statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
prod_to_prod_Top_I {a₁ a₂ : Top.of (ulift I)} {b₁ b₂ : X}
(p₁ : from_top a₁ ⟶ from_top a₂) (p₂ : from_top b₁ ⟶ from_top b₂) | @category_theory.functor.map _ _ _ _ (prod_to_prod_Top (Top.of $ ulift I) X)
(a₁, b₁) (a₂, b₂) (p₁, p₂) | abbreviation | continuous_map.homotopy.prod_to_prod_Top_I | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"Top.of",
"from_top"
] | An abbreviation for `prod_to_prod_Top`, with some types already in place to help the
typechecker. In particular, the first path should be on the ulifted unit interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_path : from_top (H (0, x₀)) ⟶ from_top (H (1, x₁)) | (πₘ H.ulift_map).map (prod_to_prod_Top_I uhpath01 p) | def | continuous_map.homotopy.diagonal_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"from_top"
] | The diagonal path `d` of a homotopy `H` on a path `p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_path' : from_top (f x₀) ⟶ from_top (g x₁) | hcast (H.apply_zero x₀).symm ≫ (H.diagonal_path p) ≫ hcast (H.apply_one x₁) | def | continuous_map.homotopy.diagonal_path' | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"from_top"
] | The diagonal path, but starting from `f x₀` and going to `g x₁` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_zero_path : (πₘ f).map p = hcast (H.apply_zero x₀).symm ≫
(πₘ H.ulift_map).map (prod_to_prod_Top_I (𝟙 (ulift.up 0)) p) ≫
hcast (H.apply_zero x₁) | begin
apply quotient.induction_on p,
intro p',
apply @eq_path_of_eq_image _ _ _ _ H.ulift_map _ _ _ _ _ ((path.refl (ulift.up _)).prod p'),
simp,
end | lemma | continuous_map.homotopy.apply_zero_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"path.refl"
] | Proof that `f(p) = H(0 ⟶ 0, p)`, with the appropriate casts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_one_path : (πₘ g).map p = hcast (H.apply_one x₀).symm ≫
((πₘ H.ulift_map).map (prod_to_prod_Top_I (𝟙 (ulift.up 1)) p)) ≫
hcast (H.apply_one x₁) | begin
apply quotient.induction_on p,
intro p',
apply @eq_path_of_eq_image _ _ _ _ H.ulift_map _ _ _ _ _ ((path.refl (ulift.up _)).prod p'),
simp,
end | lemma | continuous_map.homotopy.apply_one_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"path.refl"
] | Proof that `g(p) = H(1 ⟶ 1, p)`, with the appropriate casts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_at_eq (x : X) : ⟦H.eval_at x⟧ =
hcast (H.apply_zero x).symm ≫
(πₘ H.ulift_map).map (prod_to_prod_Top_I uhpath01 (𝟙 x)) ≫
hcast (H.apply_one x).symm.symm | begin
dunfold prod_to_prod_Top_I uhpath01 hcast,
refine (@functor.conj_eq_to_hom_iff_heq (πₓ Y) _ _ _ _ _ _ _ _ _).mpr _,
simp only [id_eq_path_refl, prod_to_prod_Top_map, path.homotopic.prod_lift, map_eq,
← path.homotopic.map_lift],
apply path.homotopic.hpath_hext, intro, refl,
end | lemma | continuous_map.homotopy.eval_at_eq | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"id_eq_path_refl",
"map_eq",
"path.homotopic.hpath_hext",
"path.homotopic.map_lift",
"path.homotopic.prod_lift"
] | Proof that `H.eval_at x = H(0 ⟶ 1, x ⟶ x)`, with the appropriate casts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_diag_path :
(πₘ f).map p ≫ ⟦H.eval_at x₁⟧ = H.diagonal_path' p ∧
(⟦H.eval_at x₀⟧ ≫ (πₘ g).map p : from_top (f x₀) ⟶ from_top (g x₁)) = H.diagonal_path' p | begin
rw [H.apply_zero_path, H.apply_one_path, H.eval_at_eq, H.eval_at_eq],
dunfold prod_to_prod_Top_I,
split; { slice_lhs 2 5 { simp [← category_theory.functor.map_comp], }, refl, },
end | lemma | continuous_map.homotopy.eq_diag_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"from_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopic_maps_nat_iso : πₘ f ⟶ πₘ g | { app := λ x, ⟦H.eval_at x⟧,
naturality' := λ x y p, by rw [(H.eq_diag_path p).1, (H.eq_diag_path p).2] } | def | fundamental_groupoid_functor.homotopic_maps_nat_iso | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [] | Given a homotopy H : f ∼ g, we have an associated natural isomorphism between the induced
functors `f` and `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_homotopy_equiv (hequiv : X ≃ₕ Y) : πₓ X ≌ πₓ Y | begin
apply equivalence.mk
(πₘ hequiv.to_fun : πₓ X ⥤ πₓ Y)
(πₘ hequiv.inv_fun : πₓ Y ⥤ πₓ X);
simp only [Groupoid.hom_to_functor, Groupoid.id_to_functor],
{ convert (as_iso (homotopic_maps_nat_iso hequiv.left_inv.some)).symm,
exacts [((π).map_id X).symm, ((π).map_comp _ _).symm] },
{ convert as_iso... | def | fundamental_groupoid_functor.equiv_of_homotopy_equiv | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"map_comp",
"map_id"
] | Homotopy equivalent topological spaces have equivalent fundamental groupoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj (i : I) : πₓ (Top.of (Π i, X i)) ⥤ πₓ (X i) | πₘ ⟨_, continuous_apply i⟩ | def | fundamental_groupoid_functor.proj | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"continuous_apply"
] | The projection map Π i, X i → X i induces a map π(Π i, X i) ⟶ π(X i). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_map (i : I) (x₀ x₁ : πₓ (Top.of (Π i, X i))) (p : x₀ ⟶ x₁) :
(proj X i).map p = (@path.homotopic.proj _ _ _ _ _ i p) | rfl | lemma | fundamental_groupoid_functor.proj_map | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"path.homotopic.proj"
] | The projection map is precisely path.homotopic.proj interpreted as a functor | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_to_pi_Top : (Π i, πₓ (X i)) ⥤ πₓ (Top.of (Π i, X i)) | { obj := λ g, g,
map := λ v₁ v₂ p, path.homotopic.pi p,
map_id' :=
begin
intro x,
change path.homotopic.pi (λ i, 𝟙 (x i)) = _,
simp only [fundamental_groupoid.id_eq_path_refl, path.homotopic.pi_lift],
refl,
end,
map_comp' := λ x y z f g, (path.homotopic.comp_pi_eq_pi_comp f g).symm, } | def | fundamental_groupoid_functor.pi_to_pi_Top | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"path.homotopic.comp_pi_eq_pi_comp",
"path.homotopic.pi",
"path.homotopic.pi_lift"
] | The map taking the pi product of a family of fundamental groupoids to the fundamental
groupoid of the pi product. This is actually an isomorphism (see `pi_iso`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso : category_theory.Groupoid.of (Π i : I, πₓ (X i)) ≅ πₓ (Top.of (Π i, X i)) | { hom := pi_to_pi_Top X,
inv := category_theory.functor.pi' (proj X),
hom_inv_id' :=
begin
change pi_to_pi_Top X ⋙ (category_theory.functor.pi' (proj X)) = 𝟭 _,
apply category_theory.functor.ext; intros,
{ ext, simp, }, { refl, },
end,
inv_hom_id' :=
begin
change (category_theory.functor.pi... | def | fundamental_groupoid_functor.pi_iso | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"category_theory.Groupoid.of",
"category_theory.functor.ext",
"category_theory.functor.pi'",
"path.homotopic.pi"
] | Shows `pi_to_pi_Top` is an isomorphism, whose inverse is precisely the pi product
of the induced projections. This shows that `fundamental_groupoid_functor` preserves products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_discrete_comp : limits.cone (discrete.functor X ⋙ π) ≌
