mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.UniformSpace.UniformEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
e : β → α
h_e : IsUniformInducing e
h_dense : DenseRange e
f : β → γ
h_f : UniformContinuous f
inst✝ : CompleteSpace γ
d : Set (γ × γ)
hd : d ∈ 𝓤 γ
s : Set (γ × γ)
hs : s ∈ 𝓤 γ
hs_comp : s ○ (s ○ s) ⊆ d
h_pnt : ∀ {a : α} {m : Set α}, m ∈ 𝓝 a → ∃ c ∈ f '' e ⁻¹' m, (c, ⋯.extend f a) ∈ s ∧ (⋯.extend f a, c) ∈ s
this : Prod.map f f ⁻¹' s ∈ 𝓤 β
⊢ Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)

Tactic:
by
    rwa [← h_e.comap_uniformity] at this

Goal state after:
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
β : Type u_4
γ : Type u_5
δ : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : One β
inst✝¹ : One γ
inst✝ : One δ
f : α → β
f₂ : α → γ
m : β → γ → δ
hf : f =ᶠ[coclosedCompact α] 1
hf₂ : f₂ =ᶠ[coclosedCompact α] 1
hm : m 1 1 = 1
x : α
hx : f x = 1 x
⊢ f₂ x = 1 x → m (f x) (f₂ x) = 1 x

Tactic:
hx₂

Goal state after:
α : Type u_2
β : Type u_4
γ : Type u_5
δ : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : One β
inst✝¹ : One γ
inst✝ : One δ
f : α → β
f₂ : α → γ
m : β → γ → δ
hf : f =ᶠ[coclosedCompact α] 1
hf₂ : f₂ =ᶠ[coclosedCompact α] 1
hm : m 1 1 = 1
x : α
hx : f x = 1 x
hx₂ : f₂ x = 1 x
⊢ m (f x) (f₂ x) = 1 x
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : MulZeroClass β
f f' : α → β
hf : f' =ᶠ[coclosedCompact α] 0
⊢ f * f' =ᶠ[coclosedCompact α] 0

Tactic:
exact hf.mono fun x hx => by simp_rw [Pi.mul_apply, hx, Pi.zero_apply, mul_zero]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α : Type u_6
β' : Type u_9
uα : UniformSpace α
u : κ → UniformSpace β'
F : ι → β' → α
k : κ
hk : UniformEquicontinuous F
⊢ UniformEquicontinuous F

Tactic:
simp_rw [uniformEquicontinuous_iff_uniformContinuous (uβ := _)] at hk ⊢
  exact uniformContinuous_iInf_dom hk

Goal state after:
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
a b c : α
⊢ (fun x ↦ a * x) ⁻¹' Ico b c = Ico (b / a) (c / a)

Tactic:
by
  simp [← Ici_inter_Iio]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
a b c : α
⊢ (fun x ↦ x * a) ⁻¹' Ico b c = Ico (b / a) (c / a)

Tactic:
by
  simp [← Ici_inter_Iio]

Goal state after:
Module: Mathlib.Topology.Algebra.MulAction

Goal state before:
pp: M : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : SMul M X
inst✝² : ContinuousSMul M X
f : Y → M
g : Y → X
b : Y
s : Set Y
inst✝¹ : SMul Mᵐᵒᵖ X
inst✝ : IsCentralScalar M X
⊢ Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ p.1 • p.2

Tactic:
suffices Continuous fun p : M × X => MulOpposite.op p.fst • p.snd from
      this.comp (MulOpposite.continuous_unop.prodMap continuous_id)

Goal state after:
M : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : SMul M X
inst✝² : ContinuousSMul M X
f : Y → M
g : Y → X
b : Y
s : Set Y
inst✝¹ : SMul Mᵐᵒᵖ X
inst✝ : IsCentralScalar M X
⊢ Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ MulOpposite.op p.1 • p.2
Module: Mathlib.Topology.ContinuousMap.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
⊢ ↑(f.symm.trans f) = ContinuousMap.id β

Tactic:
symm_trans_self,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
⊢ ↑(Homeomorph.refl β) = ContinuousMap.id β
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
β : Type u_3
G : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : ContinuousConstSMul G α
inst✝ : TopologicalSpace β
f : β → α
c : G
⊢ (Continuous[inst✝, inst✝⁴] fun x ↦ c • f x) ↔ Continuous[inst✝, inst✝⁴] f

Tactic:
by
  simp only [continuous_iff_continuousAt, continuousAt_const_smul_iff]

