isOpen_preimage]
simp only [← isOpen_compl_iff, ← preimage_compl, compl_Iic] | theorem | lowerSemicontinuous_iff_isClosed_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuous",
"isOpen_compl_iff",
"lowerSemicontinuous_iff_isOpen_preimage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.isClosed_preimage {f : α → γ} (hf : LowerSemicontinuous f) (y : γ) :
IsClosed (f ⁻¹' Iic y) | lowerSemicontinuous_iff_isClosed_preimage.1 hf y | theorem | LowerSemicontinuous.isClosed_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_iff_preimage_Iic {f : α → γ} :
LowerSemicontinuousOn f s ↔ ∀ b, ∃ v, IsClosed v ∧ s ∩ f ⁻¹' Set.Iic b = s ∩ v | by
simp only [← lowerSemicontinuous_restrict_iff, restrict_eq,
lowerSemicontinuous_iff_isClosed_preimage, preimage_comp,
isClosed_induced_iff, Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm] | theorem | lowerSemicontinuousOn_iff_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuousOn",
"Set.Iic",
"Subtype.preimage_coe_eq_preimage_coe_iff",
"isClosed_induced_iff",
"lowerSemicontinuous_iff_isClosed_preimage",
"lowerSemicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.lowerSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) :
LowerSemicontinuousWithinAt f s x | fun _y hy => h (Ioi_mem_nhds hy) | theorem | ContinuousWithinAt.lowerSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousWithinAt",
"Ioi_mem_nhds",
"LowerSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.lowerSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) :
LowerSemicontinuousAt f x | fun _y hy => h (Ioi_mem_nhds hy) | theorem | ContinuousAt.lowerSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"Ioi_mem_nhds",
"LowerSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.lowerSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) :
LowerSemicontinuousOn f s | fun x hx => (h x hx).lowerSemicontinuousWithinAt | theorem | ContinuousOn.lowerSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousOn",
"LowerSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.lowerSemicontinuous {f : α → γ} (h : Continuous f) : LowerSemicontinuous f | fun _x => h.continuousAt.lowerSemicontinuousAt | theorem | Continuous.lowerSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"LowerSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_iff_le_liminf {f : α → γ} :
LowerSemicontinuousWithinAt f s x ↔ f x ≤ liminf f (𝓝[s] x) | by
constructor
· intro h; unfold LowerSemicontinuousWithinAt at h
by_contra! hf
obtain ⟨z, ltz, y, ylt, h₁⟩ := hf.exists_disjoint_Iio_Ioi
exact ltz.not_ge
(le_liminf_of_le (by isBoundedDefault) ((h y ylt).mono fun _ h₂ =>
le_of_not_gt fun h₃ => (h₁ _ h₃ _ h₂).false))
exact fun hf y ylt =... | theorem | lowerSemicontinuousWithinAt_iff_le_liminf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt",
"le_of_not_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_iff_le_liminf {f : α → γ} :
LowerSemicontinuousAt f x ↔ f x ≤ liminf f (𝓝 x) | by
rw [← lowerSemicontinuousWithinAt_univ_iff, lowerSemicontinuousWithinAt_iff_le_liminf,
← nhdsWithin_univ]
alias ⟨LowerSemicontinuousAt.le_liminf, _⟩ := lowerSemicontinuousAt_iff_le_liminf | theorem | lowerSemicontinuousAt_iff_le_liminf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_iff_le_liminf",
"lowerSemicontinuousWithinAt_univ_iff",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_iff_le_liminf {f : α → γ} :
LowerSemicontinuous f ↔ ∀ x, f x ≤ liminf f (𝓝 x) | by
simp only [← lowerSemicontinuousAt_iff_le_liminf, lowerSemicontinuous_iff]
alias ⟨LowerSemicontinuous.le_liminf, _⟩ := lowerSemicontinuous_iff_le_liminf | theorem | lowerSemicontinuous_iff_le_liminf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"lowerSemicontinuousAt_iff_le_liminf",
"lowerSemicontinuous_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_iff_le_liminf {f : α → γ} :
LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ liminf f (𝓝[s] x) | by
simp only [← lowerSemicontinuousWithinAt_iff_le_liminf, lowerSemicontinuousOn_iff]
alias ⟨LowerSemicontinuousOn.le_liminf, _⟩ := lowerSemicontinuousOn_iff_le_liminf | theorem | lowerSemicontinuousOn_iff_le_liminf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"lowerSemicontinuousOn_iff",
"lowerSemicontinuousWithinAt_iff_le_liminf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.isCompact_inter_preimage_Iic {f : α → γ}
(hfs : LowerSemicontinuousOn f s) (ks : IsCompact s) (c : γ) :
IsCompact (s ∩ f ⁻¹' Iic c) | by
rw [lowerSemicontinuousOn_iff_preimage_Iic] at hfs
obtain ⟨v, hv, hv'⟩ := hfs c
