mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Dimension.DivisionRing

Goal state before:
pp: K : Type u
V✝ V₁ V₂ V₃ : Type v
ι : Type w
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V✝
inst✝² : Module K V✝
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
V : Type u₀
x✝¹ : AddCommGroup V
x✝ : Module K V
b : Module.Basis (Module.Free.ChooseBasisIndex K V) K V := Module.Free.chooseBasis K V
⊢ #↑(range ⇑b) = Module.rank K V

Tactic:
← lift_injective.eq_iff,

Goal state after:
K : Type u
V✝ V₁ V₂ V₃ : Type v
ι : Type w
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V✝
inst✝² : Module K V✝
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
V : Type u₀
x✝¹ : AddCommGroup V
x✝ : Module K V
b : Module.Basis (Module.Free.ChooseBasisIndex K V) K V := Module.Free.chooseBasis K V
⊢ lift.{?u.42, u₀} #↑(range ⇑b) = lift.{?u.42, u₀} (Module.rank K V)
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : HasRankNullity.{u, u_1} R
inst✝ : StrongRankCondition R
s : Set M
hs : LinearIndepOn R _root_.id s
t : Set (M ⧸ span R s)
ht : #↑t = Module.rank R (M ⧸ span R s)
ht' : LinearIndependent (ι := ↑t) R Subtype.val
sec : M ⧸ span R s → M
hsec : ∀ (b : M ⧸ span R s), (span R s).mkQ (sec b) = b
hsec' : ⇑(span R s).mkQ ∘ sec = _root_.id
⊢ s ∩ sec '' t ⊆ ∅

Tactic:
rintro _ ⟨hxs, ⟨x, hxt, rfl⟩⟩

Goal state after:
R : Type u_1
M : Type u
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : HasRankNullity.{u, u_1} R
inst✝ : StrongRankCondition R
s : Set M
hs : LinearIndepOn R _root_.id s
t : Set (M ⧸ span R s)
ht : #↑t = Module.rank R (M ⧸ span R s)
ht' : LinearIndependent (ι := ↑t) R Subtype.val
sec : M ⧸ span R s → M
hsec : ∀ (b : M ⧸ span R s), (span R s).mkQ (sec b) = b
hsec' : ⇑(span R s).mkQ ∘ sec = _root_.id
x : M ⧸ span R s
hxt : x ∈ t
hxs : sec x ∈ s
⊢ sec x ∈ ∅
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : HasRankNullity.{u, u_1} R
inst✝ : StrongRankCondition R
n : ℕ
v : Fin n → M
hv : LinearIndependent R v
h : ↑n < Module.rank R M
t : Set M
h₁ : Set.range v ⊆ t
h₂ : #↑(Set.range v) = Module.rank R M
h₃ : LinearIndepOn R _root_.id t
e : Set.range v = t
⊢ ↑n = Module.rank R M

Tactic:
← lift_injective.eq_iff,

Goal state after:
R : Type u_1
M : Type u
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : HasRankNullity.{u, u_1} R
inst✝ : StrongRankCondition R
n : ℕ
v : Fin n → M
hv : LinearIndependent R v
h : ↑n < Module.rank R M
t : Set M
h₁ : Set.range v ⊆ t
h₂ : lift.{?u.73, u} #↑(Set.range v) = lift.{?u.73, u} (Module.rank R M)
h₃ : LinearIndepOn R _root_.id t
e : Set.range v = t
⊢ ↑n = Module.rank R M
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : HasRankNullity.{u, u_1} R
N : Submodule R M
h : Module.rank R ↥N < Module.rank R M
e : Module.rank R (M ⧸ N) = 0
⊢ False

