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 |
|---|---|---|---|---|---|---|---|---|---|---|
gc_closureOperator (S : Set T) : gc.closureOperator S = hull T (kernel S) | by
simp only [toDual_sInf, GaloisConnection.closureOperator_apply, ofDual_sSup]
rw [← preimage_comp, ← OrderDual.toDual_symm_eq, Equiv.symm_comp_self, preimage_id_eq, id_eq] | lemma | PrimitiveSpectrum.gc_closureOperator | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Equiv.symm_comp_self",
"OrderDual.toDual_symm_eq",
"Set",
"ofDual_sSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderGenerates | ∀ (a : α), ∃ (S : Set T), a = kernel S | def | PrimitiveSpectrum.OrderGenerates | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Set"
] | `T` order generates `α` if, for every `a` in `α`, there exists a subset of `T` such that `a` is
the `kernel` of `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
gi (hG : OrderGenerates T) : GaloisInsertion (α := Set T) (β := αᵒᵈ)
(OrderDual.toDual ∘ kernel)
(hull T ∘ OrderDual.ofDual) | gc.toGaloisInsertion fun a ↦ by
obtain ⟨S, rfl⟩ := hG a
rw [OrderDual.le_toDual, kernel, kernel]
exact sInf_le_sInf <| image_val_mono fun c hcS => by
rw [hull, mem_preimage, mem_Ici]
exact sInf_le (mem_image_of_mem Subtype.val hcS) | def | PrimitiveSpectrum.gi | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"GaloisInsertion",
"OrderDual.le_toDual",
"OrderDual.ofDual",
"OrderDual.toDual",
"Set"
] | When `T` is order generating, the `kernel` and the `hull` form a Galois insertion | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
kernel_hull (hG : OrderGenerates T) (a : α) : kernel (hull T a) = a | by
conv_rhs => rw [← OrderDual.ofDual_toDual a, ← (gi hG).l_u_eq a]
rfl | lemma | PrimitiveSpectrum.kernel_hull | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"OrderDual.ofDual_toDual"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hull_kernel_of_isClosed [TopologicalSpace α] [IsLower α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) {C : Set T} (h : IsClosed C) :
hull T (kernel C) = C | by
obtain ⟨a, ha⟩ := (isClosed_iff hT).mp h
rw [ha, kernel_hull hG] | lemma | PrimitiveSpectrum.hull_kernel_of_isClosed | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"InfPrime",
"IsClosed",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedsGC_closureOperator [TopologicalSpace α] [IsLower α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) (S : Set T) :
(TopologicalSpace.Closeds.gc (α := T)).closureOperator S = hull T (kernel S) | by
simp only [GaloisConnection.closureOperator_apply, Closeds.coe_closure, closure, le_antisymm_iff]
constructor
· exact fun ⦃a⦄ a ↦ a (hull T (kernel S)) ⟨(isClosed_iff hT).mpr ⟨kernel S, rfl⟩,
image_subset_iff.mp (fun _ hbS => sInf_le hbS)⟩
· simp_rw [le_eq_subset, subset_sInter_iff]
intro R hR
... | lemma | PrimitiveSpectrum.closedsGC_closureOperator | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"ClosureOperator.monotone",
"InfPrime",
"Set",
"TopologicalSpace",
"TopologicalSpace.Closeds.gc",
"closure",
"le_antisymm_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f)
(hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x | by
obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x | f x ≤ g x ∧ g x ≤ f x }).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg)
(isClosed_le hg hf) (fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩
exact ⟨x, le_antisymm hfg hgf⟩ | theorem | intermediate_value_univ₂ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"PreconnectedSpace",
"isClosed_le",
"le_antisymm",
"le_total"
] | Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
intermediate_value_univ₂_eventually₁ [PreconnectedSpace X] {a : X} {l : Filter X} [NeBot l]
{f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) :
∃ x, f x = g x | let ⟨_, h⟩ := he.exists; intermediate_value_univ₂ hf hg ha h | theorem | intermediate_value_univ₂_eventually₁ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"Filter",
"PreconnectedSpace",
"intermediate_value_univ₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_univ₂_eventually₂ [PreconnectedSpace X] {l₁ l₂ : Filter X} [NeBot l₁]
[NeBot l₂] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g)
(he₂ : g ≤ᶠ[l₂] f) : ∃ x, f x = g x | let ⟨_, h₁⟩ := he₁.exists
let ⟨_, h₂⟩ := he₂.exists
intermediate_value_univ₂ hf hg h₁ h₂ | theorem | intermediate_value_univ₂_eventually₂ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"Filter",
"PreconnectedSpace",
"intermediate_value_univ₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value₂ {s : Set X} (hs : IsPreconnected s) {a b : X}
(ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
(ha' : f a ≤ g a) (hb' : g b ≤ f b) : ∃ x ∈ s, f x = g x | let ⟨x, hx⟩ :=
@intermediate_value_univ₂ s α _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ ⟨b, hb⟩ _ _
(continuousOn_iff_continuous_restrict.1 hf) (continuousOn_iff_continuous_restrict.1 hg) ha'
hb'
⟨x, x.2, hx⟩ | theorem | IsPreconnected.intermediate_value₂ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected",
"Set",
"Subtype.preconnectedSpace",
"intermediate_value_univ₂"
] | Intermediate value theorem for two functions: if `f` and `g` are two functions continuous
on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`,
then for some `x ∈ s` we have `f x = g x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x | by
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _
(comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _)
exact ⟨b, b.prop, h⟩ | theorem | IsPreconnected.intermediate_value₂_eventually₁ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"Subtype.preconnectedSpace",
"continuousOn_iff_continuous_restrict",
"intermediate_value_univ₂_eventually₁"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s)
{l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α}
(hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ x ∈ s, f x = g x | by