exact hv' ▸ ks.inter_right hv | theorem | LowerSemicontinuousOn.isCompact_inter_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsCompact",
"LowerSemicontinuousOn",
"lowerSemicontinuousOn_iff_preimage_Iic"
] | The sublevel sets of a lower semicontinuous function on a compact set are compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousOn.inter_biInter_preimage_Iic_eq_empty_iff_exists_finset
{ι : Type*} {f : ι → α → γ}
(ks : IsCompact s) {I : Set ι} {c : γ} (hfi : ∀ i ∈ I, LowerSemicontinuousOn (f i) s) :
s ∩ ⋂ i ∈ I, (f i) ⁻¹' Iic c = ∅ ↔ ∃ u : Finset I, ∀ x ∈ s, ∃ i ∈ u, c < f i x | by
refine ⟨fun H ↦ ?_, fun ⟨u, hu⟩ ↦ ?_⟩
· suffices ∀ i ∈ I, IsClosed (s ↓∩ (fun i ↦ f i ⁻¹' Iic c) i) by
simpa [Set.eq_empty_iff_forall_notMem] using
ks.elim_finite_subfamily_isClosed_subtype _ this H
exact fun i hi ↦ lowerSemicontinuous_restrict_iff.mpr (hfi i hi) |>.isClosed_preimage c
· rw [... | theorem | LowerSemicontinuousOn.inter_biInter_preimage_Iic_eq_empty_iff_exists_finset | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"IsClosed",
"IsCompact",
"LowerSemicontinuousOn",
"Set",
"Set.eq_empty_iff_forall_notMem",
"not_le"
] | An intersection of sublevel sets of a lower semicontinuous function
on a compact set is empty if and only if a finite sub-intersection is already empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerSemicontinuousOn_iff_isClosed_epigraph {f : α → γ} {s : Set α} (hs : IsClosed s) :
LowerSemicontinuousOn f s ↔ IsClosed {p : α × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} | by
simp_rw [lowerSemicontinuousOn_iff, lowerSemicontinuousWithinAt_iff,
eventually_nhdsWithin_iff, ← isOpen_compl_iff, compl_setOf, isOpen_iff_eventually, mem_setOf,
not_and, not_le]
constructor
· intro hf ⟨x, y⟩ h
by_cases hx : x ∈ s
· have ⟨y', hy', z, hz, h₁⟩ := (h hx).exists_disjoint_Iio_Ioi
... | theorem | lowerSemicontinuousOn_iff_isClosed_epigraph | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous.prodMk_left",
"IsClosed",
"LowerSemicontinuousOn",
"Set",
"by_cases",
"eventually_nhdsWithin_iff",
"isOpen_compl_iff",
"isOpen_iff_eventually",
"lowerSemicontinuousOn_iff",
"lowerSemicontinuousWithinAt_iff",
"not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_iff_isClosed_epigraph {f : α → γ} :
LowerSemicontinuous f ↔ IsClosed {p : α × γ | f p.1 ≤ p.2} | by
simp [← lowerSemicontinuousOn_univ_iff, lowerSemicontinuousOn_iff_isClosed_epigraph]
alias ⟨LowerSemicontinuous.isClosed_epigraph, _⟩ := lowerSemicontinuous_iff_isClosed_epigraph | theorem | lowerSemicontinuous_iff_isClosed_epigraph | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LowerSemicontinuous",
"lowerSemicontinuousOn_iff_isClosed_epigraph",
"lowerSemicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_lowerSemicontinuousWithinAt {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Monotone g) :
LowerSemicontinuousWithinAt (g ∘ f) s x | by
intro 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])
_ ≤ g (f a) := gmon (min_le_right _ _... | theorem | ContinuousAt.comp_lowerSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"Filter.Eventually.of_forall",
"Ioi_mem_nhds",
"LowerSemicontinuousWithinAt",
"Monotone",
"exists_Ioc_subset_of_mem_nhds",
"min_le_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_lowerSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x))
(hf : LowerSemicontinuousAt f x) (gmon : Monotone g) : LowerSemicontinuousAt (g ∘ f) x | by
simp only [← lowerSemicontinuousWithinAt_univ_iff] at hf ⊢
exact hg.comp_lowerSemicontinuousWithinAt hf gmon | theorem | ContinuousAt.comp_lowerSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"LowerSemicontinuousAt",
"Monotone",
"lowerSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_lowerSemicontinuousOn {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : LowerSemicontinuousOn f s) (gmon : Monotone g) : LowerSemicontinuousOn (g ∘ f) s | fun x hx => hg.continuousAt.comp_lowerSemicontinuousWithinAt (hf x hx) gmon | theorem | Continuous.comp_lowerSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"LowerSemicontinuousOn",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_lowerSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : LowerSemicontinuous f) (gmon : Monotone g) : LowerSemicontinuous (g ∘ f) | fun x =>
hg.continuousAt.comp_lowerSemicontinuousAt (hf x) gmon | theorem | Continuous.comp_lowerSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"LowerSemicontinuous",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_lowerSemicontinuousWithinAt_antitone {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Antitone g) :