Tactic:
rw [← rank_quotient_add_rank N, e, zero_add] at h

Goal state after:
R : Type u_1
M : Type u
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : HasRankNullity.{u, u_1} R
N : Submodule R M
h : Module.rank R ↥N < Module.rank R ↥N
e : Module.rank R (M ⧸ N) = 0
⊢ False
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : HasRankNullity.{u, u_1} R
N : Submodule R M
h : Module.rank R ↥N < Module.rank R M
this : ¬Module.rank R (M ⧸ N) = 0
⊢ ∃ m, ∀ (r : R), r ≠ 0 → r • m ∉ N

Tactic:
rank_eq_zero_iff,

Goal state after:
R : Type u_1
M : Type u
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : HasRankNullity.{u, u_1} R
N : Submodule R M
h : Module.rank R ↥N < Module.rank R M
this : ¬∀ (x : M ⧸ N), ∃ a, a ≠ 0 ∧ a • x = 0
⊢ ∃ m, ∀ (r : R), r ≠ 0 → r • m ∉ N
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
U W : Submodule K V
h : IsCompl U W
⊢ finrank K ↥⊤ = finrank K V

Tactic:
exact finrank_top _ _

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
M : Type u_5
M₁ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : CommSemiring R₁
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R₁ M₁
I₁ I₂ : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M
H : B.IsAlt
x y : M₁
h : (B x) y = 0
⊢ (B y) x = 0

Tactic:
← neg H,

Goal state after:
R : Type u_1
R₁ : Type u_2
M : Type u_5
M₁ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : CommSemiring R₁
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R₁ M₁
I₁ I₂ : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M
H : B.IsAlt
x y : M₁
h : (B x) y = 0
⊢ -(B x) y = 0
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
c : Cocone F
m : (coequalizerCocone F).pt ⟶ c.pt
J : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ m = c.ι.app j
J1 : pushoutInl F ≫ m = c.ι.app WalkingParallelPair.one
⊢ pushout.inr (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left))
        (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right)) ≫
      m =
    pushout.inr (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left))
        (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right)) ≫
      pushout.desc (c.ι.app WalkingParallelPair.one) (c.ι.app WalkingParallelPair.one) ⋯

Tactic:
colimit.ι_desc,

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
c : Cocone F
m : (coequalizerCocone F).pt ⟶ c.pt
J : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ m = c.ι.app j
J1 : pushoutInl F ≫ m = c.ι.app WalkingParallelPair.one
⊢ pushout.inr (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left))
        (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right)) ≫
      m =
    (PushoutCocone.mk (c.ι.app WalkingParallelPair.one) (c.ι.app WalkingParallelPair.one) ⋯).ι.app WalkingSpan.right
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type uK
V : Type uV
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x✝ : Nonempty (V ≃ₗ[K] Dual K V)
e : V ≃ₗ[K] Dual K V
this✝ : ℵ₀ ≤ Module.rank K V
this : lift.{uK, uV} (Module.rank K V) = lift.{uV, max uK uV} (Module.rank K (Dual K V))
⊢ lift.{uK, uV} (Module.rank K V) = Module.rank K (V →ₗ[K] K)

Tactic:
lift_id'.{uV}

Goal state after:
K : Type uK
V : Type uV
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x✝ : Nonempty (V ≃ₗ[K] Dual K V)
e : V ≃ₗ[K] Dual K V
this✝ : ℵ₀ ≤ Module.rank K V
this : lift.{uK, uV} (Module.rank K V) = Module.rank K (Dual K V)
⊢ lift.{uK, uV} (Module.rank K V) = Module.rank K (V →ₗ[K] K)
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: ι : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : Field 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Finite ι
L : ι → E →ₗ[𝕜] 𝕜
K : E →ₗ[𝕜] 𝕜
h : ⨅ i, (L i).ker ≤ K.ker
x✝ : Fintype ι
φ : E →ₗ[𝕜] ι → 𝕜 := LinearMap.pi L
⊢ K ∈ span 𝕜 (Set.range L)

Tactic:
let p := ⨅ i, ker (L i)