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₂ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) _ _
(comap_coe_neBot_of_le_principal hl₁) (comap_coe_neBot_of_le_principal hl₂) _ _ hf hg
(he₁.comap _) (he₂.comap _)
exact ⟨b, b.prop, h⟩ | theorem | IsPreconnected.intermediate_value₂_eventually₂ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"Subtype.preconnectedSpace",
"continuousOn_iff_continuous_restrict",
"intermediate_value_univ₂_eventually₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s)
(hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s) : Icc (f a) (f b) ⊆ f '' s | fun _x hx =>
hs.intermediate_value₂ ha hb hf continuousOn_const hx.1 hx.2 | theorem | IsPreconnected.intermediate_value | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected",
"Set",
"continuousOn_const"
] | **Intermediate Value Theorem** for continuous functions on connected sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
(ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s | fun _ h =>
hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1 (ht.eventually_const_le h.2) | theorem | IsPreconnected.intermediate_value_Ico | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
(ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s | fun _ h =>
(hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h.2
(ht.eventually_le_const h.1)).imp fun _ h => h.imp_right Eq.symm | theorem | IsPreconnected.intermediate_value_Ioc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) :
Ioo v₁ v₂ ⊆ f '' s | fun _ h =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const
(ht₁.eventually_le_const h.1) (ht₂.eventually_const_le h.2) | theorem | IsPreconnected.intermediate_value_Ioo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht : Tendsto f l atTop) : Ici (f a) ⊆ f '' s | fun y h =>
hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h (tendsto_atTop.1 ht y) | theorem | IsPreconnected.intermediate_value_Ici | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Iic {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht : Tendsto f l atBot) : Iic (f a) ⊆ f '' s | fun y h =>
(hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h (tendsto_atBot.1 ht y)).imp
fun _ h => h.imp_right Eq.symm | theorem | IsPreconnected.intermediate_value_Iic | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s | fun y h =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const
(ht₁.eventually_le_const h) (ht₂.eventually_ge_atTop y) | theorem | IsPreconnected.intermediate_value_Ioi | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s | fun y h =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y)
(ht₂.eventually_const_le h) | theorem | IsPreconnected.intermediate_value_Iio | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s | fun y _ =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y)
(ht₂.eventually_ge_atTop y) | theorem | IsPreconnected.intermediate_value_Iii | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Filter",
"IsPreconnected",
"Set",
"continuousOn_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) :
Icc (f a) (f b) ⊆ range f | fun _ hx => intermediate_value_univ₂ hf continuous_const hx.1 hx.2 | theorem | intermediate_value_univ | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"PreconnectedSpace",
"continuous_const",
"intermediate_value_univ₂"
] | **Intermediate Value Theorem** for continuous functions on connected spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_range_of_exists_le_of_exists_ge [PreconnectedSpace X] {c : α} {f : X → α}
(hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) : c ∈ range f | let ⟨a, ha⟩ := h₁; let ⟨b, hb⟩ := h₂; intermediate_value_univ a b hf ⟨ha, hb⟩ | theorem | mem_range_of_exists_le_of_exists_ge | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"PreconnectedSpace",
"intermediate_value_univ"
] | **Intermediate Value Theorem** for continuous functions on connected spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s | by
simpa only [image_id] using! hs.intermediate_value ha hb continuousOn_id | theorem | IsPreconnected.Icc_subset | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"Set",
"continuousOn_id"
] | If a preconnected set contains endpoints of an interval, then it includes the whole interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsPreconnected.ordConnected {s : Set α} (h : IsPreconnected s) : OrdConnected s | ⟨fun _ hx _ hy => h.Icc_subset hx hy⟩ | theorem | IsPreconnected.ordConnected | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsConnected.Icc_subset {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s | hs.2.Icc_subset ha hb | theorem | IsConnected.Icc_subset | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected",
"Set"
] | If a preconnected set contains endpoints of an interval, then it includes the whole interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsPreconnected.eq_univ_of_unbounded {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
(ha : ¬BddAbove s) : s = univ | by
refine eq_univ_of_forall fun x => ?_
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bddBelow_iff.1 hb x
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩ | theorem | IsPreconnected.eq_univ_of_unbounded | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsPreconnected",
"Set",
"le_of_lt"
] | If a preconnected set in a linear order space is unbounded below and above, then it is the whole
space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
denselyOrdered_of_preconnectedSpace [LinearOrder α] [OrderTopology α]
[PreconnectedSpace α] : DenselyOrdered α | where
dense x y hxy := by
suffices (Iio y ∩ Ioi x).Nonempty by grind [Set.inter_nonempty_iff_exists_left]
exact nonempty_inter (isOpen_Iio' y) (isOpen_Ioi' x) (Set.Iio_union_Ioi_of_lt hxy)