limits.cone (discrete.functor (λ i, πₓ (X i))) | limits.cones.postcompose_equivalence (discrete.comp_nat_iso_discrete X π) | def | fundamental_groupoid_functor.cone_discrete_comp | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [] | Equivalence between the categories of cones over the objects `π Xᵢ` written in two ways | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_discrete_comp_obj_map_cone :
(cone_discrete_comp X).functor.obj ((π).map_cone (Top.pi_fan.{u} X))
= limits.fan.mk (πₓ (Top.of (Π i, X i))) (proj X) | rfl | lemma | fundamental_groupoid_functor.cone_discrete_comp_obj_map_cone | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Top_to_pi_cone : (limits.fan.mk (πₓ (Top.of (Π i, X i))) (proj X)) ⟶
Groupoid.pi_limit_fan (λ i : I, (πₓ (X i))) | { hom := category_theory.functor.pi' (proj X) } | def | fundamental_groupoid_functor.pi_Top_to_pi_cone | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"category_theory.functor.pi'"
] | This is `pi_iso.inv` as a cone morphism (in fact, isomorphism) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_product : limits.preserves_limit (discrete.functor X) π | begin
apply limits.preserves_limit_of_preserves_limit_cone (Top.pi_fan_is_limit.{u} X),
apply (limits.is_limit.of_cone_equiv (cone_discrete_comp X)).to_fun,
simp only [cone_discrete_comp_obj_map_cone],
apply limits.is_limit.of_iso_limit _ (as_iso (pi_Top_to_pi_cone X)).symm,
exact Groupoid.pi_limit_fan_is_lim... | def | fundamental_groupoid_functor.preserves_product | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [] | The fundamental groupoid functor preserves products | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_left : πₓ (Top.of (A × B)) ⥤ πₓ A | πₘ ⟨_, continuous_fst⟩ | def | fundamental_groupoid_functor.proj_left | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of"
] | The induced map of the left projection map X × Y → X | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_right : πₓ (Top.of (A × B)) ⥤ πₓ B | πₘ ⟨_, continuous_snd⟩ | def | fundamental_groupoid_functor.proj_right | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of"
] | The induced map of the right projection map X × Y → Y | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_left_map (x₀ x₁ : πₓ (Top.of (A × B))) (p : x₀ ⟶ x₁) :
(proj_left A B).map p = path.homotopic.proj_left p | rfl | lemma | fundamental_groupoid_functor.proj_left_map | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"path.homotopic.proj_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_right_map (x₀ x₁ : πₓ (Top.of (A × B))) (p : x₀ ⟶ x₁) :
(proj_right A B).map p = path.homotopic.proj_right p | rfl | lemma | fundamental_groupoid_functor.proj_right_map | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"path.homotopic.proj_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_to_prod_Top : (πₓ A) × (πₓ B) ⥤ πₓ (Top.of (A × B)) | { obj := λ g, g,
map := λ x y p, match x, y, p with
| (x₀, x₁), (y₀, y₁), (p₀, p₁) := path.homotopic.prod p₀ p₁
end,
map_id' :=
begin
rintro ⟨x₀, x₁⟩,
simp only [category_theory.prod_id, fundamental_groupoid.id_eq_path_refl],
unfold_aux, rw path.homotopic.prod_lift, refl,
end,
map_comp' := λ... | def | fundamental_groupoid_functor.prod_to_prod_Top | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"category_theory.prod_id",
"path.homotopic.comp_prod_eq_prod_comp",
"path.homotopic.prod",
"path.homotopic.prod_lift"
] | The map taking the product of two fundamental groupoids to the fundamental groupoid of the product
of the two topological spaces. This is in fact an isomorphism (see `prod_iso`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_to_prod_Top_map {x₀ x₁ : πₓ A} {y₀ y₁ : πₓ B}
(p₀ : x₀ ⟶ x₁) (p₁ : y₀ ⟶ y₁) :
@category_theory.functor.map _ _ _ _
(prod_to_prod_Top A B) (x₀, y₀) (x₁, y₁) (p₀, p₁) = path.homotopic.prod p₀ p₁ | rfl | lemma | fundamental_groupoid_functor.prod_to_prod_Top_map | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"path.homotopic.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_iso : category_theory.Groupoid.of ((πₓ A) × (πₓ B)) ≅ (πₓ (Top.of (A × B))) | { hom := prod_to_prod_Top A B,
inv := (proj_left A B).prod' (proj_right A B),
hom_inv_id' :=
begin
change prod_to_prod_Top A B ⋙ ((proj_left A B).prod' (proj_right A B)) = 𝟭 _,
apply category_theory.functor.hext, { intros, ext; simp; refl, },
rintros ⟨x₀, x₁⟩ ⟨y₀, y₁⟩ ⟨f₀, f₁⟩,
have := and.intro ... | def | fundamental_groupoid_functor.prod_iso | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/product.lean | [
"category_theory.groupoid",
"algebraic_topology.fundamental_groupoid.basic",
"topology.category.Top.limits.products",
"topology.homotopy.product"
] | [
"Top.of",
"category_theory.Groupoid.of",
"category_theory.functor.hext",
"path.homotopic.prod_proj_left_proj_right",
"path.homotopic.proj_left_prod",
"path.homotopic.proj_right_prod"
] | Shows `prod_to_prod_Top` is an isomorphism, whose inverse is precisely the product
of the induced left and right projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_equiv_discrete_punit : fundamental_groupoid punit.{u+1} ≌ discrete punit.{v+1} | equivalence.mk (functor.star _) ((category_theory.functor.const _).obj punit.star)
(nat_iso.of_components (λ _, eq_to_iso dec_trivial) (λ _ _ _, dec_trivial))
(functor.punit_ext _ _) | def | fundamental_groupoid.punit_equiv_discrete_punit | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/punit.lean | [
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"category_theory.functor.const",
"fundamental_groupoid"
] | Equivalence of groupoids between fundamental groupoid of punit and punit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simply_connected_space (X : Type*) [topological_space X] : Prop | (equiv_unit [] : nonempty (fundamental_groupoid X ≌ discrete unit)) | class | simply_connected_space | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"fundamental_groupoid",
"topological_space"
] | A simply connected space is one whose fundamental groupoid is equivalent to `discrete unit` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simply_connected_def (X : Type*) [topological_space X] :
simply_connected_space X ↔ nonempty (fundamental_groupoid X ≌ discrete unit) | ⟨λ h, @simply_connected_space.equiv_unit X _ h, λ h, ⟨h⟩⟩ | lemma | simply_connected_def | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"fundamental_groupoid",
"simply_connected_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simply_connected_iff_unique_homotopic (X : Type*) [topological_space X] :
simply_connected_space X ↔ (nonempty X) ∧
∀ (x y : X), nonempty (unique (path.homotopic.quotient x y)) | by { rw [simply_connected_def, equiv_punit_iff_unique], refl, } | lemma | simply_connected_iff_unique_homotopic | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"path.homotopic.quotient",
"simply_connected_def",
"simply_connected_space",
"topological_space",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
paths_homotopic {x y : X} (p₁ p₂ : path x y) : path.homotopic p₁ p₂ | by simpa using @subsingleton.elim (path.homotopic.quotient x y) _ ⟦p₁⟧ ⟦p₂⟧ | lemma | simply_connected_space.paths_homotopic | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"path",