Goal state after:
ι : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : Field 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Finite ι
L : ι → E →ₗ[𝕜] 𝕜
K : E →ₗ[𝕜] 𝕜
h : ⨅ i, (L i).ker ≤ K.ker
x✝ : Fintype ι
φ : E →ₗ[𝕜] ι → 𝕜 := LinearMap.pi L
p : Submodule 𝕜 E := ⨅ i, (L i).ker
⊢ K ∈ span 𝕜 (Set.range L)
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
W : Subspace K V
v : V
⊢ v ∉ W → v ∉ (dualAnnihilator W).dualCoannihilator

Tactic:
simp only [mem_dualAnnihilator, mem_dualCoannihilator]

Goal state after:
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
W : Subspace K V
v : V
⊢ v ∉ W → ¬∀ (φ : Dual K V), (∀ w ∈ W, φ w = 0) → φ v = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
B : LinearMap.BilinForm R M
inst✝ : IsTorsionFree R M
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
hle : ∀ (z : M), 0 ≤ (B z) z
r s : R
hl : r • x + s • y = 0
h0 : r = 0 → s ≠ 0
hr : ¬r = 0
hs : ¬s = 0
this : r * r * s * s * ((B x) x * (B y) y) = r * r * s * s * ((B x) y * (B y) x)
hrs : r * r * s * s ≠ 0
⊢ (B x) x * (B y) y ≤ (B x) y * (B y) x

Tactic:
exact le_of_eq <| mul_right_injective₀ hrs this

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Pullbacks

Goal state before:
pp: C : Type u
𝒞 : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝¹ : HasColimit (pair Y Z)
inst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))
ι₁ : Y ⟶ Y ⨿ Z := coprod.inl
ι₂ : Z ⟶ Y ⨿ Z := coprod.inr
c : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)
s : PushoutCocone f g
⊢ f ≫ ι₁ ≫ coprod.desc (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right) =
    (g ≫ ι₂) ≫ coprod.desc (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)

Tactic:
colimit.ι_desc,

Goal state after:
C : Type u
𝒞 : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝¹ : HasColimit (pair Y Z)
inst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))
ι₁ : Y ⟶ Y ⨿ Z := coprod.inl
ι₂ : Z ⟶ Y ⨿ Z := coprod.inr
c : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)
s : PushoutCocone f g
⊢ f ≫ (BinaryCofan.mk (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)).ι.app { as := WalkingPair.left } =
    (g ≫ ι₂) ≫ coprod.desc (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
not_surj : f.range ≠ ⊤
inj : Function.Injective ⇑f.dualMap
⊢ False

Tactic:
← lt_top_iff_ne_top

Goal state after:
K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
not_surj : f.range < ⊤
inj : Function.Injective ⇑f.dualMap
⊢ False
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasKernels C
inst✝² : IsNormalMonoCategory C
X Y Z : C
a : X ⟶ Z
b : Y ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
P : C
f : Z ⟶ P
haf : a ≫ f = 0
i : IsLimit (KernelFork.ofι a haf)
Q : C
g : Z ⟶ Q
hbg : b ≫ g = 0
i' : IsLimit (KernelFork.ofι b hbg)
a' : kernel (prod.lift f g) ⟶ (KernelFork.ofι a haf).pt
ha' : a' ≫ Fork.ι (KernelFork.ofι a haf) = kernel.ι (prod.lift f g)
b' : kernel (prod.lift f g) ⟶ (KernelFork.ofι b hbg).pt
hb' : b' ≫ Fork.ι (KernelFork.ofι b hbg) = kernel.ι (prod.lift f g)
s : PullbackCone a b
⊢ (kernel.lift (prod.lift f g) (s.snd ≫ b) ⋯ ≫ b') ≫ b = s.snd ≫ b

Tactic:
rw [KernelFork.ι_ofι] at hb'
              simp [hb']

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasRightHomology
A : C
f₁ f₂ : S.opcycles ⟶ A
h : S.pOpcycles ≫ f₁ = S.pOpcycles ≫ f₂
⊢ f₁ = f₂

