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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upperSemicontinuousWithinAt_iff {f : α → β} {s : Set α} {x : α} :
UpperSemicontinuousWithinAt f s x ↔ ∀ y, f x < y → ∀ᶠ x' in 𝓝[s] x, f x' < y | Iff.rfl | lemma | upperSemicontinuousWithinAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iff {f : α → β} {s : Set α} :
UpperSemicontinuousOn f s ↔ ∀ x ∈ s, UpperSemicontinuousWithinAt f s x | Iff.rfl | lemma | upperSemicontinuousOn_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperSemicontinuousOn",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_iff {f : α → β} {x : α} :
UpperSemicontinuousAt f x ↔ ∀ y, f x < y → ∀ᶠ x' in 𝓝 x, f x' < y | Iff.rfl | lemma | upperSemicontinuousAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iff {f : α → β} :
UpperSemicontinuous f ↔ ∀ x, UpperSemicontinuousAt f x | Iff.rfl | lemma | upperSemicontinuous_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuous",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt.mono (h : LowerSemicontinuousWithinAt f s x) (hst : t ⊆ s) :
LowerSemicontinuousWithinAt f t x | SemicontinuousWithinAt.mono h hst | theorem | LowerSemicontinuousWithinAt.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousWithinAt",
"SemicontinuousWithinAt.mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt.congr_of_eventuallyEq {a : α}
(h : LowerSemicontinuousWithinAt f s a)
(has : a ∈ s) (hfg : f =ᶠ[𝓝[s] a] g) :
LowerSemicontinuousWithinAt g s a | SemicontinuousWithinAt.congr_of_eventuallyEq h has <| by
filter_upwards [hfg] with x hx
simp [hx] | theorem | LowerSemicontinuousWithinAt.congr_of_eventuallyEq | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousWithinAt",
"SemicontinuousWithinAt.congr_of_eventuallyEq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_univ_iff :
LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x | semicontinuousWithinAt_univ_iff | theorem | lowerSemicontinuousWithinAt_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousAt",
"LowerSemicontinuousWithinAt",
"semicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousAt.lowerSemicontinuousWithinAt (s : Set α)
(h : LowerSemicontinuousAt f x) : LowerSemicontinuousWithinAt f s x | h.semicontinuousWithinAt s | theorem | LowerSemicontinuousAt.lowerSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousAt",
"LowerSemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.lowerSemicontinuousWithinAt (h : LowerSemicontinuousOn f s)
(hx : x ∈ s) : LowerSemicontinuousWithinAt f s x | h.semicontinuousWithinAt hx | theorem | LowerSemicontinuousOn.lowerSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousOn",
"LowerSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.mono (h : LowerSemicontinuousOn f s) (hst : t ⊆ s) :
LowerSemicontinuousOn f t | SemicontinuousOn.mono h hst | theorem | LowerSemicontinuousOn.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousOn",
"SemicontinuousOn.mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_univ_iff : LowerSemicontinuousOn f univ ↔ LowerSemicontinuous f | semicontinuousOn_univ_iff | theorem | lowerSemicontinuousOn_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"LowerSemicontinuousOn",
"semicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_restrict_iff :
LowerSemicontinuous (s.restrict f) ↔ LowerSemicontinuousOn f s | semicontinuous_restrict_iff (r := (f · > ·)) | theorem | lowerSemicontinuous_restrict_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"LowerSemicontinuousOn",
"semicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.lowerSemicontinuousAt (h : LowerSemicontinuous f) (x : α) :
LowerSemicontinuousAt f x | h x | theorem | LowerSemicontinuous.lowerSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"LowerSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.lowerSemicontinuousWithinAt (h : LowerSemicontinuous f) (s : Set α)
(x : α) : LowerSemicontinuousWithinAt f s x | (h x).semicontinuousWithinAt s | theorem | LowerSemicontinuous.lowerSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"LowerSemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.lowerSemicontinuousOn (h : LowerSemicontinuous f) (s : Set α) :