Goal state after:
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul G α
c : G
s : Set α
hs : closure[inst✝³] s = univ
⊢ closure[inst✝³] (c • s) = univ

Tactic:
closure_smul,

Goal state after:
α : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul G α
c : G
s : Set α
hs : closure[inst✝³] s = univ
⊢ c • closure[inst✝³] s = univ
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
p : Filter ι
s : Set β
F : ι → α → β
f : α → β
t : Set α
hF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s
hf : ∀ x ∈ t, f x ∈ s
g : β → γ
hg : UniformContinuousOn g s
h : TendstoUniformlyOn F f p t
⊢ TendstoUniformlyOn (fun i x ↦ g (F i x)) (fun x ↦ g (f x)) p t

Tactic:
rw [tendstoUniformlyOn_iff_restrict]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
p : Filter ι
s : Set β
F : ι → α → β
f : α → β
t : Set α
hF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s
hf : ∀ x ∈ t, f x ∈ s
g : β → γ
hg : UniformContinuousOn g s
h : TendstoUniformlyOn F f p t
⊢ TendstoUniformly (fun n ↦ t.domRestrict fun x ↦ g (F n x)) (t.domRestrict fun x ↦ g (f x)) p
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: case mpr
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous[inst✝¹, inst✝] f
H : ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
⊢ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, (comap f (𝓝 y) ⊓ ℱ).NeBot → ∃ x, f x = y ∧ (𝓝 x ⊓ ℱ).NeBot

Tactic:
← exists_ultrafilter_iff,

Goal state after:
case mpr
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous[inst✝¹, inst✝] f
H : ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
⊢ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, (∃ u, ↑u ≤ comap f (𝓝 y) ⊓ ℱ) → ∃ x, f x = y ∧ ∃ u, ↑u ≤ 𝓝 x ⊓ ℱ
Module: Mathlib.Topology.Algebra.Group.Quotient

Goal state before:
pp: G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
N : Subgroup G
⊢ DiscreteTopology (G ⧸ N) ↔ IsOpen[inst✝²] ↑N

Tactic:
rw [← QuotientGroup.preimage_mk_one, MulAction.IsPretransitive.discreteTopology_iff G (mk 1),
      isOpen_coinduced]

Goal state after:
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
N : Subgroup G
⊢ IsOpen[inst✝²] (Quotient.mk' ⁻¹' {↑1}) ↔ IsOpen[inst✝²] (mk ⁻¹' {↑1})
Module: Mathlib.Order.Filter.AtTopBot.Ring

Goal state before:
pp: α : Type u_1
inst✝² : Ring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
⊢ ¬Tendsto (fun x ↦ x ^ 0) atTop atBot

Tactic:
simp [not_tendsto_const_atBot]

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : ContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
x : G₀
m : ℤ
h : x ≠ 0 ∨ 0 ≤ m
⊢ ContinuousAt (fun x ↦ x ^ m) x

Tactic:
rcases m with m | m

Goal state after:
case ofNat
G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : ContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
x : G₀
m : ℕ
h : x ≠ 0 ∨ 0 ≤ Int.ofNat m
⊢ ContinuousAt (fun x ↦ x ^ Int.ofNat m) x

case negSucc
G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : ContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
x : G₀
m : ℕ
h : x ≠ 0 ∨ 0 ≤ Int.negSucc m
⊢ ContinuousAt (fun x ↦ x ^ Int.negSucc m) x
Module: Mathlib.Order.Filter.AtTopBot.CompleteLattice

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
⊢ ⨆ a, f (g a) = ⨆ b, f b

Tactic:
by
  have : Nonempty α := nonempty_of_neBot l
  have : Nonempty β := .map g ‹_›
  rw [← csInf_upperBounds_range, ← csInf_upperBounds_range,
    ← hf.upperBounds_range_comp_tendsto_atTop hg, Function.comp_def]
  exacts [hb, hb.mono <| range_comp_subset_range _ _]

Goal state after:
Module: Mathlib.Topology.Algebra.Field

Goal state before:
pp: case inr
K✝ : Type u_1
inst✝⁴ : DivisionRing K✝
inst✝³ : TopologicalSpace K✝
α : Type u_2
inst✝² : Field α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalDivisionRing α
K : Subfield α
x : α
hx : x ∈ closure ↑K
h : x ≠ 0
⊢ x⁻¹ ∈ closure ↑K