Tactic:
simpa only [opcycles_ext_iff]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
⊢ f' ≫ Fork.ι ((KernelFork.ofι h.i ⋯).map F) = F.map h.f' ≫ Fork.ι ((KernelFork.ofι h.i ⋯).map F)

Tactic:
rw [Fork.IsLimit.lift_ι hi]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
⊢ Fork.ι (KernelFork.ofι (S.map F).f ⋯) = F.map h.f' ≫ Fork.ι ((KernelFork.ofι h.i ⋯).map F)
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
hf' : f' = F.map h.f'
wπ : f' ≫ F.map h.π = 0
e : parallelPair f' 0 ≅ parallelPair (F.map h.f') 0 :=
  parallelPair.ext (Iso.refl ((parallelPair f' 0).obj WalkingParallelPair.zero))
    (Iso.refl ((parallelPair f' 0).obj WalkingParallelPair.one)) ⋯ ⋯
⊢ Cofork.π ((CokernelCofork.ofπ h.π ⋯).map F) ≫ (Iso.refl ((CokernelCofork.ofπ h.π ⋯).map F).pt).hom =
    Cofork.π ((Cocone.precompose e.inv).obj (CokernelCofork.ofπ (F.map h.π) wπ))

Tactic:
by simp [e]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.RightHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesColimit (parallelPair S.f 0) F
this : PreservesLimit (parallelPair h.g' 0) F
wp : F.map S.f ≫ F.map h.p = 0
hp : IsColimit ((CokernelCofork.ofπ h.p ⋯).map F)
g' : F.obj h.Q ⟶ F.obj S.X₃ := hp.desc (CokernelCofork.ofπ (S.map F).g ⋯)
hg' : g' = F.map h.g'
wι : F.map h.ι ≫ g' = 0
e : parallelPair g' 0 ≅ parallelPair (F.map h.g') 0 :=
  parallelPair.ext (Iso.refl ((parallelPair g' 0).obj WalkingParallelPair.zero))
    (Iso.refl ((parallelPair g' 0).obj WalkingParallelPair.one)) ⋯ ⋯
⊢ (Iso.refl ((KernelFork.ofι h.ι ⋯).map F).pt).hom ≫
      Fork.ι ((Cone.postcompose e.hom).obj (KernelFork.ofι (F.map h.ι) wι)) =
    Fork.ι ((KernelFork.ofι h.ι ⋯).map F)

Tactic:
by simp [e]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₃ ⟶ S₁
⊢ (ψ ≫ φ₁).τ₁ = S₃.f ≫ ψ.τ₂ ≫ h.h₁ + ψ.τ₁ ≫ h.h₀ + (ψ ≫ φ₂).τ₁

Tactic:
comp_τ₁,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₃ ⟶ S₁
⊢ ψ.τ₁ ≫ φ₁.τ₁ = S₃.f ≫ ψ.τ₂ ≫ h.h₁ + ψ.τ₁ ≫ h.h₀ + (ψ ≫ φ₂).τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₃ ⟶ S₁
⊢ ψ.τ₁ ≫ φ₁.τ₁ = S₃.f ≫ ψ.τ₂ ≫ h.h₁ + ψ.τ₁ ≫ h.h₀ + (ψ ≫ φ₂).τ₁

Tactic:
comp_τ₁,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₃ ⟶ S₁
⊢ ψ.τ₁ ≫ φ₁.τ₁ = S₃.f ≫ ψ.τ₂ ≫ h.h₁ + ψ.τ₁ ≫ h.h₀ + ψ.τ₁ ≫ φ₂.τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ (φ₁ ≫ ψ).τ₁ = S₁.f ≫ h.h₁ ≫ ψ.τ₁ + h.h₀ ≫ ψ.τ₁ + (φ₂ ≫ ψ).τ₁

