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 |
|---|---|---|---|---|---|---|---|---|---|---|
isLocalExtr_const {b : β} : IsLocalExtr (fun _ => b) a | isExtrFilter_const | theorem | isLocalExtr_const | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr",
"isExtrFilter_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.comp_mono (hf : IsLocalMin f a) {g : β → γ} (hg : Monotone g) :
IsLocalMin (g ∘ f) a | hf.comp_mono hg | theorem | IsLocalMin.comp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.comp_mono (hf : IsLocalMax f a) {g : β → γ} (hg : Monotone g) :
IsLocalMax (g ∘ f) a | hf.comp_mono hg | theorem | IsLocalMax.comp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.comp_mono (hf : IsLocalExtr f a) {g : β → γ} (hg : Monotone g) :
IsLocalExtr (g ∘ f) a | hf.comp_mono hg | theorem | IsLocalExtr.comp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.comp_antitone (hf : IsLocalMin f a) {g : β → γ} (hg : Antitone g) :
IsLocalMax (g ∘ f) a | hf.comp_antitone hg | theorem | IsLocalMin.comp_antitone | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"Antitone",
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.comp_antitone (hf : IsLocalMax f a) {g : β → γ} (hg : Antitone g) :
IsLocalMin (g ∘ f) a | hf.comp_antitone hg | theorem | IsLocalMax.comp_antitone | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"Antitone",
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.comp_antitone (hf : IsLocalExtr f a) {g : β → γ} (hg : Antitone g) :
IsLocalExtr (g ∘ f) a | hf.comp_antitone hg | theorem | IsLocalExtr.comp_antitone | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"Antitone",
"IsLocalExtr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.comp_mono (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Monotone g) :
IsLocalMinOn (g ∘ f) s a | hf.comp_mono hg | theorem | IsLocalMinOn.comp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.comp_mono (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Monotone g) :
IsLocalMaxOn (g ∘ f) s a | hf.comp_mono hg | theorem | IsLocalMaxOn.comp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtrOn.comp_mono (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Monotone g) :
IsLocalExtrOn (g ∘ f) s a | hf.comp_mono hg | theorem | IsLocalExtrOn.comp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn",
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.comp_antitone (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Antitone g) :
IsLocalMaxOn (g ∘ f) s a | hf.comp_antitone hg | theorem | IsLocalMinOn.comp_antitone | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"Antitone",
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.comp_antitone (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Antitone g) :
IsLocalMinOn (g ∘ f) s a | hf.comp_antitone hg | theorem | IsLocalMaxOn.comp_antitone | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"Antitone",
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtrOn.comp_antitone (hf : IsLocalExtrOn f s a) {g : β → γ}
(hg : Antitone g) : IsLocalExtrOn (g ∘ f) s a | hf.comp_antitone hg | theorem | IsLocalExtrOn.comp_antitone | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"Antitone",
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMin f a) {g : α → γ}
(hg : IsLocalMin g a) : IsLocalMin (fun x => op (f x) (g x)) a | hf.bicomp_mono hop hg | theorem | IsLocalMin.bicomp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMax f a) {g : α → γ}
(hg : IsLocalMax g a) : IsLocalMax (fun x => op (f x) (g x)) a | hf.bicomp_mono hop hg | theorem | IsLocalMax.bicomp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMinOn f s a) {g : α → γ}
(hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => op (f x) (g x)) s a | hf.bicomp_mono hop hg | theorem | IsLocalMinOn.bicomp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMaxOn f s a) {g : α → γ}
(hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => op (f x) (g x)) s a | hf.bicomp_mono hop hg | theorem | IsLocalMaxOn.bicomp_mono | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
(hf : IsLocalMin f (g b)) (hg : ContinuousAt g b) : IsLocalMin (f ∘ g) b | hg hf | theorem | IsLocalMin.comp_continuous | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousAt",
"IsLocalMin",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
(hf : IsLocalMax f (g b)) (hg : ContinuousAt g b) : IsLocalMax (f ∘ g) b | hg hf | theorem | IsLocalMax.comp_continuous | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousAt",
"IsLocalMax",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
(hf : IsLocalExtr f (g b)) (hg : ContinuousAt g b) : IsLocalExtr (f ∘ g) b | hf.comp_tendsto hg | theorem | IsLocalExtr.comp_continuous | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousAt",