LowerSemicontinuousOn f s | h.semicontinuousOn s | theorem | LowerSemicontinuous.lowerSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"LowerSemicontinuousOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_const : LowerSemicontinuousWithinAt (fun _x => z) s x | SemicontinuousWithinAt.const | theorem | lowerSemicontinuousWithinAt_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousWithinAt",
"SemicontinuousWithinAt.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_const : LowerSemicontinuousAt (fun _x => z) x | SemicontinuousAt.const | theorem | lowerSemicontinuousAt_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousAt",
"SemicontinuousAt.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_const : LowerSemicontinuousOn (fun _x => z) s | SemicontinuousOn.const | theorem | lowerSemicontinuousOn_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousOn",
"SemicontinuousOn.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_const : LowerSemicontinuous fun _x : α => z | Semicontinuous.const | theorem | lowerSemicontinuous_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"Semicontinuous.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousWithinAt.comp
(hf : LowerSemicontinuousWithinAt f s (g x)) (hg : ContinuousWithinAt g t x)
(hg' : MapsTo g t s) :
LowerSemicontinuousWithinAt (f ∘ g) t x | SemicontinuousWithinAt.comp hf hg hg' | theorem | LowerSemicontinuousWithinAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousWithinAt",
"LowerSemicontinuousWithinAt",
"SemicontinuousWithinAt.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousAt.comp
(hf : LowerSemicontinuousAt f (g x)) (hg : ContinuousAt g x) :
LowerSemicontinuousAt (f ∘ g) x | SemicontinuousAt.comp hf hg | theorem | LowerSemicontinuousAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousAt",
"LowerSemicontinuousAt",
"SemicontinuousAt.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.comp
(hf : LowerSemicontinuousOn f s) (hg : ContinuousOn g t) (hg' : MapsTo g t s) :
LowerSemicontinuousOn (f ∘ g) t | SemicontinuousOn.comp hf hg hg' | theorem | LowerSemicontinuousOn.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"LowerSemicontinuousOn",
"SemicontinuousOn.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuous.comp
(hf : LowerSemicontinuous f) (hg : Continuous g) : LowerSemicontinuous (f ∘ g) | Semicontinuous.comp hf hg | theorem | LowerSemicontinuous.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"LowerSemicontinuous",
"Semicontinuous.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousWithinAt.mono (h : UpperSemicontinuousWithinAt f s x) (hst : t ⊆ s) :
UpperSemicontinuousWithinAt f t x | SemicontinuousWithinAt.mono h hst | theorem | UpperSemicontinuousWithinAt.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt.mono",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousWithinAt.congr_of_eventuallyEq {a : α}
(h : UpperSemicontinuousWithinAt f s a)
(has : a ∈ s) (hfg : ∀ᶠ x in nhdsWithin a s, f x = g x) :
UpperSemicontinuousWithinAt g s a | LowerSemicontinuousWithinAt.congr_of_eventuallyEq (β := βᵒᵈ) h has hfg | theorem | UpperSemicontinuousWithinAt.congr_of_eventuallyEq | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousWithinAt.congr_of_eventuallyEq",
"UpperSemicontinuousWithinAt",
"nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_univ_iff :
UpperSemicontinuousWithinAt f univ x ↔ UpperSemicontinuousAt f x | semicontinuousWithinAt_univ_iff | theorem | upperSemicontinuousWithinAt_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuousAt",
"UpperSemicontinuousWithinAt",
"semicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iff_restrict {s : Set α} :
UpperSemicontinuous (s.restrict f) ↔ UpperSemicontinuousOn f s | lowerSemicontinuous_restrict_iff (β := βᵒᵈ) | theorem | upperSemicontinuousOn_iff_restrict | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperSemicontinuous",
"UpperSemicontinuousOn",
"lowerSemicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousAt.upperSemicontinuousWithinAt (s : Set α)