⟨x, Set.mem_Iio.mpr hxy⟩ ⟨y, Set.mem_Ioi.mpr hxy⟩ | theorem | denselyOrdered_of_preconnectedSpace | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"DenselyOrdered",
"LinearOrder",
"OrderTopology",
"PreconnectedSpace",
"Set.Iio_union_Ioi_of_lt",
"Set.inter_nonempty_iff_exists_left",
"isOpen_Ioi'",
"nonempty_inter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsConnected.Ioo_csInf_csSup_subset {s : Set α} (hs : IsConnected s) (hb : BddBelow s)
(ha : BddAbove s) : Ioo (sInf s) (sSup s) ⊆ s | fun _x hx =>
let ⟨_y, ys, hy⟩ := (isGLB_lt_iff (isGLB_csInf hs.nonempty hb)).1 hx.1
let ⟨_z, zs, hz⟩ := (lt_isLUB_iff (isLUB_csSup hs.nonempty ha)).1 hx.2
hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ | theorem | IsConnected.Ioo_csInf_csSup_subset | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsConnected",
"Set",
"isLUB_csSup",
"lt_isLUB_iff"
] | A bounded connected subset of a conditionally complete linear order includes the open interval
`(Inf s, Sup s)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_Icc_csInf_csSup_of_connected_bdd_closed {s : Set α} (hc : IsConnected s)
(hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s) : s = Icc (sInf s) (sSup s) | (subset_Icc_csInf_csSup hb ha).antisymm <|
hc.Icc_subset (hcl.csInf_mem hc.nonempty hb) (hcl.csSup_mem hc.nonempty ha) | theorem | eq_Icc_csInf_csSup_of_connected_bdd_closed | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsClosed",
"IsConnected",
"Set",
"antisymm",
"subset_Icc_csInf_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.Ioi_csInf_subset {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s)
(ha : ¬BddAbove s) : Ioi (sInf s) ⊆ s | fun x hx =>
have sne : s.Nonempty := nonempty_of_not_bddAbove ha
let ⟨_y, ys, hy⟩ : ∃ y ∈ s, y < x := (isGLB_lt_iff (isGLB_csInf sne hb)).1 hx
let ⟨_z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x
hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ | theorem | IsPreconnected.Ioi_csInf_subset | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsPreconnected",
"Set",
"nonempty_of_not_bddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.Iio_csSup_subset {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
(ha : BddAbove s) : Iio (sSup s) ⊆ s | IsPreconnected.Ioi_csInf_subset (α := αᵒᵈ) hs ha hb | theorem | IsPreconnected.Iio_csSup_subset | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsPreconnected",
"IsPreconnected.Ioi_csInf_subset",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) :
s ∈
({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s),
Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) | by
rcases s.eq_empty_or_nonempty with (rfl | hne)
· apply_rules [Or.inr, mem_singleton]
have hs' : IsConnected s := ⟨hne, hs⟩
by_cases hb : BddBelow s <;> by_cases ha : BddAbove s
· refine mem_of_subset_of_mem ?_ <| mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
(hs'.Ioo_csInf_csSup_subset hb ha) (subset_Icc_c... | theorem | IsPreconnected.mem_intervals | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsConnected",
"IsPreconnected",
"Set",
"by_cases",
"le_csSup",
"subset_Icc_csInf_csSup"
] | A preconnected set in a conditionally complete linear order is either one of the intervals
`[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`,
`(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires
`α` to be densely ordered. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
setOf_isPreconnected_subset_of_ordered :
{ s : Set α | IsPreconnected s } ⊆
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) | by
intro s hs
rcases hs.mem_intervals with (hs | hs | hs | hs | hs | hs | hs | hs | hs | hs) <;> rw [hs] <;>
simp only [union_insert, union_singleton, mem_insert_iff, mem_union, mem_range, Prod.exists,
uncurry_apply_pair, exists_apply_eq_apply, true_or, or_true, exists_apply_eq_apply2] | theorem | setOf_isPreconnected_subset_of_ordered | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"Set",
"exists_apply_eq_apply2"
] | A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`,
`Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordered. Though
one can represent `∅` as `(Inf ∅, Inf ∅)`, we include it into the list of possible cases to improve
readability. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.mem_of_ge_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty) : b ∈ s | by
let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := sSup (s ∩ Icc a b)
have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd
have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2
rcases eq_or_lt_of_le c_le with hc | hc
· exact hc ▸... | theorem | IsClosed.mem_of_ge_of_forall_exists_gt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsClosed",
"Set",
"csSup_le",
"eq_or_lt_of_le",
"le_csSup",
"le_of_lt",
"le_trans",
"not_lt_of_ge"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s | by
intro y hy
have : IsClosed (s ∩ Icc a y) := by
suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y from this ▸ hs.inter isClosed_Icc
grind [inter_assoc, inter_eq_self_of_subset_right, Icc_subset_Icc_right]
exact IsClosed.mem_of_ge_of_forall_exists_gt this ha hy.1 fun x hx ↦
hgt x ⟨hx.1, Ico_subset_Ico_rig... | theorem | IsClosed.Icc_subset_of_forall_exists_gt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsClosed",
"IsClosed.mem_of_ge_of_forall_exists_gt",
"Set",
"isClosed_Icc"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]`
is not empty, then `[a, b] ⊆ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.mem_of_ge_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(hb : b ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty) : a ∈ s | by
suffices OrderDual.toDual a ∈ ofDual ⁻¹' s by aesop
have : IsClosed (OrderDual.ofDual ⁻¹' (s ∩ Icc a b)) := hs
rw [preimage_inter, ← Icc_toDual] at this
apply this.mem_of_ge_of_forall_exists_gt (by simp_all) (by simp_all) (fun x hx ↦ ?_)
