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
value |
|---|---|---|---|---|---|---|---|---|---|---|
upperSemicontinuousWithinAt_inv_iff :
UpperSemicontinuousWithinAt f⁻¹ s a ↔ LowerSemicontinuousWithinAt f s a | by
simp [← lowerSemicontinuousWithinAt_inv_iff]
@[to_additive]
alias ⟨_, LowerSemicontinuousWithinAt.inv⟩ := upperSemicontinuousWithinAt_inv_iff | theorem | upperSemicontinuousWithinAt_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt",
"UpperSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_inv_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuouAt_inv_iff :
LowerSemicontinuousAt f⁻¹ a ↔ UpperSemicontinuousAt f a | by
simp [← lowerSemicontinuousWithinAt_univ_iff, ← upperSemicontinuousWithinAt_univ_iff]
@[to_additive]
alias ⟨_, UpperSemicontinuousAt.inv⟩ := lowerSemicontinuouAt_inv_iff | theorem | lowerSemicontinuouAt_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"UpperSemicontinuousAt",
"lowerSemicontinuousWithinAt_univ_iff",
"upperSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_inv_iff :
UpperSemicontinuousAt f⁻¹ a ↔ LowerSemicontinuousAt f a | by
simp [← lowerSemicontinuousWithinAt_univ_iff, ← upperSemicontinuousWithinAt_univ_iff]
@[to_additive]
alias ⟨_, LowerSemicontinuousAt.inv⟩ := upperSemicontinuousAt_inv_iff | theorem | upperSemicontinuousAt_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt",
"UpperSemicontinuousAt",
"lowerSemicontinuousWithinAt_univ_iff",
"upperSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_inv_iff :
LowerSemicontinuousOn f⁻¹ s ↔ UpperSemicontinuousOn f s | by
simp [lowerSemicontinuousOn_iff, upperSemicontinuousOn_iff]
@[to_additive]
alias ⟨_, UpperSemicontinuousOn.inv⟩ := lowerSemicontinuousOn_inv_iff | theorem | lowerSemicontinuousOn_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_iff",
"upperSemicontinuousOn_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_inv_iff :
UpperSemicontinuousOn f⁻¹ s ↔ LowerSemicontinuousOn f s | by
simp [← lowerSemicontinuousOn_inv_iff]
@[to_additive]
alias ⟨_, LowerSemicontinuousOn.inv⟩ := upperSemicontinuousOn_inv_iff | theorem | upperSemicontinuousOn_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn",
"UpperSemicontinuousOn",
"lowerSemicontinuousOn_inv_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemiContinuous_inv_iff :
LowerSemicontinuous f⁻¹ ↔ UpperSemicontinuous f | by
simp [← upperSemicontinuousOn_univ_iff, ← lowerSemicontinuousOn_univ_iff]
@[to_additive]
alias ⟨_, UpperSemicontinuous.inv⟩ := lowerSemiContinuous_inv_iff | theorem | lowerSemiContinuous_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"UpperSemicontinuous",
"lowerSemicontinuousOn_univ_iff",
"upperSemicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemiContinuous_inv_iff :
UpperSemicontinuous f⁻¹ ↔ LowerSemicontinuous f | by
simp [← upperSemicontinuousOn_univ_iff, ← lowerSemicontinuousOn_univ_iff]
@[to_additive]
alias ⟨_, LowerSemicontinuous.inv⟩ := upperSemiContinuous_inv_iff | theorem | upperSemiContinuous_inv_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous",
"UpperSemicontinuous",
"lowerSemicontinuousOn_univ_iff",
"upperSemicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_upperSemicontinuousWithinAt {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Monotone g) :
UpperSemicontinuousWithinAt (g ∘ f) s x | ContinuousAt.comp_lowerSemicontinuousWithinAt (γ := γᵒᵈ) (δ := δᵒᵈ) hg hf gmon.dual | theorem | ContinuousAt.comp_upperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"ContinuousAt.comp_lowerSemicontinuousWithinAt",
"Monotone",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_upperSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x))
(hf : UpperSemicontinuousAt f x) (gmon : Monotone g) : UpperSemicontinuousAt (g ∘ f) x | ContinuousAt.comp_lowerSemicontinuousAt (γ := γᵒᵈ) (δ := δᵒᵈ) hg hf gmon.dual | theorem | ContinuousAt.comp_upperSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"ContinuousAt.comp_lowerSemicontinuousAt",
"Monotone",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_upperSemicontinuousOn {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : UpperSemicontinuousOn f s) (gmon : Monotone g) : UpperSemicontinuousOn (g ∘ f) s | fun x hx => hg.continuousAt.comp_upperSemicontinuousWithinAt (hf x hx) gmon | theorem | Continuous.comp_upperSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"Monotone",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_upperSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : UpperSemicontinuous f) (gmon : Monotone g) : UpperSemicontinuous (g ∘ f) | fun x =>