(h : UpperSemicontinuousAt f x) : UpperSemicontinuousWithinAt f s x | h.semicontinuousWithinAt s | theorem | UpperSemicontinuousAt.upperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperSemicontinuousAt",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.upperSemicontinuousWithinAt (h : UpperSemicontinuousOn f s)
(hx : x ∈ s) : UpperSemicontinuousWithinAt f s x | h x hx | theorem | UpperSemicontinuousOn.upperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuousOn",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.mono (h : UpperSemicontinuousOn f s) (hst : t ⊆ s) :
UpperSemicontinuousOn f t | SemicontinuousOn.mono h hst | theorem | UpperSemicontinuousOn.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn.mono",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_univ_iff : UpperSemicontinuousOn f univ ↔ UpperSemicontinuous f | semicontinuousOn_univ_iff | theorem | upperSemicontinuousOn_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuous",
"UpperSemicontinuousOn",
"semicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.upperSemicontinuousAt (h : UpperSemicontinuous f) (x : α) :
UpperSemicontinuousAt f x | h x | theorem | UpperSemicontinuous.upperSemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuous",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.upperSemicontinuousWithinAt (h : UpperSemicontinuous f) (s : Set α)
(x : α) : UpperSemicontinuousWithinAt f s x | (h x).semicontinuousWithinAt s | theorem | UpperSemicontinuous.upperSemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperSemicontinuous",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.upperSemicontinuousOn (h : UpperSemicontinuous f) (s : Set α) :
UpperSemicontinuousOn f s | h.semicontinuousOn s | theorem | UpperSemicontinuous.upperSemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperSemicontinuous",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_const : UpperSemicontinuousWithinAt (fun _x => z) s x | SemicontinuousWithinAt.const | theorem | upperSemicontinuousWithinAt_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt.const",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_const : UpperSemicontinuousAt (fun _x => z) x | SemicontinuousAt.const | theorem | upperSemicontinuousAt_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt.const",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_const : UpperSemicontinuousOn (fun _x => z) s | SemicontinuousOn.const | theorem | upperSemicontinuousOn_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn.const",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_const : UpperSemicontinuous fun _x : α => z | Semicontinuous.const | theorem | upperSemicontinuous_const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous.const",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousWithinAt.comp
(hf : UpperSemicontinuousWithinAt f s (g c)) (hg : ContinuousWithinAt g t c)
(hg' : MapsTo g t s) :
UpperSemicontinuousWithinAt (f ∘ g) t c | SemicontinuousWithinAt.comp (r := (f · < ·)) hf hg hg' | theorem | UpperSemicontinuousWithinAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousWithinAt",
"SemicontinuousWithinAt.comp",
"UpperSemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousAt.comp
(hf : UpperSemicontinuousAt f (g c)) (hg : ContinuousAt g c) :
UpperSemicontinuousAt (f ∘ g) c | SemicontinuousAt.comp (r := (f · < ·)) hf hg | theorem | UpperSemicontinuousAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousAt",
"SemicontinuousAt.comp",
"UpperSemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuousOn.comp
(hf : UpperSemicontinuousOn f s) (hg : ContinuousOn g t) (hg' : MapsTo g t s) :
UpperSemicontinuousOn (f ∘ g) t | SemicontinuousOn.comp (r := (f · < ·)) hf hg hg' | theorem | UpperSemicontinuousOn.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"SemicontinuousOn.comp",
"UpperSemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperSemicontinuous.comp
(hf : UpperSemicontinuous f) (hg : Continuous g) : UpperSemicontinuous (f ∘ g) | Semicontinuous.comp (r := (f · < ·)) hf hg | theorem | UpperSemicontinuous.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"Semicontinuous.comp",
"UpperSemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousWithinAt_iff_frequently :
LowerSemicontinuousWithinAt f s x ↔ ∀ y, (∃ᶠ x' in 𝓝[s] x, f x' ≤ y) → f x ≤ y | by