rw [Ico_toDual, ← preimage_inter, ← Equiv.image_symm_eq_preimage, mem... | theorem | IsClosed.mem_of_ge_of_forall_exists_lt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Equiv.image_symm_eq_preimage",
"IsClosed",
"OrderDual.ofDual",
"OrderDual.toDual",
"Set"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `b`, and the set `s ∩ (a, b]` has no minimal point, then `a ∈ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.Icc_subset_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(hb : b ∈ s) (hgt : ∀ x ∈ s ∩ Ioc a b, ∀ y ∈ Iio x, (s ∩ Ico y x).Nonempty) : Icc a b ⊆ s | by
intro y hy
have : IsClosed (s ∩ Icc y b) := by
suffices s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b from this ▸ hs.inter isClosed_Icc
grind [Icc_subset_Icc_left, inter_eq_self_of_subset_right, inter_assoc]
exact IsClosed.mem_of_ge_of_forall_exists_lt this hb hy.2 fun x hx ↦
hgt x ⟨hx.1, Ioc_subset_Ioc_left... | theorem | IsClosed.Icc_subset_of_forall_exists_lt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsClosed",
"IsClosed.mem_of_ge_of_forall_exists_lt",
"Set",
"isClosed_Icc"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `b`, and for any `a ≤ y < x ≤ b`, `x ∈ s`, the set `s ∩ [y, x)`
is not empty, then `[a, b] ⊆ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset {a b : α} {s : Set α}
(hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s)
(h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t) :
Icc a b ⊆ s | by
rcases lt_or_ge b a with hab | hab
· simp_all
set A := {t ∈ Icc a b | Icc a t ⊆ s}
have a_mem : a ∈ A := ⟨left_mem_Icc.mpr hab, by simp [ha]⟩
have bdd_A : BddAbove A := ⟨b, fun t ht ↦ ht.1.2⟩
set t₁ := sSup A
have t₁_mem : t₁ ∈ Icc a b := ⟨le_csSup bdd_A a_mem, csSup_le ⟨a, a_mem⟩ (fun t ht ↦ ht.1.2)⟩
... | lemma | IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BddAbove",
"IsClosed",
"Set",
"closure",
"csSup_le",
"csSup_mem_closure",
"eq_of_le_of_not_lt",
"exists_between",
"le_csSup",
"le_or_gt",
"lt_csSup_iff",
"lt_min",
"lt_or_ge",
"mem_nhdsGT_iff_exists_Ioo_subset'",
"min_le_left",
"min_le_right"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ [a, b)` such that `[a, x]` is included in `s`,
the set `s` includes some open neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.Icc_subset_of_forall_mem_nhdsWithin {a b : α} {s : Set α}
(hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) :
Icc a b ⊆ s | hs.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset ha
(fun _t ht h't ↦ hgt _ ⟨h't ⟨ht.1, le_rfl⟩, ht⟩) | theorem | IsClosed.Icc_subset_of_forall_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsClosed",
"Set"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open
neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
(ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
(Icc a b ∩ (s ∩ t)).Nonempty | by
have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
by_contra hst
suffices Icc x y ⊆ s from
hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
rintro z ⟨zs, hz⟩
have zt : z ∈ tᶜ := fun zt =>... | theorem | isPreconnected_Icc_aux | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsClosed",
"IsClosed.inter",
"Set",
"isClosed_Icc",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Icc : IsPreconnected (Icc a b) | isPreconnected_closed_iff.2
(by
rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
-- This used to use `wlog`, but it was causing timeouts.
rcases le_total x y with h | h
· exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
· rw [inter_comm s t]
rw [union_comm s t] at hab
exact i... | theorem | isPreconnected_Icc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"isPreconnected_Icc_aux",
"le_total"
] | A closed interval in a densely ordered conditionally complete linear order is preconnected. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isPreconnected_uIcc : IsPreconnected ([[a, b]]) | isPreconnected_Icc | theorem | isPreconnected_uIcc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"isPreconnected_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.OrdConnected.isPreconnected {s : Set α} (h : s.OrdConnected) : IsPreconnected s | isPreconnected_of_forall_pair fun x hx y hy =>
⟨[[x, y]], h.uIcc_subset hx hy, left_mem_uIcc, right_mem_uIcc, isPreconnected_uIcc⟩ | theorem | Set.OrdConnected.isPreconnected | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"Set",
"isPreconnected_of_forall_pair"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_iff_ordConnected {s : Set α} : IsPreconnected s ↔ OrdConnected s | ⟨IsPreconnected.ordConnected, Set.OrdConnected.isPreconnected⟩ | theorem | isPreconnected_iff_ordConnected | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Ici : IsPreconnected (Ici a) | ordConnected_Ici.isPreconnected | theorem | isPreconnected_Ici | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Iic : IsPreconnected (Iic a) | ordConnected_Iic.isPreconnected | theorem | isPreconnected_Iic | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Iio : IsPreconnected (Iio a) | ordConnected_Iio.isPreconnected | theorem | isPreconnected_Iio | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Ioi : IsPreconnected (Ioi a) | ordConnected_Ioi.isPreconnected | theorem | isPreconnected_Ioi | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Ioo : IsPreconnected (Ioo a b) | ordConnected_Ioo.isPreconnected | theorem | isPreconnected_Ioo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_uIoo : IsPreconnected (uIoo a b) | isPreconnected_Ioo | theorem | isPreconnected_uIoo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"isPreconnected_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Ioc : IsPreconnected (Ioc a b) | ordConnected_Ioc.isPreconnected | theorem | isPreconnected_Ioc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_uIoc : IsPreconnected (uIoc a b) | isPreconnected_Ioc | theorem | isPreconnected_uIoc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"isPreconnected_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreconnected_Ico : IsPreconnected (Ico a b) | ordConnected_Ico.isPreconnected | theorem | isPreconnected_Ico | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Ici : IsConnected (Ici a) | ⟨nonempty_Ici, isPreconnected_Ici⟩ | theorem | isConnected_Ici | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Iic : IsConnected (Iic a) | ⟨nonempty_Iic, isPreconnected_Iic⟩ | theorem | isConnected_Iic | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Ioi [NoMaxOrder α] : IsConnected (Ioi a) | ⟨nonempty_Ioi, isPreconnected_Ioi⟩ | theorem | isConnected_Ioi | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected",
"NoMaxOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Iio [NoMinOrder α] : IsConnected (Iio a) | ⟨nonempty_Iio, isPreconnected_Iio⟩ | theorem | isConnected_Iio | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected",
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Icc (h : a ≤ b) : IsConnected (Icc a b) | ⟨nonempty_Icc.2 h, isPreconnected_Icc⟩ | theorem | isConnected_Icc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Ioo (h : a < b) : IsConnected (Ioo a b) | ⟨nonempty_Ioo.2 h, isPreconnected_Ioo⟩ | theorem | isConnected_Ioo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_uIoo (h : a ≠ b) : IsConnected (uIoo a b) | ⟨nonempty_uIoo.2 h, isPreconnected_uIoo⟩ | theorem | isConnected_uIoo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Ioc (h : a < b) : IsConnected (Ioc a b) | ⟨nonempty_Ioc.2 h, isPreconnected_Ioc⟩ | theorem | isConnected_Ioc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_uIoc (h : a ≠ b) : IsConnected (uIoc a b) | ⟨nonempty_uIoc.2 h, isPreconnected_uIoc⟩ | theorem | isConnected_uIoc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isConnected_Ico (h : a < b) : IsConnected (Ico a b) | ⟨nonempty_Ico.2 h, isPreconnected_Ico⟩ | theorem | isConnected_Ico | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsConnected"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
setOf_isPreconnected_eq_of_ordered :
{ s : Set α | IsPreconnected s } =
-- bounded intervals
range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) | by
refine Subset.antisymm setOf_isPreconnected_subset_of_ordered ?_
simp only [subset_def, forall_mem_range, uncurry, or_imp, forall_and, mem_union,
mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true,
isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Io... | theorem | setOf_isPreconnected_eq_of_ordered | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsPreconnected",
"Set",
"forall_true_iff",
"isPreconnected_Icc",
"isPreconnected_Ici",
"isPreconnected_Ico",
"isPreconnected_Iic",
"isPreconnected_Iio",
"isPreconnected_Ioc",
"isPreconnected_Ioi",
"isPreconnected_Ioo",
"isPreconnected_empty",
"setOf_isPreconnected_subset_of_ordered"
] | In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(sInf s, sInf s)`, we include it into the list of
possible cases to improve readability. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTotallyDisconnected_iff_lt {s : Set α} :
IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y | by
simp only [IsTotallyDisconnected, isPreconnected_iff_ordConnected, ← not_nontrivial_iff,
nontrivial_iff_exists_lt, not_exists, not_and]
refine ⟨fun h x hx y hy hxy ↦ ?_, fun h t hts ht x hx y hy hxy ↦ ?_⟩
· simp_rw [← not_ordConnected_inter_Icc_iff hx hy]
exact fun hs ↦ h _ inter_subset_left hs _ ⟨hx, ... | lemma | isTotallyDisconnected_iff_lt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"IsTotallyDisconnected",
"Set",
"isPreconnected_iff_ordConnected",
"le_rfl"
] | This lemma characterizes when a subset `s` of a densely ordered conditionally complete linear
order is totally disconnected with respect to the order topology: between any two distinct points
of `s` must lie a point not in `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Icc (f a) (f b) ⊆ f '' Icc a b | isPreconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf | theorem | intermediate_value_Icc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn"
] | **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≤ t ≤ f b`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Icc (f b) (f a) ⊆ f '' Icc a b | isPreconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf | theorem | intermediate_value_Icc' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn"
] | **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≥ t ≥ f b`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f [[a, b]]) :
[[f a, f b]] ⊆ f '' uIcc a b | by
cases le_total (f a) (f b) <;> simp [*, isPreconnected_uIcc.intermediate_value] | theorem | intermediate_value_uIcc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"le_total"
] | **Intermediate Value Theorem** for continuous functions on closed intervals, unordered case. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_mem_uIcc_isFixedPt {a b : α} {f : α → α} (hf : ContinuousOn f (uIcc a b))
(ha : a ≤ f a) (hb : f b ≤ b) : ∃ c ∈ [[a, b]], IsFixedPt f c | isPreconnected_uIcc.intermediate_value₂ right_mem_uIcc left_mem_uIcc hf continuousOn_id hb ha | theorem | exists_mem_uIcc_isFixedPt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"continuousOn_id"
] | If `f : α → α` is continuous on `[[a, b]]`, `a ≤ f a`, and `f b ≤ b`,
then `f` has a fixed point on `[[a, b]]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_mem_Icc_isFixedPt {a b : α} {f : α → α} (hf : ContinuousOn f (Icc a b))
(hle : a ≤ b) (ha : a ≤ f a) (hb : f b ≤ b) : ∃ c ∈ Icc a b, IsFixedPt f c | isPreconnected_Icc.intermediate_value₂
(right_mem_Icc.2 hle) (left_mem_Icc.2 hle) hf continuousOn_id hb ha | theorem | exists_mem_Icc_isFixedPt | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"continuousOn_id"
] | If `f : α → α` is continuous on `[a, b]`, `a ≤ b`, `a ≤ f a`, and `f b ≤ b`,
then `f` has a fixed point on `[a, b]`.
