mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.ExactSequence

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ComposableArrows C 2
hS : S.IsComplex
⊢ S.Exact → (S.sc' hS 0 1 2 exact₂_iff._proof_2 exact₂_iff._proof_4 isComplex₂_iff._proof_5).Exact

pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ComposableArrows C 2
hS : S.IsComplex
⊢ (S.sc' hS 0 1 2 exact₂_iff._proof_2 exact₂_iff._proof_4 isComplex₂_iff._proof_5).Exact → S.Exact

Tactic:
· intro h
    exact h.exact 0 (by lia)

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ComposableArrows C 2
hS : S.IsComplex
⊢ (S.sc' hS 0 1 2 exact₂_iff._proof_2 exact₂_iff._proof_4 isComplex₂_iff._proof_5).Exact → S.Exact
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A₀ : C
x₃ : A₀ ⟶ S.L₀.X₃
hx₃ : x₃ ≫ S.δ = 0
A₁ : C
π₁ : A₁ ⟶ A₀
hπ₁ : Epi π₁
p : A₁ ⟶ S.P
hp : π₁ ≫ x₃ = p ≫ pullback.snd S.L₁.g S.v₀₁.τ₃
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₃ = x₁ ≫ S.L₁'.f

Tactic:
have hp' : (p ≫ S.φ₁) ≫ S.v₂₃.τ₁ = 0 := by
    rw [assoc, ← S.snd_δ, ← reassoc_of% hp, hx₃, comp_zero]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A₀ : C
x₃ : A₀ ⟶ S.L₀.X₃
hx₃ : x₃ ≫ S.δ = 0
A₁ : C
π₁ : A₁ ⟶ A₀
hπ₁ : Epi π₁
p : A₁ ⟶ S.P
hp : π₁ ≫ x₃ = p ≫ pullback.snd S.L₁.g S.v₀₁.τ₃
hp' : (p ≫ S.φ₁) ≫ S.v₂₃.τ₁ = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₃ = x₁ ≫ S.L₁'.f
Module: Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type w
inst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁵ : ConcreteCategory C FC
inst✝⁴ : HasForget₂ C Ab
inst✝³ : Preadditive C
inst✝² : (forget₂ C Ab).Additive
inst✝¹ : (forget₂ C Ab).PreservesHomology
S : ShortComplex C
inst✝ : S.HasHomology
⊢ S.Exact ↔
    ∀ (x₂ : ↑((forget₂ C Ab).obj S.X₂)),
      (ConcreteCategory.hom ((forget₂ C Ab).map S.g)) x₂ = 0 →
        ∃ x₁, (ConcreteCategory.hom ((forget₂ C Ab).map S.f)) x₁ = x₂

Tactic:
rw [S.exact_iff_exact_map_forget₂, ab_exact_iff]
  rfl

Goal state after:
Module: Mathlib.CategoryTheory.Monad.Equalizer

Goal state before:
pp: case refine_1
C : Type u₁
inst✝ : Category.{v₁, u₁} C
T : Comonad C
X : T.Coalgebra
s : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X)
h₁ : s.ι.f ≫ T.map X.a = s.ι.f ≫ T.δ.app X.A
h₂ : s.pt.a ≫ T.map s.ι.f = s.ι.f ≫ T.δ.app X.A
⊢ s.ι.f ≫ 𝟙 (T.obj X.A) = (s.ι.f ≫ T.ε.app X.A) ≫ X.a

Tactic:
rw [Category.comp_id, Category.assoc, ← T.counit_naturality,
        reassoc_of% h₁, Comonad.left_counit]

