geometry.manifold.instances.units_of_normed_algebra

Changes since the initial port

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.

Changes in mathlib3

ef901ea6

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

36938f77

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,8 +3,8 @@ Copyright © 2021 Nicolò Cavalleri. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Nicolò Cavalleri, Heather Macbeth
 -/
-import Mathbin.Geometry.Manifold.SmoothManifoldWithCorners
-import Mathbin.Analysis.NormedSpace.Units
+import Geometry.Manifold.SmoothManifoldWithCorners
+import Analysis.NormedSpace.Units
 
 #align_import geometry.manifold.instances.units_of_normed_algebra from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"

36938f77

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,15 +2,12 @@
 Copyright © 2021 Nicolò Cavalleri. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Nicolò Cavalleri, Heather Macbeth
-
-! This file was ported from Lean 3 source module geometry.manifold.instances.units_of_normed_algebra
-! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Geometry.Manifold.SmoothManifoldWithCorners
 import Mathbin.Analysis.NormedSpace.Units
 
+#align_import geometry.manifold.instances.units_of_normed_algebra from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
 /-!
 # Units of a normed algebra

36938f77

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -63,13 +63,17 @@ variable {R : Type _} [NormedRing R] [CompleteSpace R]
 instance : ChartedSpace R Rˣ :=
   openEmbedding_val.singletonChartedSpace
 
+#print Units.chartAt_apply /-
 theorem chartAt_apply {a : Rˣ} {b : Rˣ} : chartAt R a b = b :=
   rfl
 #align units.chart_at_apply Units.chartAt_apply
+-/
 
+#print Units.chartAt_source /-
 theorem chartAt_source {a : Rˣ} : (chartAt R a).source = Set.univ :=
   rfl
 #align units.chart_at_source Units.chartAt_source
+-/
 
 variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 R]

36938f77

bump to nightly-2023-06-07-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

831e78ce

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Nicolò Cavalleri, Heather Macbeth
 
 ! This file was ported from Lean 3 source module geometry.manifold.instances.units_of_normed_algebra
-! leanprover-community/mathlib commit ef901ea68d3bb1dd08f8bc3034ab6b32b2e6ecdf
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.NormedSpace.Units
 /-!
 # Units of a normed algebra
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file is a stub, containing a construction of the charted space structure on the group of units
 of a complete normed ring `R`, and of the smooth manifold structure on the group of units of a
 complete normed `𝕜`-algebra `R`.

ef901ea6

bump to nightly-2023-05-28-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

8d353152

Diff

@@ -51,7 +51,7 @@ None of this should be particularly difficult.
 
 noncomputable section
 
-open Manifold
+open scoped Manifold
 
 namespace Units

ef901ea6

bump to nightly-2023-05-12-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

e67823e3

Diff

@@ -58,7 +58,7 @@ namespace Units
 variable {R : Type _} [NormedRing R] [CompleteSpace R]
 
 instance : ChartedSpace R Rˣ :=
-  openEmbedding_coe.singletonChartedSpace
+  openEmbedding_val.singletonChartedSpace
 
 theorem chartAt_apply {a : Rˣ} {b : Rˣ} : chartAt R a b = b :=
   rfl
@@ -71,7 +71,7 @@ theorem chartAt_source {a : Rˣ} : (chartAt R a).source = Set.univ :=
 variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 R]
 
 instance : SmoothManifoldWithCorners 𝓘(𝕜, R) Rˣ :=
-  openEmbedding_coe.singleton_smoothManifoldWithCorners 𝓘(𝕜, R)
+  openEmbedding_val.singleton_smoothManifoldWithCorners 𝓘(𝕜, R)
 
 end Units

ef901ea6

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

ef901ea6

feat(Geometry/Manifold/Instances/UnitsOfNormedAlgebra): Lie group structure (#8397)

Port of [mathlib#15763](https://github.com/leanprover-community/mathlib/pull/15763)

We complete the construction of the Lie group of units of a complete normed ring.

To show that multiplication and inversion are smooth, we add some more lemmas:

For a manifold M whose chart structure is induced by an open embedding from M to H, the open embedding itself is a smooth manifold map.
The inverse of this open embedding is smooth on its range.
For a manifold map f : M → M' whose target space M' has a chart structure induced by an open embedding e' : M' → H', if e' ∘ f is smooth, then f is smooth.

