mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
m n : ℕ
⊢ ↑↑m = ↑n → m = n

Tactic:
norm_cast

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
m n : ℕ
⊢ m = n → m = n
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
⊢ ∀ (i j : (glueData hf).J),
    (ConcreteCategory.hom (F.obj.map ((glueData hf).f i j).op)) (yonedaEquiv (f i)) =
      (ConcreteCategory.hom (F.obj.map (((glueData hf).f j i).op ≫ ((glueData hf).t i j).op))) (yonedaEquiv (f j))

Tactic:
intro i

Goal state after:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j k : ι
i : (glueData hf).J
⊢ ∀ (j : (glueData hf).J),
    (ConcreteCategory.hom (F.obj.map ((glueData hf).f i j).op)) (yonedaEquiv (f i)) =
      (ConcreteCategory.hom (F.obj.map (((glueData hf).f j i).op ≫ ((glueData hf).t i j).op))) (yonedaEquiv (f j))
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
inst✝¹ : HasPullbacks C
a b : C
f : a ⟶ b
inst✝ : ∀ (c : C) (g : c ⟶ b), PreservesLimit (cospan f g) F
c : C
g : c ⟶ b
⊢ ∃ b_1 snd fst, IsPullback fst (F.map snd) (F.map f) (F.map g)

Tactic:
refine ⟨Limits.pullback f g, Limits.pullback.snd f g, F.map (Limits.pullback.fst f g), ?_⟩

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
inst✝¹ : HasPullbacks C
a b : C
f : a ⟶ b
inst✝ : ∀ (c : C) (g : c ⟶ b), PreservesLimit (cospan f g) F
c : C
g : c ⟶ b
⊢ IsPullback (F.map (pullback.fst f g)) (F.map (pullback.snd f g)) (F.map f) (F.map g)
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ ↑(padicNorm p { num := n, den := d, den_nz := hn, reduced := hd }) ≤ 1

Tactic:
norm_cast

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ padicNorm p { num := n, den := d, den_nz := hn, reduced := hd } ≤ 1
Module: Mathlib.AlgebraicTopology.DoldKan.Faces

Goal state before:
pp: case hf
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
k : ℕ
hk : k < q
⊢ ⟨a + 1, ⋯⟩.succ < Fin.cast ⋯ (Fin.natAdd (a + 3) ⟨k, hk⟩)

Tactic:
simp only [Fin.lt_def]

Goal state after:
case hf
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
k : ℕ
hk : k < q
⊢ ↑⟨a + 1, ⋯⟩.succ < ↑(Fin.cast ⋯ (Fin.natAdd (a + 3) ⟨k, hk⟩))
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 1 ≤ ↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num

Tactic:
norm_cast

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 1 ≤ p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
h : ¬↑p ∣ k
⊢ ‖↑k‖ = ‖↑↑k‖

Tactic:
norm_cast

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ ↑k = ↑↑k

Tactic:
norm_cast

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ ↑(padicNorm p ↑k) ≤ ↑↑p ^ (-↑n) ↔ ↑p ^ n ∣ k

Tactic:
norm_cast

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ padicNorm p ↑k ≤ ↑p ^ (-↑n) ↔ ↑(p ^ n) ∣ k
Module: Mathlib.AlgebraicTopology.DoldKan.PInfty

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
⊢ QInfty.f 0 = 0

Tactic:
dsimp [QInfty]
  simp only [sub_self]

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'
⊢ ∀ j ≥ N, ‖↑(f - const norm q) j‖ < ε

Tactic:
intro i

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'
i : ℕ
⊢ i ≥ N → ‖↑(f - const norm q) i‖ < ε
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
⊢ x ≠ 0 → ‖x‖ = ↑p ^ (-x.valuation)