simp [semicontinuousWithinAt_iff_frequently]
alias ⟨LowerSemicontinuousWithinAt.frequently, LowerSemicontinuousWithinAt.of_frequently⟩ :=
lowerSemicontinuousWithinAt_iff_frequently | lemma | lowerSemicontinuousWithinAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousWithinAt",
"semicontinuousWithinAt_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousOn_iff_frequently :
LowerSemicontinuousOn f s ↔ ∀ x ∈ s, ∀ y, (∃ᶠ x' in 𝓝[s] x, f x' ≤ y) → f x ≤ y | by
simp [semicontinuousOn_iff_frequently]
alias ⟨LowerSemicontinuousOn.frequently, LowerSemicontinuousOn.of_frequently⟩ :=
lowerSemicontinuousOn_iff_frequently | lemma | lowerSemicontinuousOn_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousOn",
"semicontinuousOn_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuousAt_iff_frequently :
LowerSemicontinuousAt f x ↔ ∀ y, (∃ᶠ x' in 𝓝 x, f x' ≤ y) → f x ≤ y | by
simp [semicontinuousAt_iff_frequently]
alias ⟨LowerSemicontinuousAt.frequently, LowerSemicontinuousAt.of_frequently⟩ :=
lowerSemicontinuousAt_iff_frequently | lemma | lowerSemicontinuousAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuousAt",
"semicontinuousAt_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSemicontinuous_iff_frequently :
LowerSemicontinuous f ↔ ∀ x y, (∃ᶠ x' in 𝓝 x, f x' ≤ y) → f x ≤ y | by
simp [semicontinuous_iff_frequently]
alias ⟨LowerSemicontinuous.frequently, LowerSemicontinuous.of_frequently⟩ :=
lowerSemicontinuous_iff_frequently | lemma | lowerSemicontinuous_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerSemicontinuous",
"semicontinuous_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousWithinAt_iff_frequently :
UpperSemicontinuousWithinAt f s x ↔ ∀ y, (∃ᶠ x' in 𝓝[s] x, f x' ≥ y) → f x ≥ y | by
simp [semicontinuousWithinAt_iff_frequently]
alias ⟨UpperSemicontinuousWithinAt.frequently, UpperSemicontinuousWithinAt.of_frequently⟩ :=
upperSemicontinuousWithinAt_iff_frequently | lemma | upperSemicontinuousWithinAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuousWithinAt",
"semicontinuousWithinAt_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousOn_iff_frequently :
UpperSemicontinuousOn f s ↔ ∀ x ∈ s, ∀ y, (∃ᶠ x' in 𝓝[s] x, f x' ≥ y) → f x ≥ y | by
simp [semicontinuousOn_iff_frequently]
alias ⟨UpperSemicontinuousOn.frequently, UpperSemicontinuousOn.of_frequently⟩ :=
upperSemicontinuousOn_iff_frequently | lemma | upperSemicontinuousOn_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuousOn",
"semicontinuousOn_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuousAt_iff_frequently :
UpperSemicontinuousAt f x ↔ ∀ y, (∃ᶠ x' in 𝓝 x, f x' ≥ y) → f x ≥ y | by
simp [semicontinuousAt_iff_frequently]
alias ⟨UpperSemicontinuousAt.frequently, UpperSemicontinuousAt.of_frequently⟩ :=
upperSemicontinuousAt_iff_frequently | lemma | upperSemicontinuousAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuousAt",
"semicontinuousAt_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSemicontinuous_iff_frequently :
UpperSemicontinuous f ↔ ∀ x y, (∃ᶠ x' in 𝓝 x, f x' ≥ y) → f x ≥ y | by
simp [semicontinuous_iff_frequently]
alias ⟨UpperSemicontinuous.frequently, UpperSemicontinuous.of_frequently⟩ :=
upperSemicontinuous_iff_frequently | lemma | upperSemicontinuous_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperSemicontinuous",
"semicontinuous_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousWithinAt (f : α → Set β) (s : Set α) (x : α) | SemicontinuousWithinAt (fun x t ↦ IsOpen t ∧ ((f x) ∩ t).Nonempty) s x | abbrev | LowerHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"SemicontinuousWithinAt",
"Set"
] | A function `f : α → Set β` is lower hemicontinuous at `x` within a set `s` if, whenever `t` is
an open set intersecting `f x`, then `t` also intersects `f x'` for all `x'` sufficiently close to
`x` within `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerHemicontinuousOn (f : α → Set β) (s : Set α) | SemicontinuousOn (fun x t ↦ IsOpen t ∧ ((f x) ∩ t).Nonempty) s | abbrev | LowerHemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"SemicontinuousOn",
"Set"
] | A function `f : α → Set β` is lower hemicontinuous on a set `s` if, whenever `x ∈ s` and `t` is
