state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case neg.inl
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exfalso | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case neg.inl.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X :... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact hf mf'.symm | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case neg.inr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact mf' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rw [@w' _ _ mX' mY' f' _] | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → X ∈ O → (... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply Finset.mem_of_mem_insert_of_ne mf' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → X ∈ O → (... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | contrapose! h | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → X ∈ O → (... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain ⟨rfl, h⟩ := h | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case refl
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} →... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | trivial | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.693_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
⊢ Nonempty (Cone F) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fun X : J =>
Finset.univ.biUnion fun Y : J =>
Finset.univ.image fun f : X ⟶ Y => ⟨F.obj X, F.obj Y, by simp, by simp, F.map f⟩
obtain ⟨Z, f, w⟩ := inf_exists O H
refin... | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
| Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
⊢ Nonempty (Cone F) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | let O := Finset.univ.image F.obj | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
| Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
⊢ Nonempty (Cone F) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fun X : J =>
Finset.univ.biUnion fun Y : J =>
Finset.univ.image fun f : X ⟶ Y => ⟨F.obj X, F.obj Y, by simp, by simp, F.map f⟩ | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
| Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
X Y : J
f : X ⟶ Y
⊢ F.obj X ∈ O | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
X Y : J
f : X ⟶ Y
⊢ F.obj Y ∈ O | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ :... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain ⟨Z, f, w⟩ := inf_exists O H | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | refine' ⟨⟨Z, ⟨fun X => f (by simp), _⟩⟩⟩ | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ :... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | intro j j' g | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | dsimp | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp only [Category.id_comp] | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | symm | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply w | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro.a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp only [Finset.mem_biUnion, Finset.mem_univ, Finset.mem_image,
PSigma.mk.injEq, true_and, exists_and_left] | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
case intro.intro.a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact ⟨j, rfl, j', g, by simp⟩ | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : IsCofiltered C
O✝ : Finset C
H✝ : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O✝) ×' (_ : Y ∈ O✝) ×' (X ⟶ Y))
J : Type w
inst✝¹ : SmallCategory J
inst✝ : FinCategory J
F : J ⥤ C
O : Finset C := Finset.image F.obj Finset.univ
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ :... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fu... | Mathlib.CategoryTheory.Filtered.Basic.754_0.dhnXC1TuYVuk8Vb | /-- If we have `IsCofiltered C`, then for any functor `F : J ⥤ C` with `FinCategory J`,
there exists a cone over `F`.
-/
theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ (IsFiltered.coeqHom f.unop g.unop).op ≫ f = (IsFiltered.coeqHom f.unop g.unop).op ≫ g | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rw [show f = f.unop.op by simp, show g = g.unop.op by simp, ← op_comp, ← op_comp] | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y :=
⟨op (IsFiltered.max X.unop Y.unop), (IsFiltered.leftToMax _ _).op,
(IsFiltered.rightToMax _ _).op, trivial⟩
cone_maps X Y f g :=
⟨op (IsFiltered.coeq f.unop g.unop), (IsFilte... | Mathlib.CategoryTheory.Filtered.Basic.810_0.dhnXC1TuYVuk8Vb | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ f = f.unop.op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y :=
⟨op (IsFiltered.max X.unop Y.unop), (IsFiltered.leftToMax _ _).op,
(IsFiltered.rightToMax _ _).op, trivial⟩
cone_maps X Y f g :=
⟨op (IsFiltered.coeq f.unop g.unop), (IsFilte... | Mathlib.CategoryTheory.Filtered.Basic.810_0.dhnXC1TuYVuk8Vb | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ g = g.unop.op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y :=