UpperSemicontinuousWithinAt (g ∘ f) s x | ContinuousAt.comp_lowerSemicontinuousWithinAt (δ := δᵒᵈ) hg hf gmon | theorem | ContinuousAt.comp_lowerSemicontinuousWithinAt_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"ContinuousAt",
"ContinuousAt.comp_lowerSemicontinuousWithinAt",
"LowerSemicontinuousWithinAt",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_lowerSemicontinuousAt_antitone {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousAt f x) (gmon : Antitone g) :
UpperSemicontinuousAt (g ∘ f) x | ContinuousAt.comp_lowerSemicontinuousAt (δ := δᵒᵈ) hg hf gmon | theorem | ContinuousAt.comp_lowerSemicontinuousAt_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"ContinuousAt",
"ContinuousAt.comp_lowerSemicontinuousAt",
"LowerSemicontinuousAt",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_lowerSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : LowerSemicontinuousOn f s) (gmon : Antitone g) : UpperSemicontinuousOn (g ∘ f) s | fun x hx => hg.continuousAt.comp_lowerSemicontinuousWithinAt_antitone (hf x hx) gmon | theorem | Continuous.comp_lowerSemicontinuousOn_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"Continuous",
"LowerSemicontinuousOn",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_lowerSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : LowerSemicontinuous f) (gmon : Antitone g) : UpperSemicontinuous (g ∘ f) | fun x =>
hg.continuousAt.comp_lowerSemicontinuousAt_antitone (hf x) gmon
@[deprecated (since := "2025-12-06")]
alias LowerSemicontinuousAt.comp_continuousAt := LowerSemicontinuousAt.comp
@[deprecated (since := "2025-12-06")]
alias LowerSemicontinuousAt.comp_continuousAt_of_eq := LowerSemicontinuousAt.comp
@[deprec... | theorem | Continuous.comp_lowerSemicontinuous_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"Continuous",
"LowerSemicontinuous",
"LowerSemicontinuous.comp",
"LowerSemicontinuousAt.comp",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt.add' {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
(hg : LowerSemicontinuousWithinAt g s x)
(hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
LowerSemicontinuousWithinAt (fun z => f z + g z) s x | by
intro y hy
obtain ⟨u, v, u_open, xu, v_open, xv, h⟩ :
∃ u v : Set γ,
IsOpen u ∧ f x ∈ u ∧ IsOpen v ∧ g x ∈ v ∧ u ×ˢ v ⊆ { p : γ × γ | y < p.fst + p.snd } :=
mem_nhds_prod_iff'.1 (hcont (isOpen_Ioi.mem_nhds hy))
by_cases hx₁ : ∃ l, l < f x
· obtain ⟨z₁, z₁lt, h₁⟩ : ∃ z₁ < f x, Ioc z₁ (f x) ⊆ u :... | theorem | LowerSemicontinuousWithinAt.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"Filter.Eventually.of_forall",
"IsOpen",
"LowerSemicontinuousWithinAt",
"Set",
"by_cases",
"exists_Ioc_subset_of_mem_nhds",
"min_le_left",
"not_lt"
] | 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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousAt.add' {f g : α → γ} (hf : LowerSemicontinuousAt f x)
(hg : LowerSemicontinuousAt g x)
(hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
LowerSemicontinuousAt (fun z => f z + g z) x | by
simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at *
exact hf.add' hg hcont | theorem | LowerSemicontinuousAt.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousOn.add' {f g : α → γ} (hf : LowerSemicontinuousOn f s)
(hg : LowerSemicontinuousOn g s)
(hcont : ∀ x ∈ s, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
LowerSemicontinuousOn (fun z => f z + g z) s | fun x hx =>
LowerSemicontinuousWithinAt.add' (hf x hx) (hg x hx) (hcont x hx) | theorem | LowerSemicontinuousOn.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"LowerSemicontinuousOn",
"LowerSemicontinuousWithinAt.add'"
] | 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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuous.add' {f g : α → γ} (hf : LowerSemicontinuous f)
(hg : LowerSemicontinuous g)
(hcont : ∀ x, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
LowerSemicontinuous fun z => f z + g z | fun x => LowerSemicontinuousAt.add' (hf x) (hg x) (hcont x) | theorem | LowerSemicontinuous.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"LowerSemicontinuous",
"LowerSemicontinuousAt.add'"
] | 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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousWithinAt.add {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
(hg : LowerSemicontinuousWithinAt g s x) :