Goal state after:
case refine_1
C : Type u₁
inst✝ : Category.{v₁, u₁} C
T : Comonad C
X : T.Coalgebra
s : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X)
h₁ : s.ι.f ≫ T.map X.a = s.ι.f ≫ T.δ.app X.A
h₂ : s.pt.a ≫ T.map s.ι.f = s.ι.f ≫ T.δ.app X.A
⊢ s.ι.f = s.ι.f ≫ 𝟙 (T.obj X.A)
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F✝ G✝ F G : ComposableArrows C n
app : (i : Fin (n + 1)) → F.obj i ≅ G.obj i
w : ∀ (i : ℕ) (hi : i < n), F.map' i (i + 1) ⋯ hi ≫ (app ⟨i + 1, ⋯⟩).hom = (app ⟨i, ⋯⟩).hom ≫ G.map' i (i + 1) ⋯ hi
i : ℕ
hi : i < n
⊢ F.map' i (i + 1) ⋯ hi ≫ 𝟙 (F.obj ⟨i + 1, ⋯⟩) = (app ⟨i, ⋯⟩).hom ≫ (app ⟨i, ⋯⟩).inv ≫ F.map' i (i + 1) ⋯ hi

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F✝ G✝ F G : ComposableArrows C n
app : (i : Fin (n + 1)) → F.obj i ≅ G.obj i
w : ∀ (i : ℕ) (hi : i < n), F.map' i (i + 1) ⋯ hi ≫ (app ⟨i + 1, ⋯⟩).hom = (app ⟨i, ⋯⟩).hom ≫ G.map' i (i + 1) ⋯ hi
i : ℕ
hi : i < n
⊢ F.map' i (i + 1) ⋯ hi = (app ⟨i, ⋯⟩).hom ≫ (app ⟨i, ⋯⟩).inv ≫ F.map' i (i + 1) ⋯ hi
Module: Mathlib.CategoryTheory.Monad.Adjunction

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
adj : L ⊣ R
i : L ⋙ R ≅ 𝟭 C
⊢ (adj.toMonad.transport i).μ ≫ adj.unit = i.inv ≫ 𝟙 (L ⋙ R)

Tactic:
comp_id,

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
adj : L ⊣ R
i : L ⋙ R ≅ 𝟭 C
⊢ (adj.toMonad.transport i).μ ≫ adj.unit = i.inv
Module: Mathlib.CategoryTheory.Monad.Comonadicity

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝² : HasEqualizerOfIsCosplitPair F
inst✝¹ : PreservesLimitOfIsCosplitPair F
inst✝ : ReflectsLimitOfIsCosplitPair F
⊢ (comparison adj).IsEquivalence

Tactic:
by
    have : ∀ (X : Coalgebra adj.toComonad), IsIso ((comparisonAdjunction adj).counit.app X) := by
      intro X
      apply @isIso_of_reflects_iso _ _ _ _ _ _ _ (Comonad.forget adj.toComonad) ?_ _
      · change IsIso ((comparisonAdjunction adj).counit.app X).f
        rw [comparisonAdjunction_counit_f]
        change
          IsIso
            (IsLimit.conePointUniqueUpToIso (beckEqualizer X)
                (counitLimitOfPreservesEqualizer X)).inv
        exact (IsLimit.conePointUniqueUpToIso _ _).isIso_inv
    have : ∀ (Y : C), IsIso ((comparisonAdjunction adj).unit.app Y) := by
      intro Y
      rw [comparisonAdjunction_unit_app]
      change IsIso (IsLimit.conePointUniqueUpToIso _ ?_).inv
      · infer_instance
      apply @unitEqualizerOfCoreflectsEqualizer _ _ _ _ _ _ _ _ ?_
      let _ :
        F.IsCosplitPair (G.map (F.map (adj.unit.app Y)))
          (adj.unit.app (G.obj (F.obj Y))) :=
        ComonadicityInternal.main_pair_F_cosplit _ ((comparison adj).obj Y)
      infer_instance
    exact (comparisonAdjunction adj).toEquivalence.symm.isEquivalence_inverse