an open set intersecting `f x`, then `t` also intersects `f x'` for all `x'` sufficiently close to
`x` within `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerHemicontinuousAt (f : α → Set β) (x : α) | SemicontinuousAt (fun x t ↦ IsOpen t ∧ ((f x) ∩ t).Nonempty) x | abbrev | LowerHemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"SemicontinuousAt",
"Set"
] | A function `f : α → Set β` is lower hemicontinuous at `x` if, whenever `t` is an open set
intersecting `f x`, then `t` also intersects `f x'` for all `x'` sufficiently close to `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerHemicontinuous (f : α → Set β) | Semicontinuous (fun x t ↦ IsOpen t ∧ ((f x) ∩ t).Nonempty) | abbrev | LowerHemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"Semicontinuous",
"Set"
] | A function `f : α → Set β` is lower hemicontinuous if, for any `x`, whenever `t` is an open set
intersecting `f x`, then `t` also intersects `f x'` for all `x'` sufficiently close to `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousWithinAt (f : α → Set β) (s : Set α) (x : α) | SemicontinuousWithinAt (fun x t ↦ t ∈ 𝓝ˢ (f x)) s x | abbrev | UpperHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt",
"Set"
] | A function `f : α → Set β` is upper hemicontinuous at `x` within a set `s` if, whenever `t` is
a neighborhood of `f x`, then `t` is a neighborhood of `f x'` for all `x'` sufficiently close to
`x` within `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousOn (f : α → Set β) (s : Set α) | SemicontinuousOn (fun x t ↦ t ∈ 𝓝ˢ (f x)) s | abbrev | UpperHemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"Set"
] | A function `f : α → Set β` is upper hemicontinuous on a set `s` if, whenever `x ∈ s` and `t` is
a neighborhood of `f x`, then `t` is a neighborhood of `f x'` for all `x'` sufficiently close to
`x` within `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousAt (f : α → Set β) (x : α) | SemicontinuousAt (fun x t ↦ t ∈ 𝓝ˢ (f x)) x | abbrev | UpperHemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt",
"Set"
] | A function `f : α → Set β` is upper hemicontinuous at `x` if, whenever `t` is a neighborhood of
`f x`, then `t` is a neighborhood of `f x'` for all `x'` sufficiently close to `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuous (f : α → Set β) | Semicontinuous (fun x t ↦ t ∈ 𝓝ˢ (f x)) | abbrev | UpperHemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"Set"
] | A function `f : α → Set β` is upper hemicontinuous if, for all `x`, whenever `t` is a
neighborhood of `f x`, then `t` is a neighborhood of `f x'` for all `x'` sufficiently close
to `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerHemicontinuousWithinAt_iff {f : α → Set β} {s : Set α} {x : α} :
LowerHemicontinuousWithinAt f s x ↔
∀ u, IsOpen u → ((f x) ∩ u).Nonempty → ∀ᶠ x' in 𝓝[s] x, ((f x') ∩ u).Nonempty | by
simp +contextual [SemicontinuousWithinAt] | lemma | lowerHemicontinuousWithinAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"LowerHemicontinuousWithinAt",
"SemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousOn_iff {f : α → Set β} {s : Set α} :
LowerHemicontinuousOn f s ↔ ∀ x ∈ s, LowerHemicontinuousWithinAt f s x | Iff.rfl | lemma | lowerHemicontinuousOn_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousOn",
"LowerHemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousAt_iff {f : α → Set β} {x : α} :
LowerHemicontinuousAt f x ↔
∀ u, IsOpen u → ((f x) ∩ u).Nonempty → ∀ᶠ x' in 𝓝 x, ((f x') ∩ u).Nonempty | by
simp +contextual [SemicontinuousAt] | lemma | lowerHemicontinuousAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"LowerHemicontinuousAt",
"SemicontinuousAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuous_iff {f : α → Set β} :
LowerHemicontinuous f ↔ ∀ x, LowerHemicontinuousAt f x | Iff.rfl | lemma | lowerHemicontinuous_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuous",
"LowerHemicontinuousAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousWithinAt_iff {f : α → Set β} {s : Set α} {x : α} :