⟨op (IsFiltered.max X.unop Y.unop), (IsFiltered.leftToMax _ _).op,
(IsFiltered.rightToMax _ _).op, trivial⟩
cone_maps X Y f g :=
⟨op (IsFiltered.coeq f.unop g.unop), (IsFilte... | Mathlib.CategoryTheory.Filtered.Basic.810_0.dhnXC1TuYVuk8Vb | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ (f.unop ≫ IsFiltered.coeqHom f.unop.op.unop g.unop.op.unop).op =
(g.unop ≫ IsFiltered.coeqHom f.unop.op.unop g.unop.op.unop).op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | congr 1 | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y :=
⟨op (IsFiltered.max X.unop Y.unop), (IsFiltered.leftToMax _ _).op,
(IsFiltered.rightToMax _ _).op, trivial⟩
cone_maps X Y f g :=
⟨op (IsFiltered.coeq f.unop g.unop), (IsFilte... | Mathlib.CategoryTheory.Filtered.Basic.810_0.dhnXC1TuYVuk8Vb | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
case e_f
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ f.unop ≫ IsFiltered.coeqHom f.unop.op.unop g.unop.op.unop = g.unop ≫ IsFiltered.coeqHom f.unop.op.unop g.unop.op.unop | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact IsFiltered.coeq_condition f.unop g.unop | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y :=
⟨op (IsFiltered.max X.unop Y.unop), (IsFiltered.leftToMax _ _).op,
(IsFiltered.rightToMax _ _).op, trivial⟩
cone_maps X Y f g :=
⟨op (IsFiltered.coeq f.unop g.unop), (IsFilte... | Mathlib.CategoryTheory.Filtered.Basic.810_0.dhnXC1TuYVuk8Vb | instance isCofilteredOrEmpty_op_of_isFilteredOrEmpty [IsFilteredOrEmpty C] :
IsCofilteredOrEmpty Cᵒᵖ where
cone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ f ≫ (IsCofiltered.eqHom f.unop g.unop).op = g ≫ (IsCofiltered.eqHom f.unop g.unop).op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rw [show f = f.unop.op by simp, show g = g.unop.op by simp, ← op_comp, ← op_comp] | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y :=
⟨op (IsCofiltered.min X.unop Y.unop), (IsCofiltered.minToLeft X.unop Y.unop).op,
(IsCofiltered.minToRight X.unop Y.unop).op, trivial⟩
cocone_maps X Y f g :=
⟨op (IsCofilter... | Mathlib.CategoryTheory.Filtered.Basic.825_0.dhnXC1TuYVuk8Vb | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ f = f.unop.op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y :=
⟨op (IsCofiltered.min X.unop Y.unop), (IsCofiltered.minToLeft X.unop Y.unop).op,
(IsCofiltered.minToRight X.unop Y.unop).op, trivial⟩
cocone_maps X Y f g :=
⟨op (IsCofilter... | Mathlib.CategoryTheory.Filtered.Basic.825_0.dhnXC1TuYVuk8Vb | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ g = g.unop.op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y :=
⟨op (IsCofiltered.min X.unop Y.unop), (IsCofiltered.minToLeft X.unop Y.unop).op,
(IsCofiltered.minToRight X.unop Y.unop).op, trivial⟩
cocone_maps X Y f g :=
⟨op (IsCofilter... | Mathlib.CategoryTheory.Filtered.Basic.825_0.dhnXC1TuYVuk8Vb | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ (IsCofiltered.eqHom f.unop.op.unop g.unop.op.unop ≫ f.unop).op =
(IsCofiltered.eqHom f.unop.op.unop g.unop.op.unop ≫ g.unop).op | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | congr 1 | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y :=
⟨op (IsCofiltered.min X.unop Y.unop), (IsCofiltered.minToLeft X.unop Y.unop).op,
(IsCofiltered.minToRight X.unop Y.unop).op, trivial⟩
cocone_maps X Y f g :=
⟨op (IsCofilter... | Mathlib.CategoryTheory.Filtered.Basic.825_0.dhnXC1TuYVuk8Vb | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
case e_f
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
X Y : Cᵒᵖ
f g : X ⟶ Y
⊢ IsCofiltered.eqHom f.unop.op.unop g.unop.op.unop ≫ f.unop = IsCofiltered.eqHom f.unop.op.unop g.unop.op.unop ≫ g.unop | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact IsCofiltered.eq_condition f.unop g.unop | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y :=
⟨op (IsCofiltered.min X.unop Y.unop), (IsCofiltered.minToLeft X.unop Y.unop).op,
(IsCofiltered.minToRight X.unop Y.unop).op, trivial⟩
cocone_maps X Y f g :=
⟨op (IsCofilter... | Mathlib.CategoryTheory.Filtered.Basic.825_0.dhnXC1TuYVuk8Vb | instance isFilteredOrEmpty_op_of_isCofilteredOrEmpty [IsCofilteredOrEmpty C] :
IsFilteredOrEmpty Cᵒᵖ where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
n : ℕ
⊢ range (succ n) = n ::ₘ range n | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | rw [range, List.range_succ, ← coe_add, add_comm] | @[simp]
theorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n := by
| Mathlib.Data.Multiset.Range.34_0.3IYEfNCsb0H7K9Z | @[simp]
theorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n | Mathlib_Data_Multiset_Range |
n : ℕ
⊢ ↑[n] + ↑(List.range n) = n ::ₘ range n | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | rfl | @[simp]
theorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n := by