hg.continuousAt.comp_upperSemicontinuousAt (hf x) gmon | theorem | Continuous.comp_upperSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"Monotone",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_upperSemicontinuousWithinAt_antitone {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Antitone g) :
LowerSemicontinuousWithinAt (g ∘ f) s x | ContinuousAt.comp_upperSemicontinuousWithinAt (δ := δᵒᵈ) hg hf gmon | theorem | ContinuousAt.comp_upperSemicontinuousWithinAt_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"ContinuousAt",
"ContinuousAt.comp_upperSemicontinuousWithinAt",
"LowerSemicontinuousWithinAt",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.comp_upperSemicontinuousAt_antitone {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousAt f x) (gmon : Antitone g) :
LowerSemicontinuousAt (g ∘ f) x | ContinuousAt.comp_upperSemicontinuousAt (δ := δᵒᵈ) hg hf gmon | theorem | ContinuousAt.comp_upperSemicontinuousAt_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"ContinuousAt",
"ContinuousAt.comp_upperSemicontinuousAt",
"LowerSemicontinuousAt",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_upperSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : UpperSemicontinuousOn f s) (gmon : Antitone g) : LowerSemicontinuousOn (g ∘ f) s | fun x hx => hg.continuousAt.comp_upperSemicontinuousWithinAt_antitone (hf x hx) gmon | theorem | Continuous.comp_upperSemicontinuousOn_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"Continuous",
"LowerSemicontinuousOn",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.comp_upperSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
(hf : UpperSemicontinuous f) (gmon : Antitone g) : LowerSemicontinuous (g ∘ f) | fun x =>
hg.continuousAt.comp_upperSemicontinuousAt_antitone (hf x) gmon | theorem | Continuous.comp_upperSemicontinuous_antitone | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Antitone",
"Continuous",
"LowerSemicontinuous",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousWithinAt.add' {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x)
(hg : UpperSemicontinuousWithinAt g s x)
(hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
UpperSemicontinuousWithinAt (fun z => f z + g z) s x | LowerSemicontinuousWithinAt.add' (γ := γᵒᵈ) hf hg hcont | theorem | UpperSemicontinuousWithinAt.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"LowerSemicontinuousWithinAt.add'",
"UpperSemicontinuousWithinAt"
] | 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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousAt.add' {f g : α → γ} (hf : UpperSemicontinuousAt f x)
(hg : UpperSemicontinuousAt g x)
(hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
UpperSemicontinuousAt (fun z => f z + g z) x | by
simp_rw [← upperSemicontinuousWithinAt_univ_iff] at *
exact hf.add' hg hcont | theorem | UpperSemicontinuousAt.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"UpperSemicontinuousAt",
"upperSemicontinuousWithinAt_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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousOn.add' {f g : α → γ} (hf : UpperSemicontinuousOn f s)
(hg : UpperSemicontinuousOn g s)
(hcont : ∀ x ∈ s, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
UpperSemicontinuousOn (fun z => f z + g z) s | fun x hx =>
UpperSemicontinuousWithinAt.add' (hf x hx) (hg x hx) (hcont x hx) | theorem | UpperSemicontinuousOn.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"UpperSemicontinuousOn",
"UpperSemicontinuousWithinAt.add'"
] | 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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuous.add' {f g : α → γ} (hf : UpperSemicontinuous f)
(hg : UpperSemicontinuous g)
(hcont : ∀ x, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
UpperSemicontinuous fun z => f z + g z | fun x => UpperSemicontinuousAt.add' (hf x) (hg x) (hcont x) | theorem | UpperSemicontinuous.add' | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"UpperSemicontinuous",
"UpperSemicontinuousAt.add'"
] | 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 `[ContinuousAdd]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousWithinAt.add {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x)
(hg : UpperSemicontinuousWithinAt g s x) :
UpperSemicontinuousWithinAt (fun z => f z + g z) s x | hf.add' hg continuous_add.continuousAt | theorem | UpperSemicontinuousWithinAt.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousWithinAt"
] | The sum of two upper semicontinuous functions is upper 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 |