Tactic:
rw [← inv_coe_set, ← Set.image_inv_eq_inv]
        exact mem_closure_image (continuousAt_inv₀ h) hx

Goal state after:
Module: Mathlib.Topology.Algebra.Field

Goal state before:
pp: α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : ContinuousMul 𝕜
hS : IsPreconnected S
hf : ContinuousOn f S
hg : ContinuousOn g S
hsq : EqOn (f ^ 2) (g ^ 2) S
hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0
x : α
hx : x ∈ S
⊢ ((f / g) ^ 2) x = 1 x

Tactic:
simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx)), div_pow] using hsq hx

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : ∀ (x : α), ↑e x = ↑e' x
hsymm : ∀ (x : β), ↑e.symm x = ↑e'.symm x
hs : e.source = e'.source
x : α
⊢ ↑e x = ↑e' x

Tactic:
exact h x

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
A : (e.trans e.symm).source = e.source
x : α
hx : x ∈ (e.trans e.symm).source
⊢ ↑(e.trans e.symm) x = ↑(ofSet e.source) x

Tactic:
A

Goal state after:
α : Type u_1
β : Type u_2
e : PartialEquiv α β
A : (e.trans e.symm).source = e.source
x : α
hx : x ∈ e.source
⊢ ↑(e.trans e.symm) x = ↑(ofSet e.source) x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : AddCommGroup 𝕜
p : 𝕜
n : ℕ
h0 : n ≠ 0
hn : IsSMulRegular 𝕜 n
f : ↑{x | n • x = 0} → Fin n
hf : ∀ (x : ↑{x | n • x = 0}), ∃ y, ↑y = ↑x ∧ n • y = ↑(f x) • p
x x' : ↑{x | n • x = 0}
eq : f x = f x'
y : 𝕜
hyx : ↑y = ↑x
y' : 𝕜
hy : y = y'
hyx' : ↑y' = ↑x'
hy' : n • y' = ↑(f x') • p
⊢ ↑x = ↑x'

Tactic:
rw [← hyx, hy, hyx']

Goal state after:
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen[inst✝³] a
t : Set α := {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x}
z : α → α
hz : ∀ x ∈ t, z x < x
h'z : ∀ x ∈ t, Ioc (z x) x ⊆ a
this : t.PairwiseDisjoint fun x ↦ Ioc (z x) x
x : α
hx : x ∈ t
⊢ ∀ x_1 ∈ Ioo (z x) (y x), x_1 ∈ Ioc (z x) x

Tactic:
mem_Ioo,

Goal state after:
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen[inst✝³] a
t : Set α := {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x}
z : α → α
hz : ∀ x ∈ t, z x < x
h'z : ∀ x ∈ t, Ioc (z x) x ⊆ a
this : t.PairwiseDisjoint fun x ↦ Ioc (z x) x
x : α
hx : x ∈ t
⊢ ∀ (x_1 : α), z x < x_1 ∧ x_1 < y x → x_1 ∈ Ioc (z x) x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
hu : ¬IsOfFinAddOrder u
⊢ Infinite { u // ¬IsOfFinAddOrder u }

Tactic:
← coe_ofPred,

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
hu : ¬IsOfFinAddOrder u
⊢ Infinite ↑{x | ¬IsOfFinAddOrder x}
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : n > 0
⊢ Nat.card { u // addOrderOf u = n } = n.totient

Tactic:
← coe_ofPred,

Goal state after:
case inr
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : n > 0
⊢ Nat.card ↑{x | addOrderOf x = n} = n.totient
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
a✝ : Epi f
⊢ ∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f

Tactic:
A

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
a✝ : Epi f
A : C
⊢ ∀ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles
⊢ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f

Tactic:
A

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles
A : C
⊢ ∀ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
⊢ ∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles

Tactic:
A

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
A : C
⊢ ∀ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles
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
h : S.IsComplex
⊢ S.map' 0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4 ≫
      S.map' 1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5 =
    0

Tactic:
exact h.zero 0 (by lia)

Goal state after:
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
⊢ S.IsComplex →
    S.map' 0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4 ≫
        S.map' 1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5 =
      0

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

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Refinements

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
h : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ
⊢ ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),
    y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f

Tactic:
A

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ
A : C
⊢ ∀ (y₂ : A ⟶ S₂.X₂),
    y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f
Module: Mathlib.CategoryTheory.Abelian.Refinements