rw [range, List.range_succ, ← coe_add, add_comm]; | Mathlib.Data.Multiset.Range.34_0.3IYEfNCsb0H7K9Z | @[simp]
theorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n | Mathlib_Data_Multiset_Range |
a : ℕ
m : Multiset ℕ
⊢ Disjoint (range a) (map (fun x => a + x) m) | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | intro x hxa hxb | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) := by
| Mathlib.Data.Multiset.Range.66_0.3IYEfNCsb0H7K9Z | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) | Mathlib_Data_Multiset_Range |
a : ℕ
m : Multiset ℕ
x : ℕ
hxa : x ∈ range a
hxb : x ∈ map (fun x => a + x) m
⊢ False | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | rw [range, mem_coe, List.mem_range] at hxa | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) := by
intro x hxa hxb
| Mathlib.Data.Multiset.Range.66_0.3IYEfNCsb0H7K9Z | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) | Mathlib_Data_Multiset_Range |
a : ℕ
m : Multiset ℕ
x : ℕ
hxa : x < a
hxb : x ∈ map (fun x => a + x) m
⊢ False | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | obtain ⟨c, _, rfl⟩ := mem_map.1 hxb | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) := by
intro x hxa hxb
rw [range, mem_coe, List.mem_range] at hxa
| Mathlib.Data.Multiset.Range.66_0.3IYEfNCsb0H7K9Z | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) | Mathlib_Data_Multiset_Range |
case intro.intro
a : ℕ
m : Multiset ℕ
c : ℕ
left✝ : c ∈ m
hxa : a + c < a
hxb : a + c ∈ map (fun x => a + x) m
⊢ False | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | exact (self_le_add_right _ _).not_lt hxa | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) := by
intro x hxa hxb
rw [range, mem_coe, List.mem_range] at hxa
obtain ⟨c, _, rfl⟩ := mem_map.1 hxb
| Mathlib.Data.Multiset.Range.66_0.3IYEfNCsb0H7K9Z | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) | Mathlib_Data_Multiset_Range |
a b : ℕ
⊢ range (a + b) = range a ∪ map (fun x => a + x) (range b) | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | rw [range_add, add_eq_union_iff_disjoint] | theorem range_add_eq_union (a b : ℕ) : range (a + b) = range a ∪ (range b).map (a + ·) := by
| Mathlib.Data.Multiset.Range.74_0.3IYEfNCsb0H7K9Z | theorem range_add_eq_union (a b : ℕ) : range (a + b) = range a ∪ (range b).map (a + ·) | Mathlib_Data_Multiset_Range |
a b : ℕ
⊢ Disjoint (range a) (map (fun x => a + x) (range b)) | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.List.Range
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7... | apply range_disjoint_map_add | theorem range_add_eq_union (a b : ℕ) : range (a + b) = range a ∪ (range b).map (a + ·) := by
rw [range_add, add_eq_union_iff_disjoint]
| Mathlib.Data.Multiset.Range.74_0.3IYEfNCsb0H7K9Z | theorem range_add_eq_union (a b : ℕ) : range (a + b) = range a ∪ (range b).map (a + ·) | Mathlib_Data_Multiset_Range |
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X✝ B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
X Y : WidePullbackShape J
f : X ⟶ Y
⊢ ((Functor.const (WidePullbackShape J)).obj c.pt.left).map f ≫
(fun X => Option.casesOn X c.pt.hom fun j => (c.π.app { as := j }).left) Y =
(fun X => Option.casesOn X c.... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | dsimp | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X✝ B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
X Y : WidePullbackShape J
f : X ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) Y =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) X ≫
WidePullbackShape.Hom... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases X | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case none
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
Y : WidePullbackShape J
f : none ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) Y =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none ≫
WidePull... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases Y | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
Y : WidePullbackShape J
val✝ : J
f : some val✝ ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) Y =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some v... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases Y | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case none.none
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
f : none ⟶ none
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none ≫
WidePullbackShape.Hom... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases f | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case none.some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
val✝ : J
f : none ⟶ some val✝
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some val✝) =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none ≫
... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases f | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some.none
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
val✝ : J
f : some val✝ ⟶ none
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some val✝) ≫
... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases f | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some.some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
val✝¹ val✝ : J
f : some val✝¹ ⟶ some val✝
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some val✝) =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases f | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case none.none.id
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none ≫
WidePullbackShape.Hom.rec (motive ... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rw [Category.id_comp, Category.comp_id] | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some.none.term
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
val✝ : J
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) none =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some val✝) ≫
WidePullbackShap... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rw [Over.w, Category.id_comp] | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some.some.id
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
val✝ : J
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some val✝) =
Option.rec c.pt.hom (fun val => (c.π.app { as := val }).left) (some val✝) ≫
WidePullbac... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rw [Category.id_comp, Category.comp_id] | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt := c.pt.left
π :=
{ app := fun X => Option.casesOn X c.pt.hom fun j : J => (c.π.app ⟨j⟩).left
-... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.46_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverseObj (B : C) {J : Type w} (F : Discrete J ⥤ Over B) (c : Cone F) :
Cone (widePullbackDiagramOfDiagramOver B F) where
pt | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
X✝ Y✝ : Cone F
f : X✝ ⟶ Y✝
j : WidePullbackShape J
⊢ f.hom.left ≫ (conesEquivInverseObj B F Y✝).π.app j = (conesEquivInverseObj B F X✝).π.app j | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | cases' j with j | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj := conesEquivInverseObj B F
map f :=
{ hom := f.hom.left
w := fun j => by
| Mathlib.CategoryTheory.Limits.Constructions.Over.Products.61_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case none
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
X✝ Y✝ : Cone F
f : X✝ ⟶ Y✝
⊢ f.hom.left ≫ (conesEquivInverseObj B F Y✝).π.app none = (conesEquivInverseObj B F X✝).π.app none | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | simp | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj := conesEquivInverseObj B F
map f :=
{ hom := f.hom.left
w := fun j => by
cases' j with j
... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.61_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
X✝ Y✝ : Cone F
f : X✝ ⟶ Y✝
j : J
⊢ f.hom.left ≫ (conesEquivInverseObj B F Y✝).π.app (some j) = (conesEquivInverseObj B F X✝).π.app (some j) | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | dsimp | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj := conesEquivInverseObj B F
map f :=
{ hom := f.hom.left
w := fun j => by
cases' j with j
... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.61_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
X✝ Y✝ : Cone F
f : X✝ ⟶ Y✝
j : J
⊢ f.hom.left ≫ (Y✝.π.app { as := j }).left = (X✝.π.app { as := j }).left | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rw [← f.w ⟨j⟩] | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj := conesEquivInverseObj B F
map f :=
{ hom := f.hom.left
w := fun j => by
cases' j with j
... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.61_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
J : Type w
F : Discrete J ⥤ Over B
X✝ Y✝ : Cone F
f : X✝ ⟶ Y✝
j : J
⊢ f.hom.left ≫ (Y✝.π.app { as := j }).left = (f.hom ≫ Y✝.π.app { as := j }).left | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rfl | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj := conesEquivInverseObj B F
map f :=
{ hom := f.hom.left
w := fun j => by
cases' j with j
... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.61_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivInverse (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone F ⥤ Cone (widePullbackDiagramOfDiagramOver B F) where
obj | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X✝ B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone (widePullbackDiagramOfDiagramOver B F)
x✝² x✝¹ : Discrete J
X Y : J
x✝ : { as := X } ⟶ { as := Y }
f : { as := X }.as = { as := Y }.as