Tactic:
comp_τ₁,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ φ₁.τ₁ ≫ ψ.τ₁ = S₁.f ≫ h.h₁ ≫ ψ.τ₁ + h.h₀ ≫ ψ.τ₁ + (φ₂ ≫ ψ).τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ φ₁.τ₁ ≫ ψ.τ₁ = S₁.f ≫ h.h₁ ≫ ψ.τ₁ + h.h₀ ≫ ψ.τ₁ + (φ₂ ≫ ψ).τ₁

Tactic:
comp_τ₁,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ φ₁.τ₁ ≫ ψ.τ₁ = S₁.f ≫ h.h₁ ≫ ψ.τ₁ + h.h₀ ≫ ψ.τ₁ + φ₂.τ₁ ≫ ψ.τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
inst✝² : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
inst✝¹ : IsIso φ.τ₁
inst✝ : IsIso φ.τ₃
⊢ IsIso φ.τ₂

Tactic:
by
  have := h₁.mono_f
  have := h₂.mono_f
  have := h₁.epi_g
  have := h₂.epi_g
  have := mono_τ₂_of_exact_of_mono φ h₁.exact
  have := epi_τ₂_of_exact_of_epi φ h₂.exact
  apply isIso_of_mono_of_epi

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: case mp
S : ShortComplex Ab
x₂ : ↑S.X₂
h : (ConcreteCategory.hom S.g) x₂ = 0 → ∃ x₁, (ConcreteCategory.hom S.f) x₁ = x₂
⊢ x₂ ∈ Set.range ⇑(ConcreteCategory.hom S.f) → (ConcreteCategory.hom S.g) x₂ = 0

Tactic:
rintro ⟨x₁, rfl⟩

Goal state after:
case mp
S : ShortComplex Ab
x₁ : ↑S.X₁
h :
  (ConcreteCategory.hom S.g) ((ConcreteCategory.hom S.f) x₁) = 0 →
    ∃ x₁_1, (ConcreteCategory.hom S.f) x₁_1 = (ConcreteCategory.hom S.f) x₁
⊢ (ConcreteCategory.hom S.g) ((ConcreteCategory.hom S.f) x₁) = 0
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
P : Subobject X
⊢ p P

Tactic:
by
  induction P using Quotient.inductionOn' with | _ a
  exact h a.arrow

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

Goal state before:
pp: case inr
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X A₁ A₂ : C
i₁ : A₁ ⟶ X
i₂ : A₂ ⟶ X
inst✝¹ : Mono i₁
inst✝ : Mono i₂
f : A₁ ⟶ A₂
fac : f ≫ i₂ = i₁
hf : ¬IsIso f
h : mk i₁ = mk i₂
⊢ mk i₁ < mk i₂

Tactic:
exfalso
    apply hf
    convert! (isoOfMkEqMk i₁ i₂ h).isIso_hom
    rw [← cancel_mono i₂, isoOfMkEqMk_hom, ofMkLEMk_comp, fac]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
w : φ.τ₂ ≫ S₂.g = 0
⊢ IsIso (S₂.liftCycles φ.τ₂ ⋯) →
    { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact ∧ Mono φ.τ₂

Tactic:
intro h
    have : Mono φ.τ₂ := by
      rw [← S₂.liftCycles_i φ.τ₂ w]
      apply mono_comp
    refine ⟨?_, this⟩
    apply exact_of_f_is_kernel
    exact IsLimit.ofIsoLimit S₂.cyclesIsKernel
      (Fork.ext (asIso (S₂.liftCycles φ.τ₂ w)).symm (by simp))

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : Mono f
inst✝ : Mono g
x : Subobject X
⊢ (map (f ≫ g)).obj x = (map g).obj ((map f).obj x)

Tactic:
induction x using Quotient.inductionOn' with | _ t
  exact Quotient.sound ⟨(MonoOver.mapComp _ _).app t⟩