"IsLocalExtr",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ}
(hf : IsLocalMin f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMinOn (f ∘ g) s b | hf.comp_tendsto (hg b hb) | theorem | IsLocalMin.comp_continuousOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousOn",
"IsLocalMin",
"IsLocalMinOn",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ}
(hf : IsLocalMax f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMaxOn (f ∘ g) s b | hf.comp_tendsto (hg b hb) | theorem | IsLocalMax.comp_continuousOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousOn",
"IsLocalMax",
"IsLocalMaxOn",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.comp_continuousOn [TopologicalSpace δ] {s : Set δ} (g : δ → α) {b : δ}
(hf : IsLocalExtr f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalExtrOn (f ∘ g) s b | hf.elim (fun hf => (hf.comp_continuousOn hg hb).isExtr) fun hf =>
(IsLocalMax.comp_continuousOn hf hg hb).isExtr | theorem | IsLocalExtr.comp_continuousOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousOn",
"IsLocalExtr",
"IsLocalExtrOn",
"IsLocalMax.comp_continuousOn",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} {g : δ → α}
{b : δ} (hf : IsLocalMinOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) :
IsLocalMinOn (f ∘ g) s b | hf.comp_tendsto
(tendsto_nhdsWithin_mono_right (image_subset_iff.mpr hst)
(ContinuousWithinAt.tendsto_nhdsWithin_image (hg b hb))) | theorem | IsLocalMinOn.comp_continuousOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.tendsto_nhdsWithin_image",
"IsLocalMinOn",
"Set",
"TopologicalSpace",
"tendsto_nhdsWithin_mono_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} {g : δ → α}
{b : δ} (hf : IsLocalMaxOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) :
IsLocalMaxOn (f ∘ g) s b | hf.comp_tendsto
(tendsto_nhdsWithin_mono_right (image_subset_iff.mpr hst)
(ContinuousWithinAt.tendsto_nhdsWithin_image (hg b hb))) | theorem | IsLocalMaxOn.comp_continuousOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.tendsto_nhdsWithin_image",
"IsLocalMaxOn",
"Set",
"TopologicalSpace",
"tendsto_nhdsWithin_mono_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtrOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} (g : δ → α)
{b : δ} (hf : IsLocalExtrOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s)
(hb : b ∈ s) : IsLocalExtrOn (f ∘ g) s b | hf.elim (fun hf => (hf.comp_continuousOn hst hg hb).isExtr) fun hf =>
(IsLocalMaxOn.comp_continuousOn hf hst hg hb).isExtr | theorem | IsLocalExtrOn.comp_continuousOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"ContinuousOn",
"IsLocalExtrOn",
"IsLocalMaxOn.comp_continuousOn",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.add (hf : IsLocalMin f a) (hg : IsLocalMin g a) :
IsLocalMin (fun x => f x + g x) a | hf.add hg | theorem | IsLocalMin.add | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.add (hf : IsLocalMax f a) (hg : IsLocalMax g a) :
IsLocalMax (fun x => f x + g x) a | hf.add hg | theorem | IsLocalMax.add | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.add (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) :
IsLocalMinOn (fun x => f x + g x) s a | hf.add hg | theorem | IsLocalMinOn.add | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.add (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) :
IsLocalMaxOn (fun x => f x + g x) s a | hf.add hg | theorem | IsLocalMaxOn.add | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.neg (hf : IsLocalMin f a) : IsLocalMax (fun x => -f x) a | hf.neg | theorem | IsLocalMin.neg | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.neg (hf : IsLocalMax f a) : IsLocalMin (fun x => -f x) a | hf.neg | theorem | IsLocalMax.neg | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.neg (hf : IsLocalExtr f a) : IsLocalExtr (fun x => -f x) a | hf.neg | theorem | IsLocalExtr.neg | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.neg (hf : IsLocalMinOn f s a) : IsLocalMaxOn (fun x => -f x) s a | hf.neg | theorem | IsLocalMinOn.neg | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.neg (hf : IsLocalMaxOn f s a) : IsLocalMinOn (fun x => -f x) s a | hf.neg | theorem | IsLocalMaxOn.neg | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtrOn.neg (hf : IsLocalExtrOn f s a) : IsLocalExtrOn (fun x => -f x) s a | hf.neg | theorem | IsLocalExtrOn.neg | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.sub (hf : IsLocalMin f a) (hg : IsLocalMax g a) :
IsLocalMin (fun x => f x - g x) a | hf.sub hg | theorem | IsLocalMin.sub | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.sub (hf : IsLocalMax f a) (hg : IsLocalMin g a) :
IsLocalMax (fun x => f x - g x) a | hf.sub hg | theorem | IsLocalMax.sub | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.sub (hf : IsLocalMinOn f s a) (hg : IsLocalMaxOn g s a) :
