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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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subset_closure_inter_of_is_preirreducible_of_is_open {S U : set α}
(hS : is_preirreducible S) (hU : is_open U) (h : (S ∩ U).nonempty) : S ⊆ closure (S ∩ U) | begin
by_contra h',
obtain ⟨x, h₁, h₂, h₃⟩ := hS _ (closure (S ∩ U))ᶜ hU (is_open_compl_iff.mpr is_closed_closure) h
(set.inter_compl_nonempty_iff.mpr h'),
exact h₃ (subset_closure ⟨h₁, h₂⟩)
end | lemma | subset_closure_inter_of_is_preirreducible_of_is_open | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"by_contra",
"closure",
"is_closed_closure",
"is_open",
"is_preirreducible",
"subset_closure"
] | A nonemtpy open subset of a preirreducible subspace is dense in the subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preirreducible.subset_irreducible {S U Z : set α}
(hZ : is_preirreducible Z) (hU : U.nonempty) (hU' : is_open U)
(h₁ : U ⊆ S) (h₂ : S ⊆ Z) : is_irreducible S | begin
classical,
obtain ⟨z, hz⟩ := hU,
replace hZ : is_irreducible Z := ⟨⟨z, h₂ (h₁ hz)⟩, hZ⟩,
refine ⟨⟨z, h₁ hz⟩, _⟩,
rintros u v hu hv ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩,
obtain ⟨a, -, ha'⟩ := is_irreducible_iff_sInter.mp hZ {U, u, v} (by tidy) _,
replace ha' : a ∈ U ∧ a ∈ u ∧ a ∈ v := by simpa using ha',
exac... | lemma | is_preirreducible.subset_irreducible | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finset.mem_insert",
"finset.mem_singleton",
"is_irreducible",
"is_open",
"is_preirreducible"
] | If `∅ ≠ U ⊆ S ⊆ Z` such that `U` is open and `Z` is preirreducible, then `S` is irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preirreducible.open_subset {Z U : set α} (hZ : is_preirreducible Z)
(hU : is_open U) (hU' : U ⊆ Z) :
is_preirreducible U | U.eq_empty_or_nonempty.elim (λ h, h.symm ▸ is_preirreducible_empty)
(λ h, (hZ.subset_irreducible h hU (λ _, id) hU').2) | lemma | is_preirreducible.open_subset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_open",
"is_preirreducible",
"is_preirreducible_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preirreducible.interior {Z : set α} (hZ : is_preirreducible Z) :
is_preirreducible (interior Z) | hZ.open_subset is_open_interior interior_subset | lemma | is_preirreducible.interior | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"interior",
"interior_subset",
"is_open_interior",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preirreducible.preimage {Z : set α} (hZ : is_preirreducible Z)
{f : β → α} (hf : open_embedding f) :
is_preirreducible (f ⁻¹' Z) | begin
rintros U V hU hV ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩,
obtain ⟨_, h₁, ⟨z, h₂, rfl⟩, ⟨z', h₃, h₄⟩⟩ := hZ _ _ (hf.is_open_map _ hU) (hf.is_open_map _ hV)
⟨f x, hx, set.mem_image_of_mem f hx'⟩ ⟨f y, hy, set.mem_image_of_mem f hy'⟩,
cases hf.inj h₄,
exact ⟨z, h₁, h₂, h₃⟩
end | lemma | is_preirreducible.preimage | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_preirreducible",
"open_embedding",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tsupport (f : X → α) : set X | closure (mul_support f) | def | mul_tsupport | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure"
] | The topological support of a function is the closure of its support, i.e. the closure of the
set of all elements where the function is not equal to 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_mul_tsupport (f : X → α) : mul_support f ⊆ mul_tsupport f | subset_closure | lemma | subset_mul_tsupport | topology | src/topology/support.lean | [
"topology.separation"
] | [
"mul_tsupport",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_mul_tsupport (f : X → α) : is_closed (mul_tsupport f) | is_closed_closure | lemma | is_closed_mul_tsupport | topology | src/topology/support.lean | [
"topology.separation"
] | [
"is_closed",
"is_closed_closure",
"mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tsupport_eq_empty_iff {f : X → α} : mul_tsupport f = ∅ ↔ f = 1 | by rw [mul_tsupport, closure_empty_iff, mul_support_eq_empty_iff] | lemma | mul_tsupport_eq_empty_iff | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure_empty_iff",
"mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_one_of_nmem_mul_tsupport {f : X → α} {x : X} (hx : x ∉ mul_tsupport f) : f x = 1 | mul_support_subset_iff'.mp (subset_mul_tsupport f) x hx | lemma | image_eq_one_of_nmem_mul_tsupport | topology | src/topology/support.lean | [
"topology.separation"
] | [
"mul_tsupport",
"subset_mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_subset_insert_image_mul_tsupport (f : X → α) :
range f ⊆ insert 1 (f '' mul_tsupport f) | (range_subset_insert_image_mul_support f).trans $
insert_subset_insert $ image_subset _ subset_closure | lemma | range_subset_insert_image_mul_tsupport | topology | src/topology/support.lean | [
"topology.separation"
] | [
"mul_tsupport",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_image_mul_tsupport_or (f : X → α) :
range f = f '' mul_tsupport f ∨ range f = insert 1 (f '' mul_tsupport f) | (wcovby_insert _ _).eq_or_eq (image_subset_range _ _) (range_subset_insert_image_mul_tsupport f) | lemma | range_eq_image_mul_tsupport_or | topology | src/topology/support.lean | [
"topology.separation"
] | [
"mul_tsupport",
"range_subset_insert_image_mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsupport_mul_subset_left {α : Type*} [mul_zero_class α] {f g : X → α} :
tsupport (λ x, f x * g x) ⊆ tsupport f | closure_mono (support_mul_subset_left _ _) | lemma | tsupport_mul_subset_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure_mono",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsupport_mul_subset_right {α : Type*} [mul_zero_class α] {f g : X → α} :