UpperHemicontinuousWithinAt f s x ↔ ∀ t, t ∈ 𝓝ˢ (f x) → ∀ᶠ x' in 𝓝[s] x, t ∈ 𝓝ˢ (f x') | Iff.rfl | lemma | upperHemicontinuousWithinAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_iff {f : α → Set β} {s : Set α} :
UpperHemicontinuousOn f s ↔ ∀ x ∈ s, UpperHemicontinuousWithinAt f s x | Iff.rfl | lemma | upperHemicontinuousOn_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuousOn",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousAt_iff {f : α → Set β} {x : α} :
UpperHemicontinuousAt f x ↔ ∀ t, t ∈ 𝓝ˢ (f x) → ∀ᶠ x' in 𝓝 x, t ∈ 𝓝ˢ (f x') | Iff.rfl | lemma | upperHemicontinuousAt_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_iff {f : α → Set β} :
UpperHemicontinuous f ↔ ∀ x, UpperHemicontinuousAt f x | Iff.rfl | lemma | upperHemicontinuous_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuous",
"UpperHemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousWithinAt.mono (h : LowerHemicontinuousWithinAt f s x) (hst : t ⊆ s) :
LowerHemicontinuousWithinAt f t x | SemicontinuousWithinAt.mono h hst | theorem | LowerHemicontinuousWithinAt.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousWithinAt",
"SemicontinuousWithinAt.mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousWithinAt.congr_of_eventuallyEq {a : α}
(h : LowerHemicontinuousWithinAt f s a)
(has : a ∈ s) (hfg : f =ᶠ[𝓝[s] a] g) :
LowerHemicontinuousWithinAt g s a | SemicontinuousWithinAt.congr_of_eventuallyEq h has <| by
filter_upwards [hfg] with x hx
simp [hx] | theorem | LowerHemicontinuousWithinAt.congr_of_eventuallyEq | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousWithinAt",
"SemicontinuousWithinAt.congr_of_eventuallyEq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousWithinAt_univ_iff :
LowerHemicontinuousWithinAt f univ x ↔ LowerHemicontinuousAt f x | semicontinuousWithinAt_univ_iff | theorem | lowerHemicontinuousWithinAt_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousAt",
"LowerHemicontinuousWithinAt",
"semicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousAt.lowerHemicontinuousWithinAt (s : Set α)
(h : LowerHemicontinuousAt f x) : LowerHemicontinuousWithinAt f s x | h.semicontinuousWithinAt s | theorem | LowerHemicontinuousAt.lowerHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousAt",
"LowerHemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousOn.lowerHemicontinuousWithinAt (h : LowerHemicontinuousOn f s)
(hx : x ∈ s) : LowerHemicontinuousWithinAt f s x | h.semicontinuousWithinAt hx | theorem | LowerHemicontinuousOn.lowerHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousOn",
"LowerHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousOn.mono (h : LowerHemicontinuousOn f s) (hst : t ⊆ s) :
LowerHemicontinuousOn f t | SemicontinuousOn.mono h hst | theorem | LowerHemicontinuousOn.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousOn",
"SemicontinuousOn.mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousOn_univ_iff : LowerHemicontinuousOn f univ ↔ LowerHemicontinuous f | semicontinuousOn_univ_iff | theorem | lowerHemicontinuousOn_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuous",
"LowerHemicontinuousOn",
"semicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuous_restrict_iff :
LowerHemicontinuous (s.restrict f) ↔ LowerHemicontinuousOn f s | semicontinuous_restrict_iff (r := (fun x t ↦ IsOpen t ∧ ((f x) ∩ t).Nonempty)) | theorem | lowerHemicontinuous_restrict_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"LowerHemicontinuous",
"LowerHemicontinuousOn",
"semicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuous.lowerHemicontinuousAt (h : LowerHemicontinuous f) (x : α) :
LowerHemicontinuousAt f x | h x | theorem | LowerHemicontinuous.lowerHemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuous",
"LowerHemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuous.lowerHemicontinuousWithinAt (h : LowerHemicontinuous f) (s : Set α)
(x : α) : LowerHemicontinuousWithinAt f s x | (h x).semicontinuousWithinAt s | theorem | LowerHemicontinuous.lowerHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuous",
"LowerHemicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuous.lowerHemicontinuousOn (h : LowerHemicontinuous f) (s : Set α) :