Lemmas previously not ported are now ported:

open_embedding.to_local_homeomorph_left_inv
open_embedding.to_local_homeomorph_right_inv

Co-authored-by: Michael Rothgang <rothgami@math.hu-berlin.de>

395e8530

Diff

@@ -1,48 +1,28 @@
 /-
 Copyright © 2021 Nicolò Cavalleri. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Nicolò Cavalleri, Heather Macbeth
+Authors: Nicolò Cavalleri, Heather Macbeth, Winston Yin
 -/
-import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
-import Mathlib.Analysis.NormedSpace.Units
+import Mathlib.Geometry.Manifold.Algebra.LieGroup
 
 #align_import geometry.manifold.instances.units_of_normed_algebra from "leanprover-community/mathlib"@"ef901ea68d3bb1dd08f8bc3034ab6b32b2e6ecdf"
 
 /-!
 # Units of a normed algebra
 
-This file is a stub, containing a construction of the charted space structure on the group of units
-of a complete normed ring `R`, and of the smooth manifold structure on the group of units of a
-complete normed `𝕜`-algebra `R`.
-
-This manifold is actually a Lie group, which eventually should be the main result of this file.
+We construct the Lie group structure on the group of units of a complete normed `𝕜`-algebra `R`. The
+group of units `Rˣ` has a natural smooth manifold structure modelled on `R` given by its embedding
+into `R`. Together with the smoothness of the multiplication and inverse of its elements, `Rˣ` forms
+a Lie group.
 
 An important special case of this construction is the general linear group.  For a normed space `V`
 over a field `𝕜`, the `𝕜`-linear endomorphisms of `V` are a normed `𝕜`-algebra (see
 `ContinuousLinearMap.toNormedAlgebra`), so this construction provides a Lie group structure on
-its group of units, the general linear group GL(`𝕜`, `V`).
-
-## TODO
-
-The Lie group instance requires the following fields:
+its group of units, the general linear group GL(`𝕜`, `V`), as demonstrated by:
 ```
-instance : LieGroup 𝓘(𝕜, R) Rˣ :=
-  { Units.smoothManifoldWithCorners with
-    smooth_mul := sorry,
-    smooth_inv := sorry }
+example {V : Type*} [NormedAddCommGroup V] [NormedSpace 𝕜 V] [CompleteSpace V] [Nontrivial V] :
+  LieGroup 𝓘(𝕜, V →L[𝕜] V) (V →L[𝕜] V)ˣ := by infer_instance
 ```
-
-The ingredients needed for the construction are
-* smoothness of multiplication and inversion in the charts, i.e. as functions on the normed
-  `𝕜`-space `R`:  see `contDiffAt_ring_inverse` for the inversion result, and
-  `contDiff_mul` (needs to be generalized from field to algebra) for the multiplication
-  result
-* for an open embedding `f`, whose domain is equipped with the induced manifold structure
-  `f.singleton_smoothManifoldWithCorners`, characterization of smoothness of functions to/from
-  this manifold in terms of smoothness in the target space.  See the pair of lemmas
-  `ContMDiff_coe_sphere` and `ContMDiff.codRestrict_sphere` for a model.
-None of this should be particularly difficult.
-
 -/
 
 
@@ -70,4 +50,30 @@ variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 R]
 instance : SmoothManifoldWithCorners 𝓘(𝕜, R) Rˣ :=
   openEmbedding_val.singleton_smoothManifoldWithCorners 𝓘(𝕜, R)
 