"path.homotopic",
"path.homotopic.quotient"
] | In a simply connected space, any two paths are homotopic | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_contractible (Y : Type*) [topological_space Y] [contractible_space Y] :
simply_connected_space Y | { equiv_unit :=
let H : Top.of Y ≃ₕ Top.of unit := (contractible_space.hequiv_unit Y).some in
⟨(fundamental_groupoid_functor.equiv_of_homotopy_equiv H).trans
fundamental_groupoid.punit_equiv_discrete_punit⟩, } | instance | simply_connected_space.of_contractible | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"Top.of",
"contractible_space",
"fundamental_groupoid_functor.equiv_of_homotopy_equiv",
"simply_connected_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simply_connected_iff_paths_homotopic {Y : Type*} [topological_space Y] :
simply_connected_space Y ↔ (path_connected_space Y) ∧
(∀ x y : Y, subsingleton (path.homotopic.quotient x y)) | ⟨by { introI, split; apply_instance, },
λ h, begin
casesI h, rw simply_connected_iff_unique_homotopic,
exact ⟨infer_instance, λ x y, ⟨unique_of_subsingleton ⟦path_connected_space.some_path x y⟧⟩⟩,
end⟩ | lemma | simply_connected_iff_paths_homotopic | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"path.homotopic.quotient",
"path_connected_space",
"simply_connected_iff_unique_homotopic",
"simply_connected_space",
"topological_space"
] | A space is simply connected iff it is path connected, and there is at most one path
up to homotopy between any two points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simply_connected_iff_paths_homotopic' {Y : Type*} [topological_space Y] :
simply_connected_space Y ↔ (path_connected_space Y) ∧
(∀ {x y : Y} (p₁ p₂ : path x y), path.homotopic p₁ p₂) | begin
convert simply_connected_iff_paths_homotopic,
simp [path.homotopic.quotient, setoid.eq_top_iff], refl,
end | lemma | simply_connected_iff_paths_homotopic' | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/simply_connected.lean | [
"algebraic_topology.fundamental_groupoid.induced_maps",
"topology.homotopy.contractible",
"category_theory.punit",
"algebraic_topology.fundamental_groupoid.punit"
] | [
"path",
"path.homotopic",
"path.homotopic.quotient",
"path_connected_space",
"setoid.eq_top_iff",
"simply_connected_iff_paths_homotopic",
"simply_connected_space",
"topological_space"
] | Another version of `simply_connected_iff_paths_homotopic` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evariation_on (f : α → E) (s : set α) : ℝ≥0∞ | ⨆ (p : ℕ × {u : ℕ → α // monotone u ∧ ∀ i, u i ∈ s}),
∑ i in finset.range p.1, edist (f ((p.2 : ℕ → α) (i+1))) (f ((p.2 : ℕ → α) i)) | def | evariation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"finset.range",
"monotone"
] | The (extended real valued) variation of a function `f` on a set `s` inside a linear order is
the supremum of the sum of `edist (f (u (i+1))) (f (u i))` over all finite increasing
sequences `u` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_bounded_variation_on (f : α → E) (s : set α) | evariation_on f s ≠ ∞ | def | has_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on"
] | A function has bounded variation on a set `s` if its total variation there is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_locally_bounded_variation_on (f : α → E) (s : set α) | ∀ a b, a ∈ s → b ∈ s → has_bounded_variation_on f (s ∩ Icc a b) | def | has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_bounded_variation_on"
] | A function has locally bounded variation on a set `s` if, given any interval `[a, b]` with
endpoints in `s`, then the function has finite variation on `s ∩ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_monotone_mem {s : set α} (hs : s.nonempty) :
nonempty {u // monotone u ∧ ∀ (i : ℕ), u i ∈ s} | begin
obtain ⟨x, hx⟩ := hs,
exact ⟨⟨λ i, x, λ i j hij, le_rfl, λ i, hx⟩⟩,
end | lemma | evariation_on.nonempty_monotone_mem | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"le_rfl",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_edist_zero_on {f f' : α → E} {s : set α} (h : ∀ ⦃x⦄, x ∈ s → edist (f x) (f' x) = 0) :
evariation_on f s = evariation_on f' s | begin
dsimp only [evariation_on],
congr' 1 with p : 1,
congr' 1 with i : 1,
rw [edist_congr_right (h $ p.snd.prop.2 (i+1)), edist_congr_left (h $ p.snd.prop.2 i)],
end | lemma | evariation_on.eq_of_edist_zero_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"edist_congr_left",
"edist_congr_right",
"evariation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_eq_on {f f' : α → E} {s : set α} (h : eq_on f f' s) :
evariation_on f s = evariation_on f' s | eq_of_edist_zero_on (λ x xs, by rw [h xs, edist_self]) | lemma | evariation_on.eq_of_eq_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le
(f : α → E) {s : set α} (n : ℕ) {u : ℕ → α} (hu : monotone u) (us : ∀ i, u i ∈ s) :
∑ i in finset.range n, edist (f (u (i+1))) (f (u i)) ≤ evariation_on f s | le_supr_of_le ⟨n, u, hu, us⟩ le_rfl | lemma | evariation_on.sum_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"finset.range",
"le_rfl",
"le_supr_of_le",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_of_monotone_on_Iic
(f : α → E) {s : set α} {n : ℕ} {u : ℕ → α} (hu : monotone_on u (Iic n))
(us : ∀ i ≤ n, u i ∈ s) :
∑ i in finset.range n, edist (f (u (i+1))) (f (u i)) ≤ evariation_on f s | begin
let v := λ i, if i ≤ n then u i else u n,
have vs : ∀ i, v i ∈ s,
{ assume i,
simp only [v],
split_ifs,
{ exact us i h },
{ exact us n le_rfl } },
have hv : monotone v,
{ apply monotone_nat_of_le_succ (λ i, _),
simp only [v],
rcases lt_trichotomy i n with hi|rfl|hi,
{ have : ... | lemma | evariation_on.sum_le_of_monotone_on_Iic | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"finset.mem_range",
"finset.range",
"le_rfl",
"le_zero_iff",
"monotone",
"monotone_nat_of_le_succ",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_of_monotone_on_Icc
(f : α → E) {s : set α} {m n : ℕ} {u : ℕ → α} (hu : monotone_on u (Icc m n))
(us : ∀ i ∈ Icc m n, u i ∈ s) :
∑ i in finset.Ico m n, edist (f (u (i+1))) (f (u i)) ≤ evariation_on f s | begin
rcases le_or_lt n m with hnm|hmn,
{ simp only [finset.Ico_eq_empty_of_le hnm, finset.sum_empty, zero_le'] },
let v := λ i, u (m + i),
have hv : monotone_on v (Iic (n - m)),
{ assume a ha b hb hab,
simp only [le_tsub_iff_left hmn.le, mem_Iic] at ha hb,
exact hu ⟨le_add_right le_rfl, ha⟩ ⟨le_add_r... | lemma | evariation_on.sum_le_of_monotone_on_Icc | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"finset.Ico",
"finset.Ico_eq_empty_of_le",
"finset.range",
"finset.range_eq_Ico",
"le_rfl",
"le_tsub_iff_left",
"monotone_on",
"tsub_add_cancel_of_le",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (f : α → E) {s t : set α} (hst : t ⊆ s) :
evariation_on f t ≤ evariation_on f s | begin
apply supr_le _,
rintros ⟨n, ⟨u, hu, ut⟩⟩,
exact sum_le f n hu (λ i, hst (ut i)),
end | lemma | evariation_on.mono | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.has_bounded_variation_on.mono {f : α → E} {s : set α}