LowerSemicontinuousWithinAt (fun z => f z + g z) s x | hf.add' hg continuous_add.continuousAt | theorem | LowerSemicontinuousWithinAt.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `EReal`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousAt.add {f g : α → γ} (hf : LowerSemicontinuousAt f x)
(hg : LowerSemicontinuousAt g x) : LowerSemicontinuousAt (fun z => f z + g z) x | hf.add' hg continuous_add.continuousAt | theorem | LowerSemicontinuousAt.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `EReal`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousOn.add {f g : α → γ} (hf : LowerSemicontinuousOn f s)
(hg : LowerSemicontinuousOn g s) : LowerSemicontinuousOn (fun z => f z + g z) s | hf.add' hg fun _x _hx => continuous_add.continuousAt | theorem | LowerSemicontinuousOn.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `EReal`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuous.add {f g : α → γ} (hf : LowerSemicontinuous f)
(hg : LowerSemicontinuous g) : LowerSemicontinuous fun z => f z + g z | hf.add' hg fun _x => continuous_add.continuousAt | theorem | LowerSemicontinuous.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous"
] | The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `EReal`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x | by
classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Fin... | theorem | lowerSemicontinuousWithinAt_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"Finset.induction_on",
"Finset.mem_insert_of_mem",
"Finset.mem_insert_self",
"LowerSemicontinuousWithinAt",
"LowerSemicontinuousWithinAt.add",
"lowerSemicontinuousWithinAt_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, LowerSemicontinuousAt (f i) x) :
LowerSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x | by
simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at *
exact lowerSemicontinuousWithinAt_sum ha | theorem | lowerSemicontinuousAt_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_sum",
"lowerSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, LowerSemicontinuousOn (f i) s) :
LowerSemicontinuousOn (fun z => ∑ i ∈ a, f i z) s | fun x hx =>
lowerSemicontinuousWithinAt_sum fun i hi => ha i hi x hx | theorem | lowerSemicontinuousOn_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"LowerSemicontinuousOn",
"lowerSemicontinuousWithinAt_sum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, LowerSemicontinuous (f i)) : LowerSemicontinuous fun z => ∑ i ∈ a, f i z | fun x => lowerSemicontinuousAt_sum fun i hi => ha i hi x | theorem | lowerSemicontinuous_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"LowerSemicontinuous",
"lowerSemicontinuousAt_sum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt.sup
(hf : LowerSemicontinuousWithinAt f s a) (hg : LowerSemicontinuousWithinAt g s a) :
LowerSemicontinuousWithinAt (fun x ↦ f x ⊔ g x) s a | by
intro b hb
simp only [lt_sup_iff] at hb ⊢
rcases hb with hb | hb
· filter_upwards [hf b hb] with x using Or.intro_left _
· filter_upwards [hg b hb] with x using Or.intro_right _ | theorem | LowerSemicontinuousWithinAt.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt",
"lt_sup_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousAt.sup
(hf : LowerSemicontinuousAt f a) (hg : LowerSemicontinuousAt g a) :
LowerSemicontinuousAt (fun x ↦ f x ⊔ g x) a | by
rw [← lowerSemicontinuousWithinAt_univ_iff] at *
exact hf.sup hg | theorem | LowerSemicontinuousAt.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.sup
(hf : LowerSemicontinuousOn f s) (hg : LowerSemicontinuousOn g s) :
LowerSemicontinuousOn (fun x ↦ f x ⊔ g x) s | fun a ha ↦
LowerSemicontinuousWithinAt.sup (hf a ha) (hg a ha) | theorem | LowerSemicontinuousOn.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"LowerSemicontinuousWithinAt.sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.sup
(hf : LowerSemicontinuous f) (hg : LowerSemicontinuous g) :
LowerSemicontinuous fun x ↦ f x ⊔ g x | fun a ↦
LowerSemicontinuousAt.sup (hf a) (hg a) | theorem | LowerSemicontinuous.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"LowerSemicontinuousAt.sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt.inf
(hf : LowerSemicontinuousWithinAt f s a) (hg : LowerSemicontinuousWithinAt g s a) :
LowerSemicontinuousWithinAt (fun x ↦ f x ⊓ g x) s a | by
intro b hb
simp only [lt_inf_iff] at hb ⊢
exact Eventually.and (hf b hb.1) (hg b hb.2) | theorem | LowerSemicontinuousWithinAt.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousAt.inf
(hf : LowerSemicontinuousAt f a) (hg : LowerSemicontinuousAt g a) :