Tactic:
induction x using Quotient.inductionOn with

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
⊢ ⟦f⟧ ≠ 0 → ‖⟦f⟧‖ = ↑p ^ (-valuation ⟦f⟧)
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hx : ¬x = 0
hy : ¬y = 0
⊢ (if x * y = 0 then ⊤ else ↑(x * y).valuation) =
    (if x = 0 then ⊤ else ↑x.valuation) + if y = 0 then ⊤ else ↑y.valuation

Tactic:
if_neg hx,

Goal state after:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hx : ¬x = 0
hy : ¬y = 0
⊢ (if x * y = 0 then ⊤ else ↑(x * y).valuation) = ↑x.valuation + if y = 0 then ⊤ else ↑y.valuation
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hxy : ¬x + y = 0
hx : ¬x = 0
hy : ¬y = 0
⊢ min (if x = 0 then ⊤ else ↑x.valuation) (if y = 0 then ⊤ else ↑y.valuation) ≤
    if x + y = 0 then ⊤ else ↑(x + y).valuation

Tactic:
if_neg hx,

Goal state after:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hxy : ¬x + y = 0
hx : ¬x = 0
hy : ¬y = 0
⊢ min (↑x.valuation) (if y = 0 then ⊤ else ↑y.valuation) ≤ if x + y = 0 then ⊤ else ↑(x + y).valuation
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ Mono A.e → A.EqId

Tactic:
intro
    rw [eqId_iff_len_le]
    exact len_le_of_mono A.e

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.EqId ↔ Mono A.e

Tactic:
by
  constructor
  · intro h
    dsimp at h
    subst h
    dsimp only [id, e]
    infer_instance
  · intro
    rw [eqId_iff_len_le]
    exact len_le_of_mono A.e

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.FunctorGamma

Goal state before:
pp: case neg
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)

Tactic:
have eq : Δ.len = Δ''.len + (k + k' + 2) := by lia

Goal state after:
case neg
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
Module: Mathlib.AlgebraicTopology.DoldKan.Normalized

Goal state before:
pp: case succ
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
n : ℕ
⊢ ∀ i ∈ Finset.univ, (kernelSubobject (X.δ i.succ)).Factors ((P (n + 1)).f (n + 1))

Tactic:
intro i

Goal state after:
case succ
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
n : ℕ
i : Fin (n + 1)
⊢ i ∈ Finset.univ → (kernelSubobject (X.δ i.succ)).Factors ((P (n + 1)).f (n + 1))
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
T : C
f : X _⦋n⦌ ⟶ T
hf : DegeneraciesVanish f
U : C
g : T ⟶ U
⊢ DegeneraciesVanish (f ≫ g)

Tactic:
by
  obtain _ | n := n
  · simp
  · simp [degeneraciesVanish_succ_iff, hf.σ_comp_assoc]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ ∀ (x : Fin (n + 2)),
    x ≠ 0 →
      ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫
          ((-1) ^ ↑x • (Γ₀.obj K).δ x) ≫ (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌)) =
        0

Tactic:
intro i

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
i : Fin (n + 2)
⊢ i ≠ 0 →
    ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫
        ((-1) ^ ↑i • (Γ₀.obj K).δ i) ≫ (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌)) =
      0
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case mp.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Z : C
n : ℕ
f : Z ⟶ X _⦋n + 1⦌
h : f ≫ PInfty.f (n + 1) = 0
h' : f ≫ QInfty.f (n + 1) = f
⊢ ∀ x ∈ {i | ↑i < n + 1},
    f ≫ ((P ↑x).f (n + 1) ≫ X.δ x.rev.succ ≫ X.σ x.rev) ≫ s.πSummand (IndexSet.id (op ⦋n + 1⦌)) = 0

Tactic:
intro i

Goal state after:
case mp.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Z : C
n : ℕ
f : Z ⟶ X _⦋n + 1⦌
h : f ≫ PInfty.f (n + 1) = 0
h' : f ≫ QInfty.f (n + 1) = f
i : Fin (n + 1)
⊢ i ∈ {i | ↑i < n + 1} → f ≫ ((P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev) ≫ s.πSummand (IndexSet.id (op ⦋n + 1⦌)) = 0
Module: Mathlib.Topology.Homotopy.Product

