topology.extend_from
⟷
Mathlib.Topology.ExtendFrom
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Anatole Dedecker
-/
-import Mathbin.Topology.Separation
+import Topology.Separation
#align_import topology.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Anatole Dedecker
-
-! This file was ported from Lean 3 source module topology.extend_from
-! leanprover-community/mathlib commit 0a0ec35061ed9960bf0e7ffb0335f44447b58977
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Topology.Separation
+#align_import topology.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
+
/-!
# Extending a function from a subset
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -47,12 +47,14 @@ def extendFrom (A : Set X) (f : X → Y) : X → Y := fun x => @limUnder _ ⟨f
#align extend_from extendFrom
-/
+#print tendsto_extendFrom /-
/-- If `f` converges to some `y` as `x` tends to `x₀` within `A`,
then `f` tends to `extend_from A f x` as `x` tends to `x₀`. -/
theorem tendsto_extendFrom {A : Set X} {f : X → Y} {x : X} (h : ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) :
Tendsto f (𝓝[A] x) (𝓝 <| extendFrom A f x) :=
tendsto_nhds_limUnder h
#align tendsto_extend_from tendsto_extendFrom
+-/
#print extendFrom_eq /-
theorem extendFrom_eq [T2Space Y] {A : Set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A)
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -33,7 +33,7 @@ it suffices that `f` converges within `A` at any point of `B`, provided that
noncomputable section
-open Topology
+open scoped Topology
open Filter Set
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -47,12 +47,6 @@ def extendFrom (A : Set X) (f : X → Y) : X → Y := fun x => @limUnder _ ⟨f
#align extend_from extendFrom
-/
-/- warning: tendsto_extend_from -> tendsto_extendFrom is a dubious translation:
-lean 3 declaration is
- forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {A : Set.{u1} X} {f : X -> Y} {x : X}, (Exists.{succ u2} Y (fun (y : Y) => Filter.Tendsto.{u1, u2} X Y f (nhdsWithin.{u1} X _inst_1 x A) (nhds.{u2} Y _inst_2 y))) -> (Filter.Tendsto.{u1, u2} X Y f (nhdsWithin.{u1} X _inst_1 x A) (nhds.{u2} Y _inst_2 (extendFrom.{u1, u2} X Y _inst_1 _inst_2 A f x)))
-but is expected to have type
- forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {A : Set.{u2} X} {f : X -> Y} {x : X}, (Exists.{succ u1} Y (fun (y : Y) => Filter.Tendsto.{u2, u1} X Y f (nhdsWithin.{u2} X _inst_1 x A) (nhds.{u1} Y _inst_2 y))) -> (Filter.Tendsto.{u2, u1} X Y f (nhdsWithin.{u2} X _inst_1 x A) (nhds.{u1} Y _inst_2 (extendFrom.{u2, u1} X Y _inst_1 _inst_2 A f x)))
-Case conversion may be inaccurate. Consider using '#align tendsto_extend_from tendsto_extendFromₓ'. -/
/-- If `f` converges to some `y` as `x` tends to `x₀` within `A`,
then `f` tends to `extend_from A f x` as `x` tends to `x₀`. -/
theorem tendsto_extendFrom {A : Set X} {f : X → Y} {x : X} (h : ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
have
, replace
and suffices
(#10640)
No changes to tactic file, it's just boring fixes throughout the library.
This follows on from #6964.
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@@ -71,8 +71,8 @@ theorem continuousOn_extendFrom [RegularSpace Y] {f : X → Y} {A B : Set X} (hB
have := tendsto_extendFrom (hf x x_in)
rcases (nhdsWithin_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩
exact ⟨V, IsOpen.mem_nhds V_op hxV, V_op, hV⟩
- suffices : ∀ y ∈ V ∩ B, φ y ∈ V'
- exact mem_of_superset (inter_mem_inf V_in <| mem_principal_self B) this
+ suffices ∀ y ∈ V ∩ B, φ y ∈ V' from
+ mem_of_superset (inter_mem_inf V_in <| mem_principal_self B) this
rintro y ⟨hyV, hyB⟩
haveI := mem_closure_iff_nhdsWithin_neBot.mp (hB hyB)
have limy : Tendsto f (𝓝[A] y) (𝓝 <| φ y) := tendsto_extendFrom (hf y hyB)
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -31,7 +31,7 @@ open Topology
open Filter Set
-variable {X Y : Type _} [TopologicalSpace X] [TopologicalSpace Y]
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
/-- Extend a function from a set `A`. The resulting function `g` is such that
at any `x₀`, if `f` converges to some `y` as `x` tends to `x₀` within `A`,
@@ -2,14 +2,11 @@
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Anatole Dedecker
-
-! This file was ported from Lean 3 source module topology.extend_from
-! leanprover-community/mathlib commit b363547b3113d350d053abdf2884e9850a56b205
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Topology.Separation
+#align_import topology.extend_from from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
+
/-!
# Extending a function from a subset
@@ -17,7 +17,7 @@ The main definition of this file is `extendFrom A f` where `f : X → Y`
and `A : Set X`. This defines a new function `g : X → Y` which maps any
`x₀ : X` to the limit of `f` as `x` tends to `x₀`, if such a limit exists.
-This is analoguous to the way `DenseInducing.extend` "extends" a function
+This is analogous to the way `DenseInducing.extend` "extends" a function
`f : X → Z` to a function `g : Y → Z` along a dense inducing `i : X → Y`.
The main theorem we prove about this definition is `continuousOn_extendFrom`
The unported dependencies are