IsLocalMinOn (fun x => f x - g x) s a | hf.sub hg | theorem | IsLocalMinOn.sub | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.sub (hf : IsLocalMaxOn f s a) (hg : IsLocalMinOn g s a) :
IsLocalMaxOn (fun x => f x - g x) s a | hf.sub hg | theorem | IsLocalMaxOn.sub | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.sup (hf : IsLocalMin f a) (hg : IsLocalMin g a) :
IsLocalMin (fun x => f x ⊔ g x) a | hf.sup hg | theorem | IsLocalMin.sup | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.sup (hf : IsLocalMax f a) (hg : IsLocalMax g a) :
IsLocalMax (fun x => f x ⊔ g x) a | hf.sup hg | theorem | IsLocalMax.sup | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.sup (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) :
IsLocalMinOn (fun x => f x ⊔ g x) s a | hf.sup hg | theorem | IsLocalMinOn.sup | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.sup (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) :
IsLocalMaxOn (fun x => f x ⊔ g x) s a | hf.sup hg | theorem | IsLocalMaxOn.sup | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.inf (hf : IsLocalMin f a) (hg : IsLocalMin g a) :
IsLocalMin (fun x => f x ⊓ g x) a | hf.inf hg | theorem | IsLocalMin.inf | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.inf (hf : IsLocalMax f a) (hg : IsLocalMax g a) :
IsLocalMax (fun x => f x ⊓ g x) a | hf.inf hg | theorem | IsLocalMax.inf | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.inf (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) :
IsLocalMinOn (fun x => f x ⊓ g x) s a | hf.inf hg | theorem | IsLocalMinOn.inf | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.inf (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) :
IsLocalMaxOn (fun x => f x ⊓ g x) s a | hf.inf hg | theorem | IsLocalMaxOn.inf | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.min (hf : IsLocalMin f a) (hg : IsLocalMin g a) :
IsLocalMin (fun x => min (f x) (g x)) a | hf.min hg | theorem | IsLocalMin.min | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.min (hf : IsLocalMax f a) (hg : IsLocalMax g a) :
IsLocalMax (fun x => min (f x) (g x)) a | hf.min hg | theorem | IsLocalMax.min | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.min (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) :
IsLocalMinOn (fun x => min (f x) (g x)) s a | hf.min hg | theorem | IsLocalMinOn.min | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.min (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) :
IsLocalMaxOn (fun x => min (f x) (g x)) s a | hf.min hg | theorem | IsLocalMaxOn.min | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.max (hf : IsLocalMin f a) (hg : IsLocalMin g a) :
IsLocalMin (fun x => max (f x) (g x)) a | hf.max hg | theorem | IsLocalMin.max | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.max (hf : IsLocalMax f a) (hg : IsLocalMax g a) :
IsLocalMax (fun x => max (f x) (g x)) a | hf.max hg | theorem | IsLocalMax.max | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.max (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) :
IsLocalMinOn (fun x => max (f x) (g x)) s a | hf.max hg | theorem | IsLocalMinOn.max | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.max (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) :
IsLocalMaxOn (fun x => max (f x) (g x)) s a | hf.max hg | theorem | IsLocalMaxOn.max | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyLE.isLocalMaxOn {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝[s] a] f)
(hfga : f a = g a) (h : IsLocalMaxOn f s a) : IsLocalMaxOn g s a | hle.isMaxFilter hfga h | theorem | Filter.EventuallyLE.isLocalMaxOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.congr {f g : α → β} {a : α} (h : IsLocalMaxOn f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMaxOn g s a | h.congr heq <| heq.eq_of_nhdsWithin hmem | theorem | IsLocalMaxOn.congr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyEq.isLocalMaxOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : IsLocalMaxOn f s a ↔ IsLocalMaxOn g s a | heq.isMaxFilter_iff <| heq.eq_of_nhdsWithin hmem | theorem | Filter.EventuallyEq.isLocalMaxOn_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyLE.isLocalMinOn {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝[s] a] g)