+/-- For a complete normed ring `R`, the embedding of the units `Rˣ` into `R` is a smooth map between
+manifolds. -/
+lemma contMDiff_val {m : ℕ∞} : ContMDiff 𝓘(𝕜, R) 𝓘(𝕜, R) m (val : Rˣ → R) :=
+  contMDiff_openEmbedding 𝓘(𝕜, R) Units.openEmbedding_val
+
+/-- The units of a complete normed ring form a Lie group. -/
+instance : LieGroup 𝓘(𝕜, R) Rˣ where
+  smooth_mul := by
+    apply ContMDiff.of_comp_openEmbedding Units.openEmbedding_val
+    have : (val : Rˣ → R) ∘ (fun x : Rˣ × Rˣ => x.1 * x.2) =
+      (fun x : R × R => x.1 * x.2) ∘ (fun x : Rˣ × Rˣ => (x.1, x.2)) := by ext; simp
+    rw [this]
+    have : ContMDiff (𝓘(𝕜, R).prod 𝓘(𝕜, R)) 𝓘(𝕜, R × R) ∞
+      (fun x : Rˣ × Rˣ => ((x.1 : R), (x.2 : R))) :=
+      (contMDiff_val.comp contMDiff_fst).prod_mk_space (contMDiff_val.comp contMDiff_snd)
+    refine ContMDiff.comp ?_ this
+    rw [contMDiff_iff_contDiff]
+    exact contDiff_mul
+  smooth_inv := by
+    apply ContMDiff.of_comp_openEmbedding Units.openEmbedding_val
+    have : (val : Rˣ → R) ∘ (fun x : Rˣ => x⁻¹) = Ring.inverse ∘ val := by ext; simp
+    rw [this, ContMDiff]
+    refine fun x => ContMDiffAt.comp x ?_ (contMDiff_val x)
+    rw [contMDiffAt_iff_contDiffAt]
+    exact contDiffAt_ring_inverse _ _
+
 end Units

ef901ea6

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -52,7 +52,7 @@ open scoped Manifold
 
 namespace Units
 
-variable {R : Type _} [NormedRing R] [CompleteSpace R]
+variable {R : Type*} [NormedRing R] [CompleteSpace R]
 
 instance : ChartedSpace R Rˣ :=
   openEmbedding_val.singletonChartedSpace
@@ -65,7 +65,7 @@ theorem chartAt_source {a : Rˣ} : (chartAt R a).source = Set.univ :=
   rfl
 #align units.chart_at_source Units.chartAt_source
 
-variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 R]
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 R]
 
 instance : SmoothManifoldWithCorners 𝓘(𝕜, R) Rˣ :=
   openEmbedding_val.singleton_smoothManifoldWithCorners 𝓘(𝕜, R)

ef901ea6

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,15 +2,12 @@
 Copyright © 2021 Nicolò Cavalleri. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Nicolò Cavalleri, Heather Macbeth
-
-! This file was ported from Lean 3 source module geometry.manifold.instances.units_of_normed_algebra
-! leanprover-community/mathlib commit ef901ea68d3bb1dd08f8bc3034ab6b32b2e6ecdf
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
 import Mathlib.Analysis.NormedSpace.Units
 
+#align_import geometry.manifold.instances.units_of_normed_algebra from "leanprover-community/mathlib"@"ef901ea68d3bb1dd08f8bc3034ab6b32b2e6ecdf"
+
 /-!
 # Units of a normed algebra

ef901ea6

chore: formatting issues (#4947)

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

5068808d

Diff

@@ -30,9 +30,9 @@ its group of units, the general linear group GL(`𝕜`, `V`).
 The Lie group instance requires the following fields:
 ```
 instance : LieGroup 𝓘(𝕜, R) Rˣ :=
-{ Units.smoothManifoldWithCorners with
-  smooth_mul := sorry,
-  smooth_inv := sorry }
+  { Units.smoothManifoldWithCorners with
+    smooth_mul := sorry,
+    smooth_inv := sorry }
 ```
 
 The ingredients needed for the construction are

ef901ea6

feat: port Geometry.Manifold.Instances.UnitsOfNormedAlgebra (#4681)

20cb2f3d

`geometry.manifold.instances.units_of_normed_algebra` ⟷ `Mathlib.Geometry.Manifold.Instances.UnitsOfNormedAlgebra`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 12 + 875

geometry.manifold.instances.units_of_normed_algebra ⟷ Mathlib.Geometry.Manifold.Instances.UnitsOfNormedAlgebra

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 12 + 875

`geometry.manifold.instances.units_of_normed_algebra` ⟷ `Mathlib.Geometry.Manifold.Instances.UnitsOfNormedAlgebra`