UpperSemicontinuousAt.add {f g : α → γ} (hf : UpperSemicontinuousAt f x)
(hg : UpperSemicontinuousAt g x) : UpperSemicontinuousAt (fun z => f z + g z) x | hf.add' hg continuous_add.continuousAt | theorem | UpperSemicontinuousAt.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousAt"
] | The sum of two upper semicontinuous functions is upper 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 |
UpperSemicontinuousOn.add {f g : α → γ} (hf : UpperSemicontinuousOn f s)
(hg : UpperSemicontinuousOn g s) : UpperSemicontinuousOn (fun z => f z + g z) s | hf.add' hg fun _x _hx => continuous_add.continuousAt | theorem | UpperSemicontinuousOn.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousOn"
] | The sum of two upper semicontinuous functions is upper 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 |
UpperSemicontinuous.add {f g : α → γ} (hf : UpperSemicontinuous f)
(hg : UpperSemicontinuous g) : UpperSemicontinuous fun z => f z + g z | hf.add' hg fun _x => continuous_add.continuousAt | theorem | UpperSemicontinuous.add | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuous"
] | The sum of two upper semicontinuous functions is upper 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 |
upperSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, UpperSemicontinuousWithinAt (f i) s x) :
UpperSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x | lowerSemicontinuousWithinAt_sum (γ := γᵒᵈ) ha | theorem | upperSemicontinuousWithinAt_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"UpperSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_sum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x) :
UpperSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x | by
simp_rw [← upperSemicontinuousWithinAt_univ_iff] at *
exact upperSemicontinuousWithinAt_sum ha | theorem | upperSemicontinuousAt_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"UpperSemicontinuousAt",
"upperSemicontinuousWithinAt_sum",
"upperSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, UpperSemicontinuousOn (f i) s) :
UpperSemicontinuousOn (fun z => ∑ i ∈ a, f i z) s | fun x hx =>
upperSemicontinuousWithinAt_sum fun i hi => ha i hi x hx | theorem | upperSemicontinuousOn_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"UpperSemicontinuousOn",
"upperSemicontinuousWithinAt_sum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, UpperSemicontinuous (f i)) : UpperSemicontinuous fun z => ∑ i ∈ a, f i z | fun x => upperSemicontinuousAt_sum fun i hi => ha i hi x | theorem | upperSemicontinuous_sum | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Finset",
"UpperSemicontinuous",
"upperSemicontinuousAt_sum"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousWithinAt.inf
(hf : UpperSemicontinuousWithinAt f s a) (hg : UpperSemicontinuousWithinAt g s a) :
UpperSemicontinuousWithinAt (fun x ↦ f x ⊓ g x) s a | LowerSemicontinuousWithinAt.sup (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuousWithinAt.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt.sup",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousAt.inf
(hf : UpperSemicontinuousAt f a) (hg : UpperSemicontinuousAt g a) :
UpperSemicontinuousAt (fun x ↦ f x ⊓ g x) a | LowerSemicontinuousAt.sup (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuousAt.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt.sup",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.inf
(hf : UpperSemicontinuousOn f s) (hg : UpperSemicontinuousOn g s) :
UpperSemicontinuousOn (fun x ↦ f x ⊓ g x) s | LowerSemicontinuousOn.sup (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuousOn.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn.sup",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.inf (hf : UpperSemicontinuous f) (hg : UpperSemicontinuous g) :
UpperSemicontinuous (fun x ↦ f x ⊓ g x) | LowerSemicontinuous.sup (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuous.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous.sup",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousWithinAt.sup
(hf : UpperSemicontinuousWithinAt f s a) (hg : UpperSemicontinuousWithinAt g s a) :
UpperSemicontinuousWithinAt (fun x ↦ f x ⊔ g x) s a | LowerSemicontinuousWithinAt.inf (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuousWithinAt.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousWithinAt.inf",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousAt.sup
(hf : UpperSemicontinuousAt f a) (hg : UpperSemicontinuousAt g a) :