⊢ ((Functor.const (Discrete J)).obj (mk (c.π.app none))).map { down := { down := f ... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | dsimp at f ⊢ | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivFunctor (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone (widePullbackDiagramOfDiagramOver B F) ⥤ Cone F where
obj c :=
{ pt := Over.mk (c.π.app none)
π :=
{ app := fun ⟨j⟩ => Over.homMk (c.π.app (some j)) (c.w... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.80_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivFunctor (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone (widePullbackDiagramOfDiagramOver B F) ⥤ Cone F where
obj c | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X✝ B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone (widePullbackDiagramOfDiagramOver B F)
x✝² x✝¹ : Discrete J
X Y : J
x✝ : { as := X } ⟶ { as := Y }
f : X = Y
⊢ 𝟙 (mk (c.π.app none)) ≫ homMk (c.π.app (some Y)) = homMk (c.π.app (some X)) ≫ F.map { down := { down := f ... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | aesop_cat | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivFunctor (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone (widePullbackDiagramOfDiagramOver B F) ⥤ Cone F where
obj c :=
{ pt := Over.mk (c.π.app none)
π :=
{ app := fun ⟨j⟩ => Over.homMk (c.π.app (some j)) (c.w... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.80_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simps]
def conesEquivFunctor (B : C) {J : Type w} (F : Discrete J ⥤ Over B) :
Cone (widePullbackDiagramOfDiagramOver B F) ⥤ Cone F where
obj c | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete J ⥤ Over B
x✝ : Cone (widePullbackDiagramOfDiagramOver B F)
⊢ ∀ (j : WidePullbackShape J),
((𝟭 (Cone (widePullbackDiagramOfDiagramOver B F))).obj x✝).π.app j =
(Iso.mk (𝟙 ((𝟭 (Cone (widePullbackDiagramOfDiagramOver B F))).obj x✝).pt)
... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rintro (j | j) | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simp]
def conesEquivUnitIso (B : C) (F : Discrete J ⥤ Over B) :
𝟭 (Cone (widePullbackDiagramOfDiagramOver B F)) ≅
conesEquivFunctor B F ⋙ conesEquivInverse B F :=
NatIso.ofComponents fun _ => Cones.ext
{ hom := 𝟙 _
inv := 𝟙 _ }
(by... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.98_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simp]
def conesEquivUnitIso (B : C) (F : Discrete J ⥤ Over B) :
𝟭 (Cone (widePullbackDiagramOfDiagramOver B F)) ≅
conesEquivFunctor B F ⋙ conesEquivInverse B F | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case none
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete J ⥤ Over B
x✝ : Cone (widePullbackDiagramOfDiagramOver B F)
⊢ ((𝟭 (Cone (widePullbackDiagramOfDiagramOver B F))).obj x✝).π.app none =
(Iso.mk (𝟙 ((𝟭 (Cone (widePullbackDiagramOfDiagramOver B F))).obj x✝).pt)
(𝟙 ((conesEquiv... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | aesop_cat | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simp]
def conesEquivUnitIso (B : C) (F : Discrete J ⥤ Over B) :
𝟭 (Cone (widePullbackDiagramOfDiagramOver B F)) ≅
conesEquivFunctor B F ⋙ conesEquivInverse B F :=
NatIso.ofComponents fun _ => Cones.ext
{ hom := 𝟙 _
inv := 𝟙 _ }
(by... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.98_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simp]
def conesEquivUnitIso (B : C) (F : Discrete J ⥤ Over B) :
𝟭 (Cone (widePullbackDiagramOfDiagramOver B F)) ≅
conesEquivFunctor B F ⋙ conesEquivInverse B F | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case some
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete J ⥤ Over B
x✝ : Cone (widePullbackDiagramOfDiagramOver B F)
j : J
⊢ ((𝟭 (Cone (widePullbackDiagramOfDiagramOver B F))).obj x✝).π.app (some j) =
(Iso.mk (𝟙 ((𝟭 (Cone (widePullbackDiagramOfDiagramOver B F))).obj x✝).pt)
(𝟙 ((... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | aesop_cat | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simp]
def conesEquivUnitIso (B : C) (F : Discrete J ⥤ Over B) :
𝟭 (Cone (widePullbackDiagramOfDiagramOver B F)) ≅
conesEquivFunctor B F ⋙ conesEquivInverse B F :=
NatIso.ofComponents fun _ => Cones.ext
{ hom := 𝟙 _
inv := 𝟙 _ }
(by... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.98_0.USyhUKjMzubPAtG | /-- (Impl) A preliminary definition to avoid timeouts. -/
@[simp]
def conesEquivUnitIso (B : C) (F : Discrete J ⥤ Over B) :
𝟭 (Cone (widePullbackDiagramOfDiagramOver B F)) ≅
conesEquivFunctor B F ⋙ conesEquivInverse B F | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete PEmpty.{1} ⥤ Over B
s : Cone F
m : s.pt ⟶ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.pt
x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.π.app j = s.π.app j
⊢ m = (fun s => homM... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | simp only | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.165_0.USyhUKjMzubPAtG | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete PEmpty.{1} ⥤ Over B