In particular, if `[a, b]` is forward-invariant under `f`,
then `f` has a fixed point on `[a, b]`, see `exists_mem_Icc_isFixedPt_of_mapsTo`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_mem_Icc_isFixedPt_of_mapsTo {a b : α} {f : α → α} (hf : ContinuousOn f (Icc a b))
(hle : a ≤ b) (hmaps : MapsTo f (Icc a b) (Icc a b)) : ∃ c ∈ Icc a b, IsFixedPt f c | exists_mem_Icc_isFixedPt hf hle (hmaps <| left_mem_Icc.2 hle).1 (hmaps <| right_mem_Icc.2 hle).2 | theorem | exists_mem_Icc_isFixedPt_of_mapsTo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"exists_mem_Icc_isFixedPt"
] | If a closed interval is forward-invariant under a continuous map `f : α → α`,
then this map has a fixed point on this interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
intermediate_value_Ico {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Ico (f a) (f b) ⊆ f '' Ico a b | Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_ge (he ▸ h.1))) fun hlt =>
@IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩
(right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _
((hf.continuousWithinAt ⟨hab, refl b⟩).mo... | theorem | intermediate_value_Ico | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected.intermediate_value_Ico",
"eq_or_lt_of_le",
"isPreconnected_Ico",
"not_lt_of_ge",
"refl",
"right_nhdsWithin_Ico_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_Ico' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Ioc (f b) (f a) ⊆ f '' Ico a b | Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_ge (he ▸ h.2))) fun hlt =>
@IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩
(right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _
((hf.continuousWithinAt ⟨hab, refl b⟩).mo... | theorem | intermediate_value_Ico' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected.intermediate_value_Ioc",
"eq_or_lt_of_le",
"isPreconnected_Ico",
"not_lt_of_ge",
"refl",
"right_nhdsWithin_Ico_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_Ioc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Ioc (f a) (f b) ⊆ f '' Ioc a b | Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_le_of_gt (he ▸ h.1))) fun hlt =>
@IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩
(left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _
((hf.continuousWithinAt ⟨refl a, hab⟩).mon... | theorem | intermediate_value_Ioc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected.intermediate_value_Ioc",
"eq_or_lt_of_le",
"isPreconnected_Ioc",
"left_nhdsWithin_Ioc_neBot",
"not_le_of_gt",
"refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_Ioc' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Ico (f b) (f a) ⊆ f '' Ioc a b | Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_le_of_gt (he ▸ h.2))) fun hlt =>
@IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩
(left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _
((hf.continuousWithinAt ⟨refl a, hab⟩).mon... | theorem | intermediate_value_Ioc' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected.intermediate_value_Ico",
"eq_or_lt_of_le",
"isPreconnected_Ioc",
"left_nhdsWithin_Ioc_neBot",
"not_le_of_gt",
"refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_Ioo {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Ioo (f a) (f b) ⊆ f '' Ioo a b | Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_gt (he ▸ h.1))) fun hlt =>
@IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _
(left_nhdsWithin_Ioo_neBot hlt) (right_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _
(hf.mono Ioo_subset_Icc_self) _ _
((... | theorem | intermediate_value_Ioo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected.intermediate_value_Ioo",
"eq_or_lt_of_le",
"isPreconnected_Ioo",
"left_nhdsWithin_Ioo_neBot",
"refl",
"right_nhdsWithin_Ioo_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
intermediate_value_Ioo' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Ioo (f b) (f a) ⊆ f '' Ioo a b | Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_gt (he ▸ h.2))) fun hlt =>
@IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _
(right_nhdsWithin_Ioo_neBot hlt) (left_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _
(hf.mono Ioo_subset_Icc_self) _ _
((... | theorem | intermediate_value_Ioo' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"IsPreconnected.intermediate_value_Ioo",
"eq_or_lt_of_le",
"isPreconnected_Ioo",
"left_nhdsWithin_Ioo_neBot",
"refl",
"right_nhdsWithin_Ioo_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.surjOn_Icc {s : Set α} [hs : OrdConnected s] {f : α → δ}
(hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (Icc (f a) (f b)) | hs.isPreconnected.intermediate_value ha hb hf | theorem | ContinuousOn.surjOn_Icc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Set"
] | **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `Icc (f a) (f b)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
(hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
SurjOn f s (uIcc (f a) (f b)) | by
rcases le_total (f a) (f b) with hab | hab <;> simp [hf.surjOn_Icc, *] | theorem | ContinuousOn.surjOn_uIcc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Set",
"le_total"
] | **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `[f a, f b]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.surjective {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atTop atTop)
(h_bot : Tendsto f atBot atBot) : Function.Surjective f | fun p =>
mem_range_of_exists_le_of_exists_ge hf (h_bot.eventually (eventually_le_atBot p)).exists
