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 mk.h.associator_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
⊢ (fun p => normalizeAux p (f✝ ≫ g✝ ≫ h✝)) = fun p => normalizeAux p ((f✝ ≫ g✝) ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.associator_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
⊢ (fun p => normalizeAux p (f✝ ≫ g✝ ≫ h✝)) = fun p => normalizeAux p ((f✝ ≫ g✝) ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.associator_inv.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
x✝ : Path a a✝
⊢ normalizeAux x✝ (f✝ ≫ g✝ ≫ h✝) = normalizeAux x✝ ((f✝ ≫ g✝) ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.right_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p (f✝ ≫ 𝟙 b✝)) = fun p => normalizeAux p f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.right_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p (f✝ ≫ 𝟙 b✝)) = fun p => normalizeAux p f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.right_unitor.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
x✝ : Path a a✝
⊢ normalizeAux x✝ (f✝ ≫ 𝟙 b✝) = normalizeAux x✝ f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.right_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p (f✝ ≫ 𝟙 b✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.right_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p (f✝ ≫ 𝟙 b✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.right_unitor_inv.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
x✝ : Path a a✝
⊢ normalizeAux x✝ f✝ = normalizeAux x✝ (f✝ ≫ 𝟙 b✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.left_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p (𝟙 a✝ ≫ f✝)) = fun p => normalizeAux p f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.left_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p (𝟙 a✝ ≫ f✝)) = fun p => normalizeAux p f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.left_unitor.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
x✝ : Path a a✝
⊢ normalizeAux x✝ (𝟙 a✝ ≫ f✝) = normalizeAux x✝ f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.left_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p (𝟙 a✝ ≫ f✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.left_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p (𝟙 a✝ ≫ f✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.left_unitor_inv.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
x✝ : Path a a✝
⊢ normalizeAux x✝ f✝ = normalizeAux x✝ (𝟙 a✝ ≫ f✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η : f ⟶ g
⊢ (↑(preinclusion B)).map { as := p } ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
PrelaxFunctor.map₂ (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g })) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rcases η with ⟨η'⟩ | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η : f ⟶ g
η' : Hom₂ f g
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
PrelaxFunctor.map₂ (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g })... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | clear η | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η' : Hom₂ f g
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
PrelaxFunctor.map₂ (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g })) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction η' with
| id => simp
| vcomp η θ ihf ihg =>
simp only [mk_vcomp, Bicategory.whiskerLeft_comp]
slice_lhs 2 3 => rw [ihg]
slice_lhs 1 2 => rw [ihf]
simp
-- p ≠ nil required! See the docstring of `normalizeAux`.
| whisker_left _ _ ih =>
dsimp
rw [associator_inv_naturality_right_as... | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η' : Hom₂ f g
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
PrelaxFunctor.map₂ (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g })) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction η' with
| id => simp
| vcomp η θ ihf ihg =>
simp only [mk_vcomp, Bicategory.whiskerLeft_comp]
slice_lhs 2 3 => rw [ihg]
slice_lhs 1 2 => rw [ihf]
simp
-- p ≠ nil required! See the docstring of `normalizeAux`.
| whisker_left _ _ ih =>
dsimp
rw [associator_inv_naturality_right_as... | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.id
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.id f✝) ≫ (normalizeIso p f✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) (eqToHom (_ : { as := normalizeAux p f✝ ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | id => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.id
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.id f✝) ≫ (normalizeIso p f✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) (eqToHom (_ : { as := normalizeAux p f✝ ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclus... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | vcomp η θ ihf ihg =>
simp only [mk_vcomp, Bicategory.whiskerLeft_comp]
slice_lhs 2 3 => rw [ihg]
slice_lhs 1 2 => rw [ihf]
simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclus... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp only [mk_vcomp, Bicategory.whiskerLeft_comp] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclus... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | slice_lhs 2 3 => rw [ihg] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case a
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [ihg] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case a
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [ihg] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case a
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [ihg] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclus... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | slice_lhs 1 2 => rw [ihf] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case a
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [ihf] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case a
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [ihf] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case a
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B) ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [ihf] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η : Hom₂ f✝ g✝
θ : Hom₂ g✝ h✝
ihf :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclus... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_left
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih :
∀ (p : Path a b✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η✝ ≫ (normalizeIso p h✝).hom =
(normalizeIso p g✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | whisker_left _ _ ih =>
dsimp
rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]
simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_left
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih :
∀ (p : Path a b✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η✝ ≫ (normalizeIso p h✝).hom =
(normalizeIso p g✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | dsimp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_left
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih :
∀ (p : Path a b✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η✝ ≫ (normalizeIso p h✝).hom =
(normalizeIso p g✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_left
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih :
∀ (p : Path a b✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η✝ ≫ (normalizeIso p h✝).hom =
(normalizeIso p g✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | whisker_right h η' ih =>
dsimp
rw [associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, ih, comp_whiskerRight]
have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ η'))
dsimp at this; simp [this] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | dsimp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, ih, comp_whiskerRight] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ η')) | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | dsimp at this | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (pre... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp [this] | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.associator
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.associator f✝ g✝ h✝) ≫ (normalizeIso p (f✝ ≫ g✝ ≫ h✝)).hom =
(normalizeIso p ((f✝ ≫ g✝) ≫ h✝)).hom ≫
... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.associator
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.associator f✝ g✝ h✝) ≫ (normalizeIso p (f✝ ≫ g✝ ≫ h✝)).hom =
(normalizeIso p ((f✝ ≫ g✝) ≫ h✝)).hom ≫
... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.associator_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.associator_inv f✝ g✝ h✝) ≫
(normalizeIso p ((f✝ ≫ g✝) ≫ h✝)).hom =
(normalizeIso p (f✝ ≫ g✝ ≫... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.associator_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.associator_inv f✝ g✝ h✝) ≫
(normalizeIso p ((f✝ ≫ g✝) ≫ h✝)).hom =
(normalizeIso p (f✝ ≫ g✝ ≫... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.right_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.right_unitor f✝) ≫ (normalizeIso p f✝).hom =
(normalizeIso p (f✝ ≫ 𝟙 b✝)).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
(e... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.right_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.right_unitor f✝) ≫ (normalizeIso p f✝).hom =
(normalizeIso p (f✝ ≫ 𝟙 b✝)).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
(e... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.right_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.right_unitor_inv f✝) ≫ (normalizeIso p (f✝ ≫ 𝟙 b✝)).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.right_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.right_unitor_inv f✝) ≫ (normalizeIso p (f✝ ≫ 𝟙 b✝)).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.left_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.left_unitor f✝) ≫ (normalizeIso p f✝).hom =
(normalizeIso p (𝟙 a✝ ≫ f✝)).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
(eqT... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.left_unitor
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.left_unitor f✝) ≫ (normalizeIso p f✝).hom =
(normalizeIso p (𝟙 a✝ ≫ f✝)).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
(eqT... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.left_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.left_unitor_inv f✝) ≫ (normalizeIso p (𝟙 a✝ ≫ f✝)).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.left_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
p : Path a a✝
⊢ (↑(preinclusion B)).map { as := p } ◁ Quot.mk Rel (Hom₂.left_unitor_inv f✝) ≫ (normalizeIso p (𝟙 a✝ ≫ f✝)).hom =
(normalizeIso p f✝).hom ≫
PrelaxFunctor.map₂ (preinclusion B)
... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | simp | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib.CategoryTheory.Bicategory.Coherence.160_0.scNCB7gGNV3iY0Z | /-- The 2-isomorphism `normalizeIso p f` is natural in `f`. -/
theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η... | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b c : B
f : Hom a b
g : Hom b c
⊢ normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction g generalizing a with
| id => rfl
| of => rfl
| comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc] | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
| Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b c : B
f : Hom a b
g : Hom b c
⊢ normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction g generalizing a with
| id => rfl
| of => rfl
| comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc] | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
| Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
case id
B : Type u
inst✝ : Quiver B
b c a✝ a : B
f : Hom a a✝
⊢ normalizeAux nil (Hom.comp f (Hom.id a✝)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.id a✝)) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | id => rfl | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with
| Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
case id
B : Type u
inst✝ : Quiver B
b c a✝ a : B
f : Hom a a✝
⊢ normalizeAux nil (Hom.comp f (Hom.id a✝)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.id a✝)) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with
| id => | Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
case of
B : Type u
inst✝ : Quiver B
b c a✝ b✝ : B
f✝ : a✝ ⟶ b✝
a : B
f : Hom a a✝
⊢ normalizeAux nil (Hom.comp f (Hom.of f✝)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.of f✝)) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | of => rfl | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with
| id => rfl
| Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
case of
B : Type u
inst✝ : Quiver B
b c a✝ b✝ : B
f✝ : a✝ ⟶ b✝
a : B
f : Hom a a✝
⊢ normalizeAux nil (Hom.comp f (Hom.of f✝)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.of f✝)) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with
| id => rfl
| of => | Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
case comp
B : Type u
inst✝ : Quiver B
b c a✝ b✝ c✝ : B
g : Hom a✝ b✝
g✝ : Hom b✝ c✝
ihf : ∀ {a : B} (f : Hom a a✝), normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)
ihg : ∀ {a : B} (f : Hom a b✝), normalizeAux nil (Hom.comp f g✝) = comp (normalizeAux nil f) (normalizeAux nil g✝)
a : B
f ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc] | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with
| id => rfl
| of => rfl
| Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
case comp
B : Type u
inst✝ : Quiver B
b c a✝ b✝ c✝ : B
g : Hom a✝ b✝
g✝ : Hom b✝ c✝
ihf : ∀ {a : B} (f : Hom a a✝), normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)
ihg : ∀ {a : B} (f : Hom a b✝), normalizeAux nil (Hom.comp f g✝) = comp (normalizeAux nil f) (normalizeAux nil g✝)
a : B
f ... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | erw [ihg (f.comp g), ihf f, ihg g, comp_assoc] | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with
| id => rfl
| of => rfl
| comp g _ ihf ihg => | Mathlib.CategoryTheory.Bicategory.Coherence.188_0.scNCB7gGNV3iY0Z | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b : FreeBicategory B
⊢ ∀ {X Y : a ⟶ b} (f : X ⟶ Y),
(𝟭 (a ⟶ b)).map f ≫ ((fun f => (λ_ f).symm ≪≫ normalizeIso nil f) Y).hom =
((fun f => (λ_ f).symm ≪≫ normalizeIso nil f) X).hom ≫
(Pseudofunctor.mapFunctor (normalize B) a b ⋙ inclusionPath a b).map f | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | intro f g η | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b :=
NatIso.ofComponents (fun f => (λ_ f).symm ≪≫ normalizeIso nil f)
(by
| Mathlib.CategoryTheory.Bicategory.Coherence.206_0.scNCB7gGNV3iY0Z | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b : FreeBicategory B
f g : a ⟶ b
η : f ⟶ g
⊢ (𝟭 (a ⟶ b)).map η ≫ ((fun f => (λ_ f).symm ≪≫ normalizeIso nil f) g).hom =
((fun f => (λ_ f).symm ≪≫ normalizeIso nil f) f).hom ≫
(Pseudofunctor.mapFunctor (normalize B) a b ⋙ inclusionPath a b).map η | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | erw [leftUnitor_inv_naturality_assoc, assoc] | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b :=
NatIso.ofComponents (fun f => (λ_ f).symm ≪≫ normalizeIso nil f)
(by
intro f g η
| Mathlib.CategoryTheory.Bicategory.Coherence.206_0.scNCB7gGNV3iY0Z | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b : FreeBicategory B
f g : a ⟶ b
η : f ⟶ g
⊢ (λ_ ((𝟭 (a ⟶ b)).obj f)).inv ≫ 𝟙 a ◁ (𝟭 (a ⟶ b)).map η ≫ (normalizeIso nil g).hom =
(λ_ f).symm.hom ≫ (normalizeIso nil f).hom ≫ (Pseudofunctor.mapFunctor (normalize B) a b ⋙ inclusionPath a b).map η | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | congr 1 | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b :=
NatIso.ofComponents (fun f => (λ_ f).symm ≪≫ normalizeIso nil f)
(by
intro f g η
erw [leftUnitor_inv_naturality_assoc, assoc]
... | Mathlib.CategoryTheory.Bicategory.Coherence.206_0.scNCB7gGNV3iY0Z | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b | Mathlib_CategoryTheory_Bicategory_Coherence |
case e_a
B : Type u
inst✝ : Quiver B
a b : FreeBicategory B
f g : a ⟶ b
η : f ⟶ g
⊢ 𝟙 a ◁ (𝟭 (a ⟶ b)).map η ≫ (normalizeIso nil g).hom =
(normalizeIso nil f).hom ≫ (Pseudofunctor.mapFunctor (normalize B) a b ⋙ inclusionPath a b).map η | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | exact normalize_naturality nil η | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b :=
NatIso.ofComponents (fun f => (λ_ f).symm ≪≫ normalizeIso nil f)
(by
intro f g η
erw [leftUnitor_inv_naturality_assoc, assoc]
... | Mathlib.CategoryTheory.Bicategory.Coherence.206_0.scNCB7gGNV3iY0Z | /-- Auxiliary definition for `normalizeEquiv`. -/
def normalizeUnitIso (a b : FreeBicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).mapFunctor a b ⋙ @inclusionPath B _ a b | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b : B
f : Discrete (Path a b)
⊢ (inclusionPath a b ⋙ Pseudofunctor.mapFunctor (normalize B) a b).obj f = (𝟭 (Discrete (Path a b))).obj f | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction' f with f | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
| Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk
B : Type u
inst✝ : Quiver B
a b : B
f : Path a b
⊢ (inclusionPath a b ⋙ Pseudofunctor.mapFunctor (normalize B) a b).obj { as := f } =
(𝟭 (Discrete (Path a b))).obj { as := f } | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction' f with _ _ _ _ ih | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
| Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.nil
B : Type u
inst✝ : Quiver B
a b : B
⊢ (inclusionPath a a ⋙ Pseudofunctor.mapFunctor (normalize B) a a).obj { as := nil } =
(𝟭 (Discrete (Path a a))).obj { as := nil } | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.cons
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih :
(inclusionPath a b✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a b✝).obj { as := a✝¹ } =
(𝟭 (Discrete (Path a b✝))).obj { as := a✝¹ }
⊢ (inclusionPath a c✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a c✝).obj { as := cons a✝¹ a✝... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | ext1 | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.cons.as
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih :
(inclusionPath a b✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a b✝).obj { as := a✝¹ } =
(𝟭 (Discrete (Path a b✝))).obj { as := a✝¹ }
⊢ ((inclusionPath a c✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a c✝).obj { as := cons a✝... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | injection ih with ih | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.cons.as
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih : normalizeAux nil ((inclusionPath a b✝).obj { as := a✝¹ }) = a✝¹
⊢ ((inclusionPath a c✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a c✝).obj { as := cons a✝¹ a✝ }).as =
((𝟭 (Discrete (Path a c✝))).obj { as := cons a✝¹ a✝ }).as | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | conv =>
rhs
rw [← ih] | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih : normalizeAux nil ((inclusionPath a b✝).obj { as := a✝¹ }) = a✝¹
| ((inclusionPath a c✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a c✝).obj { as := cons a✝¹ a✝ }).as =
((𝟭 (Discrete (Path a c✝))).obj { as := cons a✝¹ a✝ }).as | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rhs
rw [← ih] | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih : normalizeAux nil ((inclusionPath a b✝).obj { as := a✝¹ }) = a✝¹
| ((inclusionPath a c✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a c✝).obj { as := cons a✝¹ a✝ }).as =
((𝟭 (Discrete (Path a c✝))).obj { as := cons a✝¹ a✝ }).as | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rhs
rw [← ih] | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih : normalizeAux nil ((inclusionPath a b✝).obj { as := a✝¹ }) = a✝¹
| ((inclusionPath a c✝ ⋙ Pseudofunctor.mapFunctor (normalize B) a c✝).obj { as := cons a✝¹ a✝ }).as =
((𝟭 (Discrete (Path a c✝))).obj { as := cons a✝¹ a✝ }).as | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rhs | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b b✝ c✝ : B
a✝¹ : Path a b✝
a✝ : b✝ ⟶ c✝
ih : normalizeAux nil ((inclusionPath a b✝).obj { as := a✝¹ }) = a✝¹
| ((𝟭 (Discrete (Path a c✝))).obj { as := cons a✝¹ a✝ }).as | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rw [← ih] | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
(Discrete.natIso fun f => eqToIso (by
induction' f with f
induction' f with _ _ _ _ ih
... | Mathlib.CategoryTheory.Bicategory.Coherence.217_0.scNCB7gGNV3iY0Z | /-- Normalization as an equivalence of categories. -/
def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) | Mathlib_CategoryTheory_Bicategory_Coherence |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
⊢ IsNoetherian R ↥N ↔ ∀ s ≤ N, Submodule.FG s | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩ | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
| Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
h : ∀ s ≤ N, Submodule.FG s
s : Submodule R ↥N
⊢ Submodule.FG s | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | have f := (Submodule.equivMapOfInjective N.subtype Subtype.val_injective s).symm | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
| Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
h : ∀ s ≤ N, Submodule.FG s
s : Submodule R ↥N
f : ↥(Submodule.map (Submodule.subtype N) s) ≃ₗ[R] ↥s
⊢ Submodule.FG s | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | have h₁ := h (s.map N.subtype) (Submodule.map_subtype_le N s) | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
have f := (Submod... | Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
h : ∀ s ≤ N, Submodule.FG s
s : Submodule R ↥N
f : ↥(Submodule.map (Submodule.subtype N) s) ≃ₗ[R] ↥s
h₁ : Submodule.FG (Submodule.map (Submodule.subtype N... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | have h₂ : (⊤ : Submodule R (s.map N.subtype)).map f = ⊤ := by simp | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
have f := (Submod... | Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
h : ∀ s ≤ N, Submodule.FG s
s : Submodule R ↥N
f : ↥(Submodule.map (Submodule.subtype N) s) ≃ₗ[R] ↥s
h₁ : Submodule.FG (Submodule.map (Submodule.subtype N... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | simp | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
have f := (Submod... | Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
h : ∀ s ≤ N, Submodule.FG s
s : Submodule R ↥N
f : ↥(Submodule.map (Submodule.subtype N) s) ≃ₗ[R] ↥s
