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leanprover-community-bot committed Jul 10, 2023
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20 changes: 20 additions & 0 deletions Mathbin/Analysis/Normed/Group/SemiNormedGroup/Completion.lean
Expand Up @@ -44,6 +44,7 @@ open UniformSpace MulOpposite CategoryTheory NormedAddGroupHom

namespace SemiNormedGroupCat

#print SemiNormedGroupCat.completion /-
/-- The completion of a seminormed group, as an endofunctor on `SemiNormedGroup`. -/
@[simps]
def completion : SemiNormedGroupCat.{u} ⥤ SemiNormedGroupCat.{u}
Expand All @@ -53,11 +54,15 @@ def completion : SemiNormedGroupCat.{u} ⥤ SemiNormedGroupCat.{u}
map_id' V := completion_id
map_comp' U V W f g := (completion_comp f g).symm
#align SemiNormedGroup.Completion SemiNormedGroupCat.completion
-/

#print SemiNormedGroupCat.completion_completeSpace /-
instance completion_completeSpace {V : SemiNormedGroupCat} : CompleteSpace (completion.obj V) :=
Completion.completeSpace _
#align SemiNormedGroup.Completion_complete_space SemiNormedGroupCat.completion_completeSpace
-/

#print SemiNormedGroupCat.completion.incl /-
/-- The canonical morphism from a seminormed group `V` to its completion. -/
@[simps]
def completion.incl {V : SemiNormedGroupCat} : V ⟶ completion.obj V
Expand All @@ -66,18 +71,24 @@ def completion.incl {V : SemiNormedGroupCat} : V ⟶ completion.obj V
map_add' := Completion.coe_add
bound' := ⟨1, fun v => by simp⟩
#align SemiNormedGroup.Completion.incl SemiNormedGroupCat.completion.incl
-/

#print SemiNormedGroupCat.completion.norm_incl_eq /-
theorem completion.norm_incl_eq {V : SemiNormedGroupCat} {v : V} : ‖completion.incl v‖ = ‖v‖ := by
simp
#align SemiNormedGroup.Completion.norm_incl_eq SemiNormedGroupCat.completion.norm_incl_eq
-/

#print SemiNormedGroupCat.completion.map_normNoninc /-
theorem completion.map_normNoninc {V W : SemiNormedGroupCat} {f : V ⟶ W} (hf : f.NormNoninc) :
(completion.map f).NormNoninc :=
NormedAddGroupHom.NormNoninc.normNoninc_iff_norm_le_one.2 <|
(NormedAddGroupHom.norm_completion f).le.trans <|
NormedAddGroupHom.NormNoninc.normNoninc_iff_norm_le_one.1 hf
#align SemiNormedGroup.Completion.map_norm_noninc SemiNormedGroupCat.completion.map_normNoninc
-/

#print SemiNormedGroupCat.completion.mapHom /-
/-- Given a normed group hom `V ⟶ W`, this defines the associated morphism
from the completion of `V` to the completion of `W`.
The difference from the definition obtained from the functoriality of completion is in that the
Expand All @@ -86,11 +97,14 @@ def completion.mapHom (V W : SemiNormedGroupCat.{u}) :
(V ⟶ W) →+ (completion.obj V ⟶ completion.obj W) :=
AddMonoidHom.mk' (CategoryTheory.Functor.map completion) fun f g => f.completion_add g
#align SemiNormedGroup.Completion.map_hom SemiNormedGroupCat.completion.mapHom
-/

#print SemiNormedGroupCat.completion.map_zero /-
@[simp]
theorem completion.map_zero (V W : SemiNormedGroupCat) : completion.map (0 : V ⟶ W) = 0 :=
(completion.mapHom V W).map_zero
#align SemiNormedGroup.Completion.map_zero SemiNormedGroupCat.completion.map_zero
-/

instance : Preadditive SemiNormedGroupCat.{u}
where
Expand All @@ -104,6 +118,7 @@ instance : Preadditive SemiNormedGroupCat.{u}

instance : Functor.Additive completion where map_add' X Y := (completion.mapHom _ _).map_add