Goal state before:
pp: I : Type u_1
A : Type u_2
X : I → Type u_3
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : TopologicalSpace A
f g : (i : I) → C(A, X i)
S : Set A
homotopies : (i : I) → (f i).HomotopyRel (g i) S
t : ↑unitInterval
x : A
hx : x ∈ S
⊢ ∀ (x_1 : I), (Homotopy.pi fun i ↦ (homotopies i).toHomotopy).toContinuousMap (t, x) x_1 = (f x_1) x

Tactic:
intro i

Goal state after:
I : Type u_1
A : Type u_2
X : I → Type u_3
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : TopologicalSpace A
f g : (i : I) → C(A, X i)
S : Set A
homotopies : (i : I) → (f i).HomotopyRel (g i) S
t : ↑unitInterval
x : A
hx : x ∈ S
i : I
⊢ (Homotopy.pi fun i ↦ (homotopies i).toHomotopy).toContinuousMap (t, x) i = (f i) x
Module: Mathlib.Topology.Homotopy.Path

Goal state before:
pp: case inr
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t x : ↑I
hx : x ∈ {1}
⊢ { toFun := fun x ↦ p ⟨↑(σ (t, x).1) * ↑(t, x).2 + ↑(t, x).1 * ↑(f (t, x).2), ⋯⟩, continuous_toFun := ⋯ } x =
    p.toContinuousMap x

Tactic:
rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]

Goal state after:
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X✝
inst✝² : TopologicalSpace Y
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : SimplyConnectedSpace X
x y : X
⊢ Nonempty (Unique (Path.Homotopic.Quotient x y))

Tactic:
simply_connected_iff_unique_homotopic

Goal state after:
X✝ : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X✝
inst✝² : TopologicalSpace Y
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : Nonempty X ∧ ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y))
x y : X
⊢ Nonempty (Unique (Path.Homotopic.Quotient x y))
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected

Goal state before:
pp: case intro
Y : Type u_2
inst✝ : TopologicalSpace Y
left✝ : PathConnectedSpace Y
right✝ : ∀ (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)
⊢ SimplyConnectedSpace Y

Tactic:
simply_connected_iff_unique_homotopic

Goal state after:
case intro
Y : Type u_2
inst✝ : TopologicalSpace Y
left✝ : PathConnectedSpace Y
right✝ : ∀ (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)
⊢ Nonempty Y ∧ ∀ (x y : Y), Nonempty (Unique (Path.Homotopic.Quotient x y))
Module: Mathlib.AlgebraicTopology.ModelCategory.Over

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
S : C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).HasFactorization (trivialFibrations C)
⊢ (cofibrations (Over S)).HasFactorization (trivialFibrations (Over S))

Tactic:
by
  rw [cofibrations_over_def, trivialFibrations_over_eq]
  infer_instance

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Horn

Goal state before:
pp: n : ℕ
i : Fin (n + 1)
⊢ ∀ (i_1 : ↑{i}ᶜ), (stdSimplex.face {↑i_1}ᶜ).toSSet.HasDimensionLT n

Tactic:
intro i

Goal state after:
n : ℕ
i✝ : Fin (n + 1)
i : ↑{i✝}ᶜ
⊢ (stdSimplex.face {↑i}ᶜ).toSSet.HasDimensionLT n
Module: Mathlib.AlgebraicTopology.SimplicialSet.Boundary

Goal state before:
pp: n : ℕ
⊢ ∀ (i : Fin (n + 1)), (stdSimplex.face {i}ᶜ).toSSet.HasDimensionLT n

Tactic:
intro i

Goal state after:
n : ℕ
i : Fin (n + 1)
⊢ (stdSimplex.face {i}ᶜ).toSSet.HasDimensionLT n
Module: Mathlib.AlgebraicTopology.SimplicialSet.Boundary