s : Cone F
m : s.pt ⟶ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.pt
x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.π.app j = s.π.app j
⊢ m = homMk s.pt.hom | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | ext | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.165_0.USyhUKjMzubPAtG | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case h
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete PEmpty.{1} ⥤ Over B
s : Cone F
m : s.pt ⟶ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.pt
x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.π.app j = s.π.app j
⊢ m.left = (h... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rw [Over.homMk_left _] | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.165_0.USyhUKjMzubPAtG | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case h
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete PEmpty.{1} ⥤ Over B
s : Cone F
m : s.pt ⟶ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.pt
x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.π.app j = s.π.app j
⊢ m.left = s.... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | have := m.w | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.165_0.USyhUKjMzubPAtG | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case h
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete PEmpty.{1} ⥤ Over B
s : Cone F
m : s.pt ⟶ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.pt
x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.π.app j = s.π.app j
this :
(𝟭 ... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | dsimp at this | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.165_0.USyhUKjMzubPAtG | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
case h
J : Type w
C : Type u
inst✝ : Category.{v, u} C
X B : C
F : Discrete PEmpty.{1} ⥤ Over B
s : Cone F
m : s.pt ⟶ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.pt
x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := NatTrans.mk fun p => PEmpty.elim p.as }.π.app j = s.π.app j
this : m.left... | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.WidePu... | rwa [Category.comp_id, Category.comp_id] at this | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib.CategoryTheory.Limits.Constructions.Over.Products.165_0.USyhUKjMzubPAtG | /-- Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_widePullback` above.)
-/
theorem over_hasTerminal (B : C) : HasTermin... | Mathlib_CategoryTheory_Limits_Constructions_Over_Products |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
U : (Opens ↑X)ᵒᵖ
⊢ { obj := fun U => CommRingCat.of (Localization (obj G U)),
map := fun {U V} i =>
CommRingCat.ofHom
(IsLocalization.map (Localization (obj ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | simp_rw [F.map_id] | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U := CommRingCat.of <| Localization (G.obj U)
map {U V} i := CommRingCat.ofHom <| IsLocalization.map _ (F.map i) (G.map i)... | Mathlib.Topology.Sheaves.Operations.51_0.VAfusCW1hNuysqq | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
U : (Opens ↑X)ᵒᵖ
⊢ CommRingCat.ofHom
(IsLocalization.map (Localization (obj G U)) (𝟙 (F.obj U))
(_ : obj G U ≤ Submonoid.comap (𝟙 (F.obj U)) (obj G U))) =
𝟙 (CommRingCat.of... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | ext x | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U := CommRingCat.of <| Localization (G.obj U)
map {U V} i := CommRingCat.ofHom <| IsLocalization.map _ (F.map i) (G.map i)... | Mathlib.Topology.Sheaves.Operations.51_0.VAfusCW1hNuysqq | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U | Mathlib_Topology_Sheaves_Operations |
case w
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
U : (Opens ↑X)ᵒᵖ
x : (forget CommRingCat).obj (CommRingCat.of (Localization (obj G U)))
⊢ (CommRingCat.ofHom
(IsLocalization.map (Localization (obj G U)) (𝟙 (F.obj U))
... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | exact IsLocalization.map_id (M := G.obj U) (S := Localization (G.obj U)) x | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U := CommRingCat.of <| Localization (G.obj U)
map {U V} i := CommRingCat.ofHom <| IsLocalization.map _ (F.map i) (G.map i)... | Mathlib.Topology.Sheaves.Operations.51_0.VAfusCW1hNuysqq | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
U V W : (Opens ↑X)ᵒᵖ
i : U ⟶ V
j : V ⟶ W
⊢ { obj := fun U => CommRingCat.of (Localization (obj G U)),
map := fun {U V} i =>
CommRingCat.ofHom
(IsLocalization... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | delta CommRingCat.ofHom CommRingCat.of Bundled.of | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U := CommRingCat.of <| Localization (G.obj U)
map {U V} i := CommRingCat.ofHom <| IsLocalization.map _ (F.map i) (G.map i)... | Mathlib.Topology.Sheaves.Operations.51_0.VAfusCW1hNuysqq | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
U V W : (Opens ↑X)ᵒᵖ
i : U ⟶ V
j : V ⟶ W
⊢ { obj := fun U => Bundled.mk (Localization (obj G U)),
map := fun {U V} i =>
IsLocalization.map (Localization (obj G V)) (F.map ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | simp_rw [F.map_comp, CommRingCat.comp_eq_ring_hom_comp] | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U := CommRingCat.of <| Localization (G.obj U)
map {U V} i := CommRingCat.ofHom <| IsLocalization.map _ (F.map i) (G.map i)... | Mathlib.Topology.Sheaves.Operations.51_0.VAfusCW1hNuysqq | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
U V W : (Opens ↑X)ᵒᵖ
i : U ⟶ V
j : V ⟶ W
⊢ IsLocalization.map (Localization (obj G W)) (RingHom.comp (F.map j) (F.map i))
(_ : obj G U ≤ Submonoid.comap (RingHom.comp (F.map j) (F.map i))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | rw [IsLocalization.map_comp_map] | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U := CommRingCat.of <| Localization (G.obj U)
map {U V} i := CommRingCat.ofHom <| IsLocalization.map _ (F.map i) (G.map i)... | Mathlib.Topology.Sheaves.Operations.51_0.VAfusCW1hNuysqq | /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
S : (x : ↑X) → Submonoid ↑(stalk F x)
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
⊢ (fun U => ⨅ x, Submonoid.comap (germ F x) (S ↑x)) U ≤
Submonoid.comap (F.map i) ((fun U => ⨅ x, Submonoid.comap (germ F ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | intro s hs | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U := ⨅ x : U.unop, Subm... | Mathlib.Topology.Sheaves.Operations.86_0.VAfusCW1hNuysqq | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
S : (x : ↑X) → Submonoid ↑(stalk F x)
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
s : (forget CommRingCat).obj (F.obj U)
hs : s ∈ (fun U => ⨅ x, Submonoid.comap (germ F x) (S ↑x)) U
⊢ s ∈ Submonoid.comap (F.m... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | simp only [Submonoid.mem_comap, Submonoid.mem_iInf] at hs ⊢ | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U := ⨅ x : U.unop, Subm... | Mathlib.Topology.Sheaves.Operations.86_0.VAfusCW1hNuysqq | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
S : (x : ↑X) → Submonoid ↑(stalk F x)
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
s : (forget CommRingCat).obj (F.obj U)
hs : ∀ (i : ↥U.unop), (germ F i) s ∈ S ↑i
⊢ ∀ (i_1 : ↥V.unop), (germ F i_1) ((F.map i) ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | intro x | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U := ⨅ x : U.unop, Subm... | Mathlib.Topology.Sheaves.Operations.86_0.VAfusCW1hNuysqq | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
S : (x : ↑X) → Submonoid ↑(stalk F x)
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
s : (forget CommRingCat).obj (F.obj U)
hs : ∀ (i : ↥U.unop), (germ F i) s ∈ S ↑i
x : ↥V.unop
⊢ (germ F x) ((F.map i) s) ∈ S ↑x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | change (F.map i.unop.op ≫ F.germ x) s ∈ _ | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U := ⨅ x : U.unop, Subm... | Mathlib.Topology.Sheaves.Operations.86_0.VAfusCW1hNuysqq | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
S : (x : ↑X) → Submonoid ↑(stalk F x)
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
s : (forget CommRingCat).obj (F.obj U)
hs : ∀ (i : ↥U.unop), (germ F i) s ∈ S ↑i
x : ↥V.unop
⊢ (F.map i.unop.op ≫ germ F x) s ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | rw [F.germ_res] | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U := ⨅ x : U.unop, Subm... | Mathlib.Topology.Sheaves.Operations.86_0.VAfusCW1hNuysqq | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F : Presheaf CommRingCat X
G : SubmonoidPresheaf F
S : (x : ↑X) → Submonoid ↑(stalk F x)
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
s : (forget CommRingCat).obj (F.obj U)
hs : ∀ (i : ↥U.unop), (germ F i) s ∈ S ↑i
x : ↥V.unop
⊢ (germ F ((fun x => { val := ↑x,... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | exact hs _ | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U := ⨅ x : U.unop, Subm... | Mathlib.Topology.Sheaves.Operations.86_0.VAfusCW1hNuysqq | /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
F.SubmonoidPresheaf where
obj U | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
⊢ Mono (toTotalQuotientPresheaf (Sheaf.presheaf F)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U) | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
| Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
this : ∀ (U : (Opens ↑X)ᵒᵖ), Mono ((toTotalQuotientPresheaf (Sheaf.presheaf F)).app U)
⊢ Mono (toTotalQuotientPresheaf (Sheaf.presheaf F)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | apply NatTrans.mono_of_mono_app | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