(hfga : f a = g a) (h : IsLocalMinOn f s a) : IsLocalMinOn g s a | hle.isMinFilter hfga h | theorem | Filter.EventuallyLE.isLocalMinOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.congr {f g : α → β} {a : α} (h : IsLocalMinOn f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMinOn g s a | h.congr heq <| heq.eq_of_nhdsWithin hmem | theorem | IsLocalMinOn.congr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyEq.isLocalMinOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : IsLocalMinOn f s a ↔ IsLocalMinOn g s a | heq.isMinFilter_iff <| heq.eq_of_nhdsWithin hmem | theorem | Filter.EventuallyEq.isLocalMinOn_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtrOn.congr {f g : α → β} {a : α} (h : IsLocalExtrOn f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalExtrOn g s a | h.congr heq <| heq.eq_of_nhdsWithin hmem | theorem | IsLocalExtrOn.congr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyEq.isLocalExtrOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : IsLocalExtrOn f s a ↔ IsLocalExtrOn g s a | heq.isExtrFilter_iff <| heq.eq_of_nhdsWithin hmem | theorem | Filter.EventuallyEq.isLocalExtrOn_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyLE.isLocalMax {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝 a] f) (hfga : f a = g a)
(h : IsLocalMax f a) : IsLocalMax g a | hle.isMaxFilter hfga h | theorem | Filter.EventuallyLE.isLocalMax | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.congr {f g : α → β} {a : α} (h : IsLocalMax f a) (heq : f =ᶠ[𝓝 a] g) :
IsLocalMax g a | h.congr heq heq.eq_of_nhds | theorem | IsLocalMax.congr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyEq.isLocalMax_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
IsLocalMax f a ↔ IsLocalMax g a | heq.isMaxFilter_iff heq.eq_of_nhds | theorem | Filter.EventuallyEq.isLocalMax_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyLE.isLocalMin {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝 a] g) (hfga : f a = g a)
(h : IsLocalMin f a) : IsLocalMin g a | hle.isMinFilter hfga h | theorem | Filter.EventuallyLE.isLocalMin | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.congr {f g : α → β} {a : α} (h : IsLocalMin f a) (heq : f =ᶠ[𝓝 a] g) :
IsLocalMin g a | h.congr heq heq.eq_of_nhds | theorem | IsLocalMin.congr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyEq.isLocalMin_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
IsLocalMin f a ↔ IsLocalMin g a | heq.isMinFilter_iff heq.eq_of_nhds | theorem | Filter.EventuallyEq.isLocalMin_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.congr {f g : α → β} {a : α} (h : IsLocalExtr f a) (heq : f =ᶠ[𝓝 a] g) :
IsLocalExtr g a | h.congr heq heq.eq_of_nhds | theorem | IsLocalExtr.congr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.EventuallyEq.isLocalExtr_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
IsLocalExtr f a ↔ IsLocalExtr g a | heq.isExtrFilter_iff heq.eq_of_nhds | theorem | Filter.EventuallyEq.isLocalExtr_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMax_of_mono_anti' {α : Type*} [TopologicalSpace α] [LinearOrder α]
{β : Type*} [Preorder β] {b : α} {f : α → β}
{a : Set α} (ha : a ∈ 𝓝[≤] b) {c : Set α} (hc : c ∈ 𝓝[≥] b)
(h₀ : MonotoneOn f a) (h₁ : AntitoneOn f c) : IsLocalMax f b | have : b ∈ a := mem_of_mem_nhdsWithin (by simp) ha
have : b ∈ c := mem_of_mem_nhdsWithin (by simp) hc
mem_of_superset (nhds_of_Ici_Iic ha hc) (fun x _ => by rcases le_total x b <;> aesop) | lemma | isLocalMax_of_mono_anti' | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"AntitoneOn",
"IsLocalMax",
"LinearOrder",
"MonotoneOn",
"Preorder",
"Set",
"TopologicalSpace",
"le_total",
"mem_of_mem_nhdsWithin",
"nhds_of_Ici_Iic"
] | If `f` is monotone to the left and antitone to the right, then it has a local maximum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLocalMin_of_anti_mono' {α : Type*} [TopologicalSpace α] [LinearOrder α]
{β : Type*} [Preorder β] {b : α} {f : α → β}
{a : Set α} (ha : a ∈ 𝓝[≤] b) {c : Set α} (hc : c ∈ 𝓝[≥] b)
(h₀ : AntitoneOn f a) (h₁ : MonotoneOn f c) : IsLocalMin f b | have : b ∈ a := mem_of_mem_nhdsWithin (by simp) ha
have : b ∈ c := mem_of_mem_nhdsWithin (by simp) hc
mem_of_superset (nhds_of_Ici_Iic ha hc) (fun x _ => by rcases le_total x b <;> aesop) | lemma | isLocalMin_of_anti_mono' | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"AntitoneOn",
"IsLocalMin",
"LinearOrder",
"MonotoneOn",
"Preorder",
"Set",
"TopologicalSpace",
"le_total",
"mem_of_mem_nhdsWithin",
"nhds_of_Ici_Iic"
] | If `f` is antitone to the left and monotone to the right, then it has a local minimum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lower (α : Type*) [Preorder α] : TopologicalSpace α | generateFrom {s | ∃ a, (Ici a)ᶜ = s} | def | Topology.lower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The lower topology is the topology generated by the complements of the left-closed right-infinite
intervals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upper (α : Type*) [Preorder α] : TopologicalSpace α | generateFrom {s | ∃ a, (Iic a)ᶜ = s} | def | Topology.upper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The upper topology is the topology generated by the complements of the right-closed left-infinite