(h : has_bounded_variation_on f s) {t : set α} (ht : t ⊆ s) :
has_bounded_variation_on f t | (lt_of_le_of_lt (evariation_on.mono f ht) (lt_top_iff_ne_top.2 h)).ne | lemma | has_bounded_variation_on.mono | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on.mono",
"has_bounded_variation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.has_bounded_variation_on.has_locally_bounded_variation_on {f : α → E} {s : set α}
(h : has_bounded_variation_on f s) : has_locally_bounded_variation_on f s | λ x y hx hy, h.mono (inter_subset_left _ _) | lemma | has_bounded_variation_on.has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_bounded_variation_on",
"has_locally_bounded_variation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le (f : α → E) {s : set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
edist (f x) (f y) ≤ evariation_on f s | begin
wlog hxy : x ≤ y,
{ rw edist_comm,
exact this f hy hx (le_of_not_le hxy) },
let u : ℕ → α := λ n, if n = 0 then x else y,
have hu : monotone u,
{ assume m n hmn,
dsimp only [u],
split_ifs,
exacts [le_rfl, hxy, by linarith [pos_iff_ne_zero.2 h], le_rfl] },
have us : ∀ i, u i ∈ s,
{ as... | lemma | evariation_on.edist_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"le_rfl",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff (f : α → E) {s : set α} :
evariation_on f s = 0 ↔ ∀ (x y ∈ s), edist (f x) (f y) = 0 | begin
split,
{ rintro h x xs y ys,
rw [←le_zero_iff, ←h],
exact edist_le f xs ys, },
{ rintro h,
dsimp only [evariation_on],
rw ennreal.supr_eq_zero,
rintro ⟨n, u, um, us⟩,
exact finset.sum_eq_zero (λ i hi, h _ (us i.succ) _ (us i)), },
end | lemma | evariation_on.eq_zero_iff | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.supr_eq_zero",
"eq_zero_iff",
"evariation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_on {f : α → E} {s : set α} (hf : (f '' s).subsingleton) : evariation_on f s = 0 | begin
rw eq_zero_iff,
rintro x xs y ys,
rw [hf ⟨x, xs, rfl⟩ ⟨y, ys, rfl⟩, edist_self],
end | lemma | evariation_on.constant_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"eq_zero_iff",
"evariation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton (f : α → E) {s : set α} (hs : s.subsingleton) :
evariation_on f s = 0 | constant_on (hs.image f) | lemma | evariation_on.subsingleton | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_continuous_aux {ι : Type*} {F : ι → α → E} {p : filter ι}
{f : α → E} {s : set α} (Ffs : ∀ x ∈ s, tendsto (λ i, F i x) p (𝓝 (f x)))
{v : ℝ≥0∞} (hv : v < evariation_on f s) : ∀ᶠ (n : ι) in p, v < evariation_on (F n) s | begin
obtain ⟨⟨n, ⟨u, um, us⟩⟩, hlt⟩ :
∃ (p : ℕ × {u : ℕ → α // monotone u ∧ ∀ i, u i ∈ s}),
v < ∑ i in finset.range p.1, edist (f ((p.2 : ℕ → α) (i+1))) (f ((p.2 : ℕ → α) i)) :=
lt_supr_iff.mp hv,
have : tendsto (λ j, ∑ (i : ℕ) in finset.range n, edist (F j (u (i + 1))) (F j (u i)))
p (𝓝 ... | lemma | evariation_on.lower_continuous_aux | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"eventually_gt_of_tendsto_gt",
"filter",
"finset.range",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous (s : set α) :
lower_semicontinuous (λ f : α →ᵤ[s.image singleton] E, evariation_on f s) | begin
intro f,
apply @lower_continuous_aux _ _ _ _ (uniform_on_fun α E (s.image singleton)) id (𝓝 f) f s _,
simpa only [uniform_on_fun.tendsto_iff_tendsto_uniformly_on, mem_image, forall_exists_index,
and_imp, forall_apply_eq_imp_iff₂,
tendsto_uniformly_on_singleton_iff_tendsto] using... | lemma | evariation_on.lower_semicontinuous | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"and_imp",
"evariation_on",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"lower_semicontinuous",
"tendsto_uniformly_on_singleton_iff_tendsto",
"uniform_on_fun",
"uniform_on_fun.tendsto_iff_tendsto_uniformly_on"
] | The map `λ f, evariation_on f s` is lower semicontinuous for pointwise convergence *on `s`*.
Pointwise convergence on `s` is encoded here as uniform convergence on the family consisting of the
singletons of elements of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_uniform_on (s : set α) :
lower_semicontinuous (λ f : α →ᵤ[{s}] E, evariation_on f s) | begin
intro f,
apply @lower_continuous_aux _ _ _ _ (uniform_on_fun α E {s}) id (𝓝 f) f s _,
have := @tendsto_id _ (𝓝 f),
rw uniform_on_fun.tendsto_iff_tendsto_uniformly_on at this,
simp_rw ←tendsto_uniformly_on_singleton_iff_tendsto,
exact λ x xs, ((this s rfl).mono (singleton_subset_iff.mpr xs)),
end | lemma | evariation_on.lower_semicontinuous_uniform_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"lower_semicontinuous",
"uniform_on_fun",
"uniform_on_fun.tendsto_iff_tendsto_uniformly_on"
] | The map `λ f, evariation_on f s` is lower semicontinuous for uniform convergence on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.has_bounded_variation_on.dist_le {E : Type*} [pseudo_metric_space E]
{f : α → E} {s : set α} (h : has_bounded_variation_on f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
dist (f x) (f y) ≤ (evariation_on f s).to_real | begin
rw [← ennreal.of_real_le_of_real_iff ennreal.to_real_nonneg, ennreal.of_real_to_real h,
← edist_dist],
exact edist_le f hx hy
end | lemma | has_bounded_variation_on.dist_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"edist_dist",
"ennreal.of_real_le_of_real_iff",
"ennreal.of_real_to_real",
"ennreal.to_real_nonneg",
"evariation_on",
"has_bounded_variation_on",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.has_bounded_variation_on.sub_le
{f : α → ℝ} {s : set α} (h : has_bounded_variation_on f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
f x - f y ≤ (evariation_on f s).to_real | begin
apply (le_abs_self _).trans,
rw ← real.dist_eq,
exact h.dist_le hx hy
end | lemma | has_bounded_variation_on.sub_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"has_bounded_variation_on",
"le_abs_self",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_point (f : α → E) {s : set α} {x : α} (hx : x ∈ s)
(u : ℕ → α) (hu : monotone u) (us : ∀ i, u i ∈ s) (n : ℕ) :
∃ (v : ℕ → α) (m : ℕ), monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' (Iio m) ∧
∑ i in finset.range n, edist (f (u (i+1))) (f (u i)) ≤
∑ j in finset.range m, edist (f (v (j+1))) (f (v j)) | begin
rcases le_or_lt (u n) x with h|h,
{ let v := λ i, if i ≤ n then u i else x,
have vs : ∀ i, v i ∈ s,
{ assume i,
simp only [v],
split_ifs,
{ exact us i },
{ exact hx } },
have hv : monotone v,
{ apply monotone_nat_of_le_succ (λ i, _),
simp only [v],
rcases lt... | lemma | evariation_on.add_point | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"eq_or_lt_of_le",
"finset.Ico",
"finset.Ico_self",
"finset.Ico_subset_Ico",
"finset.mem_Ico",
"finset.mem_range",
"finset.range",
"finset.range_eq_Ico",
"finset.range_mono",
"finset.sum_eq_sum_Ico_succ_bot",
"is_empty.forall_iff",
"ite_eq_right_iff",
"le_rfl",
"le_zero_iff",
"monotone",
... | Consider a monotone function `u` parameterizing some points of a set `s`. Given `x ∈ s`, then
one can find another monotone function `v` parameterizing the same points as `u`, with `x` added.