tsupport (λ x, f x * g x) ⊆ tsupport g | closure_mono (support_mul_subset_right _ _) | lemma | tsupport_mul_subset_right | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure_mono",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsupport_smul_subset_left {M α} [topological_space X] [has_zero M] [has_zero α]
[smul_with_zero M α] (f : X → M) (g : X → α) :
tsupport (λ x, f x • g x) ⊆ tsupport f | closure_mono $ support_smul_subset_left f g | lemma | tsupport_smul_subset_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure_mono",
"smul_with_zero",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_mul_tsupport_iff_eventually_eq : x ∉ mul_tsupport f ↔ f =ᶠ[𝓝 x] 1 | by simp_rw [mul_tsupport, mem_closure_iff_nhds, not_forall, not_nonempty_iff_eq_empty,
← disjoint_iff_inter_eq_empty, disjoint_mul_support_iff, eventually_eq_iff_exists_mem] | lemma | not_mem_mul_tsupport_iff_eventually_eq | topology | src/topology/support.lean | [
"topology.separation"
] | [
"mem_closure_iff_nhds",
"mul_tsupport",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_mul_tsupport [topological_space β] {f : α → β}
(hf : ∀ x ∈ mul_tsupport f, continuous_at f x) : continuous f | continuous_iff_continuous_at.2 $ λ x, (em _).elim (hf x) $ λ hx,
(@continuous_at_const _ _ _ _ _ 1).congr (not_mem_mul_tsupport_iff_eventually_eq.mp hx).symm | lemma | continuous_of_mul_tsupport | topology | src/topology/support.lean | [
"topology.separation"
] | [
"continuous",
"continuous_at",
"continuous_at_const",
"em",
"mul_tsupport",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support (f : α → β) : Prop | is_compact (mul_tsupport f) | def | has_compact_mul_support | topology | src/topology/support.lean | [
"topology.separation"
] | [
"is_compact",
"mul_tsupport"
] | A function `f` *has compact multiplicative support* or is *compactly supported* if the closure
of the multiplicative support of `f` is compact. In a T₂ space this is equivalent to `f` being equal
to `1` outside a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_mul_support_def :
has_compact_mul_support f ↔ is_compact (closure (mul_support f)) | by refl | lemma | has_compact_mul_support_def | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure",
"has_compact_mul_support",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_compact_iff_has_compact_mul_support [t2_space α] :
(∃ K : set α, is_compact K ∧ ∀ x ∉ K, f x = 1) ↔ has_compact_mul_support f | by simp_rw [← nmem_mul_support, ← mem_compl_iff, ← subset_def, compl_subset_compl,
has_compact_mul_support_def, exists_compact_superset_iff] | lemma | exists_compact_iff_has_compact_mul_support | topology | src/topology/support.lean | [
"topology.separation"
] | [
"exists_compact_superset_iff",
"has_compact_mul_support",
"has_compact_mul_support_def",
"is_compact",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.intro [t2_space α] {K : set α}
(hK : is_compact K) (hfK : ∀ x ∉ K, f x = 1) : has_compact_mul_support f | exists_compact_iff_has_compact_mul_support.mp ⟨K, hK, hfK⟩ | lemma | has_compact_mul_support.intro | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support",
"is_compact",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.is_compact (hf : has_compact_mul_support f) :
is_compact (mul_tsupport f) | hf | lemma | has_compact_mul_support.is_compact | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support",
"is_compact",
"mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support_iff_eventually_eq :
has_compact_mul_support f ↔ f =ᶠ[coclosed_compact α] 1 | ⟨ λ h, mem_coclosed_compact.mpr ⟨mul_tsupport f, is_closed_mul_tsupport _, h,
λ x, not_imp_comm.mpr $ λ hx, subset_mul_tsupport f hx⟩,
λ h, let ⟨C, hC⟩ := mem_coclosed_compact'.mp h in
is_compact_of_is_closed_subset hC.2.1 (is_closed_mul_tsupport _) (closure_minimal hC.2.2 hC.1)⟩ | lemma | has_compact_mul_support_iff_eventually_eq | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure_minimal",
"has_compact_mul_support",
"is_closed_mul_tsupport",
"is_compact_of_is_closed_subset",
"subset_mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.is_compact_range [topological_space β]
(h : has_compact_mul_support f) (hf : continuous f) : is_compact (range f) | begin
cases range_eq_image_mul_tsupport_or f with h2 h2; rw [h2],
exacts [h.image hf, (h.image hf).insert 1]
end | lemma | has_compact_mul_support.is_compact_range | topology | src/topology/support.lean | [
"topology.separation"
] | [
"continuous",
"has_compact_mul_support",
"is_compact",
"range_eq_image_mul_tsupport_or",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.mono' {f' : α → γ} (hf : has_compact_mul_support f)
(hff' : mul_support f' ⊆ mul_tsupport f) : has_compact_mul_support f' | is_compact_of_is_closed_subset hf is_closed_closure $ closure_minimal hff' is_closed_closure | lemma | has_compact_mul_support.mono' | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closure_minimal",
"has_compact_mul_support",
"is_closed_closure",
"is_compact_of_is_closed_subset",
"mul_tsupport"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.mono {f' : α → γ} (hf : has_compact_mul_support f)
(hff' : mul_support f' ⊆ mul_support f) : has_compact_mul_support f' | hf.mono' $ hff'.trans subset_closure | lemma | has_compact_mul_support.mono | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.comp_left (hf : has_compact_mul_support f) (hg : g 1 = 1) :
has_compact_mul_support (g ∘ f) | hf.mono $ mul_support_comp_subset hg f | lemma | has_compact_mul_support.comp_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support_comp_left (hg : ∀ {x}, g x = 1 ↔ x = 1) :
has_compact_mul_support (g ∘ f) ↔ has_compact_mul_support f | by simp_rw [has_compact_mul_support_def, mul_support_comp_eq g @hg f] | lemma | has_compact_mul_support_comp_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support",
"has_compact_mul_support_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.comp_closed_embedding (hf : has_compact_mul_support f)