#print SemiNormedGroupCat.completion.lift /-
/-- Given a normed group hom `f : V → W` with `W` complete, this provides a lift of `f` to
the completion of `V`. The lemmas `lift_unique` and `lift_comp_incl` provide the api for the
universal property of the completion. -/
Expand All @@ -113,16 +128,21 @@ def completion.lift {V W : SemiNormedGroupCat} [CompleteSpace W] [SeparatedSpace
map_add' := f.extension.toAddMonoidHom.map_add'
bound' := f.extension.bound'
#align SemiNormedGroup.Completion.lift SemiNormedGroupCat.completion.lift
-/

#print SemiNormedGroupCat.completion.lift_comp_incl /-
theorem completion.lift_comp_incl {V W : SemiNormedGroupCat} [CompleteSpace W] [SeparatedSpace W]
(f : V ⟶ W) : completion.incl ≫ completion.lift f = f := by ext;
apply NormedAddGroupHom.extension_coe
#align SemiNormedGroup.Completion.lift_comp_incl SemiNormedGroupCat.completion.lift_comp_incl
-/

#print SemiNormedGroupCat.completion.lift_unique /-
theorem completion.lift_unique {V W : SemiNormedGroupCat} [CompleteSpace W] [SeparatedSpace W]
(f : V ⟶ W) (g : completion.obj V ⟶ W) : completion.incl ≫ g = f → g = completion.lift f :=
fun h => (NormedAddGroupHom.extension_unique _ fun v => ((ext_iff.1 h) v).symm).symm
#align SemiNormedGroup.Completion.lift_unique SemiNormedGroupCat.completion.lift_unique
-/

end SemiNormedGroupCat

8 changes: 4 additions & 4 deletions lake-manifest.json
Expand Up @@ -10,9 +10,9 @@
{"git":
{"url": "https://github.com/leanprover-community/lean3port.git",
"subDir?": null,
"rev": "492eaebaa2524b7c3933c05d5653b5507ca74528",
"rev": "4aca376101b0a20a9a1b01f13b4ea8b564934e91",
"name": "lean3port",
"inputRev?": "492eaebaa2524b7c3933c05d5653b5507ca74528"}},
"inputRev?": "4aca376101b0a20a9a1b01f13b4ea8b564934e91"}},
{"git":
{"url": "https://github.com/mhuisi/lean4-cli.git",
"subDir?": null,
Expand All @@ -22,9 +22,9 @@
{"git":
{"url": "https://github.com/leanprover-community/mathlib4.git",
"subDir?": null,
"rev": "02ea90a076686968ce1d3b3938193f66912f422f",
"rev": "278a8fe1522abc239723c43e7a7d24f3d36316ac",
"name": "mathlib",
"inputRev?": "02ea90a076686968ce1d3b3938193f66912f422f"}},
"inputRev?": "278a8fe1522abc239723c43e7a7d24f3d36316ac"}},
{"git":
{"url": "https://github.com/gebner/quote4",
"subDir?": null,
Expand Down
4 changes: 2 additions & 2 deletions lakefile.lean
Expand Up @@ -4,7 +4,7 @@ open Lake DSL System
-- Usually the `tag` will be of the form `nightly-2021-11-22`.
-- If you would like to use an artifact from a PR build,
-- it will be of the form `pr-branchname-sha`.
def tag : String := "nightly-2023-07-10-09"
def tag : String := "nightly-2023-07-10-11"
def releaseRepo : String := "leanprover-community/mathport"
def oleanTarName : String := "mathlib3-binport.tar.gz"

Expand Down Expand Up @@ -38,7 +38,7 @@ target fetchOleans (_pkg : Package) : Unit := do
untarReleaseArtifact releaseRepo tag oleanTarName libDir
return .nil

require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"492eaebaa2524b7c3933c05d5653b5507ca74528"
require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"4aca376101b0a20a9a1b01f13b4ea8b564934e91"

@[default_target]
lean_lib Mathbin where
Expand Down

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