UpperSemicontinuousAt (fun x ↦ f x ⊔ g x) a | LowerSemicontinuousAt.inf (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuousAt.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousAt.inf",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.sup
(hf : UpperSemicontinuousOn f s) (hg : UpperSemicontinuousOn g s) :
UpperSemicontinuousOn (fun x ↦ f x ⊔ g x) s | LowerSemicontinuousOn.inf (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuousOn.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuousOn.inf",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.sup (hf : UpperSemicontinuous f) (hg : UpperSemicontinuous g) :
UpperSemicontinuous fun x ↦ f x ⊔ g x | LowerSemicontinuous.inf (β := βᵒᵈ) hf hg | theorem | UpperSemicontinuous.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"LowerSemicontinuous.inf",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_ciInf {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝[s] x, BddBelow (range fun i => f i y))
(h : ∀ i, UpperSemicontinuousWithinAt (f i) s x) :
UpperSemicontinuousWithinAt (fun x' => ⨅ i, f i x') s x | lowerSemicontinuousWithinAt_ciSup (δ' := δ'ᵒᵈ) bdd h | theorem | upperSemicontinuousWithinAt_ciInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_iInf {f : ι → α → δ}
(h : ∀ i, UpperSemicontinuousWithinAt (f i) s x) :
UpperSemicontinuousWithinAt (fun x' => ⨅ i, f i x') s x | lowerSemicontinuousWithinAt_iSup (δ := δᵒᵈ) h | theorem | upperSemicontinuousWithinAt_iInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousWithinAt",
"lowerSemicontinuousWithinAt_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, UpperSemicontinuousWithinAt (f i hi) s x) :
UpperSemicontinuousWithinAt (fun x' => ⨅ (i) (hi), f i hi x') s x | upperSemicontinuousWithinAt_iInf fun i => upperSemicontinuousWithinAt_iInf fun hi => h i hi | theorem | upperSemicontinuousWithinAt_biInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousWithinAt",
"upperSemicontinuousWithinAt_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_ciInf {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝 x, BddBelow (range fun i => f i y)) (h : ∀ i, UpperSemicontinuousAt (f i) x) :
UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x | @lowerSemicontinuousAt_ciSup α _ x ι δ'ᵒᵈ _ f bdd h | theorem | upperSemicontinuousAt_ciInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousAt",
"lowerSemicontinuousAt_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_iInf {f : ι → α → δ} (h : ∀ i, UpperSemicontinuousAt (f i) x) :
UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x | @lowerSemicontinuousAt_iSup α _ x ι δᵒᵈ _ f h | theorem | upperSemicontinuousAt_iInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousAt",
"lowerSemicontinuousAt_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, UpperSemicontinuousAt (f i hi) x) :
UpperSemicontinuousAt (fun x' => ⨅ (i) (hi), f i hi x') x | upperSemicontinuousAt_iInf fun i => upperSemicontinuousAt_iInf fun hi => h i hi | theorem | upperSemicontinuousAt_biInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousAt",
"upperSemicontinuousAt_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_ciInf {f : ι → α → δ'}
(bdd : ∀ x ∈ s, BddBelow (range fun i => f i x)) (h : ∀ i, UpperSemicontinuousOn (f i) s) :
UpperSemicontinuousOn (fun x' => ⨅ i, f i x') s | fun x hx =>
upperSemicontinuousWithinAt_ciInf (eventually_nhdsWithin_of_forall bdd) fun i => h i x hx | theorem | upperSemicontinuousOn_ciInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousOn",
"eventually_nhdsWithin_of_forall",
"upperSemicontinuousWithinAt_ciInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iInf {f : ι → α → δ} (h : ∀ i, UpperSemicontinuousOn (f i) s) :
UpperSemicontinuousOn (fun x' => ⨅ i, f i x') s | fun x hx =>
upperSemicontinuousWithinAt_iInf fun i => h i x hx | theorem | upperSemicontinuousOn_iInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousOn",
"upperSemicontinuousWithinAt_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, UpperSemicontinuousOn (f i hi) s) :
UpperSemicontinuousOn (fun x' => ⨅ (i) (hi), f i hi x') s | upperSemicontinuousOn_iInf fun i => upperSemicontinuousOn_iInf fun hi => h i hi | theorem | upperSemicontinuousOn_biInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuousOn",
"upperSemicontinuousOn_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_ciInf {f : ι → α → δ'} (bdd : ∀ x, BddBelow (range fun i => f i x))
(h : ∀ i, UpperSemicontinuous (f i)) : UpperSemicontinuous fun x' => ⨅ i, f i x' | fun x =>