LowerHemicontinuousOn f s | h.semicontinuousOn s | theorem | LowerHemicontinuous.lowerHemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuous",
"LowerHemicontinuousOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousWithinAt_iff_frequently :
LowerHemicontinuousWithinAt f s x ↔
∀ t, IsClosed t → (∃ᶠ x' in 𝓝[s] x, f x' ⊆ t) → f x ⊆ t | by
rw [lowerHemicontinuousWithinAt_iff, compl_surjective.forall]
simp only [isOpen_compl_iff]
refine forall₂_congr fun t ht ↦ ?_
rw [← not_imp_not]
simp [not_nonempty_iff_eq_empty, ← disjoint_iff_inter_eq_empty, disjoint_compl_right_iff_subset]
alias ⟨LowerHemicontinuousWithinAt.frequently, LowerHemicontinuo... | lemma | lowerHemicontinuousWithinAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"LowerHemicontinuousWithinAt",
"isOpen_compl_iff",
"lowerHemicontinuousWithinAt_iff",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousOn_iff_frequently :
LowerHemicontinuousOn f s ↔
∀ x ∈ s, ∀ t, IsClosed t → (∃ᶠ x' in 𝓝[s] x, f x' ⊆ t) → f x ⊆ t | by
simp_rw [lowerHemicontinuousOn_iff, lowerHemicontinuousWithinAt_iff_frequently]
alias ⟨LowerHemicontinuousOn.frequently, LowerHemicontinuousOn.of_frequently⟩ :=
lowerHemicontinuousOn_iff_frequently | lemma | lowerHemicontinuousOn_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"LowerHemicontinuousOn",
"lowerHemicontinuousOn_iff",
"lowerHemicontinuousWithinAt_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuousAt_iff_frequently :
LowerHemicontinuousAt f x ↔ ∀ t, IsClosed t → (∃ᶠ x' in 𝓝 x, f x' ⊆ t) → f x ⊆ t | by
rw [← lowerHemicontinuousWithinAt_univ_iff, lowerHemicontinuousWithinAt_iff_frequently]
simp
alias ⟨LowerHemicontinuousAt.frequently, LowerHemicontinuousAt.of_frequently⟩ :=
lowerHemicontinuousAt_iff_frequently | lemma | lowerHemicontinuousAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"LowerHemicontinuousAt",
"lowerHemicontinuousWithinAt_iff_frequently",
"lowerHemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuous_iff_frequently :
LowerHemicontinuous f ↔ ∀ x t, IsClosed t → (∃ᶠ x' in 𝓝 x, f x' ⊆ t) → f x ⊆ t | by
simp_rw [lowerHemicontinuous_iff, lowerHemicontinuousAt_iff_frequently]
alias ⟨LowerHemicontinuous.frequently, LowerHemicontinuous.of_frequently⟩ :=
lowerHemicontinuous_iff_frequently | lemma | lowerHemicontinuous_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"LowerHemicontinuous",
"lowerHemicontinuousAt_iff_frequently",
"lowerHemicontinuous_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousWithinAt.const : LowerHemicontinuousWithinAt (fun _x => z) s x | SemicontinuousWithinAt.const | theorem | LowerHemicontinuousWithinAt.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousWithinAt",
"SemicontinuousWithinAt.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousAt.const : LowerHemicontinuousAt (fun _x => z) x | SemicontinuousAt.const | theorem | LowerHemicontinuousAt.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousAt",
"SemicontinuousAt.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousOn.const : LowerHemicontinuousOn (fun _x => z) s | SemicontinuousOn.const | theorem | LowerHemicontinuousOn.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuousOn",
"SemicontinuousOn.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuous.const : LowerHemicontinuous fun _x : α => z | Semicontinuous.const | theorem | LowerHemicontinuous.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"LowerHemicontinuous",
"Semicontinuous.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousWithinAt.comp
(hf : LowerHemicontinuousWithinAt f s (g x)) (hg : ContinuousWithinAt g t x)
(hg' : MapsTo g t s) :
LowerHemicontinuousWithinAt (f ∘ g) t x | SemicontinuousWithinAt.comp hf hg hg' | theorem | LowerHemicontinuousWithinAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousWithinAt",
"LowerHemicontinuousWithinAt",
"SemicontinuousWithinAt.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousAt.comp
(hf : LowerHemicontinuousAt f (g x)) (hg : ContinuousAt g x) :
LowerHemicontinuousAt (f ∘ g) x | SemicontinuousAt.comp hf hg | theorem | LowerHemicontinuousAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousAt",
"LowerHemicontinuousAt",
"SemicontinuousAt.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuousOn.comp