In particular, the variation of a function along `u` is bounded by its variation along `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_le_union (f : α → E) {s t : set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
evariation_on f s + evariation_on f t ≤ evariation_on f (s ∪ t) | begin
by_cases hs : s = ∅,
{ simp [hs] },
haveI : nonempty {u // monotone u ∧ ∀ (i : ℕ), u i ∈ s},
from nonempty_monotone_mem (nonempty_iff_ne_empty.2 hs),
by_cases ht : t = ∅,
{ simp [ht] },
haveI : nonempty {u // monotone u ∧ ∀ (i : ℕ), u i ∈ t},
from nonempty_monotone_mem (nonempty_iff_ne_empty.2... | lemma | evariation_on.add_le_union | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"add_tsub_cancel_left",
"ennreal.supr_add_supr_le",
"evariation_on",
"finset.Ico",
"finset.mem_Ico",
"finset.mem_range",
"finset.mem_union",
"finset.range",
"finset.range_eq_Ico",
"le_rfl",
"monotone",
"tsub_eq_iff_eq_add_of_le",
"tsub_le_tsub"
] | The variation of a function on the union of two sets `s` and `t`, with `s` to the left of `t`,
bounds the sum of the variations along `s` and `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
union (f : α → E) {s t : set α} {x : α} (hs : is_greatest s x) (ht : is_least t x) :
evariation_on f (s ∪ t) = evariation_on f s + evariation_on f t | begin
classical,
apply le_antisymm _ (evariation_on.add_le_union f (λ a ha b hb, le_trans (hs.2 ha) (ht.2 hb))),
apply supr_le _,
rintros ⟨n, ⟨u, hu, ust⟩⟩,
obtain ⟨v, m, hv, vst, xv, huv⟩ : ∃ (v : ℕ → α) (m : ℕ), monotone v ∧ (∀ i, v i ∈ s ∪ t) ∧
x ∈ v '' (Iio m) ∧ ∑ i in finset.range n, edist (f (u (i+1... | lemma | evariation_on.union | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"evariation_on.add_le_union",
"evariation_on.add_point",
"finset.Ico",
"finset.range",
"finset.range_eq_Ico",
"is_greatest",
"is_least",
"monotone",
"supr_le"
] | If a set `s` is to the left of a set `t`, and both contain the boundary point `x`, then
the variation of `f` along `s ∪ t` is the sum of the variations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Icc_add_Icc (f : α → E) {s : set α} {a b c : α}
(hab : a ≤ b) (hbc : b ≤ c) (hb : b ∈ s) :
evariation_on f (s ∩ Icc a b) + evariation_on f (s ∩ Icc b c) = evariation_on f (s ∩ Icc a c) | begin
have A : is_greatest (s ∩ Icc a b) b :=
⟨⟨hb, hab, le_rfl⟩, (inter_subset_right _ _).trans (Icc_subset_Iic_self)⟩,
have B : is_least (s ∩ Icc b c) b :=
⟨⟨hb, le_rfl, hbc⟩, (inter_subset_right _ _).trans (Icc_subset_Ici_self)⟩,
rw [← evariation_on.union f A B, ← inter_union_distrib_left, Icc_union_Ic... | lemma | evariation_on.Icc_add_Icc | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"evariation_on.union",
"is_greatest",
"is_least",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_le_of_monotone_on (f : α → E) {s : set α} {t : set β} (φ : β → α)
(hφ : monotone_on φ t) (φst : maps_to φ t s) :
evariation_on (f ∘ φ) t ≤ evariation_on f s | supr_le $ λ ⟨n, u, hu, ut⟩, le_supr_of_le
⟨n, φ ∘ u, λ x y xy, hφ (ut x) (ut y) (hu xy), λ i, φst (ut i)⟩ le_rfl | lemma | evariation_on.comp_le_of_monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"le_rfl",
"le_supr_of_le",
"monotone_on",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_le_of_antitone_on (f : α → E) {s : set α} {t : set β} (φ : β → α)
(hφ : antitone_on φ t) (φst : maps_to φ t s) :
evariation_on (f ∘ φ) t ≤ evariation_on f s | begin
refine supr_le _,
rintro ⟨n, u, hu, ut⟩,
rw ←finset.sum_range_reflect,
refine (finset.sum_congr rfl $ λ x hx, _).trans_le (le_supr_of_le ⟨n, λ i, φ (u $ n-i),
λ x y xy, hφ (ut _) (ut _) (hu $ n.sub_le_sub_left xy), λ i, φst (ut _)⟩ le_rfl),
dsimp only [subtype.coe_mk],
rw [edist_comm, nat.sub_sub,... | lemma | evariation_on.comp_le_of_antitone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"antitone_on",
"evariation_on",
"finset.mem_range",
"le_rfl",
"le_supr_of_le",
"subtype.coe_mk",
"supr_le",
"tsub_pos_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_of_monotone_on (f : α → E) {t : set β} (φ : β → α) (hφ : monotone_on φ t) :
evariation_on (f ∘ φ) t = evariation_on f (φ '' t) | begin
apply le_antisymm (comp_le_of_monotone_on f φ hφ (maps_to_image φ t)),
casesI is_empty_or_nonempty β,
{ convert zero_le _,
exact evariation_on.subsingleton f
((subsingleton_of_subsingleton.image _).anti (surj_on_image φ t)) },
let ψ := φ.inv_fun_on t,
have ψφs : eq_on (φ ∘ ψ) id (φ '' t)... | lemma | evariation_on.comp_eq_of_monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"evariation_on.subsingleton",
"function.monotone_on_of_right_inv_on_of_maps_to",
"is_empty_or_nonempty",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.subsingleton_Icc_of_ge {α : Type*} [partial_order α] {a b : α} (h : b ≤ a) :
set.subsingleton (Icc a b) | begin
rintros c ⟨ac,cb⟩ d ⟨ad,db⟩,
cases le_antisymm (cb.trans h) ac,
cases le_antisymm (db.trans h) ad,
refl,
end | lemma | set.subsingleton_Icc_of_ge | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"set.subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inter_Icc_eq_of_monotone_on (f : α → E) {t : set β} (φ : β → α)
(hφ : monotone_on φ t) {x y : β} (hx : x ∈ t) (hy : y ∈ t) :
evariation_on (f ∘ φ) (t ∩ Icc x y) = evariation_on f ((φ '' t) ∩ Icc (φ x) (φ y)) | begin
rcases le_total x y with h|h,
{ convert comp_eq_of_monotone_on f φ (hφ.mono (set.inter_subset_left t (Icc x y))),
apply le_antisymm,
{ rintro _ ⟨⟨u, us, rfl⟩, vφx, vφy⟩,
rcases le_total x u with xu|ux,
{ rcases le_total u y with uy|yu,
{ exact ⟨u, ⟨us, ⟨xu, uy⟩⟩, rfl⟩, },
{... | lemma | evariation_on.comp_inter_Icc_eq_of_monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"evariation_on.subsingleton",
"monotone_on",
"set.inter_subset_left",
"set.inter_subset_right",
"set.subsingleton_Icc_of_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_of_antitone_on (f : α → E) {t : set β} (φ : β → α) (hφ : antitone_on φ t) :
evariation_on (f ∘ φ) t = evariation_on f (φ '' t) | begin
apply le_antisymm (comp_le_of_antitone_on f φ hφ (maps_to_image φ t)),
casesI is_empty_or_nonempty β,
{ convert zero_le _,
exact evariation_on.subsingleton f
((subsingleton_of_subsingleton.image _).anti (surj_on_image φ t)) },
let ψ := φ.inv_fun_on t,
have ψφs : eq_on (φ ∘ ψ) id (φ '' t) := (s... | lemma | evariation_on.comp_eq_of_antitone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"antitone_on",
"evariation_on",
"evariation_on.subsingleton",
"function.antitone_on_of_right_inv_on_of_maps_to",
"is_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_of_dual (f : α → E) (s : set α) :
evariation_on (f ∘ of_dual) (of_dual ⁻¹' s) = evariation_on f s | begin
convert comp_eq_of_antitone_on f of_dual (λ _ _ _ _, id),
simp only [equiv.image_preimage],