{g : α' → α} (hg : closed_embedding g) : has_compact_mul_support (f ∘ g) | begin
rw [has_compact_mul_support_def, function.mul_support_comp_eq_preimage],
refine is_compact_of_is_closed_subset (hg.is_compact_preimage hf) is_closed_closure _,
rw [hg.to_embedding.closure_eq_preimage_closure_image],
exact preimage_mono (closure_mono $ image_preimage_subset _ _)
end | lemma | has_compact_mul_support.comp_closed_embedding | topology | src/topology/support.lean | [
"topology.separation"
] | [
"closed_embedding",
"closure_mono",
"function.mul_support_comp_eq_preimage",
"has_compact_mul_support",
"has_compact_mul_support_def",
"is_closed_closure",
"is_compact_of_is_closed_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.comp₂_left (hf : has_compact_mul_support f)
(hf₂ : has_compact_mul_support f₂) (hm : m 1 1 = 1) :
has_compact_mul_support (λ x, m (f x) (f₂ x)) | begin
rw [has_compact_mul_support_iff_eventually_eq] at hf hf₂ ⊢,
filter_upwards [hf, hf₂] using λ x hx hx₂, by simp_rw [hx, hx₂, pi.one_apply, hm]
end | lemma | has_compact_mul_support.comp₂_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support",
"has_compact_mul_support_iff_eventually_eq",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.mul (hf : has_compact_mul_support f)
(hf' : has_compact_mul_support f') : has_compact_mul_support (f * f') | by apply hf.comp₂_left hf' (mul_one 1) | lemma | has_compact_mul_support.mul | topology | src/topology/support.lean | [
"topology.separation"
] | [
"has_compact_mul_support",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.smul_left (hf : has_compact_support f') : has_compact_support (f • f') | begin
rw [has_compact_support_iff_eventually_eq] at hf ⊢,
refine hf.mono (λ x hx, by simp_rw [pi.smul_apply', hx, pi.zero_apply, smul_zero])
end | lemma | has_compact_support.smul_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"pi.smul_apply'",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.smul_right (hf : has_compact_support f) : has_compact_support (f • f') | begin
rw [has_compact_support_iff_eventually_eq] at hf ⊢,
refine hf.mono (λ x hx, by simp_rw [pi.smul_apply', hx, pi.zero_apply, zero_smul])
end | lemma | has_compact_support.smul_right | topology | src/topology/support.lean | [
"topology.separation"
] | [
"pi.smul_apply'",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.smul_left' (hf : has_compact_support f') : has_compact_support (f • f') | begin
rw [has_compact_support_iff_eventually_eq] at hf ⊢,
refine hf.mono (λ x hx, by simp_rw [pi.smul_apply', hx, pi.zero_apply, smul_zero])
end | lemma | has_compact_support.smul_left' | topology | src/topology/support.lean | [
"topology.separation"
] | [
"pi.smul_apply'",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.mul_right (hf : has_compact_support f) : has_compact_support (f * f') | begin
rw [has_compact_support_iff_eventually_eq] at hf ⊢,
refine hf.mono (λ x hx, by simp_rw [pi.mul_apply, hx, pi.zero_apply, zero_mul])
end | lemma | has_compact_support.mul_right | topology | src/topology/support.lean | [
"topology.separation"
] | [
"pi.mul_apply",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.mul_left (hf : has_compact_support f') : has_compact_support (f * f') | begin
rw [has_compact_support_iff_eventually_eq] at hf ⊢,
refine hf.mono (λ x hx, by simp_rw [pi.mul_apply, hx, pi.zero_apply, mul_zero])
end | lemma | has_compact_support.mul_left | topology | src/topology/support.lean | [
"topology.separation"
] | [
"mul_zero",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_finset_nhd_mul_support_subset
{f : ι → X → R} (hlf : locally_finite (λ i, mul_support (f i)))
(hso : ∀ i, mul_tsupport (f i) ⊆ U i) (ho : ∀ i, is_open (U i)) (x : X) :
∃ (is : finset ι) {n : set X} (hn₁ : n ∈ 𝓝 x) (hn₂ : n ⊆ ⋂ i ∈ is, U i), ∀ (z ∈ n),
mul_support (λ i, f i z) ⊆ is | begin
obtain ⟨n, hn, hnf⟩ := hlf x,
classical,
let is := hnf.to_finset.filter (λ i, x ∈ U i),
let js := hnf.to_finset.filter (λ j, x ∉ U j),
refine ⟨is, n ∩ (⋂ j ∈ js, (mul_tsupport (f j))ᶜ) ∩ (⋂ i ∈ is, U i),
inter_mem (inter_mem hn _) _, inter_subset_right _ _, λ z hz, _⟩,
{ exact (bInter_finset_mem j... | lemma | locally_finite.exists_finset_nhd_mul_support_subset | topology | src/topology/support.lean | [
"topology.separation"
] | [
"and_imp",
"finset",
"finset.mem_filter",
"is_closed.compl_mem_nhds",
"is_closed_mul_tsupport",
"is_open",
"locally_finite",
"mul_tsupport",
"set.not_mem_subset",
"subset_mul_tsupport"
] | If a family of functions `f` has locally-finite multiplicative support, subordinate to a family
of open sets, then for any point we can find a neighbourhood on which only finitely-many members of
`f` are not equal to 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuity : user_attribute | { name := `continuity,
descr := "lemmas usable to prove continuity" } | def | continuity | topology | src/topology/tactic.lean | [
"tactic.auto_cases",
"tactic.tidy",
"tactic.with_local_reducibility",
"tactic.show_term",
"topology.basic"
] | [] | User attribute used to mark tactics used by `continuity`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_id' {α : Type*} [topological_space α] : continuous (λ a : α, a) | continuous_id | lemma | continuous_id' | topology | src/topology/tactic.lean | [
"tactic.auto_cases",
"tactic.tidy",
"tactic.with_local_reducibility",
"tactic.show_term",
"topology.basic"
] | [
"continuous",
"continuous_id",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_continuous.comp : tactic unit | `[fail_if_success { exact continuous_const };
refine continuous.comp _ _;
fail_if_success { exact continuous_id }] | def | tactic.apply_continuous.comp | topology | src/topology/tactic.lean | [
"tactic.auto_cases",
"tactic.tidy",
"tactic.with_local_reducibility",
"tactic.show_term",
"topology.basic"
] | [
"continuous.comp",
"continuous_const",
"continuous_id"
] | Tactic to apply `continuous.comp` when appropriate.