LowerSemicontinuousAt (fun x ↦ f x ⊓ g x) a | by
rw [← lowerSemicontinuousWithinAt_univ_iff] at *
exact hf.inf hg | theorem | LowerSemicontinuousAt.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.inf
(hf : LowerSemicontinuousOn f s) (hg : LowerSemicontinuousOn g s) :
LowerSemicontinuousOn (fun x ↦ f x ⊓ g x) s | fun a ha ↦
LowerSemicontinuousWithinAt.inf (hf a ha) (hg a ha) | theorem | LowerSemicontinuousOn.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"LowerSemicontinuousWithinAt.inf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.inf (hf : LowerSemicontinuous f)
(hg : LowerSemicontinuous g) :
LowerSemicontinuous fun x ↦ f x ⊓ g x | fun a ↦
LowerSemicontinuousAt.inf (hf a) (hg a) | theorem | LowerSemicontinuous.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"LowerSemicontinuousAt.inf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_ciSup {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝[s] x, BddAbove (range fun i => f i y))
(h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x | by
cases isEmpty_or_nonempty ι
· simpa only [iSup_of_empty'] using lowerSemicontinuousWithinAt_const
· intro y hy
rcases exists_lt_of_lt_ciSup hy with ⟨i, hi⟩
filter_upwards [h i y hi, bdd] with y hy hy' using hy.trans_le (le_ciSup hy' i) | theorem | lowerSemicontinuousWithinAt_ciSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"BddAbove",
"LowerSemicontinuousWithinAt",
"exists_lt_of_lt_ciSup",
"iSup_of_empty'",
"isEmpty_or_nonempty",
"le_ciSup",
"lowerSemicontinuousWithinAt_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_iSup {f : ι → α → δ}
(h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x | lowerSemicontinuousWithinAt_ciSup (by simp) h | theorem | lowerSemicontinuousWithinAt_iSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, LowerSemicontinuousWithinAt (f i hi) s x) :
LowerSemicontinuousWithinAt (fun x' => ⨆ (i) (hi), f i hi x') s x | lowerSemicontinuousWithinAt_iSup fun i => lowerSemicontinuousWithinAt_iSup fun hi => h i hi | theorem | lowerSemicontinuousWithinAt_biSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_ciSup {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝 x, BddAbove (range fun i => f i y)) (h : ∀ i, LowerSemicontinuousAt (f i) x) :
LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x | by
simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at *
rw [← nhdsWithin_univ] at bdd
exact lowerSemicontinuousWithinAt_ciSup bdd h | theorem | lowerSemicontinuousAt_ciSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"BddAbove",
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_ciSup",
"lowerSemicontinuousWithinAt_univ_iff",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousAt (f i) x) :
LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x | lowerSemicontinuousAt_ciSup (by simp) h | theorem | lowerSemicontinuousAt_iSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"lowerSemicontinuousAt_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, LowerSemicontinuousAt (f i hi) x) :
LowerSemicontinuousAt (fun x' => ⨆ (i) (hi), f i hi x') x | lowerSemicontinuousAt_iSup fun i => lowerSemicontinuousAt_iSup fun hi => h i hi | theorem | lowerSemicontinuousAt_biSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"lowerSemicontinuousAt_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_ciSup {f : ι → α → δ'}
(bdd : ∀ x ∈ s, BddAbove (range fun i => f i x)) (h : ∀ i, LowerSemicontinuousOn (f i) s) :
LowerSemicontinuousOn (fun x' => ⨆ i, f i x') s | fun x hx =>
lowerSemicontinuousWithinAt_ciSup (eventually_nhdsWithin_of_forall bdd) fun i => h i x hx | theorem | lowerSemicontinuousOn_ciSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"BddAbove",
"LowerSemicontinuousOn",
"eventually_nhdsWithin_of_forall",
"lowerSemicontinuousWithinAt_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousOn (f i) s) :
LowerSemicontinuousOn (fun x' => ⨆ i, f i x') s | lowerSemicontinuousOn_ciSup (by simp) h | theorem | lowerSemicontinuousOn_iSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"lowerSemicontinuousOn_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, LowerSemicontinuousOn (f i hi) s) :
LowerSemicontinuousOn (fun x' => ⨆ (i) (hi), f i hi x') s | lowerSemicontinuousOn_iSup fun i => lowerSemicontinuousOn_iSup fun hi => h i hi | theorem | lowerSemicontinuousOn_biSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"lowerSemicontinuousOn_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_ciSup {f : ι → α → δ'} (bdd : ∀ x, BddAbove (range fun i => f i x))