end | lemma | evariation_on.comp_of_dual | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"equiv.image_preimage",
"evariation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.evariation_on_le {f : α → ℝ} {s : set α} (hf : monotone_on f s) {a b : α}
(as : a ∈ s) (bs : b ∈ s) :
evariation_on f (s ∩ Icc a b) ≤ ennreal.of_real (f b - f a) | begin
apply supr_le _,
rintros ⟨n, ⟨u, hu, us⟩⟩,
calc
∑ i in finset.range n, edist (f (u (i+1))) (f (u i))
= ∑ i in finset.range n, ennreal.of_real (f (u (i + 1)) - f (u i)) :
begin
apply finset.sum_congr rfl (λ i hi, _),
simp only [finset.mem_range] at hi,
rw [edist_dist, real.dist_... | lemma | monotone_on.evariation_on_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"abs_of_nonneg",
"edist_dist",
"ennreal.of_real",
"ennreal.of_real_le_of_real",
"ennreal.of_real_sum_of_nonneg",
"evariation_on",
"finset.mem_range",
"finset.range",
"monotone_on",
"real.dist_eq",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.has_locally_bounded_variation_on {f : α → ℝ} {s : set α} (hf : monotone_on f s) :
has_locally_bounded_variation_on f s | λ a b as bs, ((hf.evariation_on_le as bs).trans_lt ennreal.of_real_lt_top).ne | lemma | monotone_on.has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.of_real_lt_top",
"has_locally_bounded_variation_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variation_on_from_to (f : α → E) (s : set α) (a b : α) : ℝ | if a ≤ b then (evariation_on f (s ∩ Icc a b)).to_real else
- (evariation_on f (s ∩ Icc b a)).to_real | def | variation_on_from_to | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on"
] | The **signed** variation of `f` on the interval `Icc a b` intersected with the set `s`,
squashed to a real (therefore only really meaningful if the variation is finite) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self (a : α) : variation_on_from_to f s a a = 0 | begin
dsimp only [variation_on_from_to],
rw [if_pos le_rfl, Icc_self, evariation_on.subsingleton, ennreal.zero_to_real],
exact λ x hx y hy, hx.2.trans hy.2.symm,
end | lemma | variation_on_from_to.self | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.zero_to_real",
"evariation_on.subsingleton",
"le_rfl",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_of_le {a b : α} (h : a ≤ b) : 0 ≤ variation_on_from_to f s a b | by simp only [variation_on_from_to, if_pos h, ennreal.to_real_nonneg] | lemma | variation_on_from_to.nonneg_of_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.to_real_nonneg",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_neg_swap (a b : α) :
variation_on_from_to f s a b = - variation_on_from_to f s b a | begin
rcases lt_trichotomy a b with ab|rfl|ba,
{ simp only [variation_on_from_to, if_pos ab.le, if_neg ab.not_le, neg_neg], },
{ simp only [self, neg_zero], },
{ simp only [variation_on_from_to, if_pos ba.le, if_neg ba.not_le, neg_neg], },
end | lemma | variation_on_from_to.eq_neg_swap | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonpos_of_ge {a b : α} (h : b ≤ a) : variation_on_from_to f s a b ≤ 0 | begin
rw eq_neg_swap,
exact neg_nonpos_of_nonneg (nonneg_of_le f s h),
end | lemma | variation_on_from_to.nonpos_of_ge | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_le {a b : α} (h : a ≤ b) :
variation_on_from_to f s a b = (evariation_on f (s ∩ Icc a b)).to_real | if_pos h | lemma | variation_on_from_to.eq_of_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_ge {a b : α} (h : b ≤ a) :
variation_on_from_to f s a b = - (evariation_on f (s ∩ Icc b a)).to_real | by rw [eq_neg_swap, neg_inj, eq_of_le f s h] | lemma | variation_on_from_to.eq_of_ge | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add {f : α → E} {s : set α} (hf : has_locally_bounded_variation_on f s)
{a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
variation_on_from_to f s a b + variation_on_from_to f s b c = variation_on_from_to f s a c | begin
symmetry,
refine additive_of_is_total (≤) (variation_on_from_to f s) (∈s) _ _ ha hb hc,
{ rintro x y xs ys,
simp only [eq_neg_swap f s y x, subtype.coe_mk, add_right_neg, forall_true_left], },
{ rintro x y z xy yz xs ys zs,
rw [eq_of_le f s xy, eq_of_le f s yz, eq_of_le f s (xy.trans yz),
... | lemma | variation_on_from_to.add | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on.Icc_add_Icc",
"forall_true_left",
"has_locally_bounded_variation_on",
"subtype.coe_mk",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_zero_of_eq_zero
{f : α → E} {s : set α} (hf : has_locally_bounded_variation_on f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : variation_on_from_to f s a b = 0) :
edist (f a) (f b) = 0 | begin
wlog h' : a ≤ b,
{ rw edist_comm,
apply this hf hb ha _ (le_of_not_le h'),
rw [eq_neg_swap, h, neg_zero] },
{ apply le_antisymm _ (zero_le _),
rw [←ennreal.of_real_zero, ←h, eq_of_le f s h', ennreal.of_real_to_real (hf a b ha hb)],
apply evariation_on.edist_le,
exacts [⟨ha, ⟨le_rfl, h'⟩⟩... | lemma | variation_on_from_to.edist_zero_of_eq_zero | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.of_real_to_real",
"evariation_on.edist_le",
"has_locally_bounded_variation_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_left_iff
{f : α → E} {s : set α} (hf : has_locally_bounded_variation_on f s)
{a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
variation_on_from_to f s a b = variation_on_from_to f s a c ↔ variation_on_from_to f s b c = 0 | by simp only [←add hf ha hb hc, self_eq_add_right] | lemma | variation_on_from_to.eq_left_iff | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff_of_le
{f : α → E} {s : set α} (hf : has_locally_bounded_variation_on f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) (ab : a ≤ b) :
variation_on_from_to f s a b = 0 ↔
∀ ⦃x⦄ (hx : x ∈ s ∩ Icc a b) ⦃y⦄ (hy : y ∈ s ∩ Icc a b), edist (f x) (f y) = 0 | by rw [eq_of_le _ _ ab, ennreal.to_real_eq_zero_iff,
or_iff_left (hf a b ha hb), evariation_on.eq_zero_iff] | lemma | variation_on_from_to.eq_zero_iff_of_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.to_real_eq_zero_iff",
"evariation_on.eq_zero_iff",
"has_locally_bounded_variation_on",
"or_iff_left",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff_of_ge
{f : α → E} {s : set α} (hf : has_locally_bounded_variation_on f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) (ba : b ≤ a) :
variation_on_from_to f s a b = 0 ↔
∀ ⦃x⦄ (hx : x ∈ s ∩ Icc b a) ⦃y⦄ (hy : y ∈ s ∩ Icc b a), edist (f x) (f y) = 0 | by rw [eq_of_ge _ _ ba, neg_eq_zero, ennreal.to_real_eq_zero_iff,
or_iff_left (hf b a hb ha), evariation_on.eq_zero_iff] | lemma | variation_on_from_to.eq_zero_iff_of_ge | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.to_real_eq_zero_iff",
"evariation_on.eq_zero_iff",