Applying `continuous.comp` is not always a good idea, so we have some
extra logic here to try to avoid bad cases.
* If the function we're trying to prove continuous is actually
constant, and that constant is a function application `f z`, then
continuous.comp wou... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuity_tactics (md : transparency := reducible) : list (tactic string) | [
intros1 >>= λ ns, pure ("intros " ++ (" ".intercalate (ns.map (λ e, e.to_string)))),
apply_rules [] [``continuity] 50 { md := md }
>> pure "apply_rules with continuity",
apply_continuous.comp >> pure "refine continuous.comp _ _"
] | def | tactic.continuity_tactics | topology | src/topology/tactic.lean | [
"tactic.auto_cases",
"tactic.tidy",
"tactic.with_local_reducibility",
"tactic.show_term",
"topology.basic"
] | [
"continuity"
] | List of tactics used by `continuity` internally. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuity
(bang : parse $ optional (tk "!")) (trace : parse $ optional (tk "?")) (cfg : tidy.cfg := {}) :
tactic unit | let md := if bang.is_some then semireducible else reducible,
continuity_core := tactic.tidy { tactics := continuity_tactics md, ..cfg },
trace_fn := if trace.is_some then show_term else id in
trace_fn continuity_core | def | tactic.interactive.continuity | topology | src/topology/tactic.lean | [
"tactic.auto_cases",
"tactic.tidy",
"tactic.with_local_reducibility",
"tactic.show_term",
"topology.basic"
] | [
"continuity",
"tactic.tidy"
] | Solve goals of the form `continuous f`. `continuity?` reports back the proof term it found. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuity' : tactic unit | continuity none none {} | def | tactic.interactive.continuity' | topology | src/topology/tactic.lean | [
"tactic.auto_cases",
"tactic.tidy",
"tactic.with_local_reducibility",
"tactic.show_term",
"topology.basic"
] | [
"continuity"
] | Version of `continuity` for use with auto_param. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : closed_embedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.comp_continuous e) f ≤ (2 / 3) * ‖f‖ | begin
have h3 : (0 : ℝ) < 3 := by norm_num1,
have h23 : 0 < (2 / 3 : ℝ) := by norm_num1,
-- In the trivial case `f = 0`, we take `g = 0`
rcases eq_or_ne f 0 with (rfl|hf), { use 0, simp },
replace hf : 0 < ‖f‖ := norm_pos_iff.2 hf,
/- Otherwise, the closed sets `e '' (f ⁻¹' (Iic (-‖f‖ / 3)))` and `e '' (f ⁻... | lemma | bounded_continuous_function.tietze_extension_step | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"abs_le",
"abs_of_nonneg",
"abs_sub_comm",
"closed_embedding",
"disjoint",
"disjoint.preimage",
"div_lt_div_right",
"div_nonneg",
"eq_or_ne",
"exists_bounded_mem_Icc_of_closed_of_le",
"is_closed",
"neg_div",
"real.norm_eq_abs"
] | One step in the proof of the Tietze extension theorem. If `e : C(X, Y)` is a closed embedding
of a topological space into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous
function, then there exists a bounded continuous function `g : Y →ᵇ ℝ` of the norm `‖g‖ ≤ ‖f‖ / 3`
such that the distance between ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(X, Y))
(he : closed_embedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.comp_continuous e = f | begin
/- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
between the previous approximation and `f`. -/
choose F hF_norm hF_dist using λ f : X →ᵇ ℝ, tietze_extension_step f e he,
set g : ℕ → Y →ᵇ ℝ := λ n, (λ g, g + F (f - g.comp_continuous e))^[n] 0,
have g0 : g 0 =... | lemma | bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding' | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"cauchy_seq",
"cauchy_seq_of_le_geometric",
"closed_embedding",
"dist_le_of_le_geometric_of_tendsto₀",
"dist_nonneg",
"div_eq_inv_mul",
"function.iterate_succ_apply'",
"lim",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"one_div",
"pow_succ",
"squeeze_zero",
"tendsto_const_nhds",
"tendsto_n... | **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
embedding and bundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
a bounded continuous functio... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_norm_eq_of_closed_embedding (f : X →ᵇ ℝ) {e : X → Y}
(he : closed_embedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g ∘ e = f | begin
rcases exists_extension_norm_eq_of_closed_embedding' f ⟨e, he.continuous⟩ he with ⟨g, hg, rfl⟩,
exact ⟨g, hg, rfl⟩
end | lemma | bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding"
] | **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
embedding and unbundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
a bounded continuous funct... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_norm_eq_restrict_eq_of_closed {s : set Y} (f : s →ᵇ ℝ) (hs : is_closed s) :
∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.restrict s = f | exists_extension_norm_eq_of_closed_embedding' f ((continuous_map.id _).restrict s)
(closed_embedding_subtype_coe hs) | lemma | bounded_continuous_function.exists_norm_eq_restrict_eq_of_closed | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding_subtype_coe",