(h : ∀ i, LowerSemicontinuous (f i)) : LowerSemicontinuous fun x' => ⨆ i, f i x' | fun x =>
lowerSemicontinuousAt_ciSup (Eventually.of_forall bdd) fun i => h i x | theorem | lowerSemicontinuous_ciSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"BddAbove",
"LowerSemicontinuous",
"lowerSemicontinuousAt_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuous (f i)) :
LowerSemicontinuous fun x' => ⨆ i, f i x' | lowerSemicontinuous_ciSup (by simp) h | theorem | lowerSemicontinuous_iSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"lowerSemicontinuous_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, LowerSemicontinuous (f i hi)) :
LowerSemicontinuous fun x' => ⨆ (i) (hi), f i hi x' | lowerSemicontinuous_iSup fun i => lowerSemicontinuous_iSup fun hi => h i hi | theorem | lowerSemicontinuous_biSup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"lowerSemicontinuous_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_tsum {f : ι → α → ℝ≥0∞}
(h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun x' => ∑' i, f i x') s x | by
simp_rw [ENNReal.tsum_eq_iSup_sum]
refine lowerSemicontinuousWithinAt_iSup fun b => ?_
exact lowerSemicontinuousWithinAt_sum fun i _hi => h i | theorem | lowerSemicontinuousWithinAt_tsum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ENNReal.tsum_eq_iSup_sum",
"LowerSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_iSup",
"lowerSemicontinuousWithinAt_sum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousAt (f i) x) :
LowerSemicontinuousAt (fun x' => ∑' i, f i x') x | by
simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at *
exact lowerSemicontinuousWithinAt_tsum h | theorem | lowerSemicontinuousAt_tsum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"lowerSemicontinuousWithinAt_tsum",
"lowerSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousOn (f i) s) :
LowerSemicontinuousOn (fun x' => ∑' i, f i x') s | fun x hx =>
lowerSemicontinuousWithinAt_tsum fun i => h i x hx | theorem | lowerSemicontinuousOn_tsum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"lowerSemicontinuousWithinAt_tsum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuous (f i)) :
LowerSemicontinuous fun x' => ∑' i, f i x' | fun x => lowerSemicontinuousAt_tsum fun i => h i x | theorem | lowerSemicontinuous_tsum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"lowerSemicontinuousAt_tsum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.exists_isMaxOn {s : Set α} (ne_s : s.Nonempty)
(hs : IsCompact s) (hf : UpperSemicontinuousOn f s) :
∃ a ∈ s, IsMaxOn f s a | LowerSemicontinuousOn.exists_isMinOn (β := βᵒᵈ) ne_s hs hf | theorem | UpperSemicontinuousOn.exists_isMaxOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsCompact",
"IsMaxOn",
"LowerSemicontinuousOn.exists_isMinOn",
"Set",
"UpperSemicontinuousOn"
] | An upper semicontinuous function attains its upper bound on a nonempty compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousOn.bddAbove_of_isCompact [Nonempty β] {s : Set α}
(hs : IsCompact s) (hf : UpperSemicontinuousOn f s) : BddAbove (f '' s) | LowerSemicontinuousOn.bddBelow_of_isCompact (β := βᵒᵈ) hs hf | theorem | UpperSemicontinuousOn.bddAbove_of_isCompact | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"BddAbove",
"IsCompact",
"LowerSemicontinuousOn.bddBelow_of_isCompact",
"Set",
"UpperSemicontinuousOn"
] | An upper semicontinuous function is bounded above on a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsOpen.upperSemicontinuous_indicator (hs : IsOpen s) (hy : y ≤ 0) :
UpperSemicontinuous (indicator s fun _x => y) | IsOpen.lowerSemicontinuous_indicator (β := βᵒᵈ) hs hy | theorem | IsOpen.upperSemicontinuous_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"IsOpen.lowerSemicontinuous_indicator",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.upperSemicontinuousOn_indicator (hs : IsOpen s) (hy : y ≤ 0) :
UpperSemicontinuousOn (indicator s fun _x => y) t | (hs.upperSemicontinuous_indicator hy).upperSemicontinuousOn t | theorem | IsOpen.upperSemicontinuousOn_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.upperSemicontinuousAt_indicator (hs : IsOpen s) (hy : y ≤ 0) :
UpperSemicontinuousAt (indicator s fun _x => y) x | (hs.upperSemicontinuous_indicator hy).upperSemicontinuousAt x | theorem | IsOpen.upperSemicontinuousAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.upperSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : y ≤ 0) :