(hf : LowerHemicontinuousOn f s) (hg : ContinuousOn g t) (hg' : MapsTo g t s) :
LowerHemicontinuousOn (f ∘ g) t | SemicontinuousOn.comp hf hg hg' | theorem | LowerHemicontinuousOn.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"LowerHemicontinuousOn",
"SemicontinuousOn.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerHemicontinuous.comp
(hf : LowerHemicontinuous f) (hg : Continuous g) : LowerHemicontinuous (f ∘ g) | Semicontinuous.comp hf hg | theorem | LowerHemicontinuous.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"LowerHemicontinuous",
"Semicontinuous.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousWithinAt.mono (h : UpperHemicontinuousWithinAt f s x) (hst : t ⊆ s) :
UpperHemicontinuousWithinAt f t x | SemicontinuousWithinAt.mono h hst | theorem | UpperHemicontinuousWithinAt.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt.mono",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousWithinAt.congr_of_eventuallyEq {a : α}
(h : UpperHemicontinuousWithinAt f s a)
(has : a ∈ s) (hfg : ∀ᶠ x in nhdsWithin a s, f x = g x) :
UpperHemicontinuousWithinAt g s a | SemicontinuousWithinAt.congr_of_eventuallyEq h has <| by
filter_upwards [hfg] with x hx
simp [hx] | theorem | UpperHemicontinuousWithinAt.congr_of_eventuallyEq | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt.congr_of_eventuallyEq",
"UpperHemicontinuousWithinAt",
"nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousWithinAt_univ_iff :
UpperHemicontinuousWithinAt f univ x ↔ UpperHemicontinuousAt f x | semicontinuousWithinAt_univ_iff | theorem | upperHemicontinuousWithinAt_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousAt",
"UpperHemicontinuousWithinAt",
"semicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_iff_restrict {s : Set α} :
UpperHemicontinuous (s.restrict f) ↔ UpperHemicontinuousOn f s | semicontinuous_restrict_iff (r := (fun x t ↦ t ∈ 𝓝ˢ (f x))) | theorem | upperHemicontinuousOn_iff_restrict | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuous",
"UpperHemicontinuousOn",
"semicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousAt.upperHemicontinuousWithinAt (s : Set α)
(h : UpperHemicontinuousAt f x) : UpperHemicontinuousWithinAt f s x | h.semicontinuousWithinAt s | theorem | UpperHemicontinuousAt.upperHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuousAt",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousOn.upperHemicontinuousWithinAt (h : UpperHemicontinuousOn f s)
(hx : x ∈ s) : UpperHemicontinuousWithinAt f s x | h x hx | theorem | UpperHemicontinuousOn.upperHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousOn",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousOn.mono (h : UpperHemicontinuousOn f s) (hst : t ⊆ s) :
UpperHemicontinuousOn f t | SemicontinuousOn.mono h hst | theorem | UpperHemicontinuousOn.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn.mono",
"UpperHemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_univ_iff : UpperHemicontinuousOn f univ ↔ UpperHemicontinuous f | semicontinuousOn_univ_iff | theorem | upperHemicontinuousOn_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuous",
"UpperHemicontinuousOn",
"semicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuous.upperHemicontinuousAt (h : UpperHemicontinuous f) (x : α) :
UpperHemicontinuousAt f x | h x | theorem | UpperHemicontinuous.upperHemicontinuousAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuous",
"UpperHemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuous.upperHemicontinuousWithinAt (h : UpperHemicontinuous f) (s : Set α)
(x : α) : UpperHemicontinuousWithinAt f s x | (h x).semicontinuousWithinAt s | theorem | UpperHemicontinuous.upperHemicontinuousWithinAt | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuous",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuous.upperHemicontinuousOn (h : UpperHemicontinuous f) (s : Set α) :
UpperHemicontinuousOn f s | h.semicontinuousOn s | theorem | UpperHemicontinuous.upperHemicontinuousOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuous",
"UpperHemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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