upperSemicontinuousAt_ciInf (Eventually.of_forall bdd) fun i => h i x | theorem | upperSemicontinuous_ciInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuous",
"upperSemicontinuousAt_ciInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iInf {f : ι → α → δ} (h : ∀ i, UpperSemicontinuous (f i)) :
UpperSemicontinuous fun x' => ⨅ i, f i x' | fun x => upperSemicontinuousAt_iInf fun i => h i x | theorem | upperSemicontinuous_iInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuous",
"upperSemicontinuousAt_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, UpperSemicontinuous (f i hi)) :
UpperSemicontinuous fun x' => ⨅ (i) (hi), f i hi x' | upperSemicontinuous_iInf fun i => upperSemicontinuous_iInf fun hi => h i hi | theorem | upperSemicontinuous_biInf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"UpperSemicontinuous",
"upperSemicontinuous_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_iff_lower_upperSemicontinuousWithinAt {f : α → γ} :
ContinuousWithinAt f s x ↔
LowerSemicontinuousWithinAt f s x ∧ UpperSemicontinuousWithinAt f s x | by
refine ⟨fun h => ⟨h.lowerSemicontinuousWithinAt, h.upperSemicontinuousWithinAt⟩, ?_⟩
rintro ⟨h₁, h₂⟩
intro v hv
simp only [Filter.mem_map]
by_cases! Hl : ∃ l, l < f x
· rcases exists_Ioc_subset_of_mem_nhds hv Hl with ⟨l, lfx, hl⟩
by_cases! Hu : ∃ u, f x < u
· rcases exists_Ico_subset_of_mem_nhds ... | theorem | continuousWithinAt_iff_lower_upperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousWithinAt",
"Filter.Eventually.of_forall",
"Filter.mem_map",
"LowerSemicontinuousWithinAt",
"UpperSemicontinuousWithinAt",
"exists_Ioc_subset_of_mem_nhds",
"le_antisymm",
"le_or_gt",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_iff_lower_upperSemicontinuousAt {f : α → γ} :
ContinuousAt f x ↔ LowerSemicontinuousAt f x ∧ UpperSemicontinuousAt f x | by
simp_rw [← continuousWithinAt_univ, ← lowerSemicontinuousWithinAt_univ_iff, ←
upperSemicontinuousWithinAt_univ_iff, continuousWithinAt_iff_lower_upperSemicontinuousWithinAt] | theorem | continuousAt_iff_lower_upperSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousAt",
"LowerSemicontinuousAt",
"UpperSemicontinuousAt",
"continuousWithinAt_iff_lower_upperSemicontinuousWithinAt",
"continuousWithinAt_univ",
"lowerSemicontinuousWithinAt_univ_iff",
"upperSemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_iff_lower_upperSemicontinuousOn {f : α → γ} :
ContinuousOn f s ↔ LowerSemicontinuousOn f s ∧ UpperSemicontinuousOn f s | by
simp only [ContinuousOn, continuousWithinAt_iff_lower_upperSemicontinuousWithinAt]
exact
⟨fun H => ⟨fun x hx => (H x hx).1, fun x hx => (H x hx).2⟩, fun H x hx => ⟨H.1 x hx, H.2 x hx⟩⟩ | theorem | continuousOn_iff_lower_upperSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"ContinuousOn",
"LowerSemicontinuousOn",
"UpperSemicontinuousOn",
"continuousWithinAt_iff_lower_upperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff_lower_upperSemicontinuous {f : α → γ} :
Continuous f ↔ LowerSemicontinuous f ∧ UpperSemicontinuous f | by
simp_rw [← continuousOn_univ, continuousOn_iff_lower_upperSemicontinuousOn,
lowerSemicontinuousOn_univ_iff, upperSemicontinuousOn_univ_iff] | theorem | continuous_iff_lower_upperSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Continuous",
"LowerSemicontinuous",
"UpperSemicontinuous",
"continuousOn_iff_lower_upperSemicontinuousOn",
"continuousOn_univ",
"lowerSemicontinuousOn_univ_iff",
"upperSemicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt (r : α → β → Prop) (s : Set α) (x : α) | ∀ y, r x y → ∀ᶠ x' in 𝓝[s] x, r x' y | def | SemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set"
] | A relation `r : α → β → Prop` is semicontinuous within `s` at `x : α`, if whenever `r x y`
is true, it is also true for all `x'` sufficiently close to `x` within `s`.
This notion generalizes lower and upper semicontinuity of functions, as well as
lower and upper hemicontinuity of set-valued correspondences. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
SemicontinuousOn (r : α → β → Prop) (s : Set α) | ∀ x ∈ s, SemicontinuousWithinAt r s x | def | SemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt",
"Set"
] | A relation `r : α → β → Prop` is semicontinuous on `s` if it is semicontinuous within `s` at
each `x ∈ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
SemicontinuousAt (r : α → β → Prop) (x : α) : Prop | ∀ y, r x y → ∀ᶠ x' in 𝓝 x, r x' y | def | SemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [] | A relation `r : α → β → Prop` is semicontinuous at `x : α`, if whenever `r x y`
is true, it is also true for all `x'` sufficiently close to `x`.