UpperSemicontinuousWithinAt (indicator s fun _x => y) t x | (hs.upperSemicontinuous_indicator hy).upperSemicontinuousWithinAt t x | theorem | IsOpen.upperSemicontinuousWithinAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.upperSemicontinuous_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
UpperSemicontinuous (indicator s fun _x => y) | IsClosed.lowerSemicontinuous_indicator (β := βᵒᵈ) hs hy | theorem | IsClosed.upperSemicontinuous_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"IsClosed.lowerSemicontinuous_indicator",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.upperSemicontinuousOn_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
UpperSemicontinuousOn (indicator s fun _x => y) t | (hs.upperSemicontinuous_indicator hy).upperSemicontinuousOn t | theorem | IsClosed.upperSemicontinuousOn_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.upperSemicontinuousAt_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
UpperSemicontinuousAt (indicator s fun _x => y) x | (hs.upperSemicontinuous_indicator hy).upperSemicontinuousAt x | theorem | IsClosed.upperSemicontinuousAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.upperSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
UpperSemicontinuousWithinAt (indicator s fun _x => y) t x | (hs.upperSemicontinuous_indicator hy).upperSemicontinuousWithinAt t x | theorem | IsClosed.upperSemicontinuousWithinAt_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iff_isOpen_preimage :
UpperSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Iio y) | ⟨fun H y => isOpen_iff_mem_nhds.2 fun x hx => H x y hx, fun H _x y y_lt =>
IsOpen.mem_nhds (H y) y_lt⟩ | theorem | upperSemicontinuous_iff_isOpen_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"IsOpen.mem_nhds",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.isOpen_preimage (hf : UpperSemicontinuous f) (y : β) :
IsOpen (f ⁻¹' Iio y) | upperSemicontinuous_iff_isOpen_preimage.1 hf y | theorem | UpperSemicontinuous.isOpen_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iff_isClosed_preimage {f : α → γ} :
UpperSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Ici y) | by
rw [upperSemicontinuous_iff_isOpen_preimage]
simp only [← isOpen_compl_iff, ← preimage_compl, compl_Ici] | theorem | upperSemicontinuous_iff_isClosed_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuous",
"isOpen_compl_iff",
"upperSemicontinuous_iff_isOpen_preimage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.isClosed_preimage {f : α → γ} (hf : UpperSemicontinuous f) (y : γ) :
IsClosed (f ⁻¹' Ici y) | upperSemicontinuous_iff_isClosed_preimage.1 hf y | theorem | UpperSemicontinuous.isClosed_preimage | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.upperSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) :
UpperSemicontinuousWithinAt f s x | fun _y hy => h (Iio_mem_nhds hy) | theorem | ContinuousWithinAt.upperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousWithinAt",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.upperSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) :
UpperSemicontinuousAt f x | fun _y hy => h (Iio_mem_nhds hy) | theorem | ContinuousAt.upperSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.upperSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) :
UpperSemicontinuousOn f s | fun x hx => (h x hx).upperSemicontinuousWithinAt | theorem | ContinuousOn.upperSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousOn",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.upperSemicontinuous {f : α → γ} (h : Continuous f) : UpperSemicontinuous f | fun _x => h.continuousAt.upperSemicontinuousAt | theorem | Continuous.upperSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_iff_limsup_le {f : α → γ} :
UpperSemicontinuousWithinAt f s x ↔ limsup f (𝓝[s] x) ≤ f x | lowerSemicontinuousWithinAt_iff_le_liminf (γ := γᵒᵈ)
alias ⟨UpperSemicontinuousWithinAt.limsup_le, _⟩ := upperSemicontinuousWithinAt_iff_limsup_le | theorem | upperSemicontinuousWithinAt_iff_limsup_le | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_iff_le_liminf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_iff_limsup_le {f : α → γ} :
UpperSemicontinuousAt f x ↔ limsup f (𝓝 x) ≤ f x | lowerSemicontinuousAt_iff_le_liminf (γ := γᵒᵈ)
alias ⟨UpperSemicontinuousAt.limsup_le, _⟩ := upperSemicontinuousAt_iff_limsup_le | theorem | upperSemicontinuousAt_iff_limsup_le | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousAt",
"lowerSemicontinuousAt_iff_le_liminf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iff_limsup_le {f : α → γ} :