⊢ ∀ (U : (Opens ↑X)ᵒᵖ), Mono ((toTotalQuotientPresheaf (Sheaf.presheaf F)).app U) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | intro U | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
⊢ Mono ((toTotalQuotientPresheaf (Sheaf.presheaf F)).app U) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | apply ConcreteCategory.mono_of_injective | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
⊢ Function.Injective ⇑((toTotalQuotientPresheaf (Sheaf.presheaf F)).app U) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | dsimp [toTotalQuotientPresheaf, CommRingCat.ofHom] | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
⊢ Function.Injective
⇑(algebraMap (↑((Sheaf.presheaf F).obj U))
(Localization (⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | set m := _ | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : ?m.64484 := ?m.64485
⊢ Function.Injective
⇑(algebraMap (↑((Sheaf.presheaf F).obj U))
(Localization (⨅ x, Submonoid.comap (... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | change Function.Injective (algebraMap _ (Localization m)) | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sheaf.... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | change Function.Injective (algebraMap (F.presheaf.obj U) _) | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sheaf.... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | haveI : IsLocalization _ (Localization m) := Localization.isLocalization | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sheaf.... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | refine IsLocalization.injective (M := m) (S := Localization m) ?_ | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sheaf.... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | intro s hs t e | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sheaf.... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | apply section_ext F (unop U) | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i.h
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Shea... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | intro x | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i.h
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Shea... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | rw [map_zero] | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i.h
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Shea... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | apply Submonoid.mem_iInf.mp hs x | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i.h.a
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sh... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | dsimp | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
case this.i.h.a
X : TopCat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ConcreteCategory C
F✝ : Presheaf CommRingCat X
G : SubmonoidPresheaf F✝
F : Sheaf CommRingCat X
U : (Opens ↑X)ᵒᵖ
m : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=
⨅ x, Submonoid.comap (germ (Sheaf.presheaf F) x) (↑(stalk (Sh... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Topology.Sh... | rw [← map_mul, e, map_zero] | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf := by
-- Porting note : was an `apply (config := { instances := false })`
-- See https://github.com/leanprover/lean4/issues/2273
suffices : ∀ (U : (Opens ↑X)ᵒᵖ), Mono (F.presheaf.toTotalQuotientPresheaf.app U)
· apply NatTrans.mono... | Mathlib.Topology.Sheaves.Operations.117_0.VAfusCW1hNuysqq | instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf | Mathlib_Topology_Sheaves_Operations |
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace β
l l' : Filter α
f✝ g : α → β
μ ν : Measure α
f : α → β
⊢ AEStronglyMeasurable f (Measure.restrict μ ∅) | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | simp | @[simp]
theorem stronglyMeasurableAt_bot {f : α → β} : StronglyMeasurableAtFilter f ⊥ μ :=
⟨∅, mem_bot, by | Mathlib.MeasureTheory.Integral.IntegrableOn.42_0.qIpN2P2TD1gUH4J | @[simp]
theorem stronglyMeasurableAt_bot {f : α → β} : StronglyMeasurableAtFilter f ⊥ μ | Mathlib_MeasureTheory_Integral_IntegrableOn |
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace β
l l' : Filter α
f g : α → β
μ ν : Measure α
h : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable f (Measure.restrict μ univ) | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | rwa [Measure.restrict_univ] | protected theorem MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter
(h : AEStronglyMeasurable f μ) : StronglyMeasurableAtFilter f l μ :=
⟨univ, univ_mem, by | Mathlib.MeasureTheory.Integral.IntegrableOn.58_0.qIpN2P2TD1gUH4J | protected theorem MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter
(h : AEStronglyMeasurable f μ) : StronglyMeasurableAtFilter f l μ | Mathlib_MeasureTheory_Integral_IntegrableOn |
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