Goal state after:
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 2
app₀ : f.obj' 0 _proof_245✝ ≅ g.obj' 0 _proof_245✝
app₁ : f.obj' 1 _proof_246✝ ≅ g.obj' 1 _proof_246✝
app₂ : f.obj' 2 _proof_247✝ ≅ g.obj' 2 _proof_247✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_246✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_246✝
w₁ : f.map' 1 2 _proof_246✝ _proof_247✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_247✝
⊢ f.map' 0 1 homMk₁._proof_4 _proof_246✝ ≫ 𝟙 (f.obj' 1 _proof_246✝) =
    app₀.hom ≫ app₀.inv ≫ f.map' 0 1 homMk₁._proof_4 _proof_246✝

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 2
app₀ : f.obj' 0 _proof_245✝ ≅ g.obj' 0 _proof_245✝
app₁ : f.obj' 1 _proof_246✝ ≅ g.obj' 1 _proof_246✝
app₂ : f.obj' 2 _proof_247✝ ≅ g.obj' 2 _proof_247✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_246✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_246✝
w₁ : f.map' 1 2 _proof_246✝ _proof_247✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_247✝
⊢ f.map' 0 1 homMk₁._proof_4 _proof_246✝ = app₀.hom ≫ app₀.inv ≫ f.map' 0 1 homMk₁._proof_4 _proof_246✝
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 2
app₀ : f.obj' 0 _proof_245✝ ≅ g.obj' 0 _proof_245✝
app₁ : f.obj' 1 _proof_246✝ ≅ g.obj' 1 _proof_246✝
app₂ : f.obj' 2 _proof_247✝ ≅ g.obj' 2 _proof_247✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_246✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_246✝
w₁ : f.map' 1 2 _proof_246✝ _proof_247✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_247✝
⊢ f.map' 1 2 _proof_246✝ _proof_247✝ ≫ 𝟙 (f.obj' 2 _proof_247✝) =
    app₁.hom ≫ app₁.inv ≫ f.map' 1 2 _proof_246✝ _proof_247✝

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 2
app₀ : f.obj' 0 _proof_245✝ ≅ g.obj' 0 _proof_245✝
app₁ : f.obj' 1 _proof_246✝ ≅ g.obj' 1 _proof_246✝
app₂ : f.obj' 2 _proof_247✝ ≅ g.obj' 2 _proof_247✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_246✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_246✝
w₁ : f.map' 1 2 _proof_246✝ _proof_247✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_247✝
⊢ f.map' 1 2 _proof_246✝ _proof_247✝ = app₁.hom ≫ app₁.inv ≫ f.map' 1 2 _proof_246✝ _proof_247✝
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 3
app₀ : f.obj' 0 _proof_297✝ ≅ g.obj' 0 _proof_297✝
app₁ : f.obj' 1 _proof_298✝ ≅ g.obj' 1 _proof_298✝
app₂ : f.obj' 2 _proof_299✝ ≅ g.obj' 2 _proof_299✝
app₃ : f.obj' 3 _proof_300✝ ≅ g.obj' 3 _proof_300✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_298✝
w₁ : f.map' 1 2 _proof_246✝ _proof_299✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_299✝
w₂ : f.map' 2 3 _proof_299✝ _proof_300✝ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 _proof_299✝ _proof_300✝
⊢ f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ 𝟙 (f.obj' 1 _proof_298✝) =
    app₀.hom ≫ app₀.inv ≫ f.map' 0 1 homMk₁._proof_4 _proof_298✝