"continuous_map.id",
"is_closed"
] | **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
set. If `f` is a bounded continuous real-valued function defined on a closed set in a normal
topological space, then it can be extended to a bounded continuous function of the same norm defined
on the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_forall_mem_Icc_of_closed_embedding (f : X →ᵇ ℝ) {a b : ℝ} {e : X → Y}
(hf : ∀ x, f x ∈ Icc a b) (hle : a ≤ b) (he : closed_embedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, g y ∈ Icc a b) ∧ g ∘ e = f | begin
rcases exists_extension_norm_eq_of_closed_embedding (f - const X ((a + b) / 2)) he
with ⟨g, hgf, hge⟩,
refine ⟨const Y ((a + b) / 2) + g, λ y, _, _⟩,
{ suffices : ‖f - const X ((a + b) / 2)‖ ≤ (b - a) / 2,
by simpa [real.Icc_eq_closed_ball, add_mem_closed_ball_iff_norm]
using (norm_coe_le_... | lemma | bounded_continuous_function.exists_extension_forall_mem_Icc_of_closed_embedding | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding",
"div_nonneg",
"real.Icc_eq_closed_ball",
"zero_le_two"
] | **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
embedding and a bounded continuous function that takes values in a non-trivial closed interval.
See also `exists_extension_forall_mem_of_closed_embedding` for a more general statement that works
for any interval (finite or infi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_forall_exists_le_ge_of_closed_embedding [nonempty X] (f : X →ᵇ ℝ) {e : X → Y}
(he : closed_embedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f | begin
inhabit X,
-- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
obtain ⟨a, ha⟩ : ∃ a, is_glb (range f) a,
from ⟨_, is_glb_cinfi (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).1⟩,
obtain ⟨b, hb⟩ : ∃ b, is_lub (range f) b,
from ⟨_, is_lub_csupr (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).... | lemma | bounded_continuous_function.exists_extension_forall_exists_le_ge_of_closed_embedding | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding",
"disjoint",
"disjoint.preimage",
"em",
"exists_bounded_mem_Icc_of_closed_of_le",
"function.funext_iff",
"is_glb",
"is_glb_cinfi",
"is_lub",
"is_lub_csupr",
"ring",
"set.disjoint_left"
] | **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
topological space `Y`. Let `f` be a bounded continuous real-valued function on `X`. Then there
exists a bounded continuous function `g :... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_forall_mem_of_closed_embedding (f : X →ᵇ ℝ) {t : set ℝ} {e : X → Y}
[hs : ord_connected t] (hf : ∀ x, f x ∈ t) (hne : t.nonempty) (he : closed_embedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, g y ∈ t) ∧ g ∘ e = f | begin
casesI is_empty_or_nonempty X,
{ rcases hne with ⟨c, hc⟩,
refine ⟨const Y c, λ y, hc, funext $ λ x, is_empty_elim x⟩ },
rcases exists_extension_forall_exists_le_ge_of_closed_embedding f he with ⟨g, hg, hgf⟩,
refine ⟨g, λ y, _, hgf⟩,
rcases hg y with ⟨xl, xu, h⟩,
exact hs.out (hf _) (hf _) h
end | lemma | bounded_continuous_function.exists_extension_forall_mem_of_closed_embedding | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding",
"is_empty_elim",
"is_empty_or_nonempty"
] | **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
topological space `Y`. Let `f` be a bounded continuous real-valued function on `X`. Let `t` be
a nonempty convex set of real numbers (we... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_mem_restrict_eq_of_closed {s : set Y} (f : s →ᵇ ℝ) (hs : is_closed s)
{t : set ℝ} [ord_connected t] (hf : ∀ x, f x ∈ t) (hne : t.nonempty) :
∃ g : Y →ᵇ ℝ, (∀ y, g y ∈ t) ∧ g.restrict s = f | begin
rcases exists_extension_forall_mem_of_closed_embedding f hf hne (closed_embedding_subtype_coe hs)
with ⟨g, hg, hgf⟩,
exact ⟨g, hg, fun_like.coe_injective hgf⟩
end | lemma | bounded_continuous_function.exists_forall_mem_restrict_eq_of_closed | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding_subtype_coe",
"fun_like.coe_injective",
"is_closed"
] | **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
set. Let `s` be a closed set in a normal topological space `Y`. Let `f` be a bounded continuous
real-valued function on `s`. Let `t` be a nonempty convex set of real numbers (we use
`ord_connected` instead of `convex` to automa... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_forall_mem_of_closed_embedding (f : C(X, ℝ)) {t : set ℝ} {e : X → Y}
[hs : ord_connected t] (hf : ∀ x, f x ∈ t) (hne : t.nonempty) (he : closed_embedding e) :
∃ g : C(Y, ℝ), (∀ y, g y ∈ t) ∧ g ∘ e = f | begin
have h : ℝ ≃o Ioo (-1 : ℝ) 1 := order_iso_Ioo_neg_one_one ℝ,
set F : X →ᵇ ℝ :=
{ to_fun := coe ∘ (h ∘ f),
continuous_to_fun := continuous_subtype_coe.comp (h.continuous.comp f.continuous),
map_bounded' := bounded_range_iff.1 ((bounded_Ioo (-1 : ℝ) 1).mono $
forall_range_iff.2 $ λ x, (h (f x)).... | lemma | continuous_map.exists_extension_forall_mem_of_closed_embedding | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding",
"lift",
"order_iso_Ioo_neg_one_one",
"subtype.ext_iff"
] | **Tietze extension theorem** for real-valued continuous maps, a version for a closed
embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
topological space `Y`. Let `f` be a continuous real-valued function on `X`. Let `t` be a nonempty
convex set of real numbers (we use `ord_connec... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_of_closed_embedding (f : C(X, ℝ)) (e : X → Y) (he : closed_embedding e) :
∃ g : C(Y, ℝ), g ∘ e = f | (exists_extension_forall_mem_of_closed_embedding f (λ x, mem_univ _) univ_nonempty he).imp $
λ g, and.right | lemma | continuous_map.exists_extension_of_closed_embedding | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding"
] | **Tietze extension theorem** for real-valued continuous maps, a version for a closed
embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
topological space `Y`. Let `f` be a continuous real-valued function on `X`. Then there exists a
continuous real-valued function `g : C(Y, ℝ)` su... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_restrict_eq_forall_mem_of_closed {s : set Y} (f : C(s, ℝ)) {t : set ℝ}
[ord_connected t] (ht : ∀ x, f x ∈ t) (hne : t.nonempty) (hs : is_closed s) :
∃ g : C(Y, ℝ), (∀ y, g y ∈ t) ∧ g.restrict s = f | let ⟨g, hgt, hgf⟩ := exists_extension_forall_mem_of_closed_embedding f ht hne
(closed_embedding_subtype_coe hs)
in ⟨g, hgt, coe_injective hgf⟩ | lemma | continuous_map.exists_restrict_eq_forall_mem_of_closed | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"closed_embedding_subtype_coe",
"is_closed"
] | **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
`s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
on `s`. Let `t` be a nonempty convex set of real numbers (we use `ord_connected` instead of `convex`
to automatically deduce t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_restrict_eq_of_closed {s : set Y} (f : C(s, ℝ)) (hs : is_closed s) :
∃ g : C(Y, ℝ), g.restrict s = f | let ⟨g, hg, hgf⟩ := exists_restrict_eq_forall_mem_of_closed f (λ _, mem_univ _) univ_nonempty hs
in ⟨g, hgf⟩ | lemma | continuous_map.exists_restrict_eq_of_closed | topology | src/topology/tietze_extension.lean | [
"analysis.specific_limits.basic",
"data.set.intervals.iso_Ioo",
"topology.algebra.order.monotone_continuity",
"topology.urysohns_bounded"
] | [
"is_closed"
] | **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
`s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
on `s`. Then there exists a continuous real-valued function `g : C(Y, ℝ)` such that
`g.restrict s = f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_interval : set ℝ | set.Icc 0 1 | abbreviation | unit_interval | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"set.Icc"
] | The unit interval `[0,1]` in ℝ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mem : (0 : ℝ) ∈ I | ⟨le_rfl, zero_le_one⟩ | lemma | unit_interval.zero_mem | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem : (1 : ℝ) ∈ I | ⟨zero_le_one, le_rfl⟩ | lemma | unit_interval.one_mem | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem {x y : ℝ} (hx : x ∈ I) (hy : y ∈ I) : x * y ∈ I | ⟨mul_nonneg hx.1 hy.1, (mul_le_mul hx.2 hy.2 hy.1 zero_le_one).trans_eq $ one_mul 1⟩ | lemma | unit_interval.mul_mem | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"mul_le_mul",
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mem {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y) : x / y ∈ I | ⟨div_nonneg hx hy, div_le_one_of_le hxy hy⟩ | lemma | unit_interval.div_mem | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"div_le_one_of_le",
"div_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_mem (x : ℝ) : fract x ∈ I | ⟨fract_nonneg _, (fract_lt_one _).le⟩ | lemma | unit_interval.fract_mem | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_one_sub_mem {t : ℝ} : t ∈ I ↔ 1 - t ∈ I | begin
rw [mem_Icc, mem_Icc],
split ; intro ; split ; linarith
end | lemma | unit_interval.mem_iff_one_sub_mem | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero : has_zero I | ⟨⟨0, zero_mem⟩⟩ | instance | unit_interval.has_zero | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_one : has_one I | ⟨⟨1, by split ; norm_num⟩⟩ | instance | unit_interval.has_one | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_zero {x : I} : (x : ℝ) ≠ 0 ↔ x ≠ 0 | not_iff_not.mpr coe_eq_zero | lemma | unit_interval.coe_ne_zero | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_one {x : I} : (x : ℝ) ≠ 1 ↔ x ≠ 1 | not_iff_not.mpr coe_eq_one | lemma | unit_interval.coe_ne_one | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_left {x y : I} : x * y ≤ x | subtype.coe_le_coe.mp $ (mul_le_mul_of_nonneg_left y.2.2 x.2.1).trans_eq $ mul_one x | lemma | unit_interval.mul_le_left | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"mul_le_mul_of_nonneg_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_right {x y : I} : x * y ≤ y | subtype.coe_le_coe.mp $ (mul_le_mul_of_nonneg_right x.2.2 y.2.1).trans_eq $ one_mul y | lemma | unit_interval.mul_le_right | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"mul_le_mul_of_nonneg_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm : I → I | λ t, ⟨1 - t, mem_iff_one_sub_mem.mp