Goal state before:
pp: n : ℕ
⊢ ∀ (x : Fin (n + 1)), objMk OrderHom.id ∉ (face {x}ᶜ).obj (op ⦋n⦌)

Tactic:
intro i

Goal state after:
n : ℕ
i : Fin (n + 1)
⊢ objMk OrderHom.id ∉ (face {i}ᶜ).obj (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Horn

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 2))
j : Fin (n + 2)
⊢ stdSimplex.face S ≤ Λ[n + 1, j] ↔ S ≠ Finset.univ ∧ S ≠ {j}ᶜ

Tactic:
subcomplex_le_horn_iff,

Goal state after:
n : ℕ
S : Finset (Fin (n + 2))
j : Fin (n + 2)
⊢ ¬stdSimplex.face {j}ᶜ ≤ stdSimplex.face S ↔ S ≠ Finset.univ ∧ S ≠ {j}ᶜ
Module: Mathlib.CategoryTheory.Limits.Set

Goal state before:
pp: case hinj
J : Type w
inst✝¹ : Category.{w', w} J
X : Type u
inst✝ : IsFilteredOrEmpty J
F : J ⥤ Set X
⊢ ∀ (i j : J) (xi : (fun X ↦ X) ((F ⋙ functorToTypes).obj i)) (xj : (fun X ↦ X) ((F ⋙ functorToTypes).obj j)),
    (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app i)) xi =
        (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app j)) xj →
      ∃ k f g,
        (ConcreteCategory.hom ((F ⋙ functorToTypes).map f)) xi = (ConcreteCategory.hom ((F ⋙ functorToTypes).map g)) xj

Tactic:
intro i

Goal state after:
case hinj
J : Type w
inst✝¹ : Category.{w', w} J
X : Type u
inst✝ : IsFilteredOrEmpty J
F : J ⥤ Set X
i : J
⊢ ∀ (j : J) (xi : (fun X ↦ X) ((F ⋙ functorToTypes).obj i)) (xj : (fun X ↦ X) ((F ⋙ functorToTypes).obj j)),
    (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app i)) xi =
        (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app j)) xj →
      ∃ k f g,
        (ConcreteCategory.hom ((F ⋙ functorToTypes).map f)) xi = (ConcreteCategory.hom ((F ⋙ functorToTypes).map g)) xj
Module: Mathlib.CategoryTheory.MorphismProperty.FunctorCategory

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
K : Type u'
inst✝⁶ : LinearOrder K
inst✝⁵ : SuccOrder K
inst✝⁴ : OrderBot K
inst✝³ : WellFoundedLT K
inst✝² : W.IsStableUnderTransfiniteCompositionOfShape K
J : Type u''
inst✝¹ : Category.{v'', u''} J
inst✝ : HasIterationOfShape K C
⊢ (W.functorCategory J).transfiniteCompositionsOfShape K ≤ W.functorCategory J

Tactic:
rintro X Y f ⟨hf⟩ j
    have : W.functorCategory J ≤ W.inverseImage ((evaluation _ _).obj j) := fun _ _ _ h ↦ h _
    exact W.transfiniteCompositionsOfShape_le K _ ⟨(hf.ofLE this).map⟩

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

Goal state before:
pp: case mpr
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
x₁ y₁ : X₁
⊢ (x₁ = y₁ ∨
      ∃ x₀ y₀,
        ∃ (_ : (ConcreteCategory.hom g) x₀ = (ConcreteCategory.hom g) y₀),
          x₁ = (ConcreteCategory.hom f) x₀ ∧ y₁ = (ConcreteCategory.hom f) y₀) →
    Rel' f g (Sum.inl x₁) (Sum.inl y₁)

Tactic:
· rintro (rfl | ⟨_, _, h, rfl, rfl⟩)
    · apply Rel'.refl
    · exact Rel'.inl_inl _ _ h