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 3
app₀ : f.obj' 0 _proof_297✝ ≅ g.obj' 0 _proof_297✝
app₁ : f.obj' 1 _proof_298✝ ≅ g.obj' 1 _proof_298✝
app₂ : f.obj' 2 _proof_299✝ ≅ g.obj' 2 _proof_299✝
app₃ : f.obj' 3 _proof_300✝ ≅ g.obj' 3 _proof_300✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_298✝
w₁ : f.map' 1 2 _proof_246✝ _proof_299✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_299✝
w₂ : f.map' 2 3 _proof_299✝ _proof_300✝ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 _proof_299✝ _proof_300✝
⊢ f.map' 0 1 homMk₁._proof_4 _proof_298✝ = app₀.hom ≫ app₀.inv ≫ f.map' 0 1 homMk₁._proof_4 _proof_298✝
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 3
app₀ : f.obj' 0 _proof_297✝ ≅ g.obj' 0 _proof_297✝
app₁ : f.obj' 1 _proof_298✝ ≅ g.obj' 1 _proof_298✝
app₂ : f.obj' 2 _proof_299✝ ≅ g.obj' 2 _proof_299✝
app₃ : f.obj' 3 _proof_300✝ ≅ g.obj' 3 _proof_300✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_298✝
w₁ : f.map' 1 2 _proof_246✝ _proof_299✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_299✝
w₂ : f.map' 2 3 _proof_299✝ _proof_300✝ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 _proof_299✝ _proof_300✝
⊢ f.map' 1 2 _proof_246✝ _proof_299✝ ≫ 𝟙 (f.obj' 2 _proof_299✝) =
    app₁.hom ≫ app₁.inv ≫ f.map' 1 2 _proof_246✝ _proof_299✝

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 3
app₀ : f.obj' 0 _proof_297✝ ≅ g.obj' 0 _proof_297✝
app₁ : f.obj' 1 _proof_298✝ ≅ g.obj' 1 _proof_298✝
app₂ : f.obj' 2 _proof_299✝ ≅ g.obj' 2 _proof_299✝
app₃ : f.obj' 3 _proof_300✝ ≅ g.obj' 3 _proof_300✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_298✝
w₁ : f.map' 1 2 _proof_246✝ _proof_299✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_299✝
w₂ : f.map' 2 3 _proof_299✝ _proof_300✝ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 _proof_299✝ _proof_300✝
⊢ f.map' 1 2 _proof_246✝ _proof_299✝ = app₁.hom ≫ app₁.inv ≫ f.map' 1 2 _proof_246✝ _proof_299✝
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 3
app₀ : f.obj' 0 _proof_297✝ ≅ g.obj' 0 _proof_297✝
app₁ : f.obj' 1 _proof_298✝ ≅ g.obj' 1 _proof_298✝
app₂ : f.obj' 2 _proof_299✝ ≅ g.obj' 2 _proof_299✝
app₃ : f.obj' 3 _proof_300✝ ≅ g.obj' 3 _proof_300✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_298✝
w₁ : f.map' 1 2 _proof_246✝ _proof_299✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_299✝
w₂ : f.map' 2 3 _proof_299✝ _proof_300✝ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 _proof_299✝ _proof_300✝
⊢ f.map' 2 3 _proof_299✝ _proof_300✝ ≫ 𝟙 (f.obj' 3 _proof_300✝) =
    app₂.hom ≫ app₂.inv ≫ f.map' 2 3 _proof_299✝ _proof_300✝

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F G : ComposableArrows C n
f g : ComposableArrows C 3
app₀ : f.obj' 0 _proof_297✝ ≅ g.obj' 0 _proof_297✝
app₁ : f.obj' 1 _proof_298✝ ≅ g.obj' 1 _proof_298✝
app₂ : f.obj' 2 _proof_299✝ ≅ g.obj' 2 _proof_299✝
app₃ : f.obj' 3 _proof_300✝ ≅ g.obj' 3 _proof_300✝
w₀ : f.map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 homMk₁._proof_4 _proof_298✝
w₁ : f.map' 1 2 _proof_246✝ _proof_299✝ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 _proof_246✝ _proof_299✝
w₂ : f.map' 2 3 _proof_299✝ _proof_300✝ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 _proof_299✝ _proof_300✝
⊢ f.map' 2 3 _proof_299✝ _proof_300✝ = app₂.hom ≫ app₂.inv ≫ f.map' 2 3 _proof_299✝ _proof_300✝
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
h : n + 1 ≤ m
f g : ComposableArrows C m
app : f.obj' n ⋯ ≅ g.obj' n ⋯
app' : f.obj' (n + 1) h ≅ g.obj' (n + 1) h
w : f.map' n (n + 1) ⋯ h ≫ app'.hom = app.hom ≫ g.map' n (n + 1) ⋯ h
⊢ f.map' n (n + 1) ⋯ h ≫ 𝟙 (f.obj' (n + 1) h) = app.hom ≫ app.inv ≫ f.map' n (n + 1) ⋯ h