intervals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
WithLower (α : Type*) | α | def | Topology.WithLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | Type synonym for a preorder equipped with the lower set topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLower : α ≃ WithLower α | Equiv.refl _ | def | Topology.WithLower.toLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Equiv.refl"
] | `toLower` is the identity function to the `WithLower` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofLower : WithLower α ≃ α | Equiv.refl _ | def | Topology.WithLower.ofLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Equiv.refl"
] | `ofLower` is the identity function from the `WithLower` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLower_symm : (@toLower α).symm = ofLower | rfl | lemma | Topology.WithLower.toLower_symm | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLower_symm : (@ofLower α).symm = toLower | rfl | lemma | Topology.WithLower.ofLower_symm | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLower_ofLower (a : WithLower α) : toLower (ofLower a) = a | rfl | lemma | Topology.WithLower.toLower_ofLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLower_toLower (a : α) : ofLower (toLower a) = a | rfl | lemma | Topology.WithLower.ofLower_toLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLower_inj {a b : α} : toLower a = toLower b ↔ a = b | Iff.rfl | lemma | Topology.WithLower.toLower_inj | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLower_inj {a b : WithLower α} : ofLower a = ofLower b ↔ a = b | Iff.rfl | theorem | Topology.WithLower.ofLower_inj | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rec {motive : WithLower α → Sort*} (toLower : ∀ a, motive (toLower a)) :
∀ a, motive a | fun a =>
toLower (ofLower a) | def | Topology.WithLower.rec | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | A recursor for `WithLower`. Use as `induction x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLower_le_toLower {x y : α} : toLower x ≤ toLower y ↔ x ≤ y | .rfl | lemma | Topology.WithLower.toLower_le_toLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLower_lt_toLower {x y : α} : toLower x < toLower y ↔ x < y | .rfl | lemma | Topology.WithLower.toLower_lt_toLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLower_le_ofLower {x y : WithLower α} : ofLower x ≤ ofLower y ↔ x ≤ y | .rfl | lemma | Topology.WithLower.ofLower_le_ofLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLower_lt_ofLower {x y : WithLower α} : ofLower x < ofLower y ↔ x < y | .rfl | lemma | Topology.WithLower.ofLower_lt_ofLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_preimage_ofLower : IsOpen (ofLower ⁻¹' s) ↔ IsOpen[lower α] s | Iff.rfl | lemma | Topology.WithLower.isOpen_preimage_ofLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_def (T : Set (WithLower α)) : IsOpen T ↔ IsOpen[lower α] (WithLower.toLower ⁻¹' T) | Iff.rfl | lemma | Topology.WithLower.isOpen_def | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_toLower [TopologicalSpace α] [ClosedIciTopology α] :
Continuous (toLower : α → WithLower α) | continuous_generateFrom_iff.mpr <| by rintro _ ⟨a, rfl⟩; exact isClosed_Ici.isOpen_compl | theorem | Topology.WithLower.continuous_toLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"ClosedIciTopology",
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithUpper (α : Type*) | α | def | Topology.WithUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | Type synonym for a preorder equipped with the upper topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUpper : α ≃ WithUpper α | Equiv.refl _ | def | Topology.WithUpper.toUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Equiv.refl"
] | `toUpper` is the identity function to the `WithUpper` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofUpper : WithUpper α ≃ α | Equiv.refl _ | def | Topology.WithUpper.ofUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Equiv.refl"
] | `ofUpper` is the identity function from the `WithUpper` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUpper_symm {α} : (@toUpper α).symm = ofUpper | rfl | lemma | Topology.WithUpper.toUpper_symm | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpper_symm : (@ofUpper α).symm = toUpper | rfl | lemma | Topology.WithUpper.ofUpper_symm | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpper_ofUpper (a : WithUpper α) : toUpper (ofUpper a) = a | rfl | lemma | Topology.WithUpper.toUpper_ofUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a | rfl | lemma | Topology.WithUpper.ofUpper_toUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | 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.