UpperSemicontinuous f ↔ ∀ x, limsup f (𝓝 x) ≤ f x | lowerSemicontinuous_iff_le_liminf (γ := γᵒᵈ)
alias ⟨UpperSemicontinuous.limsup_le, _⟩ := upperSemicontinuous_iff_limsup_le | theorem | upperSemicontinuous_iff_limsup_le | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuous",
"lowerSemicontinuous_iff_le_liminf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iff_limsup_le {f : α → γ} :
UpperSemicontinuousOn f s ↔ ∀ x ∈ s, limsup f (𝓝[s] x) ≤ f x | lowerSemicontinuousOn_iff_le_liminf (γ := γᵒᵈ)
alias ⟨UpperSemicontinuousOn.limsup_le, _⟩ := upperSemicontinuousOn_iff_limsup_le | theorem | upperSemicontinuousOn_iff_limsup_le | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_iff_le_liminf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.isCompact_inter_preimage_Ici {f : α → γ}
(hfs : UpperSemicontinuousOn f s) (ks : IsCompact s) (c : γ) :
IsCompact (s ∩ f ⁻¹' Ici c) | LowerSemicontinuousOn.isCompact_inter_preimage_Iic (γ := γᵒᵈ) hfs ks c | theorem | UpperSemicontinuousOn.isCompact_inter_preimage_Ici | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsCompact",
"LowerSemicontinuousOn.isCompact_inter_preimage_Iic",
"UpperSemicontinuousOn"
] | The overlevel sets of an upper semicontinuous function on a compact set are compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousOn.inter_biInter_preimage_Ici_eq_empty_iff_exists_finset
{ι : Type*} {f : ι → α → γ}
(ks : IsCompact s) {I : Set ι} {c : γ} (hfi : ∀ i ∈ I, UpperSemicontinuousOn (f i) s) :
s ∩ ⋂ i ∈ I, (f i) ⁻¹' Ici c = ∅ ↔ ∃ u : Finset I, ∀ x ∈ s, ∃ i ∈ u, f i x < c | LowerSemicontinuousOn.inter_biInter_preimage_Iic_eq_empty_iff_exists_finset ks hfi (γ := γᵒᵈ) | theorem | UpperSemicontinuousOn.inter_biInter_preimage_Ici_eq_empty_iff_exists_finset | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"IsCompact",
"LowerSemicontinuousOn.inter_biInter_preimage_Iic_eq_empty_iff_exists_finset",
"Set",
"UpperSemicontinuousOn"
] | An intersection of overlevel sets of a lower semicontinuous function
on a compact set is empty if and only if a finite sub-intersection is already empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upperSemicontinuousOn_iff_isClosed_hypograph {f : α → γ} (hs : IsClosed s) :
UpperSemicontinuousOn f s ↔ IsClosed {p : α × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} | lowerSemicontinuousOn_iff_isClosed_epigraph hs (γ := γᵒᵈ) | theorem | upperSemicontinuousOn_iff_isClosed_hypograph | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_iff_isClosed_epigraph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iff_IsClosed_hypograph {f : α → γ} :
UpperSemicontinuous f ↔ IsClosed {p : α × γ | p.2 ≤ f p.1} | lowerSemicontinuous_iff_isClosed_epigraph (γ := γᵒᵈ)
alias ⟨UpperSemicontinuous.IsClosed_hypograph, _⟩ := upperSemicontinuous_iff_IsClosed_hypograph | theorem | upperSemicontinuous_iff_IsClosed_hypograph | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"UpperSemicontinuous",
"lowerSemicontinuous_iff_isClosed_epigraph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iff_preimage_Iio [Preorder β] :
UpperSemicontinuousOn f s ↔ ∀ b, ∃ u : Set α, IsOpen u ∧ s ∩ f ⁻¹' Set.Iio b = s ∩ u | lowerSemicontinuousOn_iff_preimage_Ioi (β := βᵒᵈ) | theorem | upperSemicontinuousOn_iff_preimage_Iio | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsOpen",
"Preorder",
"Set",
"Set.Iio",
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_iff_preimage_Ioi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iff_preimage_Ici [LinearOrder β] :
UpperSemicontinuousOn f s ↔ ∀ b, ∃ v : Set α, IsClosed v ∧ s ∩ f ⁻¹' Set.Ici b = s ∩ v | lowerSemicontinuousOn_iff_preimage_Iic (γ := βᵒᵈ) | theorem | upperSemicontinuousOn_iff_preimage_Ici | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsClosed",
"LinearOrder",
"Set",
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_iff_preimage_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_inv_iff :
LowerSemicontinuousWithinAt f⁻¹ s a ↔ UpperSemicontinuousWithinAt f s a | by
rw [lowerSemicontinuousWithinAt_iff, inv_surjective.forall, upperSemicontinuousWithinAt_iff]
simp
@[to_additive]
alias ⟨_, UpperSemicontinuousWithinAt.inv⟩ := lowerSemicontinuousWithinAt_inv_iff | theorem | lowerSemicontinuousWithinAt_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt",
"UpperSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_iff",
"upperSemicontinuousWithinAt_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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