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
h : n + 1 ≤ m
f g : ComposableArrows C m
app : f.obj' n ⋯ ≅ g.obj' n ⋯
app' : f.obj' (n + 1) h ≅ g.obj' (n + 1) h
w : f.map' n (n + 1) ⋯ h ≫ app'.hom = app.hom ≫ g.map' n (n + 1) ⋯ h
⊢ f.map' n (n + 1) ⋯ h = app.hom ≫ app.inv ≫ f.map' n (n + 1) ⋯ h
Module: Mathlib.CategoryTheory.Limits.Shapes.Reflexive

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : WalkingReflexivePair ⥤ C
G : ReflexiveCofork F
⊢ ((Cocone.precompose (diagramIsoParallelPair (WalkingParallelPair.inclusionWalkingReflexivePair ⋙ F)).inv).obj
        (Cocone.whisker WalkingParallelPair.inclusionWalkingReflexivePair G)).π =
    G.π

Tactic:
rw [ReflexiveCofork.π, Cofork.π]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : WalkingReflexivePair ⥤ C
G : ReflexiveCofork F
⊢ ((Cocone.precompose (diagramIsoParallelPair (WalkingParallelPair.inclusionWalkingReflexivePair ⋙ F)).inv).obj
            (Cocone.whisker WalkingParallelPair.inclusionWalkingReflexivePair G)).ι.app
      WalkingParallelPair.one =
    G.ι.app zero
Module: Mathlib.RingTheory.IsPrimary

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S T : Submodule R M
h : (S.colon Set.univ).radical = (T.colon Set.univ).radical
left✝¹ : S ≠ ⊤
left✝ : T ≠ ⊤
r✝ : R
x✝¹ : M
x✝ : r✝ • x✝¹ ∈ S ⊓ T
hS' : r✝ • x✝¹ ∈ ↑S
hT' : r✝ • x✝¹ ∈ ↑T
hS : ∀ {r : R} {x : M}, r • x ∈ S → x ∈ S ∨ r ∈ (S.colon ↑⊤).radical
hT : ∀ {r : R} {x : M}, r • x ∈ T → x ∈ T ∨ r ∈ (T.colon ↑⊤).radical
⊢ x✝¹ ∈ S ⊓ T ∨ r✝ ∈ (S.colon ↑⊤ ⊓ T.colon ↑⊤).radical

Tactic:
Ideal.radical_inf,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S T : Submodule R M
h : (S.colon Set.univ).radical = (T.colon Set.univ).radical
left✝¹ : S ≠ ⊤
left✝ : T ≠ ⊤
r✝ : R
x✝¹ : M
x✝ : r✝ • x✝¹ ∈ S ⊓ T
hS' : r✝ • x✝¹ ∈ ↑S
hT' : r✝ • x✝¹ ∈ ↑T
hS : ∀ {r : R} {x : M}, r • x ∈ S → x ∈ S ∨ r ∈ (S.colon ↑⊤).radical
hT : ∀ {r : R} {x : M}, r • x ∈ T → x ∈ T ∨ r ∈ (T.colon ↑⊤).radical
⊢ x✝¹ ∈ S ⊓ T ∨ r✝ ∈ (S.colon ↑⊤).radical ⊓ (T.colon ↑⊤).radical
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M' : Submodule R M
m : M
hm : m ∈ M'
s : ↥p
⊢ IsLocalization.mk' S 1 s • IsLocalizedModule.mk' f m 1 ∈ span S (⇑f '' ↑M')

Tactic:
exact smul_mem _ _ (subset_span ⟨m, hm, by simp⟩)

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M' M'' : Submodule R M
⊢ ∀ (y : ↥(localized₀ p f M')), ∃ x, x.2 • y = (toLocalized₀ p f M') x.1