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

Goal state before:
pp: case inr_inl.inl.inr
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
inst✝ : Mono f
s✝ w✝¹ w✝ : S
h : (ConcreteCategory.hom g) w✝¹ = (ConcreteCategory.hom g) w✝
h' : (ConcreteCategory.hom f) s✝ = (ConcreteCategory.hom f) w✝¹
⊢ Rel' f g (Sum.inr ((ConcreteCategory.hom g) s✝)) (Sum.inl ((ConcreteCategory.hom f) w✝))

Tactic:
obtain rfl := (mono_iff_injective f).1 inferInstance h'
          rw [h]
          apply Rel'.inr_inl

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X : SSet
d : ℕ
x : ↑(X.nonDegenerate d)
n : ℕ
⊢ ↑x ∈ (X.skeleton n).obj (op ⦋d⦌) ↔ d < n

Tactic:
refine ⟨fun h ↦ ?_, fun _ ↦ X.mem_skeleton _⟩
  obtain ⟨x, hx⟩ := x
  simp only [skeleton, OrderHom.coe_mk, Subfunctor.iSup_obj,
    Set.iUnion_coe_set, Set.mem_iUnion, exists_prop] at h
  obtain ⟨⟨i, hi⟩, y, hy, ⟨f⟩, rfl⟩ := h
  have := X.mono_of_nonDegenerate ⟨_, hx⟩ f y rfl
  have : d ≤ i := SimplexCategory.len_le_of_mono f
  lia

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
⊢ (skeletonOfMono i) (n + 1) =
    (skeletonOfMono i) n ⊔ ⨆ x, ⨆ (_ : ↑x ∉ (Subcomplex.range i).obj (op ⦋n⦌)), Subcomplex.ofSimplex ↑x

Tactic:
dsimp only [skeletonOfMono, OrderHom.coe_mk]

Goal state after:
X Y : SSet
i : X ⟶ Y
n : ℕ
⊢ Subcomplex.range i ⊔ Y.skeleton (n + 1) =
    Subcomplex.range i ⊔ Y.skeleton n ⊔ ⨆ x, ⨆ (_ : ↑x ∉ (Subcomplex.range i).obj (op ⦋n⦌)), Subcomplex.ofSimplex ↑x
Module: Mathlib.AlgebraicTopology.SimplicialSet.Path

Goal state before:
pp: n : ℕ
X✝ : Truncated (n + 1)
m✝ : ℕ
X Y : Truncated (n + 1)
m : ℕ
f : X.Path m
σ : X ⟶ Y
i : Fin m
⊢ (ConcreteCategory.hom (((trunc (n + 1) 1 ⋯).obj Y).map (tr (SimplexCategory.δ 0) Path₁._proof_1 Path₁._proof_5).op))
      ((ConcreteCategory.hom (σ.app (op { obj := ⦋1⦌, property := ⋯ }))) (f.arrow i)) =
    (ConcreteCategory.hom (σ.app (op { obj := ⦋0⦌, property := ⋯ }))) (f.vertex i.succ)

Tactic:
by
    simp only [← f.arrow_tgt i]
    exact ConcreteCategory.congr_hom (σ.naturality (tr (δ 0)).op) _ |>.symm

Goal state after:
Module: Mathlib.CategoryTheory.Category.Cat.Terminal

Goal state before:
pp: T : Type u
inst✝² : Category.{v, u} T
inst✝¹ : Unique T
inst✝ : IsDiscrete T
x✝¹ : Cat
x✝ : x✝¹ ⟶ of T
⊢ ∀ (X : ↑x✝¹), x✝.toFunctor.obj X = ((const ↑x✝¹).obj default).toCatHom.toFunctor.obj X

Tactic:
simp [eq_iff_true_of_subsingleton]