(h_top.eventually (eventually_ge_atTop p)).exists | theorem | Continuous.surjective | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"mem_range_of_exists_le_of_exists_ge"
] | A continuous function which tends to `Filter.atTop` along `Filter.atTop` and to `Filter.atBot`
along `Filter.atBot` is surjective. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.surjective' {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atBot atTop)
(h_bot : Tendsto f atTop atBot) : Function.Surjective f | Continuous.surjective (α := αᵒᵈ) hf h_top h_bot | theorem | Continuous.surjective' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"Continuous.surjective"
] | A continuous function which tends to `Filter.atBot` along `Filter.atTop` and to `Filter.atTop`
along `Filter.atBot` is surjective. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.surjOn_of_tendsto {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty)
(hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atBot)
(htop : Tendsto (fun x : s => f x) atTop atTop) : SurjOn f s univ | haveI := Classical.inhabited_of_nonempty hs.to_subtype
surjOn_iff_surjective.2 <| hf.restrict.surjective htop hbot | theorem | ContinuousOn.surjOn_of_tendsto | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"Set"
] | If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `Filter.atBot : Filter β` along `Filter.atBot : Filter ↥s` and tends to
`Filter.atTop : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is
surjective. We formulate the conclusion as `Functi... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.surjOn_of_tendsto' {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty)
(hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atTop)
(htop : Tendsto (fun x : s => f x) atTop atBot) : SurjOn f s univ | ContinuousOn.surjOn_of_tendsto (δ := δᵒᵈ) hs hf hbot htop | theorem | ContinuousOn.surjOn_of_tendsto' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"ContinuousOn.surjOn_of_tendsto",
"Set"
] | If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `Filter.atTop : Filter β` along `Filter.atBot : Filter ↥s` and tends to
`Filter.atBot : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is
surjective. We formulate the conclusion as `Functi... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.strictMono_of_inj_boundedOrder [BoundedOrder α] {f : α → δ}
(hf_c : Continuous f) (hf : f ⊥ ≤ f ⊤) (hf_i : Injective f) : StrictMono f | by
intro a b hab
by_contra! h
have H : f b < f a := lt_of_le_of_ne h <| hf_i.ne hab.ne'
by_cases! ha : f a ≤ f ⊥
· obtain ⟨u, hu⟩ := intermediate_value_Ioc le_top hf_c.continuousOn ⟨H.trans_le ha, hf⟩
have : u = ⊥ := hf_i hu.2
simp_all
· by_cases! hb : f ⊥ < f b
· obtain ⟨u, hu⟩ := intermediate_... | theorem | Continuous.strictMono_of_inj_boundedOrder | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BoundedOrder",
"Continuous",
"StrictMono",
"intermediate_value_Ioc",
"intermediate_value_Ioo",
"intermediate_value_Ioo'",
"le_top",
"lt_of_le_of_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.strictAnti_of_inj_boundedOrder [BoundedOrder α] {f : α → δ}
(hf_c : Continuous f) (hf : f ⊤ ≤ f ⊥) (hf_i : Injective f) : StrictAnti f | hf_c.strictMono_of_inj_boundedOrder (δ := δᵒᵈ) hf hf_i | theorem | Continuous.strictAnti_of_inj_boundedOrder | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BoundedOrder",
"Continuous",
"StrictAnti"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.strictMono_of_inj_boundedOrder' [BoundedOrder α] {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f | (le_total (f ⊥) (f ⊤)).imp
(hf_c.strictMono_of_inj_boundedOrder · hf_i)
(hf_c.strictAnti_of_inj_boundedOrder · hf_i) | theorem | Continuous.strictMono_of_inj_boundedOrder' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"BoundedOrder",
"Continuous",
"StrictAnti",
"StrictMono",
"le_total"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.strictMonoOn_of_inj_rigidity {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b)
(hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f | by
intro x y hxy
let s := min a x
let t := max b y
have hsa : s ≤ a := min_le_left a x
have hbt : b ≤ t := le_max_left b y
have hf_mono_st : StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t) := by
have : Fact (s ≤ t) := ⟨hsa.trans <| hbt.trans' hab.le⟩
have := Continuous.strictMono_of_inj_boundedO... | theorem | Continuous.strictMonoOn_of_inj_rigidity | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"Continuous.strictMono_of_inj_boundedOrder'",
"Fact",
"IsAntichain",
"IsAntichain.of_strictMonoOn_antitoneOn",
"Set.restrict",
"StrictAntiOn",
"StrictAntiOn.mono",
"StrictMono",
"StrictMonoOn",
"StrictMonoOn.mono",
"le_of_lt",
"min_le_left",
"min_le_right"
] | Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is
continuous and injective. Then `f` is strictly monotone (increasing) if
it is strictly monotone (increasing) on some closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.strictMonoOn_of_injOn_Icc {a b : α} {f : α → δ}
(hab : a ≤ b) (hfab : f a ≤ f b)
(hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
StrictMonoOn f (Icc a b) | by
have : Fact (a ≤ b) := ⟨hab⟩
refine StrictMono.of_restrict ?_
set g : Icc a b → δ := Set.restrict (Icc a b) f
have hgab : g ⊥ ≤ g ⊤ := by aesop
exact Continuous.strictMono_of_inj_boundedOrder (f := g) hf_c.restrict hgab hf_i.injective | theorem | ContinuousOn.strictMonoOn_of_injOn_Icc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous.strictMono_of_inj_boundedOrder",