"has_locally_bounded_variation_on",
"or_iff_left",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff
{f : α → E} {s : set α} (hf : has_locally_bounded_variation_on f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
variation_on_from_to f s a b = 0 ↔
∀ ⦃x⦄ (hx : x ∈ s ∩ uIcc a b) ⦃y⦄ (hy : y ∈ s ∩ uIcc a b), edist (f x) (f y) = 0 | begin
rcases le_total a b with ab|ba,
{ rw uIcc_of_le ab,
exact eq_zero_iff_of_le hf ha hb ab, },
{ rw uIcc_of_ge ba,
exact eq_zero_iff_of_ge hf ha hb ba, },
end | lemma | variation_on_from_to.eq_zero_iff | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"eq_zero_iff",
"has_locally_bounded_variation_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on (hf : has_locally_bounded_variation_on f s)
{a : α} (as : a ∈ s) : monotone_on (variation_on_from_to f s a) s | begin
rintro b bs c cs bc,
rw ←add hf as bs cs,
exact le_add_of_nonneg_right (nonneg_of_le f s bc),
end | lemma | variation_on_from_to.monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"monotone_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on (hf : has_locally_bounded_variation_on f s)
{b : α} (bs : b ∈ s) : antitone_on (λ a, variation_on_from_to f s a b) s | begin
rintro a as c cs ac,
dsimp only,
rw ←add hf as cs bs,
exact le_add_of_nonneg_left (nonneg_of_le f s ac),
end | lemma | variation_on_from_to.antitone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"antitone_on",
"has_locally_bounded_variation_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_monotone_on {f : α → ℝ} {s : set α}
(hf : has_locally_bounded_variation_on f s) {a : α} (as : a ∈ s) :
monotone_on (variation_on_from_to f s a - f) s | begin
rintro b bs c cs bc,
rw [pi.sub_apply, pi.sub_apply, le_sub_iff_add_le, add_comm_sub, ← le_sub_iff_add_le'],
calc f c - f b
≤ |f c - f b| : le_abs_self _
... = dist (f b) (f c) : by rw [dist_comm, real.dist_eq]
... ≤ variation_on_from_to f s b c : by
{ rw [eq_of_le f s bc, dist_edist],
appl... | lemma | variation_on_from_to.sub_self_monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"dist_comm",
"dist_edist",
"ennreal.to_real_mono",
"evariation_on.edist_le",
"has_locally_bounded_variation_on",
"le_abs_self",
"le_rfl",
"monotone_on",
"real.dist_eq",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_of_monotone_on (f : α → E) {t : set β} (φ : β → α) (hφ : monotone_on φ t)
{x y : β} (hx : x ∈ t) (hy : y ∈ t) :
variation_on_from_to (f ∘ φ) t x y = variation_on_from_to f (φ '' t) (φ x) (φ y) | begin
rcases le_total x y with h|h,
{ rw [eq_of_le _ _ h, eq_of_le _ _ (hφ hx hy h),
evariation_on.comp_inter_Icc_eq_of_monotone_on f φ hφ hx hy], },
{ rw [eq_of_ge _ _ h, eq_of_ge _ _ (hφ hy hx h),
evariation_on.comp_inter_Icc_eq_of_monotone_on f φ hφ hy hx], },
end | lemma | variation_on_from_to.comp_eq_of_monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on.comp_inter_Icc_eq_of_monotone_on",
"monotone_on",
"variation_on_from_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_locally_bounded_variation_on.exists_monotone_on_sub_monotone_on {f : α → ℝ} {s : set α}
(h : has_locally_bounded_variation_on f s) :
∃ (p q : α → ℝ), monotone_on p s ∧ monotone_on q s ∧ f = p - q | begin
rcases eq_empty_or_nonempty s with rfl|⟨c, cs⟩,
{ exact ⟨f, 0, subsingleton_empty.monotone_on _, subsingleton_empty.monotone_on _,
(sub_zero f).symm⟩ },
{ exact ⟨_, _, variation_on_from_to.monotone_on h cs,
variation_on_from_to.sub_self_monotone_on h cs, (sub_sub_cancel _ _).symm⟩ },
end | lemma | has_locally_bounded_variation_on.exists_monotone_on_sub_monotone_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"monotone_on",
"variation_on_from_to.monotone_on",
"variation_on_from_to.sub_self_monotone_on"
] | If a real valued function has bounded variation on a set, then it is a difference of monotone
functions there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with.comp_evariation_on_le {f : E → F} {C : ℝ≥0} {t : set E}
(h : lipschitz_on_with C f t) {g : α → E} {s : set α} (hg : maps_to g s t) :
evariation_on (f ∘ g) s ≤ C * evariation_on g s | begin
apply supr_le _,
rintros ⟨n, ⟨u, hu, us⟩⟩,
calc
∑ i in finset.range n, edist (f (g (u (i+1)))) (f (g (u i)))
≤ ∑ i in finset.range n, C * edist (g (u (i+1))) (g (u i)) :
finset.sum_le_sum (λ i hi, h (hg (us _)) (hg (us _)))
... = C * ∑ i in finset.range n, edist (g (u (i+1))) (g (u i)) : by rw... | lemma | lipschitz_on_with.comp_evariation_on_le | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"evariation_on",
"evariation_on.sum_le",
"finset.mul_sum",
"finset.range",
"lipschitz_on_with",
"mul_le_mul_left'",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with.comp_has_bounded_variation_on {f : E → F} {C : ℝ≥0} {t : set E}
(hf : lipschitz_on_with C f t) {g : α → E} {s : set α} (hg : maps_to g s t)
(h : has_bounded_variation_on g s) :
has_bounded_variation_on (f ∘ g) s | ne_top_of_le_ne_top (ennreal.mul_ne_top ennreal.coe_ne_top h) (hf.comp_evariation_on_le hg) | lemma | lipschitz_on_with.comp_has_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"ennreal.coe_ne_top",
"ennreal.mul_ne_top",
"has_bounded_variation_on",
"lipschitz_on_with",
"ne_top_of_le_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with.comp_has_locally_bounded_variation_on {f : E → F} {C : ℝ≥0} {t : set E}
(hf : lipschitz_on_with C f t) {g : α → E} {s : set α} (hg : maps_to g s t)
(h : has_locally_bounded_variation_on g s) :
has_locally_bounded_variation_on (f ∘ g) s | λ x y xs ys, hf.comp_has_bounded_variation_on (hg.mono_left (inter_subset_left _ _)) (h x y xs ys) | lemma | lipschitz_on_with.comp_has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.comp_has_bounded_variation_on {f : E → F} {C : ℝ≥0}
(hf : lipschitz_with C f) {g : α → E} {s : set α} (h : has_bounded_variation_on g s) :
has_bounded_variation_on (f ∘ g) s | (hf.lipschitz_on_with univ).comp_has_bounded_variation_on (maps_to_univ _ _) h | lemma | lipschitz_with.comp_has_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_bounded_variation_on",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.comp_has_locally_bounded_variation_on {f : E → F} {C : ℝ≥0}
(hf : lipschitz_with C f) {g : α → E} {s : set α} (h : has_locally_bounded_variation_on g s) :
has_locally_bounded_variation_on (f ∘ g) s | (hf.lipschitz_on_with univ).comp_has_locally_bounded_variation_on (maps_to_univ _ _) h | lemma | lipschitz_with.comp_has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with.has_locally_bounded_variation_on {f : ℝ → E} {C : ℝ≥0} {s : set ℝ}
(hf : lipschitz_on_with C f s) : has_locally_bounded_variation_on f s | hf.comp_has_locally_bounded_variation_on (maps_to_id _)