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

Goal state before:
pp: case g.refine_2
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : HasPullbacks C
X Y Z W : C
f : X ⟶ Y
g : X ⟶ Z
h : Y ⟶ W
k : Z ⟶ W
inst✝¹ : Mono h
inst✝ : Mono g
comm : f ≫ h = g ≫ k
t : IsLimit (PullbackCone.mk f g comm)
a : MonoOver Y
⊢ pullback.lift (pullback.fst ((MonoOver.forget W).obj ((MonoOver.map h).obj a)).hom k)
        (PullbackCone.IsLimit.lift t (pullback.fst ((MonoOver.forget W).obj ((MonoOver.map h).obj a)).hom k ≫ a.arrow)
          (pullback.snd ((MonoOver.forget W).obj ((MonoOver.map h).obj a)).hom k) ⋯)
        ⋯ ≫
      ((MonoOver.map g).obj ((MonoOver.pullback f).obj a)).arrow =
    ((MonoOver.pullback k).obj ((MonoOver.map h).obj a)).arrow

Tactic:
· dsimp
      rw [pullback.lift_snd_assoc]
      apply PullbackCone.IsLimit.lift_snd

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
I : Type u_1
A B : C
s : Finset I
P : I → Subobject B
f : A ⟶ B
⊢ (s.inf P).Factors f ↔ ∀ i ∈ s, (P i).Factors f

Tactic:
classical
  induction s using Finset.induction_on with
  | empty => simp [top_factors]
  | insert _ _ _ ih => simp [ih]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Filtered

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
⊢ IsCofiltered C ↔
    ∀ {J : Type v} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), ∃ X, Nonempty (limit (F ⋙ coyoneda.obj (op X)))

Tactic:
by
  rw [IsCofiltered.iff_cone_nonempty.{v}]
  refine ⟨fun h J _ _ F => ?_, fun h J _ _ F => ?_⟩
  · obtain ⟨c⟩ := h F
    exact ⟨c.pt, ⟨(limitCompCoyonedaIsoCone F c.pt).inv c.π⟩⟩
  · obtain ⟨pt, ⟨π⟩⟩ := h F
    exact ⟨⟨pt, (limitCompCoyonedaIsoCone F pt).hom π⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
hX : IsInitial X
A : C
i : A ⟶ X
w✝ : Mono i
fac : hX.to A ≫ i = 𝟙 X
e : A ≅ X := { hom := i, inv := hX.to A, hom_inv_id := ⋯, inv_hom_id := fac }
⊢ e.hom ≫ 𝟙 X = i

Tactic:
by simp [e]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
H : Type u_3
inst✝¹ : Category.{v_3, u_3} H
inst✝ : ∀ (F : C ⥤ H), L.HasRightKanExtension F
F : D ⥤ H
G₁ G₂ : C ⥤ H
β : ((whiskeringLeft C D H).obj L).obj F ⟶ G₁
f : G₁ ⟶ G₂
⊢ L.whiskerLeft (((L.ran.obj G₂).homEquivOfIsRightKanExtension (L.ranCounit.app G₂) F).symm (β ≫ f)) ≫
      L.ranCounit.app G₂ =
    L.whiskerLeft (((L.ran.obj G₁).homEquivOfIsRightKanExtension (L.ranCounit.app G₁) F).symm β ≫ L.ran.map f) ≫
      L.ranCounit.app G₂

Tactic:
ext X
        dsimp [homEquivOfIsRightKanExtension]
        rw [liftOfIsRightKanExtension_fac_app, NatTrans.comp_app, assoc]
        have h := congr_app (L.ranCounit.naturality f) X
        dsimp at h ⊢
        rw [h, liftOfIsRightKanExtension_fac_app_assoc]

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : ObjectProperty C
inst✝ : Balanced C
hP : P.IsSeparating
⊢ P.IsDetecting

Tactic:
intro X Y f hf

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : ObjectProperty C
inst✝ : Balanced C
hP : P.IsSeparating
X Y : C
f : X ⟶ Y
hf : ∀ (G : C), P G → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h
⊢ IsIso f