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
h : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ
A : C
y₂ : A ⟶ S₂.X₂
hy₂ : y₂ ≫ S₂.g = 0
A₁ : C
π₁ : A₁ ⟶ A
hπ₁ : Epi π₁
γ : A₁ ⟶ S₁.homology
hγ : π₁ ≫ S₂.liftCycles y₂ hy₂ ≫ S₂.homologyπ = γ ≫ homologyMap φ
A₂ : C
π₂ : A₂ ⟶ A₁
hπ₂ : Epi π₂
x₂ : A₂ ⟶ S₁.X₂
hx₂ : x₂ ≫ S₁.g = 0
fac : π₂ ≫ γ = S₁.liftCycles x₂ hx₂ ≫ S₁.homologyπ
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f

Tactic:
replace hγ := π₂ ≫= hγ

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ
A : C
y₂ : A ⟶ S₂.X₂
hy₂ : y₂ ≫ S₂.g = 0
A₁ : C
π₁ : A₁ ⟶ A
hπ₁ : Epi π₁
γ : A₁ ⟶ S₁.homology
A₂ : C
π₂ : A₂ ⟶ A₁
hπ₂ : Epi π₂
x₂ : A₂ ⟶ S₁.X₂
hx₂ : x₂ ≫ S₁.g = 0
fac : π₂ ≫ γ = S₁.liftCycles x₂ hx₂ ≫ S₁.homologyπ
hγ : π₂ ≫ π₁ ≫ S₂.liftCycles y₂ hy₂ ≫ S₂.homologyπ = π₂ ≫ γ ≫ homologyMap φ
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h :
  ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),
    y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f
⊢ ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ

Tactic:
A

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h :
  ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),
    y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f
A : C
⊢ ∀ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ
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
A : C
x₂ : A ⟶ S.L₀.X₂
⊢ x₂ ≫ S.L₀.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.L₀.f

Tactic:
hx₂

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A : C
x₂ : A ⟶ S.L₀.X₂
hx₂ : x₂ ≫ S.L₀.g = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.L₀.f
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
A : C
x₂ : A ⟶ S.L₀'.X₂
⊢ x₂ ≫ S.L₀'.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.L₀'.f

Tactic:
hx₂

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A : C
x₂ : A ⟶ S.L₀'.X₂
hx₂ : x₂ ≫ S.L₀'.g = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.L₀'.f
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
S₁ S₂ : SnakeInput C
f : S₁ ⟶ S₂
⊢ S₁.δ ≫ f.f₃.τ₁ = f.f₀.τ₃ ≫ S₂.δ

Tactic:
← cancel_epi (pullback.snd _ _ : S₁.P ⟶ _),

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : SnakeInput C
f : S₁ ⟶ S₂
⊢ pullback.snd S₁.L₁.g S₁.v₀₁.τ₃ ≫ S₁.δ ≫ f.f₃.τ₁ = pullback.snd S₁.L₁.g S₁.v₀₁.τ₃ ≫ f.f₀.τ₃ ≫ S₂.δ
Module: Mathlib.CategoryTheory.Limits.Types.Pullbacks

Goal state before:
pp: X Y Z : Type u
f : X ⟶ Z
g : Y ⟶ Z
⊢ (pullbackIsoPullback f g).inv ≫ pullback.snd f g = ↾fun p ↦ (↑p).2

Tactic:
by
  ext
  exact PullbackCone.IsLimit.equivPullbackObj_symm_apply_snd (pullbackIsPullback f g) _

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

Goal state before:
pp: X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h : IsPullback t l r b
x₂ : X₂
x₃ : X₃
hx : (hom r) x₂ = (hom b) x₃
x₁ : h.cone.pt
hx₁ : ↑((PullbackCone.IsLimit.equivPullbackObj h.isLimit) x₁) = ↑⟨(x₂, x₃), hx⟩
⊢ ∃ x₁, (hom t) x₁ = x₂ ∧ (hom l) x₁ = x₃

Tactic:
exact ⟨x₁, congr_arg _root_.Prod.fst hx₁,
    congr_arg _root_.Prod.snd hx₁⟩

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

Goal state before:
pp: R : Type u
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
x : ↑S.X₂
⊢ (ConcreteCategory.hom S.pOpcycles) x = 0 ↔ x ∈ (ModuleCat.Hom.hom S.f).range