t.prop⟩ | def | unit_interval.symm | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | Unit interval central symmetry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_zero : σ 0 = 1 | subtype.ext $ by simp [symm] | lemma | unit_interval.symm_zero | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_one : σ 1 = 0 | subtype.ext $ by simp [symm] | lemma | unit_interval.symm_one | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (x : I) : σ (σ x) = x | subtype.ext $ by simp [symm] | lemma | unit_interval.symm_symm | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_eq (x : I) : (σ x : ℝ) = 1 - x | rfl | lemma | unit_interval.coe_symm_eq | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_symm : continuous σ | by continuity! | lemma | unit_interval.continuous_symm | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg (x : I) : 0 ≤ (x : ℝ) | x.2.1 | lemma | unit_interval.nonneg | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_minus_nonneg (x : I) : 0 ≤ 1 - (x : ℝ) | by simpa using x.2.2 | lemma | unit_interval.one_minus_nonneg | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one (x : I) : (x : ℝ) ≤ 1 | x.2.2 | lemma | unit_interval.le_one | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_minus_le_one (x : I) : 1 - (x : ℝ) ≤ 1 | by simpa using x.2.1 | lemma | unit_interval.one_minus_le_one | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pos {t : I} {x : ℝ} (hx : 0 < x) : 0 < (x + t : ℝ) | add_pos_of_pos_of_nonneg hx $ nonneg _ | lemma | unit_interval.add_pos | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg' {t : I} : 0 ≤ t | t.2.1 | lemma | unit_interval.nonneg' | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | like `unit_interval.nonneg`, but with the inequality in `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_one' {t : I} : t ≤ 1 | t.2.2 | lemma | unit_interval.le_one' | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [] | like `unit_interval.le_one`, but with the inequality in `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ set.Icc (0 : ℝ) (1/a) | begin
split; rintros ⟨h₁, h₂⟩; split,
{ exact nonneg_of_mul_nonneg_right h₁ ha },
{ rwa [le_div_iff ha, mul_comm] },
{ exact mul_nonneg ha.le h₁ },
{ rwa [le_div_iff ha, mul_comm] at h₂ }
end | lemma | unit_interval.mul_pos_mem_iff | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"le_div_iff",
"mul_comm",
"nonneg_of_mul_nonneg_right",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_mul_sub_one_mem_iff {t : ℝ} : 2 * t - 1 ∈ I ↔ t ∈ set.Icc (1/2 : ℝ) 1 | by split; rintros ⟨h₁, h₂⟩; split; linarith | lemma | unit_interval.two_mul_sub_one_mem_iff | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_Icc_eq_zero {x : ℝ} : proj_Icc (0 : ℝ) 1 zero_le_one x = 0 ↔ x ≤ 0 | proj_Icc_eq_left zero_lt_one | lemma | proj_Icc_eq_zero | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_Icc_eq_one {x : ℝ} : proj_Icc (0 : ℝ) 1 zero_le_one x = 1 ↔ 1 ≤ x | proj_Icc_eq_right zero_lt_one | lemma | proj_Icc_eq_one | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_interval : tactic unit | `[apply unit_interval.nonneg] <|> `[apply unit_interval.one_minus_nonneg] <|>
`[apply unit_interval.le_one] <|> `[apply unit_interval.one_minus_le_one] | def | tactic.interactive.unit_interval | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"unit_interval",
"unit_interval.le_one",
"unit_interval.nonneg",
"unit_interval.one_minus_le_one",
"unit_interval.one_minus_nonneg"
] | A tactic that solves `0 ≤ ↑x`, `0 ≤ 1 - ↑x`, `↑x ≤ 1`, and `1 - ↑x ≤ 1` for `x : I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_homeomorph_image_I (a b : 𝕜) (h : 0 < a) :
affine_homeomorph a b h.ne.symm '' set.Icc 0 1 = set.Icc b (a + b) | by simp [h] | lemma | affine_homeomorph_image_I | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"affine_homeomorph",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_homeo_I (a b : 𝕜) (h : a < b) : set.Icc a b ≃ₜ set.Icc (0 : 𝕜) (1 : 𝕜) | begin
let e := homeomorph.image (affine_homeomorph (b-a) a (sub_pos.mpr h).ne.symm) (set.Icc 0 1),
refine (e.trans _).symm,
apply homeomorph.set_congr,
simp [sub_pos.mpr h],
end | def | Icc_homeo_I | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"affine_homeomorph",
"homeomorph.image",
"homeomorph.set_congr",
"set.Icc"
] | The affine homeomorphism from a nontrivial interval `[a,b]` to `[0,1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Icc_homeo_I_apply_coe (a b : 𝕜) (h : a < b) (x : set.Icc a b) :
((Icc_homeo_I a b h) x : 𝕜) = (x - a) / (b - a) | rfl | lemma | Icc_homeo_I_apply_coe | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"Icc_homeo_I",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_homeo_I_symm_apply_coe (a b : 𝕜) (h : a < b) (x : set.Icc (0 : 𝕜) (1 : 𝕜)) :
((Icc_homeo_I a b h).symm x : 𝕜) = (b - a) * x + a | rfl | lemma | Icc_homeo_I_symm_apply_coe | topology | src/topology/unit_interval.lean | [
"topology.instances.real",
"topology.algebra.field",
"data.set.intervals.proj_Icc",
"data.set.intervals.instances"
] | [
"Icc_homeo_I",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_bounded_zero_one_of_closed {X : Type*} [topological_space X] [normal_space X]