Goal state after:
Module: Mathlib.CategoryTheory.Enriched.Ordinary.Basic

Goal state before:
pp: V✝ : Type u'
inst✝⁷ : Category.{v', u'} V✝
inst✝⁶ : MonoidalCategory V✝
C✝ : Type u
inst✝⁵ : Category.{v, u} C✝
inst✝⁴ : EnrichedOrdinaryCategory V✝ C✝
V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
P : ObjectProperty C
X✝ Y✝ Z✝ : P.FullSubcategory
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (P.fullyFaithfulι.homEquiv.trans (eHomEquiv V)) (f ≫ g) =
    (λ_ (𝟙_ V)).inv ≫
      ((P.fullyFaithfulι.homEquiv.trans (eHomEquiv V)) f ⊗ₘ (P.fullyFaithfulι.homEquiv.trans (eHomEquiv V)) g) ≫
        eComp V X✝ Y✝ Z✝

Tactic:
simp only [ObjectProperty.ι_obj]
    change (eHomEquiv V) (P.ι.map (f ≫ g)) = _
    rw [Functor.map_comp, eHomEquiv_comp]
    rfl

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat

Goal state before:
pp: V : Truncated 2
x : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
⊢ (quotientFunctor V).map (𝟙 (Cat.FreeRefl.mk x)) = 𝟙 (mk x)

Tactic:
CategoryTheory.Functor.map_id

Goal state after:
V : Truncated 2
x : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
⊢ 𝟙 ((quotientFunctor V).obj (Cat.FreeRefl.mk x)) = 𝟙 (mk x)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal

Goal state before:
pp: case succ
X : SSet
sx : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)
x : s.pt
k : ℕ
hk :
  ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + k = j →
      (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =
        (ConcreteCategory.hom (s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) fac_aux₁._proof_1))) x
⊢ ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + (k + 1) = j →
      (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =
        (ConcreteCategory.hom (s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) fac_aux₁._proof_1))) x

Tactic:
intro i

Goal state after:
case succ
X : SSet
sx : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)
x : s.pt
k : ℕ
hk :
  ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + k = j →
      (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =
        (ConcreteCategory.hom (s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) fac_aux₁._proof_1))) x
i : ℕ
⊢ ∀ (j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + (k + 1) = j →
      (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =
        (ConcreteCategory.hom (s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) fac_aux₁._proof_1))) x
Module: Mathlib.AlgebraicTopology.Reedy.Basic

Goal state before:
pp: case neg
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
W₁ W₂ : MorphismProperty C
inst✝⁵ : W₁.IsMultiplicative
inst✝⁴ : W₂.IsMultiplicative
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : OrderBot α
inst✝¹ : SuccOrder α
inst✝ : WellFoundedLT α
r : ReedyStructure W₁ W₂ α
X Y : C
f : X ⟶ Y
hf : W₂ f
hf' : ¬MorphismProperty.identities C f
⊢ r.deg X ≤ r.deg Y

Tactic:
exact (r.lt₂ f hf hf').le

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal

Goal state before:
pp: X X' Y Y' Z : Truncated 2
x : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
y : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
z₀ z₁ : Z.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e : Edge z₀ z₁
⊢ (mapHomotopyCategory (α_ X Y Z).hom).map
      ((inverse (X ⊗ Y) Z).map (Prod.mkHom ((inverse X Y).map (𝟙 (mk x, mk y))) (homMk e))) =
    homMk ((Edge.id x).tensor ((Edge.id y).tensor e))

Tactic:
CategoryTheory.Functor.map_id

Goal state after:
X X' Y Y' Z : Truncated 2
x : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
y : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
z₀ z₁ : Z.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e : Edge z₀ z₁
⊢ (mapHomotopyCategory (α_ X Y Z).hom).map
      ((inverse (X ⊗ Y) Z).map (Prod.mkHom (𝟙 ((inverse X Y).obj (mk x, mk y))) (homMk e))) =
    homMk ((Edge.id x).tensor ((Edge.id y).tensor e))