Tactic:
by
  simpa using moduleCat_pOpcycles_eq_iff _ x 0

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Biproducts

Goal state before:
pp: R : Type u
A M B : Type v
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup A
inst✝⁴ : Module R A
inst✝³ : AddCommGroup B
inst✝² : Module R B
inst✝¹ : AddCommGroup M
inst✝ : Module R M
j : A →ₗ[R] M
g : M →ₗ[R] B
f : B →ₗ[R] M
hj : Function.Injective ⇑j
exac : j.range = g.ker
h : g ∘ₗ f = LinearMap.id
⊢ Mono (ShortComplex.moduleCatMkOfKerLERange (↟j) (↟g) ⋯).f

Tactic:
by simpa only [ModuleCat.mono_iff_injective]

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

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : ∀ (A : adj.toComonad.Coalgebra), HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))
B : C
⊢ (comparisonAdjunction adj).unit.app B =
    limit.lift (parallelPair (G.map (F.map (adj.unit.app B))) (adj.unit.app (G.obj (F.obj B)))) (unitFork adj B)

Tactic:
by
  apply equalizer.hom_ext
  change
    equalizer.lift ((adj.homEquiv B _) (𝟙 _)) _ ≫ equalizer.ι _ _ =
      equalizer.lift _ _ ≫ equalizer.ι _ _
  simp [Adjunction.homEquiv_unit]

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ComposableArrows C 2
f : F ⟶ G
⊢ IsIso f ↔ IsIso (f.app 0) ∧ IsIso (f.app 1) ∧ IsIso (f.app 2)

Tactic:
rw [NatTrans.isIso_iff_isIso_app]
  exact ⟨fun h ↦ ⟨h 0, h 1, h 2⟩, fun _ i ↦ by fin_cases i <;> tauto⟩

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

Goal state before:
pp: A : Type u_2
inst✝⁶ : CommSemiring A
B : Type u_3
inst✝⁵ : CommSemiring B
C : Type u_4
inst✝⁴ : Semiring C
inst✝³ : Algebra A B
inst✝² : Algebra B C
inst✝¹ : Algebra A C
inst✝ : IsScalarTower A B C
𝔓 : Ideal C
P : Ideal B
p : Ideal A
hp : 𝔓.LiesOver p
hP : 𝔓.LiesOver P
⊢ p = under A P

Tactic:
𝔓.over_def p,

Goal state after:
A : Type u_2
inst✝⁶ : CommSemiring A
B : Type u_3
inst✝⁵ : CommSemiring B
C : Type u_4
inst✝⁴ : Semiring C
inst✝³ : Algebra A B
inst✝² : Algebra B C
inst✝¹ : Algebra A C
inst✝ : IsScalarTower A B C
𝔓 : Ideal C
P : Ideal B
p : Ideal A
hp : 𝔓.LiesOver p
hP : 𝔓.LiesOver P
⊢ under A 𝔓 = under A P
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
n : ℕ
obj : Fin (n + 1) → C
mapSucc : (i : Fin n) → obj i.castSucc ⟶ obj i.succ
⊢ ∃ F e, ∀ (i : ℕ) (hi : i < n), mapSucc ⟨i, hi⟩ = (e ⟨i, ⋯⟩).inv ≫ F.map' i (i + 1) ⋯ hi ≫ (e ⟨i + 1, ⋯⟩).hom

Tactic:
induction n with

Goal state after:
Module: Mathlib.RingTheory.IsPrimary

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submodule R M
x✝ : S ≠ ⊤
⊢ (∀ {r : R} {x : M}, r • x ∈ S → x ∈ S ∨ ∃ n, r ^ n • ⊤ ≤ S) ↔
    ∀ (r : R), ∀ x ∉ S.mkQ.ker, r • x ∈ S.mkQ.ker → ∃ n, r ^ n • ⊤ = ⊥

Tactic:
ker_mkQ

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submodule R M
x✝ : S ≠ ⊤
⊢ (∀ {r : R} {x : M}, r • x ∈ S → x ∈ S ∨ ∃ n, r ^ n • ⊤ ≤ S) ↔ ∀ (r : R), ∀ x ∉ S, r • x ∈ S → ∃ n, r ^ n • ⊤ = ⊥
Module: Mathlib.RingTheory.IsPrimary

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submodule R M
x✝ : S ≠ ⊤
a✝¹ : R
a✝ : M
⊢ (a✝¹ • a✝ ∈ S → a✝ ∈ S ∨ ∃ n, a✝¹ ^ n • ⊤ ≤ S) ↔ a✝¹ • a✝ ∈ S → a✝ ∈ S ∨ ∃ n, a✝¹ ^ n • ⊤ ≤ S.mkQ.ker

Tactic:
ker_mkQ

Goal state after: