category_theory.abelian.diagram_lemmas.four

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

d34cbcf6

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

6cf59007

bump to nightly-2023-12-23-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

b8c74971

Diff

@@ -83,7 +83,7 @@ section
 
 variable (hfg : Exact f g) (hgh : Exact g h) (hf'g' : Exact f' g')
 
-#print CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono /-
+#print CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono' /-
 /-- The four lemma, mono version. For names of objects and morphisms, refer to the following
     diagram:
 
@@ -96,7 +96,7 @@ v         v         v         v
 A' --f'-> B' --g'-> C' --h'-> D'
 ```
 -/
-theorem mono_of_epi_of_mono_of_mono (hα : Epi α) (hβ : Mono β) (hδ : Mono δ) : Mono γ :=
+theorem mono_of_epi_of_mono_of_mono' (hα : Epi α) (hβ : Mono β) (hδ : Mono δ) : Mono γ :=
   mono_of_zero_of_map_zero _ fun c hc =>
     have : h c = 0 :=
       suffices δ (h c) = 0 from zero_of_map_zero _ (pseudo_injective_of_mono _) _ this
@@ -122,7 +122,7 @@ theorem mono_of_epi_of_mono_of_mono (hα : Epi α) (hβ : Mono β) (hδ : Mono 
             c = g b := hb.symm
             _ = g (f a) := by rw [this]
             _ = 0 := (pseudo_exact_of_exact hfg).1 _
-#align category_theory.abelian.mono_of_epi_of_mono_of_mono CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono
+#align category_theory.abelian.mono_of_epi_of_mono_of_mono CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'
 -/
 
 end
@@ -131,7 +131,7 @@ section
 
 variable (hgh : Exact g h) (hf'g' : Exact f' g') (hg'h' : Exact g' h')
 
-#print CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono /-
+#print CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono' /-
 /-- The four lemma, epi version. For names of objects and morphisms, refer to the following
     diagram:
 
@@ -144,7 +144,7 @@ v         v         v         v
 A' --f'-> B' --g'-> C' --h'-> D'
 ```
 -/
-theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ) : Epi β :=
+theorem epi_of_epi_of_epi_of_mono' (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ) : Epi β :=
   Preadditive.epi_of_cancel_zero _ fun R r hβr =>
     have hf'r : f' ≫ r = 0 :=
       Limits.zero_of_epi_comp α <|
@@ -181,7 +181,7 @@ theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ)
       _ = g' ≫ h' ≫ w := by rw [hzv]
       _ = 0 ≫ w := (hg'h'.w_assoc _)
       _ = 0 := HasZeroMorphisms.zero_comp _ _
-#align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono
+#align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono'
 -/
 
 end

6cf59007

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
 -/
-import Mathbin.CategoryTheory.Abelian.Pseudoelements
+import CategoryTheory.Abelian.Pseudoelements
 
 #align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"6cf5900728239efa287df7761ec2a1ac9cf39b29"

6cf59007

bump to nightly-2023-08-02-01

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

7aca3f15

Diff

@@ -196,7 +196,7 @@ variable (hf'g' : Exact f' g') (hg'h' : Exact g' h') (hh'i' : Exact h' i')
 
 variable [IsIso α] [IsIso β] [IsIso δ] [IsIso ε]
 
-#print CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso /-
+#print CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono /-
 /-- The five lemma. For names of objects and morphisms, refer to the following diagram:
 
 ```
@@ -208,13 +208,13 @@ v         v         v         v         v
 A' --f'-> B' --g'-> C' --h'-> D' --i'-> E'
 ```
 -/
-theorem isIso_of_isIso_of_isIso_of_isIso_of_isIso : IsIso γ :=
+theorem isIso_of_epi_of_isIso_of_isIso_of_mono : IsIso γ :=
   have : Mono γ := by
     apply mono_of_epi_of_mono_of_mono comm₁ comm₂ comm₃ hfg hgh hf'g' <;> infer_instance
   have : Epi γ := by
     apply epi_of_epi_of_epi_of_mono comm₂ comm₃ comm₄ hhi hg'h' hh'i' <;> infer_instance
   is_iso_of_mono_of_epi _
-#align category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso
+#align category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
 -/
 
 end Five

6cf59007

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,14 +2,11 @@
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
-
-! This file was ported from Lean 3 source module category_theory.abelian.diagram_lemmas.four
-! leanprover-community/mathlib commit 6cf5900728239efa287df7761ec2a1ac9cf39b29
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Abelian.Pseudoelements
 
+#align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"6cf5900728239efa287df7761ec2a1ac9cf39b29"
+
 /-!
 # The four and five lemmas

6cf59007

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -82,8 +82,6 @@ variable {α : A ⟶ A'} {β : B ⟶ B'} {γ : C ⟶ C'} {δ : D ⟶ D'}
 
 variable (comm₁ : α ≫ f' = f ≫ β) (comm₂ : β ≫ g' = g ≫ γ) (comm₃ : γ ≫ h' = h ≫ δ)
 
-include comm₁ comm₂ comm₃
-
 section
 
 variable (hfg : Exact f g) (hgh : Exact g h) (hf'g' : Exact f' g')
@@ -201,8 +199,6 @@ variable (hf'g' : Exact f' g') (hg'h' : Exact g' h') (hh'i' : Exact h' i')
 
 variable [IsIso α] [IsIso β] [IsIso δ] [IsIso ε]
 
-include comm₄ hfg hgh hhi hf'g' hg'h' hh'i'
-
 #print CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso /-
 /-- The five lemma. For names of objects and morphisms, refer to the following diagram:

6cf59007

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -109,14 +109,12 @@ theorem mono_of_epi_of_mono_of_mono (hα : Epi α) (hβ : Mono β) (hδ : Mono 
         δ (h c) = h' (γ c) := by rw [← comp_apply, ← comm₃, comp_apply]
         _ = h' 0 := by rw [hc]
         _ = 0 := apply_zero _
-        
     Exists.elim ((pseudo_exact_of_exact hgh).2 _ this) fun b hb =>
       have : g' (β b) = 0 :=
         calc
           g' (β b) = γ (g b) := by rw [← comp_apply, comm₂, comp_apply]
           _ = γ c := by rw [hb]
           _ = 0 := hc
-          
       Exists.elim ((pseudo_exact_of_exact hf'g').2 _ this) fun a' ha' =>
         Exists.elim (pseudo_surjective_of_epi α a') fun a ha =>
           have : f a = b :=
@@ -125,12 +123,10 @@ theorem mono_of_epi_of_mono_of_mono (hα : Epi α) (hβ : Mono β) (hδ : Mono 
               β (f a) = f' (α a) := by rw [← comp_apply, ← comm₁, comp_apply]
               _ = f' a' := by rw [ha]
               _ = β b := ha'
-              
           calc
             c = g b := hb.symm
             _ = g (f a) := by rw [this]
             _ = 0 := (pseudo_exact_of_exact hfg).1 _
-            
 #align category_theory.abelian.mono_of_epi_of_mono_of_mono CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono
 -/
 
@@ -161,7 +157,6 @@ theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ)
           α ≫ f' ≫ r = f ≫ β ≫ r := by rw [reassoc_of comm₁]
           _ = f ≫ 0 := by rw [hβr]
           _ = 0 := HasZeroMorphisms.comp_zero _ _
-          
     let y : R ⟶ pushout r g' := pushout.inl
     let z : C' ⟶ pushout r g' := pushout.inr
     have : Mono y :=
@@ -174,7 +169,6 @@ theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ)
         _ = β ≫ r ≫ y := by rw [← pushout.condition]
         _ = 0 ≫ y := by rw [reassoc_of hβr]
         _ = 0 := HasZeroMorphisms.zero_comp _ _
-        
     let v : pushout r g' ⟶ pushout (γ ≫ z) (h ≫ δ) := pushout.inl
     let w : D' ⟶ pushout (γ ≫ z) (h ≫ δ) := pushout.inr
     have : Mono v :=
@@ -186,14 +180,12 @@ theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ)
         calc
           γ ≫ z ≫ v = h ≫ δ ≫ w := by rw [← category.assoc, pushout.condition, category.assoc]
           _ = γ ≫ h' ≫ w := by rw [reassoc_of comm₃]
-          
     suffices (r ≫ y) ≫ v = 0 from zero_of_comp_mono _ (zero_of_comp_mono _ this)
     calc
       (r ≫ y) ≫ v = g' ≫ z ≫ v := by rw [pushout.condition, category.assoc]
       _ = g' ≫ h' ≫ w := by rw [hzv]
       _ = 0 ≫ w := (hg'h'.w_assoc _)
       _ = 0 := HasZeroMorphisms.zero_comp _ _
-      
 #align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono
 -/

6cf59007

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -68,7 +68,7 @@ variable {V : Type u} [Category.{v} V] [Abelian V]
 
 attribute [local instance] preadditive.has_equalizers_of_has_kernels
 
-open Pseudoelement
+open scoped Pseudoelement
 
 namespace CategoryTheory.Abelian

6cf59007

bump to nightly-2023-05-14-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

fbb256bf

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
 
 ! This file was ported from Lean 3 source module category_theory.abelian.diagram_lemmas.four
-! leanprover-community/mathlib commit d34cbcf6c94953e965448c933cd9cc485115ebbd
+! leanprover-community/mathlib commit 6cf5900728239efa287df7761ec2a1ac9cf39b29
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.CategoryTheory.Abelian.Pseudoelements
 /-!
 # The four and five lemmas
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Consider the following commutative diagram with exact rows in an abelian category:
 
 ```

d34cbcf6

bump to nightly-2023-05-12-10

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

489c7f8c

Diff

@@ -85,6 +85,7 @@ section
 
 variable (hfg : Exact f g) (hgh : Exact g h) (hf'g' : Exact f' g')
 
+#print CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono /-
 /-- The four lemma, mono version. For names of objects and morphisms, refer to the following
     diagram:
 
@@ -128,6 +129,7 @@ theorem mono_of_epi_of_mono_of_mono (hα : Epi α) (hβ : Mono β) (hδ : Mono 
             _ = 0 := (pseudo_exact_of_exact hfg).1 _
             
 #align category_theory.abelian.mono_of_epi_of_mono_of_mono CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono
+-/
 
 end
 
@@ -135,6 +137,7 @@ section
 
 variable (hgh : Exact g h) (hf'g' : Exact f' g') (hg'h' : Exact g' h')
 
+#print CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono /-
 /-- The four lemma, epi version. For names of objects and morphisms, refer to the following
     diagram:
 
@@ -189,6 +192,7 @@ theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ)
       _ = 0 := HasZeroMorphisms.zero_comp _ _
       
 #align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono
+-/
 
 end
 
@@ -204,6 +208,7 @@ variable [IsIso α] [IsIso β] [IsIso δ] [IsIso ε]
 
 include comm₄ hfg hgh hhi hf'g' hg'h' hh'i'
 
+#print CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso /-
 /-- The five lemma. For names of objects and morphisms, refer to the following diagram:
 
 ```
@@ -222,6 +227,7 @@ theorem isIso_of_isIso_of_isIso_of_isIso_of_isIso : IsIso γ :=
     apply epi_of_epi_of_epi_of_mono comm₂ comm₃ comm₄ hhi hg'h' hh'i' <;> infer_instance
   is_iso_of_mono_of_epi _
 #align category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso
+-/
 
 end Five

d34cbcf6

bump to nightly-2023-03-01-20

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

20cadee0

Diff

@@ -185,7 +185,7 @@ theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ)
     calc
       (r ≫ y) ≫ v = g' ≫ z ≫ v := by rw [pushout.condition, category.assoc]
       _ = g' ≫ h' ≫ w := by rw [hzv]
-      _ = 0 ≫ w := hg'h'.w_assoc _
+      _ = 0 ≫ w := (hg'h'.w_assoc _)
       _ = 0 := HasZeroMorphisms.zero_comp _ _
       
 #align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono

d34cbcf6

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

d34cbcf6

chore: Rename mul-div cancellation lemmas (#11530)

Lemma names around cancellation of multiplication and division are a mess.

This PR renames a handful of them according to the following table (each big row contains the multiplicative statement, then the three rows contain the GroupWithZero lemma name, the Group lemma, the AddGroup lemma name).

4ea4a86a

Diff

@@ -117,7 +117,7 @@ theorem epi_of_epi_of_epi_of_mono'
   rw [assoc, assoc, assoc, add_comp, assoc, assoc, assoc, NatTrans.naturality,
     ← reassoc_of% h₆, ← h₅, comp_sub]
   dsimp
-  rw [add_sub_cancel'_right]
+  rw [add_sub_cancel]
 #align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono'
 
 theorem epi_of_epi_of_epi_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)

d34cbcf6

perf (CategoryTheory.ComposableArrows): swap out linarith for omega (#9594)

This file now has the lead in the number of CPU instructions amongst all of mathlib. The majority come from linarith being an autoparam for many definitions, often in multiple arguments. This uses omega in place of linarith to improve performance.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

f34cd078

Diff

@@ -106,7 +106,7 @@ theorem epi_of_epi_of_epi_of_mono'
       ← R₂.map'_comp 1 2 3, hR₂', comp_zero, comp_zero]
   obtain ⟨A₂, π₂, _, f₁, h₃⟩ := (hR₁.exact 0).exact_up_to_refinements _ h₂
   dsimp at f₁ h₃
-  have h₄ : (π₂ ≫ π₁ ≫ g₁ - f₁ ≫ app' φ 1) ≫ R₂.map' 1 2 = 0 := by
+  have h₄ : (π₂ ≫ π₁ ≫ g₁ - f₁ ≫ app' φ 1 _) ≫ R₂.map' 1 2 = 0 := by
     rw [sub_comp, assoc, assoc, assoc, ← NatTrans.naturality, ← reassoc_of% h₃, h₁, sub_self]
   obtain ⟨A₃, π₃, _, g₀, h₅⟩ := (hR₂.exact 0).exact_up_to_refinements _ h₄
   dsimp at g₀ h₅

d34cbcf6

refactor: use the new homology API for the four and the five lemmas (#9022)

This PR refactors the four and the five lemmas so as to use the new homology API. The files Algebra.Homology.ShortExact.Abelian and Algebra.Homology.ShortExact.Preadditive are also removed because the content of these files has become redundant with the new homology API.

Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>

850c6404

Diff

@@ -1,16 +1,17 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel
+Authors: Markus Himmel, Joël Riou
 -/
-import Mathlib.CategoryTheory.Abelian.Pseudoelements
+import Mathlib.Algebra.Homology.ExactSequence
+import Mathlib.CategoryTheory.Abelian.Refinements
 
 #align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd"
 
 /-!
 # The four and five lemmas
 
-Consider the following commutative diagram with exact rows in an abelian category:
+Consider the following commutative diagram with exact rows in an abelian category `C`:
 
 ```
 A ---f--> B ---g--> C ---h--> D ---i--> E
@@ -30,19 +31,13 @@ We show:
 
 ## Implementation details
 
-To show the mono version, we use pseudoelements. For the epi version, we use a completely different
-arrow-theoretic proof. In theory, it should be sufficient to have one version and the other should
-follow automatically by duality. In practice, mathlib's knowledge about duality isn't quite at the
-point where this is doable easily.
+The diagram of the five lemmas is given by a morphism in the category `ComposableArrows C 4`
+between two objects which satisfy `ComposableArrows.Exact`. Similarly, the two versions of the
+four lemma are stated in terms of the category `ComposableArrows C 3`.
 
-However, one key duality statement about exactness is needed in the proof of the epi version of the
-four lemma: we need to know that exactness of a pair `(f, g)`, which we defined via the map from
-the image of `f` to the kernel of `g`, is the same as "co-exactness", defined via the map from the
-cokernel of `f` to the coimage of `g` (more precisely, we only need the consequence that if `(f, g)`
-is exact, then the factorization of `g` through the cokernel of `f` is monomorphic). Luckily, in the
-case of abelian categories, we have the characterization that `(f, g)` is exact if and only if
-`f ≫ g = 0` and `kernel.ι g ≫ cokernel.π f = 0`, and the latter condition is self dual, so the
-equivalence of exactness and co-exactness follows easily.
+The five lemmas is deduced from the two versions of the four lemma. Both of these versions
+are proved separately. It would be easy to deduce the epi version from the mono version
+using duality, but this would require lengthy API developments for `ComposableArrows` (TODO).
 
 ## Tags
 
@@ -50,169 +45,109 @@ four lemma, five lemma, diagram lemma, diagram chase
 -/
 
 
-open CategoryTheory hiding comp_apply
-
-open CategoryTheory.Abelian.Pseudoelement
-
-open CategoryTheory.Limits
-
-universe v u
-
-variable {V : Type u} [Category.{v} V] [Abelian V]
-
-attribute [local instance] Preadditive.hasEqualizers_of_hasKernels
-
-open Pseudoelement
-
-namespace CategoryTheory.Abelian
-
-variable {A B C D A' B' C' D' : V}
-
-variable {f : A ⟶ B} {g : B ⟶ C} {h : C ⟶ D}
-
-variable {f' : A' ⟶ B'} {g' : B' ⟶ C'} {h' : C' ⟶ D'}
-
-variable {α : A ⟶ A'} {β : B ⟶ B'} {γ : C ⟶ C'} {δ : D ⟶ D'}
-
-variable (comm₁ : α ≫ f' = f ≫ β) (comm₂ : β ≫ g' = g ≫ γ) (comm₃ : γ ≫ h' = h ≫ δ)
-
-section
-
-variable (hfg : Exact f g) (hgh : Exact g h) (hf'g' : Exact f' g')
-
-/-- The four lemma, mono version. For names of objects and morphisms, refer to the following
-    diagram:
-
-```
-A ---f--> B ---g--> C ---h--> D
-|         |         |         |
-α         β         γ         δ
-|         |         |         |
-v         v         v         v
-A' --f'-> B' --g'-> C' --h'-> D'
-```
--/
-theorem mono_of_epi_of_mono_of_mono (hα : Epi α) (hβ : Mono β) (hδ : Mono δ) : Mono γ :=
-  mono_of_zero_of_map_zero _ fun c hc =>
-    have : h c = 0 :=
-      suffices δ (h c) = 0 from zero_of_map_zero _ (pseudo_injective_of_mono _) _ this
-      calc
-        δ (h c) = h' (γ c) := by rw [← Pseudoelement.comp_apply, ← comm₃, Pseudoelement.comp_apply]
-        _ = h' 0 := by rw [hc]
-        _ = 0 := apply_zero _
-    Exists.elim ((pseudo_exact_of_exact hgh).2 _ this) fun b hb =>
-      have : g' (β b) = 0 :=
-        calc
-          g' (β b) = γ (g b) := by rw [← Pseudoelement.comp_apply, comm₂, Pseudoelement.comp_apply]
-          _ = γ c := by rw [hb]
-          _ = 0 := hc
-      Exists.elim ((pseudo_exact_of_exact hf'g').2 _ this) fun a' ha' =>
-        Exists.elim (pseudo_surjective_of_epi α a') fun a ha =>
-          have : f a = b :=
-            suffices β (f a) = β b from pseudo_injective_of_mono _ this
-            calc
-              β (f a) = f' (α a) := by
-                rw [← Pseudoelement.comp_apply, ← comm₁, Pseudoelement.comp_apply]
-              _ = f' a' := by rw [ha]
-              _ = β b := ha'
-          calc
-            c = g b := hb.symm
-            _ = g (f a) := by rw [this]
-            _ = 0 := (pseudo_exact_of_exact hfg).1 _
-#align category_theory.abelian.mono_of_epi_of_mono_of_mono CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono
-
-end
-
-section
-
-variable (hgh : Exact g h) (hf'g' : Exact f' g') (hg'h' : Exact g' h')
-
-/-- The four lemma, epi version. For names of objects and morphisms, refer to the following
-    diagram:
-
-```
-A ---f--> B ---g--> C ---h--> D
-|         |         |         |
-α         β         γ         δ
-|         |         |         |
-v         v         v         v
-A' --f'-> B' --g'-> C' --h'-> D'
-```
--/
-theorem epi_of_epi_of_epi_of_mono (hα : Epi α) (hγ : Epi γ) (hδ : Mono δ) : Epi β :=
-  Preadditive.epi_of_cancel_zero _ fun {R} r hβr => by
-    have hf'r : f' ≫ r = 0 :=
-      Limits.zero_of_epi_comp α <|
-        calc
-          α ≫ f' ≫ r = f ≫ β ≫ r := by rw [reassoc_of% comm₁]
-          _ = f ≫ 0 := by rw [hβr]
-          _ = 0 := HasZeroMorphisms.comp_zero _ _
-    let y : R ⟶ pushout r g' := pushout.inl
-    let z : C' ⟶ pushout r g' := pushout.inr
-    -- Porting note: Added instance for `Mono (cokernel.desc f' g' hf'g'.w)`
-    have : Mono (cokernel.desc f' g' hf'g'.w) := mono_cokernel_desc_of_exact _ _ hf'g'
-    have : Mono y :=
-      mono_inl_of_factor_thru_epi_mono_factorization r g' (cokernel.π f')
-        (cokernel.desc f' g' hf'g'.w) (by simp) (cokernel.desc f' r hf'r) (by simp) _
-        (colimit.isColimit _)
-    have hz : g ≫ γ ≫ z = 0 :=
-      calc
-        g ≫ γ ≫ z = β ≫ g' ≫ z := by rw [← reassoc_of% comm₂]
-        _ = β ≫ r ≫ y := by rw [← pushout.condition]
-        _ = 0 ≫ y := by rw [reassoc_of% hβr]
-        _ = 0 := HasZeroMorphisms.zero_comp _ _
-    let v : pushout r g' ⟶ pushout (γ ≫ z) (h ≫ δ) := pushout.inl
-    let w : D' ⟶ pushout (γ ≫ z) (h ≫ δ) := pushout.inr
-    -- Porting note: Added instance for `Mono (cokernel.desc g h hgh.w)`
-    have : Mono (cokernel.desc g h hgh.w) := mono_cokernel_desc_of_exact _ _ hgh
-    have : Mono v :=
-      mono_inl_of_factor_thru_epi_mono_factorization _ _ (cokernel.π g)
-        (cokernel.desc g h hgh.w ≫ δ) (by simp) (cokernel.desc _ _ hz) (by simp) _
-        (colimit.isColimit _)
-    have hzv : z ≫ v = h' ≫ w :=
-      (cancel_epi γ).1 <|
-        calc
-          γ ≫ z ≫ v = h ≫ δ ≫ w := by rw [← Category.assoc, pushout.condition, Category.assoc]
-          _ = γ ≫ h' ≫ w := by rw [reassoc_of% comm₃]
-    suffices (r ≫ y) ≫ v = 0 from zero_of_comp_mono _ (zero_of_comp_mono _ this)
-    calc
-      (r ≫ y) ≫ v = g' ≫ z ≫ v := by rw [pushout.condition, Category.assoc]
-      _ = g' ≫ h' ≫ w := by rw [hzv]
-      _ = 0 ≫ w := (hg'h'.w_assoc _)
-      _ = 0 := HasZeroMorphisms.zero_comp _ _
-#align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono
-
-end
+namespace CategoryTheory
+
+open Category Limits Preadditive
+
+namespace Abelian
+
+variable {C : Type*} [Category C] [Abelian C]
+
+open ComposableArrows
+
+section Four
+
+variable {R₁ R₂ : ComposableArrows C 3} (φ : R₁ ⟶ R₂)
+
+theorem mono_of_epi_of_mono_of_mono' (hR₁ : R₁.map' 0 2 = 0)
+    (hR₁' : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
+    (hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact)
+    (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) (h₃ : Mono (app' φ 3)) :
+    Mono (app' φ 2) := by
+  apply mono_of_cancel_zero
+  intro A f₂ h₁
+  have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
+    rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
+      zero_comp, zero_comp]
+  obtain ⟨A₁, π₁, _, f₁, hf₁⟩ := (hR₁'.exact 0).exact_up_to_refinements f₂ h₂
+  dsimp at hf₁
+  have h₃ : (f₁ ≫ app' φ 1) ≫ R₂.map' 1 2 = 0 := by
+    rw [assoc, ← NatTrans.naturality, ← reassoc_of% hf₁, h₁, comp_zero]
+  obtain ⟨A₂, π₂, _, g₀, hg₀⟩ := (hR₂.exact 0).exact_up_to_refinements _ h₃
+  obtain ⟨A₃, π₃, _, f₀, hf₀⟩ := surjective_up_to_refinements_of_epi (app' φ 0 _) g₀
+  have h₄ : f₀ ≫ R₁.map' 0 1 = π₃ ≫ π₂ ≫ f₁ := by
+    rw [← cancel_mono (app' φ 1 _), assoc, assoc, assoc, NatTrans.naturality,
+      ← reassoc_of% hf₀, hg₀]
+    rfl
+  rw [← cancel_epi π₁, comp_zero, hf₁, ← cancel_epi π₂, ← cancel_epi π₃, comp_zero,
+    comp_zero, ← reassoc_of% h₄, ← R₁.map'_comp 0 1 2, hR₁, comp_zero]
+#align category_theory.abelian.mono_of_epi_of_mono_of_mono CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'
+
+theorem mono_of_epi_of_mono_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
+    (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) (h₃ : Mono (app' φ 3)) :
+    Mono (app' φ 2) :=
+  mono_of_epi_of_mono_of_mono' φ
+    (by simpa only [R₁.map'_comp 0 1 2] using hR₁.toIsComplex.zero 0)
+    (hR₁.exact 1).exact_toComposableArrows (hR₂.exact 0).exact_toComposableArrows h₀ h₁ h₃
+
+attribute [local instance] epi_comp
+
+theorem epi_of_epi_of_epi_of_mono'
+    (hR₁ : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
+    (hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact) (hR₂' : R₂.map' 1 3 = 0)
+    (h₀ : Epi (app' φ 0)) (h₂ : Epi (app' φ 2)) (h₃ : Mono (app' φ 3)) :
+    Epi (app' φ 1) := by
+  rw [epi_iff_surjective_up_to_refinements]
+  intro A g₁
+  obtain ⟨A₁, π₁, _, f₂, h₁⟩ :=
+    surjective_up_to_refinements_of_epi (app' φ 2 _) (g₁ ≫ R₂.map' 1 2)
+  have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
+    rw [← cancel_mono (app' φ 3 _), assoc, zero_comp, NatTrans.naturality, ← reassoc_of% h₁,
+      ← R₂.map'_comp 1 2 3, hR₂', comp_zero, comp_zero]
+  obtain ⟨A₂, π₂, _, f₁, h₃⟩ := (hR₁.exact 0).exact_up_to_refinements _ h₂
+  dsimp at f₁ h₃
+  have h₄ : (π₂ ≫ π₁ ≫ g₁ - f₁ ≫ app' φ 1) ≫ R₂.map' 1 2 = 0 := by
+    rw [sub_comp, assoc, assoc, assoc, ← NatTrans.naturality, ← reassoc_of% h₃, h₁, sub_self]
+  obtain ⟨A₃, π₃, _, g₀, h₅⟩ := (hR₂.exact 0).exact_up_to_refinements _ h₄
+  dsimp at g₀ h₅
+  rw [comp_sub] at h₅
+  obtain ⟨A₄, π₄, _, f₀, h₆⟩ := surjective_up_to_refinements_of_epi (app' φ 0 _) g₀
+  refine' ⟨A₄, π₄ ≫ π₃ ≫ π₂ ≫ π₁, inferInstance,
+    π₄ ≫ π₃ ≫ f₁ + f₀ ≫ (by exact R₁.map' 0 1), _⟩
+  rw [assoc, assoc, assoc, add_comp, assoc, assoc, assoc, NatTrans.naturality,
+    ← reassoc_of% h₆, ← h₅, comp_sub]
+  dsimp
+  rw [add_sub_cancel'_right]
+#align category_theory.abelian.epi_of_epi_of_epi_of_mono CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono'
+
+theorem epi_of_epi_of_epi_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
+    (h₀ : Epi (app' φ 0)) (h₂ : Epi (app' φ 2)) (h₃ : Mono (app' φ 3)) :
+    Epi (app' φ 1) :=
+  epi_of_epi_of_epi_of_mono' φ (hR₁.exact 1).exact_toComposableArrows
+    (hR₂.exact 0).exact_toComposableArrows
+    (by simpa only [R₂.map'_comp 1 2 3] using hR₂.toIsComplex.zero 1) h₀ h₂ h₃
+
+end Four
 
 section Five
 
-variable {E E' : V} {i : D ⟶ E} {i' : D' ⟶ E'} {ε : E ⟶ E'} (comm₄ : δ ≫ i' = i ≫ ε)
-
-variable (hfg : Exact f g) (hgh : Exact g h) (hhi : Exact h i)
-
-variable (hf'g' : Exact f' g') (hg'h' : Exact g' h') (hh'i' : Exact h' i')
-
-variable [Epi α] [IsIso β] [IsIso δ] [Mono ε]
-
-/-- The five lemma. For names of objects and morphisms, refer to the following diagram:
-
-```
-A ---f--> B ---g--> C ---h--> D ---i--> E
-|         |         |         |         |
-α         β         γ         δ         ε
-|         |         |         |         |
-v         v         v         v         v
-A' --f'-> B' --g'-> C' --h'-> D' --i'-> E'
-```
--/
-theorem isIso_of_epi_of_isIso_of_isIso_of_mono : IsIso γ :=
-  have : Mono γ := by
-    apply mono_of_epi_of_mono_of_mono comm₁ comm₂ comm₃ hfg hgh hf'g' <;> infer_instance
-  have : Epi γ := by
-    apply epi_of_epi_of_epi_of_mono comm₂ comm₃ comm₄ hhi hg'h' hh'i' <;> infer_instance
-  isIso_of_mono_of_epi _
+variable {R₁ R₂ : ComposableArrows C 4} (hR₁ : R₁.Exact) (hR₂ : R₂.Exact) (φ : R₁ ⟶ R₂)
+
+/-- The five lemma. -/
+theorem isIso_of_epi_of_isIso_of_isIso_of_mono (h₀ : Epi (app' φ 0)) (h₁ : IsIso (app' φ 1))
+    (h₂ : IsIso (app' φ 3)) (h₃ : Mono (app' φ 4)) : IsIso (app' φ 2) := by
+  dsimp at h₀ h₁ h₂ h₃
+  have : Mono (app' φ 2) := by
+    apply mono_of_epi_of_mono_of_mono (δlastFunctor.map φ) (R₁.exact_iff_δlast.1 hR₁).1
+      (R₂.exact_iff_δlast.1 hR₂).1 <;> dsimp <;> infer_instance
+  have : Epi (app' φ 2) := by
+    apply epi_of_epi_of_epi_of_mono (δ₀Functor.map φ) (R₁.exact_iff_δ₀.1 hR₁).2
+      (R₂.exact_iff_δ₀.1 hR₂).2 <;> dsimp <;> infer_instance
+  apply isIso_of_mono_of_epi
 #align category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
 
 end Five
 
-end CategoryTheory.Abelian
+end Abelian
+
+end CategoryTheory

d34cbcf6

chore: trivial generalization of the five lemma (#6181)

05363324

Diff

@@ -192,7 +192,7 @@ variable (hfg : Exact f g) (hgh : Exact g h) (hhi : Exact h i)
 
 variable (hf'g' : Exact f' g') (hg'h' : Exact g' h') (hh'i' : Exact h' i')
 
-variable [IsIso α] [IsIso β] [IsIso δ] [IsIso ε]
+variable [Epi α] [IsIso β] [IsIso δ] [Mono ε]
 
 /-- The five lemma. For names of objects and morphisms, refer to the following diagram:
 
@@ -205,13 +205,13 @@ v         v         v         v         v
 A' --f'-> B' --g'-> C' --h'-> D' --i'-> E'
 ```
 -/
-theorem isIso_of_isIso_of_isIso_of_isIso_of_isIso : IsIso γ :=
+theorem isIso_of_epi_of_isIso_of_isIso_of_mono : IsIso γ :=
   have : Mono γ := by
     apply mono_of_epi_of_mono_of_mono comm₁ comm₂ comm₃ hfg hgh hf'g' <;> infer_instance
   have : Epi γ := by
     apply epi_of_epi_of_epi_of_mono comm₂ comm₃ comm₄ hhi hg'h' hh'i' <;> infer_instance
   isIso_of_mono_of_epi _
-#align category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso
+#align category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
 
 end Five

d34cbcf6

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

depends on: #5966

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

89aa3a3b

Diff

@@ -2,14 +2,11 @@
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
-
-! This file was ported from Lean 3 source module category_theory.abelian.diagram_lemmas.four
-! leanprover-community/mathlib commit d34cbcf6c94953e965448c933cd9cc485115ebbd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.CategoryTheory.Abelian.Pseudoelements
 
+#align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd"
+
 /-!
 # The four and five lemmas

d34cbcf6

feat: port CategoryTheory.Abelian.DiagramLemmas.Four (#3935)

30768e35

`category_theory.abelian.diagram_lemmas.four` ⟷ `Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 524

category_theory.abelian.diagram_lemmas.four ⟷ Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 524

`category_theory.abelian.diagram_lemmas.four` ⟷ `Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four`