(@monotone_on_id ℝ _ s).has_locally_bounded_variation_on | lemma | lipschitz_on_with.has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"lipschitz_on_with",
"monotone_on_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.has_locally_bounded_variation_on {f : ℝ → E} {C : ℝ≥0}
(hf : lipschitz_with C f) (s : set ℝ) : has_locally_bounded_variation_on f s | (hf.lipschitz_on_with s).has_locally_bounded_variation_on | lemma | lipschitz_with.has_locally_bounded_variation_on | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"has_locally_bounded_variation_on",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_differentiable_within_at_of_mem_real
{f : ℝ → ℝ} {s : set ℝ} (h : has_locally_bounded_variation_on f s) :
∀ᵐ x, x ∈ s → differentiable_within_at ℝ f s x | begin
obtain ⟨p, q, hp, hq, fpq⟩ : ∃ p q, monotone_on p s ∧ monotone_on q s ∧ f = p - q,
from h.exists_monotone_on_sub_monotone_on,
filter_upwards [hp.ae_differentiable_within_at_of_mem, hq.ae_differentiable_within_at_of_mem]
with x hxp hxq xs,
have fpq : ∀ x, f x = p x - q x, by simp [fpq],
refine ((hx... | theorem | has_locally_bounded_variation_on.ae_differentiable_within_at_of_mem_real | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_within_at",
"has_locally_bounded_variation_on",
"monotone_on"
] | A bounded variation function into `ℝ` is differentiable almost everywhere. Superseded by
`ae_differentiable_within_at_of_mem`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_differentiable_within_at_of_mem_pi {ι : Type*} [fintype ι]
{f : ℝ → (ι → ℝ)} {s : set ℝ} (h : has_locally_bounded_variation_on f s) :
∀ᵐ x, x ∈ s → differentiable_within_at ℝ f s x | begin
have A : ∀ (i : ι), lipschitz_with 1 (λ (x : ι → ℝ), x i) := λ i, lipschitz_with.eval i,
have : ∀ (i : ι), ∀ᵐ x, x ∈ s → differentiable_within_at ℝ (λ (x : ℝ), f x i) s x,
{ assume i,
apply ae_differentiable_within_at_of_mem_real,
exact lipschitz_with.comp_has_locally_bounded_variation_on (A i) h },... | theorem | has_locally_bounded_variation_on.ae_differentiable_within_at_of_mem_pi | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_within_at",
"fintype",
"has_locally_bounded_variation_on",
"lipschitz_with",
"lipschitz_with.comp_has_locally_bounded_variation_on",
"lipschitz_with.eval"
] | A bounded variation function into a finite dimensional product vector space is differentiable
almost everywhere. Superseded by `ae_differentiable_within_at_of_mem`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_differentiable_within_at_of_mem
{f : ℝ → V} {s : set ℝ} (h : has_locally_bounded_variation_on f s) :
∀ᵐ x, x ∈ s → differentiable_within_at ℝ f s x | begin
let A := (basis.of_vector_space ℝ V).equiv_fun.to_continuous_linear_equiv,
suffices H : ∀ᵐ x, x ∈ s → differentiable_within_at ℝ (A ∘ f) s x,
{ filter_upwards [H] with x hx xs,
have : f = (A.symm ∘ A) ∘ f,
by simp only [continuous_linear_equiv.symm_comp_self, function.comp.left_id],
rw this,
... | theorem | has_locally_bounded_variation_on.ae_differentiable_within_at_of_mem | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"basis.of_vector_space",
"continuous_linear_equiv.symm_comp_self",
"differentiable_within_at",
"has_locally_bounded_variation_on"
] | A real function into a finite dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_differentiable_within_at
{f : ℝ → V} {s : set ℝ} (h : has_locally_bounded_variation_on f s) (hs : measurable_set s) :
∀ᵐ x ∂(volume.restrict s), differentiable_within_at ℝ f s x | begin
rw ae_restrict_iff' hs,
exact h.ae_differentiable_within_at_of_mem
end | theorem | has_locally_bounded_variation_on.ae_differentiable_within_at | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_within_at",
"has_locally_bounded_variation_on",
"measurable_set"
] | A real function into a finite dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_differentiable_at {f : ℝ → V} (h : has_locally_bounded_variation_on f univ) :
∀ᵐ x, differentiable_at ℝ f x | begin
filter_upwards [h.ae_differentiable_within_at_of_mem] with x hx,
rw differentiable_within_at_univ at hx,
exact hx (mem_univ _),
end | theorem | has_locally_bounded_variation_on.ae_differentiable_at | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_at",
"differentiable_within_at_univ",
"has_locally_bounded_variation_on"
] | A real function into a finite dimensional real vector space with bounded variation
is differentiable almost everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with.ae_differentiable_within_at_of_mem
{C : ℝ≥0} {f : ℝ → V} {s : set ℝ} (h : lipschitz_on_with C f s) :
∀ᵐ x, x ∈ s → differentiable_within_at ℝ f s x | h.has_locally_bounded_variation_on.ae_differentiable_within_at_of_mem | lemma | lipschitz_on_with.ae_differentiable_within_at_of_mem | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_within_at",
"lipschitz_on_with"
] | A real function into a finite dimensional real vector space which is Lipschitz on a set
is differentiable almost everywhere in this set . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with.ae_differentiable_within_at
{C : ℝ≥0} {f : ℝ → V} {s : set ℝ} (h : lipschitz_on_with C f s) (hs : measurable_set s) :
∀ᵐ x ∂(volume.restrict s), differentiable_within_at ℝ f s x | h.has_locally_bounded_variation_on.ae_differentiable_within_at hs | lemma | lipschitz_on_with.ae_differentiable_within_at | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_within_at",
"lipschitz_on_with",
"measurable_set"
] | A real function into a finite dimensional real vector space which is Lipschitz on a set
is differentiable almost everywhere in this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_with.ae_differentiable_at
{C : ℝ≥0} {f : ℝ → V} (h : lipschitz_with C f) :
∀ᵐ x, differentiable_at ℝ f x | (h.has_locally_bounded_variation_on univ).ae_differentiable_at | lemma | lipschitz_with.ae_differentiable_at | analysis | src/analysis/bounded_variation.lean | [
"analysis.calculus.deriv.add",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.prod",
"analysis.calculus.monotone",
"data.set.function",
"algebra.group.basic",
"tactic.wlog"
] | [
"differentiable_at",
"lipschitz_with"
] | A real Lipschitz function into a finite dimensional real vector space is differentiable
almost everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_constant_speed_on_with | ∀ ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s), evariation_on f (s ∩ Icc x y) = ennreal.of_real (l * (y - x)) | def | has_constant_speed_on_with | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real",
"evariation_on"
] | `f` has constant speed `l` on `s` if the variation of `f` on `s ∩ Icc x y` is equal to
`l * (y - x)` for any `x y` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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