Tactic:
by
    rintro ⟨y, x, hx, s, rfl⟩
    exact ⟨⟨⟨x, hx⟩, s⟩, by ext; simp⟩

Goal state after:
Module: Mathlib.RingTheory.Localization.Ideal

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommSemiring R
S : Finset R
hS : Ideal.span ↑S = ⊤
I : Ideal R
x : R
hx : x ∈ ⨅ f ∈ S, Ideal.under R (Ideal.map (algebraMap R (Localization.Away f)) I)
s : R
hs : s ∈ ↑S
⊢ ∃ n, ↑⟨s, hs⟩ ^ n • x ∈ I

Tactic:
simp only [Ideal.mem_iInf, Ideal.mem_comap] at hx

Goal state after:
case a
R : Type u_1
inst✝ : CommSemiring R
S : Finset R
hS : Ideal.span ↑S = ⊤
I : Ideal R
x s : R
hs : s ∈ ↑S
hx : ∀ i ∈ S, (algebraMap R (Localization.Away i)) x ∈ Ideal.map (algebraMap R (Localization.Away i)) I
⊢ ∃ n, ↑⟨s, hs⟩ ^ n • x ∈ I
Module: Mathlib.LinearAlgebra.Matrix.MvPolynomial

Goal state before:
pp: m : Type u_1
R : Type u_3
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : CommRing R
inst✝ : Nontrivial R
⊢ (mvPolynomialX m m R).det ≠ 0

Tactic:
intro h_det

Goal state after:
m : Type u_1
R : Type u_3
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : CommRing R
inst✝ : Nontrivial R
h_det : (mvPolynomialX m m R).det = 0
⊢ False
Module: Mathlib.LinearAlgebra.Matrix.Adjugate

Goal state before:
pp: case pos
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
i j : n
h : j = i
⊢ (Aᵀ.updateCol j (A i)).det = A.det

pp: case neg
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
i j : n
h : ¬j = i
⊢ (Aᵀ.updateCol j (A i)).det = 0

Tactic:
· -- i = j: this entry should be `A.det`
    subst h
    simp only [updateCol_transpose, det_transpose, updateRow_eq_self]

Goal state after:
case neg
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
i j : n
h : ¬j = i
⊢ (Aᵀ.updateCol j (A i)).det = 0
Module: Mathlib.LinearAlgebra.Dimension.Localization

Goal state before:
pp: R : Type uR
M : Type uM
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDomain R
w : Free.ChooseBasisIndex (FractionRing R) (LocalizedModule R⁰ M) → M
hw : LinearIndependent R w
⊢ lift.{uM, max uM uR} (Module.rank (FractionRing R) (LocalizedModule R⁰ M)) = lift.{max uR uM, uM} (Module.rank R M)

Tactic:
IsLocalization.rank_eq (FractionRing R) R⁰ le_rfl,

Goal state after:
R : Type uR
M : Type uM
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDomain R
w : Free.ChooseBasisIndex (FractionRing R) (LocalizedModule R⁰ M) → M
hw : LinearIndependent R w
⊢ lift.{uM, max uM uR} (Module.rank R (LocalizedModule R⁰ M)) = lift.{max uR uM, uM} (Module.rank R M)
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: case pos
n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
i j : n
inst✝ : Finite n
c : R
val✝ : Fintype n
a b : n
ha : i = a
⊢ updateRow 1 i (1 i + c • 1 j) a b = transvection i j c a b

pp: case neg
n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
i j : n
inst✝ : Finite n
c : R
val✝ : Fintype n
a b : n
ha : ¬i = a
⊢ updateRow 1 i (1 i + c • 1 j) a b = transvection i j c a b

Tactic:
· by_cases hb : j = b
    · simp only [ha, updateRow_self, Pi.add_apply, one_apply, Pi.smul_apply, hb, ↓reduceIte,
        smul_eq_mul, mul_one, transvection, add_apply, single_apply_same]
    · simp only [ha, updateRow_self, Pi.add_apply, one_apply, Pi.smul_apply, hb, ↓reduceIte,
        smul_eq_mul, mul_zero, add_zero, transvection, add_apply, and_false, not_false_eq_true,
        single_apply_of_ne]