"ContinuousOn",
"Fact",
"Set.restrict",
"StrictMonoOn"
] | Suppose `f : [a, b] → δ` is
continuous and injective. Then `f` is strictly monotone (increasing) if `f(a) ≤ f(b)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.strictAntiOn_of_injOn_Icc {a b : α} {f : α → δ}
(hab : a ≤ b) (hfab : f b ≤ f a)
(hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
StrictAntiOn f (Icc a b) | ContinuousOn.strictMonoOn_of_injOn_Icc (δ := δᵒᵈ) hab hfab hf_c hf_i | theorem | ContinuousOn.strictAntiOn_of_injOn_Icc | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"ContinuousOn.strictMonoOn_of_injOn_Icc",
"StrictAntiOn"
] | Suppose `f : [a, b] → δ` is
continuous and injective. Then `f` is strictly antitone (decreasing) if `f(b) ≤ f(a)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.strictMonoOn_of_injOn_Icc' {a b : α} {f : α → δ} (hab : a ≤ b)
(hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
StrictMonoOn f (Icc a b) ∨ StrictAntiOn f (Icc a b) | (le_total (f a) (f b)).imp
(ContinuousOn.strictMonoOn_of_injOn_Icc hab · hf_c hf_i)
(ContinuousOn.strictAntiOn_of_injOn_Icc hab · hf_c hf_i) | theorem | ContinuousOn.strictMonoOn_of_injOn_Icc' | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"ContinuousOn",
"ContinuousOn.strictAntiOn_of_injOn_Icc",
"ContinuousOn.strictMonoOn_of_injOn_Icc",
"StrictAntiOn",
"StrictMonoOn",
"le_total"
] | Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone
or antitone (increasing or decreasing). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.strictMono_of_inj {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f | by
have H {c d : α} (hcd : c < d) : StrictMono f ∨ StrictAnti f :=
(hf_c.continuousOn.strictMonoOn_of_injOn_Icc' hcd.le hf_i.injOn).imp
(hf_c.strictMonoOn_of_inj_rigidity hf_i hcd)
(hf_c.strictMonoOn_of_inj_rigidity (δ := δᵒᵈ) hf_i hcd)
cases subsingleton_or_nontrivial α with
| inl h => exact Or.i... | theorem | Continuous.strictMono_of_inj | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous",
"StrictAnti",
"StrictMono",
"Subsingleton.strictMono",
"exists_pair_lt",
"subsingleton_or_nontrivial"
] | Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is
continuous and injective. Then `f` is strictly monotone or antitone (increasing or decreasing). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.strictMonoOn_of_injOn_Ioo {a b : α} {f : α → δ} (hab : a < b)
(hf_c : ContinuousOn f (Ioo a b)) (hf_i : InjOn f (Ioo a b)) :
StrictMonoOn f (Ioo a b) ∨ StrictAntiOn f (Ioo a b) | by
haveI : Inhabited (Ioo a b) := Classical.inhabited_of_nonempty (nonempty_Ioo_subtype hab)
let g : Ioo a b → δ := Set.restrict (Ioo a b) f
have : StrictMono g ∨ StrictAnti g :=
Continuous.strictMono_of_inj hf_c.restrict hf_i.injective
exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr | theorem | ContinuousOn.strictMonoOn_of_injOn_Ioo | Topology.Order | Mathlib/Topology/Order/IntermediateValue.lean | [] | [
"Continuous.strictMono_of_inj",
"ContinuousOn",
"Set.restrict",
"StrictAnti",
"StrictAntiOn",
"StrictMono",
"StrictMonoOn"
] | Every continuous injective `f : (a, b) → δ` is strictly monotone
or antitone (increasing or decreasing). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLocallyClosed_Icc [Preorder X] [OrderClosedTopology X] :
IsLocallyClosed (Set.Icc a b) | isClosed_Icc.isLocallyClosed | theorem | isLocallyClosed_Icc | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"IsLocallyClosed",
"OrderClosedTopology",
"Preorder",
"Set.Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocallyClosed_Ioo [LinearOrder X] [OrderClosedTopology X] :
IsLocallyClosed (Set.Ioo a b) | isOpen_Ioo.isLocallyClosed | theorem | isLocallyClosed_Ioo | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"IsLocallyClosed",
"LinearOrder",
"OrderClosedTopology",
"Set.Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocallyClosed_Ici [Preorder X] [ClosedIciTopology X] :
IsLocallyClosed (Set.Ici a) | isClosed_Ici.isLocallyClosed | theorem | isLocallyClosed_Ici | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"ClosedIciTopology",
"IsLocallyClosed",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocallyClosed_Iic [Preorder X] [ClosedIicTopology X] :
IsLocallyClosed (Set.Iic a) | isClosed_Iic.isLocallyClosed | theorem | isLocallyClosed_Iic | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"ClosedIicTopology",
"IsLocallyClosed",
"Preorder",
"Set.Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocallyClosed_Ioi [LinearOrder X] [ClosedIicTopology X] :
IsLocallyClosed (Set.Ioi a) | isOpen_Ioi.isLocallyClosed | theorem | isLocallyClosed_Ioi | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"ClosedIicTopology",
"IsLocallyClosed",
"LinearOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocallyClosed_Iio [LinearOrder X] [ClosedIciTopology X] :
IsLocallyClosed (Set.Iio a) | isOpen_Iio.isLocallyClosed | theorem | isLocallyClosed_Iio | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"ClosedIciTopology",
"IsLocallyClosed",
"LinearOrder",
"Set.Iio"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocallyClosed_Ioc [LinearOrder X] [ClosedIicTopology X] :
IsLocallyClosed (Set.Ioc a b) | by
rw [← Set.Iic_inter_Ioi]
exact isLocallyClosed_Iic.inter isLocallyClosed_Ioi | theorem | isLocallyClosed_Ioc | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"ClosedIicTopology",
"IsLocallyClosed",
"LinearOrder",
"Set.Ioc",
"isLocallyClosed_Ioi"
] | 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.