{s t : set X} (hs : is_closed s) (ht : is_closed t)
(hd : disjoint s t) :
∃ f : X →ᵇ ℝ, eq_on f 0 s ∧ eq_on f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 | let ⟨f, hfs, hft, hf⟩ := exists_continuous_zero_one_of_closed hs ht hd
in ⟨⟨f, 1, λ x y, real.dist_le_of_mem_Icc_01 (hf _) (hf _)⟩, hfs, hft, hf⟩ | lemma | exists_bounded_zero_one_of_closed | topology | src/topology/urysohns_bounded.lean | [
"topology.urysohns_lemma",
"topology.continuous_function.bounded"
] | [
"disjoint",
"exists_continuous_zero_one_of_closed",
"is_closed",
"normal_space",
"real.dist_le_of_mem_Icc_01",
"topological_space"
] | Urysohns lemma: if `s` and `t` are two disjoint closed sets in a normal topological space `X`,
then there exists a continuous function `f : X → ℝ` such that
* `f` equals zero on `s`;
* `f` equals one on `t`;
* `0 ≤ f x ≤ 1` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_bounded_mem_Icc_of_closed_of_le {X : Type*} [topological_space X] [normal_space X]
{s t : set X} (hs : is_closed s) (ht : is_closed t) (hd : disjoint s t)
{a b : ℝ} (hle : a ≤ b) :
∃ f : X →ᵇ ℝ, eq_on f (const X a) s ∧ eq_on f (const X b) t ∧ ∀ x, f x ∈ Icc a b | let ⟨f, hfs, hft, hf01⟩ := exists_bounded_zero_one_of_closed hs ht hd
in ⟨bounded_continuous_function.const X a + (b - a) • f,
λ x hx, by simp [hfs hx], λ x hx, by simp [hft hx],
λ x, ⟨by dsimp; nlinarith [(hf01 x).1], by dsimp; nlinarith [(hf01 x).2]⟩⟩ | lemma | exists_bounded_mem_Icc_of_closed_of_le | topology | src/topology/urysohns_bounded.lean | [
"topology.urysohns_lemma",
"topology.continuous_function.bounded"
] | [
"disjoint",
"exists_bounded_zero_one_of_closed",
"is_closed",
"normal_space",
"topological_space"
] | Urysohns lemma: if `s` and `t` are two disjoint closed sets in a normal topological space `X`,
and `a ≤ b` are two real numbers, then there exists a continuous function `f : X → ℝ` such that
* `f` equals `a` on `s`;
* `f` equals `b` on `t`;
* `a ≤ f x ≤ b` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CU (X : Type*) [topological_space X] | (C U : set X)
(closed_C : is_closed C)
(open_U : is_open U)
(subset : C ⊆ U) | structure | urysohns.CU | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"is_closed",
"is_open",
"topological_space"
] | An auxiliary type for the proof of Urysohn's lemma: a pair of a closed set `C` and its
open neighborhood `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left (c : CU X) : CU X | { C := c.C,
U := (normal_exists_closure_subset c.closed_C c.open_U c.subset).some,
closed_C := c.closed_C,
open_U := (normal_exists_closure_subset c.closed_C c.open_U c.subset).some_spec.1,
subset := (normal_exists_closure_subset c.closed_C c.open_U c.subset).some_spec.2.1 } | def | urysohns.CU.left | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"normal_exists_closure_subset"
] | Due to `normal_exists_closure_subset`, for each `c : CU X` there exists an open set `u`
such chat `c.C ⊆ u` and `closure u ⊆ c.U`. `c.left` is the pair `(c.C, u)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right (c : CU X) : CU X | { C := closure (normal_exists_closure_subset c.closed_C c.open_U c.subset).some,
U := c.U,
closed_C := is_closed_closure,
open_U := c.open_U,
subset := (normal_exists_closure_subset c.closed_C c.open_U c.subset).some_spec.2.2 } | def | urysohns.CU.right | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"closure",
"is_closed_closure",
"normal_exists_closure_subset"
] | Due to `normal_exists_closure_subset`, for each `c : CU X` there exists an open set `u`
such chat `c.C ⊆ u` and `closure u ⊆ c.U`. `c.right` is the pair `(closure u, c.U)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_U_subset_right_C (c : CU X) : c.left.U ⊆ c.right.C | subset_closure | lemma | urysohns.CU.left_U_subset_right_C | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_U_subset (c : CU X) : c.left.U ⊆ c.U | subset.trans c.left_U_subset_right_C c.right.subset | lemma | urysohns.CU.left_U_subset | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_right_C (c : CU X) : c.C ⊆ c.right.C | subset.trans c.left.subset c.left_U_subset_right_C | lemma | urysohns.CU.subset_right_C | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx : ℕ → CU X → X → ℝ | | 0 c x := indicator c.Uᶜ 1 x
| (n + 1) c x := midpoint ℝ (approx n c.left x) (approx n c.right x) | def | urysohns.CU.approx | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"midpoint"
] | `n`-th approximation to a continuous function `f : X → ℝ` such that `f = 0` on `c.C` and `f = 1`
outside of `c.U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
approx_of_mem_C (c : CU X) (n : ℕ) {x : X} (hx : x ∈ c.C) :
c.approx n x = 0 | begin
induction n with n ihn generalizing c,
{ exact indicator_of_not_mem (λ hU, hU $ c.subset hx) _ },
{ simp only [approx],
rw [ihn, ihn, midpoint_self],
exacts [c.subset_right_C hx, hx] }
end | lemma | urysohns.CU.approx_of_mem_C | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"midpoint_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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