Goal state after:
case neg
n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
i j : n
inst✝ : Finite n
c : R
val✝ : Fintype n
a b : n
ha : ¬i = a
⊢ updateRow 1 i (1 i + c • 1 j) a b = transvection i j c a b
Module: Mathlib.LinearAlgebra.Matrix.Adjugate

Goal state before:
pp: n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
r : α
A : Matrix n n α
⊢ (r • A).adjugate = r ^ (Fintype.card n - 1) • A.adjugate

Tactic:
rw [adjugate, adjugate, transpose_smul, cramer_smul]
  rfl

Goal state after:
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
S : Type u_3
inst✝ : CommSemiring S
f : R →+* S
x : Sˣ
r : R
n : ℕ
⊢ (eval₂ f x) (C r * T (-↑n)) = f r * ↑x⁻¹ ^ n

Tactic:
by
  rw [map_mul, eval₂_T_neg_n, eval₂_C]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid S
inst✝¹ : SMulWithZero R S
inst✝ : Monoid S
x : Sˣ
⊢ smeval 0 x = 0

Tactic:
by simp [smeval]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: n : Type u_1
p : Type u_2
R : Type u₂
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : Fintype p
e : n ≃ p
t : TransvectionStruct n R
⊢ (reindexEquiv e t).toMatrix = (reindexAlgEquiv R R e) t.toMatrix

Tactic:
rcases t with ⟨t_i, t_j, _⟩

Goal state after:
n : Type u_1
p : Type u_2
R : Type u₂
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : Fintype p
e : n ≃ p
t_i t_j : n
hij✝ : t_i ≠ t_j
c✝ : R
⊢ (reindexEquiv e { i := t_i, j := t_j, hij := hij✝, c := c✝ }).toMatrix =
    (reindexAlgEquiv R R e) { i := t_i, j := t_j, hij := hij✝, c := c✝ }.toMatrix
Module: Mathlib.LinearAlgebra.Dimension.Localization

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsDomain R
r : R
s : ↥R⁰
h : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s
n : ℕ
IH : ∀ (g : ℕ → R) (x : ℕ), ∑ i ∈ Finset.range n, g i • (r * ↑s ^ (i + x)) = 0 → ∀ i < n, g i = 0
g : ℕ → R
x : ℕ
hg : ∑ k ∈ Finset.range n, g (k + 1) • (r * ↑s ^ (k + 1 + x)) + g 0 • (r * ↑s ^ (0 + x)) = 0
i : ℕ
hin : i < n + 1
hg0 : g 0 = 0
⊢ g i = 0

Tactic:
simp only [hg0, zero_smul, add_zero, add_assoc] at hg
      cases i; exacts [hg0, IH _ _ hg _ (Nat.succ_lt_succ_iff.mp hin)]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: case pos
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
k n : ℕ
hn : n < r
IH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())
hn' : n < (listTransvecCol M).length
n' : Fin r := ⟨n, hn⟩
A : (listTransvecCol M)[n] = transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))
h : n' = i
⊢ (transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ())) *
        ((List.drop (n + 1) (listTransvecCol M)).prod * M))
      (inl i) (inr ()) =
    if n ≤ ↑i then 0 else M (inl i) (inr ())

Tactic:
have hni : n = i := by
        cases i
        simp only [n', Fin.mk_eq_mk] at h
        simp [h]

Goal state after:
case pos
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
k n : ℕ
hn : n < r
IH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())
hn' : n < (listTransvecCol M).length
n' : Fin r := ⟨n, hn⟩
A : (listTransvecCol M)[n] = transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))
h : n' = i
hni : n = ↑i
⊢ (transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ())) *
        ((List.drop (n + 1) (listTransvecCol M)).prod * M))
      (inl i) (inr ()) =
    if n ≤ ↑i then 0 else M (inl i) (inr ())