This notion generalizes lower and upper semicontinuity of functions, as well as
lower and upper hemicontinuity of set-valued correspondences. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Semicontinuous (r : α → β → Prop) : Prop | ∀ x, SemicontinuousAt r x | def | Semicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt"
] | A relation `r : α → β → Prop` is semicontinuous if it is semicontinuous within `s` at each
`x : α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
semicontinuousWithinAt_iff_frequently :
SemicontinuousWithinAt r s x ↔ ∀ y, (∃ᶠ x' in 𝓝[s] x, ¬ r x' y) → ¬ r x y | by
simp only [← not_eventually, not_imp_not, SemicontinuousWithinAt] | lemma | semicontinuousWithinAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuousOn_iff_frequently :
SemicontinuousOn r s ↔ ∀ x ∈ s, ∀ y, (∃ᶠ x' in 𝓝[s] x, ¬ r x' y) → ¬ r x y | by
simp only [← not_eventually, not_imp_not, SemicontinuousWithinAt, SemicontinuousOn] | lemma | semicontinuousOn_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"SemicontinuousWithinAt",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuousAt_iff_frequently :
SemicontinuousAt r x ↔ ∀ y, (∃ᶠ x' in 𝓝 x, ¬ r x' y) → ¬ r x y | by
simp only [← not_eventually, not_imp_not, SemicontinuousAt] | lemma | semicontinuousAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuous_iff_frequently :
Semicontinuous r ↔ ∀ x y, (∃ᶠ x' in 𝓝 x, ¬ r x' y) → ¬ r x y | by
simp only [← not_eventually, not_imp_not, Semicontinuous, SemicontinuousAt] | lemma | semicontinuous_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousAt",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt.mono (h : SemicontinuousWithinAt r s x) (hst : t ⊆ s) :
SemicontinuousWithinAt r t x | fun y hy =>
Filter.Eventually.filter_mono (nhdsWithin_mono _ hst) (h y hy) | theorem | SemicontinuousWithinAt.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually.filter_mono",
"SemicontinuousWithinAt",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt.congr_of_eventuallyEq {a : α}
(h : SemicontinuousWithinAt r s a)
(has : a ∈ s) (hfg : ∀ᶠ x in 𝓝[s] a, ∀ y, r x y ↔ r' x y) :
SemicontinuousWithinAt r' s a | by
intro b hb
simp_rw [← propext_iff, ← funext_iff] at hfg
rw [← Filter.EventuallyEq.eq_of_nhdsWithin hfg has] at hb
filter_upwards [hfg, h b hb] with x hx hxb
exact hx ▸ hxb | theorem | SemicontinuousWithinAt.congr_of_eventuallyEq | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.EventuallyEq.eq_of_nhdsWithin",
"SemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuousWithinAt_univ_iff :
SemicontinuousWithinAt r univ x ↔ SemicontinuousAt r x | by
simp [SemicontinuousWithinAt, SemicontinuousAt, nhdsWithin_univ] | theorem | semicontinuousWithinAt_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt",
"SemicontinuousWithinAt",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousAt.semicontinuousWithinAt (s : Set α)
(h : SemicontinuousAt r x) : SemicontinuousWithinAt r s x | fun y hy =>
Filter.Eventually.filter_mono nhdsWithin_le_nhds (h y hy) | theorem | SemicontinuousAt.semicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually.filter_mono",
"SemicontinuousAt",
"SemicontinuousWithinAt",
"Set",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousOn.semicontinuousWithinAt (h : SemicontinuousOn r s)
(hx : x ∈ s) : SemicontinuousWithinAt r s x | h x hx | theorem | SemicontinuousOn.semicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"SemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousOn.mono (h : SemicontinuousOn r s) (hst : t ⊆ s) :
SemicontinuousOn r t | fun x hx => (h x (hst hx)).mono hst | theorem | SemicontinuousOn.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuousOn_univ_iff : SemicontinuousOn r univ ↔ Semicontinuous r | by
simp [SemicontinuousOn, Semicontinuous, semicontinuousWithinAt_univ_iff] | theorem | semicontinuousOn_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousOn",
"semicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuous_restrict_iff :
Semicontinuous (s.restrict r) ↔ SemicontinuousOn r s | by
rw [SemicontinuousOn, Semicontinuous, SetCoe.forall]
refine forall₂_congr fun a ha ↦ forall₂_congr fun b _ ↦ ?_
simp only [nhdsWithin_eq_map_subtype_coe ha, eventually_map, restrict] | theorem | semicontinuous_restrict_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousOn",
"SetCoe.forall",
"nhdsWithin_eq_map_subtype_coe"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.semicontinuousAt (h : Semicontinuous r) (x : α) :
SemicontinuousAt r x | h x | theorem | Semicontinuous.semicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.semicontinuousWithinAt (h : Semicontinuous r) (s : Set α)