h₁ : Submodule.FG (Submodule.map (Submodule.subtype N... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | have h₃ := ((Submodule.fg_top _).2 h₁).map (↑f : _ →ₗ[R] s) | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
have f := (Submod... | Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
N : Submodule R M
h : ∀ s ≤ N, Submodule.FG s
s : Submodule R ↥N
f : ↥(Submodule.map (Submodule.subtype N) s) ≃ₗ[R] ↥s
h₁ : Submodule.FG (Submodule.map (Submodule.subtype N... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact (Submodule.fg_top _).1 (h₂ ▸ h₃) | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
have f := (Submod... | Mathlib.RingTheory.Noetherian.88_0.5UPGNrmhtW81IjE | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
⊢ IsNoetherian R ↥⊤ ↔ IsNoetherian R M | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | constructor | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by
| Mathlib.RingTheory.Noetherian.135_0.5UPGNrmhtW81IjE | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M | Mathlib_RingTheory_Noetherian |
case mp
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
⊢ IsNoetherian R ↥⊤ → IsNoetherian R M | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | intro h | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by
constructor <;> | Mathlib.RingTheory.Noetherian.135_0.5UPGNrmhtW81IjE | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M | Mathlib_RingTheory_Noetherian |
case mpr
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
⊢ IsNoetherian R M → IsNoetherian R ↥⊤ | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | intro h | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by
constructor <;> | Mathlib.RingTheory.Noetherian.135_0.5UPGNrmhtW81IjE | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M | Mathlib_RingTheory_Noetherian |
case mp
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
h : IsNoetherian R ↥⊤
⊢ IsNoetherian R M | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl) | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by
constructor <;> intro h
· | Mathlib.RingTheory.Noetherian.135_0.5UPGNrmhtW81IjE | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M | Mathlib_RingTheory_Noetherian |
case mpr
R : Type u_1
M : Type u_2
P : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid P
inst✝¹ : Module R M
inst✝ : Module R P
h : IsNoetherian R M
⊢ IsNoetherian R ↥⊤ | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl).symm | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by
constructor <;> intro h
· exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl)
· | Mathlib.RingTheory.Noetherian.135_0.5UPGNrmhtW81IjE | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M | Mathlib_RingTheory_Noetherian |
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Is... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | cases nonempty_fintype ι | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
| Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case intro
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | haveI := Classical.decEq ι | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
| Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case intro
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i) | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
| Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case intro
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | let coe_e := Equiv.subtypeUnivEquiv <| @Finset.mem_univ ι _ | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case intro
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | letI : IsNoetherian R (∀ i : Finset.univ, M (coe_e i)) := on_finset Finset.univ | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case intro
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact isNoetherian_of_linearEquiv (LinearEquiv.piCongrLeft R M coe_e) | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case on_finset
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | intro s | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case on_finset
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | induction' s using Finset.induction with a s has ih | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
case on_finset.empty
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact ⟨fun s => by
have : s = ⊥ := by simp only [eq_iff_true_of_subsingleton]
rw [this]
apply Submodule.fg_bot⟩ | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Is... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | have : s = ⊥ := by simp only [eq_iff_true_of_subsingleton] | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Is... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | simp only [eq_iff_true_of_subsingleton] | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
R✝ : Type u_1
M✝ : Type u_2
P : Type u_3
inst✝⁹ : Ring R✝
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R✝ M✝
inst✝⁵ : Module R✝ P
R : Type u_4
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Ring R
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Is... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [this] | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
cases nonempty_fintype ι
haveI := Classical.decEq ι
suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i)
... | Mathlib.RingTheory.Noetherian.205_0.5UPGNrmhtW81IjE | instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
[Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
[∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) | Mathlib_RingTheory_Noetherian |
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