(x : α) : SemicontinuousWithinAt r s x | (h x).semicontinuousWithinAt s | theorem | Semicontinuous.semicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.semicontinuousOn (h : Semicontinuous r) (s : Set α) :
SemicontinuousOn r s | fun x _hx => h.semicontinuousWithinAt s x | theorem | Semicontinuous.semicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
semicontinuous_iff_isOpen : Semicontinuous r ↔ ∀ b, IsOpen {x | r x b} | by
exact ⟨fun h b ↦ by simpa [isOpen_iff_mem_nhds, Filter.Eventually] using fun x hx ↦ h x b hx,
fun h x b hbx ↦ (h b).mem_nhds hbx⟩ | theorem | semicontinuous_iff_isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually",
"IsOpen",
"Semicontinuous",
"isOpen_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.isOpen (h : Semicontinuous r) (b : β) : IsOpen {x | r x b} | semicontinuous_iff_isOpen.mp h b | theorem | Semicontinuous.isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"Semicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt.inf {r' : α → β → Prop}
(h : SemicontinuousWithinAt r s x) (h' : SemicontinuousWithinAt r' s x) :
SemicontinuousWithinAt (r ⊓ r') s x | fun b ⟨hb, hb'⟩ ↦
(h b hb).and (h' b hb') | theorem | SemicontinuousWithinAt.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt.sup {r' : α → β → Prop}
(h : SemicontinuousWithinAt r s x) (h' : SemicontinuousWithinAt r' s x) :
SemicontinuousWithinAt (r ⊔ r') s x | by
intro b hab
obtain hb | hb' := hab
· exact (h b hb).mono fun _ hx ↦ Or.inl hx
· exact (h' b hb').mono fun _ hx ↦ Or.inr hx | theorem | SemicontinuousWithinAt.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousAt.inf {r' : α → β → Prop}
(h : SemicontinuousAt r x) (h' : SemicontinuousAt r' x) :
SemicontinuousAt (r ⊓ r') x | fun b ⟨hb, hb'⟩ ↦
(h b hb).and (h' b hb') | theorem | SemicontinuousAt.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousAt.sup {r' : α → β → Prop}
(h : SemicontinuousAt r x) (h' : SemicontinuousAt r' x) :
SemicontinuousAt (r ⊔ r') x | by
intro b hab
obtain hb | hb' := hab
· exact (h b hb).mono fun _ hx ↦ Or.inl hx
· exact (h' b hb').mono fun _ hx ↦ Or.inr hx | theorem | SemicontinuousAt.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousOn.inf {r' : α → β → Prop}
(h : SemicontinuousOn r s) (h' : SemicontinuousOn r' s) :
SemicontinuousOn (r ⊓ r') s | fun x hx ↦ (h x hx).inf (h' x hx) | theorem | SemicontinuousOn.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousOn.sup {r' : α → β → Prop}
(h : SemicontinuousOn r s) (h' : SemicontinuousOn r' s) :
SemicontinuousOn (r ⊔ r') s | fun x hx ↦ (h x hx).sup (h' x hx) | theorem | SemicontinuousOn.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.inf {r' : α → β → Prop} (h : Semicontinuous r) (h' : Semicontinuous r') :
Semicontinuous (r ⊓ r') | fun a ↦ (h a).inf (h' a) | theorem | Semicontinuous.inf | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.sup {r' : α → β → Prop} (h : Semicontinuous r) (h' : Semicontinuous r') :
Semicontinuous (r ⊔ r') | fun a ↦ (h a).sup (h' a) | theorem | Semicontinuous.sup | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt.const {f : β → Prop} : SemicontinuousWithinAt (fun _x => f) s x | fun _y hy => Filter.Eventually.of_forall fun _x => hy | theorem | SemicontinuousWithinAt.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually.of_forall",
"SemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousAt.const {f : β → Prop} : SemicontinuousAt (fun _x => f) x | fun _y hy =>
Filter.Eventually.of_forall fun _x => hy | theorem | SemicontinuousAt.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually.of_forall",
"SemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousOn.const {f : β → Prop} : SemicontinuousOn (fun _x => f) s | fun _x _hx =>
SemicontinuousWithinAt.const | theorem | SemicontinuousOn.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"SemicontinuousWithinAt.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.const {f : β → Prop} : Semicontinuous fun _x : α => f | fun _x =>
SemicontinuousAt.const | theorem | Semicontinuous.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"SemicontinuousAt.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousWithinAt.comp (h : SemicontinuousWithinAt r t (g x))
(hg : ContinuousWithinAt g s x) (hst : Set.MapsTo g s t) :
SemicontinuousWithinAt (r ∘ g) s x | (hg.tendsto_nhdsWithin hst <| h · ·) | lemma | SemicontinuousWithinAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousWithinAt",
"SemicontinuousWithinAt",
"Set.MapsTo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousOn.comp {r : α → β → Prop} {γ : Type*}
[TopologicalSpace γ] {g : γ → α} {s : Set γ} {t : Set α}
(h : SemicontinuousOn r t) (hg : ContinuousOn g s)
(hst : Set.MapsTo g s t) :
SemicontinuousOn (r ∘ g) s | fun x hx ↦ h (g x) (hst hx) |>.comp (hg x hx) hst | lemma | SemicontinuousOn.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"SemicontinuousOn",
"Set",
"Set.MapsTo",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemicontinuousAt.comp {r : α → β → Prop} {γ : Type*} [TopologicalSpace γ]
{x : γ} {g : γ → α} (h : SemicontinuousAt r (g x)) (hg : ContinuousAt g x) :
SemicontinuousAt (r ∘ g) x | (hg <| h · ·) | lemma | SemicontinuousAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousAt",
"SemicontinuousAt",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Semicontinuous.comp {r : α → β → Prop} {γ : Type*} [TopologicalSpace γ]
{g : γ → α} (h : Semicontinuous r) (hg : Continuous g) :
Semicontinuous (r ∘ g) | fun _ ↦ (h.semicontinuousAt _).comp hg.continuousAt | lemma | Semicontinuous.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"Semicontinuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) | SemicontinuousWithinAt (f · > ·) s x | abbrev | LowerSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt",
"Set"
] | A real function `f` is lower semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all
`x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general
preordered space, using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousOn (f : α → β) (s : Set α) | SemicontinuousOn (f · > ·) s | abbrev | LowerSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"Set"
] | A real function `f` is lower semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`,
for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in
a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousAt (f : α → β) (x : α) | SemicontinuousAt (f · > ·) x | abbrev | LowerSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt"
] | A real function `f` is lower semicontinuous at `x` if, for any `ε > 0`, for all `x'` close
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space,
using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuous (f : α → β) | Semicontinuous (f · > ·) | abbrev | LowerSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous"
] | A real function `f` is lower semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space,
using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) | SemicontinuousWithinAt (f · < ·) s x | abbrev | UpperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt",
"Set"
] | A real function `f` is upper semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all
`x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general
preordered space, using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousOn (f : α → β) (s : Set α) | SemicontinuousOn (f · < ·) s | abbrev | UpperSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"Set"
] | A real function `f` is upper semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`,
for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a
general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuousAt (f : α → β) (x : α) | SemicontinuousAt (f · < ·) x | abbrev | UpperSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt"
] | A real function `f` is upper semicontinuous at `x` if, for any `ε > 0`, for all `x'` close
enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space,
using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperSemicontinuous (f : α → β) | Semicontinuous (f · < ·) | abbrev | UpperSemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous"
] | A real function `f` is upper semicontinuous if, for any `ε > 0`, for any `x`, for all `x'`
close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered
space, using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerSemicontinuousWithinAt_iff {f : α → β} {s : Set α} {x : α} :
LowerSemicontinuousWithinAt f s x ↔ ∀ y, y < f x → ∀ᶠ x' in 𝓝[s] x, y < f x' | Iff.rfl | lemma | lowerSemicontinuousWithinAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_iff {f : α → β} {s : Set α} :
LowerSemicontinuousOn f s ↔ ∀ x ∈ s, LowerSemicontinuousWithinAt f s x | Iff.rfl | lemma | lowerSemicontinuousOn_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousOn",
"LowerSemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_iff {f : α → β} {x : α} :
LowerSemicontinuousAt f x ↔ ∀ y, y < f x → ∀ᶠ x' in 𝓝 x, y < f x' | Iff.rfl | lemma | lowerSemicontinuousAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_iff {f : α → β} :
LowerSemicontinuous f ↔ ∀ x, LowerSemicontinuousAt f x | Iff.rfl | lemma | lowerSemicontinuous_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"LowerSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.